Particles and Fields — Draft

Technical University of Clausthal WS 1986/87 – SS 1998

W. Lücke

Preface

Quantum field theory originally ment the theory of quantizing classical fields. Nowadays this notion is used for the general theory of quantum systems with infinitly many degrees of freedom. This is a vast subject, but we can cover only a few topics.

Warning: This manuscript is just a condensed set of notes for the lecturer. Obviously, the explanations given here are insufficient for the non-expert. Maybe they will be extended in the future. For the time being these notes should be considered no more than an outline of the stuff to be elaborated in the lectures.

Purely mathematical proofs will mainly be skipped and replaced by suitable references.

Recommended Literature: (Araki, 1999; Baumgärtel, 1995; Bogolubov et al., 1990; Borchers, 1996; Buchholz, 1985; Fredenhagen, 1995; Haag, 1992; Horuzhy, 1990; Jost, 1965; Streater and Wightman, 1989)

4

Contents

1 General Quantum Theory 7

1.1 Basic Logical Structur . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Classical Logic and General Notions . . . . . . . . . . . . . . . 7 1.1.2 Quantum Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 Quantum Reasoning . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.4 Symmetries and Dynamics . . . . . . . . . . . . . . . . . . . . 14

1.2 Orthodox Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1 Logic and Observables . . . . . . . . . . . . . . . . . . . . . . 15 1.2.2 Symmetries and Dynamics . . . . . . . . . . . . . . . . . . . . 20 1.2.3 Algebras of Bounded Observables . . . . . . . . . . . . . . . . 20 1.2.4 State Functionals . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3 Algebraic Formulation of General Quantum Theory . . . . . . . . . . 25 1.3.1 Partial States . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3.2 GNS-Representation . . . . . . . . . . . . . . . . . . . . . . . 28 1.3.3 Canonical Quantization . . . . . . . . . . . . . . . . . . . . . 31 1.3.4 Spontaneously Broken Symmetries . . . . . . . . . . . . . . . 36

2 Massive Scalar Fields 39

2.1 Free Neutral Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . 39 2.1.1 1-Particle Space . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.1.2 Fock Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.1.3 The Free Field . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.2 Wightman Theory for Neutral Scalar Fields . . . . . . . . . . . . . 54 2.2.1 Wightman Axioms . . . . . . . . . . . . . . . . . . . . . . . 54 2.2.2 Remarks on the Choice of the Space of Test Functions …. 56 2.2.3 Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . . 57 2.2.4 Some Standard Results . . . . . . . . . . . . . . . . . . . . . . 62 2.2.5 PCT Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.3 S-Matrix for Self-interacting Neutral Scalar Field . . . . . . . . . . . 75 2.3.1 General Scattering Theory . . . . . . . . . . . . . . . . . . . . 75 2.3.2 Asymptotic condition for massive neutral scalar particles … 78 2.3.3 Evaluation of the Asymptotic Condition . . . . . . . . . . . . 81 2.3.4 Cluster Properties of the S-Matrix . . . . . . . . . . . . . . . 89

2.4 Charged Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.4.1 Free Charged Scalar Fields . . . . . . . . . . . . . . . . . . . . 93 2.4.2 Wightman Theory for Charged Scalar Fields . . . . . . . . . 94 2.4.3 Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . 99

3 Scalar Perturbation Theory 101

3.1 General Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.1.1 Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . 101 3.1.2 Canonical Field Quantization . . . . . . . . . . . . . . . . . . 104

3.2 Canonical Perturbation Theory . . . . . . . . . . . . . . . . . . . . . 110 3.2.1 Dyson Series and Wick’s Theorem . . . . . . . . . . . . . . 110 3.2.2 Counter Terms and Renormalization . . . . . . . . . . . . . . 116 3.2.3 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.3 Bogoliubov-Shirkov Theory . . . . . . . . . . . . . . . . . . . . . 123 3.3.1 Basic Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 123 3.3.2 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.3.3 Generalization to Nonlocalizable Test Spaces . . . . . . . . . . 131

4 Quantum Electrodynamics 139

4.1 The Free Electromagnetic Field Operators . . . . . . . . . . . . . . . 139 4.1.1 Wightman Theory . . . . . . . . . . . . . . . . . . . . . . . 139 4.1.2 Problems With the Quantized Potentials . . . . . . . . . . . . 141 4.1.3 Gupta-Bleuler Construction . . . . . . . . . . . . . . . . . 144 4.1.4 Gupta-Bleuler Observables . . . . . . . . . . . . . . . . . . 149

4.2 The Quantized Free Dirac Field . . . . . . . . . . . . . . . . . . . . 153

4.2.1 Lorentz Transformations Characterized via Complex 2 × 2-

Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.2.2 Relativistic Covariance in General . . . . . . . . . . . . . . . . 157 4.2.3 Dirac Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.2.4 Quantized Dirac Field . . . . . . . . . . . . . . . . . . . . . . 168

4.3 The S-Matrix of Quantum Electrodynamics (QED) . . . . . . . . . . 174 4.3.1 Naive Interaction Picture of QED . . . . . . . . . . . . . . . . 174 4.3.2 General Perturbation Theory . . . . . . . . . . . . . . . . . . 177 4.3.3 The Feynman Rules of QED . . . . . . . . . . . . . . . . . . 180 4.3.4 Example: Compton Scattering . . . . . . . . . . . . . . . . . 184

Bibliography 189

Index 199

Chapter 1

General Quantum Theory

1.1 Basic Logical Structure

1.1.1 Classical Logic and General Notions^{1}

Let L be a set of propositions fulfilling

E_{1}, E_{2} ∈ L ⟹ E_{1} ∧ E_{2}, ¬E_{1} ∈ L .

In common sense logic E_{1} ∧ E_{2} holds if and only if both E_{1} and E_{2} hold while ¬E_{1} holds if and only if E_{1} does not hold. Let us identify E_{1} with E_{2} whenever E_{1} holds if and only if E_{2} holds. Under these circumstances

E_{1} ≼ E_{2} ⟺^{def} E_{1} ∧ E_{2} = E_{1}  (1.1)

defines a semi-ordering ≼ on L, i.e. the ordinary logical implication ≼ is

reflexive:  E ≼ E ,
transitive:  E_{1} ≼ E_{2}, E_{2} ≼ E_{3} ⟹ E_{1} ≼ E_{3} ,
anti-symmetric:  E_{1} ≼ E_{2}, E_{2} ≼ E_{1} ⟹ E_{1} = E_{2} .

The semi-ordered set (L, ≼) is even a lattice, i.e. for any two elements E_{1}, E_{2} ∈ L there is an infimum as well as a supremum, namely

E_{1} ∧ E_{2} = inf_{L} {E_{1}, E_{2}} , E_{1} ∨ E_{2} = sup_{L} {E_{1}, E_{2}} ,  (1.2)

where E_{1} ∨ E_{2} holds iff at least one of the propositions E_{1} , E_{2} holds.^{2} This lattice has a universal lower bound

ˇ0 =^{def} inf_{L} L

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1The definitions and notions introduced here are in agreement with those of (Birkhoff, 1967) and (Varadarajan, 2007).

2For lattices in general (1.2) serves as a definition for ∧ and ∨, consistent with (1.1).

as well as a universal upper bound

ˇ1 =^{def} sup_{L} L .

The negation ¬ is an orthocomplementation, i.e. for all E , E_{1} , E_{2} ∈ L we have

\begin{matrix}(O_{1}) : & E ∧ ¬E = ˇ0 , \\ (O_{2}) : & E ∨ ¬E = ˇ1, , \\ (O_{3}) : & ¬(¬E) = E , \\ (O_{4}) : & E_{1} ≼ E_{2} ⟹ ¬E_{2} ≼ ¬E_{1} . \end{matrix}

The orthocomplemented lattice (L, ≼, ¬) is weakly modular, i.e.:

E_{1} ≼ E_{2} ⟹ E_{2} = (E_{2} ∧ E_{1}) ∨ (E_{2} ∧ ¬E_{1}) .

A logic is defined to be a weakly modular orthocomplemented lattice (L, ≼, ¬) which is σ-complete; i.e. in which inf_{L} A exists for every countable subset A of L. A logic (L, ≼, ¬) is called classical if – as in the above example – it is distributive; i.e.:^{3}

\begin{matrix}(D_{1}) : & E_{1} ∧ (E_{2} ∨ E_{3}) = (E_{1} ∧ E_{2}) ∨ (E_{1} ∧ E_{3}) , \\ (D_{2}) : & E_{1} ∨ (E_{2} ∧ E_{3}) = (E_{1} ∨ E_{2}) ∧ (E_{1} ∨ E_{3}) . \end{matrix}

An ordered pair (E_{1}, E_{2}) ∈ L × L is called compatible if^{4}

E_{1} = (E_{1} ∧ E_{2}) ∨ (E_{1} ∧ ¬E_{2}) .  (1.3)

A probability measure on a logic (L, ≼, ¬) is defined to be a mapping w from L into the interval [0, 1] fulfilling the following two conditions:^{5}

(W_{1}) : w(1) = 1 .
(W_{2}) : w is σ -additive; i.e.:
w(sup{E_{j} : j = 1, 2, …}) = ∑_{j} w(E_{j}) if E_{j} ≼ ¬E_{k} for j ≠ k .

1.1.2 Quantum Logic

Every known concrete quantum theory is a statistical theory of the following type. It is affiliated with

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3Thanks to orthocomplementation, (D_{1}) and (D_{2}) are equivalent.

4An orthocomplemented lattice (L, ≼, ¬) is weakly modular if and only if

(E_{1}, E_{2}) compatible ⟺ (E_{2}, E_{1}) compatible

holds for all E_{1} , E_{2} ∈ L (Birkhoff, 1967, Theorem 21, p. 53).

5Actually, one should also make sure that w(E_{1}) = w(E_{2}) = 1 ⟹ w(E_{1} ∧ E_{2}) = 1 holds for all E_{1}, E_{2} ∈ L.

1. A set Q of macroscopic prescriptions for preparing a ‘state’ of the system under consideration.

2. A set X of macroscopic prescriptions for performing idealized simple tests (called questions by Piron) on the system under consideration^{6} with only two possible outcomes referred to as ‘yes’ or ‘no’.

3. A mapping^{7}

w : Q × X ⟶ [0, 1]

with the following interpretation:^{8}

w(S, T) is the probability^{9} for the outcome ‘yes’ when performing a simple ‘test’ corresponding to T on the system in a ‘state’ corresponding to S.

Obviously, the ‘tests’ T ∈ X cannot discriminate elements S_{1}, S_{2} ∈ Q, which are equivalent in the following sense:

S_{1} ∼ S_{2} ⟺^{def} w(S_{1}, T) = w(S_{2}, T) ∀ T ∈ X .

Similarly the ‘states’ S ∈ Q cannot discriminate ‘tests’ T_{1}, T_{2} ∈ X , which are equivalent in the sense that

T_{1} ∼ T_{2} ⟺^{def} w(S, T_{1}) = w(S, T_{2}) ∀ S ∈ Q .

Therefore the appropriate mathematical formalism deals with the equivalence classes [S] (also called states) and [T] (also called propositions or questions) together with the (consistent) assignment

ω\binom{}{P̂} =^{def} w(S, T) for ω = [S] , P̂ = [T]

rather than the specific prescriptions S, T and the mapping w.

“What we call a state nowadays might turn out to be an equivalence class of states at later times. But this is only possible after having discovered new observables and new states at the same time because states and observables must be mutually separating.” (Borchers, 1996, p. 2)

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6In relativistic quantum field theory we cannot assume that these tests can be performed within arbitrarily small time intervals. Therefore, as becomes evident by Theorem 1.2.2, we work in the so-called Heisenberg picture, in which time evolution is attributed to the ‘tests’ rather than to the ‘states’.

7Actually – as well known for open systems (Davies, 1976) – the probability for the outcome ‘yes’ or ‘no’ in a test performed before the ‘state’ is prepared need not have any meaning. However for all known models of closed quantum systems the ‘states’ can be imagined as having been prepared as early as one likes. This is essential for standard scattering theory.

8We do not claim that S uniquely characterizes a microscopic state, nor do we claim that T fixes the microscopic details of a test!

9Compare, e.g. (Peres, 1995, p. 25).

Q and X are always (more or less implicitly) chosen such that the following three conditions are fulfilled:^{10}

(I_{1}): For every P̂ ∈ L =^{def} {[T] : T ∈ X } there is also an element ¬P̂ ∈ L fulfilling^{11}

ω(¬P̂) = 1 − ω(P̂) for all ω ∈ S =^{def} {[S] : S ∈ Q} .

(I_{2}): Let P̂_{1}, P̂_{2}, … ∈ L. Then there is an element Î ∈ L such that for all ω ∈ S

ω(Î) = 1 if and only if ω(P̂_{j}) = 1 for j = 1, 2, …

(I_{3}): Let P̂_{1}, P̂_{2}, … ∈ L be such that

ω(P̂_{j}) = 1 ⟹ ω(P̂_{k}) = 0 for all ω ∈ S whenever j < k .

Then there is an element Ŝ ∈ L fulfilling

ω(Ŝ) = ω(P̂_{1}) + ω(P̂_{2}) + … ∀ ω ∈ S .

Note that (I_{1}) defines a mapping ¬ : L ⟶ L. Moreover, there is always a natural semi-ordering of the elements of L given by

P̂_{1} ≼ P̂_{2} ⟺^{def} ω(P̂_{1}) ≤ ω(P̂_{2}) ∀ ω ∈ S .  (1.4)

Theorem 1.1.1 (Structure Theorem) If L ≠ ∅ and S fulfill conditions (I_{1})— (I_{3}), then (L, ≼, ¬) , with ≼ given by (1.4) and ¬ given by (I_{1}), is a logic, i.e. a σcomplete weakly modular lattice (L, ≼) . Moreover, under these conditions, every ω ∈ S is a probability measure over (L, ≼, ¬) fulfilling the Jauch-Piron condition

( ω \binom{}{P̂_{1}} = 1 = ω \binom{}{P̂_{2}} ⟹ ω (P̂_{1} ∧ P̂_{2}) = 1 ) ∀ P̂_{1} , P̂_{2} ∈ L .  (1.5)

Proof: See (Doebner and Lücke, 1991, appendix) (see also (Maczyński, 1974) for related results).

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10These conditions are designed to allow for classical reasoning as far as possible. Implicit in (I_{3}) and (I_{1}) is the following standardization postulate: For every P̂ ∈ L \ \left\{{ \atop ˆ0}\right\} there exist a state ω ∈ S with ω(P̂) = 1 . Therefore semi-transparent windows, e.g., cannot be used for simple tests.

11A more general framework, allowing for nonlinear time evolution, was suggested in (Mielnik,

1.1.3 Quantum Reasoning

It seems natural to assign ‘actual’ properties EP̂ to the elements P̂ of L in the sense that:

A system in the state ω ∈ S has property EP̂ with certainty if and

only if^{12} ω(P̂) = 1 . (1.6)

We are used giving names to these properties like ‘spin up’, ‘positive energy’ and so on. However, there is no evidence for the assumption that under all circumstances – independent of any test – the system has either property EP̂ or property E_{¬P̂} – even though

ω(¬P̂) = 1 − ω(P̂) ∀ ω ∈ S , P̂ ∈ L

and even though tests corresponding to P̂ and ¬P̂ can typically be performed jointly.^{13} This also becomes clear by the following lemma.

Lemma 1.1.2 (D. Pfeil) For every finite set Ľ there is a classical logic (B, ≼_{B} , ¬_{B}) and a mapping M : Ľ ⟶ B for which the following holds:

For every mapping ω : Ľ ⟶ [0, 1] there is a probability measure μ on (B, ≼_{B} , ¬_{B}) fulfilling

ω(P̂) = μ(M(P̂)) ∀ P̂ ∈ Ľ

and

P̂_{1} ≠ P̂_{2} ⟹ μ(M(P̂_{1}) ∩ M(P̂_{2})) = μ(M(P̂_{1}))μ(M(P̂_{2})) ∀ P̂_{1} , P̂_{2} ∈ Ľ.

Proof: See (Lücke, 1996, Proof of Lemma 2.3).

Now we should no longer be surprised^{14} if, in orthodox quantum theory, we encounter quantum peculiarities such as^{15}

ω(P̂) = 1 ≠⇒ (ω(P̂ ∧ P̂′) = ω(P̂′) ∀ P̂′ ∈ L)  (1.7)

or^{16}

P̂_{1} ∧ P̂_{2} = 0 ≠⇒ P̂_{1} ≼ ¬P̂_{2} .  (1.8)

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12Recall Footnote 10.

13An example for the latter is given by the Stern-Gerlach experiment, where P̂ corresponds to ‘spin up’ and ¬P̂ to ‘spin down’.

14According to Lemma 1.1.2, the (quantum logical) relations between (equivalence classes of) tests may be just a consequence of the (experimental) restrictions on the set of ‘states’.

15See also (Szabó, 1996, Sect. 3.1).

16By (1.5), P̂_{1} ∧ P̂_{2} = 0 means that there is no \underline{preparable} property guaranteeing both EP̂_{1} and EP̂_{2} .

Nevertheless, simple quantum reasoning according to the following rules is consistent:

• Choose a classical sublogic (L_{c}, ≼, ¬) of (L, ≼, ¬) and forget about all the other elements of L.

• Then imagine that every individual – in whatever situation – has either property EP̂ or E_{¬P̂} if P̂ ∈ L_{c} .

• For ω ∈ S, imagine that ω(P̂) is the relative number of individuals having property EP̂ in an ensemble corresponding to ω if P̂ ∈ L_{c} .

• Imagine that

≼ corresponds to common sense logical implication , ¬ corresponds to common sense logical negation, ∧ corresponds to common sense logical ‘and’ , ∨ corresponds to common sense logical ‘or’ .

This way all quantum peculiarities are avoided. For instance, in spite of (1.7), we may conclude

\begin{matrix}ω(P̂_{1}) = 1 , \\ P̂_{1} & compatible with & P̂_{2} \end{matrix} \} ⟹ ω(P̂_{1} ∧ P̂_{2}) = ω(P̂_{2}) ∀ ω

or even

P̂_{1} compatible^{17} with P̂_{2}
⟹ ω(P̂_{1} ∨ P̂_{2}) = ω(P̂_{1} ∧ ¬P̂_{2}) + ω(¬P̂_{1} ∧ P̂_{2}) + ω(P̂_{1} ∧ P̂_{2}) ∀ ω

— if P̂_{1}, P̂_{2} ∈ L_{c} .

Simple quantum reasoning naturally leads to the notion of observable:^{18}

Definition 1.1.3 An observable A of a physical system modeled by the logic (L, ≼, ¬) is a σ-morphism Ê_{A} of the Borel ring on the real line^{19} into (L, ≼, ¬) which is unitary, i.e. Ê_{A}(ℝ) = ˆ1 . It is called bounded iff Ê_{A}(Δ) = 1 for suitable compact Δ ∈ ℝ.

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17For compatible P̂_{1}, P̂_{2} :

\begin{aligned}P̂_{1} ∨ P̂_{2} = (P̂_{1} ∨ P̂_{2}) ∧ (¬P̂_{2} ∨ P̂_{2}) & = & (P̂_{1} ∧ ¬P̂_{2}) ∨ P̂_{2} \\ & = & (P̂_{1} ∧ ¬P̂_{2}) ∨ (¬P̂_{1} ∧ P̂_{2}) ∨ (P̂_{1} ∧ P̂_{2}) \end{aligned}

18The Borel ring on ℝ^{1} could be replaced by an arbitrary classical logic; possibly associated with some physical dimension.

19The Borel sets on a locally compact space X form the smallest family of sets containing all compact subsets of X and being closed with respect to forming relative complements and countable unions.

The physical interpretation of Ê_{A} in the sense of quantum reasoning is as follows:

Given ω ∈ S and a Borel subset Δ of ℝ^{1} then ω(Ê_{A}(Δ)) can be imagined as the relative number of individuals, in an ensemble corresponding to ω, for which A ∈ Δ .

Consequently, the expectation value^{20} for A in an ensemble corresponding to ω is given by

\overline{A}(ω) = ∫ λ dμ_{ω}^{A}(λ) .  (1.9)

with Borel measure

μ_{ω}^{A}(B) =^{def} ω (Ê_{A}(B)) for Borel subsets B ⊂ ℝ^{1} .  (1.10)

Simple quantum reasoning can be applied to a whole family of observables A_{1}, A_{2}, … if and only if all the pairs

(Ê_{A_{j}}(Δ_{j}), Ê_{A_{k}}(Δ_{k})) , Δ_{j} , Δ_{k} ∈ ℝ

are compatible.^{21}

In order to make predictions for multiple tests one has to know how states change as a result of a simple test. Here we assume^{22}

Lüders postulate: For every P̂ ∈ L there is at least one corresponding measurement of first kind , i.e. a simple test T with [T] = P̂ causing a transition^{23} ω ↦ ω_{,P̂} whenever the result is ‘yes’. Here, if ω(P̂) > 0 , ω_{,P̂} ∈ S is assumed to be uniquely characterized by the condition

ω_{,P̂}(P̂′) = ω(P̂′)/ω(P̂) ∀ P̂′ ≼ P̂ .

Remark: Aktual measurements of first kind can be implemented using, e.g., CNOTgates; see Footnote 23 of (Lücke, qip).

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20Of course, the expectation value may be infinite!

21In a logic (L, ≼, ¬) the sublogic generated by P̂_{1} , P̂_{2} , … ∈ L is classical if and only if all the pairs (P̂_{j}, P̂_{k}) are compatible (Piron, 1976, §2-2).

22A first kind measurement corresponding to P̂ does not destroy any of the properties EP̂′ with P̂′ , P̂ compatible (Lücke, 1996). Usually, a test causes a much more drastic change of the state or even ends by absorbing the corresponding individual. Intermediate measurements of first kind, typically, would be approximately realized by means of highly efficient filters.

23Naively interpreted, ω_{,P̂}(P̂′) describes the conditional probability in the state ω for EP̂′ – defined by (1.6) – being true provided EP̂ is true. In ordinary quantum theory the new statistical operator T̂_{ω_{,P̂}} is given by P̂T̂_{ω}P̂/ω(P̂) .

By Lüders’ postulate,^{24} given the initial state ω ∈ S, the probability for the homogeneous history (P̂_{1}, …, P̂_{n}) – i.e. for getting the answer ‘yes’ for all subsequent first kind measurements of a series corresponding to P̂_{1}, …, P̂_{n} ∈ L – should be^{25}

ω(P̂_{1})ω_{,P̂_{1}}(P̂_{2}) ⋯ ω_{,P̂_{1},...,P̂_{n−1}}(P̂_{n}) .

Consistent quantum reasoning with respect to histories leads to the modern notion of decoherent histories.

Given a history (P̂_{1}, …, P̂_{n}) not corresponding to a simple test, we can no longer be sure that there is an initial state for which (P̂_{1}, …, P̂_{n}) is certain, i.e., for which ω(P̂_{1})ω_{,P̂_{1}}(P̂_{2}) ⋯ ω_{,P̂_{1},...,P̂_{n−1}}(P̂_{n}) = 1 . Therefore the ‘logic’ of histories is weaker than that for simple tests and may provide a useful basis for generalizing quantum theory (Isham, 1995).

1.1.4 Symmetries and Dynamics

Just for simplicity we always use the following assumption, fulfilled in ordinary quantum theory:

S = S_{(L,≼,¬)} =^{def} set of all probability measures on (L, ≼, ¬) .

(1.11)

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24In the relativistic theory Lüders’ postulate causes interesting problems (Schlieder, 1971) (see also (Mittelstaedt, 1983),(Mittelstaedt and Stachow, 1983)).

25Naively interpreted, ω(P̂_{1})ω_{,P̂_{1}}(P̂_{2}) ⋯ ω_{,P̂_{1},...,P̂_{n−1}}(P̂_{n}) is the probability for joint validity of the properties EP̂_{1}, …, EP̂_{1} in the state ω. Usually (see, e.g., (Omnès, 1994), (Griffith, 1995)), unfortunately, this is formulated in the Schrödinger picture, thus imposing unnecessary restrictions.

Definition 1.1.4 A symmetry of a physical system modeled by the logic (L, ≼, ¬) is an automorphism of (L, ≼, ¬) , i.e. a bijection of L onto itself preserving the least upper bound and the orthocomplementation. A dynamical semi-group for such a system is a family {α_{t}}_{t∈ℝ_{+}} of symmetries α_{t} fulfilling the following three conditions:^{26}

(i) α_{0}(P̂) = P̂ ∀ P̂ ∈ L ,

(ii) α_{t_{1}} ∘ α_{t_{2}} = α_{t_{1}+t_{2}} ∀ t_{1} , t_{2} ∈ ℝ_{+} =^{def} {t ∈ ℝ : t ≥ 0} ,

(iii) t ⟼ α_{t} is weakly continuous, i.e., for fixed P̂ ∈ L and ω ∈ S the probability ω (α_{t}(P̂)) is a continuous function of t ∈ ℝ_{+} .

The most important symmetries are the time-translations α_{t} , t ∈ ℝ_{+} , for systems which are homogeneous in time:

Let T be a macroscopic prescription for performing a simple test corresponding to P̂ ∈ L. Then the prescription T_{t} to do everything prescribed by T just with time delay t characterizes a test corresponding to α_{t}(P̂) .

Then the family {α_{t}}_{t∈ℝ_{+}} of time-translations, determining the dynamics of the system, is naturally assumed to be a dynamical semi-group.

1.2 Orthodox Quantum Mechanics and Algebraic Formulation

1.2.1 Logic and Observables

In ‘pioneer quantum mechanics’ (Primas, 1981) (without superselection rules^{27}) the logic (L, ≼, ¬) described in Section 1.1.2 is realized as follows (standard quantum logic):

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26If S ≠ S_{(L,≼,¬)} one should also require α^{∗}(S) = S for the dual α^{∗} of a symmetry α with respect to S, defined by

(α^{∗}ω) (P̂) =^{def} ω (α(P̂)) ∀ ω ∈ S , P̂ ∈ L .

Then the inverse of a symmetry need not be a symmetry.

27A system modeled by (L, ≼, ¬) is said to possess superselection rules if the center

C_{(L,≼,¬)} =^{def} {P̂ ∈ L : (P̂, P̂′) compatible ∀ P̂′ ∈ L}

of (L, ≼, ¬) is nontrivial (L ≠ C ≠ {ˆ0, ˆ1}). (L, ≼, ¬) is called irreducible if C = {ˆ0, ˆ1}.

• L is given as the set of all projection operators^{28} in some separable complex Hilbert space H of dimension ≥ 2 .

• For arbitrary P̂_{1}, P̂_{2} ∈ L we have

P̂_{1} ≼ P̂_{2} ⟺^{def} P̂_{1} ≤ P̂_{2}
⟺ (⟨Ψ | P̂_{1}Ψ⟩ ≤ ⟨Ψ | P̂_{1}Ψ⟩ ∀ Ψ ∈ H) .

• For every P̂ ∈ H we have

¬P̂ =^{def} ˆ1 − P̂ .

Then, if dim (H) ≥ 3 , Gleason’s theorem (Gleason, 1957) tells us that for every ω ∈ S_{(L,≼,¬)} there is a unique positive trace class operators^{29} T̂_{ω} ∈ L(H) fulfilling

ω(P̂) = Tr (T̂_{ω}P̂) ∀ P̂ ∈ L .

Every positive trace class operator T̂ of trace 1 can be written in the form

T̂ = ∑_{ν=0}^{∞} \underbrace{λ_{ν}}_{≥0} P̂_{Ψ_{ν}} , ∑_{ν=0}^{∞} λ_{ν} = 1 , H ∋ Ψ_{ν} ≠ 0 ∀ ν .

Here we use the standard notation

P̂_{Ψ}Φ =^{def} ⟨\frac{Ψ}{∥Ψ∥}|Φ⟩ \frac{Ψ}{∥Ψ∥} ∀ Φ ∈ H , Ψ ∈ H \ \{0\} .

Hence

ω_{Ψ}(P̂) =^{def} Tr (P̂_{Ψ}P̂) = ⟨\frac{Ψ}{∥Ψ∥}|P̂ \frac{Ψ}{∥Ψ∥}⟩ ∀ Ψ ∈ H \ \{0\} , P̂ ∈ L .

Now we have the following form of the Lüders postulate:

For every P̂ ∈ L there is at least one corresponding measurement of first kind , i.e. a simple test T with [T] ˆ= P̂ causing a transition ω ↦ ω_{,P̂} whenever the result is ‘yes’, where

T̂_{ω_{,P̂}} = \begin{matrix}P̂ & T̂_{ω} & P̂ \\ Tr & (P̂ T̂_{ω} P̂) \end{matrix} .

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and (Lücke, 1995). quantum logic.

28Their specific physical identification depends on the dynamics, as discussed in (Mielnik, 1974)

29Conversely, every trace class operator of trace 1 induces a probability measure on standard

Especially, if ω = ω_{Ψ} for some Ψ ∈ H \ {0}, a first kind measurement corresponding to P̂_{Φ} , Φ ∈ H \ {0}, causes the transition ω_{Ψ} ⟶ ω_{Φ} whenever the result is ‘yes’:

ω_{Ψ} \binom{}{P̂_{Φ}} > 0 ⟹ P̂_{Φ}P̂_{Ψ}P̂_{Φ} ; Tr (P̂_{Φ}P̂_{Ψ}P̂_{Φ}) = P̂_{Φ} .

The corresponding transition probability is

ω_{Ψ} \binom{}{P̂_{Φ}} = |⟨\frac{Φ}{∥Φ∥}|\frac{Ψ}{∥Ψ∥}⟩|^{2} .

Exercise 1 Prove the following:^{30}

P̂_{1} ≼ P̂_{2} ⟺ P̂_{1}H ⊂ P̂_{2}H ,  (1.12)
P̂_{1} ≼ P̂_{2} ⟺ P̂_{1} = P̂_{2} P̂_{1} ,  (1.13)
¬P̂H = H ⊖ P̂H =^{def} {Ψ ∈ H : ⟨Φ|Ψ⟩ = 0 for all Φ ∈ P̂H} ,  (1.14)

P̂_{1} ∧ P̂_{2} ∧ … =^{def} inf{P̂_{1}, P̂_{2}, …} = orthogonal projection onto ⋂_{j=1}^{∞} P̂_{j}H , (1.15)

sup\{P̂_{1}, P̂_{2}, …\} = orthogonal projection onto \overline{span (⋃_{j=1}^{∞} P̂_{j}H)} ,  (1.16)

P̂_{1} ∧ P̂_{2} = s- lim_{n→∞}(P̂_{1}P̂_{2})^{n} ≠^{i.a.} P̂_{1}P̂_{2} ,  (1.17)
(P̂_{1} , P̂_{2}) compatible ⟺ P̂_{1}P̂_{2} = P̂_{2}P̂_{1} ,  (1.18)

P̂_{1} Atom ⟺^{def} (P̂_{1} ≠ 0 and P̂ ≼ P̂_{1} ⟹ P̂ ∈ {0, P̂_{1}})
⟺ P̂_{1}H 1-dimensional ,  (1.19)

P̂_{1} ∧ ¬P̂_{2} = 0 and P̂_{1} Atom ⟹^{“covering law”} (P̂_{1} ∨ ¬P̂_{2}) ∧ P̂_{2} Atom ,  (1.20)
ω_{Ψ}(P̂) > 0 ⟹ \begin{matrix}P̂ & P̂_{Ψ} & P̂ \\ Tr & (P̂P̂_{Ψ}P̂) \end{matrix} = \frac{P̂_{P̂Ψ}}{Tr \binom{}{P̂_{P̂Ψ}}} .  (1.21)

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2.4), however.

30Assuming irreducibility, atomicity and (1.20) one may prove that – apart from some exceptional cases – (L, ≼, ¬) is isomorphic to the logic of all projection operators on some generalized Hilbert space (Piron, 1976, Section 3–1). According to (1.21) the probability for the homogeneous history (P̂_{1}, P̂_{2}) is ω_{Ψ}(P̂_{2}P̂_{1}P̂_{2}) , where ˆ0 ≤ P̂_{2}P̂_{1}P̂_{2} ≤ ˆ1 but P̂_{2}P̂_{1}P̂_{2} ∉^{i.g.} L. Note (Davies, 1976, Lemma

Here, according to Definition 1.1.3, an observable A corresponds to a projection valued measure, i.e. a mapping Ê_{A} from the ring of Borel sets (over ℝ^{1}) into L such that:

(PVM_{1}) : Ê_{A}(ℝ^{1}) = ˆ1 ,  (Normalization )
(PVM_{1}) : Ê_{A}(⋃_{j=1}^{∞} B_{j}) = s − lim_{n→∞} ∑_{j=1}^{n} Ê_{A}(B_{j}) ,
whenever the B_{j} are mutually disjoint Borel sets . (σ-Additivity )

Ê_{A} gives rise to a self-adjoint operator Â, uniquely characterized by^{31}

\overline{A}(ω_{Ψ}) = ⟨\frac{Ψ}{∥Ψ∥}|Â\frac{Ψ}{∥Ψ∥}⟩ ∀ Ψ ∈ DÂ \ \{0\}  (1.22)

(remember (1.9)/(1.10)), where

DÂ = {Ψ : ∫_{λ∈ℝ^{1}} λ^{2} ω_{Ψ} (Ê_{A}(dλ)) < ∞}  (1.23)

is the domain of Â (see e.g. (Achieser and Glasmann, 1965)). For this operator one usually writes

Â = ∫_{λ∈ℝ^{1}} λ Ê_{A}(dλ) .  (1.24)

With this operator we also have

\overline{A}(ω) = Tr (T̂_{ω}Â) ∀ ω ∈ S_{(L,≼,¬)}
if Â is bounded

(1.25)

resp.

\overline{A}(ω) = lim_{Λ_{+}→+∞} Tr (T̂_{ω}ÂÊÂ ((0, Λ_{+}]))
+ lim_{Λ_{−}→−∞} Tr (T̂_{ω}ÂÊÂ ((Λ_{−}, 0])) ∀ ω ∈ S_{(L,≼,¬)}

if Â is unbounded, where the l.h.s. is defined iff the r.h.s is.

According to the well known spectral!theorem (see e.g. (Reed und Simon, 1972)), for every self-adjoint operator Â there is a unique regular^{32} projector-valued measure Ê_{A} fulfilling (1.22)/(1.23), called the spectral!measure of Â. In this sense, according to Definition 1.1.3, the observables of orthodox quantum mechanics may be identified with the self-adjoint operators.^{33}

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31This is very useful for heuristic physical identification of observables.

32A projector-valued measure Ê_{A} is called regular if

Ê_{A}(B) = sup {Ê_{A}(C) : B ⊃ C compact}
= inf {Ê_{A}(O) : B ⊂ O open}

holds for all Borel subsets B of ℝ (not only for B = (−∞, λ] , λ ∈ ℝ^{1}).

33We skip physical dimensions to allow for addition of these operators.

Exercise 2 Determine the spectral measures for the following operators of elementary L^{2}(ℝ^{1}, dx)-quantum mechanics:

(i) position operator,
(ii) linear momentum operator,
(iii) energy operator of the harmonic oscillator,
(iv) zero operator,
(v) identity operator.

Let Â be a self-adjoint operator on the complex Hilbert space H with spectral measure Ê_{A} and let f be some complex-valued Borel function^{34} on ℝ^{1} . Then there is a unique operator f(Â) , written

f(Â) = ∫ f(λ) Ê_{A}(dλ) ,  (1.26)

with domain

D_{f(Â)} =^{def} {Ψ ∈ H : ∫ |f(λ)|^{2} ω_{Ψ} (Ê_{A}(dλ)) < ∞}  (1.27)

fulfilling

⟨\frac{Ψ}{∥Ψ∥}|f(Â)\frac{Ψ}{∥Ψ∥}⟩ = ∫ f(λ) ω_{Ψ}(Ê_{A}(dλ)) ∀ Ψ ∈ D_{f(Â)} \ \{0\} .

If f is real-valued then f(Â) is self-adjoint on this domain and its spectral measure is characterized by

Ê_{f(A)}(J) = Ê_{A} (f^{−1}(J)) for all intervals J ⊂ ℝ .  (1.28)

If f is bounded then the operator f(Â) is bounded and

∥f(Â)∥ =^{def} sup_{Ψ∈H\\{0\}} ∥Â\frac{Ψ}{∥Ψ∥}∥ ≤ sup_{λ∈ℝ^{1}} |f(λ)| .  (1.29)

Exercise 3 Prove the following:

f(λ) = f_{0} for all λ ∈ ℝ^{1} ⟹ f(Â) = f_{0}ˆ1 ,  (1.30)
f(λ) = g(λ) + h(λ) for all λ ∈ ℝ^{1} ⟹ f(Â) = \overline{g(Â) + h(Â)} ,  (1.31)
f(λ) = g(λ)h(λ) for all λ ∈ ℝ^{1} ⟹ f(Â) = \overline{g(Â)h(Â)} ,  (1.32)
g (f(Â)) = h(Â) if f is real-valued and g (f(λ̂)) = h(λ̂) for all λ ∈ ℝ^{1} , (1.33)
|f(λ)| = 1 for all λ ∈ ℝ^{1} ⟹ f(Â) unitary ,  (1.34)
Ê_{A}(B) = χ_{B}(Â) for all Borel sets B .  (1.35)

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34If B is a Borel subset of ℂ then f^{−1}(B) has to be a Borel subset of ℝ^{1}.

1.2.2 Symmetries and Dynamics

If H is a complex Hilbert space, let us denote by (L_{H}, ≼, ¬) the standard quantum logic described in 1.2.1.

Theorem 1.2.1 (Wigner) Let H a be a complex Hilbert space of dimension ≥ 3 . Then a map α : L_{H} ⟶ L_{H} is a symmetry of (L_{H}, ≼, ¬) iff there is either a unitary or an anti-unitary operator Û with^{35}

α(P̂) = ÛP̂Û^{∗} ∀ P̂ ∈ L_{H} .

Proof: See (Piron, 1976, §3–2).

For the elements of a dynamical semi-group of (L_{H}, ≼, ¬) the choice of anti-unitary Û is excluded:

Theorem 1.2.2 Let H a be a complex Hilbert space of dimension ≥ 3 and let {α_{t}}_{t∈ℝ_{+}} be a dynamical semi-group of (L_{H}, ≼, ¬) . Then there is a unique selfadjoint operator Ĥ fulfilling

α_{t}(P̂) = e^{\frac{i}{ℏ} Ĥt}P̂e^{− \frac{i}{ℏ} Ĥt} ∀P̂ ∈ L_{H} , t ∈ ℝ_{+} .

Proof: See (Lücke, 1996, Sect. 3.3).

1.2.3 Algebras of Bounded Observables

Unbounded observables Â have the unpleasant feature that their domain is always smaller than H, by the Hellinger-Toeplitz theorem (see (Reed und Simon, 1972, corollary to the Closed Graph Theorem II.12).) This causes lots of technical complications. Fortunately, from the principal point of view, it is sufficient to know the spectral operators Ê_{A}(J) , which are always bounded, for all intervals J ⊂ ℝ. This allows for taking advantage of the powerful mathematical theory of algebras of bounded operators.

The set L(H) of all bounded operators Â in H (with DÂ = H) is a complex Banach algebra , i.e. a complex Banach space^{36} with associative^{37} and distributive multiplication^{38} fulfilling the so-called product inequality

∥ÂB̂∥ ≤ ∥Â∥ ∥B̂∥ .

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35Especially, we have α(P̂_{Φ}) = P̂_{ÛΦ} .

36In this case the set of all bounded operators with ordinary addition, multiplication by complex numbers, and operator norm.

37This means: Â (B̂Ĉ) = (ÂB̂) Ĉ for all elements Â , B̂ , Ĉ of the Algebra, and (αβ)(ÂB̂) = (αÂ)(βB̂) for all complex Numbers α , β and all elements Â , B̂ , Ĉ of the Algebra .

38In this case ordinary multiplication with operators and/or numbers.

Transition to the adjoint operator is a involution, i.e. a mapping Â → Â^{∗} fulfilling the following three conditions:

(I_{1}): \binom{}{Â^{∗}}^{∗} = Â

(I_{2}): (ÂB̂)^{∗} = B̂^{∗}Â^{∗}

(I_{3}): (αÂ + βB̂)^{∗} = \overline{α}Â^{∗} + \overline{β}B̂^{∗}

L(H) is also a C^{∗}-algebra^{39}, i.e. a complex Banach algebra with involution ^{∗}, obeying the condition

∥Â^{∗}Â∥ = ∥Â∥^{2} .

Exercise 4 Proof that ∥Â^{∗}∥ = ∥Â∥, hence also ∥ÂÂ^{∗}∥ = ∥Â∥^{2}, holds for every C^{∗}-algebra.

L(H) is even a von Neumann algebra, i.e. a subalgebra M of L(H) , H some complex Hilbert space, that is given by the commutant

N′ =^{def} {Â ∈ L(H) : [Â, B̂]_{−} = 0 ∀ B̂ ∈ N }

of some ∗-invariant subset N ⊂ L(H) :

M = (N ∪ N^{∗})′ (and hence M = M′′) .

Exercise 5 Let B be a set and r a binary relation on B. Show that

A_{1} ⊂ A_{2} ⟹ A_{1}^{r} ⊃ A_{2}^{r}

holds for all subsets A_{1} , A_{2} of B, where

A^{r} =^{def} {B̂ ∈ B : r(B̂, Â) ∀ Â ∈ A} ∀A ⊂ B .

Moreover, show for symmetric r that A ⊂ A^{r}^{r} and therefore also

A^{r}^{r}^{r} = A^{r}

holds for all A ⊂ B.

Another important Banach algebra with involution is the set T_{1} ⊂ L(H) of trace class operators with the norm

∥Â∥_{Tr} =^{def} Tr\sqrt{Â^{∗}Â} ≥ ∥Â∥ .

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39Called a completely regular algebra by Neumark (Neumark, 1959).

Identifying Â ∈ L(H) with the mapping

T̂ → Tr (T̂Â)

we get

L(H) = T_{1}(H)^{∗} =^{def} { linear continuous mappings T_{1}(H) → C}  (1.36)

(see (Bratteli and Robinson, 1979, Proposition 2.4.3)). Similarly,^{40} we have

T_{1}(H) = C_{1}(H)^{∗}  (1.37)

((Gaal, 1973, pp 100/101)) by this identification, where C_{1}(H) denotes the C^{∗}-sub-algebra^{41} of L(H) consisting of all compact (=completely continuous) operators^{42} in H.

1.2.4 State Functionals

According to Gleason’s theorem, if H is separable and of dimension > 2 , every probability measure on the standard quantum logic is the restriction (to the projection operators) of a (unique) mapping ω : L(H) ⟶ ℂ of the form

ω(Â) = Tr(T̂_{ω}Â), T̂_{ω} ⊂ T_{1}(H), Tr(T̂_{ω}) = 1, T̂_{ω} = T̂_{ω}^{∗} ≥ 0 .  (1.38)

Exercise 6

(i) Given P̂_{1}, …, P̂_{n} ∈ L_{H} and Ψ ∈ H \ {0}, show that the probability for the homogeneous history (P̂_{1}, …, P̂_{n}) in a state prepared according to ω_{Ψ} is

\begin{matrix}∥P̂_{n} ⋯ P̂_{1}Ψ∥ & 2 \\ ∥Ψ∥ & 2 \end{matrix} = ω_{Ψ}(P̂_{1} ⋯ P̂_{n}P̂_{n−1} ⋯ P̂_{1})

and that

ˆ0 ≤ P̂_{n} ⋯ P̂_{1}P̂_{n−1} ⋯ P̂_{1} ≤ ˆ1

even though P̂_{n} ⋯ P̂_{1}P̂_{n−1} ⋯ P̂_{1} ∉^{i.g.} L_{H}.

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40Note that (1.36) resp. (1.37) is analogous to l^{∞} = (l^{1})^{∗}, resp. l^{1} = (c_{0})^{∗}, where: l^{∞} =^{def} {bounded sequences}, c_{0} =^{def} {null sequences}, l^{1} =^{def} {absolutely convergent sequences}

41See (Gaal, 1973, Theorem 3 and Proposition 4 of Section II.2)

42Compact operators are uniform limits of increasing countable sequences of finite rank operators (Gaal, 1973, Theorem 7 of Section II.2).

(ii) Show that, contrary to L_{H}, the set {F̂ ∈ L(H) : ˆ0 ≤ F̂ ≤ ˆ1} of all effects with its natural semi-ordering and ‘orthocomplementation’ is not a logic.^{43}

The modern notion of state is as follows:^{44} A state on a C^{∗}-algebra A with unit is a mapping Â → ω(Â) of A into the complex numbers fulfilling the following three conditions for all Â , B̂ ∈ A and α ∈ ℂ :

(S_{1}) : ω(Â + αB̂) = ω(Â) + αω(B̂) (linearity )
(S_{2}) :  ω(ˆ1) = 1  (normalization )
(S_{3}) :  ω(Â^{∗}Â) ≥ 0  (positivity^{45} )

Exercise 7 Show that the following three conditions are fulfilled for every state ω on a C^{∗}-algebra with unit element:

(i) |ω(Â^{∗}B̂)|^{2} ≤ ω(Â^{∗}Â)ω(B̂^{∗}B̂) (Cauchy Schwarz inequality )
(ii) ω(Â^{∗}) = \overline{ω(Â)} (hermiticity )
(iii) ω(Â_{n}) → ω(Â) if ∥Â − Â_{n}∥ → 0 (continuity )

However, due to T_{1}(H) = C(H)^{∗}, there are also so-called singular states,^{46} for which ω(P̂) = 0 whenever P̂ has finite rank. Of course, the restrictions of such functionals do not define probability measures on the standard quantum logic since they cannot be σ-additive. Therefore the state functionals need an additional characterization which relies on the following.

Lemma 1.2.3 Let M be a von Neumann subalgebra of L(H) , let I be some ordered index set, and let \left\{{ \atop Â_{i}}\right\}_{i∈I} ⊂ M be an increasing net of positive operators

with sup_{ℝ} {∥Â_{i}∥ : i ∈ I} < ∞. Then sup_{L(H)} {Â_{i} : i ∈ I} exists and is an element of the algebra M.

Proof: See (Bratteli and Robinson, 1979, Lemma 2.4.19).

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43Hint: Show that an effect F̂ is a projection if and only if ˆ0 = inf_{{effects}} {F̂ , (ˆ1 − F̂)}.

44This way explicit use of the Hilbert space becomes unnecessary, in principle (see Sect. 1.3.3 for details).

45It would be quite tedious to show in general for C^{∗}-algebras that Â^{∗}Â = −B̂^{∗}B̂ ⟹ Â^{∗}Â = 0 . 46In general, a state ω on von Neumann algebra is called singular, iff for every nonzero

projection operator P̂ there is another nonzero projection operator P̂′ for which ω(P̂′) = 0 and P̂′ ≼ P̂.

Definition 1.2.4 A state ω on a von Neumann algebra M is said to be normal, iff

ω (sup_{L(H)} {Â_{i} : i ∈ I}) = sup_{ℝ} {ω(Â_{i}) : i ∈ I}

holds for every net fulfilling the requirements of Lemma 1.2.3.

Theorem 1.2.5 Let M be a von Neumann subalgebra of L(H). Then the normal states ω on M are exactly those of the form (1.38).

Proof: See (Bratteli and Robinson, 1979, Theorem 2.4.21).

Exercise 8 Prove Theorem 1.2.5 for M = L(H) , where H is separable.^{47}

According to Theorem 1.2.5 (and Gleason’s theorem) the probability measures on the standard quantum logic correspond to restrictions of normal states on L(H) to the projection operators, if dim(H) > 2 .

Definition 1.2.6 A state ω on a C^{∗}-algebra A is said to be mixed, iff there are states ω_{1} ≠ ω_{2} on A and a real number λ ∈ (0, 1) fulfilling

ω(Â) = λω_{1}(Â) + (1 − λ)ω_{2}(Â) for every Â ∈ A .

Otherwise ω is said to be pure.

Exercise 9

(i) Show that the states ω_{1}, ω_{2} of Definition 1.2.6 must both be normal, if ω is normal.

(ii) Prove that a normal state ω on L(H) is pure iff there is a normed vector Ω ∈ H

fulfilling

ω(Â) = ⟨Ω | Â Ω⟩ = Tr(P̂_{Ω} Â) for all Â ∈ L(H) .

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47Hint: If \left\{{ \atop Â_{i}}\right\}_{i∈I} is an increasing bounded net then so is \{\sqrt{T̂}Â_{i}\sqrt{T̂}\}_{i∈I} .

1.3. ALGEBRAIC FORMULATION OF GENERAL QUANTUM THEORY 25

1.3 Algebraic Formulation of General Quantum Theory^{48}

1.3.1 Partial States

If one is interested only in a certain subset^{49} A of physical entities A, it is sufficient to know the partial state ω = ω_{total}/^{\}M of the proper normal state ω_{total} on L(H_{total}) with respect to the smallest von Neumann subalgebra M of L(H_{total}) that contains all the projection operators Ê_{A} ((−∞, λ]) with A ∈ A, λ ∈ ℝ^{1} .

An immediate consequence of Theorem 1.2.5 is the following.

Corollary 1.3.1 Let M_{2} be a von Neumann subalgebra of the von Neumann algebra M_{1} . Then the set of all normal states on M_{2} coincides with the set of all partial states of normal states on M_{1} .

We conclude that in orthodox quantum theory the relevant states are the normal states on the von Neumann subalgebra M ⊂ L(H_{total}) of interest.

Exercise 10 Show that even if ω_{total} is pure the partial state ω = ω_{total}/M may be mixed.

From now on we consider only quantum logics (L, ≼, ¬) of the following type:^{50}

There is a separable complex Hilbert space H and a von Neumann subalgebra M of L(H) by which (L, ≼, ¬) is realized in the following way:

• L = L_{M} =^{def} {P̂ ∈ M : P̂^{∗} = P̂ = P̂^{2}},

• P̂_{1} ≼ P̂_{2} ⟺^{def} P̂_{1} ≤ P̂_{2} ∀ P̂_{1} , P̂_{2} ∈ L,

• ¬P̂ =^{def} ˆ1 − P̂ ∀ P̂ ∈ L.

Exercise 11 For bounded self-adjoint operators it is known that they commute (in the naive sense) if and only if all their spectral projections commute^{51} (Neumark,

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48See (Roberts and Roepstorff, 1969).

49A typical case would be the description of extended structures via their centers of mass.

50That the projection operators of a von Neumann algebra always form a sublogic (L_{M}, ≼, ¬) of the corresponding standard quantum logic follows from (1.17) (note P̂_{1} ∨ P̂_{2} = ¬ (¬P̂_{1} ∧ ¬P̂_{2})) and Lemma 1.2.3. Necessary and sufficient conditions for a given quantum logic to be isomorphic to a sublogic of standard quantum logic are given in (Gudder, 1979).

51For unbounded self-adjoint operators the latter is the appropriate definition for commutativity (Reed und Simon, 1972, Sect. VIII.5).

1959, Theorem VII of §17.4) (or (Riesz and Sz.-Nagy, 1982, Theorem on Page 335)). Use this to show that

Â = Â^{∗} ∈ M ⟹ Ê_{A}(J) ∈ M

holds for all Â ∈ L(H) and all intervals J ⊂ ℝ.

Theorem 1.3.2 (Generalized Gleason Theorem) Let M be a von Neumann algebra with no type I_{2} summand.^{52} Every finitely additive probability measure ω on L_{M} can be extended to a state on M. This state is normal if and only if the corresponding probability measure is completely additive.

Proof: See (Maeda, 1989).

We conclude^{53} that the physically relevant states will always be the normal states on the corresponding von Neumann algebra M.

Definition 1.3.3 A ∗-morphism of a C^{∗}-algebra A_{1} into a C^{∗}-algebra A_{2} is a mapping γ of A_{1} into A_{2} respecting linearity, multiplication, and involution:

\begin{matrix}(M_{1}) : & γ(Â + β B̂) = γ(Â) + β γ(B̂) \\ (M_{2}) : & γ(Â B̂) = γ(Â) γ(B̂) \\ (M_{3}) : & γ(Â^{∗}) = γ(Â)^{∗} \end{matrix}

A ∗-morphism of a C^{∗}-algebra A_{1} into a C^{∗}-algebra A_{2} is called a ∗-isomorphism if it is a bijection (one-one and onto). A ∗-automorphism of a C^{∗}-algebra A is a ∗-isomorphism of A onto itself.

Now Theorem 1.3.2 has the following consequence (remember Definition 1.1.4):

Corollary 1.3.4 Let M be a von Neumann algebra with no type I_{2} summand and let {α_{t}}_{t∈ℝ} be a dynamical semi-group for some system modeled by (L_{M}, ≼, ¬) . Then {α_{t}}_{t≥0} is the restriction to L_{M} of a weakly^{∗} continuous^{54} 1-parameter semi-group of ∗-automorphisms of M.

Proof: See (Lücke, 1996, Appendix A).

Draft, March 26, 2009

52The latter means, if M is a von Neumann subalgebra of L(H) , that there is no projection P̂ ∈ L_{H} onto a 2-dimensional subspace of H for which M = P̂ L(H) P̂ + (ˆ1 − P̂) M (ˆ1 − P̂) (see (Neumark, 1959, §38 Nr. 3 Theorem 2)).

53Remember our assumption (1.11).

54{α_{t}}_{t≥0} is weakly^{∗} continuous iff ω (α_{t}(Â)) is continuous in t for all normal states ω and all Â ∈ M (Bratteli and Robinson, 1979, Propositon 2.4.3).

In the Haag-Doplicher-Roberts theory (see (Haag, 1992)) the relevant von Neumann algebra M is to be constructed by weak closure of a suitable representation of some C^{∗}-algebra. Therefore we have to discuss the latter concept.

Definition 1.3.5 A representation of a C^{∗}-algebra A in the complex Hilbert space H is a ∗-morphism π of A into L(H) . The representation is said to be faithful iff π is an injection.

Theorem 1.3.6 Let γ be a ∗-morphism of the C^{∗}-algebra A_{1} into the C^{∗}-algebra A_{2}. Then γ(A_{1}) is a C^{∗}-subalgebra of A_{2} and:^{55}

∥γ(Â)∥_{A_{2}} ≤ ∥Â∥_{A_{1}} ∀ Â ∈ A_{1} .

Proof: See (Bratteli and Robinson, 1979, Lemma 2.3.1) ((Dixmier, 1969, sections 1.3.7 and 1.8.3)).

Exercise 12 Prove Theorem 1.3.6 for the case that π(A_{1}) is known to be a C^{∗}- subalgebra of A_{2} .

An immediate consequence of Theorem 1.3.6 is the following

Corollary 1.3.7 Let π be a representation of the C^{∗}-algebra A in H. Then π(A) is a C^{∗}-subalgebra of L(H) . If the representation is faithful, then:

∥π(Â)∥_{L(H)} = ∥Â∥_{A} ∀ Â ∈ A .

Warning: Even if A is a von Neumann algebra and π is faithful it may happen^{56} that sup_{L(H)} {π(Â_{i}) : i ∈ I} ∉ π(A) for some net {Â^{i}}_{i∈I} ⊂ A of the type considered in Lemma 1.2.3. Then π(A) is not a von Neumann subalgebra of L(H) and π will not map (L_{A}, ≼, ¬) onto a sublogic of (L, ≼, ¬) .

Draft, March 26, 2009

55Thus application to the special case π =identity shows that the norm of a C^{∗}-algebra is uniquely fixed by the algebraic structure.

56Just consider the GNS representation π_{ω} (Theorem 1.3.8) for the mixture ω of a separating state ω on L(H) (see (Bratteli and Robinson, 1979, Prop. 2.5.6)) with a non-normal state on L(H) (recall Statement (i) of Exercise 9).

1.3.2 GNS-Representation

Theorem 1.3.8 Let A be a C^{∗}-algebra with ˆ1 and let ω be a state on A. Then the set of equivalence classes

[Â_{1}] =^{def} {Â_{2} ∈ A, ω((Â_{1} − Â_{2})^{∗}(Â_{1} − Â_{2})) = 0} , Â_{1} ∈ A

with linear structure

α [Â] + β [B̂] =^{def} [α Â + β B̂]

and inner product

⟨[Â]| [B̂]⟩ =^{def} ω(Â^{∗}B̂)

is a complex pre-Hilbert space. Moreover, continuous extension of the operators

π_{ω}(Â)[B̂] =^{def} [ÂB̂]

onto the completion H_{ω} of this pre-Hilbert space^{57}yields a representation π_{ω} of A in H_{ω}, the so-called GNS-representation of A given by ω. With Ω_{ω} =^{def} [ˆ1] we have

ω(Â) = ⟨Ω_{ω} | π_{ω}(Â) Ω_{ω}⟩ ∀ Â ∈ A

and the vector Ω_{ω} is cyclic with respect to π_{ω}(A) ; i.e. \overline{π_{ω}(A)Ω_{ω}} = H_{ω}.

Exercise 13 Prove Theorem 1.3.8.^{58}

Theorem 1.3.9 Let ω be a normal state on the von Neumann algebra M. Then the GNS-representation π_{ω} is normal, i.e.

sup_{L(H)} {π_{ω}(Â_{i}) : i ∈ I} = π_{ω} (sup_{M} {Â_{i} : i ∈ I})

holds^{59} for every increasing uniformly bounded net {Â_{i}}_{i∈I} ⊂ M. Moreover, π_{ω}(M) is a von Neumann subalgebra of L(H_{ω}) .

Proof: See (Bratteli and Robinson, 1979, Theorem 2.4.24).

Draft, March 26, 2009

57For general *-algebras this extension should be skipped; see also (Antoine and Ôta, 1989). 58Hint: First of all show, using Schwarz’s inequality (statement (i) of Exercise 7), that [0] is

a left ideal of A, i.e. that [0] is a linear subspace of A and that Â [0] ⊂ [0] ∀Â ∈ A (see (Bratteli and Robinson, 1979, Sect. 2.2.3)).

59This means, if ω′ is a normal state on π_{ω}(M) then ω̂(Â) =^{def} ω′(π_{ω}(Â) is a normal state on M.

Definition 1.3.10 Let A be a C^{∗}-algebra and let π_{1}, π_{2} be representations of A in H_{1} resp. H_{2} . Then π_{2} is said to be unitarily equivalent to π_{1} iff there is a unitary mapping Û of H_{1} onto H_{2} fulfilling

Û π_{1}(Â) = π_{2}(Â) Û ∀Â ∈ A .

Corollary 1.3.11 Let ω be a state on the C^{∗}-algebra A with ˆ1, let π be a representation of A in H, and let Ω be a cyclic Vector (with respect to π(A)) fulfilling

ω(Â) = ⟨Ω | Â Ω⟩ ∀ Â ∈ A

Then, according to Theorem 1.3.8, π is unitarily equivalent to the GNS-representation π_{ω}.

Exercise 14 Prove Corollary 1.3.11 and show^{60} that – contrary to what Bratteli and Robinson claim (Bratteli and Robinson, 1979, beginning of Section 2.4.4) – equality of the sets of vector states belonging to the representations π_{1}, π_{2} of A does not imply unitary equivalence of π_{1} and π_{2}, in general.^{61}

Definition 1.3.12 Let π be a representation in H of the C^{∗}-algebra A. Then π is said to be (topologically) irreducible iff H and {0} are the only closed subspaces of H that are mapped into themselves by all Â ∈ A.^{62} Otherwise π is said to be reducible.

Exercise 15 Let H_{1}, H_{2} be complex Hilbert spaces and let ω be a pure normal state on the von Neumann subalgebra

M =^{def} {Â_{1} ⊗ ˆ1 Â_{1} ∈ L(H_{1})}

of L(H_{1} ⊗ H_{2}) . Show that the GNS-representation π_{ω} of M is irreducible and unitarily equivalent to the representation

π(Â_{1} ⊗ ˆ1) =^{def} Â_{1} ∀Â_{1} ∈ L(H_{1})

of M in H_{1} .

Draft, March 26, 2009

60Inspect the direct sums π_{1} =^{def} ⊕_{t∈R^{1}}π and π_{2} =^{def} ⊕_{n=1}^{∞}π of one and the same representation π.

61In case both π_{1} and π_{2} are irreducible Bratteli’s and Robinson’s statement is a consequence of Corollary 1.3.11 and Theorem 1.3.13, below.

62That cyclicity does not imply irreducibility, in general, can be clearly seen by inspecting the algebra of all functions of the position observable in the Schrödinger representation (compare Section 1.3.3).

Theorem 1.3.13 Let ω be a state on the C^{∗}-algebra A with ˆ1 and let π_{ω} be the corresponding GNS-representation of A in H_{ω}. Then the following four statements are equivalent:

1. π_{ω} is irreducible.

2. ω is pure.

3. Every Ψ ∈ H_{ω} \ {0} is cyclic with respect to π_{ω}(A) .

4. (π_{ω}(A))′ = {αˆ1 : α ∈ ℂ}.

Proof: See (Bratteli and Robinson, 1979, Prop. 2.3.8 and Theorem 2.3.19.)

Exercise 16 Let A be a C^{∗}-algebra with ˆ1. Prove the following three statements:^{63}

(i) A is the linear span of all its unitary^{64} elements Û.

(ii) Every representation of A is unitarily equivalent to a suitable direct sum of either cyclic or trivial representations of A.

(iii) A cyclic representation π of A is unitarily equivalent to a suitable GNSrepresentation iff it is nontrivial, i.e. iff π(Â) ≠ 0 for at least one Â ∈ A.

Concluding remark: Let A be a C^{∗}-algebra (with ˆ1). Then one may prove (Bratteli and Robinson, 1979, Lemma 2.3.23) that for every Â ∈ A \ {0} there is a state ω on A for which ω(Â) ≠ 0. Hence, if E_{A} denotes the set of all states on A,

π = ⨁_{ω∈E_{A}} π_{ω}

is a faithful representation of A. This shows that every C^{∗}-algebra is *- isomorphic to a suitable C^{∗}-subalgebra of L(H), for suitable H !

Draft, March 26, 2009

63Hint for (i): First show that the formal Taylor expansion of B̂ ± i \sqrt{ˆ1 − B̂^{2}} with respect to the ‘variable’ B̂ converges in A for self-adjoint B̂ ∈ A with ∥B̂∥ < 1 and the limit is unitary.

Hint for (ii): Use (i) and Zorn’s lemma. 64Here, unitarity of Û means Û^{∗} = Û^{−1} .

1.3.3 Canonical Quantization

In elementary quantum mechanics of n ‘1-dimensional’ distinguishable particles without inner degrees of freedom and without (further) constraints one uses the state space

H_{n} =^{def} L^{2}(ℝ^{n}, dx_{1}, …, dx_{n})

and the time-zero position operators

(x̂_{ν}Ψ)(x_{1}, …, x_{n}) =^{def} x_{ν}Ψ(x_{1}, …, x_{n})  (1.39)

with obvious domains.^{65}

Exercise 17 Let M_{pos} denote the smallest von Neumann subalgebra of L(H_{n}) containing all bounded functions of every x̂_{ν} . Prove the following four statements:

(i) M_{pos} is maximally abelian, i.e. M_{pos} coincides with its commutant^{66}

M′_{pos} = {B̂ ∈ L(H_{n}) : [B̂, Â] = 0 ∀Â ∈ M_{pos}}

(ii) H_{n} contains a dense set of vectors which are all cyclic with respect to the identical representation of M_{pos} .

(iii) The identical representation of M_{pos} is reducible.

(iv) The von Neumann logic of M_{pos} does not contain any atom.

Translation of particle ν at time zero by a_{ν} corresponds to a symmetry of the time zero standard logic. The corresponding ∗-automorphism α_{a_{ν}} is implemented by the unitary operator Û_{ν}(−a_{ν}) defined by

[Û_{ν}(−a_{ν})ψ](x_{1}, …, x_{n}) =^{def} ψ(x_{1}, …, x_{ν} − a_{ν}, …, x_{n}) ,  (1.40)

i.e.:

α_{a_{ν}}(Â) = Û_{ν}(−a_{ν}) Â Û_{ν}(−a_{ν})^{−1} .

By Equation (1.39), the operators Û_{ν}(τ) for fixed ν fulfill the relation

Û_{ν}(τ_{1})Û_{ν}(τ_{2}) = Û_{ν}(τ_{1} + τ_{2})

Draft, March 26, 2009

65Compare Exercise 2, Statement (i), and Equation (1.23).

66First show that

\begin{matrix}∫ \overline{ϕ(x_{1}, …, x_{n})}(T̂ψ)(x_{1}, …, x_{n})dx_{1} … dx_{n} \\ = ∫ \overline{ϕ(x_{1}, …, x_{n})}(T̂χ_{M})(x_{1}, …, x_{n})ψ(x_{1}, …, x_{n})dx_{1} … dx_{n} \end{matrix}

holds for T̂ ∈ M′_{pos} and ϕ, ψ ∈ H_{n} whenever suppϕ ⊂ M .

and depend continuously on the parameter τ , i.e. they form a continuous 1parameter unitary group. Hence, by Stone’s theorem, there exist unique selfadjoint operators p̂_{1}, …, p̂_{n} with

Û_{ν}(τ) = e^{+ \frac{i}{ℏ} p̂_{ν}τ} for ν = 1, …, n .  (1.41)

p̂_{ν} is interpreted as time zero momentum operator for particle ν. If ψ is sufficiently regular we have the Taylor expansion

ψ(x_{1}, …, x_{ν} + τ, …, x_{n}) = e^{τ∂_{xν}} ψ(x_{1}, …, x_{n})

which, by (1.40) and (1.41), gives

(p̂_{ν}ψ)(x_{1}, …, x_{n}) = \frac{ℏ}{i}∂_{x_{ν}}ψ(x_{1}, …, x_{n}) .  (1.42)

This, finally, yields the canonical commutation relations

\begin{matrix}[p̂_{ν}, x̂_{μ}] = \frac{ℏ}{i}δ_{νμ}ˆ1 \\ [x̂_{ν}, x̂_{μ}] = [p̂_{ν}, p̂_{μ}] \end{matrix} \} on a suitable domain.  (1.43)

Exercise 18 Explain why Heisenberg’s uncertainty relations are not valid for the angular momentum L_{3} and its corresponding angular variable φ ∈ [0, 2π) of an ordinary quantum mechanical particle (in ℝ^{3}) even though their observables obey the canonical commutation relations

[L̂_{3}, φ̂] = \frac{ℏ}{i}ˆ1 on some invariant dense domain.

In order to avoid domain problems one replaces the fundamental relations (1.43) by the corresponding ones obeyed by the continuous 1-parameter groups (1.41) and

V̂_{μ}(s) =^{def} e^{i x̂_{μ} s}  (1.44)

i.e. by the so-called Weyl relations:

Û_{ν}(τ) V̂_{μ}(s) = e^{i τ s δ_{νμ}} V̂_{μ}(s) Û_{ν}(τ) ,  (1.45)
Û_{ν}(τ_{1}) Û_{ν}(τ_{2}) = Û_{ν}(τ_{1} + τ_{1}) , V̂_{μ}(s_{1}) V̂_{μ}(s_{2}) = V̂_{μ}(s_{1} + s_{1}) ,
Û_{ν}(−τ) = Û_{ν}(τ)^{∗} ≠ 0 ≠ V̂_{μ}(−s) = V̂_{μ}(s)^{∗} .

Remark: (1.45) shows that

(p_{1}, …, p_{n}; x_{1}, …, x_{n}; t) ⟼ e^{2 π i h t}e^{π i h ∑_{ν=1}^{n} p_{ν}x_{ν}} ∏_{μ=1}^{n} V̂_{μ}(2πx_{μ})Û_{μ}(hp_{μ})

is a representation of the Heisenberg group H_{n}, i.e. of ℝ^{2n+1} with multiplication

\begin{matrix}(p_{1}, …, p_{n}; x_{1}, …, x_{n}; t)(p′_{1}, …, p′_{n}; x′_{1}, …, x′_{n}; t′) \\ =^{def} & (p_{1} + p′_{1}, …, p_{n} + p′_{n}; x_{1} + x′_{1}, …, x_{n} + x′_{n}; t + t′ + \frac{1}{2} ∑_{ν=1}^{n} (p_{ν}x′_{ν} − p′_{ν}x_{ν})) \end{matrix}

(compare (Folland, 1989, Sect. 1.2)).

Theorem 1.3.14 There is a C^{∗}-algebra A_{n}^{B} , unique up to *-isometry, that is generated by elements Û_{ν}(τ) , V̂_{μ}(s) (ν, μ ∈ {1, …, n} ; τ, s ∈ ℝ) fulfilling the Weyl relations.

Proof: See (Bratteli and Robinson, 1981, Theorem 5.2.8).

The representation of the so-called CCR-algebra A_{n}^{B} given by the C^{∗}-subalgebra of L(H_{n}) generated by all Û_{ν}(τ) and V̂_{μ}(s) defined above is called the Schrödinger representation . From now on, let us identify A_{n}^{B} with its Schrödinger representation.

Exercise 19 Prove^{67} that the Schrödinger representation of A_{n}^{B} is irreducible.

Every state ω on A_{n}^{B} is uniquely fixed by^{68} its generating functional

E_{ω}(z_{1}, …, z_{n}) =^{def} ω (Ŵ_{1}(z_{1}) … Ŵ_{n}(z_{n})) ,  (1.46)

where: Ŵ_{ν}(s + iτ) =^{def} Û_{ν}(\frac{τ}{2}) V̂_{ν}(s) Û_{ν}(\frac{τ}{2}) .

If one takes for ω the so-called Fock ground state^{69}

ω_{F}(Â) =^{def} ⟨Ω_{F} | Â Ω_{F}⟩ , where: Ω_{F}(x_{1}, …, x_{n}) = π^{− \frac{n}{4}} e^{− \frac{1}{2} (x_{1}^{2}+...x_{n}^{2})} ,

one gets the so-called Fock functional

E_{F}(s_{1}, τ_{1}; …) =^{def} E_{ω_{F}}(s_{1} + iτ_{1}; …) ,

which uniquely characterizes the so-called Fock representation, i.e. the GNSrepresentation π_{F} =^{def} π_{ω_{F}} . According to Exercise 19 and Corollary 1.3.11 the Fock representation is unitary equivalent to the Schrödinger representation.

Exercise 20 Show that^{70}

E_{F}(s_{1}, τ_{1}; …; s_{n}, τ_{n}) = exp (−\frac{1}{4}(s_{1}^{2} + τ_{1}^{2} + … s_{n}^{2} + τ_{n}^{2})) .

Draft, March 26, 2009

67Recall Statement (i) of Exercise 17 and Theorem 1.3.13.

68Recall (1.45) and Exercise 13.

69The ground state of n independent harmonic oscillators with mass and circular frequency equal to 1.

70Recall (1.40) and note that ∫_{−∞}^{+∞} e^{−(z+z_{0})^{2}} dx = \sqrt{π} for arbitrary complex z_{0} .

Definition 1.3.15 A representation π of A_{n}^{B} is called regular if the π (Û_{ν}(τ))

π (V̂_{μ}(τ)) depend strongly continuously on the parameter τ and coincide with the unit operator for τ = 0 .

Obviously, the Fock representation of A_{n}^{B} is regular.

Theorem 1.3.16 (Stone - von Neumann) Let π be a regular^{71} representation of A_{n}^{B} (with finite n) in H. Then π is unitarily equivalent to some direct sum of Fock representations (resp. Schrödinger representations) of A_{n}^{B} .

Proof: See (Bratteli and Robinson, 1981, Corollary 5.2.15) or (Folland, 1989, pp 35–36).

Exercise 21 Using the results of Exercise 20, prove Theorem 1.3.16.^{72}

Let M denote the von Neumann subalgebra of L(H_{tot}) generated by those nonrelativistic n-particle observables which refer only to the motion of identifiable particles with respect to one space dimension but not to inner degrees of freedom. If – as usual – we assume M to be ∗-isomorphic to L (L^{2}(ℝ^{3n}, dx_{1}, …, dx_{n})), interpreting (1.41) and (1.44) in the standard way (compare Exercise 15), we see from Theorem 1.3.16:

The partial states of physically realizable ensembles of the kind described above correspond to regular states^{73} ω on the C^{∗}-algebra A_{n}^{B} characterized by the canonical commutation relations in Weyl form. Here the self-adjoint elements of A_{n}^{B} may be interpreted as time-zero observables (‘quantum kinematics’) in accord with 1.41, 1.44, and 1.45. Time evolution in the sense of Section 1.1.4 has to be unitary, according to Theorem 1.1.4.

In this sense replacement of the complex numbers (c-numbers) p_{ν}, q_{ν} by operators p̂_{ν}, q̂_{ν} (q-numbers) fulfilling the commutation Relations (1.43) (as a consequence of (1.45)) is called a quantization of the system described above.

Draft, March 26, 2009

71For the classification of strongly measurable (not necessarily regular) representations in nonseparable Hilbert spaces see (Cavallaro et al., 1998).

72Show, first of all, that

Ω =^{def} lim_{ϵ→0} π (∏_{ν=1}^{n} (∫ ds e^{− \frac{s^{2}}{2}} V̂_{ν}(s)) (∫ dτ e^{−(ϵτ)^{2}} Û_{ν}(τ))) ψ

is normalized for suitable ψ and that the generating functional E_{ω}, corresponding to the vector state

ω(Â) =^{def} ⟨Ω | π(Â) Ω⟩ ∀ Â ∈ A_{n}^{B} ,

coincides with the Fock functional.

73I.e. π_{ω} is regular.

For infinitely many degrees of freedom (n → ∞) the situation is much more complicated:

Let {Ψ_{j}}_{j∈ℤ_{+}} be complete orthonormal system of H_{1} = L^{2}(ℝ^{1}, dx) with Ψ_{0} = Ω_{F}(x_{1}) and consider

{(Ψ_{j_{1}}, Ψ_{j_{2}}, …, Ψ_{0}, Ψ_{0}, …)}_{j_{1},j_{2},...∈ℤ_{+}}

as a complete orthonormal system of a complex Hilbert space (⨂_{ν=1}^{∞} H_{1})^{Ω_{F}^{∞}}, where Ω_{F}^{∞} =^{def} (Ψ_{0}, Ψ_{0}, …) , called an infinite tensor product of H_{1} with itself. Then, given n, the definition

π_{n}(\underbrace{Ŵ_{ν}(z)}_{∈A_{n}^{B}})(Ψ_{j_{1}}, …, Ψ_{j_{ν}}, …) =^{def} (Ψ_{j_{1}}, …, Ŵ_{1}(z)Ψ_{j_{ν}}, …) ,

for ν = 1, …, n and complex z (compare (1.46)), fixes a representation π_{n} of A_{n}^{B} that is unitary equivalent to the corresponding Schrödinger representation.

The C^{∗}-subalgebra A_{∞}^{B} of A_{∞}^{B} of L ((⨂_{n=1}^{∞} H_{1})^{Ω_{F}^{∞}}) generated by ⋃_{n=1}^{∞} π_{n}(A_{n}^{B}) is called the CCR-algebra. It is considered to be the algebra of time zero observables of a Bose system with ‘infinitely many degrees of freedom’ (see also (Yurtsever, 1994) and (Borchers, 1996, Sect. I.2)). Because of

π_{ν}(A_{ν}^{B}) ⊂ π_{ν+1}(A_{ν+1}^{B})

the Weyl relations 1.45 hold also for

Û_{ν}^{∞}(τ) = π_{ν} (Û_{ν}(τ)) V̂_{ν}^{∞}(s) = π_{ν} (V̂_{ν}(s)) ,
where Û_{ν}(τ), V̂_{ν}(s) ∈ A_{ν}^{B} ,

instead of Û_{ν}(τ), V̂_{ν}(s) for arbitrary ν, μ ∈ ℕ.

The regular state ω_{F} on the CCR-algebra with generating functional

E_{F}(z_{1}, …, z_{n}, 0, 0, …) =^{def} ⟨Ω_{F}| Ŵ_{1}^{∞}(z_{1}), …, Ŵ_{n}^{∞}(z_{n}) Ω_{F}^{∞}⟩

(compare (1.46)) is called the Fock vacuum. The corresponding GNS-representation π_{F} = π_{ω_{F}} is called the Fock representation of A_{∞}^{B} .

Exercise 22 Show that the identical representation of A_{∞}^{B} is unitary equivalent to the Fock representation and irreducible.

1.3.4 Spontaneously Broken Symmetries

Let Û_{1}, Û_{2}, … be unitary operators in H_{1} and define

Ω_{N}^{∞} =^{def} (Û_{1}Ψ_{0}, …, Û_{N}Ψ_{0}, Ψ_{0}, Ψ_{0}, …) for N = 1, 2, ….

Then, for finite N, the GNS-representation π_{ω_{N}} of A_{∞}^{B} for the vector state

ω_{N}(Â) =^{def} ⟨Ω_{N}^{∞}| ÂΩ_{N}^{∞}⟩

is unitary equivalent to the Fock representation. In the limit N → ∞, however, we get a state ω_{∞} the GNS-representation π_{ω_{∞}} of which is regular, irreducible, and faithful^{74} but, in general,^{75} not unitary equivalent to the Fock representation.

Exercise 23 Show, by Theorem 1.3.13, that the von Neumann completion of π_{ω_{N_{1}}}(A_{∞}^{B}) in H_{ω_{N_{1}}} is isomorphic to the von Neumann completion of π_{F}(A_{∞}^{B}) in H_{ω_{∞}}. Explain why, nevertheless, π_{ω_{N_{1}}} may be unitary inequivalent to π_{F} .

The above construction shows us that Theorem 1.3.16 does not hold for n = ∞ but that there is a myriad of – up to know unclassified – physically relevant, regular, irreducible representations of A_{∞}^{B} which are inequivalent to the Fock representation !

Problem: Which is the von Neumann algebra M corresponding to A_{∞}^{B} in the sense of Section 1.3.1 and how is A_{∞}^{B} embedded into M ?

As already pointed out, one does not always know the von Neumann algebra M of the considered partial theory in the sense of Section 1.3.1, but only – up to C^{∗}-algebra isometry – a C^{∗}-subalgebra A of M generating M.

This is the reason for using also C^{∗}-algebras which are not von Neumann algebras, in quantum statistical mechanics and relativistic quantum field theory.

Regarding the physical relevance of states^{76} we then need suitable criteria (such as regularity of states over A_{N}^{B}).

Draft, March 26, 2009

74Limit states of a similar kind play an important role in statistical quantum mechanics (Araki and Woods, 1963) and constructive quantum field theory (Wightman, 1967).

75One may prove that

(Φ_{1}, Φ_{2}, …) ∈ (⨂_{ν=1}^{∞} H_{1})^{Ω_{F}^{∞}} ⟺ ∑_{ν=1}^{∞} |1 − ⟨Ψ_{0}| Φ_{ν}⟩| < ∞ .

76The main problem is to characterize those those partial states on A which are restrictions of normal states on M.

Definition 1.3.17 A state ω on the C^{∗}-algebra A is called normal ^{77} with respect to the representation π on A in H, if there is an operator T̂_{ω} ∈ T_{1}(H) with

ω(Â) = Tr (T̂_{ω}π(Â)) ∀Â ∈ A .

The set of all states which are normal with respect to π is called the folium S_{π} corresponding to π.

Definition 1.3.18 Two representations π_{1}, π_{2} of a C^{∗}-algebra A are called quasi equivalent if their folia coincide, i.e. if S_{π_{1}} = S_{π_{2}} .

Theorem 1.3.19 Let π_{1}, π_{2} be non-degenerate (i.e. π_{j}(A)Ψ = {0} ⟹ Ψ = 0) representations of the C^{∗}-algebra A. Then π_{1}, π_{2} are quasi equivalent if and only if a suitable direct sum of π_{1} is unitary equivalent to a suitable sum of π_{2} .

Proof: See (Bratteli and Robinson, 1981, Theorem 2.4.26).

Definition 1.3.20 A physical symmetry corresponding to the ∗-automorphism (or ∗-anti-automorphism) φ of the C^{∗}-algebra A is said to be spontaneously broken by the state ω on A if the GNS-representations corresponding to ω and φ_{∗}ω are not quasi-equivalent, i.e. if S_{π_{ω}} ≠ S_{π_{φ∗ω}} .

Exercise 24 Let A be a C^{∗}-subalgebra of L(H) and let Û be an anti-unitary operator on H.

(i) Show that

γ(Â) =^{def} ÛÂ^{∗}Û^{∗} for Â ∈ A

defines a ∗-antiautomorphism, i.e. for all Â , B̂ ∈ A and all z ∈ ℂ :

\begin{matrix}(A_{1}) : & γ(Â + zB̂) = γ(Â) + z γ(B̂) , \\ (A_{2}) : & γ(ÂB̂) = γ(B̂)γ(Â) , \\ (A_{3}) : & γ(Â^{∗}) = γ(Â)^{∗} . \end{matrix}

(ii) Show that γ, as defined above, is the only ∗-antiautomorphism of A into L(H)

with

γ(P̂) = ÛP̂Û^{∗} ∀ P̂ ∈ L_{H} .

Draft, March 26, 2009

77In view of Theorem 1.2.5

38 CHAPTER 1. GENERAL QUANTUM THEORY

Exercise 25 Show that symmetries may be broken by regular states on A_{∞}^{B} but not by regular states on A_{N}^{B}, N < ∞.

Remarks:

(i) In physically relevant theories (like QED) the symmetries corresponding to homogeneous Lorentz transformations are spontaneously broken (see e.g. (Buchholz, 1986)).

(ii) For interesting speculations regarding spontaneous breaking of time translation symmetry see (Rieckers, 1985).

(iii) Spontaneous breaking of gauge symmetries plays a decisive role in the Weinberg-Salam theory of electroweak interactions (see e.g. (Mohapatra, 1986) and (Watkins, 1986)).

Exercise 26 Let ω be a state and φ a ∗-automorphism of the C^{∗}-algebra A. Show^{78} that the representation π of A, defined by

π(Â) =^{def} π_{ω} (φ^{−1}(Â)) for Â ∈ A ,

is unitary equivalent to π_{φ_{∗}ω} but not necessarily to π_{ω}.

Draft, March 26, 2009

78Hint: First, show that

Ûπ_{φ_{∗}ω}(Â)Ω_{φ_{∗}ω} =^{def} π_{ω} (φ^{−1}(Â)) Ω_{ω} for Â ∈ A

(consistently!) defines a unitary mapping Û from H_{φ_{∗}ω} onto H_{ω}.

Massive Scalar Fields

2.1 Free Neutral Scalar Fields

We are going to describe systems of noninteracting, indistinguishable, relativistic point ‘particles’ on Minkowski space^{1} with rest mass m > 0 having no internal degrees of freedom and no charge. We use natural units throughout, especially

c = ℏ = 1 .

2.1.1 1-Particle Space

Momentum Space Representation

The three-momentum of a classical relativistic point particle is

p = m_{v} v ,

where v is its (three-)velocity and

m_{v} = \frac{m}{\sqrt{1 − |v|^{2}}}

its inertial mass coinciding^{2} – thanks to natural units – with its energy

p^{0} = ω_{p} =^{def} \sqrt{m^{2} + |p|^{2}} > 0 .

Therefore,^{3}

v = \frac{p}{ω_{p}} .

Draft, March 26, 2009

1For generalization to curved space-time see, e.g. (Verch, 1997) and references given there. 2Just compare the squares of m_{v} and p^{0} . 3This, obviously, is consistent with \frac{d}{dt}ω_{p(t)} = v(t) · \frac{d}{dt}p(t) .

[figure: ^{123}; _{−3 −2 −1  1 2 3}; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; p^{0}/m; p^{1}/m]

Figure 2.1: 1-particle mass-shell, restricted to the p^{0}-p^{1}-plane

Moreover, we have

p · p =^{def} p^{0}p^{0} − p · p = m^{2} , p^{0} > 0  (2.1)

for its four-momentum

p = (p^{0}, p^{1}, p^{2}, p^{3}) = (p^{0}, p) .

When changing the inertial system (but not the origin of the resp. system) p has to be transformed by the same Lorentz matrix as x = (x^{0}, x):

x, p ⟶^{change of ref.syst.} x′ = Λx, p′ = Λp  (2.2)

Exercise 27 Let \{\underline{p}(s)\}_{s∈ℝ} be some (sufficiently well behaved) curve on the 1-

particle mass shell M_{m} =^{def} {p ∈ ℝ^{4} : p^{0} = ω_{p}}. Let (p^{0}(s), p^{1}(s), p^{2}(s), p^{3}(s)) resp. (p′^{0}(s), p′^{1}(s), p′^{2}(s), p′^{3}(s)) be the coordinates of \underline{p}(s) in the inertial System L resp. L′, related to each other by a special Lorentz transformation:

p′^{0} = \frac{p^{0} − u p^{1}}{\sqrt{1 − u^{2}}} , p′^{1} = \frac{p^{1} − u p^{0}}{\sqrt{1 − u^{2}}} , p′^{2} = p^{2} , p′^{3} = p^{3}

(u fixed). Show that

\frac{\frac{dp′^{1}}{ds}}{ω_{p′}} = \frac{\frac{dp^{1}}{ds}}{ω_{p}} .

Quantum mechanical 1-particle state space:

The pure 1-particle states are given – in the sense of orthodox quantum mechanics – by the vectors of the separable complex Hilbert space

H_{0}^{(1)} =^{def} L^{2} (ℝ^{3}, \frac{dp}{2ω_{p}} )

with inner product

⟨f̌| ǧ⟩ =^{def} ∫ \overline{f̌(p)} ǧ(p) \frac{dp}{2ω_{p}} ∀ f̌, ǧ ∈ H_{0}^{(1)} .  (2.3)

Obviously,

(Û_{0}(a, Λ)f̌) (p) =^{def} (e^{ipa}f̌(\overrightarrow{Λ^{−1}p}))_{|_{p^{0}=ωp}}  (2.4)

defines a representation of P_{+}^{↑} (‘restricted’ Poincaré group), i.e.

Û_{0}(a_{2}, Λ_{2}) Û_{0}(a_{1}, Λ_{1}) = Û_{0}(Λ_{2}a_{1} + a_{2}, Λ_{2}Λ_{1}) ∀ (a_{2}, Λ_{2}) , (a_{1}, Λ_{1}) ∈ P_{+}^{↑} ,

which is (strongly) continuous. By Exercise 27 the measure \frac{dp}{2ω_{p}} is Lorentz in-

variant. Hence the representation (2.4) is unitary, i.e.:

⟨Û_{0}(a, Λ)f̌| Û_{0}(a, Λ)ǧ⟩ = ⟨f̌| ǧ⟩ ∀f̌, ǧ ∈ H_{0}^{(1)} .  (2.5)

It is to be interpreted as follows:

Û_{0}(a, Λ)f̌ corresponds to an ensemble that, with respect to the coordinates x′ =^{def} Λ^{−1}(x − a) , is to be described in exactly the same way as an ensemble corresponding to f̌ is to be described with respect to the coordinates x.

(2.6)

According to (2.4), with the projection-valued measure

Ê_{0}(J) f̌(p) =^{def} \{ \begin{matrix}f̌(p) & if (ω_{p}, p) ∈ J \\ 0 & otherwise \end{matrix} for Borel sets J ⊂ ℝ^{4}  (2.7)

we have^{4}

Û_{0}(a) =^{def} Û_{0}(a, 1l_{4}) = exp (i\overline{P̂_{0} a}) = ∫ e^{ipa} Ê_{0}(dp) ,
where P̂_{0} =^{def} ∫ p Ê_{0}(dp) .

(2.8)

Draft, March 26, 2009

4As usual, \overline{P̂_{0} a} denotes the closure of the (essentially self-adjoint) operator P̂_{0}a.

42 CHAPTER 2. MASSIVE SCALAR FIELDS Ê_{0} is to be interpreted as the spectral measure for 4-momentum, i.e. for ∥f̌∥ = 1 :

⟨f̌| Ê_{0}(J)f̌⟩ =_{(2.7)} ∫_{(ω_{p},p)∈J} |f̌(p)|^{2} \frac{dp}{2ω_{p}}
=_{Interpret.} probability for: p ∈ J in a state ˆ= f̌
= probability for: p ∈ J ∩ M_{m} in a state ˆ= f̌.

(2.9)

This is equivalent to interpreting P̂_{0} as energy-momentum operator (= observable of 4-momentum):

⟨f̌| P̂_{0}^{μ} f̌⟩ = ∫_{p^{0}=ω_{p}} p^{μ} |f̌(p)|^{2} \frac{dp}{2p^{0}}
= \{ \begin{matrix}expectation value for the μ-component of the \\ (time-independent) 4-momentum in a state & ˆ= & f̌ \end{matrix} (2.10)

(if ∥f̌∥ = 1). This interpretation is also suggested by the relations

Û_{0}(a, Λ)^{−1}P̂_{0}^{μ} Û_{0}(a, Λ) = Λ_{ν}^{μ} P̂_{0}^{ν}  (2.11)

and

P̂_{0}^{0} ≥ mˆ1 , P̂_{0} · P̂_{0} = m^{2} ˆ1 .  (2.12)

Space-Time Representation Instead of the f̌(p) one may also use the corresponding wave functions^{5}

f^{+}(x) =^{def} (2π)^{−3/2} ∫_{p^{0}=ω_{p}} f̌(p) e^{−ipx} \frac{dp}{2p^{0}} ,  (2.13)

which uniquely characterize the f̌(p) due to

f̌(p) = (2π)^{−3/2} ∫_{p^{0}=ω_{p}} e^{ipx} i∂^{↔}_{0} f^{+}(x) dx ∀ x^{0} ∈ ℝ ,  (2.14)

where

g(x)∂^{↔}_{0}f(x) =^{def} g(x)\frac{∂}{∂x^{0}} f(x) − (\frac{∂}{∂x^{0}} g(x)) f(x) .  (2.15)

Exercise 28 Show, for sufficiently well behaved f̌(p) , that

ȷ_{μ}(x) =^{def} \overline{f^{+}(x)} i∂^{↔}_{μ}f^{+}(x)

Draft, March 26, 2009

5Assume the f̌(p) to be sufficiently well behaved, for the moment.

is a conserved Lorentz vector field^{6} with

∫ j^{0}(x) dx = ∥f̌∥^{2} ∀ x^{0} ∈ ℝ ,

but that j^{0}(x) is not nonnegative,^{7} in general.

The f^{+}(x) transform according to

ǧ(p) = Û_{0}(a, Λ)f̌(p) ⟹_{(2.4)} g^{+}(x) = f^{+} (Λ^{−1}(x − a))  (2.16)

and are solutions of the Klein-Gordon equation

(□ + m^{2}) f^{+}(x) = 0  (2.17)

(□ =^{def} (∂_{0})^{2} − ∇ · ∇). The idea is that |f^{+}(x)|^{2} describes at least roughly the localization in space-time. There are fundamental obstructions^{8} for defining a position operator in the sense of Sect. 1.2.2 (Hegerfeldt, 1974),(Hegerfeldt and Ruijsenaars, 1980). The most natural definition^{9} would be the one given by Newton and Wigner, according to which – in full analogy to nonrelativistic quantum mechanics – the 3-dimensional Fourier transform of the time-dependent momentum amplitude

e^{−iω_{p}x^{0}} \frac{f̌(p)}{\sqrt{2ω_{p}}}

(compare (2.9)) is interpreted as position amplitude; i.e.

∫_{V} |f_{N.W.}(x)|^{2} dx = probability for “x ∈ V at time x^{0} in a state ˆ= f̌” (2.18)

(if ∥f∥ = 1), where^{10}

f̌_{N.W.}(p) = \frac{f̌(p)}{\sqrt{2ω_{p}}} ,
f_{N.W.}(x) =_{(2.13)} (2π)^{−3/2} ∫_{p^{0}=ω_{p}} f̌_{N.W.}(p)e^{−ipx} dp .

(2.19)

Here one easily realizes the following problem:^{11}

Draft, March 26, 2009

6I.e.: ∂_{μ}ȷ^{μ} = 0 , \overline{f^{+}(Λ^{−1}(x − a))} i∂^{↔}_{μ}f^{+}(Λ^{−1}(x − a)) = Λ_{ν}^{μ} ȷ^{ν}(Λ^{−1}(x − a)).

f̌(p) ∼ ω_{p} (δ_{ϵ}(p − p_{1}) − \frac{1}{2}δ_{ϵ}(p − p_{2}))

7Check ȷ^{0}(0) for where δ_{ϵ} is sufficiently close to the delta function and p_{1} , p_{2} are fixed momenta with |p_{1}| < |p_{2}|.

8From the relativistic point of view this is quite satisfactory (see (Crewther, 1995, Sect. 1)).

9See (Wightman, 1962) for a very detailed discussion.

10Note that ∥f̌∥ = ∫ |f_{N.W.}(x)|^{2} dx.

11See Exercise 29, below. A consistent definition of strict localization for relativistic quantum field theory was given in (Knight, 1961) – not on the 1-particle level, of course.

Even though – in the relativistic theory – the velocity is bounded by |v| ≤ 1, there can exist at most one instant of time at which the ‘particle’ is localized within a bounded space region in the sense of Newton and Wigner!

As usual, let us denote, for n ∈ ℕ, by S(ℝ^{n}) the linear topological space of all complex-valued C^{∞} functions φ(ξ_{1}, …, ξ_{n}) on ℝ^{n} for which all the norms

∥φ∥_{N} =^{def} sup_{(ξ_{1},...,ξ_{n})∈ℝ^{n}} |1 + ∑_{ν=1}^{n} ξ_{ν}^{2}|^{N} sup_{(α_{1},...,α_{n})∈ℤ_{+}^{n}_{α_{1}+...+αn<N}} |(\frac{∂}{∂ξ_{1}} )^{α_{1}} ⋯ (\frac{∂}{∂ξ_{n}} )^{α_{n}} φ(ξ_{1}, …, ξ_{n})|

(N ∈ ℕ) are finite; the sets

U_{N,ϵ} =^{def} {φ ∈ S(ℝ^{n}) : ∥φ∥_{N} < ϵ}

(N ∈ ℕ , ϵ > 0) forming a basis of open neighborhoods of 0, by definition.

A function f^{+}(x) is called a positive frequency smooth Klein-Gordon solution iff it is of the form (2.13) with f̌ ∈ S (ℝ^{3}). The function f^{−}(x) is called a negative frequency smooth Klein-Gordon solution iff it is the complex conjugate of a positive frequency Klein-Gordon solution: f^{−}(x) = \overline{f^{+}(x)}. Finally, the function f(x) is called a smooth Klein-Gordon solution iff it is of the form

f(x) = f^{+}(x) + f^{−}(x)

with f^{+}(x) resp. f^{−}(x) a positive resp. negative frequency smooth Klein-Gordon solution.^{12}

Exercise 29 Let f^{+} be a positive frequency smooth Klein-Gordon solution. Using the easy part of the Paley-Wiener theorem (see, e.g., (Gårding and Lions, 1959, Theorem 7^{.}1.5.)), saying that the Fourier transform of a (generalized) function with compact support is an entire analytic function, prove the following statements:

(i) There is at most one instant of time x^{0} for which f^{+}(x) , considered as a function of x , vanishes outside some bounded subset of ℝ^{3} .

(ii) There is no instant of time x^{0} for which both f^{+}(x) and \frac{∂}{∂x^{0}} f^{+}(x) vanish outside some bounded subset of ℝ^{3} .

As mentioned above, in spite of the obstructions for defining a fully satisfactory position operator, the transformation rule (2.16) suggests that |f^{+}(x)|^{2} describes at

Lemma 2.1.1 (Ruelle) Draft, March 26, 2009

12Note that this decomposition of f(x) is unique.

2.1. FREE NEUTRAL SCALAR FIELDS 45

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; Kf̌; p^{0}; p^{1}; p^{2} = m^{2}; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; supp f̌]

Figure 2.2: Velocity cone Kf̌ ( = asymptotic localization region for f(x)), restricted to the p^{0}-p^{1}-plane

least roughly the localization in space-time for a particle with momentum amplitude f̌(p). This expectation is confirmed by the following Lemma.^{13}

Let f^{+}(x) be a positive frequency smooth Klein-Gordon solution. Then for every N ∈ ℤ_{+} there is a constant C for which

∥(t, vt)∥^{N} |f^{+}(t − x^{0}, vt − x)|
≤ (1 + ∥x∥)^{N} C ∀ x ∈ ℝ^{4} , t ∈ ℝ^{1}, v ∈ ℝ^{3} \ \{\frac{p}{ω_{p}} : p ∈ suppf̌\} .

Proof: See (Lücke, 1974b, Appendix 2).

Exercise 30 Proof Lemma 2.1.1 for the special case

v = (v, 0, 0) , v ∉ \{\frac{p^{1}}{ω_{p}} : p ∈ suppf̌\} , x = 0

by substitution of variables

p^{1} ⟶ ξ =^{def} ω_{p} − p · v

and N-fold partial integration^{14} with respect to ξ.

Draft, March 26, 2009

13A weaker version of Lemma 2.1.1 was originally proved in (Ruelle, 1962). 14Note that dp^{1} = \frac{ω_{p(ξ)}}{p^{1}(ξ)−vω_{p(ξ)}} dξ and rewrite differentiation with respect to ξ as differentiation

with respect to p^{1} after partial integration.

2.1.2 Fock Space

Free n-Particle System (Momentum Representation)

As in nonrelativistic quantum mechanics, n-particle states are described by functions of n-times as many variables as in the 1-particle case. Since the particles cannot be distinguished and have spin 0, we require these functions to be symmetric with respect to exchange of 3-momenta:^{15}

H_{0}^{(n)} = \{f̌_{n} (p_{1}, …, p_{n}) ∈ L^{2} (ℝ^{3n}, \frac{dp_{1}···dp_{n}}{2ω_{p_{1}}···2ω_{pn}} ) :
f̌_{n} (p_{π1}, …, p_{πn}) = f̌_{n} (p_{1}, …, p_{n}) ∀ π ∈ S_{n}} ,
⟨f̌_{n} | ǧ_{n}⟩ =^{def} ∫ \overline{f̌_{n} (p_{1}, …, p_{n})}ǧ_{n} (p_{1}, …, p_{n}) \frac{dp_{1} ⋯ dp_{n}}{2ω_{p_{1}} ⋯ 2ω_{p_{n}}} .  (2.20)

Again, the corresponding representation

(Û_{0}(a, Λ)f̌_{n}) (p_{1}, …, p_{n}) =^{def} (e^{i(p_{1}+...+p_{n})a}f̌_{n} (\overrightarrow{Λ^{−1}p_{1}}, …, \overrightarrow{Λ^{−1}p_{n}}))_{|_{p^{0}_{j}=ωp_{j}}} (2.21)

of P_{+}^{↑} fulfills (2.5) (unitarity), (2.8) (4-dim. spectral representation), and (2.11) (transformation behaviour of P̂_{0}).

Exercise 31 Determine the spectral measure Ê_{0} of P̂_{0} on ℝ^{4} (generalization of (2.7)).

Exercise 32 Show that for every n ∈ ℕ and every function w on {0, 1}^{n} the equation

∑_{ν=1}^{n} ν ∑_{(b_{1},...,b_{n})∈M_{ν}} w(b_{1}, …, b_{n}) = ∑_{μ=1}^{n} ∑_{(b_{1},...,b_{n})∈{0,1}^{n}_{bμ=1}} w(b_{1}, …, b_{n})

holds where

M_{ν} =^{def} {(b_{1}, …, b_{n}) ∈ {0, 1}^{n} : ∑_{μ=1}^{n} b_{μ} = ν} for ν = 1, …, n .

If the particles could be distinguished we had

∫_{p_{ν}∈B_{3}} |f̌_{n}(p_{1}, …, p_{n})|^{2} \frac{dp_{1} ⋯ p_{n}}{2ω_{p_{1}} ⋯ 2ω_{p_{n}}} = \{ \begin{matrix}probability for particle ν having a \\ 3-momentum p_{ν} ∈ B_{3} in a state ˆ= f̌_{n} \end{matrix}

Draft, March 26, 2009

15As usual, we denote by S_{n} the group of all permutation of n elements. We do not consider the – much more complicated – possibility of para-Bose statistics (see e.g. (Ohnuki and Kamefuchi, 1982) and references given there; especially (Stolt and Taylor, 1970),(Hartle and Taylor, 1969)).

2.1. FREE NEUTRAL SCALAR FIELDS 47 under obvious conditions. Therefore, according to Exercise 32, (2.9) becomes

∫_{B_{3n}} ∥â_{0}(p)f̌_{n}∥^{2} \frac{dp}{2ω_{p}} = \{ \begin{matrix}expectation value for the \\ number of particles with p ∈ B_{3n} \end{matrix}  (2.22)

for normalized f̌_{n} ∈ S (ℝ^{3n}) ⊂ H_{0}^{(n)} , n > 0 , if â_{0}(p) denotes the linear mapping

(â_{0}(p)f̌_{n}) (p_{1}, …, p_{n−1})

absent for n≤1

=^{def} \{ \begin{matrix}0 & for n = 0 \\ \sqrt{n}f̌_{n}(p \underbrace{, p_{1}, …, p_{n−1}}_{absent for n=1}) ∈ H_{0}^{(n−1)} & for n > 0 \end{matrix}

(2.23)

from S (ℝ^{3n}) ⊂ H_{0}^{(n)} into S (ℝ^{3(n−1)}) ⊂ H_{0}^{(n−1)} ; where

S (ℝ^{0}) =^{def} H_{0}^{(0)} =^{def} ℂ , ⟨f̌_{0} | ǧ_{0}⟩ =^{def} \overline{f̌_{0}} ǧ_{0} .

(2.10) becomes

⟨f̌_{n} | P̂_{0}^{μ}f̌_{n}⟩ = ∫_{p^{0}=ω_{p}} p^{μ} ∥â_{0}(p)f̌_{n}∥^{2} \frac{dp}{2p^{0}}
= ∫_{p_{j}^{0}=ω_{p_{j}}} (p_{1}^{μ} + … + p_{n}^{μ}) |f̌_{n} (p_{1}, …, p_{n})|^{2} \frac{dp_{1} ⋯ dp_{n}}{2p_{1}^{0} ⋯ 2p_{n}^{0}} (2.24)
= \{ \begin{matrix}expectation value for the μ-component of the \\ total 4-momentum in a state & ˆ= & f̌_{n} \end{matrix}

for normalized f̌_{n} ∈ S (ℝ^{3n}) ∩ H_{0}^{(n)} , but instead of (2.12) we have

P̂_{0}^{0} ≥ nmˆ1 , P̂_{0} · P̂_{0} ≥ (nm)^{2}ˆ1 .  (2.25)

Total State Space If one does not want – or even cannot – fix the particle number, it is convenient to identify H_{0} with the so-called Fock-space :

H_{0} = ⨁_{n=0}^{∞} H_{0}^{(n)} .

Here the elements of H_{0} are sequences

\underline{f̌} =^{def} \{f̌_{0}, f̌_{1}, f̌_{2}, …\}

with f̌_{n} ∈ H_{0}^{(n)} for n = 0, 1, … and ∥\underline{f̌}∥ < ∞, where^{16}

⟨\underline{f̌} | \underline{ǧ}⟩ =^{def} ∑_{n=0}^{∞} ⟨f̌_{n} | ǧ_{n}⟩ .

Draft, March 26, 2009

16To be absolutely precise, one should use different symbols for the various inner products.

48 CHAPTER 2. MASSIVE SCALAR FIELDS

The corresponding unitary representation Û_{0}(a, Λ) of P_{+}^{↑} in H_{0} is given by^{17}

Û_{0}(a, Λ)\underline{f̌} =^{def} \{f̌_{0}, Û_{0}(a, Λ)f̌_{1}, Û_{0}(a, Λ)f̌_{2}, …\} .

Again, (2.8) and

P̂_{0}^{0} ≥ 0 , P̂_{0} · P̂_{0} ≥ 0 .

hold, while (2.22) becomes

∫_{G} ∥â_{0}(p)\underline{f̌}∥^{2} \frac{dp}{2ω_{p}} = \{ \begin{matrix}expectation value for the \\ number of particles with p ∈ G \end{matrix}  (2.26)

for normalized \underline{f̌} ∈ D_{0} . Here, the domain D_{0} is defined by

D_{0} =^{def} \{\underline{f̌} ∈ H_{0} : f̌_{n} ∈ S (ℝ^{3n}) ∀n , f̌_{n} = 0 ∀ n > n_{0} (\underline{f̌})\}  (2.27)

and the annihilation operator (field) â_{0}(p) by

â_{0}(p)\underline{f̌} =^{def} \{â_{0}(p)f̌_{1}, â_{0}(p)f̌_{2}, …\} ∀\underline{f̌} ∈ D_{0} .  (2.28)

As a consequence of (2.23) and (2.4) we thus have

Û_{0}(a, Λ)^{−1}â_{0}(p)Û_{0}(a, Λ) = e^{+ipa}â_{0} (\overrightarrow{Λ^{−1}p})_{|_{p^{0}=ωp}} ,
Û_{0}(a, Λ)â_{0}(p)Û_{0}(a, Λ)^{−1} = e^{−iΛpa}â_{0} (\overrightarrow{Λp})_{|_{p^{0}=ωp}} .

(2.29)

According to (2.28), â_{0}(p) annihilates the so-called vacuum vector Ω_{0} =^{def} {1, 0, …} just as the energy-momentum operator does:

â_{0}(p)Ω_{0} = (ˆ1 − Û_{0}(a, Λ)) Ω_{0} = P̂_{0}^{μ}Ω_{0} = 0 .  (2.30)

2.1.3 The Free Field

Creation Operators in Momentum Space From (2.24)/(2.28) we conclude formally:

P̂_{0}^{μ} = ∫_{p^{0}=ω_{p}} â_{0}(p)^{∗}p^{μ}â(p) \frac{dp}{2p^{0}} .  (2.31)

Unfortunately, however, the adjoint â_{0}(p)^{∗} of â_{0}(p) does not exist for fixed p :

Draft, March 26, 2009

17Similarly, to be precise, different symbols should be used for the various subrepresentations.

[figure: ^{1234}; _{−3 −2 −1  1 2 3}; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; 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.; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; •; p^{0}/m; p^{1}/m]

Figure 2.3: Spectrum of P̂_{0} ( = support of Ê_{0}), restricted to the p^{0}-p^{1}-plane

The restriction to D_{0} of the adjoint â_{0}(χ̌)^{∗} of

â_{0}(χ̌) =^{def} ∫ â_{0}(p)χ̌(p) dp for χ̌ ∈ S(ℝ^{3})

is given by^{18}

(â_{0}(χ̌)^{∗}\underline{f̌})_{0} = 0 ,

(â_{0}(χ̌)^{∗}\underline{f̌})_{1} (p_{1}) = 2ω_{p_{1}} \overline{χ̌(p_{1})} f_{0} ,

(â_{0}(χ̌)^{∗}\underline{f̌})_{n+1} (p_{1}, …, p_{n+1}) = \frac{1}{\sqrt{n + 1}} ∑_{ν=1}^{n+1} 2ω_{p_{ν}} \overline{χ̌(p_{ν})}f̌_{n} (p_{1}, …, p_{ν} \, …, p_{n+1}) ,

for \underline{f̌} ∈ D_{0} ; hence, formally, by

(â_{0}(p)^{∗}\underline{f̌})_{n+1} (p_{1}, …, p_{n+1}) = \frac{1}{\sqrt{n + 1}} ∑_{ν=1}^{n+1} 2ω_{p} δ(p − p_{ν})f̌_{n} (p_{1}, …, p_{ν} \, …, p_{n+1})

(2.32)

for \underline{f̌} ∈ D_{0} . This means,

â_{0}^{∗}(χ̌) =^{def} ∫ â_{0}^{∗}(p)χ̌(p) dp =^{def} (∫ â_{0}(p)\overline{χ̌(p)} dp)^{∗} /^{\}D_{0}  (2.33)

creates a particle with momentum space wave function 2ω_{p} χ̌(p) . As a simple consequence of (2.23) we get the canonical commutation relations

[â_{0}(p), â_{0}(p′)]_{−} = [â_{0}^{∗}(p), â_{0}^{∗}(p′)]_{−} = 0 ,

[â_{0}(p), â_{0}^{∗}(p′)]_{−} = 2ω_{p} δ(p − p′) . (2.34)

Draft, March 26, 2009 18We write p_{ν} \ when p_{ν} has to be skipped.

The precise meaning of (2.34) is^{19}

\begin{matrix}[Âf̌, Âǧ]_{−} = [Â∗ ; f̌, Â∗ ; ǧ]_{−} = 0 \\ [Âf̌, Â∗ ; ǧ]_{−} = ⟨f̌ | ǧ⟩_{H_{0}^{(1)}} ˆ1 \end{matrix} \} ∀ f̌, ǧ ∈ S(ℝ^{3})

on D_{0} , where

Âf̌ =^{def} ∫ â_{0}(p) \overline{\frac{f̌(p)}{2ω_{p}}} dp , hence Â∗ ; f̌/^{\}D_{0} = ∫ â_{0}^{∗}(p) \frac{f̌(p)}{2ω_{p}} dp .

The present representation of the canonical commutation relations has the Fock property

Âf̌Ω = 0 ∀ f̌ ∈ S(ℝ^{3}) .

Moreover, according to (2.29) and (2.33), we have

Û_{0}(a, Λ)^{−1}â_{0}^{∗}(p)Û_{0}(a, Λ) = e^{−ipa}â_{0}^{∗} (\overrightarrow{Λ^{−1}p})_{|_{p^{0}=ωp}} ,
Û_{0}(a, Λ)â_{0}^{∗}(p)Û_{0}(a, Λ)^{−1} = e^{+iΛpa}â_{0}^{∗} (\overrightarrow{Λp})_{|_{p^{0}=ωp}} .

(2.35)

Expressions like

⟨Ψ_{1} | â_{0}^{∗}(p)Ψ_{2}⟩ , Ψ_{1} ∈ H , Ψ_{2} ∈ D_{0} ,

being well-defined only when smeared by some test function χ̌,

∫ ⟨Ψ_{1} | â_{0}^{∗}(p)Ψ_{2}⟩ χ̌(p) dp =^{def} ⟨Ψ_{1} | â_{0}^{∗}(χ̌)Ψ_{2}⟩ ,

(χ̌ ∈ S(ℝ^{3}) , here), are called generalized functions (if linearly and continuously depending on the test function from a suitable topological test space). This name indicates that many operations, defined for ordinary functions, may be generalized to these functionals:

Let K̂, K̂′ be ‘sufficiently well-behaved’ linear operators fulfilling

∫ (K̂χ̌_{1}) (p)χ̌_{2}(p) dp = ∫ χ̌_{1}(p)K̂′χ̌_{2}(p) dp  (2.36)

Draft, March 26, 2009

19Note that, if \left\{{ \atop f̌_{ν}}\right\}_{ν∈ℕ} is a complete orthonormal system in H_{0}^{(1)} , the operators

p̂_{ν} =^{def} \frac{Â∗ ; f̌_{ν} /^{\}D_{0} + Âf̌_{ν}}{\sqrt{2}} , x̂_{ν} =^{def} \frac{Â∗ ; f̌_{ν} /^{\}D_{0} − Âf̌_{ν}}{i\sqrt{2}}

P̂_{0}/^{\}D_{0} = ∑_{ν=1}^{∞} ⟨f_{ν} | P̂_{0}f_{ν}⟩ Â∗ ; f̌_{ν} Âf̌_{ν} .

fulfill the relations (1.43) with ℏ = 1 and that

for all test functions χ̌_{1}, χ̌_{2} . then it is natural to define, e.g.,

∫ (K̂F ) (p) χ̌(p) dp =^{def} ∫ F (p) K̂′χ̌(p) dp ∀ χ̌ ∈ S(ℝ^{3})  (2.37)

for continuous linear functionals F on S(ℝ^{3}) . Correspondingly, we then define

∫ (K̂â_{0}^{∗}) (p)χ̌(p) dp = ∫ â_{0}^{∗}(p)K̂′χ̌(p) dp  (2.38)

for test functions χ̌. Generalizations of these prescriptions are obvious and will be used without special explanation.

Exercise 33 Determine the following operations on â_{0}^{∗}(p) :

(i) partial differentiation
(ii) multiplication by suitable functions
(iii) transformation of variables (e.g. Poincaré transformations)
(iv) Fourier transformation

Field Operators in Minkowski Space

Similarly to the wave functions f(x) (see (2.13)) one defines the field operators^{20}

Φ̂_{0}^{+}(x) =^{def} (2π)^{−3/2} ∫_{p^{0}=ω_{p}} â_{0}(p) e^{−ipx} \frac{dp}{2ω_{p}} ,  (2.39)

(positive frequency part or creation part) and

Φ̂_{0}^{−}(x) =^{def} (Φ̂_{0}^{+}(x))^{∗} =^{def} (2π)^{−3/2} ∫_{p^{0}=−ω_{p}} â_{0}^{∗}(−p) e^{−ipx} \frac{dp}{2ω_{p}} ,  (2.40)

(negative frequency part or annihilation part) on D_{0} , which are both solutions of the Klein-Gordon equation^{21}

(□ + m^{2}) Φ̂_{0}^{±}(x) = 0  (2.41)

(in the sense of generalized functions) and, thanks to (2.29), transform according to

Û_{0}(a, Λ)Φ̂_{0}^{±}(x)Û_{0}(a, Λ)^{−1} = Φ̂_{0}^{±}(Λx + a) .  (2.42)

Draft, March 26, 2009

20The exact definition is by multiplication with exp(−iω_{p}x^{0})/2ω_{p} and subsequent 3-Fourier transform w.r.t. p, both in the sense of generalized functions.

21Note that, for \underline{f̌} ∈ D_{0} , ⟨\underline{f̌}| Φ̂_{0}^{+}(x)⟩ resp. ⟨\underline{f̌}| Φ̂_{0}^{−}(x)⟩ is a positive resp. negative frequency smooth Klein Gordon solution.

It it convenient to add both parts to get a hermitian field operator

Φ̂_{0}(x) =^{def} Φ̂_{0}^{+}(x) + Φ̂_{0}^{−}(x) .  (2.43)

Note that it it is sufficient to smear Φ̂_{0}(x) in the space variables, i.e.

Φ̂_{ψ}(x^{0}) =^{def} ∫ Φ̂_{0}(x) ψ(x) dx

is well defined for x^{0} ∈ ℝ and ψ ∈ S(ℝ^{3}) .

Exercise 34 Show for arbitrary x^{0} ∈ ℝ and ψ ∈ S(ℝ^{3}) :

(i) ψ = \overline{ψ} ⟹ (Φ̂_{ψ}(x^{0}))^{∗} /^{\}D_{0} = Φ̂_{ψ}(x^{0}) .
(ii) Every \underline{f̌} ∈ D_{0} is an entire analytic vector for Φ̂_{ψ}(x^{0}) , i.e.:
∑_{ν=0}^{∞} \frac{1}{ν!} ∥(λΦ̂_{ψ}(x^{0}))^{ν} \underline{f̌}∥ < ∞ ∀ λ > 0 .
(iii) Φ̂_{ψ}(x^{0})D_{0} ⊂ D_{0} .

According to Exercise 34 and a well-known theorem by Nelson (Reed und Simon, 1972, Sect. X.4) Φ̂_{ψ}(x^{0}) has a unique self-adjoint extension if ψ is real-valued. By (2.42), Φ̂(x) transforms like the observable of the field strength of some Lorentz invariant scalar field:

Û_{0}(a, Λ)Φ̂_{0}(x)Û_{0}(a, Λ)^{−1} =_{(2.42)} Φ̂_{0}(Λx + a) .

This suggests the interpretation

Φ̂_{ψ}(x^{0}) = \{ \begin{matrix}observable of the mean value ∫ Φ_{0}(x) ψ(x) dx \\ of the classical field^{22} Φ_{0}(x) at time zero . \end{matrix}  (2.44)

Since, for real-valued ψ, Φ̂_{ψ}(x_{0}) should correspond to some measurement performable within supp ψ at time x^{0} the condition of local commutativity^{23}

[Φ̂_{0}(x), Φ̂_{0}(y)]_{−} = 0 for x×y .  (2.45)

(also called microcausality for observable fields) should be fulfilled (compare Footnote 29). Indeed, (2.45) is a consequence of

[Φ̂_{0}(x), Φ̂_{0}(y)]_{−} = iΔ_{m}(x − y) =^{def} ⟨Ω_{0} | [Φ̂_{0}(x), Φ̂_{0}(y)]_{−} Ω_{0}⟩  (2.46)

Draft, March 26, 2009

22As we will learn from the Haag-Ruelle scattering theory in Section 2.3, it is not that important to give a concrete physical interpretation for Φ_{0}(x) .

23We use x×y as a shorthand notation for (x − y)^{2} < 0 ; i.e. for x being space-like to y.

and the fact that

Δ_{m}(x) = −i(2π)^{−3} ∫ \frac{p^{0}}{|p^{0}|}δ(p^{2} − m^{2})e^{−ipx} dp ,  (2.47)

being an odd Lorentz invariant distribution, vanishes for x×0 (Güttinger and Rieckers, 1968) (see also Footnote 60).

Exercise 35 Show, for arbitrary φ ∈ S(ℝ^{4}) , that

∫ Φ̂_{0}(x) φ(x) dx = \sqrt{2π} (â_{0}(φ̌_{−}) + â_{0}^{∗}(φ̌_{+})) ,

where

φ̌_{±}(p) =^{def} (\frac{φ̃(±p)}{2p^{0}})_{|_{p^{0}=ωp}}

and

φ̃(p) =^{def} (2π)^{−1} ∫ φ(x) e^{+ipx} dx (Fourier transform) .

By Exercise 35 we easily see that Ω_{0} is a cyclic vector for the algebra P_{0} generated by ˆ1/^{\}D_{0} and the smeared field field operators Φ̂_{0}(φ) , φ ∈ S(ℝ^{4}) , i.e.:

P_{0} Ω is dense in H_{0} .

Obviously, the common domain D_{0} has the following invariance properties:^{24}

Φ̂_{0}(φ)D_{0} ⊂ D_{0} ⊃ Û_{0}(a, Λ)D_{0}  (2.48)

(hence P_{0}Ω ⊂ D_{0}).

Exercise 36

(i) Determine the observable for the particle density according to Newton

and Wigner.

(ii) What changes will arise for Φ̂_{0}(x) , if one defines â_{0}(p) by (2.23)/(2.28)

using anti-symmetric f̌_{n}(p_{1}, …, p_{n}) (and â_{0}^{∗}(p) by (2.33), again)?

(iii) Determine the norm^{25} of \overline{Φ̂_{0}(φ)}/^{\}H_{0}^{(n)} as a function of n ∈ ℤ_{+} .

Draft, March 26, 2009

24We even have D_{0} = Û_{0}(a, Λ)D_{0} .

25Hint: Recall Exercise 35 and study the operators considered in Footnote 19 first.

2.2 Wightman Theory for Neutral Scalar Fields

2.2.1 Wightman Axioms

A Wightman theory of a single neutral scalar field Φ̂(x) is characterized by the following assumptions (Wightman axioms ):

0. Assumptions of Relativistic Quantum Theory:

The pure states are given – in the sense of orthodox quantum theory

– by the vectors Ψ of some separable,^{26} complex Hilbert space H on which a (strongly) continuous, unitary representation Û(a, Λ) of P_{+}^{↑} acts according to the following interpretation:

An ensemble corresponding to Û(a, Λ)Ψ is to be described with respect to the coordinates x′ = Λ^{−1}(x − a) in exactly the same way as an ensemble corresponding to Ψ ∈ H is to be described with respect to the coordinates x.

This representation fulfills the following spectrum condition:

For the spectral measure Ê of the observable

P̂ = ∫ p Ê(dp)

of 4-momentum, uniquely characterized by^{27}

⟨Ψ | Û(a, 1l_{4})Ψ⟩ = ∫ e^{ipa} ⟨Ψ | Ê(dp)Ψ⟩ ∀ Ψ ∈ H ,

we have

Ê(B) = 0 for all Borel B ⊂ ℝ^{4} \ \overline{V_{+}} ,

where

V_{±} =^{def} {x ∈ ℝ^{4} : xx > 0 , ±x^{0} > 0} .

There is a normed vector Ω , unique up to a phase factor, that is invariant under the representation of P_{+}^{↑} :

Û(a, Λ)Ω = Ω ∀ (a, Λ) ∈ P_{+}^{↑} .

This vector describes the vacuum state of the theory.

Draft, March 26, 2009

26Actually, separability is a consequence of the separability of the test function space and cyclicity of the vacuum state.

27That such a projection-valued measure Ê on ℝ^{4} exists is guaranteed by the so-called SNAG theorem (see (Streater and Wightman, 1989, p. 92) and references given there).

I. Assumptions about the Domain and Continuity of the Field:

The field Φ̂(x) is a hermitian operator-valued, tempered generalized function with invariant domain D ⊂ H; i.e. a linear mapping^{28}

Φ̂ : S(ℝ^{4}) ⟶ L(D, D)
φ ⟼ Φ̂(φ) = \underbrace{∫ Φ̂(x)φ(x) dx}_{formal}

for which all the

∫ ⟨Ψ | Φ̂(x)Ψ⟩ φ(x) dx =^{def} ⟨Ψ | Φ̂(φ)Ψ⟩ , Ψ ∈ D ,

are continuous in φ ∈ S(ℝ^{4}) , where D has to fulfill the following conditions for φ ∈ S(ℝ^{4}) and (a, Λ) ∈ P_{+}^{↑} :

Ω ⊂ D , Û(a, Λ)D ⊂ D , Φ̂(φ)D ⊂ D , Φ̂(\overline{φ}) = Φ̂(φ)^{∗}/^{\}D .

II. Transformation Law of the Field:

The field transforms according to

Û(a, Λ)Φ̂(x)Û(a, Λ)^{−1} = Φ̂(Λx + a) ∀ (a, Λ) ∈ P_{+}^{↑} .

III. Local Commutativity (Microscopic Causality):

The smeared fields Φ̂(φ_{1}) , Φ̂(φ_{2}) commute whenever the supports of the test functions φ_{1} , φ_{2} ∈ S(ℝ^{4}) are spacelike with respect to each other.^{29} Formally:

x×y ⟹ [Φ̂(x), Φ̂(y)]_{−} = 0 .

Finally, the vacuum vector Ω is required to be cyclic with respect to the algebra F_{0} generated by ˆ1/^{\}D and the smeared field operators Φ̂(φ) , φ ∈ S(ℝ^{4}) . This means:

F_{0} Ω is dense in H .

Obviously, all the Wightman axioms are fulfilled for the free field Φ̂_{0}(x) .

Draft, March 26, 2009

28As usual, if X and Y are linear spaces, we denote by L(X, Y) the set of all linear mappings from X into Y . We do no longer assume smearing of the field in the space variables to be sufficient. Actually it can be shown that smearing in x^{0} would be sufficient (Borchers, 1964).

29That this condition has to be fulfilled in order to avoid acausal effects even at the microscopic scale, if the field Φ̂(x) is fully observable, is shown in (Schlieder, 1971).

2.2.2 Remarks on the Choice of the Space of Test Functions

Originally (Wightman, 1956), Wightman used the Schwartz space

D(ℝ^{4}) =^{def} {φ ∈ S(ℝ^{4}) : suppφ compact}

with the usual topology:

φ_{ν} → φ in D(ℝ^{4}) if and only if^{30}

D_{x}^{α}φ_{ν}(x) → D_{x}^{α}φ(x) uniformly in x ∈ ℝ^{4} ∀ α ∈ ℤ_{+}^{4}

and if there is a compact subset K ⊂ ℝ^{4} outside which all the φ_{ν} vanish.

Since the Fourier transform φ̃ of a test function φ ∈ D(ℝ^{4}) is always an entire analytic function, the generalized functions on D(ℝ^{4}) are nonlocalizable, in general, i.e. there is no notion of support in the ordinary sense for this class of Distributions. In his original program (Wightman, 1956), Wightman indicated corresponding problems by the remark:

“We shall assume that F^{(n)} has a Fourier transform.”

Now, the Fourier transform F̃^{(n)} of a generalized function F^{(n)} on D(ℝ^{4}) is always well defined on the Fourier dual D̃(ℝ^{4}) of D(ℝ^{4}) . What Wightman ment was that his use of the Laplace transform of F̃^{(n)} should be justified. Using the notion of quasi-support for nonlocalizable generalized functions, introduced in (Bümmerstede and Lücke, 1974) (via local continuity , as explained in 3.3.3), and the corresponding definition of Laplace transform, introduced in (Lücke, 1984, Sect. 4), there is no problem at all. Another method to justify Wightman’s results without additional assumptions was presented earlier in (Borchers, 1964, Sect. 5).

The choice of test function space may well be crucial (Wightman, 1981), since:

Nobody could construct a Wightman field (for the test space S(ℝ^{4}))

with nontrivial interaction.

If physical space-time ℝ^{1} × ℝ^{3} is replaced by a toy space-time ℝ^{1} × ℝ^{d} with d < 3 then the test space S(ℝ^{d+1}) is known to be suitable.^{31}

E.P. Osipov (hep-th/9608115) claims to be able to construct a field Φ̂(x) on the physical space-time with nontrivial S-matrix fulfilling all the Wightman axioms with S(ℝ^{4}) replaced by some Jaffe space^{32} J (ℝ^{4}) , where

D(ℝ^{4}) ∩ J (ℝ^{4}) is dense in J (ℝ^{4}) ⊂ S(ℝ^{4}) .

Draft, March 26, 2009

30We use standard multi-index notation: D_{x}^{α} =^{def} (\frac{∂}{∂x^{0}} )^{α_{0}} ⋯ (\frac{∂}{∂x^{3}} )^{α_{3}} .

31See (Streater and Wightman, 1989, Appendix) for a neat review of the construction of corresponding models. For more details see (Glimm and Jaffe, 1981).

32See (Jaffe, 1967) for Jaffe’s class of test spaces for localizable fields.

Examples of nonlocalizable fields are carefully studied in (Rieckers, 1971). How the condition of microcausality has to be modified for these examples is shown in (Lücke, 1974a).

2.2.3 Mathematical Tools

Theorem 2.2.1 (Schwartz’s Nuclear Theorem) Let n_{1}, n_{2} ∈ ℕ and let

\begin{matrix}F : S(ℝ^{n_{1}}) × S(ℝ^{n_{2}}) & ⟶ & ℂ \\ (φ_{1}, φ_{2}) & ⟼ & F (φ_{1}, φ_{2}) \end{matrix}

be linear and continuous in each argument separately. Then there is a one and only one generalized function F (x_{1}, x_{2}) on S(ℝ^{n_{1}} × ℝ^{n_{2}}) with

∫ F (x_{1}, x_{2}) φ_{1}(x_{1})φ_{2}(x_{2}) dx_{1}dx_{2} = F (φ_{1}, φ_{2}) ∀ (φ_{1}, φ_{2}) ∈ S(ℝ^{n_{1}}) × S(ℝ^{n_{2}}) .

Proof:^{33} See (Gelfand and Wilenkin, 1964, Chapter I §1 Nr. 2).

Theorem 2.2.1 (together with the Hahn-Banach theorem) implies that^{34}

S(ℝ^{n_{1}}) ⊗ S(ℝ^{n_{2}}) is dense in S(ℝ^{n_{1}+n_{1}}) ∀ n_{1} , n_{2} ∈ ℕ .  (2.49)

Therefore the following Lemma allows iteration (Corollary 2.2.3, below) of the nuclear theorem.

Lemma 2.2.2 Let n ∈ ℕ and let {F_{ν}}_{ν∈ℤ_{+}} ⊂ S(ℝ^{n})′ be such that lim_{ν→∞} F_{ν}(φ) exists for all φ ∈ S(ℝ^{n}) . Then

F (φ) =^{def} lim_{ν→∞} F_{ν}(φ) for φ ∈ S(ℝ^{n})

defines a tempered generalized function on S(ℝ^{n}) ; i.e. F ∈ S(ℝ^{n})′.

Proof: See (Gelfand and Schilow, 1962, Ch. I §5 No. 6).

Draft, March 26, 2009

33See also (Bümmerstede and Lücke, 1974, Appendix).

34As usual, we denote by S(ℝ^{n_{1}}) ⊗ S(ℝ^{n_{2}}) the algebraic tensor product of S(ℝ^{n_{1}}) and S(ℝ^{n_{2}}) realized as the linear span of the set of all φ ∈ S(ℝ^{n_{1}+n_{2}}) of the form

φ(x_{1}, x_{2}) = φ_{1}(x_{1})φ_{2}(x_{2}) , φ_{j} ∈ S(ℝ^{n_{j}}) for j = 1, 2 .

Corollary 2.2.3 Let k, n_{1}, …, n_{k} ∈ ℕ and let

\begin{matrix}F : S(ℝ^{n_{1}}) × … × S(ℝ^{n_{k}}) & ⟶ & ℂ \\ (φ_{1}, …, φ_{k}) & ⟼ & F (φ_{1}, …, φ_{k}) \end{matrix}

be linear and continuous in each argument separately. Then there is a one and only one generalized function F ∈ S(ℝ^{n_{1}+...+n_{k}})′ with

∫ F (x_{1}, …, x_{k}) φ_{1}(x_{1}) ⋯ φ_{k}(x_{k}) dx_{1} ⋯ dx_{k} = F (φ_{1}, …, φ_{k})

for all (φ_{1}, …, φ_{k}) ∈ S(ℝ^{n_{1}}) × … × S(ℝ^{n_{k}}) .

Given n ∈ ℕ and φ ∈ S(ℝ^{4n}) , let us define

(K̂_{Δ}φ) (\frac{x_{1} + … + x_{n}}{n} , x_{1} − x_{2}, …, x_{n−1} − x_{n}) =^{def} φ(x_{1}, …, x_{n})

for x_{1}, …, x_{n} ∈ ℝ^{4} . Then K̂_{Δ} and its inverse K̂_{Δ}^{−1} are linear continuous mappings of S(ℝ^{4n}) into itself with

\begin{matrix}∫ (K̂_{Δ}φ) (X, ξ_{1}, …, ξ_{n−1}) ψ(X, ξ_{1}, …, ξ_{n−1}) dXdξ_{1} … dξ_{n−1}) \\ = ∫ φ(x_{1}, …, x_{n}) (K̂_{Δ}^{−1}ψ) (x_{1}, …, x_{n}) dx_{1} ⋯ dx_{n} ∀ φ , ψ ∈ S(ℝ^{4n}) . \end{matrix}

Therefore, as explained in 2.1.3, K̂_{Δ}F is well defined for tempered generalized functions F on ℝ^{4n}.

Lemma 2.2.4 Let n ∈ ℝ^{4} and let W ∈ S(ℝ^{4n})′ fulfill

W (x_{1} + a, …, x_{n} + a) = W (x_{1}, …, x_{n}) ∀ a ∈ ℝ^{4} .

Then there is a unique generalized function W ∈ S(ℝ^{4(n−1)})′ with

W (x_{1}, …, x_{n}) = W(x_{1} − x_{2}, …, x_{n−1} − x_{n}) ,

i.e.:

\begin{matrix}∫ W (x_{1}, …, x_{n}) φ(x_{1}, …, x_{n}) dx_{1} ⋯ dx_{n} \\ = ∫ W(ξ_{1}, …, ξ_{n−1}) (∫ (K̂_{Δ} φ) (a, ξ_{1}, …, ξ_{n−1}) da) dξ_{1} ⋯ dξ_{n−1} \end{matrix}

for all φ ∈ S(ℝ^{4n}) .

Sketch of proof:^{35} Let us choose some χ ∈ S(ℝ^{4}) with

∫ χ(a) da = 1

and use the short-hand notation

x̌ =^{def} (x_{1}, …, x_{n}) ,  ξ̌ =^{def} (ξ_{1}, …, ξ_{n−1}) ,
dx̌ =^{def} dx_{1} ⋯ dx_{n} ,  dξ̌ =^{def} dξ_{1} ⋯ dξ_{n−1} .

Then

∫ W (x̌) φ(x̌) dx̌
= ∫ W (x̌) φ(x_{1} − a, …, x_{n} − a) dx̌
= ∫ (K̂_{Δ}W ) (X, ξ̌) (K̂_{Δ}φ) (X − a, ξ̌)dXdξ̌
= ∫ (∫ (K̂_{Δ}W ) (X, ξ̌) (K̂_{Δ}φ) (X − a, ξ̌)dXdξ̌) χ(a) da
= ∫ (K̂_{Δ}W ) (X, ξ̌) (∫ (K̂_{Δ}φ) (X − a, ξ̌) χ(a) da) dXdξ̌
= ∫ (K̂_{Δ}W ) (X, ξ̌) (∫ (K̂_{Δ}φ) (a, ξ̌) χ(X − a) da) dXdξ̌
= ∫ (K̂_{Δ}W ) (X, ξ̌) (∫ (K̂_{Δ}φ) (a, ξ̌) χ(X) da) dXdξ̌
= ∫ (K̂_{Δ}W ) (X, ξ̌)χ(X) (∫ (K̂_{Δ}φ) (a, ξ̌) da) dXdξ̌

and hence

W(ξ_{1}, …, ξ_{n−1}) = ∫ (K̂_{Δ} W ) (X, ξ_{1}, …, ξ_{n−1}) χ(X) dX .

Theorem 2.2.5 (Bochner-Schwartz) Let W ∈ D(ℝ^{4}) be positive semi definite, i.e. fulfill

∫ W(x − y) \overline{φ(x)}φ(y) dxdy ≥ 0 ∀ φ ∈ D(ℝ^{4}) .

Then there are a (unique) positive Borel measure μ on ℝ^{4} and some k ∈ ℤ_{+} with

∫ (1 + ∥p∥)^{−k} μ(dp) < ∞

and^{36}

∫ W̃(q) ψ(q) dq = ∫ ψ(q) μ(dq) ∀ ψ ∈ D̃(ℝ^{4}) .

Hence W is the restriction to D(ℝ^{4}) of a \underline{tempered} generalized function.

Draft, March 26, 2009

35The interchanges of (ordinary and/or formal) integration may be easily justified by intermediate regularization of the involved generalized functions. An alternative, more indirect, proof can be found in (Streater and Wightman, 1989, p. 39/40).

36Of course, D̃(ℝ^{4}) means the Fourier dual of D(ℝ^{4}) : D̃(ℝ^{4}) =^{def} {φ̃: φ ∈ D(ℝ^{4})}.

60 CHAPTER 2. MASSIVE SCALAR FIELDS

Proof: See (Gelfand and Wilenkin, 1964, Ch. II §3).

Lemma 2.2.6 Let n ∈ ℝ^{4} and let W ∈ S(ℝ^{4n})′ fulfill

W (x_{1} + a, …, x_{n} + a) = W (x_{1}, …, x_{n}) ∀ a ∈ ℝ^{4}

as well as

suppχ̃ ⊂ ℝ^{4} \ \overline{V_{+}} ⟹ ∫ W (x_{1}, …, x_{ν} + a, …, x_{n} + a) χ(a) da = 0

for all χ ∈ S(ℝ^{4}) and ν = 2, …, n. Then for the Fourier transform

W̃(q_{1}, …, q_{n−1}) =^{def} (2π)^{−2(n−1)} ∫ W(ξ_{1}, …, ξ_{n−1}) e^{i(ξ_{1}q_{1}+...+ξ_{n−1}q_{n−1})} dξ_{1} ⋯ dξ_{n−1}

of the generalized function

W(ξ_{1}, …, ξ_{n−1}) = W (ξ_{1} + … + ξ_{n−1}, ξ_{2} + … + ξ_{n−1}, …, ξ_{n−1}, 0)

given by Lemma 2.2.4 we have:

suppW̃ ⊂ \overline{V_{+}} × ⋯ × \overline{V_{+}} .

Sketch of proof: By Lemma 2.2.4 we have for all ν ∈ {2, …, n}

W (x_{1}, …, x_{ν+1} + a, …, x_{n} + a) = W(ξ_{1}, …, ξ_{ν} − a, …, ξ_{ν−1}) ∀ a ∈ ℝ^{4} and hence for all χ ∈ S(ℝ^{4}) with suppχ̃ ⊂ ℝ^{4} \ \overline{V_{+}}

0 = ∫ (∫ W(ξ_{1}, …, ξ_{ν} − a, …, ξ_{n−1}) e^{i(ξ_{1}q_{1}+...+ξ_{n−1}q_{n−1})} dξ_{1} ⋯ dξ_{n−1}) χ(a) da
= (∫ W(ξ_{1}, …, ξ_{n−1}) e^{i(ξ_{1}q_{1}+...+ξ_{n−1}q_{n−1})} dξ_{1} ⋯ dξ_{n−1}) ∫ χ(a)e^{iq_{ν}a} da
= (2π)^{2(n−1)} W̃(q_{1}, …, q_{n−1}) χ̃(q_{ν}) .

Lemma 2.2.7 Let 1 < n ∈ ℕ and let W(ξ_{1}, …, ξ_{n−1}) ∈ S(ℝ^{4(n−1)}) be such that

suppW̃ ⊂ \overline{V_{+}} × ⋯ × \overline{V_{+}} .

Then there is one and only one holomorphic function (LW̃) (z_{1}, …, z_{n−1}) on

T_{n−1} =^{def} {(z_{1}, …, z_{n−1}) ∈ ℂ : ℑ(z_{ν}) ∈ V_{−} for ν = 1, …, n − 1} ,

fulfilling the condition^{37}

\begin{matrix}∫ (LW̃) (ξ_{1} + iη_{1}, …, ξ_{n−1} + iη_{n−1}) φ̃(ξ_{1}, …, ξ_{n−1}) dξ_{1} ⋯ dξ_{n−1} \\ = ∫ W̃(q_{1}, …, q_{n−1}) (e^{q_{1}η_{1}+...+q_{n−1}η_{n−1}}φ(q_{1}, …, q_{n−1})) dq_{1} ⋯ dq_{n−1} . \end{matrix}

Draft, March 26, 2009

37Note that ∫ F (q_{1}, …, q_{n−1}) φ̃(q_{1}, …, q_{n−1}) dq_{1} ⋯ dq_{n−1} = ∫ F̃(q_{1}, …, q_{n−1}) φ(q_{1}, …, q_{n−1}) dq_{1} ⋯ dq_{n−1} for all F ∈ S(ℝ^{4(n−1)})′ and φ ∈ S(ℝ^{4(n−1)}) .

2.2. WIGHTMAN THEORY FOR NEUTRAL SCALAR FIELDS 61 This so-called Laplace transform

(LW̃) (ξ_{1} + iη_{1}, …, ξ_{n−1} + iη_{n−1})

= \underbrace{∫ W̃(q_{1}, …, q_{n−1})e^{−i(q_{1}(ξ_{1}+iη_{1})+...+q_{n−1}(ξ_{n−1}+iη_{n−1}))} dq_{1} ⋯ dq_{n−1}}_{formal}

of W̃ fulfills^{38}

\begin{matrix}lim_{V_{−}∋η_{1},...,η_{n−1}→0} ∫ (LW̃) (ξ_{1} + iη_{1}, …, ξ_{n−1} + iη_{n−1}) ψ(ξ_{1}, …, ξ_{n−1}) dξ_{1} ⋯ dξ_{n−1} \\ = ∫ W(ξ_{1}, …, ξ_{n−1}) ψ(ξ_{1}, …, ξ_{n−1}) dξ_{1} ⋯ dξ_{n−1} ∀ ψ ∈ S(ℝ^{4(n−1)}) . \end{matrix}

Sketch of proof: Choose some χ ∈ C^{∞}(ℝ^{4(n−1)}) with

χ(ξ̌) = \{ \begin{matrix}1 & if ∥ξ̌− ξ̌′∥ < 1 for all ξ̌′ ∈ V_{+} × ⋯ × V_{+} \\ 0 & if ∥ξ̌− ξ̌′∥ > 2 for some ξ̌′ ∈ V_{+} × ⋯ × V_{+} \end{matrix}

and define

\begin{matrix}(LW̃) (ξ_{1} + iη_{1}, …, ξ_{n−1} + iη_{n−1}) \\ =^{def} ∫ W̃(q_{1}, …, q_{n−1})(\underbrace{e^{−i(q_{1}(ξ_{1}+iη_{1})+...+q_{n−1}(ξ_{n−1}+iη_{n−1}))}χ(q_{1}, …, q_{n−1})}_{∈S(ℝ^{4(n−1)})}) dq_{1} ⋯ dq_{n−1} \end{matrix}

for η_{1}, …, η_{n−1} ∈ V_{−}. The following theorem shows that (LW̃) (ξ_{1} + iη_{1}, …, ξ_{n−1} + iη_{n−1}) is already fixed by its (distributional) boundary values on any open subset O of ℝ^{n−1} . Theorem 2.2.8 (Edge-of-the-Wedge) Let n ∈ ℕ, let Ǒ be an open subset of ℂ^{n} for which O =^{def} Ǒ ∩ ℝ^{n} ≠ 0 , let C be an open convex cone in ℝ^{n} with apex at 0 , and let L be a holomorphic function on

B =^{def} (ℝ^{n} + i C) ∩ Ǒ

such that

lim_{C∋y→0} ∫ L(x + iy) φ(x) dx = 0 for all φ ∈ D(ℝ^{n}) with suppφ ⊂ O .

Then

L(x + iy) = 0 ∀ x + iy ∈ B .

Draft, March 26, 2009

38Compare (Lücke, 1984, Theorem 4).

Proof: See (Streater and Wightman, 1989, Theorem 2-17).

Corollary 2.2.9 Let 1 < n ∈ ℕ and W ∈ S(ℝ^{4(n−1)})′. If

suppW̃ ⊂ \overline{V_{+}} × … × \overline{V_{+}}

then either W = 0 or suppW = ℝ^{4(n−1)} .

Proof: Direct consequence of Lemma 2.2.7 and Theorem 2.2.8.

2.2.4 Some Standard Results

By Corollary 2.2.3, for every n ∈ ℕ and every Ψ ∈ D there is a unique generalized function

⟨Ψ | Φ̂(x_{1}) ⋯ Φ̂(x_{n})Ψ⟩ ∈ S(ℝ^{4n})′

with

∫ ⟨Ψ | Φ̂(x_{1}) ⋯ Φ̂(x_{n})Ψ⟩ φ_{1}(x_{1}) ⋯ φ_{n}(x_{n}) dx_{1} ⋯ dx_{n} = ⟨Ψ | Φ̂(φ_{1}) ⋯ Φ̂(φ_{n})Ψ⟩

for all φ_{1}, …, φ_{n} ∈ S(ℝ^{4}) . Thus, especially, the so-called n-point functions

W (x_{1}, …, x_{n}) = ⟨Ω | Φ̂(x_{1}) ⋯ Φ̂(x_{n})Ω⟩ ,  (2.50)

are well defined as generalized functions on S(ℝ^{4n}) . The relativistic transformation law for Φ̂(x) and the invariance of the vacuum imply

∫ W (Λx_{1} + a, …, Λx_{n} + a)φ_{1}(x_{1}) ⋯ φ_{n}(x_{n}) dx_{1} ⋯ dx_{n}

= ⟨Ω | Û(a, Λ)Φ̂(φ_{1}) \underbrace{Û(a, Λ)^{−1}Û(a, Λ)}_{=ˆ1} ⋯ Û(a, Λ)Φ̂(φ_{n}) \underbrace{Û(a, Λ)^{−1}Ω}_{=Ω}⟩

and, by the same reasoning,

\begin{matrix}∫ W (x_{1}, …, x_{ν} + a, …, x_{n} + a)φ_{1}(x_{1}) ⋯ φ_{n}(x_{n}) dx_{1} ⋯ dx_{n} \\ = ⟨Ω | Φ̂(φ_{1}) ⋯ Û(a)Φ̂(φ_{2}) ⋯ Φ̂(φ_{n})Ω⟩ \end{matrix}

for all (a, Λ) ∈ P_{+}^{↑} φ_{1}, …, φ_{n} ∈ S(ℝ^{4}) , and ν = 2, …, n. By (2.49), this implies

W (x_{1}, …, x_{n}) =^{def} W (Λx_{1} + a, …, Λx_{n} + a) ∀ n ∈ ℕ , (a, Λ) ∈ P_{+}^{↑}  (2.51)

2.2. WIGHTMAN THEORY FOR NEUTRAL SCALAR FIELDS 63 and, thanks to the spectrum condition,^{39}

suppχ̃ ⊂ ℝ^{4} \ \overline{V_{+}} ⟹ ∫ W (x_{1}, …, x_{ν} + a, …, x_{n} + a) χ(a) da = 0 . Therefore, according to Lemma 2.2.4 and Lemma 2.2.6, for every natural number n > 1 there is a generalized function W ∈ S(ℝ^{4(n−1)}) with

W (x_{1}, …, x_{n}) = W(x_{1} − x_{2}, …, x_{n−1} − x_{n})  (2.52)

and

suppW̃ ⊂ \overline{V_{+}} × ⋯ × \overline{V_{+}} .  (2.53)

Exercise 37 Show the following:

(i) For every φ ∈ S(ℝ^{4}) there is a unique^{40} operator

∫ Φ̂(x_{1}) ⋯ Φ̂(x_{n}) φ(x_{1}, …, x_{n}) dx_{1} ⋯ dx_{n} ∈ L(D, H)

with

⟨Ψ| (∫ Φ̂(x_{1}) ⋯ Φ̂(x_{n}) φ(x_{1}, …, x_{n}) dx_{1} ⋯ dx_{n}) Ψ⟩
= ∫ ⟨Ψ | Φ̂(x_{1}) ⋯ Φ̂(x_{n})Ψ⟩ φ(x_{1}, …, x_{n}) dx_{1} ⋯ dx_{n} ∀ Ψ ∈ D .

(ii) For every Ψ ∈ D

∫ Φ̂(x_{1}) ⋯ Φ̂(x_{n}) φ(x_{1}, …, x_{n}) dx_{1} ⋯ dx_{n}Ψ

depends strongly continuously on φ ∈ S(ℝ^{4n}) . (iii) For all φ_{1}, …, φ_{n} ∈ S(ℝ^{4})

∫ Φ̂(x_{1}) ⋯ Φ̂(x_{n}) φ_{1} ⋯ φ_{n} dx_{1} ⋯ dx_{n} = Φ̂(φ_{1}) ⋯ Φ̂(φ_{n}) .

Draft, March 26, 2009

39Note that ∫ Û(a) χ(a) da = ∫ χ̃(p) Ê(dp) .

40Recall that ⟨Φ | ÂΨ⟩ = \frac{1}{4} ∑_{k=0}^{3} ⟨Φ + i^{k}Ψ | Â (Φ + i^{k}Ψ)⟩ ∀ Φ, Ψ ∈ D , Â ∈ L(D, H).

By Exercise 37, without any restriction of generality, we may assume^{41}

∫ Φ̂(x_{1}) ⋯ Φ̂(x_{n}) φ(x_{1}, …, x_{n}) dx_{1} ⋯ dx_{n}D ⊂ D

and

(∫ Φ̂(x_{1}) ⋯ Φ̂(x_{n}) φ(x_{1}, …, x_{n}) dx_{1} ⋯ dx_{n})^{∗} /^{\}D
= ∫ Φ̂(x_{1}) ⋯ Φ̂(x_{n}) \overline{φ(x_{n}, …, x_{1}}) dx_{1} ⋯ dx_{n}

(2.54)

for all n ∈ ℕ and all φ ∈ S(ℝ^{4n}) .

Obviously, by the GNS technique (recall Footnote 57 of Chapter 1), a Wightman theory with D = F_{0}Ω can be reconstructed from its n-point functions, up to unitary equivalence.^{42}

Lemma 2.2.10 In a field theory of the type described in 2.2.1, with dim H > 1 , the field operator Φ̂(x) cannot be defined pointwise for x ∈ ℝ^{4} .

Proof:^{43} Obviously, the generalized function W(ξ) ∈ S(ℝ^{4})′ associated with the 2-point function is positive semi-definite. Hence, by Theorem 2.2.5, there is a polynomially bounded positive Borel measure μ on ℝ^{4} with

W (x_{1}, x_{2}) = W(x_{1} − x_{2}) = ∫ e^{ip(x_{1}−x_{2})}μ(dp) .

By (2.51), μ has to be Lorentz covariant:^{44}

μ(B) = μ(ΛB) ∀ Borel B ∈ ℝ^{4} , Λ ∈ L_{+}^{↑} .

Let us assume that Φ̂(x) is well defined for every x ∈ ℝ^{4} . Then μ(ℝ^{4}) = W(0) must be finite. Hence there is a number μ_{0} ≥ 0 with

μ(B) = \{ \begin{matrix}μ_{0} & if 0 ∈ B \\ 0 & else \end{matrix} ∀ Borel B ∈ ℝ^{4} .

This implies

⟨Φ̂(x_{1})Ω | Φ̂(x_{2})Ω⟩ = W (x_{1}, x_{2}) = W(x_{1} − x_{2}) = μ_{0} ∀ x_{1} , x_{2} ∈ ℝ^{4}

Draft, March 26, 2009

41Here we use

(∫ Φ̂(x_{1}) ⋯ Φ̂(x_{n_{1}}) φ_{1}(x_{1}, …, x_{n_{1}}) dx_{1} ⋯ dx_{n_{1}}) (∫ Φ̂(y_{1}) ⋯ Φ̂(y_{n_{2}}) φ_{2}(y_{1}, …, y_{n_{2}}) dy_{1} ⋯ dy_{n_{2}})

= ∫ Φ̂(x_{1}) ⋯ Φ̂(x_{n_{1}})Φ̂(y_{1}) ⋯ Φ̂(y_{n_{2}}) φ_{1}(x_{1}, …, x_{n_{1}})φ_{2}(y_{1}, …, y_{n_{2}}) dx_{1} ⋯ dx_{n_{1}}dy_{1} ⋯ dy_{n_{2}} of course.

42See (Streater and Wightman, 1989, Sect. 3-4) for full details.

43The main part of this proof, which uses neither microcausality nor the spectrum condition, was presented in (Jaffe, 1967, Introduction).

44As usual we denote by L_{+}^{↑} the set of all Lorentz transformations Λ with (0, Λ) ∈ P_{+}^{↑}.

even though

∥Φ̂(x)Ω∥ = ∥Û(x)Φ̂(0)Ω∥ = ∥Φ̂(0)Ω∥ ∀ x ∈ ℝ^{4} .

This, in turn, implies

Φ̂(x)Ω = Φ̂(0)Ω ∀ x ∈ ℝ^{4}

and hence

Û(a, Λ)Φ̂(x)Ω = Φ̂(x)Ω ∀ (a, Λ) ∈ P_{+}^{↑} , x ∈ ℝ^{4} .

Thanks to uniqueness of the vacuum state, therefore,

Φ̂(x)Ω ∼ Ω ∀ x ∈ ℝ^{4} .

By cyclicity of the Ω , finally, this gives dim H = 1 . Since this contradicts the assumptions of the lemma, Φ̂(x) cannot be well defined for every x ∈ ℝ^{4} .

For the theory of a single neutral scalar field the theorem on the connection between spin and statistics becomes especially simple:^{45}

Theorem 2.2.11 There is no quantum field Φ̂(x) ≠ 0 , not necessarily fulfilling microcausality but all the other assumptions of 2.2.1, for which

x×y ⟹ Φ̂(x)Φ̂(y) + Φ̂(y)Φ̂(x) = 0 .

Sketch of proof: Thanks to (2.51) and Lemma 2.2.4 there are L_{+}^{↑}-invariant generalized functions F (ξ) , G(ξ) ∈ S(ℝ^{4})′ with

F (x − y) = ⟨Ω | Φ̂(x)Φ̂(y)⟩ − ⟨Ω | Φ̂(y)Φ̂(x)⟩

and

G(x − y) = ⟨Ω | Φ̂(x)Φ̂(y)⟩ + ⟨Ω | Φ̂(y)Φ̂(x)⟩ .

Since F is also odd, we have^{46}

ξ×0 ⟹ F (ξ) = 0 .

Now, assume local anticommutativity:

x×y ⟹ Φ̂(x)Φ̂(y) + Φ̂(y)Φ̂(x) = 0 .

Draft, March 26, 2009

45Actually, strict local anti-commutativity is not necessary (Lücke, 1979). See also (Guido and Longo, 1995) and (Davidson, 1997) for a purely algebraic version using strict locality.

46Recall the reasoning for (2.45). For general fields one has to exploit the BHW theorem (Theorem 2.2.19, below).

Then, because of

W(x − y) = ⟨Ω | Φ̂(x)Φ̂(y)⟩ = \frac{1}{2} (F (x − y) + G(x − y)) ,

we have

ξ×0 ⟹ W(ξ) = 0 .

By (2.53) and Corollary 2.2.9, therefore, W = 0 and hence Φ̂(φ)Ω = 0 for all φ ∈ S(ℝ^{4}) . By cyclicity of Ω , this implies Φ̂(x) = 0 .

Theorem 2.2.12 (Reeh-Schlieder) Let Φ̂(x) fulfill the assumptions of 2.2.1 with possible exception of microcausality^{47} and let O ≠ ∅ be an open subset of ℝ^{4} . Then the vacuum state Ω is cyclic with respect to the algebra F_{0}(O) ⊂ L(D, D) generated by ˆ1/^{\}D and all Φ̂(φ) with φ ∈ S(ℝ^{4}) .

Sketch of proof: By cyclicity of Ω with respect to F_{0} = F_{0}(ℝ^{4}) it is sufficient to prove^{48}

⟨Ψ | Φ̂(φ_{1}) ⋯ Φ̂(φ_{n})Ω⟩ = 0 ∀ φ_{1}, …, φ_{n} ∈ S(O)
⟹ ⟨Ψ | Φ̂(φ_{1}) ⋯ Φ̂(φ_{n})Ω⟩ = 0 ∀ φ_{1}, …, φ_{n} ∈ S(ℝ^{4})

(2.55)

for all Ψ ∈ D and n ∈ ℕ :

By appropriate transformation of coordinates we get a generalized function L(ξ_{1}, …, ξ_{n}) ∈ S(ℝ^{4n})′ with

L(−x_{1}, x_{1} − x_{2}, …, x_{n−1} − x_{n}) = ⟨Ψ | Φ̂(x_{1}) ⋯ Φ̂(x_{n})Ω⟩

and

suppL ≠ ℝ^{4n} ,

thanks to the assumption in (2.55). On the other hand, we have

suppL̃ ⊂ \overline{V_{+}} × … × \overline{V_{+}} ,

by essentially the same reasoning as for (2.53). Therefore^{49} Corollary 2.2.9 implies L = 0 , hence the r.h.s. of (2.55).

Theorem 2.2.12 “can be interpreted as meaning that it is difficult to isolate a system described by fields from outside effects.” (Streater and Wightman, 1989, p. 139)

Draft, March 26, 2009

47Strict localizability is essential, however.

48As usual, S(O) denotes the (topological) subspace of all φ ∈ S(ℝ^{4}) with suppφ ⊂ O.

49Note that translation of the Fourier transform corresponds to multiplication of the original function by some function with constant modulus one.

Corollary 2.2.13 Let all the assumptions of Theorem 2.2.12 be fulfilled. Moreover, let Â ∈ L(D, H) commute with F(O) in the sense that^{50}

⟨Ψ | ÂĈΨ⟩ = ⟨Ĉ^{∗}Ψ | ÂΨ⟩ ∀ Ψ ∈ D , Ĉ ∈ F(O)

and assume D ⊂ DÂ^{∗}. Then

ÂΩ = 0 ⟹ Â = 0 .

Proof: If ÂΩ = 0 then we have

⟨Ψ | ÂĈΩ⟩ = ⟨Ĉ^{∗}Ψ | ÂΩ⟩ = 0 ,

for all Ψ ∈ D and Ĉ ∈ F(O) . By the Reeh-Schlieder theorem and thanks to D ∈ DÂ^{∗} this implies Â = 0 .

Corollary 2.2.14 Let all the assumptions of Corollary 2.2.13 be fulfilled and let Â be positive in the sense that

⟨Ψ | ÂΨ⟩ ≥ 0 ∀ Ψ ∈ D .

Then

⟨Ω | ÂΩ⟩ = 0 ⟹ Â = 0 .

Sketch of proof: Assume ⟨Ω | ÂΩ⟩ = 0 . Since Â is positive, it has a positive

self-adjoint extension ˆ Â (Friedrichs’ theorem, see e.g. (Yosida, 1971, Ch. XI §7)). With this extension we have

⟨\sqrt{ˆÂ}Ω | \sqrt{ˆÂ}Ω⟩ = ⟨Ω | ˆ ÂΩ⟩ = 0

and hence

ÂΩ = \sqrt{ˆÂ} (\sqrt{ˆÂ}Ω) = 0 .

By Corollary 2.2.13, this implies Â = 0 .

Remark: Corollary 2.2.14 shows that there cannot be any local positive energy density or positive 0-component of a local current density with vanishing vacuum expectation value in a field theory of the type described in 2.2.1.

Draft, March 26, 2009

50This condition is obviously fulfilled for Â ∈ F_{0}(\underline{O′}) with \underline{O′}×O.

Theorem 2.2.15 (Borchers) Let Φ̂(x) fulfill the assumptions of 2.2.1 and let Ψ ∈ H. Then^{51}

(Û(a)Ψ = Ψ ∀ a ∈ ℝ^{4}) ⟹ Ψ ∼ Ω .

Proof: See (Borchers, 1962, Theorem 3).

Corollary 2.2.16 Let Φ̂(x) fulfill the assumptions of 2.2.1. Then F_{0} is irreducible in the sense that every Ĉ ∈ L(H) with

⟨Ψ | ĈΦ̂(φ)Ψ⟩ = ⟨Φ̂(φ)^{∗}Ψ | ĈΨ⟩ ∀ Ψ ∈ D , φ ∈ S(ℝ^{4})  (2.56)

must be a multiple of ˆ1 .

Sketch of proof: Let Ĉ ∈ L(H) fulfill (2.56). Then it is sufficient to prove

ĈΩ = cΩ  (2.57)

for some c ∈ ℂ since the latter implies

\begin{aligned}⟨Ψ | ĈÂΩ⟩ & =_{(2.56)} & ⟨Â^{∗}Ψ | ĈΩ⟩ \\ & =_{(2.57)} & c ⟨Ψ | ÂΩ⟩ ∀Ψ ∈ D , Â ∈ F_{0} \end{aligned}

and hence Ĉ = cˆ1 , thanks to cyclicity of Ω . In order to prove (2.57) it is sufficient, by Theorem 2.2.15, to show

Û(a)ĈΩ = ĈΩ ∀ a ∈ ℝ^{4}

which, by cyclicity and translation invariance^{52} of Ω , is equivalent to

⟨Φ̂(φ_{1}) ⋯ Φ̂(φ_{n})Ω | ˆ Û(a)ĈΩ⟩

being independent of a ∈ ℝ^{4} for all n ∈ ℕ and φ_{1} … φ_{n} ∈ S(ℝ^{4}) . The latter,

Draft, March 26, 2009

51Let T denote the subspace of all translational invariant vectors. Obviously, then, T is invariant under P_{+}^{↑}. Therefore, if we already new T to be finite dimensional the statement of Theorem 2.2.15 were a simple consequence of the fact that there are no non-trivial unitary representations of L_{+}^{↑}.

52Translation invariance of Ω ensures that also ⟨ˆ1Ω | Û(a)Ω⟩ is a-independent.

2.2. WIGHTMAN THEORY FOR NEUTRAL SCALAR FIELDS 69 however, is an easy consequence of (2.56) and the Wightman axioms:

∫\overline{V_{+}} e^{+iap} ⟨Φ̂(φ_{1}) ⋯ Φ̂(φ_{n})Ω | Ê(dp)ĈΩ⟩

= ⟨Φ̂(φ_{1}) ⋯ Φ̂(φ_{n})Ω | Û(a)ĈΩ⟩
= ⟨(Û(−a)Φ̂(φ_{1})Û(a)) ⋯ (Û(−a)Φ̂(φ_{n})Û(a)) Ω | ĈΩ⟩
= ⟨Ω | Ĉ (Û(−a)Φ̂(φ_{n})Û(a))^{∗} ⋯ (Û(−a)Φ̂(φ_{1})Û(a))^{∗} Ω⟩
= ⟨Ĉ^{∗}Ω | Û(−a) (Φ̂(φ_{1}) ⋯ Φ̂(φ_{n}))^{∗} Ω⟩
= ∫\overline{V_{+}} e^{−iap} ⟨Ĉ^{∗}Ω | Ê(dp) (Φ̂(φ_{1}) ⋯ Φ̂(φ_{n}))^{∗} Ω⟩ .

In the following we shall denote by F(O) , O any open subset of ℝ^{4} , the algebra generated by ˆ1/^{\}D and all operators Â of the form

Â = ∫ Φ̂(x_{1}) ⋯ Φ̂(x_{n}) φ(x_{1}, …, x_{n}) dx_{1} ⋯ dx_{n}

(recall Exercise 37) with n ∈ ℕ and φ ∈ S(O × ⋯ × O) . Exercise 38 Show for the free field Φ̂(x) = Φ̂_{0}(x) , described in 2.1, that

\begin{matrix}Â ∈ F(O) \\ [Â, F(O)]_{−} = 0 \end{matrix} \} ⟹ Â ∼ ˆ1/^{\}D

holds for every bounded open set O ⊂ ℝ^{4} . Lemma 2.2.17 Let all the assumption of 2.2.1 be fulfilled.^{53} Then

Φ̂(φ_{1}) ⋯ Φ̂(φ_{n})Ω ∈ Ê (\overline{V_{+}} ∩ (suppφ̃_{1} + … suppφ̃_{n})) H

holds for all n ∈ ℕ and φ_{1}, …, φ_{n} ∈ S(ℝ^{4}) .

Sketch of proof: By

Û(a) = ∫ e^{ipa} Ê(dp)  (2.58)

and the basic relation

δ(p − p′) = (2π)^{−4} ∫ e^{i(p−p′)} da  (2.59)

Draft, March 26, 2009

53Actually, microcausality will not be used in the proof of Lemma 2.2.17.

of Fourier calculus we have

Ê(B) = (2π)^{−4} ∫_{B} (∫ Û(a) e^{−ipa} da) dp ∀ Borel B ⊂ ℝ^{4} .  (2.60)

By the transformation behaviour of Φ̂(x) and translation invariance of Ω this implies

Ê(B)Φ̂(φ_{1}) ⋯ Φ̂(φ_{n})Ω = (2π)^{−4} ∫ Φ̂(x_{1}) ⋯ Φ̂(x_{n}) (∫_{B} (∫ φ_{1}(x_{1} − a) ⋯ φ_{1}(x_{n} − a) e^{−ipa} da) d) dx_{1} ⋯ dx_{n}Ω

∀ Borel B ⊂ ℝ^{4} .

Since ∫ φ_{1}(x_{1} − a) ⋯ φ_{1}(x_{n} − a) e^{−ipa} da can only be nonzero for p ∈ suppφ̃_{1} + … + suppφ̃_{n}, this together with the spectrum condition implies the statement of the lemma.

Theorem 2.2.18 (Jost-Schroer) Let H Û(a, Λ) , Ω , D and Φ̂(x) fulfill the assumptions of 2.2.1 with dim H > 1 and let H_{0} Û_{0}(a, Λ) , Ω_{0} , D_{0} , Φ̂_{0}(x) be the corresponding objects of the free field theory described in 2.1. If also Φ̂(x) fulfills the free Klein-Gordon equation

(□ + m^{2}) Φ̂(x) = 0

and if D = F(ℝ^{4})Ω then there are a unitary mapping V̂ : H_{0} ⟶ H and a constant λ > 0 with:

D = V̂ D_{0} , Ω = V̂ Ω
Φ̂(x) = λ V̂ Φ̂_{0}(x)V̂^{−1} ,
Û(a, Λ) = V̂ Û_{0}(a, Λ)V̂^{−1} ∀ (a, Λ) ∈ P_{+}^{↑} .

Sketch of proof: Assume that Φ̂(x) fulfills the Klein-Gordon equation. This implies (□_{ξ} + m^{2}) W(ξ) = 0 and hence, by (2.53),

suppW̃ ⊂ M_{m} = {P ∈ ℝ^{4} : p^{0} = ω_{p}} .  (2.61)

From the proof of Lemma 2.2.10 we also know that there is a positive L_{+}^{↑}-invariant measure μ with

⟨Ω | Φ̂(x)Φ̂(y)Ω⟩ = W(x − y) = ∫ e^{ip(x−y)}μ(dp)

Therefore, there must exist a λ^{2} ≥ 0 with

μ(dp) = \frac{λ^{2}}{(2π)^{3}} θ(p^{0})δ(p^{2} − m^{2}) dp .

Without restriction of generality we may assume λ = 1 . Then we have^{54}

⟨Ω | Φ̂(x)Φ̂(y)Ω⟩ = iΔ_{m}^{(+)}(x − y) =^{def} (2π)^{−3} ∫ θ(p^{0})δ(p^{2} − m^{2})e^{−ip(x−y)} dp . (2.62)

The Klein-Gordon equation allows us to define

Φ̂^{(±)}(x) =^{def} (2π)^{−2} ∫_{±p^{0}≥0} ˜ Φ̂(p) e^{−ipx} dp .  (2.63)

With this definition we have

Φ̂(x) = Φ̂^{(+)}(x) + Φ̂^{(−)}(x) ,  (2.64)
Φ̂^{(+)}(x)Ω = 0 ,  (2.65)

and hence

⟨Ω | Φ̂^{(+)}(x)Φ̂^{(+)}(y)Ω⟩ = ⟨Ω | Φ̂^{(−)}(x)Φ̂^{(+)}(y)Ω⟩ = ⟨Ω | Φ̂^{(−)}(x)Φ̂^{(−)}(y)Ω⟩ = 0 ,

(2.66)

the latter because of Φ̂^{(−)}(x)^{∗} = Φ̂^{(+)}(x) . Moreover, by (2.62), it is clear that

[Φ̂(x), Φ̂(y)]_{−} = ⟨Ω | [Φ̂(x), Φ̂(y)]_{−}Ω⟩ ˆ1/^{\}D  (2.67)

would imply

[Φ̂(x), Φ̂(y)]_{−} = iΔ_{m}(x − y)ˆ1/^{\}D  (2.68)

(recall (2.47)). Since statements corresponding to (2.63), (2.65), and (2.68) hold for the free theory, this shows that (2.67) would imply that the n-point functions of Φ̂(x) are the same^{55} as those of Φ̂_{0}(x) . In view of the GNS representation, described by Theorem 1.3.8 (recall Footnote 57 of Chapter 1), therefore, it is sufficient to prove (2.67). By Corollary 2.2.13, (2.67) follows from

[Φ̂(x), Φ̂(y)]_{−}Ω ∼ Ω .  (2.69)

To prove (2.69), let us consider states of the form

Ψ̌ = Φ̂^{(+)}(φ_{+})Φ̂^{(−)}(φ_{−})Ω , φ_{±} ∈ S(ℝ^{4}) .

Draft, March 26, 2009

54Vice versa, (2.62) implies the Klein-Gordon equation, by Corollary 2.2.13. Therefore, assumption (2.61) would have been sufficient. In this sense the results of (Baumann, 1986) are much stronger than the Jost-Schroer theorem. For generalization in a different direction see (Steinmann, 1982).

55By (2.64) it is sufficient to check the vacuum expectation values ⟨Ω | Φ̂^{(σ_{1})}(x_{1}) ⋯ Φ̂^{(σ_{n})}(x_{n})Ω⟩ for arbitrary n ∈ ℕ and σ_{ν} ∈ {+, −}. By (2.68), then, it is sufficient to check the expectation values of the form ⟨Ω | Φ̂^{(+)}(x_{1}) ⋯ Φ̂^{(+)}(x_{n_{+}})Φ̂^{(−)}(x_{n_{+}+1}) ⋯ Φ̂^{(−)}(x_{n_{+}+n_{−}})Ω⟩. By (2.65), however, all of them vanish, as for the free field.

For such Ψ̌ Lemma 2.2.17 implies

Ψ̌ = Ê (\overline{V_{+}} ∩ (−suppΦ̂^{(+)} − suppΦ̂^{(−)})) Ψ̌ .

Since^{56}

\overline{V_{+}} ∩ (−suppΦ̂^{(+)} − suppΦ̂^{(−)}) = \{0\} ,

this means Ψ̌ = Ê({0})Ψ̌ . Thus, by (2.58), Ψ̌ is translational invariant and thus a multiple of Ω , by Borchers’ theorem. Therefore:

Φ̂^{(+)}(x)Φ̂^{(−)}(y)Ω  =  ⟨Ω | Φ̂^{(+)}(x)Φ̂^{(−)}(y)Ω⟩ Ω
=_{(2.64),(2.66)} ⟨Ω | Φ̂(x)Φ̂(y)Ω⟩
=_{(2.62)} iΔ_{m}^{(+)}(x − y)Ω .

By (2.65), (2.64), and

Δ_{m}(x − y) = Δ_{m}^{(+)}(x − y) − Δ_{m}^{(+)}(y − x)

this implies

[Φ̂(x), Φ̂(y)]_{−}Ω = iΔ_{m}(x − y)Ω + [Φ̂^{(−)}(x), Φ̂^{(−)}(y)]_{−}Ω .

Therefore, it is sufficient to prove

F_{Ψ}(x, y) =^{def} ⟨Ψ | [Φ̂^{(−)}(x), Φ̂^{(−)}(y)]_{−}Ω⟩ = 0

for all Ψ ∈ D. This, however, is guaranteed by Corollary 2.2.9 since, obviously, suppF_{Ψ} ≠ ℝ^{8} and suppF̃_{Ψ} ⊂ \overline{V_{+}} × \overline{V_{+}} (recall Footnote 56).

2.2.5 PCT Theorem

The defining representation of the Lorentz group L = L(ℝ) is well known to be given by the real 4 × 4-matrices Λ fulfilling

Λ^{T} η Λ = η =^{def} ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} ) .  (2.70)

Similarly, the defining representation of the complex Lorentz group L = L(ℂ) is given by the complex 4 × 4-matrices Λ fulfilling (2.70). Its subgroup

L_{+}(ℂ) =^{def} {Λ ∈ L(ℂ) : det Λ = +1}

is called the proper complex Lorentz group. Obviously:

L_{+}^{↑} ⊂ L_{+}(ℂ) ∋ +1l_{4} , −1l_{4} .  (2.71)

Draft, March 26, 2009

56By the Klein-Gordon equation we have supp˜Φ̂ ⊂ M_{m}∪(−M_{m}) and hence supp ; Φ̂^{(±)} ⊂ ±M_{m}, by (2.63).

Theorem 2.2.19 (Bargmann-Hall-Wightman) Let n, N ∈ ℕ, n > 1 , let Λ ⟶ S(Λ) be an irreducible N×N-matrix representation of L_{+}^{↑}, and let W_{1}, …, W_{N} be holomorphic functions on T′_{n−1} with

W_{μ}(z_{1}, …, z_{n−1}) = ∑_{ν=1}^{N} S_{μν}(Λ^{−1})W_{ν}(Λz_{1}, …, Λz_{n−1}) ∀ μ ∈ {1, …, N} (2.72)

for all (z_{1}, …, z_{n−1}) ∈ T_{n−1} and all Λ ∈ L_{+}^{↑}. Then the W_{ν} have unique single-valued analytic continuations onto the extended tube

T′_{n−1} =^{def} {(Λz_{1}, …, Λz_{n−1}) : Λ ∈ L_{+}(ℂ) , (z_{1}, …, z_{n−1}) ∈ T_{n−1}}

fulfilling (2.72) for all (z_{1}, …, z_{n−1}) ∈ T′_{n−1} and all Λ ∈ L_{+}(ℂ) , where S(Λ) is to be extended to the corresponding irreducible representation of L_{+}(ℂ) .

Proof: See (Bogush and Fedorov, 1977, Sect. 3.4).

Corollary 2.2.20 Let all the assumptions of 2.2.1, with possible exception of microcausality,^{57} be fulfilled and let 1 < n ∈ ℕ. Then the Laplace transform (LW̃)

of W̃ has a single valued analytic continuation onto the extended tube T′_{n−1} fulfilling the conditions

\begin{matrix}(LW̃) (Λz_{1}, …, Λz_{n−1}) = (LW̃) (z_{1}, …, z_{n−1}) \\ ∀ Λ ∈ L_{+}(ℂ) , (z_{1}, …, z_{n−1}) ∈ T′_{n−1} \end{matrix}  (2.73)

and^{58}

\begin{matrix}W(ψ) = ∫ (LW̃) (ξ_{1}, …, ξ_{n−1}) ψ(ξ_{1}, …, ξ_{n−1})dξ_{1} ⋯ dξ_{n−1} \\ ∀ ψ ∈ S(T′_{n−1} ∩ ℝ^{4(n−1)}) . \end{matrix} (2.74)

Sketch of proof: Note, first of all, that (2.52) and (2.51) imply

W(Λξ_{1}, …, Λξ_{n−1}) = W(ξ_{1}, …, ξ_{n−1}) ∀ Λ ∈ L_{+}^{↑} .

This together with (2.53) and Lemma 2.2.7 gives

(LW̃) (ξ_{1}+iη_{1}, …, ξ_{n−1}+iη_{n−1}) = (LW̃) (Λξ_{1}+iΛη_{1}, …, Λξ_{n−1}+iΛη_{n−1}) ∀ Λ ∈ L_{+}^{↑}

Draft, March 26, 2009

57Microcausality would allow further holomorphic continuation when n > 2 (see, e.g., (Tomozawa, 1963)).

58Note that T′_{n−1} ∩ ℝ^{4(n−1)} is an open subset of ℝ^{4(n−1)} , since T′_{n−1} is open in ℂ^{4(n−1)} .

74 CHAPTER 2. MASSIVE SCALAR FIELDS as an equation for holomorphic functions on T_{n−1} . Therefore, by Theorem 2.2.19, (LW̃) has a single valued analytic continuation onto the extended tube T′_{n−1} fulfilling (2.73). From Lemma 2.2.7 we also know that

W(ψ) = lim_{V_{−}∋η_{1},...,η_{n−1}→0} ∫ (LW̃) (ξ_{1}+iη_{1}, …, ξ_{n−1}+iη_{n−1}) ψ(ξ_{1}, …, ξ_{n−1}) dξ_{1} ⋯ dξ_{n−1}

for all ψ ∈ S(ℝ^{4(n−1)} . Since every Jost point^{59}

\underline{ξ̌} =^{def} (\underline{ξ}_{1}, …, \underline{ξ}_{n−1}) ∈ J_{n−1} =^{def} T′_{n−1} ∩ ℝ^{4(n−1)}

has a complex open neighborhood O\underline{ξ̌} ⊂ J_{n−1} , this implies

W(ψ) = ∫ (LW̃) (ξ̌) ψ(ξ̌) dξ̌ ∀ ψ ∈ S (O\underline{ξ̌} ∩ ℝ^{4(n−1)}) .

From this (2.74) follows by standard distribution theoretical techniques (choice of a suitable partition of unity). Theorem 2.2.21 (Jost) Let 1 < n ∈ ℕ and (ξ_{1}, …, ξ_{n−1}) ∈ ℝ^{4(n−1)} . Then

(ξ_{1}, …, ξ_{n−1}) ∈ T′_{n−1}

⟺ (∑_{ν=1}^{n−1} λ_{ν}ξ_{ν}) (∑_{ν=1}^{n−1} λ_{ν}ξ_{ν}) < 0 for all λ_{1}, …, λ_{n−1} ≥ 0 with ∑_{ν=1}^{n−1} λ_{ν} > 0 .

Proof: See (Streater and Wightman, 1989, Theorem 2-12). Corollary 2.2.22 (PCT Theorem) Let all the assumptions of 2.2.1, with possible exception of microcausality, be fulfilled and let 1 < n ∈ ℕ. Then the PCT condition

W (x_{1}, …, x_{n}) = W (−x_{n}, …, −x_{1})  (2.75)

is equivalent to the condition of weak local commutativity

W (x_{1}, …, x_{n}) = W (x_{n}, …, x_{1}) for (x_{1} − x_{2}, …, x_{n−1} − x_{n}) ∈ J_{n−1} . (2.76)

Proof: By Corollary 2.2.20, since −1l_{4} ∈ L_{+}(ℂ) , (2.76) is equivalent to^{60} W (x_{1}, …, x_{n}) = W (−x_{n}, …, −x_{1}) for (x_{1} − x_{2}, …, x_{n−1} − x_{n}) ∈ J_{n−1} ,

Draft, March 26, 2009

59The elements of T′_{n−1} ∩ ℝ^{4(n−1)} are usually called Jost points.

60Since J_{1} = {ξ ∈ ℝ^{4} : ξ×0}, this shows once more that

⟨Ω | [Φ̂(x), Φ̂(y)]_{−}Ω⟩ = 0 for x×y

holds even if microcausality is not assumed, as a consequence of the other Wightman axioms.

2.3. S-MATRIX FOR SELF-INTERACTING NEUTRAL SCALAR FIELD 75

i.e. to

F (ξ̌) =^{def} W(ξ_{1}, …, ξ_{n−1}) − W(ξ_{n−1}, …, ξ_{1}) = 0 ∀ ξ̌ ∈ J_{n−1} .  (2.77)

By Corollary 2.2.9, since (2.53) implies

supp F ⊂ \overline{V_{+}} × ⋯ × \overline{V_{+}} ,

(2.77) and hence (2.76) is equivalent to F = 0 , i.e. to (2.75).

Exercise 39 Show that (2.75) is equivalent to existence of an anti-unitary Operator θ̂ fulfilling the conditions

θ̂θ̂ = ˆ1 , θ̂Ω = Ω ,
θ̂Φ̂(φ) θ̂ = (∫ Φ̂(−x) φ(x) dx)^{∗} /^{\}D ∀ φ ∈ S(ℝ^{4}) ,

and

θ̂Û(a) θ̂ = Û(−a) ∀ a ∈ ℝ^{4} .

2.3 S-Matrix for Self-interacting Neutral Scalar Fields

2.3.1 General Scattering Theory

The main concern of scattering theory is asymptotic (for t → ±∞) identification of an interacting system (IS) with a suitable “free” system (FS):

Ȟ =^{def} set of all (pure) states of the IS ,
Ȟ_{0} =^{def} set of all (pure) states of the FS .

Here, we use the Heisenberg picture, i.e. the states describe the corresponding system for all times until some “measurement” is taking place. Whether the considered systems are classical or quantum does not matter, so far. Thus, given Ψ̌ ∈ Ȟ, the basic problem is to find the (hopefully unique) “free” states Ψ̌_{±} ∈ Ȟ_{0} such that

for t → ±∞ Ψ̌ “looks like” Ψ̌_{±} .

The precise meaning of the latter has to be specified by some asymptotic condition (AC) as sketched in Fig. 2.4. For example, in potential scattering of classical particles, sketched in Fig. 2.5, the Heisenberg states are given by the solutions x(t) of the classical equations of motion and x_{±}(t) being free means:

x_{±}(t) = x_{±} + v_{±}t .

[figure: inter- action; Ψ̌; final config.; initial config.; AC; AC; Ψ̌_{−}; Ψ̌_{+}; ˆŜ_{0}]

Figure 2.4: Asymptotic identification of the IS (left) with the FS (right) via the AC

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Figure 2.5: Asymptotics for a classical particle

For potentials of sufficiently short range forces the asymptotic condition is:^{61}

v_{±} = lim_{t→±∞} ẋ(t) , x_{±} = lim_{t→±∞} x(t) − v_{±}t .

Ψ̌ ∈ Ȟ is called a scattering state, if there are Ψ̌_{−} ∈ Ȟ_{0} and Ψ̌_{+} ∈ Ȟ_{0} fulfilling the AS for Ψ̌ . Obviously, this condition need not always be fulfilled (bounded states, particle capture etc.).

FS and AC have to meet the following requirement:

For every Φ̌_{0} ∈ Ȟ_{0} there is exactly one Ψ̌ ∈ Ȟ with Ψ̌_{−} = Φ̌_{0} ; and

Draft, March 26, 2009

61See (Reed und Simon, 1972, Vol. III, Thm. XI.1). For the Coulomb potential the “free states” have to be taken of the form

x_{±}(t) = x_{±} + v_{±}t + d_{±} ln t

(see (Reed und Simon, 1972, Vol. III, Sect. 9)).

similarly for ‘+’ instead of ‘-’.

Then we can define the following generalized wave operators:

ˆV̂_{in}Φ̌_{0} =^{def} the state Ψ̌ ∈ Ȟ for which Ψ̌_{−} = Φ̌_{0} ,
ˆV̂_{out}Φ̌_{0} =^{def} the state Ψ̌′ ∈ Ȟ for which Ψ̌′_{+} = Φ̌_{0} .

(2.78)

Now, the subject of scattering theory is to study the relation between Ψ̌_{−} = ˆ V̂ _{in}^{−1}Ψ̌

and Ψ̌_{+} = ˆ V̂ _{out}^{−1}Ψ̌ for arbitrary scattering states Ψ̌ .

The scattering operator in the Heisenberg picture is

ˆŜ =^{def} ˆ V̂_{in} ˆV̂ _{out}^{−1}  (2.79)

and maps Ȟ_{out} =^{def} ˆ V̂_{out}Ȟ_{0} one-to-one onto Ȟ_{in} =^{def} ˆ V̂_{in}Ȟ_{0} :

ˆŜ \underbrace{ˆV̂_{out}Φ̌_{0}}_{looks like Φ̌_{0} for t→+∞} = \underbrace{ˆV̂_{in}Φ̌_{0}}_{looks like Φ̌_{0} for t→−∞} .  (2.80)

Definition (2.79) has the advantage of being applicable even in case Ȟ_{in} ⊄ Ȟ_{out} , contrary to definition (2.82) below, and of being independent of the special choice for the realization of the free system. Its drawback is that ˆ Ŝ describes the relation between Ψ̌_{−} and Ψ̌_{+} only indirectly (via ˆ V̂_{out} or ˆ V̂_{in}):

\begin{matrix}ˆŜ ˆV̂_{out}Ψ̌_{−} & ( =_{(2.79)} ˆ V̂_{in}Ψ̌_{−} = Ψ̌) & = ˆ V̂_{out}Ψ̌_{+} \\ ˆŜ ˆV̂_{in}Ψ̌_{−} & (= ˆ ŜΨ̌ = ˆ Ŝ ˆV̂_{out}Ψ̌_{+}) & = ˆ V̂_{in}Ψ̌_{+} \end{matrix} \} for scattering states Ψ̌ . (2.81)

If Ȟ_{in} ⊂ Ȟ_{out} , this relation may be described directly via the scattering operator in the interaction picture

ˆŜ_{0} =^{def} ˆ V̂ _{out}^{−1} ˆV̂_{in} ,  (2.82)

namely:

ˆŜ_{0}Ψ̌_{−} = Ψ̌_{+} .  (2.83)

In this case, i.e. when Ȟ_{in} contains only scattering states (no capture),

ˆŜ = ˆ V̂_{out}ˆŜ_{0} ˆV̂ _{out}^{−1} .  (2.84)

In case of weak asymptotic completeness, i.e. if Ȟ_{in} = Ȟ_{out} , we also have

ˆŜ = ˆ V̂_{in}ˆŜ_{0} ˆV̂ _{in}^{−1} ,  (2.85)

since then D ˆV̂ _{in}^{−1} = D ˆV̂ _{out}^{−1} . For potential scattering of classical particles the action

of ˆ Ŝ and ˆ Ŝ_{0} is sketched in Fig. 2.6.

78 CHAPTER 2. MASSIVE SCALAR FIELDS

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Ψ̌_{−} Φ̌_{−} Ψ̌

Φ̌ ˆŜ

ˆŜ^{0}

∗ Φ̌ = V̂_{out}Ψ̌_{−}

Ψ̌ = V̂_{in}Ψ̌_{−}

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Figure 2.6: Two versions of the S-matrix

2.3.2 Asymptotic condition for massive neutral scalar particles

In quantum mechanics without superselection rules, the pure states of the WS resp. FS are given by the 1-dimensional subspaces Ψ̌ resp. Ψ̌_{0} of some complex Hilbert space H resp. H_{0} :

Ȟ_{(0)} = {Ψ̌_{(0)} = {λΨ_{(0)} : λ ∈ ℂ} : Ψ_{(0)} ∈ H_{(0)}} .

The generalized wave operators ˆ V̂ out_{(in)} are given by isometric (linear) mappings

V̂out_{(in)} : H_{0} ⟶ Hout_{(in)} =^{def} V̂out_{(in)}H_{0} ⊂ H

via:

ˆV̂ out_{(in)}Ψ̌_{0} =^{def} {λV̂out_{(in)}Ψ_{0} : λ ∈ ℂ} for Ψ_{0} ∈ H_{0} .

Let us consider the theory of a neutral scalar field as specified in 2.2.1, describing the IS, and assume:

1. The restriction of the representation Û(a, Λ) of P_{+}^{↑} to Ê(M_{m})H is irreducible.

2. The free field theory described in 2.1.3 represents a FS suitable for asymptotic description of the IS.

3. The corresponding scattering theory fulfills the condition of asymptotic com-

pleteness:

H_{in} = H_{out} = H .

By asymptotic completeness, then, the scattering isometries V̂out_{(in)} are even unitary

mappings from H_{0} onto H. The chosen FS can only be suitable if

Û(a, Λ) V̂out_{(in)} = V̂out_{(in)} Û_{0}(a, Λ) ∀ (a, Λ) ∈ P_{+}^{↑} .  (2.86)

This implies

suppÊ = suppÊ_{0} = {0} ∪ M_{m} ∪ {p ∈ ℝ^{4} : p^{2} ≥ (2m)^{2} , p^{0} > 0}  (2.87)

and

V̂out_{(in)}H_{0}^{(0)} = Ê ({0}) H , V̂out_{(in)}H_{0}^{(1)} = Ê(M_{m})H .

Since Û(a, Λ)/^{\}Ê (M_{m}) H is irreducible and since the vacuum state is uniquely characterized by P_{+}^{↑}-invariance, we may assume without loss of generality:

H_{0}^{(0)} ⊕ H_{0}^{(1)} = Ê ({0} ∪ M_{m}) H , V̂out_{(in)}/^{\} (H_{0}^{(0)} ⊕ H_{0}^{(1)}) = ˆ1/^{\} (H_{0}^{(0)} ⊕ H_{0}^{(1)}) .

(2.88)

By unitarity of V̂out_{(in)} , then, it is sufficient to determine the vectors

Ψout_{(in)}(χ̌_{1}, …, χ̌_{n}) =^{def} V̂out_{(in)} (â_{0}^{∗}(χ̌_{1}) ⋯ â_{0}^{∗}(χ̌_{n})Ω)  (2.89)

(recall (2.33)) for all n > 1 and functions χ̌_{1}, …, χ̌_{n} ∈ D(ℝ^{3}) that are nonoverlapping, i.e.:

ν ≠ μ ⟹ supp χ̌_{ν} ∩ supp χ̌_{μ} = ∅ .

We want to characterize the states corresponding to vectors of the form (2.89) by their expectation values for localized measurements corresponding to bounded observables.^{62} The use of the field ϕ̂(x) for this is just to associate with every open region O ⊂ ℝ^{4} the corresponding algebra of local observables^{63} A(O) , i.e. the von Neumann algebra generated be all bounded observables corresponding to measurements performable within O. Once the local algebras A(O) are specified, we may forget about the field ϕ̂(x) as far as the S matrix is concerned.

In order to be able to interpret the closed smeared field operators \overline{∫ dx Φ̂(x)φ(x)} , where φ = \overline{φ} ∈ S(ℝ^{4}) , as observables of the φ-weighted ‘field strength’ ∫ dx Φ(x)φ(x) , let us assume that before closure the operators Φ̂(φ) =^{def} ∫ dx Φ̂(x)φ(x) on D are

Draft, March 26, 2009

62A more general formalism, suitable also for nonlocalizable fields, was developed in (Lücke, 1983).

63See (Thomas and Wichmann, 1998) and references given there for further details.

essentially self-adjoint and that in case supp φ_{1}×supp φ_{2} also the spectral projection operators of Φ̂(φ_{1}) commute^{64} with those of Φ̂(φ_{2}) . Then,^{65} given an open region O ⊂ ℝ^{4} , A(O) should be identified with the von Neumann subalgebra of L(H) generated by all spectral operators of the selfadjoint operators \overline{Φ̂(φ)} , φ = \overline{φ} ∈ S(O) . This also ensures that the following four conditions are fulfilled:^{66}

O_{1} ⊂ O_{2} ⟹ A(O_{1}) ⊂ A(O_{2}) ,  (2.90)
O_{1}×O_{2} ⟹ [A(O_{1}), A(O_{2}]_{−} = \left\{{ \atop ˆ0}\right\} ,  (2.91)
Û(a, Λ)A(O))Û(a, Λ)^{−1} = A(ΛO + a) ,  (2.92)
( ⋃_{R>0} A (U_{R}(0)))′ = {λˆ1 : λ ∈ ℂ} .  (2.93)

By Ruelle’s lemma (Lemma 2.1.1) the influence of an asymptotic particle on measurements outside its velocity cone should fall off very rapidly. This statement has to be made precise in order to get a suitable asymptotic condition:

Asymptotic Condition:^{67} Let χ̌_{1}, … χ̌_{n} ∈ D(ℝ^{3}) be nonoverlapping and let the vectors â_{0}^{∗}(χ_{1})Ω_{0} , …, â_{0}^{∗}(χ_{n})Ω_{0} ∈ H_{0}^{(1)} be normalized. Moreover, let Ǩ = −Ǩ be a closed cone for which

Ǩ ∩ Kχ̌_{n} = \{0\} , Kχ̌_{n} =^{def} \{(t, \frac{p}{ω_{p}} t) : t ∈ ℝ , p ∈ suppχ_{n}\} .

Then, for sufficiently small^{68} ϵ > 0 there exists a sequence of positive numbers C_{1}, C_{2}, … for which

|t|^{N} |⟨Ψout_{(in)}(χ̌_{1}, … χ̂_{n})|Â_{t}Ψout_{(in)}(χ̌_{1}, … χ̌_{n})⟩

− ⟨Ψout_{(in)}(χ̌_{1}, … χ̌_{n−1})|Â_{t}Ψout_{(in)}(χ̌_{1}, … χ̌_{n−1})⟩|

≤ ∥Â_{t}∥ C_{N} ∀ t >_{(<)} 0 , N ∈ ℕ , Â_{t} ∈ A (Ǩ ∩ U_{ϵ|t|}(Σ_{t})) ,

where:

Σ_{t} =^{def} {X ∈ ℝ^{4} : x^{0} = t} .

Draft, March 26, 2009

64This is a stronger version of the Wightman axiom (v); for an interesting sufficient condition see (Borchers and Zimmermann, 1964), again.

65For more general considerations concerning the connection between local algebras of bounded operators and local Wightman fields see (Wollenberg, 1985) and (Driessler et al., 1986b).

66Statement (2.93) is a simple consequence of Corollary 2.2.16 and the spectral theorem.

[figure: x^{1}; x^{0}; t; Kχ̌_{n}; K̂; K̂; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .]

Figure 2.7: The regions (dotted) of measurement Ǩ ∩ U_{ϵt}(Σ_{t}) , t > 0

2.3.3 Evaluation of the Asymptotic Condition

The evaluation of the asymptotic condition depends crucially on the following representation of 1-particle states: Definition 2.3.1 Given χ̌ ∈ D(ℝ^{3}) , a sequence \left\{{ \atop B̂_{t}}\right\}_{t∈ℝ} ⊂ L(H) is called a Haag-Ruelle-Kastler sequence (HRK sequence) for

Ψ = â_{0}^{∗}(χ̌)Ω_{0} ∈ H_{0}^{(1)} ⊂ H

if the following three conditions are fulfilled: (i)

lim_{t→±∞} |t|^{N} ∥B̂_{t}Ω_{0} − Ψ∥ = 0 ∀ N ∈ ℕ .

(ii) For every N ∈ ℕ there is a sequence of local operators

Â_{t} ∈ A (U_{\frac{|t|}{N}} (Kχ̌ ∩ Σ_{t}))

with

lim_{t→±∞} |t|^{N} ∥Â_{t} − B̂_{t}∥ = 0 .

Draft, March 26, 2009

67The factor |t|^{N} with arbitrary N is appropriate for short-range forces, only. For long-range forces there are only very limited results (Buchholz, 1977).

68The causal completion O′_{t} =^{def} {x ∈ ℝ^{4} : x×O_{t}} of O_{t} = Ǩ ∩ U_{ϵ|t|}(Σ_{t}) must not intersect Kχ̌_{n} .

(iii)

lim_{t→±∞} |t|^{−N} ∥B̂_{t}∥ = 0 for sufficiently large N ∈ ℕ .

In order to prove existence of HRK sequences with the additional properties formulated in Lemma 2.3.3, below, we need the following variant of Lemma 2.2.17:

Lemma 2.3.2 Let Â ∈ L(H) and let φ ∈ S(ℝ^{4}) . Then for every Borel B ⊂ ℝ^{4} :

Ψ ∈ Ê(B)H ⟹ ∫ Â(x) φ(x) dx Ψ ∈ Ê (\overline{V_{+}} ∩ (B + suppφ̃)) H ,

where

Â(x) =^{def} Û(x) Â Û(x)^{−1} for x ∈ ℝ^{4} .

Sketch of proof: Let B and B̂ be arbitrary Borel subsets of ℝ^{4} and let Ψ ∈ Ê(B)H. Then, since (2.60) implies

Ê(B) = (2π)^{−2} ∫ Û(a) χ̃_{B}(−a) da ,

where χ_{B} denotes the characteristic function of B :

(2π)^{4}Ê(B̂) ∫ Â(x) φ(x) dx Ψ
= ∫ Û(â) Â(x) Û(a) χ̃B̂(−â) χ̃_{B}(−a)φ(x) dâdadx Ψ
= ∫ Â(x + â) Û(â + a) χ̃B̂(−â) χ̃_{B}(−a)φ(x) dâdadx Ψ
= ∫ Â(x) Û(a) χ̃B̂(−â) χ̃_{B}(â − a)φ(x − â) dâdadx Ψ .

Since

∫ χ̃B̂(−â) χ̃_{B}(â − a)φ(x − â) dâ = 0 for B̂ ∩ (B + suppφ̃) = ∅ ,

this implies

∫ Â(x) φ(x) dx Ψ ∈ Ê (B + suppφ̃) H

and hence, by the spectrum condition, the statement of the lemma.

Lemma 2.3.3 For every χ̌ ∈ D(ℝ^{3}) and ϵ > 0 there is a HKR sequence \left\{{ \atop B̂_{t}}\right\}_{t∈ℝ}

for â_{0}^{∗}(χ̌)Ω_{0} fulfilling the following two conditions:^{69}

(i)

lim_{t→±∞} |t|^{N} ∥B̂_{t}^{∗}B̂_{t}Ω_{0} − ⟨Ω_{0} | B̂_{t}^{∗}B̂_{t}Ω_{0}⟩ Ω_{0}∥ = 0 ∀ N ∈ ℕ .

Draft, March 26, 2009

69Actually, by Lemma 2.3.2 and Borchers’ theorem, (i) is a consequence of (ii) for sufficiently small ϵ > 0 .

(ii)

B̂_{t}^{∗}Ê(B)H ⊂ Ê(B − U_{ϵ} (Mχ̌))H ∀ t ∈ ℝ , Borel B ⊂ ℝ^{4} .

where

Mχ̌ =^{def} {(ω_{p}, p) : p ∈ suppχ̌} .

Sketch of proof: Let χ̌ ∈ D(ℝ^{4}) and ϵ > 0 . The essential point is to show that there is an almost local operator Â, i.e. a bounded operator Â with

lim_{0<R→∞} R^{N} inf {∥Â − B̂∥ : B̂ ∈ A (U_{R}(0))} = 0 ∀ N ∈ ℕ ,

for which

f̌Â(p) = 1 ∀ p ∈ suppχ̌ ,  (2.94)

where

f̌Â =^{def} (Ê(M_{m})ÂΩ_{0})_{1} .

By (2.93), {0} and H itself are the only closed subspaces of H which are invariant with respect to^{70}

A_{loc} =^{def} ⋂_{R<0} A (U_{R}(0)) .

Therefore

\overline{A_{loc}Ω_{0}} = H

and consequently

f̌Â ≠ 0 for some Â ∈ A_{loc} .

Moreover, from

\begin{aligned}Ê(M_{m})Û(0, Λ)ÂÛ(0, Λ)^{−1}Ω_{0} & = & Û(0, Λ)Ê(M_{m})ÂΩ_{0} \\ & =_{(2.86),(2.88)} & Û_{0}(0, Λ)Ê(M_{m})ÂΩ_{0} \end{aligned}

and (2.4) we conclude

f̌_{Û(0,Λ)ÂÛ(0,Λ)^{−1}}(p) = f̌Â (\overrightarrow{Λ^{−1}p})_{|p^{0}=ω_{p}} ∀ Â ∈ A_{loc} , p ∈ ℝ^{3} .  (2.95)

Therefore, choosing some Haar measure μ on L_{+}^{↑}, we get

f̌Âδ̌(p) = ∫ f̌Â (\overrightarrow{Λ^{−1}p})_{|_{p^{0}=ωp}} δ̌(Λ) μ (dΛ) ,

for sufficiently well-behaved δ̌, where

Âδ̌ =^{def} ∫ Û(0, Λ)ÂÛ(0, Λ)^{−1}δ̌(Λ) μ (dΛ) .

Draft, March 26, 2009

70The orthogonal projection onto a nontrivial invariant subspace would commute with all elements of A_{loc} , in contradiction to (2.93).

Now, for suitable δ̌

g(Λ′) =^{def} f̌Âδ̌ (\overrightarrow{Λ′(m, 0)})
= ∫ f̌Â (\overrightarrow{Λ^{−1}(m, 0)}) δ̌(Λ′Λ) μ (dΛ)
∈ C^{∞}(ℝ^{3}) \ {0}

and therefore

f̌Â ∈ C^{∞}(ℝ^{3}) \ {0} for suitable Â ∈ A_{loc} .

By (2.4), we also have

f̌_{Û(x)ÂÛ(x)^{−1}}(p) = (e^{ipx}f̌Â(p))_{|_{p^{0}=ωp}}

and hence

f̌_{∫ Û(x)ÂÛ(x)^{−1}φ(x) dx}(p) = (2π)^{2}f̌Â(p)φ̃(ω_{p}, p) .  (2.96)

Therefore

0 ≤ f̌Â ∈ C^{∞}(ℝ^{3}) \ {0} for suitable almost local Â .

From this, using first (2.95) and then (2.96) again, we easily get (2.94) for some almost local Â.

Note that for φ ∈ S(ℝ^{4}) and

f^{+}(x) =^{def} (2π)^{−3/2} ∫_{p^{0}=ω_{p}} χ̌(p) e^{−ipx} dp  (2.97)

we have

φ_{t}(x) =^{def} ∫_{y^{0}=t} φ(x − y) f^{+}(y) dy  (2.98)
= (2π)^{−2} ∫ ((2π)^{3}φ̃(p) e^{i(ω_{p}−p^{0})t}χ̌(p)) e^{−ipx} dp .

By (2.96) this implies

\begin{aligned}f̌_{∫ Û(x)ÂÛ(x)^{−1}φ_{t}(x) dx}(p) & = & 2ω_{p} χ̌(p) \\ & = & (â_{0}^{∗}(χ̌)Ω_{0})_{1} (p) \end{aligned}

for all t ∈ ℝ if

f̌Â(p) = 1 ∀ p ∈ suppχ̌

and

φ̃(ω_{p}, p) = (2π)^{−7/2}2ω_{p} ∀ p ∈ suppχ̌ .  (2.99)

In other words:

There is an almost local operator Â with

(2.97)–(2.99) ⟹ Ê(M_{m}) ∫ Û(x)ÂÛ(x)^{−1}φ_{t}(x) dxΩ_{0} = â_{0}^{∗}(χ̌)Ω_{0} .

Choosing φ such that

suppφ̃ ∩ suppÊ ⊂ M_{m}  (2.100)

we even have

∫ Û(x)ÂÛ(x)^{−1}φ_{t}(x) dxΩ_{0} = â_{0}^{∗}(χ̌)Ω_{0} ∀ t ∈ ℝ ,

by Lemma 2.3.2. Since, obviously,

sup_{x∈ℝ^{4}} |f^{+}(x)| < ∞

this together with Ruelle’s Lemma (Lemma 2.1.1) shows that (2.97) – (2.100) guarantee {B̂_{t} =^{def} ∫ Û(x)ÂÛ(x)^{−1}φ_{t}(x) dx}_{t∈ℝ} to be a HRK sequence for â_{0}^{∗}(χ̌)Ω_{0} . In order to fulfill also conditions (i) and (ii) of the lemma, it is sufficient – thanks to Lemma 2.3.2, spectrum condition and Borchers’ theorem – to choose φ such that also

suppφ̃ ⊂ U_{δ} (Mχ̌)

holds with δ > 0 sufficiently small to ensure

U_{δ} (Mχ̌) ∩ U_{δ} (−Mχ̌) ⊂ U_{m}(0) .

Exercise 40 Show that the HRK sequence of Lemma 2.3.3 may be constructed in the form

B̂_{t} = ∫_{x^{0}=t} B̂(x) i∂^{↔}_{0} f^{+}(x) dx , B̂(x) =^{def} Û(x) B̂ Û(x)^{−1} ,

with f^{+} defined by (2.97), where B̂ is an almost local operator fulfilling

(□_{x} + m^{2})B̂(x)Ω_{0} = 0

and hence

B̂_{Σ} Ω_{0} = â_{0}^{∗}(χ̌)Ω_{0} , B̂_{Σ} =^{def} ∫_{Σ} B̂(x) i∂^{↔}_{μ} χ^{+}(x) dσ_{μ} ,

for every (sufficiently well-behaved) spacelike hypersurface Σ (without finite boundary points). Show that B̂ may be chosen such that also

B̂_{Σ}^{∗}B̂_{Σ} Ω_{0} ∼ Ω_{0}

holds for all these surfaces.

Theorem 2.3.4 Let χ̌_{1}, …, χ̌_{n} ∈ D(ℝ^{3}) be nonoverlapping and let

∥â_{0}^{∗}(χ̌_{ν})Ω_{0}∥ = 1 ∀ ν ∈ {1, …, n} .

Then the assumptions made for the considered theory and the AB imply existence of real numbers φ_{±} with

lim_{t→ +_{(−)} ∞} |t|^{N} ∥Ψout_{(in)}(χ̌_{1}, …, χ̌_{n}) − exp (i φ +_{(−)}) Â_{1,t} ⋯ Â_{n,t}Ω_{0}∥ = 0  (2.101)

for all N ∈ ℕ and HRK sequences \left\{{ \atop Â_{ν,t}}\right\}_{t∈ℝ} for the â_{0}^{∗}(χ̌_{ν})Ω_{0} .

Sketch of proof: Obviously, the (2.101) holds for n = 1 with φ_{±} = 1 . Therefore, it is sufficient to prove the theorem for n = n′ assuming it to be valid for n < n′ :

Without restriction of generality we may assume existence of some ϵ >= with

Mχ̌_{1} + … + Mχ̌_{n′} − U_{ϵ} (Mχ̌_{1}) … − U_{ϵ} (Mχ̌_{n′}) ⊂ U_{m}(0) .  (2.102)

Let us first consider HRK sequences of the type given by Lemma 2.3.3 and prove the the lemma by induction w.r.t. n. Exploiting the AB and the definition of HRK sequences we easily see that

lim_{t→ +_{(−)} ∞} |t|^{N} (∥Â_{n′−1,t}Â_{n′−1,t}^{∗}Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′}) − Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′})∥^{2}

− ∥Â_{n′−1,t}Â_{n′−1,t}^{∗}Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′−1}) − Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′−1})∥^{2}) = 0

holds for all N ∈ ℕ. Since we already know that there are real φ′_{+} , , φ′_{−} with

lim_{t→ +_{(−)} ∞} |t|^{N} ∥exp (i φ′ ; +_{(−)}) Â_{1,t} ⋯ Â_{n′−1,t}Ω_{0} − Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′−1})∥ = 0

for all N ∈ ℕ, we conclude:

lim_{t→ +_{(−)} ∞} |t|^{N} ∥Â_{n′−1,t}Â_{n′−1,t}^{∗}Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′}) − Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′})∥

= lim_{t→ +_{(−)} ∞} |t|^{N} ∥Â_{n′−1,t}Â_{n′−1,t}^{∗}Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′−1}) − Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′−1})∥

= lim_{t→ +_{(−)} ∞} |t|^{N} ∥Â_{n′−1,t}Â_{n′−1,t}^{∗} exp (i φ′ ; +_{(−)}) Â_{1,t} ⋯ Â_{n′−1,t}Ω_{0} − Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′−1})∥

= lim_{t→ +_{(−)} ∞} |t|^{N} ∥exp (i φ′ ; +_{(−)}) Â_{1,t} ⋯ Â_{n′−1,t}Â_{n′−1,t}^{∗}Â_{n′−1,t}Ω_{0} − Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′−1})∥

= lim_{t→ +_{(−)} ∞} |t|^{N} ∥exp (i φ′ ; +_{(−)}) Â_{1,t} ⋯ Â_{n′−1,t}Ω_{0} − Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′−1})∥

= 0 ∀ N ∈ ℕ .

Since, obviously,

Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′}) = Ψout_{(in)}(χ̌_{π1}, …, χ̌_{πn′}) ∀ π ∈ S′_{n}

this implies

lim_{t→ +_{(−)} ∞} |t|^{N} ∥Â_{ν,t}Â_{ν,t}^{∗}Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′}) − Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′})∥ = 0

for all ν ∈ {1, …, n′} , N ∈ ℕ and hence, by iteration,

lim_{t→ +_{(−)} ∞} |t|^{N} ∥Â_{1,t}Â_{1,t}^{∗} ⋯ Â_{n′,t}Â_{n′,t}^{∗}Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′}) − Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′})∥ = 0

for all N ∈ ℕ. Exploiting local commutativity, once again, this gives

lim_{t→ +_{(−)} ∞} |t|^{N} ∥Â_{1,t} ⋯ Â_{n′,t}Â_{1,t}^{∗} ⋯ Â_{n′,t}^{∗}Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′}) − Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′})∥ = 0

(2.103)

for all N ∈ ℕ. Now, for the HRK sequences of the type specified in Lemma 2.3.3 we have

Â_{1,t}^{∗} ⋯ Â_{n′,t}^{∗}Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′}) ∈ Ê (Mχ̌_{1} + … + Mχ̌_{n′} − U_{ϵ} (Mχ̌_{1}) … − U_{ϵ} (Mχ̌_{n′})) H ,

since

Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′}) ∈ Ê (Mχ̌_{1} + … + Mχ̌_{n′}) H .

By (2.102), the spectrum condition, and Borchers’ theorem, therefore,

Â_{1,t}^{∗} ⋯ Â_{n′,t}^{∗}Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′}) ∼ Ω ∼ Ω_{0} .

This together with (2.103) shows that there is a complex-valued function ρ(t) with

lim_{t→ +_{(−)} ∞} |t|^{N} ∥ρ(t)Â_{1,t} ⋯ Â_{n′,t}Ω_{0} − Ψout_{(in)}(χ̌_{1}, …, χ̌_{n′})∥ = 0 ∀ n ∈ ℕ .

Since, obviously,

lim_{t→ +_{(−)} ∞} ∥ρ(t)Â_{1,t} ⋯ Â_{n′,t}Ω_{0}∥ = 1 lim_{t→ +_{(−)} ∞} ∥Â_{1,t} ⋯ Â_{n′,t}Ω_{0}∥ ,

we are left to prove

lim_{t→ +_{(−)} ∞} |t|^{N} sup_{s>t} ∥Â_{1,t} ⋯ Â_{n′,t}Ω_{0} − Â_{1,s} ⋯ Â_{n′,s}Ω_{0}∥ = 0 ∀ n ∈ ℕ ,

as far as the special HRK sequences are concerned. This however, is a simple consequence of

lim_{t→ +_{(−)} ∞} |t|^{N} sup_{s,s′∈(t,t+1)} ∥Â_{1,s} ⋯ Â_{ν,s′} ⋯ Â_{n′,s}Ω_{0} − Â_{1,s} ⋯ Â_{n′,s}Ω_{0}∥ = 0 ∀ n ∈ ℕ .

Finally, it is an easy consequence of local commutativity and Definition 2.3.1 that

lim_{t→ +_{(−)} ∞} |t|^{N} ∥Â_{1,t} ⋯ Â_{n,t}Ω_{0} − Â_{1,t} ⋯ Â′_{ν,t} ⋯ Â_{n,t}Ω_{0}, t∥

holds for all N ∈ ℕ and ν ∈ \{1, …, n\}, if \left\{{ \atop Â′_{ν,t}}\right\}_{t∈ℝ} is any other HRK sequence for â_{0}^{∗}(χ̌_{ν})Ω_{0} ; i.e.:

(2.101) does not dependent on the special choice for the HRK sequences.

Exercise 41 Show the following:

(i) Unitary mappings V̂out_{(in)} from H_{0} onto H fulfilling (2.86), (2.88), and the asymp-

totic condition do exist.

(ii) The numbers φ_{±} may^{71} depend on n but not on the functions χ̌_{ν} .

Exercise 42 Assuming that the local algebras are given by a neutral scalar Wightman field as described above, show the following for the PCT operator θ̂ considered in Exercise 39 and the corresponding operator θ̂_{0} of the FS:^{72}

(i)

θ̂Û(a, Λ) = Û(−a, Λ)θ̂ ∀ (a, Λ) ∈ P_{+}^{↑} ,

(ii)

θ̂V̂_{out} = V̂_{in}θ̂_{0} , θ̂V̂_{in} = V̂_{out}θ̂_{0} ,

(iii) The described scattering theory is PCT-invariant in the sense that^{73}

θ̂ˆŜ = ˆ Ŝ^{−1}θ̂.

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71Suitable extension of the asymptotic condition implies φ_{±} = 1 for all n (see (Lücke, 1983)). 72Note that (i) implies θ̂A(O) = A(−O)θ̂ for all open O ⊂ ℝ^{4} . 73For a (complicated) proof not using this assumption see (Epstein, 1967).

2.3.4 Cluster Properties of the S-Matrix

Definition 2.3.5 Let \{M_{t}\}_{t_{∈}ℝ} ⊂ ℝ^{4} and \left\{{ \atop B̂_{t}}\right\}_{t_{∈}ℝ} ⊂ L(H) . Then B̂_{t} is called asymptotically localized in M_{t} if the following two conditions are fulfilled:

(i) For every N ∈ ℕ there is a sequence of local operators

Â_{t} ∈ A (U_{\frac{|t|}{N}} (M_{t}))

with

lim_{t→±∞} |t|^{N} ∥Â_{t} − B̂_{t}∥ = 0 .

(ii)

lim_{t→±∞} |t|^{−N} ∥B̂_{t}∥ = 0 for sufficiently large N ∈ ℕ .

An immediate consequence of Definition 2.3.5 is the following

Corollary 2.3.6 For ν = 1, 2 let B̂_{j,t} be asymptotically localized in M_{j,t}. Then B̂_{t} =^{def} B̂_{1,t}B̂_{2,t} is asymptotically localized in M_{t} =^{def} M_{1,t} ∪ M_{2,t}.

Definition 2.3.7 Let \{M_{t}\}_{t_{∈}ℝ} ⊂ ℝ^{4} and Ψ ∈ H. Then \left\{{ \atop B̂_{t}}\right\}_{t_{∈}ℝ} ⊂ L(H) is called a M_{t}-sequence for Ψ if the following two conditions are fulfilled:^{74}

(i) lim_{t→±∞} |t|^{N} ∥B̂_{t}Ω_{0} − Ψ∥ = 0 ∀ N ∈ ℕ .

(ii) B̂_{t} is asymptotically localized in M_{t}.

Theorem 2.3.8 Let χ̌_{1}, …, χ̌_{n} ∈ D(ℝ^{3}) be non-overlapping and let Σ be a smooth spacelike hypersurface (without finite boundary points) above resp. below^{75} all the sets

(Kχ̌_{ν} + a_{ν}) ∩ (Kχ̌_{μ} + a_{μ}) , ν ≠ μ ,

for given a_{1}, …, a_{n} ∈ ℝ^{4} \ Σ . Then there are (tΣ − ta_{ν})-sequences \left\{{ \atop B̂_{ν,t}}\right\}_{t∈ℝ} for the 1-particle states â_{0}^{∗}(χ̌_{ν})Ω_{0} and a real number φ_{out} resp. φ_{in} with^{76}

lim_{λ→+∞} |t|^{N} ∥Ψ_{ex} (Û_{0}(λa_{1})χ̌_{1}, …, Û_{0}(λa_{n})χ̌_{n})

− e^{iφ_{ex}} B̂_{1,λ}(λa_{1}) ⋯ B̂_{n,λ}(λa_{n})Ω_{0}∥ = 0 ∀ N ∈ ℕ

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74Note that for Ψ = â_{0}^{∗}(χ̌)Ω_{0} and M_{t} = Kχ̌ ∩ Σ_{t} the M_{t}-sequences are just HRK-sequences. 75We say Σ is above_{(below)} a set M ⊂ ℝ^{4} if M ⊂ {(x^{0} −_{(+)} t, x) : x ∈ Σ , t > 0}. 76The special choice for the (tΣ − ta_{ν})-sequences is not essential. As usual, we use the notation

B̂(x) = Û(x)B̂ Û(x)^{−1} for B̂ ∈ L(H) , x ∈ ℝ^{4} .

where ex=out resp. ex=in .

Proof: Straightforward application of the described standard techniques.

Theorem 2.3.9 (Fredenhagen) Let τ > 0 and let Â , B̂ ∈ L(H) fulfill the condition^{77}

[Û(t)Â Û(t)^{−1}, B̂]_{−} = 0 ∀ t ∈ [−τ, +τ] .

Then

|⟨Ω_{0} | ÂB̂Ω_{0}⟩ − ⟨Ω_{0} | ÂΩ_{0}⟩ ⟨Ω_{0} | B̂Ω_{0}⟩| ≤ e^{−mτ} \sqrt{∥Â^{∗}Ω_{0}∥ ∥B̂Ω_{0}∥ ∥ÂΩ_{0}∥ ∥B̂^{∗}Ω_{0}∥} .

Proof: See (Fredenhagen, 1985).

All we need for the derivation of cluster properties of the S-matrix is the following immediate consequence of Theorem 2.3.9.

Corollary 2.3.10 For ν = 1, 2 let B̂_{j,t} be asymptotically localized in M_{j,t}. If there is some ϵ for which

λ > \frac{1}{ϵ} ⟹ M_{1,λ}×U_{ϵλ}(M_{2,λ}) ,

then

lim_{λ→+∞} λ^{N} ∥⟨Ω_{0} | B̂_{1,λ}B̂_{2,λ}Ω_{0}⟩ − ⟨Ω_{0} | B̂_{1,λ}Ω_{0}⟩ ⟨Ω_{0} | B̂_{2,λ}Ω_{0}⟩∥ = 0 ∀ N ∈ ℕ .

Now the cluster properties of the S-matrix are an immediate consequence:

Corollary 2.3.11 Let χ̌_{1}, …, χ̌_{n} a_{1}, …, a_{n} and Σ be given as in Theorem 2.3.8 for ‘above’. Moreover, let χ̌′_{1}, …, χ̌′_{n′} a′_{1}, …, a′_{n} and Σ′ be given as in Theorem 2.3.8 for ‘below’. Finally, let I ⊂ {1, …, n} and I′ ⊂ {1, …, n′} be such that

((Kχ̌_{ν_{1}} + a_{ν_{1}}) ∩ Σ) ∪ ((K_{}_{χ̌′_{ν′_{1}}} + a′_{ν′_{1}}) ∩ Σ′)

is spacelike relative to

((Kχ̌_{ν_{2}} + a_{ν_{2}}) ∩ Σ) ∪ ((K_{}_{χ̌′_{ν′_{2}}} + a′_{ν′_{2}}) ∩ Σ′)

for all ν_{1} ∈ I , ν′_{2} ∈ I′ and ν_{2} ∈ {1, ⋯, n} \ I , ν′_{2} ∈ {1, …, n′} \ I′.

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77As usual, we write Û(t) for Û ((t, 0, 0, 0), 1l_{4}) .

Then there is a real number ρ with

lim_{λ→+∞} λ^{N} |e^{iρ} ⟨Ψ_{out}(λ) | Ψ_{in}(λ)⟩ − ⟨Ψ_{1,out}(λ) | Ψ_{1,in}(λ)⟩ ⟨Ψ_{2,out}(λ) | Ψ_{2,in}(λ)⟩| = 0

for all N ∈ ℕ, where

Ψ_{out}(λ) =^{def} V̂_{out} ( ∏_{ν′∈{1,...,n′}} (Û(λa′_{ν′})â_{0}^{∗} (χ̌′_{ν′}) (Û(λa′_{ν′})^{−1}) Ω_{0}) ,

Ψ_{1,out}(λ) =^{def} V̂_{out} ( ∏_{ν′∈I′} (Û(λa′_{ν′})â_{0}^{∗} (χ̌′_{ν′}) (Û(λa′_{ν′})^{−1}) Ω_{0}) ,

Ψ_{2,out}(λ) =^{def} V̂_{out} ( ∏_{ν′∈{1,...,n′}\I′} (Û(λa′_{ν′})â_{0}^{∗} (χ̌′_{ν′}) (Û(λa′_{ν′})^{−1}) Ω_{0}) ,

Ψ_{in}(λ) =^{def} V̂_{in} ( ∏_{ν∈\{1,...,n\}} (Û(λa_{ν})â_{0}^{∗} \binom{}{χ̌_{ν}} (Û(λa_{ν})^{−1}) Ω_{0}) ,

Ψ_{1,in}(λ) =^{def} V̂_{in} (∏_{ν∈I} (Û(λa_{ν})â_{0}^{∗} \binom{}{χ̌_{ν}} (Û(λa_{ν})^{−1}) Ω_{0}) ,

Ψ_{2,in}(λ) =^{def} V̂_{in} ( ∏_{ν∈\{1,...,n\}\I} (Û(λa_{ν})â_{0}^{∗} \binom{}{χ̌_{ν}} (Û(λa_{ν})^{−1}) Ω_{0}) .

Typical consequences of Corollary 2.3.11 are illustrated by Figures 2.9 and 2.8 (see also (Lücke, 8384)).^{78} Roughly speaking, Corollary 2.3.11 shows that, from the macroscopic point of view, the S-matrix does not violate the causality principle.

It should have become clear from the evaluation of the asymptotic condition, that the field Φ̂(x) itself is not necessary to determine the S-matrix once the set of almost local operators is determined. Note that the representation Û(a, Λ) of P_{+}^{↑} is already fixed, up to unitary equivalence, by the FS. Of course, one cannot expect that the Hamiltonian P̂^{0} itself determines the physical picture of the dynamics unless the physical interpretation of the other observables is sufficiently well established. The above considerations show that it is sufficient to specify the macroscopically localized observables consistently, in order to select the S-matrix.

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78Figure 2.8 illustrates the case n = 4 , n′ = 6 , I = I′ = {1, 2}; using the notation K_{ν} =^{def}

Kχ̌_{ν} + λa_{ν} , K′_{ν} =^{def} K_{}_{χ̌′_{ν}} + λa′_{ν}

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; 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Figure 2.8: Macroscopically independent: ‘1+2 ⟶ 1+2’ and ‘3+4 ⟶ 3+…+6’

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; 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.; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; Kχ̌_{1} + a_{1}; Kχ̌_{2} + a_{2}; Σ; Σ′; ‘Start’; ‘End’; K′_{χ̌′_{1}} + a′_{1}; K′_{χ̌′_{2}} + a′_{2}]

Figure 2.9: Macroscopically forbidden: ‘output before input’

2.4 Charged Scalar Fields

2.4.1 Free Charged Scalar Fields

Fields Operators

It is known, nowadays, that for every charged particle there is an antiparticle with opposite charge. Let (2.39) describe such a particle in the sense of 2.1.3 and, similarly,

˘Φ̂_{0}^{(+)}(x) =^{def} (2π)^{−3/2} ∫_{p^{0}=ω_{p}} ˘ â_{0}(p) e^{−ipx} \frac{dp}{2p^{0}}

the corresponding antiparticle on its Fock space H̆_{0} with domain D̆_{0} and representation ˘ Û_{0}(a, Λ) of P_{+}^{↑}. Then both particles may be described simultaneously by the charged scalar field

Φ̂_{q}(x) =^{def} Φ̂_{0}^{(+)}(x) ⊗ 1l + (1l ⊗ ˘ Φ̂_{0}^{(+)}(x))^{∗}  (2.104)

on

D_{q} =^{def} D_{0} ⊗ D̆_{0} ⊂ H_{q} =^{def} H_{0} ⊗ H̆_{0} ,

generalizing (2.43). As for the neutral scalar field, the relations^{79}

(□ + m^{2}) Φ̂_{q}(x) = 0 ,  (2.105)
Û_{q}(a, Λ)Φ̂_{q}(x)Û_{q}(a, Λ)^{−1} = Φ̂_{q}(Λx + a) ,  (2.106)
[Φ̂_{q}(x), Φ̂_{q}(y)]_{−} = 0 for x×y ,  (2.107)

are fulfilled for the charged field, where

Û_{q}(a, Λ) =^{def} Û_{0}(a, Λ) ⊗ ˘ Û_{0}(a, Λ) .

Nevertheless, the charged field cannot be interpreted as observable of whatever field strength since, according to (2.104), it is not hermitian. This is the price to be paid for the important commutation relation

[Q̂, Φ̂_{q}(x)]_{−} = −q Φ̂_{q}(x) , where:
Q̂ = observable of the additive charge,
q = charge of the particle,
−q = charge of the antiparticle,

(2.108)

valid on a suitable domain. The main purpose of the charged scalar field is to create a dense set of well-interpreted states (for scattering theory) out of the vacuum.

Now it is important to supplement (2.107) by^{80}

[Φ̂_{q}(x), Φ̂_{q}^{∗}(y)]_{−} = 0 for x×y .  (2.109)

Draft, March 26, 2009

79(2.107) holds for all x, y.

80More precisely, (2.46) becomes [Φ̂_{q}(x), Φ̂_{q}^{∗}(y)]_{−} = iΔ(x − y) .

Local Gauge Transformations

Since the charged scalar field Φ̂_{q}(x) is non-hermitian, anyway, it may well be replaced by

Φ̂_{q}^{λ}(x) =^{def} e^{iQ̂λ(x)}Φ̂_{q}(x)e^{−iQ̂λ(x)} =_{(2.108)} e^{−iqλ(x)}Φ̂_{q}(x)  (2.110)

if, simultaneously, the Klein-Gordon equation (2.105) is replaced by

((∂_{μ} + iq(∂_{μ}λ)(x)) (∂^{μ} + iq(∂^{μ}λ)(x)) + m^{2}) Φ̂_{q}^{λ}(x) = 0 ,  (2.111)

where λ(x) denotes an arbitrary, but sufficiently smooth, real-valued function on ℝ^{4} . As a direct generalization this leads to the Klein-Gordon equation

((∂_{μ} + iqA_{μ}(x)) (∂^{μ} + iqA^{μ}(x)) + m^{2}) Φ̂^{λ}(x) = 0  (2.112)

for the quantized scalar field Φ̂(x) interacting with the external (classical) electromagnetic field A^{μ}(x) , being invariant under gauge transformations of second kind

Φ̂(x) ⟶ e^{−iqλ(x)}Φ̂ , A_{μ}(x) ⟶ A_{μ}(x) + ∂_{μ}λ(x) .  (2.113)

Consistency Considerations

Exchanging the roles of particles and antiparticles results, according to (2.104), in the transition

Φ̂_{(q)}(x) ⟶ Φ̂_{(q)}^{∗}(x) , q ⟶ −q .

Obviously, (2.112) is invariant under this transformation (since A^{μ}(x) is real).

For sufficiently well-behaved external fields A^{μ}(x) there exist solutions Φ̂ of (2.112) respecting the interaction picture (Seiler, 1978).

Interpreting classical solutions of the Klein-Gordon equation as expectation values of the quantized Klein-Gordon field solves the well-known problem raised by creation of negative frequency contributions in certain scattering problems (see e.g. (Baym, 1969, Chapt. 22)).

Replacing the classical electromagnetic potential A^{μ}(x) in (2.112) by the corresponding quantum field Â^{μ}(x) leads to the difficult problem of defining products of quantized fields (basic problems of renormalization theory).

2.4.2 Wightman Theory for Charged Scalar Fields

Wightman Axioms

A Wightman theory of a single charged scalar field Φ̂(x) is characterized by the following assumptions (Wightman axioms ):

0. Assumptions of Relativistic Quantum Theory:

Exactly the same as those for the Wightman Theory of a single neutral scalar field, as formulated in Section 2.2.1.

I. Assumptions about the Domain and Continuity of the Field:

There are two fields Φ̂(x) and Φ̂^{∗}(x) defined as operator-valued, tempered, generalized functions with invariant domain D ⊂ H; i.e. linear mappings

Φ̂ : S(ℝ^{4}) ⟶ L(D, D)
φ ⟼ Φ̂(φ) = \underbrace{∫ Φ̂(x)φ(x) dx}_{formal}

and

Φ̂^{∗} : S(ℝ^{4}) ⟶ L(D, D)
φ ⟼ Φ̂^{∗}(φ) = \underbrace{∫ Φ̂^{∗}(x)φ(x) dx}_{formal}

for which all the

∫ ⟨Ψ | Φ̂(x)Ψ⟩ φ(x) dx =^{def} ⟨Ψ | Φ̂(φ)Ψ⟩ , Ψ ∈ D ,

and

∫ ⟨Ψ | Φ̂^{∗}(x)Ψ⟩ φ(x) dx =^{def} ⟨Ψ | Φ̂^{∗}(φ)Ψ⟩ , Ψ ∈ D ,

are continuous in φ ∈ S(ℝ^{4}) , where D has to fulfill the following conditions for φ ∈ S(ℝ^{4}) and (a, Λ) ∈ P_{+}^{↑} :

Ω ⊂ D , Û(a, Λ)D ⊂ D , Φ̂(φ)D ⊂ D ⊃ Φ^{∗}(φ)D .

The fields Φ̂(x) and Φ̂^{∗}(x) are related by

⟨Ψ | Φ̂^{∗}(φ)Ψ⟩ = ⟨Ψ | (Φ̂(\overline{φ}))^{∗} Ψ⟩ ∀Ψ ∈ D , φ ∈ S(ℝ^{4}) . (2.114)

II. Transformation Law of the Field:

The field operators Φ̂(x) and Φ̂^{∗}(x) transform according to

Û(a, Λ)Φ̂(x)Û(a, Λ)^{−1} = Φ̂(Λx + a) ∀ (a, Λ) ∈ P_{+}^{↑}

and^{81}

Û(a, Λ)Φ̂^{∗}(x)Û(a, Λ)^{−1} = Φ̂^{∗}(Λx + a) ∀ (a, Λ) ∈ P_{+}^{↑} .

Draft, March 26, 2009

81By (2.114) the transformation law of Φ̂(x) implies that of Φ̂^{∗}(x) and vice versa.

III. Local Commutativity:

The smeared fields Φ̂(φ_{1}) and Φ̂(φ_{2}) resp. Φ̂^{∗}(φ_{2}) commute whenever the supports of the test functions φ_{1} , φ_{2} ∈ S(ℝ^{4}) are spacelike with respect to each other.^{82} Formally:

x×y ⟹ [Φ̂(x), Φ̂(y)]_{−} = [Φ̂(x), Φ̂^{∗}(y)]_{−} = 0 .

Finally, the vacuum vector Ω is required to be cyclic with respect to the algebra F_{0} generated by ˆ1/^{\}D and the smeared field field operators Φ̂(φ) and Φ̂^{∗}(φ) with φ ∈ S(ℝ^{4}) :

D_{0} =^{def} F_{0} Ω is dense in H .

Obviously, all these axioms are fulfilled for the free charged field Φ̂(x) = Φ̂_{q}(x) , if we define D =^{def} D_{q} and

Φ̂^{∗}(φ) =^{def} (Φ̂(\overline{φ}))^{∗} /^{\}D for φ ∈ S(ℝ^{4}) .

PCT and Spin-Statistics Theorem

The ‘connection between spin and statistics’ for the theory of a single charged field is given by the following two theorems.

Theorem 2.4.1 There is no charged field Φ̂(x) ≠ 0 , fulfilling all the Wightman axioms with the possible exception of local commutativity, for which

x×y ⟹ Φ̂(x)Φ̂^{∗}(y) + Φ̂^{∗}(y)Φ̂(x) = 0 .

Sketch of proof: The techniques used in Section 2.2.4 show that the expectation values of products of field operators exist as generalized functions and that there are L_{+}^{↑}-invariant generalized functions W, W̌ ∈ S(ℝ^{4})′ with

⟨Ω | Φ̂(x)Φ̂^{∗}(y)Ω⟩ = W (x − y) ,
⟨Ω | Φ̂^{∗}(x)Φ̂(y)Ω⟩ = W̌ (x − y) ,

and

suppW̃ ⊂ \overline{V_{+}} ⊃ supp˜W̌ .  (2.115)

Draft, March 26, 2009

82Note that that this condition can be shown to be necessary to avoid acausal effects.

Now, assume

x×y ⟹ Φ̂(x)Φ̂^{∗}(y) + Φ̂^{∗}(y)Φ̂(x) = 0 ,

i.e.

W (ξ) + W̌ (−ξ) = 0 for ξ×0 .

Since W̌ (ξ)−W̌ (−ξ) , being an odd L_{+}^{↑}-invariant distribution, vanishes for spacelike ξ this implies

F (ξ) =^{def} W (ξ) + W̌ (ξ) = 0 for ξ×0 .

Since, by (2.115),

suppF̃ ⊂ \overline{V_{+}} ⊃ supp˜W̌

Corollary 2.2.9 tells us that F = 0 , i.e:

⟨Ω | Φ̂(x)Φ̂^{∗}(y)Ω⟩ + ⟨Ω | Φ̂^{∗}(−y)Φ̂(−x)Ω⟩ = 0 .

Therefore, we have

0 = ∫ (⟨Ω | Φ̂(x)Φ̂^{∗}(y)Ω⟩ + ⟨Ω | Φ̂^{∗}(−y)Φ̂(−x)Ω⟩) φ(x)\overline{φ(y)} dxdy
= ∥(Φ̂(φ))^{∗} Ω∥^{2} + ∥∫ Φ̂(x)φ(−x) dx Ω∥^{2}

for all φ ∈ S(ℝ^{4}) and hence

Φ̂^{∗}(x)Ω = Φ̂(x)Ω = 0 .

By cyclicity of Ω , however, this would imply Φ̂(x) = 0 .

Theorem 2.4.2 There is no charged field Φ̂(x) ≠ 0 , fulfilling all the Wightman axioms with the possible exception of local commutativity, for which the conditions

x×y ⟹ Φ̂(x)Φ̂^{∗}(y) − Φ̂^{∗}(y)Φ̂(x) = 0

and

x×y ⟹ Φ̂(x)Φ̂(y) + Φ̂(y)Φ̂(x) = 0

hold.

Sketch of Proof:^{83} Assume

x×y ⟹ Φ̂(x)Φ̂^{∗}(y) − Φ̂^{∗}(y)Φ̂(x) = 0

and

x×y ⟹ Φ̂(x)Φ̂(y) + Φ̂(y)Φ̂(x) = 0 .

Draft, March 26, 2009

83This proof may be applied to a much more general situation (see (Streater and Wightman, 1989, Theorem 4.8)).

Then for all φ , ψ ∈ D(ℝ^{4}) with

suppφ×suppψ

we have

0 ≤ ∥Φ̂^{∗}(φ)Φ̂(ψ)Ω∥^{2}
= ⟨Ω | (Φ̂(ψ))^{∗} (Φ̂^{∗}(φ))^{∗} Φ̂^{∗}(φ)Φ̂(ψ)Ω⟩
= − ⟨Ω | (Φ̂^{∗}(φ))^{∗} Φ̂^{∗}(φ) (Φ̂(ψ))^{∗} Φ̂(ψ)Ω⟩ .

Since

Û(a)Φ̂(ψ)Û(a)^{−1} = ∫ Φ̂(x)ψ(x − a) dx ∀ a ∈ ℝ^{4} ,

this implies

⟨Ω | (Φ̂^{∗}(φ))^{∗} Φ̂^{∗}(φ)Û(λa) (Φ̂(ψ))^{∗} Φ̂(ψ)Ω⟩ ≤ 0

for spacelike a and sufficiently large λ = λ(a, φ, ψ) . On the other hand, however, one may prove^{84} that

lim_{λ→∞} ⟨Ω | (Φ̂^{∗}(φ))^{∗} Φ̂^{∗}(φ)Û(λa) (Φ̂(ψ))^{∗} Φ̂(ψ)Ω⟩
= ⟨Ω | (Φ̂^{∗}(φ))^{∗} Φ̂^{∗}(φ)Ω⟩ ⟨Ω | (Φ̂(ψ))^{∗} Φ̂(ψ)Ω⟩
= ∥Φ̂^{∗}(φ)Ω∥^{2} ∥Φ̂(ψ)Ω∥^{2} .

Therefore

Φ̂^{∗}(x)Ω = Φ̂(x)Ω = 0 ,

which, by cyclicity of Ω , would imply Φ̂(x) = 0 .

Now, of course, the PCT theorem has to involve both Φ̂(x) and Φ̂^{∗}(x) :

Theorem 2.4.3 Let n ∈ ℕ and let Φ̂(x) be a charged scalar field fulfilling all the Wightman axioms with the possible exception of local commutativity. Then the PCT condition

⟨Ω | Φ̂_{1}(x_{1}) ⋯ Φ̂_{n}(x_{n})Ω⟩ = ⟨Ω | Φ̂_{n}(−x_{n}) ⋯ Φ̂_{1}(−x_{1})Ω⟩
∀ x ∈ ℝ^{4n} , Φ̂_{ν} ∈ {Φ̂, Φ̂^{∗}}

(2.116)

is equivalent to the condition of weak local commutativity

⟨Ω | Φ̂_{1}(x_{1}) ⋯ Φ̂_{n}(x_{n})Ω⟩ = ⟨Ω | Φ̂_{n}(x_{n}) ⋯ Φ̂_{1}(x_{1})Ω⟩
for (x_{1} − x_{2}, …, x_{n−1} − x_{n}) ∈ J_{n−1} , Φ̂_{ν} ∈ {Φ̂, Φ̂^{∗}} .

(2.117)

Exercise 43 Draft, March 26, 2009

84See (Araki et al., 1962, Theorem 3) for a proof of this cluster property not depending on 0 being an isolated point of the energy-momentum spectrum. See also (Maison, 1968) and, for a C^{∗} algebraic version, (Baumgärtel, 1995, Theorem 1.2.5).

Proof: Analogous to that of Corollary 2.2.22

Show that (2.116) is equivalent to existence of an anti-unitary Operator θ̂ fulfilling the conditions

θ̂θ̂ = ˆ1 , θ̂Ω = Ω ,
θ̂ ˆΦ̂(φ) θ̂ = (∫ ˆ Φ̂(−x) φ(x) dx)^{∗} /^{\}D ∀ φ ∈ S(ℝ^{4}) , ˆ Φ̂ ∈ {Φ̂, Φ̂^{∗}}

and

θ̂Û(a, Λ) θ̂ = Û(−a, Λ) ∀ (a, Λ) ∈ P_{+}^{↑} .

2.4.3 Scattering Theory

Now assume that the IS is described by a charged scalar field Φ̂(x) fulfilling all the Wightman axioms and that the corresponding FS may be described by the free charged scalar field Φ̂_{q}(x) .

Since the 1-particle states are charged they can no longer be approximated by states of the form B̂Ω , B̂ ∈ A_{loc} . Here the local observable algebras have to be replaced by some net of local field algebras F_{b}(O) . It is not evident how to define these algebras of bounded operators.^{85} This problem can be be avoided by working with unbounded operators (Lücke, 1983). For simplicity, however, let us assume that the F_{b}(O) are specified and fulfill the conditions of isotony

O_{1} ⊂ O_{2} ⟹ F_{b}(O_{1}) ⊂ F_{b}(O_{2}) ,

local commutativity

O_{1}×O_{2} ⟹ [F_{b}(O_{1}), F_{b}(O_{2})]_{−} = \left\{{ \atop ˆ0}\right\} ,

Poincaré covariance

Û(a, Λ)F_{b}(O)Û(a, Λ)^{−1} = F_{b}(ΛO + a) ,

and irreducibility

( ⋃_{R>0} F_{b}(U_{R}(o)))′ = {λˆ1 : λ ∈ ℂ}

(compare (2.90)–(2.93)). Then the corresponding net of local observable algebras is given by

A(O) =^{def} {Â ∈ F_{b}(O) : [Â, Q̂]_{−} = 0} ,

where Q̂ is the charge operator (uniquely) defined by

\begin{matrix}[Q̂, Φ̂(φ)]_{−} & = & −qΦ̂(φ) \\ [Q̂, Φ̂^{∗}(φ)]_{−} & = & +qΦ̂^{∗}(φ) \end{matrix} \} ∀φ ∈ S(ℝ^{4}) , Q̂Ω = 0 , Q̂^{∗} = Q̂ ,

Draft, March 26, 2009

85See (Driessler et al., 1986a) for a detailed discussion of this point.

and the scattering theory described for the neutral field can be applied to the charged field with the following modifications:

1. Everywhere, except in the asymptotic condition, the field algebras F_{b}(O) have to be used instead of the observable algebras.

2. Asymptotic states with arbitrary number of particles and antiparticles have to be considered.

3. The HRK sequences have to be defined for both particle and antiparticle states.

Now, of course, the resulting PCT invariance of the transition probabilities also involves interchange of particles and antiparticles.

λΦ_{4}^{4} Perturbation Theory

“Renormalization theory has a history of egregious errors by distinguished savants. It has a justified reputation for perversity; a method that works up to 13^{th} order in the perturbation series fails in the 14^{th} order. Arguments that sound plausible often dissolve into mush when examined closely. The worst that can happen often happens. The prudent student would do well to distinguish sharply between what has been proved and what has been made plausible, and in general he should watch out!””

A. S. Wightman (Velo and Wightman, 1976, p. 16)

3.1 General Aspects

3.1.1 Interaction Picture

General Definition

Let us formally assume that the ‘same’ instant measurements can be performed on the IS as well as on the FS and that the expectation values for all such measurements performable at a fixed time t determine the corresponding state uniquely. Then we say

“Ψ̌ ∈ Ȟ resp. Ψ̌_{0} ∈ Ȟ_{0} looks like Φ̌_{0} ∈ Ȟ_{0} at time t”

if all the expectation values for identical measurements to be performed at time t predicted by Ψ̌ resp. Ψ̌_{0} are the same as those predicted by Φ̌_{0} .

Moreover, let us assume that for every state Ψ̌ and for every instant of time t there is a state Ψ̌_{I}(t) of the FS such that Ψ̌ ∈ Ȟ looks like Ψ̌_{I}(t) ∈ Ȟ_{0} at time t. Then we call Ψ̌_{I}(t) the instantaneous state at time t of the IS in the interaction picture, if the IS is in the actual state Ψ̌ . See Fig. 3.1 for the example of a classical particle moving in an external potential.

In general, if the interaction picture exists, the AC should be of the form

Ψ̌_{I}(t) ⟶_{t→±∞} Ψ̌_{±}  (3.1)

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•  Ψ̌ ˆ= x(t) Ψ̌_{I}(t_{0}) ˆ= x_{0}(t_{0}) + (t − t_{0})ẋ(t_{0}) loc. at t_{0}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} 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Figure 3.1: Interaction picture for a classical particle

for suitable specification of the type of convergence.

Formalization (in view of quantum theory)

Assuming that the interaction picture exists, in the sense described above, let us introduce the following notation:

Ǎ_{0}  =^{def} set of all instantaneous measurement performable at time 0 ,
α̌_{Δt}(A)  =^{def} measurement A ∈ Ǎ_{0} time-shifted by Δt ,
Ǎ  =^{def} ⋃_{t∈ℝ} {α̌_{t}(A) : A ∈ Ǎ_{0}} ,
E (A, Ψ̌) =^{def} expectation value of A ∈ Ǎ for the IS in state Ψ̌ ,
E_{0} (A, Ψ̌_{0}) =^{def} expectation value of A ∈ Ǎ for the FS in state Ψ̌_{0} .

While the set of all instantaneous measurement procedures is the same for the IS and the FS, the instantaneous states develop differently in time:^{1}

ˆÛ(t)Ψ̌ =^{def} state of the IS , at time t looking like Ψ̌ at time 0 ,
ˆÛ_{0}(t)Ψ̌_{0} =^{def} state of the FS , at time t looking like Ψ̌_{0} at time 0 .

(3.2)

Consistency requires

E_{(0)} (α̌_{t}(A), ˆ Û_{(0)}(t)Ψ̌_{(0)}) = E_{(0)} (A, Ψ̌_{(0)}) ∀ A ∈ Ǎ_{0} .  (3.3)

The instantaneous state Ψ̌_{I}(t) in the interaction picture at time t of the IS in the actual state Ψ̌ is determined by

E_{0} (α̌_{t}(A), Ψ̌_{I}(t)) = E (α̌_{t}(A), Ψ̌) ∀ A ∈ Ǎ_{0} .  (3.4)

Defining

ˆŴ (t)Ψ̌ =^{def} Ψ̌_{I}(t) ,

Draft, March 26, 2009

1Warning: In general, ˆ Û(t) and ˆ Û_{0}(t) depend on the choice for the origin of the time-scale.

3.1. GENERAL ASPECTS 103 by (3.1) we formally get

ˆŴ (t)Ψ̌ ⟶_{t→ +_{(−)} ∞} Ψ̌ +_{(−)} =_{(2.78)} ˆ V̂ −1 ; out_{(in)}Ψ̌ ∀ Ψ̌ ∈ Ȟout_{(in)} ,

i.e.:

ˆŴ (t)^{−1} ⟶_{t→ +_{(−)} ∞} ˆ V̂ out_{(in)} .  (3.5)

By (3.3) and (3.4), assuming (without restriction of generality)

ˆŴ (0) = identity mapping ,

we have

ˆŴ (t) = ˆ Û_{0}(t) ˆÛ(t)^{−1} , ˆ Ŵ (t)^{−1} = ˆ Û(t) ˆÛ_{0}(t)^{−1}  (3.6)

and therefore, by (2.82), (3.5) implies ˆŜ_{0} = lim_{t_{±}→±∞} ˆ Ω̂(t_{+}, t_{−}) , where:

ˆΩ̂(t_{+}, t_{−}) =^{def} ˆ Û_{0}(t_{+}) ˆÛ(t_{+})^{−1} ˆÛ(t_{−}) ˆÛ_{0}(t_{−})^{−1} .

(3.7)

The type of limit, depending on the model, has to be suitably defined, of course. If, in addition, we also have homogeneity in time, i.e.^{2}

ˆÛ_{(0)}(t_{1}) ˆÛ_{(0)}(t_{2}) = ˆ Û_{(0)}(t_{1} + t_{2}) ,

(3.5) and (3.6) imply

ˆÛ(t) ˆV̂ out_{(in)} = ˆ V̂ out_{(in)} ˆÛ_{0}(t) , ˆ V̂ −1 ; out_{(in)} ˆÛ(t) = ˆ Û_{0}(t) ˆV̂ −1 ; out_{(in)} .  (3.8)

and, by (2.79)/(2.82), therefore:

[ˆŜ_{(0)}, ˆ Û_{(0)}(t)]_{−} = 0 .

Application to Quantum Theory In quantum theory the states

Ψ̌_{(0)} = ω_{Ψ_{(0)}}

are given by state vectors Ψ_{(0)} from the corresponding Hilbert space H_{(0)} and the time translations ˆ Û_{(0)}(t) are given by unitary operators Û_{(0)}(t) in H_{(0)} ,

ˆÛ_{(0)}(t) Ψ̌_{(0)} = ω_{Û_{(0)}(t)Ψ_{(0)}} ,

Draft, March 26, 2009

2In this case ˆ Û(t) and ˆ Û_{0}(t) are independent of the choice for the origin of the time-scale.

depending strongly continuously on t. Therefore, according to Stone’s theorem, we may define the Hamiltonian Ĥ_{(0)}(t) by^{3}

Ĥ_{(0)}(t) =^{def} (i\frac{d}{dt} (Û_{(0)}(t)^{−1})) Û_{(0)}(t) = −Û_{(0)}(t)^{−1} i\frac{d}{dt}Û_{(0)}(t) ,

even if we do not have homogeneity in time. This implies

i∂_{t}Ω̂(t, t_{−}) = Ĥ_{I}(t)Ω̂(t, t_{−}) , where:

Ĥ_{I}(t) =^{def} Û_{0}(t) (Ĥ(t) − Ĥ_{0}(t)) Û_{0}(t)^{−1} , (3.9)

which, formally, is equivalent to^{4}

Ω̂(t, t_{−}) = ˆ1 − i ∫_{t_{−}}^{t} Ĥ_{I}(t_{+})Ω̂(t_{+}, t_{−}) dt_{+} ,  (3.10)

since Ω̂(t_{−}, t_{−}) = ˆ1 , by (3.7). As usual, this integral equation may be formally solved by iteration giving the so-called Dyson series.

3.1.2 Canonical Field Quantization

Field Equations

Hoping to convert the free Klein-Gordon theory into a model with nontrivial S-matrix, one adds a local self-interaction term of the form^{5} λ_{b}⋮F (Φ̂(x))⋮ with coupling constant λ_{b} > 0 as perturbation to the Klein-Gordon equation:^{6}

(□ + m_{b}^{2}) Φ̂(x) = −λ_{b}⋮F (Φ̂(x))⋮ .  (3.11)

Best studied is the so-called λ (Φ^{4})_{4}-theory given formally by F (Φ) = 2Φ^{3} . Nevertheless, nobody succeeded up to now in giving this theory a precise meaning by rigorous construction. This is due to tremendous technical difficulties connected with 4-dimensionality of physical space-time.^{7} In 2- or 3-dimensional model spacetime these difficulties are much less severe and have already been overcome (See

Draft, March 26, 2009

3Note that (\frac{d}{dt} Û^{−1}) Û + Û^{−1} \frac{d}{dt} Û = 0 .

4In naive quantum electrodynamics (before renormalization) one has:

Ĥ_{I}(t) = ∫_{x^{0}=t} g_{μν}ĵ_{free}^{μ}(x)Â_{free}^{ν}(x) dx .

5Locality, i.e. dependence of the interaction term on the field values at the space-time point x, corresponds to the point particle picture.

6For the corresponding classical field theory see (Reed, 1976). For the problem of defining the operator function F (Φ̂(x)) via ⋮ ⋮ (in the sense of (3.20)) see (Segal, 1962; Segal, 1983). Negative values of m_{b}^{2} lead to spontaneous symmetry breaking.

7Actually there are even indications that such a construction is not possible; see (Bég and Furlong, 1985) and references given there.

e.g. (Glimm and Jaffe, 1981; Constantinescu, 1980) and (Streater and Wightman, 1989, Appendix).)

Time-Zero Fields

Let us assume that there is a rigorous construction for the interacting theory formally described above. Moreover assume – in spite of all knowledge to the contrary – that the interaction picture, as described in Sect. 3.1.1, is applicable to this theory with the corresponding “free” system being described by the neutral scalar field Φ̂_{0}(x) with physical mass^{8} m :

Φ̂(0, x) = Φ̂_{0}(0, x) ,
Π̂(0, x) =^{def} ∂_{0}Φ̂(x)_{|_{x^{0}=0}} = Π̂_{0}(0, x) =^{def} ∂_{0}Φ̂_{0}(x)_{|_{x^{0}=0}} .

(3.12)

Under these conditions

⋮G(Φ̂(x), Π̂(x))⋮ = Û(x^{0}):G (Φ̂_{0}(0, x), Π̂_{0}(0, x)):Û(x^{0})^{−1}  (3.13)

might be a good definition^{9} for well-behaved functionals G of Φ(x) and Π(x) . Here : : denotes normal ordering, i.e. the factors of monomials have to be interchanged – as if they were commuting – such that no creation operator is on the right of any annihilation operator.

In any case, by (2.39), (2.40), and (2.43), we have

Φ̂_{0}(x) = (2π)^{−3/2} ∫_{p^{0}=ω_{p}} \frac{dp}{2p^{0}} (â(p)e^{−ip^{0}x^{0}} + â^{∗}(−p)e^{+ip^{0}x^{0}}) e^{+ipx}  (3.14)

and hence for real-valued φ ∈ S(ℝ^{3}) :

∫ dx Φ̂_{0}(0, x)φ(x) = Âφ̃ + Â∗ ; φ̃ , ∫ dx Π̂_{0}(0, x)φ(x) = −iÂ_{2ωφ̃} + iÂ_{2ωφ̃}^{∗} ,

where: φ̃(p) =^{def} (2π)^{−3/2} ∫ dx φ(x)e^{−ipx} , Âf̌ =^{def} ∫ \frac{dp}{2ω_{p}} â(p)\overline{f̌(p)} .

(3.15)

By (3.14) and (2.34), (3.12) implies the canonical commutation relations

[Φ̂(0, x), Π̂(0, y)]_{−} = iδ(x − y) ,
[Φ̂(0, x), Φ̂(0, y)]_{−} = [Π̂(0, x), Π̂(0, y)]_{−} = 0

(3.16)

on D_{0} , as defined by (2.27).

Draft, March 26, 2009

8In case m coincided with the bare mass m_{b} there would be no chance for e^{iĤt} = V̂_{out}e^{iĤ_{0}t}V̂_{out}^{−1} to hold together with (3.21), since the latter implies coincidence of the spectra of Ĥ and Ĥ_{0} .

9Note, however, that :G (Φ̂_{0}(0, x), Π̂_{0}(0, x)): is well defined as quadratic form on D_{0} (see, e.g., (Reed und Simon, 1972, Sect. VIII.6) for the definition of quadratic forms) but not necessarily as L(D_{0}, H)-valued generalized function of x .

Let \left\{{ \atop f̌_{μ}}\right\}_{μ} be a complete orthonormal system of scalar 1-particle wave functions fulfilling

f̌_{μ}(p) = \overline{f̌_{μ}(−p)}  (3.17)

and define

φ_{μ}(x) =^{def} (2π)^{−3/2} ∫ dp f̌_{μ}(p)e^{ipx} , ψ_{μ}(x) =^{def} (2π)^{−3/2} ∫ dp \frac{f̌_{μ}(p)}{4ω_{p}}e^{ipx} . (3.18)

Then, by (3.12)/(3.15) and in agreement with (3.16),

π (Û_{ν}(τ)) =^{def} exp (i\overline{∫ dxΠ̂(0, x)τψ_{ν}(x)}) ,
π (V̂_{μ}(τ)) =^{def} exp (i\overline{∫ dxΦ̂(0, x)τφ_{μ}(x)}) ,

(3.19)

defines a regular representation^{10} of the Weyl commutation relations (1.45):

Û_{ν}(τ)V̂_{μ}(s) = e^{iτsδ_{νμ}} V̂_{μ}(s)Û_{ν}(τ) etc.

Even if (3.12) is not fulfilled, the field theory is called canonical, whenever (3.19) is a representation of the Weyl commutation relations. Then π has a unique extension to a true representation of the CCR algebra.

Exercise 44 Show that, provided (3.12) holds, the representation of the CCR algebra resulting from (3.19) is equivalent to the Fock representation (discussed in Sect. 1.3.3).^{11}

Exercise 45 Using (3.15), show the following:

(i) The (identical representation of the) C^{∗}-algebra (in H_{0}) generated by the bounded functions of smeared time-zero fields Φ̂_{0}(0, x), Π̂_{0}(0, x) is irreducible.

(ii) Ω_{0} is cyclic w.r.t. (the restriction of this representation to) the abelian subalgebra generated by bounded functions of the smeared field Φ̂_{0}(0, x) .

Draft, March 26, 2009

10Note the nontrivial dependence on the mass value m.

11For comparison with the theory of ordinary independent quantum oscillators, note that

π(â_{ν}) =^{def} \frac{1}{2}π(x̂_{ν}) + iπ(p̂_{ν}) = \frac{\sqrt{mD}π(x̂_{ν}) + ip̂_{ν}}{\sqrt{2m}} for m = D = \frac{1}{2} .

3.1. GENERAL ASPECTS 107

Use of the Canonical Commutation Relations

If the canonical commutation relations hold and if ⋮ ⋮ is defined such that^{12}

[⋮Â(x′)^{N} ⋮, B̂(x)]_{−}

= \{ \begin{matrix}⋮NÂ(x′)^{N−1}[Â(x′), B̂(x)]_{−}⋮ & if \{Â(.), B̂(.)\} = \{Φ̂(0, .), Π̂(0, .)\} \\ & or \{Â(.), B̂(.)\} = \{∂_{j}Φ̂(0, .), Π̂(0, .)\} \\ 0 & if Â(.) ∈ \{Φ̂(0, .), ∂_{j}Φ̂(0, .)\} ∋ B̂(.) \\ & or Â(.) = B̂(.) = Π̂(0, .) , \end{matrix}

(3.20)

holds for j ∈ {1, 2, 3} we get^{13} a formal solution of the field equation (3.11) in the form

Φ̂(t, x) = e^{iĤt}Φ̂(0, x)e^{−iĤt} , where:

Ĥ =^{def} \frac{1}{2} ∫ ⋮ ((Π̂(0, x))^{2} + (∇Φ̂(0, x))^{2} + m_{b}^{2} (Φ̂(0, x))^{2} + 2λ_{b}U (Φ̂(0, x))) ⋮ dx ,

+ const. U =^{def} indefinite integral of F ,

(3.21)

which – in a cutoff version^{14} – was the original starting point for constructive field theory.

Reminder: In classical Lagrange field theory the Hamiltonian

H (Φ_{t}, Π_{t}) = \frac{1}{2} ∫ (Π_{t}(x)^{2} + (∇Φ_{t}(x))^{2} + m_{b}^{2}Φ_{t}(x)^{2} + 2λ_{b}U (Φ_{t}(x))) dx ,

corresponds to the Lagrangian

L (Φ_{t}, Φ̇_{t}) = \frac{1}{2} ∫ ((Φ̇_{t}(x))^{2} − (∇Φ_{t}(x))^{2} − m_{b}^{2}Φ_{t}(x)^{2} − 2λ_{b}U (ϕ_{t}(x))) dx

with

Π_{t}(x) =^{def} \frac{δL}{δΦ̇_{t}(x)} = Φ̇_{t}(x) .

The Euler-Lagrange equation

\frac{d}{dt} \frac{δL}{δΦ̇_{t}(x)} − \frac{δL}{δΦ_{t}(x)} = 0

Draft, March 26, 2009

12Note that, for bounded operators, [ÂB̂, Ĉ]_{−} = Â[B̂, Ĉ]_{−} + [Â, Ĉ]_{−}B̂ and, consequently:

[Â, B̂]_{−} ∼ ˆ1 ⟹ [Â^{N}, B̂]_{−} = NÂ^{N−1}[Â, B̂]_{−} .

For (3.12)/(3.13), (3.20) is (formally) a consequence of Wick’s theorem (Theorem 3.2.1).

13Thanks to

[⋮Â(x′)^{N} ⋮, Π̂(0, x)]_{−} = iδ(x − x′) ⋮\frac{∂}{∂Φ̂(0, x′)}Â(x′)^{N} ⋮ ,
[⋮Â(x′)^{N} ⋮, Φ̂(0, x)]_{−} = −iδ(x − x′) ⋮\frac{∂}{∂Π̂(0, x′)}Â(x′)^{N} ⋮ .

14Note that, e.g., even :Φ̂_{0}(0, x)^{4} : is not well-defined (except on 2-dimensional space-time).

108 CHAPTER 3. SCALAR PERTURBATION THEORY

(more precisely: family of equations, parameterized by x) for this Lagrangian is the field equation

(□ + m_{b}^{2}) Φ(x) = −λ_{b}F (Φ(x)) , Φ(t, x) =^{def} Φ_{t}(x) .

Note that the Euler-Lagrange equation is equivalent to the Hamilton equations

Φ̇_{t}(x) = − \{H (Φ_{t}, Π_{t}) , Φ_{t}(x)\} = \frac{δH}{δΠ_{t}(x)} ,
Π̇_{t}(x) = − \{H (Φ_{t}, Π_{t}) , Π_{t}(x)\} = −\frac{δH}{δΦ_{t}(x)} ,

where the Poisson bracket {, } is defined by

\{F, G\} =^{def} ∫ (\frac{δF}{δΦ_{t}(x′)} \frac{δG}{δΠ_{t}(x′)} − \frac{δF}{δΠ_{t}(x′)} \frac{δG}{δΦ_{t}(x′)}) dx′

for functionals F, G of Φ_{t}, Π_{t}.

Indeed,^{15}

˙Φ̂(t, x)  =  e^{iĤt} [iĤ, Φ̂(0, x)]_{−} e^{−iĤt}
=_{(3.21)_{(3.13),(3.16)}} e^{iĤt} \frac{i}{2} ∫ [⋮Π(0, x′)^{2}⋮, Φ̂(0, x)]_{−} dx′ e^{−iĤt}

=_{(3.20)} e^{iĤt} i ∫ ⋮Π(0, x′)[Π(0, x′), Φ̂(0, x)]_{−}⋮ dx′ e^{−iĤt}
=_{(3.16)} e^{iĤt} Π(0, x) e^{−iĤt} (3.22)

and a similar formal calculation verifies

¨Φ̂(t, x) =_{(3.22)} e^{iĤt}[iĤ, Π(0, x)]_{−}e^{−iĤt}

to be in agreement with the field equation (3.11).

However, locality and relativistic covariance of the formal solution (3.21) are not so easy to establish.^{16}

Exercise 46 Show for arbitrary φ ∈ S(ℝ^{3}) that

∫ \begin{matrix}|φ̃(p_{1} + p_{2})| & 2 \\ (ω_{p_{1}}ω_{p_{2}}) & 3 \end{matrix} dp_{1}dp_{2} < ∞

and therefore

∫ :(Π̂(0, x))^{2} + (∇Φ̂(0, x))^{2} + m_{b}^{2} (Φ̂(0, x))^{2} : φ(x) dx ∈ L(D_{0}, H) .

Draft, March 26, 2009

15Note that we do not care about the domain of definition for Ĥ. For :F (Φ̂_{0}(0, x)):Ω_{0} ≠ 0 , the case of interest, there are obvious difficulties with the Fock representation.

16Note that [Â, B̂]_{−} = 0 ≠⇒^{i.g.} [e^{iÂ}, e^{iB̂}]_{−} = 0 (see e.g. (Reed und Simon, 1972, Vol. I, Sect. VIII.5) and (Fröhlich, 1977)).

Inapplicability of the Fock Representation

Let us assume (3.12) to be valid. Then

Û(0, a)Φ_{0}(0, x)Û(0, a)^{−1} = Û(0, a)Φ(0, x)Û(0, a)^{−1}
= Φ(0, x + a)
= Φ_{0}(0, x + a)
= Û_{0}(0, a)Φ_{0}(0, x)Û_{0}(0, a)^{−1}

and the corresponding conclusion for Π_{0}(0, a) imply (compare Exercise 45(i)):

Û(0, a) = e^{iφ(a)}Û_{0}(0, a)

for suitable real-valued φ(a) . There is only one 1-dimensional subspace which is invariant under all Û_{0}(0, a) , and this contains the Fock vacuum Ω_{0} , characterized up to a factor by P̂_{0}Ω_{0} = 0 . For the physical vacuum state vector Ω , characterized up to a factor by

Û(a) Ω = Ω ∀ a ∈ ℝ^{4} ,

we have

ˆÛ_{0}(0, a)Ω = e^{−iφ(a)} ˆÛ(0, a)Ω = e^{−iφ(a)}Ω .

Therefore also Ω is an element of the invariant 1-dimensional subspace, i.e.:

Ω = e^{iα}Ω_{0} .

This means that both Û(0, a) and Û_{0}(0, a) leave Ω_{0} (and Ω) invariant and thus have to coincide:

Û_{0}(0, a) = Û(0, a) ∀ a ∈ ℝ^{3} .

Since, for obvious physical reasons, (3.8) should be supplemented by

Û(0, a) = V̂_{out}Û_{0}(0, a)V̂_{out}^{−1} ∀ a ∈ ℝ^{3} ,

we get commutativity of V̂_{out} with all Û_{0}(0, a) . Now, since P̂_{0}^{0} is a function of P̂_{0} , commutativity with all Û_{0}(0, a) implies commutativity with all Û_{0}(t) . Thus, by (3.8) we have

Û_{0}(t, 0) = Û(t, 0) ∀ t ∈ ℝ

and hence

Φ̂_{0}(x) = Φ̂(x) ∀ x ∈ ℝ^{4} .

Haag’s theorem^{17} says that this conclusion is correct even without (3.8) and its generalization and without specification of Ĥ, if the theory fulfills the Wightman

Draft, March 26, 2009

17A rigorous proof is given in (Streater and Wightman, 1989, Sect. 4.5). In a first step, relativistic covariance is used to show that

⟨Ω | Φ̂(x)Φ̂(y)Ω⟩ = ⟨Ω_{0} | Φ̂_{0}(x)Φ̂_{0}(y)Ω_{0}⟩  (∗)

holds for x×y, since the equations holds for x^{0} = y^{0} = 0 . This together with the spectrum condition implies that (∗) holds for all x, y ∈ ℝ^{4} , as can be shown by standard techniques of axiomatic field theory (for a stronger result see (Lücke, 1979, Corollary)). Then the Jost-Schroer theorem (Theorem 2.2.18) says that (∗) can only hold for all x, y ∈ ℝ^{4} if Φ̂_{0}(x) and Φ̂(x) are unitarily equivalent.

axioms formulated in 2.2.1.

So, unfortunately, it is not possible to define the Hamiltonian on a suitable domain to make the formal solution (3.21) of (3.11) a true one – as long as one insists in (3.12) for all x ∈ ℝ^{3} .

In spite of Haag’s theorem there is still hope (compare (Baumann, )) that a nontrivial canonical λ (Φ^{4})_{4}-theory might exist.^{18} For such a quantum field theory, of course, the representation π given by (3.19) must be inequivalent to the Fock representation. All this was illustrated in constructive field theory by several models living on space-time of dimension < 4 .

For the free field, of course, nothing is wrong with the canonical quantization procedure (3.21) (for λ_{b} = 0).

3.2 Canonical Perturbation Theory

3.2.1 Dyson Series and Wick’s Theorem

Let us consider an IS for which the interaction picture works as described in Section 3.1.1:

Ŝ_{0} =_{(3.7)} lim_{t_{±}→±∞} Ω̂(t_{+}, t_{−}) ,
Ω̂(t_{+}, t_{−}) =_{(3.10)} ˆ1 − i ∫_{t_{−}}^{t_{+}} Ĥ_{I}(t)Ω̂(t, t_{−}) dt ,
Ĥ_{I}(t) =_{(3.9)} Û_{0}(t) (Ĥ(t) − Ĥ_{0}(t)) Û_{0}(t)^{−1} .

(3.23)

Let us assume

Ĥ_{I}(t) = λ ˆĤ_{I}(t)

and that – on a suitable domain – Ω̂(t, t_{−}) depends sufficiently smoothly on λ. Then the Leibniz rule gives

∂_{λ}^{n}Ω̂(t, t_{−}) = −i ∫_{t_{−}}^{t} ∑_{ν=0}^{n} ( \begin{matrix}n \\ ν \end{matrix} ) (∂_{λ}^{ν}Ĥ_{I}(t′)) ∂_{λ}^{n−ν}Ω̂(t′, t_{−}) dt′ for n > 0

and, because of

(∂_{λ}^{ν}Ĥ_{I}(t))_{|_{λ=0}} = \{ \begin{matrix}ˆĤ_{I}(t) & for ν = 1 , \\ 0 & else , \end{matrix}

Draft, March 26, 2009

18Note that existence of a unitary operator Û_{R} fulfilling the conditions

Φ̂(0, x) = Û_{R}Φ̂_{0}(0, x)Û_{R}^{−1} , Φ̂(0, x) = Û_{R}Π̂_{0}(0, x)Û_{R}^{−1} ,
⋮F (Φ̂(x)) ⋮ = Û(x^{0})Û_{R}:F (Φ̂_{0}(0, x)):Û_{R}^{−1}Û(x^{0})^{−1}

for |x| < R would have been sufficient for the formal proof of (3.11) in this region.

the iteration formula

∂_{λ}^{n}Ω̂(t, t_{−})_{|_{λ=0}} = −i ∫_{t_{−}}^{t} n ˆĤ_{I}(t′) (∂_{λ}^{n−1}Ω̂(t′, t_{−}))_{|_{λ=0}} dt′ .

Since,

Ω̂(t, t_{−})_{|_{λ=0}} = ˆ1 ,

this gives the (not necessarily converging) Taylor expansion

Ω̂(t_{+}, t_{−}) = ˆ1 + ∑_{n=1}^{∞}(−i)^{n} \underbrace{∫ … ∫_{t_{−}<t_{1}<...t_{n}<t_{+}} Ĥ_{I}(t_{n}) ⋯ Ĥ_{I}(t_{1}) dt_{1} ⋯ dt_{n}}_{\frac{1}{n!} ∑_{π∈S_{n}} ∫ … ∫_{t_{−}<t_{π1}<...<t_{πn}<t_{+}} T (Ĥ_{I}(t_{n}) ⋯ Ĥ_{I}(t_{1})) dt_{1} ⋯ dt_{n}}

= ˆ1 + ∑_{n=1}^{∞} \frac{(−i)^{n}}{n!} ∫_{t_{−}}^{t_{+}} ⋯ ∫_{t_{−}}^{t_{+}} T (Ĥ_{I}(t_{n}) ⋯ Ĥ_{I}(t_{1})) dt_{1} ⋯ dt_{n} ,

where^{19}

T (Ĥ_{I}(t_{n}) ⋯ Ĥ_{I}(t_{1})) =^{def} Ĥ_{I}(t_{πn}) ⋯ Ĥ_{I}(t_{π1}) for π ∈ S_{n} with t_{π1} < … < t_{πn}

is the so-called chronological product of Ĥ_{I}(t_{n}), …, Ĥ_{I}(t_{1}) . The usual shorthand notation for the resulting formal perturbation expansion of the S-matrix is:

Ŝ_{0} = T exp (−i ∫_{−∞}^{+∞} Ĥ_{I}(t)dt) .  (3.24)

In view of λΦ_{4}^{4}-theory let us formally assume

Ĥ_{I}(x^{0}) = i ∫ g(x)Ŝ_{1}(x) dx ,  (3.25)

where Ŝ_{1}(x) is a normal ordered function of the free field Φ̂_{0}(x) and its derivatives at the space-time point x. Then (3.24) becomes:

Ŝ_{0} = ˆ1 + ∑_{n=1}^{∞} \frac{1}{n!} ∫ ⋯ ∫ Ŝ_{n}(x_{1}, …, x_{n})g(x_{1}) ⋯ g(x_{n}) dx_{1} ⋯ dx_{n} ,

where Ŝ_{n}(x_{1}, …, x_{n}) =^{def} T (Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{n})) .

(3.26)

In order to facilitate evaluation of the S-matrix elements the chronological products should be expressed by normal ordered products. Formally this may be done by applying Wick’s theorem.

Draft, March 26, 2009

19This is the definition for Bose fields, only. However, even if the theory contained Fermi fields the Hamiltonian ought to be of Bose type.

112 CHAPTER 3. SCALAR PERTURBATION THEORY Theorem 3.2.1 (Wick’s theorem) Let^{20}

χ̂_{ν}(x) ∈ {Φ̂_{0}(x), ∂_{0}Φ̂_{0}(x), …, ∂_{3}Φ̂_{0}(x)} for ν = 1, …, j_{N}

and let j_{1} < j_{2} < … < j_{N} . Then

:χ̂_{1}(x_{1}) ⋯ χ̂_{j_{1}}(x_{j_{1}})::χ̂_{j_{1}+1}(x_{j_{1}+1}) ⋯ χ̂_{j_{2}}(x_{j_{2}}): ⋯:χ̂_{j_{N−1}+1}(x_{j_{N−1}+1}) ⋯ χ̂_{j_{N}} (x_{j_{N}}):

=:exp( ∑_{ν≤j_{k}<μ_{for suit. k}} ∫ ∫ dxdy ⟨Ω_{0} | χ̂_{ν}(x)χ̂_{μ}(y)Ω_{0}⟩ \frac{δ}{δχ̂_{ν}(x)} \frac{δ}{δχ̂_{μ}(y)})

χ̂_{1}(x_{1}) ⋯ χ̂_{j_{N}} (x_{j_{N}}): ,

if – formally – the fields χ̂_{ν}(x) are considered as independent functional variables.

Proof: From the simple chain of equations Φ̂_{0}^{+}(x_{k}) ⋯ Φ̂_{0}^{+}(x_{n})Φ̂_{0}^{−}(x_{n+1}) = Φ̂_{0}^{+}(x_{k}) ⋯ Φ̂_{0}^{+}(x_{n−1})Φ̂_{0}^{−}(x_{n+1})Φ̂_{0}^{+}(x_{n})

+Φ̂_{0}^{+}(x_{k}) ⋯ Φ̂_{0}^{+}(x_{n−1})[Φ̂_{0}^{+}(x_{n}), Φ̂_{0}^{−}(x_{n+1})]_{−}

= Φ̂_{0}^{+}(x_{k}) ⋯ Φ̂_{0}^{+}(x_{n−2})Φ̂_{0}^{−}(x_{n+1})Φ̂_{0}^{+}(x_{n−1})Φ̂_{0}^{+}(x_{n})

+Φ̂_{0}^{+}(x_{k}) ⋯ Φ̂_{0}^{+}(x_{n−2})[Φ̂_{0}^{+}(x_{n−1}), Φ̂_{0}^{−}(x_{n+1})]_{−}Φ̂_{0}^{+}(x_{n})
+Φ̂_{0}^{+}(x_{k}) ⋯ Φ̂_{0}^{+}(x_{n−1})[Φ̂_{0}^{+}(x_{n}), Φ̂_{0}^{−}(x_{n+1})]_{−}

⋮

= Φ̂_{0}^{−}(x_{n+1})Φ̂_{0}^{+}(x_{k}) ⋯ Φ̂_{0}^{+}(x_{n}) + ∑_{ν=k}^{n}[Φ̂_{0}^{+}(x_{ν}), Φ̂_{0}^{−}(x_{n+1})]_{−} ∏_{μ=k_{μ≠ν}}^{n} Φ̂_{0}^{+}(x_{μ})

we easily conclude that Φ̂_{0}^{−}(x_{1}) ⋯ Φ̂_{0}^{−}(x_{k−1})Φ̂_{0}^{+}(x_{k}) ⋯ Φ̂_{0}^{+}(x_{n})χ̂_{n+1}(x_{n+1})

= :Φ̂_{0}^{−}(x_{1}) ⋯ Φ̂_{0}^{−}(x_{k−1})Φ̂_{0}^{+}(x_{k}) ⋯ Φ̂_{0}^{+}(x_{n})χ̂_{n+1}(x_{n+1}):

+ ∑_{ν=k}^{n}[Φ̂_{0}^{+}(x_{ν}), χ̂_{n+1}(x_{n+1})]_{−}Φ̂_{0}^{−}(x_{1}) ⋯ Φ̂_{0}^{−}(x_{k−1}) ∏_{μ=k_{μ≠ν}}^{n} Φ̂_{0}^{+}(x_{μ})

holds for χ̂_{n+1}(x) ∈ {Φ̂_{0}^{−}(x), Φ̂_{0}^{+}(x)}. This implies

\begin{matrix}:χ̂_{1}(x_{1}) ⋯ χ̂_{n}(x_{n}):χ̂_{n+1}(x_{n+1}) \\ = :χ̂_{1}(x_{1}) ⋯ χ̂_{n+1}(x_{n+1}): + ∑_{ν=1}^{n} ⟨Ω_{0} | χ̂_{ν}(x_{ν})χ̂_{n+1}(x_{n+1})Ω_{0}⟩: ∏_{μ=1_{μ≠ν}}^{n} χ̂_{μ}(x_{μ}): \end{matrix}

or, written in a suggestive formal way, :χ̂_{1}(x_{1}) ⋯ χ̂_{n}(x_{n}):χ̂_{n+1}(x_{n+1})

=:exp (∑_{ν=1}^{n} ∫ ∫ dxdy ⟨Ω_{0} | χ̂_{ν}(x)χ̂_{n+1}(y)Ω_{0}⟩ \frac{δ}{δχ̂_{ν}(x)} \frac{δ}{δχ̂_{n+1}(y)})

χ̂_{1}(x_{1}) ⋯ χ̂_{n+1}(x_{n+1}):

Draft, March 26, 2009

20Here, the χ̂_{ν} could also be partial derivatives of various Bose fields. For the more general case, where some of the χ̂_{ν}(x) are Fermi fields, see Chapter 4.

3.2. CANONICAL PERTURBATION THEORY 113

for

χ̂_{1}(x_{1}), …, χ̂_{n+1}(x_{n+1}) ∈ {Φ̂_{0}^{+}(x), Φ̂_{0}^{−}(x)} .  (∗)

Induction w.r.t. n, finally, gives χ̂_{1}(x_{1}) ⋯ χ̂_{n}(x_{n})χ̂_{n+1}(x_{n+1})

=:exp ( ∑_{1≤ν<μ≤n+1} ∫ ∫ dxdy ⟨Ω_{0} | χ̂_{ν}(x)χ̂_{μ}(y)Ω_{0}⟩ \frac{δ}{δχ̂_{ν}(x)} \frac{δ}{δχ̂_{μ}(y)})

χ̂_{1}(x_{1}) ⋯ χ̂_{n+1}(x_{n+1}): ,

provided (∗) holds. From this the statement of the theorem follows easily for

χ̂_{1}(x) = … = χ̂_{n+1}(x) = Φ̂_{0}(x)

and then, by just forming derivatives, the full statement.

Corollary 3.2.2 Let P_{1}, …, P_{n} be polynomials. Then^{21}

:P_{1}(Φ̂_{0}(x_{1})): ⋯:P_{n}(Φ̂_{0}(x_{n})):

= : ∏_{ν<μ} exp (⟨Ω_{0} | Φ̂_{0}(x_{ν})Φ̂_{0}(x_{μ})Ω_{0}⟩ \frac{∂}{∂Φ̂_{0}(x_{ν})} \frac{∂}{∂Φ̂_{0}(x_{μ})}) P_{1}(Φ̂_{0}(x_{1})) ⋯ P_{n}(Φ̂_{0}(x_{n})):

if – formally – Φ̂_{0}(x_{1}), …, Φ̂_{0}(x_{n}) are considered as independent scalar variables. Exercise 47 Use Corollary 3.2.2 to give a formal proof of^{22}

:e^{Φ̂_{0}(x)} ::e^{Φ̂_{0}(y)} : = e^{⟨Ω_{0}|Φ̂_{0}(x)Φ̂_{0}(y)Ω_{0}⟩}:e^{Φ̂_{0}(x)+Φ̂_{0}(y)} :

and the corresponding formula for the time-ordered product of the two (normal ordered) exponentials. For simplicity let us assume

Ŝ_{1} = :P (Φ̂_{0}(x)):

for some polynomial P . Then Corollary 3.2.2 implies formally

Draft, March 26, 2009

21For the definition of powers of 2-point functions see Equation (3.32) and Exercise 49. Actually, a rigorous proof of Corollary 3.2.2 is not straightforward.

22For a rigorous definition of expressions of the form ∑_{n=0}^{∞} c_{n}:Φ̂_{0}(x)^{n}: with arbitrary c_{n} ∈ ℝ see (Rieckers, 1971).

114 CHAPTER 3. SCALAR PERTURBATION THEORY

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4)^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

Figure 3.2: Wick’s theorem interpreted in terms of diagrams

f_{νμ}

f_{ν_{1}μ_{1}}
f_{ν_{2}μ_{2}}

T (Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{n}))

= :P (Φ̂_{0}(x_{1})) ⋯ P (Φ̂_{0}(x_{n})):

+ ∑_{ν<μ} ∑_{f_{νμ}=1}^{deg P}:P (Φ̂_{0}(x_{1})) ⋯ P (Φ̂_{0}(x_{ν})) ⋯ P (Φ̂_{0}(x_{μ})) ⋯ P (Φ̂_{0}(x_{n})):

+\frac{1}{2} ∑_{ν_{1}<μ_{1}} ∑ ν_{2}<μ_{2}_{(ν_{2},μ_{2})≠(ν_{1},μ_{1})} ∑_{f_{ν_{1}μ_{1}},f_{ν_{2}μ_{2}}=1}^{deg P} :P (Φ̂_{0}(x_{1})) ⋯

⋯ P (Φ̂_{0}(x_{ν_{1}})) ⋯ P (Φ̂_{0}(x_{ν_{2}})) ⋯ P (Φ̂_{0}(x_{μ_{1}})) ⋯ P (Φ̂_{0}(x_{μ_{2}})) ⋯ P (Φ̂_{0}(x_{n})):

+ …

(3.27)

where

f_{νμ}  ˆ= \frac{1}{f_{νμ}!} (iΔ_{F}(x_{ν} − x_{μ})\frac{∂}{∂Φ̂_{0}(x_{ν})} \frac{∂}{∂Φ̂_{0}(x_{μ})})^{f_{νμ}}

and

Δ_{F}(x − y) =^{def} lim_{ϵ→+0} (2π)^{−4} ∫ \frac{1}{p^{2} − m^{2} + iϵ}e^{−ip(x−y)} dp
= −i ⟨Ω_{0} | T (Φ̂_{0}(x)Φ̂_{0}(y)) Ω_{0}⟩ for x ≠ y .

(3.28)

This sum over all contraction schemes with f-fold contraction lines may be easily identified with a corresponding sum of diagrams.^{23}

In Fig. 3.2 this is sketched for the special case P (ξ) = ξ^{4} , n = 2 . Here the

Draft, March 26, 2009

23Subdiagrams of the type _{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.} (tadpoles) do never occur – thanks to Wick ordering of Ŝ_{1}(x) .

3.2. CANONICAL PERTURBATION THEORY 115 concrete meaning of the sum of diagrams is

T (:Φ̂_{0}(x)^{4} ::Φ̂_{0}(y)^{4} :) = :Φ̂_{0}(x)^{4}Φ̂_{0}(y)^{4} :
+:(iΔ_{F}(x − y)\frac{∂}{∂Φ̂_{0}(x)} \frac{∂}{∂Φ̂_{0}(y)}) Φ̂_{0}(x)^{4}Φ̂_{0}(y)^{4} :
+\frac{1}{2}:(iΔ_{F}(x − y)\frac{∂}{∂Φ̂_{0}(x)} \frac{∂}{∂Φ̂_{0}(y)})^{2} Φ̂_{0}(x)^{4}Φ̂_{0}(y)^{4} :
+\frac{1}{3!}:(iΔ_{F}(x − y)\frac{∂}{∂Φ̂_{0}(x)} \frac{∂}{∂Φ̂_{0}(y)})^{3} Φ̂_{0}(x)^{4}Φ̂_{0}(y)^{4} :
+\frac{1}{4!}:(iΔ_{F}(x − y)\frac{∂}{∂Φ̂_{0}(x)} \frac{∂}{∂Φ̂_{0}(y)})^{4} Φ̂_{0}(x)^{4}Φ̂_{0}(y)^{4} : ,

giving the formal result

T (:Φ̂_{0}(x)^{4} ::Φ̂_{0}(y)^{4} :) = :Φ̂_{0}(x)^{4}Φ̂_{0}(y)^{4} :
+16iΔ_{F}(x − y):Φ̂_{0}(x)^{3}Φ̂_{0}(y)^{3} :
−72Δ_{F}(x − y)^{2}:Φ̂_{0}(x)^{2}Φ̂_{0}(y)^{2} :
−96iΔ_{F}(x − y)^{3}:Φ̂_{0}(x)Φ̂_{0}(y):
+24Δ_{F}(x − y)^{4} .

(3.29)

Exercise 48 Evaluate

x^{1} x^{2}  x^{3} x^{4}  x^{5}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

considered as a single diagram (not a sum of diagrams).

Recall that, according to (3.28),

Δ_{F}(x) = \{ \begin{matrix}+Δ_{+}(x) & for x ∈ ℝ^{4} \ \overline{V_{−}}, \\ −Δ_{−}(x) & for x ∈ ℝ^{4} \ \overline{V_{+}} , \end{matrix}

where^{24}

Δ_{+}(x − y) =^{def} −i ⟨Ω_{0} | Φ̂_{0}(x)Φ̂_{0}(y)Ω_{0}⟩
= −i(2π)^{−3} ∫ dp θ(p^{0})δ(p^{2} − m^{2})e^{−ip(x−y)}

(3.30)

and

Δ_{−}(x) =^{def} −Δ_{+}(−x) .  (3.31)

Since supp Δ̃_{±}(p) ⊂ \overline{V_{±}}, we may recursively define

{ \atop Δ_{±}^{n+1}}(p) =^{def} (2π)^{−2} (Δ̃_{±}^{n} ∗ Δ̃_{±}) (p) ,  (3.32)

(see Exercise 49) in spite of the singularities^{25} of Δ_{±}(x) on the light cone. Actually, this definition has to be used in Corollary 3.2.2 and to fix Δ_{F}^{n}(x) for x ≠ 0 :

Δ_{F}^{n}(x) =^{def} \{ \begin{matrix}Δ_{+}^{n}(x) & for x ∈ ℝ^{4} \ \overline{V_{−}}, \\ (−1)^{n}Δ_{−}^{n}(x) & for x ∈ ℝ^{4} \ \overline{V_{+}} . \end{matrix}  (3.33)

Draft, March 26, 2009

24In (2.62) we wrote Δ_{m}^{(+)} instead of Δ_{+} . 25See (Bogoliubov and Shirkov, 1959, Sect. 15.2).

Supplemented by (3.33), (3.27) becomes a rigorous equation on {(x_{1}, …, x_{n}) ∈ ℝ^{4n} : x_{ν} ≠ x_{μ} for ν ≠ μ}.

Exercise 49 Let M = \overline{M} ⊂ ℝ^{4} and let F, G ∈ S(ℝ^{4})′ fulfill the conditions

supp F̃ ⊂ \overline{V_{+}} ⊃ supp G̃

and

supp F ⊂ M .

Show that convolution of F̃ with G̃ is well-defined and that the Fourier transform F G of (2π)^{−2}F̃ ∗ G̃ fulfills the conditions

supp (F G) ⊂ M , supp F̃ G ⊂ \overline{V_{+}}

and, if both F and G are sufficiently regular:

(F G)(x) = F (x)G(x) pointwise .

The nontrivial problem is extension of (3.33) to all of ℝ^{4} for n > 1 . No doubt, (3.33) may be extended to a Lorentz invariant tempered distribution on all of ℝ^{4} (see Sect. 3.2.2). This extension is unique up to addition of a Lorentz invariant distribution with point-like support at the origin. However, without further restrictions there is no hope to extract physically relevant information.

3.2.2 Counter Terms and Renormalization

The guiding heuristic principle for minimizing the arbitrariness in the definition of Δ_{F}(x)^{n} is to make it no more singular at the origin than necessary.^{26} For n = 1 this means to take (3.28). For n > 1 the allowed (tempered) solutions may be constructed as follows:

One introduces a suitable covariant regularization Δ_{F,M} of Δ_{F} depending on a parameter M such that for finite M the naive definition

{ \atop Δ_{F,M}^{n}}(p) =^{def} (2π)^{−2(n−1)} ({ \atop Δ_{F,M}} ∗ ⋯ ∗ { \atop Δ_{F,M}}) (p)  (3.34)

works and:

∫ Δ_{F,M}(x)φ(x) dx ⟶_{M→∞} ∫ Δ_{F}(x)φ(x) dx ∀ φ ∈ S(ℝ^{4}) ,
∫ Δ_{F,M}^{n}(x)φ(x) dx ⟶_{M→∞} \{ \begin{matrix}∫ Δ_{+}^{n}(x)φ(x) dx & ∀ φ ∈ S (ℝ^{4} \ \overline{V_{−}}) , \\ (−1)^{n} ∫ Δ_{−}^{n}(x)φ(x) dx & ∀ φ ∈ S (ℝ^{4} \ \overline{V_{+}}) . \end{matrix}

For given n this regularization has to fulfill the requirement that

Δ_{F,M}^{n}(x) − ∑_{ν=1}^{N} A_{ν,M} □ _{x}^{ν−1}δ(x)

Draft, March 26, 2009

26See also (Epstein and Glaser, 1973, Sect. 5).

has a (M → ∞)-limit Δ_{F,reg}^{n}(x) in S(ℝ^{4})′ for a minimal number N = N(n) of suitable sequences A_{1,M}, …, A_{N,M} . This way Δ_{F,reg}^{n}(x) is fixed up to addition of a distribution of the form

∑_{ν=1}^{N} c_{ν} □ _{x}^{ν−1}δ(x)

with finite coefficients c_{1}, …, c_{N} . A suitable regularization is for example Pauli-Villars regularization

\begin{aligned}Δ_{F,M}(x) & =^{def} & (2π)^{−4} ∫ (\frac{1}{p^{2} − m^{2} + iϵ} − \frac{1}{p^{2} − M^{2} + iϵ}) e^{−ipx} dp \\ & = & (2π)^{−4} ∫ \frac{m^{2} − M^{2}}{(p^{2} − m^{2} + iϵ)(p^{2} − M^{2} + iϵ)}e^{−ipx} dp . \end{aligned}

This gives^{27}

N(2) = 1 , N(3) = 2 ;

i.e.:

Δ_{F,reg}^{2}(x) = lim_{M→∞} (Δ_{F,M}^{2}(x) − A_{M}δ(x)) ,
Δ_{F,reg}^{3}(x) = lim_{M→∞} (Δ_{F,M}^{3}(x) − B_{M}δ(x) − C_{M} □_{x}δ(x))

in the topology of S(ℝ^{4})′ for suitable A_{M}, B_{M}, C_{M} .

Of course, these sequences have to diverge for M → ∞ in order to compensate the so-called ultraviolet divergences appearing in (3.34) for M → ∞. This is what is meant by the usual saying:

The ultraviolet infinities introduced by formal use of

Δ_{F}^{n}(x) = Δ_{F,∞}^{n}(x)
= (2π)^{−4(n+1)} ∫ \frac{1}{p_{1}^{2} − m^{2} + iϵ} \frac{1}{(p_{2} − p_{1})^{2} − m^{2} + iϵ} ⋯

⋯ \frac{1}{(p_{n} − p_{n−1})^{2} − m^{2} + iϵ}e^{−ip_{n}x} dp_{n} ⋯ dp_{1}

can be removed by infinite counter terms, e.g. A_{∞} δ(x) for n = 2 resp.

B_{∞} δ(x) + C_{∞} □_{x}δ(x) for n = 3 .

The essential result of the above considerations is the following:

For (3.26) in the limit^{28} g → 1 with Ŝ_{1}(x) = λ:Φ̂_{0}(x)^{4} : any deviation of the working definition for Δ_{F}^{n}, used in (3.29), from the physically correct one (if such exists at all) may be compensated – at least up to second order in λ – by adding suitable counter terms C_{ν} of higher order in λ :

\begin{aligned}λ:Φ̂_{0}(x)^{4} : & ⟶ & λ:Φ̂_{0}(x)^{4} : + C_{1}(λ) :Φ̂_{0}(x)^{4} : + C_{2}(λ) :Φ̂_{0}(x)^{2} : \\ & & + C_{3}(λ) :Φ̂_{0}(x) □_{x}Φ̂_{0}(x): + C_{4}(λ, x)ˆ1 . \end{aligned}  (3.35)

Draft, March 26, 2009

27See (Bogoljubov and Šhirkov, 1984, Sects. 23.2 and 25.2).

28We ignore the subtleties indicated by Exercise 52. See Section 3.3.2 in this connection.

The highly nontrivial^{29} result of renormalization theory is that this compensation works for all orders of canonical perturbation theory for λ:Φ̂_{0}(x)^{4} : .

Let us now indicate why such counter terms may be unavoidable in the construction of solutions to interacting field equations (see e.g. (Feldman and Raczka, 1977)).

We had already seen in Sect. 3.1.2 that the formal success of canonical quantization does not depend on the choice of the ‘physical’ mass m. It does not even depend on the normalization of the interacting field. More precisely, let Φ̂(x) be a solution of

(□ + m̌_{b}) Φ̂(x) = −4λ̌⋮Φ̂(x)^{3}⋮  (3.36)

for m̌ = m_{b} and λ̌ = Zλ_{b} fulfilling the canonical commutation relations (3.16), the Hamiltonian being

Ĥ = \frac{1}{2} ∫_{x^{0}=0} ⋮ ( ˙Φ̂(x)^{2} + (∇Φ̂(x))^{2} + m_{b}^{2} Φ̂(x)^{2} + 2Z λ_{b}Φ̂(x)^{4}) ⋮ dx + δE (3.37)

(compare Sect. 3.1.2). Then

Φ̂_{Z}(x) =^{def} \sqrt{Z}Φ̂(x)

fulfills (3.36) for m̌ = m_{b} and λ̌ = λ_{b} . The Hamiltonian for Φ̂_{Z}(x) is the same as for Φ̂(x) , of course, but in terms of Φ̂_{Z}(x) it is given by

Ĥ = \frac{1}{2Z} ∫_{x^{0}=0} ⋮ ( ˙Φ̂_{Z}(x)^{2} + (∇Φ̂_{Z}(x))^{2} + m_{b}^{2} Φ̂_{Z}(x)^{2} + 2λ_{b} Φ̂_{Z}(x)^{4}) ⋮ dx + const .

(3.38)

Exercise 50 Show that (3.38) and

Π̂_{Z}(x) =^{def} \frac{1}{Z} ˙ Φ̂_{Z}(x) = \frac{1}{\sqrt{Z}} ˙ Φ̂(x)

correspond to the canonical formalism for the (classical) Lagrangian

L(Φ_{Z}, Φ̇_{Z}) = \frac{1}{2Z} ∫_{x^{0}=0} (Φ̇_{Z}(x)^{2} − (∇Φ_{Z}(x))^{2} − m_{b}^{2} Φ_{Z}(x)^{2} − 2λ_{b} Φ_{Z}(x)^{4}) dx .

Therefore, given a solution Φ̂_{Z}(x) of (3.36) for m̌ = m_{b} and λ̌ = λ_{b} with asymptotic ‘free’ time evolution governed by^{30}

Ĥ_{0} = \frac{1}{2} ∫_{x^{0}=0} :( ˙Φ̂_{0}(x)^{2} + (∇Φ̂_{0}(x))^{2} + m^{2} Φ̂_{0}(x)^{2}): dx

Draft, March 26, 2009

29A rough idea of how to proceed (in the Bogoliubov-Parasiuk-Hepp sense) may be extracted from Sect. 3.3.2. Considerable complication is caused by so-called overlapping divergences.

30In principle, of course, Φ̂_{Z}(x) could be associated with several (asymptotic) particles of different masses.

(compare (3.21)), it may be necessary to scale Φ̂_{Z}(x) to Φ̂(x) in order to have (3.37) and

Φ̂(0, x) = Φ̂_{0}(0, x) , ˙ Φ̂(0, x) = ˙ Φ̂_{0}(0, x) ,

(at least up to local equivalence, if possible at all) where Φ̂_{0}(x) denotes the free field with mass m (not m_{b}). Then in (3.23), formally assuming (3.13), we get

Ĥ_{I}(t) = ∫_{x^{0}=t}:(−\frac{1}{2}δm^{2} Φ̂_{0}(x)^{2} + Zλ_{b} Φ̂_{0}(x)^{4}): dx + δE
= ∫_{x^{0}=t}:(−\frac{1}{2}δm^{2} Φ̂_{0}(x)^{2} + λ Φ̂_{0}(x)^{4} + δλ Φ̂_{0}(x)^{4}): dx + δE ,

(3.39)

where

δm^{2} =^{def} m^{2} − m_{b}^{2} , δλ =^{def} Zλ_{b} − λ ,

λ denoting the physical (i.e. the renormalized) coupling constant to be fixed by some convention. Introduction of the counter terms^{31}

−\frac{1}{2}δm^{2}:Φ̂_{0}(x)^{2} : , δλ:Φ̂_{0}(x)^{4} : , δE

in (3.39) corresponds^{32} to (3.35). Correct choice of the counter terms is called renormalization.

In perturbation theory the coefficients δm^{2}, δλ, δE are considered as power series in λ with coefficients depending on m. Their choice has to be adapted to the working definition of time ordering to meet physical requirements, especially stability of the vacuum and 1-particle states and, formally:^{33}

δm^{2}_{|_{λ=0}} = δλ_{|_{λ=0}} = δE_{|_{λ=0}} = 0 .  (3.40)

Exercise 51 For s > 0 show^{34} that also

ˆΦ̂(x) =^{def} s Φ̂_{1}(sx)

fulfills the canonical commutation relations (3.16) with Π̂(x) = ∂_{0}Φ̂(x) and the field equation (3.36) for m̂ = sm_{b} , λ̌ = λ_{b} . Moreover, show that the Hamiltonian for

Draft, March 26, 2009

31Thanks to the counter terms there is now some chance that the r.h.s. of (3.37) may be rigorously defined by suitable limiting procedures as in lower-dimensional constructive field theory. However, the counter terms cannot all be finite, because of Haag’s theorem.

32Recall that :Φ̂_{0}(x) □ Φ̂_{0}(x): = −m^{2} :Φ̂_{0}(x)^{2} : . Therefore an arbitrary term proportional to :Φ̂_{0}(x) □ Φ̂_{0}(x): can be extracted from δm^{2} :Φ̂_{0}(x)^{2} : and compensated by change of δm^{2} .

33Actually, as pointed out for Equation (3.35), δm^{2} and δλ are assumed to be of higher order in λ.

34This exercise indicates that – as far as perturbative calculations of S elements are concerned – taking a mass different from the physical mass m for the free field could be balanced by suitable change of the counter terms.

ˆΦ̂(x) is

ˆĤ = \frac{1}{2} ∫_{x^{0}=0} ⋮ (∑_{μ=0}^{3} (∂_{μ} ˆΦ̂(x))^{2} + (sm_{b})^{2} ˆ Φ̂(x)^{2} + 2λ_{b} ˆ Φ̂(x)^{4}) ⋮ dx + const .

Exercise 52 Show that ∫ :Φ̂_{0}(x)^{4} : dx exists as a quadratic form but not as an operator on D_{0} . Moreover, show that ∫ :Φ̂_{0}(x)^{2} : dx does not even exist as a quadratic form on D_{0} .

3.2.3 Feynman Rules

The perturbative expansion (3.24) for (3.39) – evaluated by Coroll. 3.2.2 – may be represented by diagrams. For these diagrams we need three types of vertices which we will draw as •, ∘, and ⊙ •. Each vertex of type • or ⊙ • is connected to exactly four solid lines, each vertex of type ∘ to exactly two solid lines. These lines may either have free ends (external lines) or connect to another vertex (internal lines). Finally, the vertices of such a diagram G have to be indexed from 1 to V_{G}, where

V_{G} =^{def} number of vertices of G .

diagrams of this kind will be called admitted (for λΦ̂_{4}^{4}-perturbation theory in xspace).

To write down the formal operator Â_{G} represented by an admitted diagram G one has to apply the following Feynman rules:

1. Write down a factor

−iλ for each vertex • ,
\frac{i}{2}δm for each vertex ∘ ,
−iδλ for each vertex ⊙ • .

2. For every pair of vertices with indices ν and μ, if directly connected by at least one internal line, write down a factor

i^{f_{νμ}}Δ_{F}^{f_{νμ}}(x_{ν} − x_{μ}) ,

where f_{νμ} is the number of internal lines directly connecting these vertices.

3. Multiply by the symmetry factor^{35}

∏_{1≤ν<μ≤V_{G}} \frac{1}{f_{νμ}!} ∏_{1≤α≤V_{G}} \frac{l_{α}!}{(l_{α} − ∑_{β=1}^{V_{G}} f_{αβ})!} ,

Draft, March 26, 2009

35Recall the evaluation of Fig. 3.2.

where

l_{α} =^{def} \{ \begin{matrix}number of lines \\ attached to vertex α . \end{matrix}

4. For every external line write down a field operator Φ̂_{0}(x_{ν}) , where ν is the index of the vertex to which the line is attached.

5. Normal order the resulting monomial and integrate all field variables over ℝ^{4} .

Any two diagrams have to be considered as equal if they differ only by their diagramatical realization.^{36} For example,

[figure: 1; 2]

is to be considered as equal to

[figure: 2; 1] .

Then, for suitable^{37} φ, we have^{38}

e^{iφ}Ŝ_{0} = ˆ1 + ∑_{G admitted} \frac{1}{V_{G}!}Â_{G}  (3.41)

as an equation for formal power series in λ of quadratic forms on D_{0} .

Any two diagrams G_{1}, G_{2} are called equivalent (G_{1} ≅ G_{2}) if they differ only by the distribution of their vertex indices; e.g.:

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; ≅; ≠; 2; 2; 2; 3; 4; 3; 1; 4; 1; 3; 1; 4; 5; 5; 5]

Then the number of elements in the equivalence class [G] of a diagram G is \frac{V_{G}!}{I_{G}} , where

I_{G} =^{def} \{ \begin{matrix}number of permutations of the vertex indices \\ that do not change the diagram G . \end{matrix}

Therefore^{39} (3.41) is equivalent to

e^{iφ}Ŝ_{0} = ˆ1 + ∑_{[G]} \frac{1}{I_{G}} Â_{G} .  (3.42)

Draft, March 26, 2009

36Consider the lines as elastic strings. Then all elastic deformations leave the diagrams unchanged.

37Actually, everything should be defined with suitable cutoffs first. When removing these limits φ becomes infinite and compensates the infinite contributions of the vacuum diagrams, i.e. those diagrams G for which Â_{G} ∼ ˆ1 . This will be used in (3.44).

38Thanks to (3.40), only a finite number of diagrams contributes to each order in λ on the r.h.s. of (3.41).

39Note that G_{1} ≅ G_{2} ⟹ Â_{G_{1}} = Â_{G_{2}} , thanks to integration over all field variables, even though G_{1} ≅ G_{2} does not imply G_{1} = G_{2} , in general.

Denote by G_{1} ⋯ G_{N} the diagram consisting of the disjoint subdiagrams G_{1}, …, G_{N} with natural renumbering of their vertices. Then

Â_{G_{1}···G_{N}} = :Â_{G_{1}} ⋯ Â_{G_{N}}:

and

I_{G_{1}···G_{N}} = I_{G_{1}} ⋯ I_{G_{N}} E_{G_{1}···G_{N}} ,

where

E_{G_{1}···G_{N}} =^{def} \{ \begin{matrix}number of permutations π ∈ S_{N} with \\ ([G_{π1}], …, [G_{πN}]) = ([G_{1}], … [G_{N}]) . \end{matrix}

Thus we have, formally,

\frac{1}{I_{G_{1}···G_{N}}} Â_{G_{1}···G_{N}} = \frac{1}{E_{G_{1}···G_{N}}}: \frac{1}{I_{G_{1}}} Â_{G_{1}} ⋯ \frac{1}{I_{G_{N}}} Â_{G_{N}}:  (3.43)

for every set of diagrams {G_{1}, …, G_{N}}.

An admitted diagram is called connected if any two vertices are connected by a chain of internal lines. Obviously, for every diagram G there is a unique N-tuple of connected diagrams (G_{1}, …, G_{N}) with G = G_{1} ⋯ G_{N} . Consequently, by (3.43), (3.42) may be written as

e^{iφ}Ŝ_{0} = :exp( ∑_{[G]_{G connected}} \frac{1}{I_{G}} Â_{G}): .

With the physically natural requirement^{40}

⟨Ω_{0} | Ŝ_{0}Ω_{0}⟩ = 1

(stability of the vacuum) this implies:

Ŝ_{0} = :exp( ∑_{[G]∈G} \frac{1}{I_{G}} Â_{G}): = :exp( ∑_{G admitted_{[G]∈G}} \frac{1}{V_{G}!}Â_{G}): ,

where:

G =^{def} {[G] : G connected , Â_{g} ≁ ˆ1} .

(3.44)

In order to evaluate (3.44) one has to fix, first of all, δm as a power series in λ depending on m and δλ. For this the physical requirement

⟨Ψ | Ŝ_{0}Ψ⟩ = ⟨Ψ | Ψ⟩ ∀ Ψ ∈ H_{0}^{(1)}

(stability of 1-particle states) is sufficient. Finally, one has to fix δλ as a power series in λ depending only on m. For this some convention concerning the 2-2-scattering amplitude – depending on the preferred technical interpretation of the coupling constant – is necessary.

Draft, March 26, 2009

40Actually, only ⟨Ω_{0} | Ŝ_{0}Ω_{0}⟩ = ˆ1 has to be required.

Exercise 53 Calculate the total cross section^{41}

σ(p_{1}, p_{2}) = \frac{(2π)^{2}}{4\sqrt{(p_{1}p_{2})^{2} − m^{4}}} ∫_{q_{j}^{0}=ω_{q_{j}}} |\frac{⟨â_{0}^{∗}(q_{1})â_{0}^{∗}(q_{2})Ω_{0} |(Ŝ_{0} − ˆ1)â_{0}^{∗}(p_{1})â_{0}^{∗}(p_{2})Ω_{0}⟩}{δ(q_{1} + q_{2} − p_{1} − p_{2})}|^{2}

×δ(q_{1} + q_{2} − p_{1} − p_{2}) \frac{dq_{1}dq_{2}}{2q_{1}^{0}2q_{2}^{0}}

of elastic scattering of two particles with initial 4-momenta p_{1}, p_{2} to first order in the renormalized coupling constant λ.

Remark: The derivation of (3.44) also shows, that one may set φ = 0 in (3.41) if summation is restricted to those diagrams which do not have disjoint parts without any external line.

3.3 Bogoliubov-Shirkov Theory

3.3.1 Basic Assumptions

Bogoliubov and Shirkov (Bogoliubov and Shirkov, 1959) assume that there is a whole family of (interaction picture) S-matrices Ŝ_{0}(g) depending sufficiently smoothly on g ∈ S(ℝ^{4}, ℝ) , where S(ℝ^{4}, ℝ) denotes the subspace of real-valued elements of S(ℝ^{4}) . If

0 ≤ g(x) ≤ 1 ∀ x ∈ ℝ^{4}  (3.45)

then g(x) is interpreted as a degree to which the interaction is ‘switched on’ at x (formally by replacing the renormalized coupling constant λ in (3.39) by g(x)λ). Ŝ_{0}(g) is assumed to fulfill the following conditions for all g ∈ S(ℝ^{4}) fulfilling (3.45):

1. Ŝ_{0}(g) is unitary.

2. Ŝ_{0}(0) = ˆ1 .

3. Relativistic Covariance:^{42}

Û_{0}(a, Λ)Ŝ_{0}(g)Û_{0}(a, Λ)^{−1} = Ŝ_{0} ({a, Λ}g) ∀ (a, Λ) ∈ P_{+}^{↑} ,

where

({a, Λ}g) (x) =^{def} g (Λ^{−1}(x − a)) .

Draft, March 26, 2009

41For a derivation of this formula see e.g. (Itzykson and Zuber, 1980a, Sect. 5-1-1). For its evaluation see also 4.3.2.

42For theories with fermions a representation of iSL(2, ℂ) has to be used; see 4.2.2.

4. Bogoliubov-Shirkov causality:

(x^{0} < y^{0} ∀(x, y) ∈ supp g_{1} × supp g_{2}) ⟹ Ŝ_{0}(g_{1} + g_{2}) = Ŝ_{0}(g_{2})Ŝ_{0}(g_{1}) .

Moreover, the functional derivatives

Ŝ_{n}(x_{1}, …, x_{n}) =^{def} (\frac{δ}{δg(x_{1})} ⋯ \frac{δ}{δg(x_{n})}Ŝ_{0}(g))_{|_{g=0}}

are assumed to exist as operator-valued distributions on S(ℝ^{4n}) with invariant dense domain D_{0} (defined by (2.27)) for the ‘smeared’ Ŝ_{n}(x_{1}, …, x_{n}) and their adjoints. With this definition we have the following formal Taylor expansion:^{43}

\begin{aligned}Ŝ_{0}(g) & = & (exp (∫ dx g(x)\frac{δ}{δǧ(x)}) Ŝ_{0}(ǧ))_{|_{ǧ=0}} \\ & = & \underbrace{ˆ1}_{=Ŝ_{0}(0)} + ∑_{n=1}^{∞} \frac{1}{n!} ∫ Ŝ_{n}(x_{1}, …, x_{n})g(x_{1}) ⋯ g(x_{n}) dx_{1} ⋯ dx_{n} . \end{aligned}

The Ŝ_{n}(x_{1}, …, x_{n}) are the central objects of the Bogoliubov-Shirkov theory.^{44} Its advantage is that it does not depend on the full-fledged interaction picture.^{45}

An immediate consequence of their definition is the permutation symmetry of the Ŝ_{n} :

Ŝ_{n}(x_{1}, …, x_{n}) = Ŝ_{n}(x_{π1}, …, x_{πn}) ∀ π ∈ S_{n} .  (3.46)

Therefore

Ŝ(M) =^{def} \{ \begin{matrix}Ŝ_{n}(ξ_{1}, …, ξ_{n}) & for M = \{ξ_{1}, …, ξ_{n}\} ≠ ∅ \\ ˆ1 & for M = ∅ \end{matrix}

is a consistent definition (if the M are considered as sets of ℝ^{4}-variables).

Exercise 54 Given the formal power series

\begin{aligned}S(g) & = & S(∅) + ∑_{n=1}^{∞} \frac{1}{n!} ∫ S (\{x_{1}, …, x_{n}\}) g(x_{1}) ⋯ g(x_{n}) dx_{1} ⋯ dx_{n} , \\ R(g) & = & R(∅) + ∑_{n=1}^{∞} \frac{1}{n!} ∫ R (\{x_{1}, …, x_{n}\}) g(x_{1}) ⋯ g(x_{n}) dx_{1} ⋯ dx_{n} \end{aligned}

in g, show that

S(g) R(g) = (S ∗ R)(∅) + ∑_{n=1}^{∞} \frac{1}{n!} ∫ (S ∗ R) (\{x_{1}, …, x_{n}\}) g(x_{1}) ⋯ g(x_{n}) dx_{1} ⋯ dx_{n}

Draft, March 26, 2009

43Many properties of the Ŝ_{n} may be easily read off from this formal power series.

44See (Stora, 1971; Epstein and Glaser, 1973) for a more elaborated version.

45This does not mean, however, that it is physically better motivated.

3.3. BOGOLIUBOV-SHIRKOV THEORY 125 holds (in the sense of formal power series) with the convolution product^{46}

(S ∗ R)(X) =^{def} ∑_{M⊂X} S(M)R(X \ M) for X ⊂ {x_{1}, x_{2}, x_{3}, …} .

Relativistic covariance implies formal covariance:

Û_{0}(a, Λ)Ŝ_{n}(x_{1}, …, x_{n})Û_{0}(a, Λ)^{−1} = Ŝ_{n} (Λx_{1} + a, …, Λx_{n} + a) .  (3.47)

Unitarity of Ŝ_{0}(g) implies that all functional derivatives of V̂ (g) = Ŝ_{0}(g)Ŝ_{0}(g)^{∗} vanish at g fulfilling (3.45). For g = 0 this gives

\begin{aligned}0 & = & (\frac{δ}{δg(x_{1})} ⋯ \frac{δ}{δg(x_{n})} (Ŝ_{0}(g)Ŝ_{0}(g)^{∗}))_{|_{g=0}} \\ & = & ∑_{M⊂X_{n}} \underbrace{(( ∏_{x∈M} \frac{δ}{δg(x)}) Ŝ_{0}(g))_{|_{g=0}}}_{=ˆ1 for M=∅} (( ∏_{x′∈X_{n}\M} \frac{δ}{δg(x′)}) Ŝ_{0}(g)^{∗})_{|_{g=0}} , \\ & & X_{n} =^{def} \{x_{1}, …, x_{n}\} . \end{aligned} (3.48)

Therefore, since

(\frac{δ}{δg(x_{1})} ⋯ \frac{δ}{δg(x_{n})}Ŝ(g)^{∗})_{|_{g=0}} = Ŝ_{ν}(x_{1}, …, x_{n})^{∗} ,

we have formal unitarity:^{47}

0 = ∑_{M⊂X_{n}} Ŝ(M)Ŝ(X_{n} \ M)^{∗} for X_{n} = {x_{1}, …, x_{n}} , n > 0 .  (3.49)

Let g_{1}, g_{2} ∈ S(ℝ^{4}, ℝ) fulfill the conditions

g_{1}(x), g_{2}(x) ∈ [0, 1] ∀ x ∈ ℝ^{4} ,

x^{0} < y^{0} ∀(x, y) ∈ supp g_{1} × supp g_{2} . (3.50)

Then Bogoliubov-Shirkov causality and unitarity imply

Ŝ_{0}(g_{1} + g_{2})Ŝ_{0}(g_{1})^{∗} = Ŝ_{0}(g_{2})  (3.51)

and therefore

(3.50) ⟹ \frac{δ}{δg_{1}(x)} (\frac{δŜ_{0}(g)}{δg(y)}Ŝ_{0}(g_{1})^{∗})_{|_{g=g_{1}+g_{2}}} = 0 for (x, y) ∈ supp g_{1} × supp g_{2} .

(3.52)

Draft, March 26, 2009

46This convolution product has many useful applications; see e.g. (Stora, 1971; Borchers, 1972; Doebner and Lücke, 1977; Hegerfeldt, 1985).

47Since unitarity means Ŝ_{0}(g)Ŝ_{0}(g)^{∗} = ˆ1 = Ŝ_{0}(g)^{∗}Ŝ_{0}(g) , also 0 = ∑_{M⊂X_{n}} Ŝ(M)^{∗}Ŝ(X_{n} \M) holds.

126 CHAPTER 3. SCALAR PERTURBATION THEORY

Remark: Conversely, (3.52) implies

∂_{λ_{1}}∂_{λ_{2}} (Ŝ_{0}(λ_{1}g_{1} + λ_{2}g_{2})Ŝ_{0}(λ_{1}g_{1})^{∗}) = 0 ∀ λ_{1}, λ_{2} ∈ [0, 1]

hence constancy of

∂_{λ_{2}} (Ŝ_{0}(λ_{1}g_{1} + λ_{2}g_{2})Ŝ_{0}(λ_{1}g_{1})^{∗})

in λ_{1} and thus^{48}

Ŝ_{0}(g_{1} + g_{2})Ŝ_{0}(g_{1})^{∗} = Ŝ_{0}(g_{2}) \underbrace{Ŝ_{0}(0)^{∗}}_{=ˆ1}

under the assumption (3.50). In other words: (3.52), thanks to unitarity and Ŝ_{0}(0) = ˆ1 , implies Bogoliubov-Shirkov causality.

Applying Bogoliubov-Shirkov causality once more we get from (3.52)

(3.50) ⟹ (\frac{δ}{δg(x)} (\frac{δŜ_{0}(g)}{δg(y)}Ŝ_{0}(g)^{∗}))_{|_{g=g_{1}+g_{2}}} = 0 for (x, y) ∈ supp g_{1} × supp g_{2} .

Evaluating this — or directly (3.52) — at g = 0 gives

K̂(y; X_{n}) = 0 for max {y^{0} − x_{1}^{0}, …, y^{0} − x_{n}^{0}} > 0 ,

where

K̂(y; X_{n}) =^{def} \frac{δ}{δg(x_{1})} ⋯ \frac{δ}{δg(x_{n})} (\frac{δŜ_{0}(g)}{δg(y)}Ŝ_{0}(g)^{∗})_{|_{g=0}} ,

i.e.

K̂(y; X_{n}) = ∑_{M⊂X_{n}} Ŝ({y} ∪ M)Ŝ(X_{n} \ M)^{∗} .

This way, by Bogoliubov-Shirkov causality and relativistic covariance, we get formal causality:

∑_{M⊂X_{n}} Ŝ(\{y\} ∪ M)Ŝ(X_{n} \ M)^{∗} = 0 if y − x ∈ ℝ^{4} \ \overline{V_{−}}

for some x ∈ X_{n} , n > 0 .

(3.53)

Exercise 55 Show that (3.53) implies

\frac{δ}{δg(x)} (\frac{δŜ_{0}(g)}{δg(y)}Ŝ_{0}(g)^{∗}) = 0 for y − x ∈ ℝ^{4} \ \overline{V_{−}}

in the sense of formal power series in g.

Draft, March 26, 2009

48Since then \frac{d}{dλ} (Ŝ_{0}(g_{1} + λg_{2})Ŝ_{0}(g_{1})^{∗} − Ŝ_{0}(λg_{2})Ŝ_{0}(0)^{∗}) = 0 .

3.3.2 General Solution

For y = x_{n+1} (3.53) implies

Ŝ(X_{n+1}) = − ∑_{M⊂X_{n}_{M≠Xn}} Ŝ ({x_{n+1}} ∪ M) Ŝ(X_{n} \ M)^{∗}

if x_{n+1} − x ∈ ℝ^{4} \ \overline{V_{−}} for some x ∈ X_{n} .

(3.54)

This shows that the Ŝ_{n}(x_{1}, …, x_{n}) are fixed on

ℝ_{≠}^{4n} =^{def} {(x_{1}, …, x_{n}) ∈ ℝ^{4n} : x_{ν} ≠ x_{μ} for ν ≠ μ}  (3.55)

for n = 2, 3, … once

L̂_{I}(x) =^{def} −iŜ_{1}(x)  (3.56)

is given. Of course, L̂_{I}(x) has to meet certain requirements. (3.49) for n = 1 means

L̂_{I}(x) = L̂_{I}(x)^{∗} .  (3.57)

Therefore evaluation of (3.54) for n = 1 gives

Ŝ(\{x_{1}, x_{2}\}) = Ŝ_{1}(x_{2})Ŝ_{1}(x_{1}) = −L̂_{I}(x_{2})L̂_{I}(x_{1}) for x_{2} − x_{1} ∈ ℝ^{4} \ \overline{V_{−}} . (3.58)

Since this does not depend on the choice of indices, we conclude that^{49}

[L̂_{I}(x), L̂_{I}(y)]_{−} = 0 for x×y .  (3.59)

By (3.47), finally, we have

Û(a, Λ)L̂_{I}(x)Û(a, Λ)^{−1} = L̂_{I}(Λx + a) .  (3.60)

Summing up:

L̂_{I}(x) must be a hermitian, scalar, local operator field.

Conversely, these properties of L̂_{I}(x) guarantee that (3.46), (3.47), (3.49), and (3.53) are fulfilled on ℝ_{≠}^{4n} by^{50}

Ŝ_{n}^{T}(x_{1}, …, x_{ν}) =^{def} T (Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{ν})) for ν = 2, 3, …  (3.61)

Draft, March 26, 2009

49Note that, x×y ⟹ x_{2} − x_{1} ∈ ℝ^{4} \ \overline{V_{−}} ∋ x_{1} − x_{2} .

50(3.49), for example, is obvious for n = 1 and therefore follows for n = 2, 3, … – if restricted to ℝ_{≠}^{4n} – because

\begin{matrix}∑_{M′⊂X_{n+1}} Ŝ^{T} (M′)Ŝ^{T} (X_{n+1} \ M′)^{∗} \\ = Ŝ_{1}(x_{n+1}) ∑_{M⊂X_{n}} Ŝ^{T} (M)Ŝ^{T} (X_{n} \ M)^{∗} + ∑_{M⊂X_{n}} Ŝ^{T} (M)Ŝ^{T} (X_{n} \ M)^{∗}Ŝ_{1}(x_{n+1})^{∗} \end{matrix}

holds on the subregion of ℝ_{≠}^{4(n+1)} characterized by: x_{n+1}^{0} > x_{j}^{0} for j = 1, …, n. For (3.53) see also (Bogoliubov and Shirkov, 1959, Sect. 18.5).

if T is a (linear) covariant time ordering operation, i.e. fulfills the conditions

T (Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{ν})) = Ŝ_{1}(x_{π1}) ⋯ Ŝ_{1}(x_{πν}) for π ∈ S_{ν} with x_{π1}^{0} > … > x_{πν}^{0}

(3.62)

and

Û_{0}(a, A)T (Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{ν})) Û_{0}(a, A)^{−1} = T (Ŝ_{1}(Λx_{1} + a) ⋯ Ŝ_{1}(Λx_{ν} + a)) .

(3.63)

(on all of ℝ^{4ν} for ν = 2, 3, …). Therefore:^{51}

On the restricted region ℝ_{≠}^{4n} there is no other choice for Ŝ_{n}(x_{1}, …, x_{n})

than (3.61).

The difficult problem is physically correct extension of Ŝ_{n}(x_{1}, …, x_{n}) to all of ℝ^{4n} (n = 2, 3, …).

If the Ŝ_{ν}(x_{1}, …, x_{ν}) are known for ν ≤ n then Ŝ_{n+1}(x_{1}, …, x_{n+1}) is fixed by (3.54) even on the complement of

ℝ_{=}^{4(n+1)} =^{def} {(x_{1}, …, x_{n+1}) ∈ ℝ^{4(n+1)} : x_{1} = x_{2} = … = x_{n+1}} ,

since there must be a pair x_{ν}, x_{μ} ∈ \{x_{1}, …, x_{n+1}\} for which x_{ν} −x_{μ} ∈ ℝ^{4} \\overline{V_{−}} unless x_{1} = x_{2} = … = x_{n}. In other words:

If Ŝ_{1}(x), …, Ŝ_{n}(x_{1}, …, x_{n}) are fixed then Ŝ_{n+1}(x_{1}, …, x_{n+1}) is unique up to addition of an operator field Â_{n+1}(x_{1}, …, x_{n+1}) which is quasilocal, i.e.

supp Â_{n+1}(x_{1}, …, x_{n+1}) ⊂ ℝ_{=}^{4(n+1)} .

Further details: Similarly to (3.52), starting from

Ŝ_{0}(g_{2})^{∗}Ŝ_{0}(g_{1} + g_{2}) = Ŝ_{0}(g_{1})

instead of (3.51), we get

(3.50) ⟹ \frac{δ}{δg_{1}(x)} (Ŝ_{0}(g_{2})^{∗} \frac{δŜ_{0}(g)}{δg(x)})_{|_{g=g_{1}+g_{2}}} = 0 for (x, y) ∈ supp g_{1} × supp g_{2}

and hence

∑_{M⊂X_{n}} Ŝ(X_{n} \ M)^{∗}Ŝ(\{y\} ∪ M) = 0 if y − x ∈ ℝ^{4} \ \overline{V_{−}} for some x ∈ X_{n} , n > 0 .

Therefore, the difference

D(X_{n}; y) =^{def} R(X_{n}; y) − A(X_{n}; y)

Draft, March 26, 2009

51Recall that (3.53) implies (3.54). of the retarded function

R(X_{n}; y) =^{def} ∑_{M⊂X_{n}} Ŝ(X_{n} \ M)^{∗}Ŝ({y} ∪ M)

and the advanced function

A(X_{n}; y) =^{def} ∑_{M⊂X_{n}} Ŝ({y} ∪ M)Ŝ(X_{n} \ M)^{∗}

vanishes whenever one of the arguments x_{ν} is spacelike w.r.t. y. If all the S(X_{ν}) are known for all ν ≤ n then also D(X_{n}; x_{n+1}) and Ŝ(X_{n+1})−A(X_{n}; x_{n+1}) are known and determination of Ŝ(X_{n+1}) is equivalent to physically correct splitting of D(X_{n}; x_{n+1}) into an advanced function A(X_{n}; x_{n+1}) and a retarded function A(X_{n}; x_{n+1}) . The main difficulty of such a splitting is to get A(X_{n}; x_{n+1}) and R(X_{n}; x_{n+1}) Lorentz covariant (see (Steinmann, 1963; Epstein, 1966)).

Note that one may work with Ŝ_{0}(x)^{−1} instead of Ŝ_{0}(x)^{∗} and use (3.57) instead of formal unitarity (see (Epstein and Glaser, 1973, Sections 1.2 and 2)).

Let us use this to analyze a given formal power series Ŝ_{0}(g) that fulfills all the Bogoliubov-Shirkov requirements.

Let Â_{2}, …, Â_{n} be quasilocal operators and let T be a (linear) time ordering operation, defined for all direct products Ĉ_{j} composed of elements from {Ŝ_{1}, Â_{2}, …, Â_{n}}, fulfilling the following requirements:

T (Ĉ_{1}(x_{1}, …, x_{j}) Ĉ_{2}(y_{1}, …, y_{k})) = T (Ĉ_{1}(x_{1}, …, x_{j}))(Ĉ_{2}(y_{1}, …, y_{k}))

if sup {x_{1}^{0}, …, x_{j}^{0}} > sup {y_{1}^{0}, …, y_{k}^{0}} ,

(3.64)

T (Ĉ_{1}(x_{1}, …, x_{j}) T (Ĉ_{2}(y_{1}, …, y_{k}))) = T (T (Ĉ_{1}(x_{1}, …, x_{j}))Ĉ_{2}(y_{1}, …, y_{k}))
= T (Ĉ_{1}(x_{1}, …, x_{j}) Ĉ_{2}(y_{1}, …, y_{k}))
= T (Ĉ_{2}(y_{1}, …, y_{k}) Ĉ_{1}(x_{1}, …, x_{j}) ) ,

(3.65)

Û_{0}(a, A)T (Ĉ(x_{1}, …, x_{ν}))Û_{0}(a, A)^{−1} = T (Ĉ(Λx_{1} + a, …, Λx_{ν} + a)) . (3.66)

Exercise 56 Show the following:

1. The Â_{ν} are local relative to Ŝ_{1} in the sense that

x×y_{1} ⟹ [Ŝ_{1}(x), Â_{ν}(y_{π1}, …, y_{πν})]_{−} = 0 ∀ π ∈ S_{ν} .

2. The Â_{ν} are local relative w.r.t. each other in the sense that

x_{1}×y_{1} ⟹ [Â_{μ}(x_{1}, …, x_{μ}), Â_{ν}(y_{π1}, …, y_{πν})]_{−} = 0 ∀ π ∈ S_{ν} .

130 CHAPTER 3. SCALAR PERTURBATION THEORY

3. The Â_{ν} transform according to

Û(a, Λ) Â_{ν}(x_{1}, …, x_{ν}) Û(a, Λ)^{−1} = Â_{ν} (Λx_{1} + a, …, Λx_{ν} + a) ∀ (a, Λ) ∈ P_{+}^{↑} . Then we define

Ŝ_{Â_{2},...,Â_{n}}^{T}(\{x_{1}, …, x_{ν}\}) =^{def} \frac{δ}{δg(x_{1})} ⋯ \frac{δ}{δg(x_{ν})}Ŝ_{Â_{2},...,Â_{n}}^{T}(g)_{|_{g=0}} ,

where Ŝ_{Â_{2},...,Â_{n}}^{T}(g)

=^{def} T (exp (∫ Ŝ_{1}(x)g(x) dx + ∑_{ν=2}^{n} ∫ Â_{ν}(x_{1}, …, x_{ν})g(x_{1}) ⋯ g_{ν}(x_{ν}) dx_{1} ⋯ dx_{ν}))

(in the sense of formal power series). Exercise 57 Show that Ŝ_{Â_{2},...,Â_{n}}^{T}(g) fulfills all the Bogoliubov-Shirkov requirements if also

\begin{matrix}Â_{ν}(x_{π1}, …, x_{πν}) = Â_{ν}(x_{1}, …, x_{ν}) = −Â_{ν}(x_{1}, …, x_{ν})^{∗} \\ ∀ π ∈ S_{ν} , ν ≤ n \end{matrix}  (3.67)

holds. Now assume

Ŝ({x_{1}, …, x_{ν}}) = Ŝ_{Â_{2},...,Â_{n}}^{T}({x_{1}, …, x_{ν}}) for ν ≤ n .  (3.68)

Then, according to the above considerations,

Â_{n+1}(x_{1}, …, x_{n+1}) =^{def} Ŝ({x_{1}, …, x_{n+1}}) − Ŝ_{Â_{2},...,Â_{n}}^{T}({x_{1}, …, x_{n+1}})  (3.69)

must be a quasilocal operator field, since (3.54) holds for both Ŝ and Ŝ_{Â_{2},...,Â_{n}}^{T}. Exercise 58 Show that, with Â_{n+1} defined by (3.69), the statements of Exercise 56 and (3.67) hold also for n replaced by n + 1 . If we can extend^{52} the T -operation to all direct products Ĉ_{j} composed of elements from {Ŝ_{1}, Â_{2}, …, Â_{n+1}} in a way respecting (3.64)–(3.66) we get (3.68) for n replaced by n + 1 . This way we are lead to the following Bogoliubov-Shirkov

Draft, March 26, 2009

52In general, it is not at all obvious that such an extension is possible. Up to now, nobody could provide a proof for existence of the covariant T -products for interacting Wightman fields (compare (Steinmann, 1963; Epstein, 1966) to see the difficulties).

conjecture:

Let Ŝ_{0}(g) be a formal power series fulfilling all the Bogoliubov-Shirkov requirements and let T be a (linear) covariant time ordering operation fulfilling (3.64)–(3.66) for arbitrary muliple direct products Ĉ_{j} of Ŝ_{1} with itself. Then there is a sequence of quasilocal operators Â_{2}, Â_{3}, … and a suitable extension of T for which

Ŝ_{0}(g) = T (exp(∫ dxŜ_{1}(x)g(x) + Â(g))) , where

Â(g) =^{def} ∑_{ν=2}^{∞} ∫ Â_{ν}(x_{1}, …, x_{ν})g(x_{1}) ⋯ g(x_{ν}) dx_{1} ⋯ dx_{ν} ,

holds in the sense of formal power series in g.

The transition

∫ Ŝ_{1}(x)g(x) dx ⟶ ∫ Ŝ_{1}(x)g(x) dx + Â(g)

is to be considered as renormalization from the Bogoliubov-Shirkov point of view. Without correct choice of the counter terms Â(g) transition to the adiabatic limit (g(x) ⟶ 1) will typically not be possible (recall Exercise 52).

3.3.3 Generalization to Nonlocalizable Test Spaces

As already pointed out in 2.2.2, there is no physical justification for the technical requirement that Φ̂_{0}(g) be defined for all tempered g. Therefore other test spaces should be taken under consideration. Convenient families of test spaces, parameterized by s ≥ 0 , are the following:^{53}

S^{s}(ℝ^{4n}) =^{def} ⋃_{A>0} S^{s,A} (inductive limit) ,
J^{s}(ℝ^{4n}) =^{def} ⋂_{A>0} S^{s,A} (projective limit) ,

where

S^{s,A}(ℝ^{4n}) =^{def} {φ ∈ S(ℝ^{4n}) : ∥φ∥_{A,N}^{s} < ∞ ∀ N ∈ ℕ} ,

the topology of S^{s,A}(ℝ^{4n}) being given by the family of norms^{54}

∥φ∥_{A,N}^{s} =^{def} sup_{x̌∈ℝ^{4n}} (1 + \left∥{ \atop x̌}\right∥)^{N} sup_{α̌∈ℤ_{+}^{4n}}(A + \frac{1}{N})^{−\left|{ \atop α̌}\right|}α̌^{−sα̌} |φ^{(α̌)}(x̌)| .

Draft, March 26, 2009

53Efimov (see (Efimov, 1968) and references given there) used slightly different test spaces. 54We write

x̌ = (x_{1}, …, x_{n}) , \left∥{ \atop x̌}\right∥ = \sqrt{∑_{ν=1}^{n} ∑_{μ=0}^{3}(x_{ν}^{μ})^{2}} , p̌x̌ = ∑_{ν=1}^{n} ∑_{μ=0}^{3} p_{ν}^{μ}x_{ν}^{μ}

Note that

s_{1} < s_{2} ⟹ S^{s_{1}} ≼ S^{s_{2}} ,

where S^{s_{1}} ≼ S^{s_{2}} means that S^{s_{1}} is contained in S^{s_{2}} as a set and that the topology of S^{s_{1}} is finer than that induced by S^{s_{2}} .

The elements of S^{s}(ℝ^{4n}) resp. J^{s}(ℝ^{4n}) may be characterized by their Fourier transforms

φ̃(p̌) =^{def} (2π)^{−2n} ∫ φ(x̌) e^{+ip̌x̌} dx̌

as follows:

\begin{matrix}φ ∈ S^{s}(ℝ^{4n}) ⟺ (∃ A > 0 : \left∥{ \atop φ̃}\right∥_{s}^{A,N} < ∞ ∀ N ∈ ℕ) , \\ φ ∈ J^{s}(ℝ^{4n}) ⟺ (∀ A > 0 : \left∥{ \atop φ̃}\right∥_{s}^{A,N} < ∞ ∀ N ∈ ℕ) . \end{matrix}

Here

\left∥{ \atop φ̃}\right∥_{s}^{A,N} =^{def} sup_{p̌∈ℝ^{4n}} sup_{α̌∈ℤ_{+}^{4n}_{\left|{ \atop α̌}\right|≤N}} g_{s} ((A + \frac{1}{N})^{−1} \left∥{ \atop p̌}\right∥) |φ̃^{(α̌)}(p̌)| ,

where

g_{s}(t) =^{def} sup_{μ∈ℤ_{+}} μ^{−sμ} |t|^{μ} for t ∈ ℝ^{1} .

Note that

e^{− \frac{se}{2}} e^{^{\frac{s}{e}} |t|^{1/s}} ≤ g_{s}(|t|) ≤ e^{^{\frac{s}{e}} |t|^{1/s}} ∀ t ∈ ℝ^{1} ,

if s > 0 , while

g_{0}(t) = \{ \begin{matrix}1 & for |t| ≤ 1 . \\ ∞ & else . \end{matrix}

This implies that S^{s}(ℝ^{4n}) contains only entire analytic functions if s < 0 and that

φ ∈ S^{0}(ℝ^{4n}) ⟺ φ̃ ∈ D(ℝ^{4n}) , J^{0}(ℝ^{4}) = ∅ .

Moreover, the elements φ of S^{1}(ℝ^{4n}) are seen to be analytic in a complex neighborhood of ℝ^{4n} (depending on φ). Finally, as shown by Roumieu (Roumieu, 1960; Roumieu, 1963) (see also (Gelfand and Schilow, 1962, Kapitel IV)):

s > 1 ⟹ S^{s}(ℝ^{4n}) ∩ D(ℝ^{4n}) dense in S^{s}(ℝ^{4n}) .

Draft, March 26, 2009

and use standard multi-index notation:

\begin{matrix}\left|{ \atop α̌}\right| = ∑_{ν=1}^{n} ∑_{μ=0}^{3} α_{ν}^{μ} , α̌β̌ = ∏_{ν=1}^{n} ∏_{μ=0}^{3} (α_{ν}^{μ})^{β_{ν}^{μ}} , α̌^{−sβ̌} = (α̌β̌)^{−s} , \\ φ^{(α̌)}(x̌) = Dα̌ ; x̌φ(x̌) , Dα̌ ; x̌ = ∏_{ν=1}^{n} ∏_{μ=0}^{3} (\frac{∂}{∂x_{ν}^{μ}} )^{α_{ν}^{μ}} . \end{matrix}

Therefore the standard definition of support may be applied to generalized functions on S^{s}(ℝ^{4n}) if and only if s > 1 . For the spaces J^{s}(ℝ^{4n}) the situation is quite similar.

For s < 1 , if one wants to test by functions φ ∈ S^{s}(ℝ^{4n}) whether a given F ∈ S^{s}(ℝ^{4n})′ is ‘localized’ within a closed subset M of ℝ^{4n}, there is essentially only one possibility:

Check whether

F (φ_{ν}) ⟶_{ν→∞} 0

holds for every sequence φ_{1}, φ_{2}, … with
φ_{ν} ⟶_{ν→∞}^{suitably} 0 on M .

The questions is just how to specify what ‘suitably’ should mean, here.

Let us interpret ‘suitably’ as ‘S^{s}-like’ in the sense that all the φ_{ν} are elements of S^{s}(ℝ^{4n}) and^{55}

(φ_{ν} ⟶_{ν→∞}^{S^{s}-like} 0 on M) ⟺^{def} (∃A > 0 : ∥φ_{ν}∥_{A,N}^{s,M} ⟶_{ν→∞} 0 ∀ N ∈ ℕ) ,

where^{56}

∥φ∥_{A,N}^{s,M} =^{def} sup_{x̌∈M} (1 + \left∥{ \atop x̌}\right∥)^{N} sup_{α̌∈ℤ_{+}^{4n}}(A + \frac{1}{N})^{−\left|{ \atop α̌}\right|}α̌^{−sα̌} |φ^{(α̌)}(x̌)| .

Let F ∈ S^{s}(ℝ^{4n})′ and M = \overline{M} ⊂ ℝ^{4n}. Then M is called a quasi-support of F with respect to S^{s}(ℝ^{4n}) if and only if

(φ_{ν} ⟶_{ν→∞}^{S^{s}-like} 0 on M) ⟹ F (φ_{ν}) ⟶_{ν→∞} 0

holds for arbitrary φ_{ν} ∈ S^{s}(ℝ^{4n}) . A neutral scalar quantum field Φ̂(x) with domain D is called essentially local with respect to S^{s}(ℝ^{4n}) if and only if {(x, y) ∈ ℝ^{4}×ℝ^{4} : x − y ∈ \overline{V} \} is a quasi-support of ⟨Ψ_{1} | [Φ̂(x), Φ̂(y)]_{−}Ψ_{2}⟩ with respect to S^{s}(ℝ^{4n}) for all Ψ_{1}, Ψ_{2} ∈ D.

For s > 1 every quasi-support w.r.t. S^{s} contains the (ordinary) support as a subset, whereas for s < 1 :

\begin{matrix}M_{1} quasi-support of F \\ ∃ϵ > 0 : M_{1} ⊂ U_{ϵ}(M_{2}) \end{matrix} \} ⟹ M_{2} quasi-support of F .

Draft, March 26, 2009

55Efimov required the φ_{1}, φ_{2}, … to be what he called projecting sequences with support M (see (Alebastrov and Efimov, 1974) and references given there). This especially means that the analytic continuations φ_{ν}(ž) of the φ_{ν}(x̌) have to converge uniformly to zero in the region M + iℝ^{4n} and uniformly to one in every region of the form K + iℝ^{4n} with K a compact subset of ℝ^{4n} \ M .

56Note that ∥φ∥_{A,N}^{s,ℝ^{4n}} = ∥φ∥_{A,N}^{s} , so that every element of S^{s}(ℝ^{4n})′ is localized in ℝ^{4n}, at least.

This is why we speak about ‘quasi’-supports. For instance, {0} is a quasi-support of F (x) = e^{−∥x∥^{2}} with respect to S^{1/4}(ℝ^{4}) . Thus, obviously, a quasi-support is not a domain of strict localization, in general. Nevertheless the P CT theorem and the spin-statistics theorem could be proved for fields which are only essentially local^{57} with respect to S^{0}(ℝ^{4}) (see (Lücke, 1984; Lücke, 1986)).

Now it is evident how to generalize the Bogoliubov-Shirkov theory:

Formulate everything – except Bogoliubov-Shirkov causality – with S^{s} instead of S and replace Bogoliubov-Shirkov causality by the requirement of generalized Bogoliubov-Shirkov causality:^{58}

\{(x, y) ∈ ℝ^{4} × ℝ^{4} : x − y ∈ \overline{V_{+}}\} is a quasi-support

of \frac{δ}{δg(x)} (\frac{δŜ_{0}(g)}{δg(y)}Ŝ_{0}(g)^{∗})_{|_{g=0}} with respect to S^{s}(ℝ^{8}) .

Naturally, we call M = \overline{M} a quasi-support of F ∈ J^{s}(ℝ^{4n})′ with respect to J^{s}(ℝ^{4n}) if and only if

(φ_{ν} ⟶_{ν→∞}^{J^{s}-like} 0 on M) ⟹ F (φ_{ν}) ⟶_{ν→∞} 0 ,

where

(φ_{ν} ⟶_{ν→∞}^{J^{s}-like} 0 on M) ⟺^{def} (∀ A, N > 0 : ∥φ_{ν}∥_{A,N}^{s,M} ⟶_{ν→∞} 0) .

Now, one would like to have a convenient criterion for \overline{V_{+}} being a quasi-support. For tempered distributions we have the following.

Theorem 3.3.1 Let L(p + iq) be an analytic function on the tube ℝ^{4} + iV_{+} having the following property:

For every η ∈ V_{+} there is a polynomial P_{η} for fulfilling

|L (p + i(η + a))| ≤ |P_{η}(p + ia)| ∀ p ∈ ℝ^{4} , a ∈ V_{+} .

Then there is a tempered distribution F (x) with

F (φ) = lim_{ϵ→+0} ∫ L (p + i(ϵ, 0, 0, 0)) φ̃(−p) dp ∀ φ ∈ S(ℝ^{4}) .

For this distribution

supp F ⊂ \overline{V_{+}} .

Draft, March 26, 2009

57In spite of the – generally misinterpreted – result of (Borchers and Pohlmeyer, 1968) there are examples of nonlocal tempered fields which are essentially local w.r.t. S^{0}(ℝ^{4}) , as shown in (Bümmerstede and Lücke, 1975, Sect. 5).

58We choose the spaces J^{s} since we are primarily interested in the case s = 1 , here. For the general study of axiomatic field theory the case s = 0 was the real challenge.

3.3. BOGOLIUBOV-SHIRKOV THEORY 135 Conversely, for every tempered distribution with such support there is a unique function L of the type specified above, namely the Laplace transform

L(p + iq) = (2π)^{−2} ∫ F (x) e^{+i(p+iq)x} dx .

Proof: See (Streater and Wightman, 1989, Theorem 2.8). An immediate consequence of Theorem 3.3.1 is

supp Δ_{ret}(x) ⊂ \overline{V_{+}}

for the retarded commutator

Δ_{ret}(x) =^{def} lim_{ϵ→+0}(2π)^{−4} ∫ \frac{e^{−ipx}}{(p^{0} + iϵ)^{2} − p^{2} − m^{2}} dp .

Note that, by Cauchy’s integral theorem,

Δ_{F}(x) = Δ_{ret}(x) − Δ_{−}(x)

for Δ_{F}(x) as defined in (3.28) and Δ_{−}(x) as defined in (3.30)/(3.31). Exercise 59 Let {μ_{α}}_{α∈ℤ_{+}^{4}} be a family of complex-valued Borel measures μ_{α} on ℝ^{4} and assume

sup_{α∈ℤ_{+}^{4}} A^{−|α|}α^{+sα} ∫ (1 + ∥x∥)^{−A} |μ_{α}|(dx) < ∞ for A > 0 large enough ,

where |μ| denotes the variation of the complex-valued measure μ :

|μ|(E) =^{def} sup {|∫_{E} f dμ| : f : E → ℂ measurable and |f| ≤ 1}  (3.70)

(compare (Halmos, 1950, p. 124)). Show^{59} that

F (x) = ∑_{α∈ℤ_{+}^{4}} D_{x}^{α}μ_{α}(x)

converges in the weak topology of J^{s}(ℝ^{4})′ and that

M = ⋂_{α∈ℤ_{+}^{4}} supp μ_{α}

is a quasi-support of F with respect to J^{s}(ℝ^{4}) .

Draft, March 26, 2009

59See (Lücke, 1984) for a proof of the converse of the corresponding statement for S^{s}(ℝ^{4}) .

In the nonlocalizable case, a useful substitute of Theorem 3.3.1 is the following.

Theorem 3.3.2 Let s > 0 , let L(p + iq) be an analytic function on ℝ^{4} + iV_{+} , and assume that

sup_{p+iq∈ℝ^{4}+iV_{+}} (∥q∥ dist (q, ∂V_{+}))^{A} e^{−(\frac{∥p∥+∥q∥}{A})^{1/s}} |L(p + iq)| < ∞

holds for sufficiently large A > 0 . Then

F̃(p) = lim_{ϵ→+0} L(p + iq)

converges in the weak topology of J̃^{s}(ℝ^{4})′. Moreover, every closed cone with apex at x = 0 and containing \overline{V_{+}} \ \{0\} in its interior is a quasi-support^{60} of F (x) with respect to J^{s}(ℝ^{4}) .

Proof: See (Fainberg and Soloviev, 1992, Theorem 4).

Moreover the following result is useful:

Theorem 3.3.3 Let^{61} s > 0 and let {x ∈ ℝ^{4} : x^{0} ≥ 0} be a quasi-support (w.r.t. J^{s}(ℝ^{4})) of F (x) ∈ J^{s}(ℝ^{4})′. If F (x) is Lorentz invariant then also \overline{V_{+}} is a quasisupport of F (x) .

Proof: See (Bümmerstede, 1976, Theorem 4.7).

Let V(t) be an entire analytic function of order \frac{1}{2s} , i.e.:

∃ρ > 0 : sup_{z∈ℂ^{1}} e^{−ρ|z|^{\frac{1}{2s}}} |V(z)| < ∞ .  (3.71)

Then, by Theorems 3.3.2 and 3.3.3,

Δ_{ret}^{V}(x) =^{def} lim_{ϵ→+0}(2π)^{−4} ∫ \frac{V (\frac{p^{2}}{m^{2}} ) e^{−ipx}}{(p^{0} + iϵ)^{2} − p^{2} − m^{2}} dp  (3.72)

is an element of J^{s}(ℝ^{4}) with quasi-support \overline{V_{+}} . If, in addition,

V(1) = 1  (3.73)

Draft, March 26, 2009

60Presumably, \overline{V_{+}} itself is a quasi-support of F (x) — as obvious for s > 1 . A generalization of Theorem 3.3.2 for s = 0 was proved in (Soloviev, 1997).

61Actually, Theorem 3.3.3 is also valid for s = 0 .

then also

Δ_{F}^{V}(x) =^{def} Δ_{ret}^{V}(x) − Δ_{−}(x)
= lim_{ϵ→+0}(2π)^{−4} ∫ \frac{V (\frac{p^{2}}{m^{2}} ) e^{−ipx}}{p^{2} − m^{2} − iϵ} dp

(3.74)

is a well defined element of J^{s}(ℝ^{4}) . If V(t) is even of sufficiently fast decrease, e.g.

sup_{t∈ℝ^{1}} t^{2} |V(t)| < ∞ ,  (3.75)

then there is a ‘canonical’ definition for powers of the modified Feynman propagator Δ_{F}^{V}(x) using Feynman parameterization

\begin{aligned}\frac{1}{c_{1} ⋯ c_{n}} & = & (n − 1)! ∫_{0}^{1} dξ_{1} ⋯ ∫_{0}^{1} dξ_{n} \frac{δ(1 − ξ_{1} − … − ξ_{n})}{(c_{1}ξ_{1} + … c_{n}ξ_{n})^{n}} \\ & & if 1 + c_{1}ξ_{1} + … + c_{n}ξ_{n} ≠ 0 ∀ξ_{1}, …, ξ_{n} ≥ 0 \end{aligned}  (3.76)

(see (Alebastrov and Efimov, 1973, Sect. 4.3)).

Proof of (3.76):

\frac{1}{n!c_{1} ⋯ c_{n}} = ∫_{0}^{∞} dξ_{1} ⋯ ∫_{0}^{∞} dξ_{n} \frac{1}{(1 + c_{1}ξ_{1} + … + c_{n}ξ_{n})^{n+1}}
= ∫_{0}^{∞} dξ_{1} ⋯ ∫_{0}^{∞} dξ_{n} ∫_{0}^{∞} dλ \frac{δ(λ − ξ_{1} − … − ξ_{n})}{(1 + c_{1}ξ_{1} + … + c_{n}ξ_{n})^{n+1}}
= ∫_{0}^{∞} dλ ∫_{0}^{∞} dξ′_{1} ⋯ ∫_{0}^{∞} dξ′_{n} \frac{δ ((1 − ξ′_{1} − … − ξ′_{n})λ)}{(1 + (c_{1}ξ′_{1} + … + c_{n}ξ′_{n})λ)^{n+1}} λ^{n}
= ∫_{0}^{∞} dξ′_{1} ⋯ ∫_{0}^{∞} dξ′_{n} ∫_{0}^{∞} dλ \frac{δ(1 − ξ′_{1} − … − ξ′_{n})}{(λ^{−1} + c_{1}ξ′_{1} + … + c_{n}ξ′_{n})^{n+1}} λ^{−2}
= ∫_{0}^{∞} dξ′_{1} ⋯ ∫_{0}^{∞} dξ′_{n} ∫_{0}^{∞} dλ′ \frac{δ(1 − ξ′_{1} − … − ξ′_{n})}{(λ′ + c_{1}ξ′_{1} + … + c_{n}ξ′_{n})^{n+1}}
= \frac{1}{n} ∫_{0}^{1} dξ′_{1} ⋯ ∫_{0}^{1} dξ′_{n} \frac{δ(1 − ξ′_{1} − … − ξ′_{n})}{(c_{1}ξ′_{1} + … + c_{n}ξ′_{n})^{n}}

Therefore, if we denote by T_{V} the time-ordering resulting by replacing – after formal application of Wick’s theorem – the products of the ordinary Feynman propagator Δ_{F}(x) by the ‘canonical’ products of the modified Feynman propagator Δ_{F}^{V}(x) ,

Ŝ_{0}^{V}(g) =^{def} T^{V} (exp (∫ Ŝ_{1}(x) g(x) dx)) for g ∈ J^{s}(ℝ^{4})  (3.77)

fulfills all the requirements of the generalized Bogoliubov-Shirkov theory.^{62}

Draft, March 26, 2009

62Generalized Bogoliubov-Shirkov causality can be proved by Theorem 3.3.2 and the

Alebastrov-Efimov analysis of \frac{δ}{δg(x)} (\frac{δŜ_{0}(g)}{δg(y)}Ŝ_{0}(g)^{∗})_{|_{g=0}} in (Alebastrov and Efimov, 1974,

Sect. 7). Of course, counterterms have still to be added if the adiabatic limit g ⟶ 1 is to exist.

Unfortunately, as a consequence of the Phragmén-Lindelöf theorem (see, e.g., Theorem 4.1.1 of (Lücke, ftm)), (3.71) and (3.75) are incompatible with each other for s ≥ 1/2 . For s < 1/2 there are plenty of entire functions V(t) fulfilling both (3.71) and (3.75) (see (Gelfand and Schilow, 1962, Kap. IV §8)). This is why Efimov (see (König, 1993) and references given there) suggested to use test spaces of entire functions in order to ‘regularize’ the perturbative expansion of the S-matrix.

For s = 1 we may still have

sup_{t∈ℝ_{+}} t^{2}V(−t) < ∞  (3.78)

in addition to (3.73). Take

V(t) = (\begin{matrix}sin & \sqrt{−t} \\ √ & \overline{−t} \end{matrix})^{4} ,

for instance. Now, the expressions resulting from the ‘canonical’ definition of powers of Δ_{F}^{V}(x) – thanks to transition to Euclidean momenta – are well defined, if only (3.78) holds. Efimov used this for the definition of ‘regularized’ solutions of the generalized Bogoliubov-Shirkov theory^{63} for J^{1}(ℝ^{4}) – motivated by suitable quantization of formally nonlocal field theories (Efimov, 1974).

Draft, March 26, 2009

63For generalization to gauge theories see, e.g., (Moffat, 1990; Cornish, 1992) and references given there.

Quantum Electrodynamics

4.1 The Free Electromagnetic Field Operators

4.1.1 Wightman Theory

Axioms

The Wightman axioms for the free electromagnetic field F̂^{μν}(x) follow from those for the neutral scalar field by straightforward adaption:

0. Assumptions of Relativistic Quantum Theory:

The same as in 2.2.1.

I. Assumptions about the Domain and Continuity of the Fields:

The field operators F̂^{μν}(x) are hermitian operator-valued, tempered generalized functions with invariant domain D ⊂ H; i.e. linear mappings

\begin{matrix}F̂^{μν} : S(ℝ^{4}) & ⟶ & L(D, D) \\ φ & ⟼ & ˆF̂^{μν}(φ) = \underbrace{∫ F̂^{μν}(x)φ(x) dx}_{formal} \end{matrix}

for which all the

∫ ⟨Ψ | F̂^{μν}(x)Ψ⟩ φ(x) dx =^{def} ⟨Ψ | F̂^{μν}(φ)Ψ⟩ , Ψ ∈ D ,

are continuous in φ ∈ S(ℝ^{4}) , where D has to fulfill the following conditions for φ ∈ S(ℝ^{4}) and (a, Λ) ∈ P_{+}^{↑} :

Ω ∈ D , Û(a, Λ)D ⊂ D , F̂^{μν}(φ)D ⊂ D , F̂^{μν}(\overline{φ}) = F̂^{μν}(φ)^{∗}/^{\}D .

II. Transformation Law of the Fields: The fields transform according to

Û(a, Λ)^{−1}F̂^{μν}(x)Û(a, Λ) = Λ_{α}^{μ}Λ_{β}^{ν}F̂^{αβ} (Λ^{−1}(x − a)) ∀ (a, Λ) ∈ P_{+}^{↑} .

Remark: Recall that Û(a, Λ)Ψ is to be described w.r.t. the coordinates x′ = Λ^{−1}(x − y) in exactly the same way as Ψ is to be described w.r.t. the coordinates x and that the field expectation values should transform like the classical fields (see, e.g., Eq. (2.9) of (Lücke, edyn). Therefore:

⟨Û(a, Λ)Ψ | F̂^{μν} (Λx + a) Û(a, Λ)⟩ = Λ_{α}^{μ}Λ_{β}^{ν} ⟨Ψ | F̂^{αβ}(x) Ψ⟩ .

III. Local Commutativity (Microscopic Causality):

The smeared fields F̂^{μν}(φ_{1}) , F̂^{αβ}(φ_{2}) commute whenever the supports of the test functions φ_{1} , φ_{2} ∈ S(ℝ^{4}) are spacelike with respect to each other. Formally:

x×y ⟹ [F̂^{μν}(x), F̂^{αβ}(y)]_{−} = 0 .

Again, the vacuum vector Ω is required to be cyclic with respect to the algebra F_{0} generated by ˆ1/^{\}D and the smeared field operators F̂^{μν}(φ) , φ ∈ S(ℝ^{4}) , μ, ν ∈ {0, 1, 2, 3}. Finally, the field operators have to fulfill the free Maxwell equations:^{1}

∂_{ν}F̂^{μν}(x) = 0 ,  (4.1)
∂^{ν}ϵ_{μναβ}F̂^{αβ}(x) = 0 .  (4.2)

Essential Uniqueness of the Wightman Theory

Using the results of (Oksak and Todorov, 1969) and (Pohlmeyer, 1969) one man prove^{2} the following variant of the Jost-Schroer theorem (Theorem 2.2.18):

Theorem 4.1.1 The Wightman theory of the free electromagnetic field, as described above^{3} is unique up to unitary equivalence and up to some common constant factor of the field operators, if D is chosen to be the smallest linear subspace of H containing Ω and being invariant under all the smeared field operators.

A realization of the Wightman theory of the free electromagnetic field will be given in 4.1.3.

Draft, March 26, 2009

1As usual, we define

ϵ_{μναβ} =^{def} \{ \begin{matrix}+1 & if (μ, ν, α, β) is an even permutation of (0, 1, 2, 3) , \\ −1 & if (μ, ν, α, β) is an odd permutation of (0, 1, 2, 3) , \\ 0 & else . \end{matrix}

2See also (Fredenhagen, 2001, III.4).

3As pointed out by D. Buchholz (private communication) it is sufficient to require the validity of the wave equation for every component of the field tensor rather than the full set of Maxwell equations.

Vacuum Fluctuations

Although we will see that

⟨Ω | F̂^{μν}(x)Ω⟩ = 0 ,

as to be expected, cyclicity of Ω implies that the 2-point functions

⟨Ω | F̂^{μν}(x)F̂^{μν}(y)Ω⟩

cannot all vanish.^{4} Therefore the variance of the field strength is nonzero already in the vacuum state. This is due to vacuum fluctuations, which also cause spontaneous emission of photons from atoms in excited states (see, e.g. (Baym, 1969, S. 278 ff.)) and may be experimentally verified by the so-called Casimir effect (see, e.g. (Itzykson and Zuber, 1980a, Section 3-2-4)).

The permanent presence of perturbations like those connected with vacuum fluctuations is one of the main problems of quantum field theory.

4.1.2 Problems With the Quantized Potentials

Desirable Properties

Just as in classical electrodynamics it is convenient – and for coupling to the charged matter fields also necessary – to introduce potentials Â^{μ}(x) which we should like to fulfill the following requirements:^{5}

(i) The Â(x, t) are operator-valued tempered generalized functions with invariant dense domain D_{A} ⊂ H.

(ii) There is a nondegenerate^{6} continuous sesquilinear form (. | .) on H w.r.t. which

Draft, March 26, 2009

4This fact implies interesting restrictions on the joint measurability of different field components (Bohr and Rosenfeld, 1933) — a necessary supplement to the Heisenberg uncertainty relations of ordinary quantum mechanics.

5See also (Strocchi, 1977) and (Lücke, edyn, Sect. 2.3.2). Of course, it would be nice to have D_{A} = D_{F} = D and (. |) = ⟨. | .⟩. By Strocchi’s Theorem (Theorem 4.1.2), however, this is not possible. While it is easy to see that quantized potentials can always be constructed in some bigger space with indefinite metric – even in the noninteracting case (Bongaarts, 1977) – there are almost no physical hints on which properties can be expected for such auxiliary field operators.

6In general a quadratic form (. | .) on H × H is called nondegenerate if the mapping

H ∋ Ψ ⟼ (Ψ | .) ∈ H′

is a bijection. For Hilbert spaces H this is equivalent to:

(Φ | Φ′) = 0 ∀ Φ′ ∈ H ⟹ Φ = 0 .

142 CHAPTER 4. QUANTUM ELECTRODYNAMICS

the Â(x, t) are hermitian:^{7}

(Φ_{1} | Â(x, t)Φ_{2}) = (Â(x, t)Φ_{1} | Φ_{2}) ∀Φ_{1}, Φ_{2} ∈ D_{A} .  (4.3)

(iii) There is a representation V̂ (a, Λ) of P_{+}^{↑} with

D_{V̂ (a,Λ)} = R_{V̂ (a,Λ)} = D_{A}  (4.4)

which is strongly continuous w.r.t. ⟨. | .⟩, unitary w.r.t. (. | .) , i.e.

(Φ_{1} | V̂ (a, Λ)Φ_{2}) = (V̂ (a, Λ)^{−1}Φ_{1} | Φ_{2}) ∀ Φ_{1}, Φ_{2} ∈ D_{A} ,  (4.5)

and transforming the Â(x, t) according to

V̂ (a, Λ)^{−1}Â(x, t)V̂ (a, Λ) = (Λ_{A})_{ν}^{μ} Â^{ν} (Λ_{A}^{−1}(x − a)) .  (4.6)

(iv) There is a vector Ω ∈ D_{A}, unique up to a phase factor, fulfilling

(Ω | Ω) = 1 and V̂ (a, Λ)Ω = Ω ∀ (a, Λ) ∈ P_{+}^{↑} .  (4.7)

(v)

x×y ⟹ [Â(x, t), Â^{ν}(y)]_{−} = 0 .  (4.8)

(vi)

□ Â^{ν}(x) = 0 ∀ ν ∈ {0…. , 3} .  (4.9)

(vii) There is a linear subspace D_{F} of D_{A} fulfilling the conditions

(Φ_{1} | ∂_{μ}Â(x, t)Φ_{2}) = 0 ∀ Φ_{1}, Φ_{2} ∈ D_{F} ,  (4.10)
F̂ μν ; Â(x)D_{F} ⊂ D_{F} ⊃ V̂ (a, Λ)D_{F} ∀ μ, ν ∈ {0, …, 3} , (a, Λ) ∈ P_{+}^{↑} , (4.11)

and

Ω ∈ D_{F} (⊂ D_{A} ⊂ H) .  (4.12)

where

F̂ μν ; Â(x) =^{def} ∂^{μ}Â^{ν}(x) − ∂^{ν}Â(x, t) ,  (4.13)

(viii)

∫ (Φ_{1} | V̂ (a, 1l_{4})Φ_{2}) e^{−ip^{μ}a_{μ}}da = 0 ∀ p ∈ ℝ^{4} \ \overline{V_{+}} , Φ_{1}, Φ_{2} ∈ D_{A} . (4.14)

Draft, March 26, 2009

7Of course, we should like to take (. | .) = ⟨. | .⟩. However, as will be shown in Corollary 4.1.4, this would be in contradiction to the other assumptions made below.

(ix)

\overline{Z_{A}}^{⟨.|.⟩} = H ,

where Z_{A} denotes the smallest linear subspace of D_{A} containing Ω and being invariant under all Â(x, t) .

(x) For every Φ ∈ D_{F} and every ϵ > 0 there is a Φ′ ∈ Z_{F} with

(Φ − Φ′ | Φ − Φ′) < ϵ ,

where Z_{F} denotes the smallest linear subspace of D_{F} containing Ω and being invariant under all F̂ μν ; Â(x) .

If (. | .) were positive definite ,

H_{F} =^{def} \overline{D_{F}}^{(.|.)} , \overline{V̂ (a, Λ)}^{(.|.)} , Ω and F̂^{μν}

would give a Wightman theory of the free electromagnetic field.

No-Go Theorem

Theorem 4.1.2 (Strocchi) Unless all F̂ μν ; Â(x) vanish, conditions (i)–(x) imply

∂_{μ}F̂ μν ; Â(x)Ω ≠ 0 .  (4.15)

Proof: See (Strocchi, 1970).

Lemma 4.1.3 Let D be a (complex) linear space with positive semi-definite sesquilinear form (. | .) and let B̂_{1}, B̂_{2} ∈ L(D, D) fulfill the condition

(Φ′ | B̂_{1}Φ) = (B̂_{2}Φ′ | Φ) ∀ Φ, Φ′ ∈ D .

Then

B̂_{1}D_{00} ⊂ D_{00} =^{def} {Φ ∈ D : (Φ | Φ) = 0} .

Proof: By means of Schwartz’ inequality (see (Strocchi and Wightman, 1974); Lemma 2.2 and application).

Corollary 4.1.4 Unless all F̂ μν ; Â(x) vanish, conditions (i)–(x) imply that (. | .) can be neither^{8} positive definite on D_{F} nor positive semi-definite on D_{A} if

(Φ | ∂_{μ}F̂ μν ; Â(x) Φ) = 0 ∀ Φ ∈ D_{F} .

Draft, March 26, 2009

8By (4.9)–(4.13) positive definiteness on D_{F} would imply ∂_{μ}F̂ μν ; Â(x) = 0 , in contradiction to Theorem 4.1.2. The case of positive semi-definiteness on D_{A} may be reduced to the case of positive definiteness via factorization, by Lemma 4.1.3.

Gupta-Bleuler Generalization

Gupta (Gupta, 1950) and Bleuler (Bleuler, 1950) realized that, fortunately, it is sufficient to postulate positive semi-definiteness of (. | .) on D_{F} :

(Φ | Φ) ≥ 0 ∀ Φ ∈ D_{F} .  (4.16)

Then assumptions (i)–(ix) guarantee that factorization^{9} of D_{F} , V̂ (a, Λ)/^{\}D_{F} and

F̂ μν ; Â(x)/^{\}D_{F} by

D_{00} =^{def} {Φ ∈ D_{F} : (Φ | Φ) = 0} ,  (4.17)

gives a Wightman theory of the free electromagnetic field (compare last part of 4.1.3).

4.1.3 Gupta-Bleuler Construction

Quantized Electromagnetic Potentials

As domain D_{A} for the quantized potentials we choose the set of all truncated sequences

A_{GB} = {A_{0}, A_{1}, …, A_{n}, 0, 0, …} ,

where

A_{0} ∈ ℂ , A_{n} ˆ= {a^{μ_{1},...,μ_{n}}(p_{1}, …, p_{n})}_{μ_{j}=0,1,2,3} ,

and

a^{μ_{1},...,μ_{n}}(p_{1}, …, p_{n}) = a^{μ_{π1},...,μ_{πn}}(p_{π1}, …, p_{πn}) ∈ S (ℝ^{3n}) ∀ π ∈ S_{n} . (4.18)

As inner product we choose

⟨A_{GB} | Ǎ_{GB}⟩

=^{def} \overline{A_{0}}Ǎ_{0} + ∑_{n=1}^{∞} ∑_{μ_{1},...,μ_{n}=0}^{3} ∫ \overline{a^{μ_{1},...,μ_{n}}(p_{1}, …, p_{n})}ǎ^{μ_{1},...,μ_{n}}(p_{1}, …, p_{n}) \frac{dp_{1}}{2 |p_{1}|} ⋯ \frac{dp_{n}}{2 |p_{n}|} .

(4.19)

Then H is chosen to be the completion of D_{A} with respect to this inner product. Similarly to (2.23) resp. (2.32) we define annihilation operators â^{μ}(p) by^{10}

(â^{μ}(p)A_{GB})_{0} =^{def} a^{μ}(p)
(â^{μ}(p)A_{GB})^{μ_{1},...,μ_{n−1}}(p_{1}, …, p_{n−1}) =^{def} \sqrt{n} a^{μ,μ_{1},...,μ_{n−1}}(p, p_{1}, …, p_{n−1}) for n > 1

(4.20)

Draft, March 26, 2009

9Factorization is possible thanks to Lemma 4.1.3.

10We use the identification (A_{GB})^{μ_{1},...μ_{ν}} (p_{1}, …, p_{ν}) = a^{μ_{1},...μ_{ν}} (p_{1}, …, p_{ν}) .

resp. creation operators â^{μ}(p)^{†} by^{11}

(â^{μ}(p)^{†}A_{GB})_{0} =^{def} 0 ,
(â^{μ}(p)^{†}A_{GB})^{μ_{1}} (p_{1}) =^{def} −2 |p_{1}| g^{μμ_{1}}δ(p − p_{1})A_{0} ,
(â^{μ}(p)^{†}A_{GB})^{μ_{1},...,μ_{n+1}}(p_{1}, …, p_{n+1}) =^{def} −\sqrt{\frac{1}{n+1}} ∑_{j=1}^{n+1} 2 |p_{j}| g^{μμ_{j}}δ(p − p_{j})·
·a^{μ_{1},...,μ_{j}\,...,μ_{n+1}}(p_{1}, …, p_{j} \ , …, p_{n+1}) .

(4.21)

Then, thanks to the commutation relations^{12}

[â^{μ}(p), â^{μ′}(p′)^{†}]_{−} = −2 |p| g^{μμ′}δ(p − p′) ,

[â^{μ}(p), â^{μ′}(p′)]_{−} = [â^{μ}(p)^{†}, â^{μ′}(p′)^{†}]_{−} = 0 (4.22)

the quantized potentials^{13}

Â_{GB}^{μ}(x) =^{def} Â_{GB}^{μ (+)}(x) + Â_{GB}^{μ (−)}(x) ,  (4.23)
Â_{GB}^{μ (+)}(x) =^{def} \sqrt{ζ} (2π)^{−3/2} ∫_{p_{0}=+|p|} â^{μ}(p)e^{− \frac{i}{ℏ}p^{ν}x_{ν}} \frac{dp}{2 |p |} ,  (4.24)

Â_{GB}^{μ (−)}(x) =^{def} \sqrt{ζ} (2π)^{−3/2} ∫_{p_{0}=−|p|} â^{μ}(−p)^{†}e^{− \frac{i}{ℏ}p^{ν}x_{ν}} \frac{dp}{2 |p |} ,  (4.25)

obeying condition (i) of 4.1.2, fulfill the commutation relations^{14}

[Â_{GB}^{μ (+)}(x), Â_{GB}^{μ′ (−)}(x′)]_{−} = −ζ g^{μμ′}iΔ_{0}^{(+)}(x − x′) ,
[Â_{GB}^{μ (+)}(x), Â_{GB}^{μ′ (+)}(x′)]_{−} = [Â_{GB}^{μ (−)}(x), Â_{GB}^{μ′ (−)}(x′)]_{−} = 0 ,

(4.26)

[Â_{GB}^{μ}(x), Â_{GB}^{μ′}(x′)]_{−} = −ζ g^{μμ′}iΔ_{0}(x − x′) .  (4.27)

(4.27) directly implies condition (v) of 4.1.2. By (4.23)–(4.25) also condition (vi) of 4.1.2 is fulfilled. According to (4.20)–(4.25) we have to define

(A_{GB} | A′_{GB}) =^{def} ⟨A_{GB} | η̂A′_{GB}⟩ ,  (4.28)

where

(η̂A_{GB})^{μ_{1},...,μ_{n}} (p_{1}, …, p_{n}) =^{def} (−g_{μ_{1}ν_{1}}) ⋯ (−g_{μ_{n}ν_{n}})a^{ν_{1},...,ν_{n}}(p_{1}, …, p_{n}) ,
(η̂A_{GB})_{0} =^{def} A_{0} ,

(4.29)

in order to fulfill (4.3) and (Ω | Ω) = 1 for the vector

Ω =^{def} {1, 0, 0, …} .  (4.30)

Draft, March 26, 2009

11The use of −g^{μμ_{1}} instead of δ_{μμ_{j}} is necessary to yield the vector transformation property (4.6) with (4.31) for the Gupta-Bleuler potentials Â^{μ}(x) = Â_{GB}^{μ}(x) .

12Compare (2.34).

13Compare (2.39), (2.40), and (2.43). For the physically correct choice of ζ see Exercise 61. 14Compare (2.46), (2.47), and (2.62).

Since η̂ is unitary^{15} w.r.t. ⟨. | .⟩ the sesquilinear form (. | .) , defined this way, fulfills condition (ii) of 4.1.2, indeed. By

(V̂ (a, Λ)A_{GB})_{0} =^{def} A_{0} ,

(V̂ (a, Λ)A_{GB})^{μ_{1},...,μ_{n}} (p_{1}, …, p_{n})

=^{def} e^{i(p_{1}+...+p_{n})a}Λ_{ν_{1}}^{μ_{1}} ⋯ Λ_{ν_{n}}^{μ_{n}}a^{ν_{1},...,ν_{n}} (\overrightarrow{Λ^{−1}p_{1}}, …, \overrightarrow{Λ^{−1}p_{n}})_{|_{p^{0}_{j}=|p_{j}|}}

(4.31)

(compare (2.21)), since (4.19) and (4.28) imply

(A_{GB} | A′_{GB}) = \overline{A_{0}}A′_{0}+∑_{n=1}^{∞} ∫ \overline{a^{μ_{1},...,μ_{n}}(p_{1}, …, p_{n})}a′_{μ_{1},...,μ_{n}}(p_{1}, …, p_{n}) \frac{dp_{1}}{2 |p_{1}|} ⋯ \frac{dp_{n}}{2 |p_{n}|} ,

(4.32)

we get a representation of P_{+}^{↑} fulfilling conditions (iii), (iv), and (viii) of 4.1.2.

Warning: For v ≠ 0 , the operators V̂ (Λ_{v}, 0) are unbounded w.r.t. ⟨. | .⟩.

Proof For every v ≠ 0 there is a 1-photon state vector Â_{GB} fulfilling

⟨Â_{GB} | Â_{GB}⟩ = 1 ,

∥V̂ (Λ_{v}, 0) A_{GB}∥^{2} =^{def} ⟨V̂ (Λ_{v}, 0) Â_{GB} | V̂ (Λ_{v}, 0) Â_{GB}⟩ = 1 + ϵ , ϵ > 0 .

Then, for

(A_{GB}^{N})^{μ_{1},...,μ_{n}} (p_{1}, …, p_{n})
=^{def} \{ \begin{matrix}\sqrt{\frac{ϵ}{N} \frac{1}{(1 + ϵ)^{n+1}}} â^{μ_{1}}(p_{1}) ⋯ â^{μ_{n}}(p_{n}) & if n < 1 + N^{2} \frac{1 + ϵ}{ϵ} , \\ 0 & else , \end{matrix}

we have^{16}

∥A_{GB}^{N}∥^{2} < \frac{1}{N} , ∥V̂ (Λ_{v}, 0) A_{GB}^{N}∥^{2} > N .

Field Operators

In view of (4.15) the simplest possibility to fulfill also condition (vii) of 4.1.2 is^{17}

D_{F} =^{def} {Φ ∈ D_{A} : ∂_{μ}Â_{GB}^{μ (+)}(x)Φ = 0} .  (4.33)

16Note that ∑_{n=0}^{∞} \frac{ϵ}{(1 + ϵ)^{n+1}} = 1 .

Draft, March 26, 2009 15W.r.t. (. | .) , of course, η̂ is unbounded.

17Then ∂_{μ}Â_{GB}^{μ}(φ)D_{F} ⊂ D_{00} ∀ φ ∈ S(ℝ^{4}) .

Then^{18}

D_{F} = D_{Coul} + D_{00} , where
D_{Coul} =^{def} {Φ ∈ D_{F} : Â_{GB}^{(+)0}(x)Φ = 0} ,

(4.34)

and

(Φ | Φ′) = 0 ∀ Φ ∈ D_{F} , Φ′ ∈ D_{00} .  (4.35)

Since (. | .) and ⟨. | .⟩ coincide on D_{Coul} this shows that (. | .) is positive semi-definite on D_{F} .

Sketch of proof for (4.34):

A_{GB} − P̂_{Coul}A_{GB} ∈ D_{00} with P̂_{Coul} = ∏_{j=1}^{∞}(1 − T̂_{j}) ,

where

(T̂_{j}A_{GB})^{μ_{1},...,μ_{n}} (p_{1}, …, p_{n})
=^{def} \{ \begin{matrix}\frac{p_{j}^{μ_{j}}}{p_{j}^{0}} a^{μ_{1},..., _{0}^{j. place},...,μ_{n}}(p_{1}, …, p_{n})_{|_{p^{0}_{j}=|p_{j}|}} & for j = 1, …, n, \\ 0 & for j > n. \end{matrix}

Sketch of proof for (4.35):

⟨A_{GB} | A_{GB}⟩ = (P̂_{Coul}A_{GB} | P̂_{Coul}A_{GB}) ∀ A_{GB} ∈ D_{F} ,
⇝ A_{GB} ∈ D_{00} ⟺ A_{GB} = (1 − P̂_{Coul})A_{GB} .

Exercise 60 Show that ∂_{μ}Â_{GB}^{μ (+)}(x)Φ commutes with all the smeared field operators F̂_{BG}^{αβ}(x) on D_{A} and hence the latter leave D_{F} invariant.

One easily checks^{19} that for every Φ ∈ D_{Coul} and for every ϵ > 0 there is a Φ′ ∈ Z_{F}^{Coul} with (Φ − Φ′ | Φ − Φ′) < ϵ, where Z_{F}^{Coul} denotes the smallest linear subspace of D_{Coul} containing Ω and being invariant under all F̂ jk ; Â_{GB}(x) with j, k ∈ {1, 2, 3}. This is because in D_{Coul} the topology induced by (. | .) is equivalent to that induced by

| ; ∥A_{GB}∥ ; |^{2} =^{def} |A_{0}|^{2} + ∑_{n=1}^{∞} ∑_{μ_{1},...,μ_{n}=1}^{3} ∫ |a^{μ_{1},...,μ_{n}}(p_{1}, …, p_{n})|^{2} \frac{dp_{1}}{2 |p_{1}|} ⋯ \frac{dp_{n}}{2 |p_{n}|} .

Therefore (4.34)/(4.35) imply that also condition (x) of 4.1.2 is fulfilled for F̂ μν ; Â_{GB}(x) .

Draft, March 26, 2009

18D_{00} was defined in (4.17).

19Note that

p · a ⟹ a = p × \frac{a × p}{|p|^{2}} .

Transition to the Wightman Theory

Let D′ denote the set of all equivalence classes

D′ =^{def} {[A_{GB}] : A_{GB} ∈ D_{F} }  (4.36)

of D_{F} corresponding to the equivalence relation

A_{GB} ∼ A′_{GB} ⟺^{def} A_{GB} − A′_{GB} ∈ D_{00} .  (4.37)

Then, according to (4.35),

⟨[A_{GB}] | [A′_{GB}]⟩ =^{def} (A_{GB} | A′_{GB})  (4.38)

does not depend on the choice of representatives A_{GB}, A′_{GB} ∈ D_{F} and, by condition (ii) of 4.1.2 and (4.34)/(4.35), and defines a positive definite inner product on D′. Hence, the completion H′ of D′ w.r.t. ⟨. | .⟩ is a Hilbert space. The appropriate representation of P_{+}^{↑} results from continuous extension of the operators^{20}

V̂′(a, Λ) [A_{GB}] =^{def} [V̂ (a, Λ)A_{GB}]  (4.39)

onto all of H′. Then, up to a constant factor,

Ω′ =^{def} [Ω]  (4.40)

is the only element of H′ that is invariant under all V̂′(a, Λ) . The spectrum condition follows from condition (viii) of 4.1.2. To summarize:

For H′ , V̂′(a, Λ) , and Ω′ all requirements of the zeroth Wightman axiom are fulfilled.

Next, we define the field operators:^{21}

F̂^{′μν}(φ)Φ =^{def} [F̂ μν ; Â_{GB}(φ)A_{GB}] for Φ = [A_{GB}] ∈ D′ .  (4.41)

Now all the Wightman axioms for the free electromagnetic field are fulfilled, as may be easily derived from conditions (i)–(x) of 4.1.2.

Corollary 4.1.5 There is a Wightman theory of the free electromagnetic field, given by H′ , V̂′(a, Λ) , Ω′, and F̂^{′μν}(x) , with

⟨Ω′ | F̂^{′μ_{1}ν_{1}}(x_{1}) ⋯ F̂^{′μ_{n}ν_{n}}(x_{n})Ω′⟩ = (Ω | F̂ μ_{1}ν_{1} ; Â_{GB}(x_{1}) ⋯ F̂ μ_{n}ν_{n} ; Â_{GB}(x_{n})Ω)

for all μ_{1}, ν_{1}, …, μ_{n}, ν_{n} ∈ {0, …, 3}.

Draft, March 26, 2009

20Definition (4.39) is allowed by (4.11, (4.5) and Lemma 4.1.3.

21By (4.11), (4.3), and Lemma 4.1.3 this definition is allowed. The F̂ μν ; Â_{GB}(x) were defined in (4.13).

Final remark: ∂_{μ}Â_{GB}^{μ (−)}(x) = 0 does not hold on all of D_{F} . Hence

∂F̂ μν ; Â_{GB}(x) ≠ 0

(compare Theorem 4.1.2). However, the following statements are true:

∂^{ν}ϵ_{μναβ}F̂ αβ ; Â_{GB}(x) =_{(4.13)} 0 on D_{A} ,
∂_{μ}F̂ μν ; Â_{GB}(x) =_{(4.9)/(4.13)} ∂^{ν}∂_{μ}Â_{GB}^{μ}(x) ,
∂_{μ} ⟨A_{GB} | F̂ μν ; Â_{GB}(x) | A′_{GB}⟩ =_{(4.10)} 0 ∀ A_{GB}, A′_{GB} ∈ D_{F} .

4.1.4 Gupta-Bleuler Observables

Agreement: By Gupta-Bleuler observable we will always mean a (. | .)-hermitian operator Â ∈ L(D_{A}, D_{A}) leaving D_{F} invariant for which Â′, defined by

Â′ [A_{GB}] =^{def} [Â A_{GB}] ∀ A_{GB} ∈ D_{F}

(recall Lemma 4.1.3), is an essentially selfadjoint operator of the Wightman theory constructed as described in 4.1.3. We will call Gupta-Bleuler observables Â and B̂ equivalent if they induce the same transformation of the equivalence classes:

Â ∼ B̂ ⟺^{def} ([Â A_{GB}] = [B̂ A_{GB}] ∀ A_{GB} ∈ D_{F} ) .

Gauge Transformations

By gauge transformation in^{22} {H, ⟨. | .⟩ , η̂, D_{A}, Ω} we will mean transition from one quantized potential Â(x, t) to another quantized potential Â′^{μ}(x) without changing the n-point functions of the corresponding field strength operators:

(Ω | F̂ μ_{1}ν_{1} ; Â(x_{1}) ⋯ F̂ μ_{n}ν_{n} ; Â(x_{n})) = (Ω | F̂ μ_{1}ν_{1} ; Â′(x_{1}) ⋯ F̂ μ_{n}ν_{n} ; Â′(x_{n})) .

Certainly, the latter is guaranteed if the connection between Â(x, t) and Â^{′μ}(x) is given by Â^{′μ}(x) = Â(x, t) + ∂^{μ}χ̂(x) , where χ̂(x) is a (. | .)-hermitian tempered field in {H, ⟨. | .⟩} with invariant domain D_{A}. Not quite that simple is the gauge transformation

Â_{GB}^{μ}(x) ⟶ Â_{Coul}^{μ}(x) =^{def} Â_{GB}^{μ}(x) + χ̂^{μ}(x)  (4.42)

Draft, March 26, 2009

22For a more general classification of gauge transformations see (Strocchi and Wightman, 1974).

with

χ̂^{μ}(x) =^{def} \sqrt{ζ} (2π)^{−3/2} ∫_{p^{0}=|p|} (ĉ^{μ}(p)e^{−ip^{μ}x_{μ}} + ĉ^{μ}(p)^{†}e^{+ip^{μ}x_{μ}}) \frac{dp}{2p^{0}} ,

ĉ^{μ}(p) =^{def} \{ \begin{matrix}−â^{0}(p) & for μ = 0 , \\ −\frac{p^{μ}}{|p|^{2}} ∑_{j=1}^{3} p^{j}â^{j}(p) & for μ = 1, 2, 3 . \end{matrix}

(4.43)

A simple calculation shows that

F̂ jk ; Â_{GB}(x) = F̂ jk ; Â_{Coul}(x) for j, k ∈ {1, 2, 3}

and^{23}

(F̂ 0j ; Â_{GB}(φ) − F̂ 0j ; Â_{Coul}(φ)) D_{F} ⊂ D_{00} ∀ φ ∈ S(ℝ^{4}) .  (4.44)

Hence, indeed, (4.42)/(4.43) is a gauge transformation. The index “Coul” is to indicate validity of the equations

Â_{Coul}^{0}(x) = 0 ,  (4.45)
∑_{j=1}^{3} ∂_{j} Â_{Coul}^{j}(x) = 0 .  (4.46)

Restricting the Â_{Coul}^{j}(x) to the domain D_{Coul} (compare (4.34)), which they leave invariant and on which (. | .) = ⟨. | .⟩, we get quantized potentials of the free electromagnetic field in a pre-Hilbert space with positive definite metric (radiation gauge, the special case (4.45) of the Coulomb gauge, characterized by (4.46)).

The price to be paid for positive definiteness of the metric is invalidity of the covariance condition (4.6) – for every representation V̂ (a, Λ) – as well as of the locality condition (4.8). This follows from (4.45) and the structure of the 2-point function^{24}

(Ω | Â_{Coul}^{j}(x)Â_{Coul}^{k}(y)) = iζ (δ_{jk} − \frac{∂^{j}∂^{k}}{Δ} ) Δ_{0}^{(+)}(x − y) .  (4.47)

(4.44) shows that, for j ∈ {1, 2, 3} and real-valued φ ∈ S(ℝ^{4}) , F̂_{GB}^{0j}(φ) and F̂_{Coul}^{0j}(φ) are equivalent Gupta-Bleuler observables, although not identical.

Draft, March 26, 2009

23Note that p^{0} ĉ^{k}(p) − p^{k} ĉ^{0}(p) = −\frac{p^{k}}{p^{0}} p_{μ}â^{μ}(p) for k ∈ \{1, 2, 3\} and that, by (4.33) and (4.26),

∂_{μ}Â_{GB}^{μ}(φ)D_{F} ⊂ D_{00} .

24The \frac{∂^{j}∂^{k}}{Δ} -term spoils commutativity.

Special Observables

Using the commutation relations (4.22), one easily checks that

P̂_{GB}^{α} = ∫_{p^{0}=|p|} p^{α} (−â^{μ}(p)^{†} â_{μ}(p)) \frac{dp}{2p^{0}} .  (4.48)

is the generator of space-time translations:

i [Â_{GB}^{α}, Â_{GB}^{(±)}^{μ}(x)]_{−} = ∂^{α}Â_{GB}^{(±)}^{μ}(x) .  (4.49)

Therefore, P̂_{GB}^{α} is the Gupta-Bleuler observable of linear 4-momentum specified by

P̂_{GB}^{α} Ω = 0 .  (4.50)

This is consistent with the interpretation of

\frac{−1}{2 |p|}â^{μ}(p)^{†} â_{μ}(p)

as a Gupta-Bleuler observable for

lim_{Ṽ_{p}→\{p\}} \frac{number of physical photons with momentum p′ ∈ Ṽ_{p}}{momentum space volume Ṽ_{p}} .

Remark: Given φ ∈ S(ℝ^{3}) , we always have

∫ â^{μ}(p)^{†}â_{μ}(p) φ(p) dp D ⊂ D for D = D_{F} as well as for D = D_{00} ,

but not â^{μ}(p)^{†} â_{μ}(p) = 0 on D_{00} .

Exercise 61 Show that the Gupta-Bleuler observables c P̂_{GB}^{0} and

Ê_{GB}(x^{0}) =^{def} \frac{ϵ′_{0} c^{′2}}{2} ∫ :(g_{αβ}g_{γδ}F̂ αγ ; Â_{GB}(x)F̂ βδ ; Â_{GB}(x) + 4 g_{μν}F̂ 0μ ; Â_{GB}(x)F̂ 0ν ; Â_{GB}(x)): dx

are equivalent (not identical) if

ζ = \frac{1}{ϵ′_{0}} (\frac{ℏ}{c′} )^{2} .  (4.51)

In the following ζ will always be assumed given by (4.51), with ϵ′_{0} and c′ specified by the used system of units^{25} (see Appendix A.3 of (Lücke, edyn)).

Draft, March 26, 2009

25In Heaviside units: ϵ′_{0} = c′ = 1 .

Similarly to (4.49) one may show that

Ĵ_{GB} = L̂_{GB} + Ŝ_{GB} , where
L̂_{GB} =^{def} ∫_{p^{0}=|p|} −â^{μ}(p)^{†} p × \frac{1}{i}∇_{p} â_{μ}(p) \frac{dp}{2p^{0}} ,
Ŝ_{GB} =^{def} i ∫_{p^{0}=|p|} â(p)^{†} × â(p) \frac{dp}{2p^{0}} ,

(4.52)

is a generator of spatial rotations:

[L̂_{GB}, Â_{GB}^{(±)}^{μ}(x)] = i x × ∇_{x}Â_{GB}^{(±)}^{μ}(x) ,
[e · Ŝ_{GB}, \overrightarrow{Â_{GB}^{(±)}}(x)] = i e × \overrightarrow{Â_{GB}^{(±)}}(x) .

(4.53)

Hence Ĵ_{GB} is the Gupta-Bleuler observable of total angular momentum specified by

Ĵ_{GB}Ω = 0 .

Warning: L̂_{GB} and Ŝ_{GB} themselves are not Gupta-Bleuler observables since they do not leave D_{F} invariant!

This is consistent with the interpretation of

\frac{1}{2 |p|} (−â^{μ}(p)^{†} p × \frac{1}{i}∇_{p} â_{μ}(p) + iâ(p)^{†} × â(p))

as Gupta-Bleuler observable of

lim_{Ṽ_{p}→\{p\}} \frac{total angular momentum of all photons with momentum p′ ∈ Ṽ_{p}}{momentum space volume Ṽ_{p} .}

Hence, the so-called helicity operator

λ̂_{GB} =^{def} i ∫_{p^{0}=|p|} \frac{p}{|p|} · (â(p)^{†} × â(p)) \frac{dp}{2p^{0}}  (4.54)

is a Gupta-Bleuler observable for the component of angular momentum along three-momentum.

n-photon states are called those represented by the elements of D_{A} which are of the form

A_{GB} = {0, …, 0, A_{n}, 0, …}

(compare 4.1.3). The Gupta-Bleuler observable for the number of physical photons is

N̂_{GB} =^{def} − ∫_{p^{0}=|p|} â^{μ}(p)^{†}â_{μ}(p) \frac{dp}{2p^{0}}  (4.55)

(recall the comment to (4.48)).

Remark: The expectation values for the field strengths in physical nphoton states all vanish!

Every 1-photon state A_{GB} = {0, A_{1}, 0, 0, …} has a unique decomposition

A_{GB} = A_{GB}^{tr} + A_{GB}^{lo} + A_{GB}^{ti}

with:

A_{GB}^{tr} ∈ D_{Coul} , i.e. p^{j} (A_{GB}^{tr})^{j} (p) = (A_{GB}^{tr})^{0} (p) = 0 ,
A_{GB}^{lo} ∈ D_{00} ,  i.e. (A_{GB}^{lo})^{j} (p) = \frac{p^{j}}{|p|} (A_{GB}^{lo})^{0}(p) ,
(A_{GB}^{ti})^{j} (p) = 0 for j = 1, 2, 3 .

One says:

A_{GB}^{tr} describes a transverse photon, A_{GB}^{lo} a longitudinal photon, and A_{GB}^{ti} a time-like photon.

Of course, the 1-photon state vector A_{GB} corresponds to a physical state only if A_{GB}^{ti} = 0 . In this case A_{GB} and A_{GB}^{tr} correspond to the same physical state.

The commutation relations

i [â^{0}(p)^{†}, λ̂_{GB}]_{−} = 0 , i [e · â(p)^{†}, λ̂_{GB}]_{−} = \frac{p}{|p|} · (e × â(p)^{†})  (4.56)

show^{26} that the longitudinal (as well as the time-like) 1-photon state vectors are eigenvectors of λ̂_{GB} with eigenvalue 0 . Moreover, we see that the transversal 1photon state vectors of the form

∫_{p^{0}=|p|} φ(p) (e_{1}(p) · â(p)^{†} − iσe_{2}(p) · â(p)^{†}) \frac{dp}{2p^{0}} Ω ,

where: \{e_{1}(p), e_{2}(p), \frac{p}{|p|}\} right-handed orthonormal basis of ℝ^{3} ∀ p ≠ 0 ,

(4.57)

with σ ∈ {+1, −1} are eigenvectors of λ̂_{GB} with Eigenvalue σ, hence correspond to physical 1-photon states with helicity σ. Obviously, every transversal 1-photon state vector may be written as a linear combination of vectors of the form (4.57).

4.2 The Quantized Free Dirac Field

4.2.1 Lorentz Transformations Characterized via Complex 2 × 2-Matrices

Every selfadjoint complex 2×2-matrix X̃ may be written in the form

X̃ = x^{μ}τ_{μ} = ( \begin{matrix}x^{0} − x^{3} & −x^{1} + ix^{2} \\ −x^{1} − ix^{2} & x^{0} + x^{3} \end{matrix} )  (4.58)

Draft, March 26, 2009

26Note that λ̂_{GB} Ω = 0 .

154 CHAPTER 4. QUANTUM ELECTRODYNAMICS

with suitable x ∈ ℝ^{4} , where

τ_{0} =^{def} ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , τ_{1} =^{def} ( \begin{matrix}0 & −1 \\ −1 & 0 \end{matrix} ) , τ_{2} =^{def} ( \begin{matrix}0 & i \\ −i & 0 \end{matrix} ) , τ_{3} =^{def} ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) . (4.59)

Exercise 62 Show that^{27}

A^{2} = Tr(A) A − det(A) 1l_{2}

holds for all complex 2 × 2-matrices A.

Since^{28}

Tr(τ_{μ}τ_{ν}) = 2 δ_{μν}  (4.60)

the coefficients in (4.58) are

x^{μ} = \frac{1}{2} Tr(τ_{μ}X̃) .  (4.61)

Moreover,^{29}

det (AX̃A^{∗}) = det (X̃) = x^{μ}x_{μ} ∀ A ∈ SL(2, ℂ)  (4.62)

implies

x′^{μ}x′_{μ} = x^{μ}x_{μ} if X̃′ = AX̃A^{∗} .

Hence, by (4.61), for every A ∈ SL(2, ℂ)

x^{μ} ⟼ x′^{μ} = (Λ_{A})_{ν}^{μ} x^{ν} ,  (4.63)

where

(Λ_{A})_{ν}^{μ} =^{def} \frac{1}{2} Tr(τ_{μ}Aτ_{ν}A^{∗}) ,  (4.64)

is a restricted Lorentz transformation,^{30} Λ_{A} ∈ L_{+}^{↑}, depending continuously on A. Note that A ↦ Λ_{A} is a representation of SL(2, ℂ) :

Λ_{A}Λ_{B} = Λ_{AB} ∀A, B ∈ SL(2, ℂ) .  (4.65)

Draft, March 26, 2009

27This is a special case of a well-known theorem by Caley stating for arbitrary n ∈ ℕ that

(c_{M}(z) =^{def} det (M − λ 1l_{n}) for z ∈ ℂ) ⟹ c_{M}(M) = 0

holds for all n × n-matrices M .

28This is a simple consequence of:

\frac{1}{2} [τ^{μ}, τ^{ν}]_{+} = \{ \begin{matrix}δ_{μν} & if μ, ν ∈ \{1, 2, 3\}, \\ τ^{μ} & if ν = 0 \end{matrix}

(compare (4.68)).

29As usual, SL(2, ℂ) denotes the group of all complex 2 × 2-matrices X with det X = 1 .

30Here we identify the active Lorentz transformations with their matrix realizations w.r.t. some fixed orthonormal inertial system. The A ∈ SL(2, ℂ) with Λ_{A} = Λ for given Λ ∈ L_{+}^{↑} are determined in (Macfarlane, 1962). That (4.64) defines restricted Lorentz transformations follows from the fact that for every A ∈ SL(2, ℂ) there is a continuous path connecting Λ_{A} with Λ_{τ_{0}} = 1l_{4} .

4.2. THE QUANTIZED FREE DIRAC FIELD 155

The subgroup of all unitary elements in SL(2, ℂ) is^{31}

SU(2) = \{U_{φ} : φ ∈ ℝ^{3} , |φ| ≤ 2π\} , U_{φ} =^{def} exp (−i τ · \frac{φ}{2}) ,  (4.66)

where the components of τ are the Pauli matrices

τ^{j} =^{def} −τ_{j} for j ∈ {1, 2, 3} ,  (4.67)

fulfilling^{32}

τ^{j}τ^{k} = δ_{jk} + i ∑_{l=1}^{3} ϵ_{jkl}τ^{l}  (4.68)

and hence

(τ · φ)^{2} = |φ|^{2} ∀ φ ∈ ℝ^{3} .

Exercise 63 Prove

U_{φ} = 1l_{2} cos \frac{|φ|}{2} − i\frac{τ · φ}{|φ|} sin \frac{|φ|}{2} ,

(4.66), and:^{33}

U_{φ}_{′}′ = U_{φ} ⟺ (φ′ = φ ∨ |φ′| = |φ| = 2π) .

Now we may easily show (recall Exercise 63), that for the unitary elements

U_{φ} ∈ SU(2) ⊂ SL(2, ℂ)

Draft, March 26, 2009

31Note that det (e^{iJ}) = e^{i Tr(J)} for J = J^{∗}, and: Tr(J) = 0 ⟹_{(4.67)} J = −τ · \frac{φ}{2} .

32As usual, we define

ϵ_{jkl} =^{def} \{ \begin{matrix}+1 & if (j, k, l) is an even permutation of (1, 2, 3) , \\ −1 & if (j, k, l) is an odd permutation of (1, 2, 3) , \\ 0 & else . \end{matrix}

Hence

{x^{0} + x^{1}(iτ^{1}) + x^{2}(iτ^{2}) + x^{3}(iτ^{3}) : x ∈ ℝ^{4}} ,

considered as algebra over ℝ^{1}, is isomorphic to the algebra of quaternions, generated by î =^{def} iτ^{1} and ĵ =^{def} iτ^{2} (as Clifford algebra; see (Choquet-Bruhat et al., 1978, S. 63/64)).

33This shows that SU(2) (w.r.t. its natural topology) – contrary to the rotation group – is simply connected.

the associated transformations Λ_{Uφ} are spatial rotations:^{34}

Λ_{Uφ} = exp (\frac{1}{2} ∑_{j,k,l=1}^{3} ϵ_{jkl}𝕋^{jk}φ^{l}) , where:

𝕋^{23} =^{def} ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 1 & 0 \end{matrix} ) = \frac{d}{dφ} ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & cos φ & − sin φ \\ 0 & 0 & sin φ & cos φ \end{matrix} )_{|_{φ=0}} ,

𝕋^{13} =^{def} ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ) , 𝕋^{12} =^{def} ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) ,

𝕋^{jk} =^{def} −𝕋^{jk} for j > k .

(4.69)

Obviously, Λ_{Uφ} is the matrix (w.r.t. the right-handed basis in which φ = (φ^{1}, φ^{2}, φ^{3})) of a right-handed rotation, by the angle |φ|, around an axis oriented along φ. Therefore:

Λ_{Uφ} = Λ_{Uφ_{′}} ⟺ U_{φ}′ ∈ {+U_{φ}, −U_{φ}} .

Exercise 64 Prove that every positive hermitian element of SL(2, ℂ) is of the form

H_{v} =^{def} exp (−\frac{χ_{v}}{2} τ · \frac{v}{|v|}) = cosh (\frac{χ_{v}}{2} )ˆ1 − sinh (\frac{χ_{v}}{2} ) τ · \frac{v}{|v|}
= \sqrt{\frac{𝔳^{0} + 1}{2}} (τ^{0} − τ · \frac{v}{1 + 1/𝔳^{0}} )

(4.70)

with suitable v ∈ ℝ^{3} , |v| < 1 , where:^{35}

χ_{v} =^{def} tanh^{−1} |v| ≥ 0 , 𝔳^{0} =^{def} \frac{1}{\sqrt{1 − |v|^{2}}} .

Moreover, show for arbitrary v, v′ ∈ ℝ^{3} with |v| , |v′| < 1 that

H_{v′} = H_{v} ⟺ v′ = v .

Draft, March 26, 2009

34It is sufficient to check the generators. The mapping Uφ ↦ Λ_{Uφ} has all the properties of a covering mapping (Pontrjagin, 1958, Def. 45). Therefore (recall Footnote 33), SU(2) is the universal covering group of the rotation group.

35Note that tanh \frac{χ_{v}}{2} = \frac{tanh χ_{v}}{1 + \sqrt{1 − tanh^{2} χ_{v}}} = \frac{|v|}{1 + 1/𝔳^{0}} , cosh = \frac{1}{\sqrt{1 − tanh^{2}}} .

Similarly (recall Exercise 64), for positive hermitian A ∈ SL(2, ℂ) the Λ_{A} correspond to Lorentz boosts:

Λ_{H_{v}} = exp (χ_{v} ∑_{j=1}^{3} 𝕋^{0j} \frac{v^{j}}{|v|}) , where:

𝕋^{01} =^{def} ( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) = \frac{d}{dχ} ( \begin{matrix}cosh χ & sinh χ & 0 & 0 \\ sinh χ & cosh χ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )_{|_{χ=0}} ,

𝕋^{02} =^{def} ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) , 𝕋^{03} =^{def} ( \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ) .

(4.71)

Exercise 65 Prove (4.69) and (4.71).

The so-called polar decomposition (for invertible^{36} A)

A = \underbrace{A\sqrt{A^{−1}A^{∗}^{−1}}}_{unitary} \underbrace{\sqrt{A^{∗}A}}_{pos. herm.}  (4.72)

together with (4.69) and (4.71) shows:^{37}

{Λ_{A} : A ∈ SL(2, ℂ)} = {Λ_{Aφ}Λ_{H_{v}} : φ, v ∈ ℝ^{3} , |v| < 1} = L_{+}^{↑}  (4.73)

(for the last equality see, e.g., equations (2.39) and (2.37) in (Lücke, rel)).

4.2.2 Relativistic Covariance in General

Consider any relativistic quantum theory with state space H. Even if the relativistic symmetries are realized as Wigner symmetries, there is no reason, why these should correspond to a true representation of P_{+}^{↑}. In this case, by Wigner’s theorem (Theorem 1.2.1), one may choose for every (a, Λ) a unitary operator Û(a, Λ) such that^{38}

{λ Û(a_{1}, Λ_{1})Û(a_{2}, Λ_{2})Ψ : λ ∈ ℂ} = {λ Û((a_{1}, Λ_{1}) ∘ (a_{2}, Λ_{2}))Ψ : λ ∈ ℂ}

Draft, March 26, 2009

36For singular A the unitary operator A\sqrt{A^{−1}A^{∗}^{−1}} has to be replaced by a suitable isometric operator (see, e.g. (Bratteli and Robinson, 1979, p. 39).)

37Note that det(A) = 1 ⟹ det (\sqrt{A^{∗}A}) = 1 .

38Here (a_{1}, Λ_{1}) ∘ (a_{2}, Λ_{2}) =^{def} (a_{1} + Λ_{1}a_{2}, Λ_{1}Λ_{2}) is the group operation of P_{+}^{↑}.

holds for every pair of Poincaré transformations (a_{1}, Λ_{1}), (a_{2}, Λ_{2}) ∈ P_{+}^{↑} and all Ψ ∈ H, since the action of Û(a, Λ) is to be interpreted in the sense of (2.6) (with f̂ replaced by an arbitrary element of H). But this only implies existence of a phase function φ((a_{1}, Λ_{1}), (a_{2}, Λ_{2})) ∈ ℝ with

Û ((a_{1}, Λ_{1}) ∘ (a_{2}, Λ_{2})) = e^{iφ((a_{1},Λ_{1}),(a_{2},Λ_{2}))}Û(a_{1}, Λ_{1})Û(a_{2}, Λ_{2}) .  (4.74)

Exercise 66 Show that the phase function φ in (4.74) is a 2-cocycle w.r.t. the trivial representation π(a, Λ) = 1 of P_{+}^{↑} in ℝ, i.e. it fulfills the condition δ^{(2)}φ = 0 , where δ^{(n)} denotes the coboundary operator defined by

\begin{matrix}(δ^{(n)}f)(g_{1}, …, g_{n+1}) =^{def} π(g_{1})f(g_{2}, …, g_{n+1})+ \\ + ∑_{ν=1}^{n}(−1)^{ν}f(g_{1}, …, g_{ν} ∘ g_{ν+1}, …, g_{n+1}) + (−1)^{n+1}f(g_{1}, …, g_{n}) \end{matrix}

((Van Est, 1953, Eq. 25)). Moreover, show that φ may be eliminated by suitable change of phase (Û(g) ⟶ e^{ih(g)}Û(g)) iff φ is a 1-coboundary, i.e. of the form φ = δ^{(1)}h.

In this case, we still have

{λ Û(a_{1}, Λ_{1})Û(a_{2}, Λ_{2}) : λ ∈ ℂ} = {λ Û((a_{1}, Λ_{1}) ∘ (a_{2}, Λ_{2})) : λ ∈ ℂ}

for every pair of Poincaré transformations (a_{1}, Λ_{1}), (a_{2}, Λ_{2}) ∈ P_{+}^{↑}, i.e.

(a, Λ) ⟼ U(a, Λ) =^{def} {λ Û(a, Λ) : λ ∈ ℂ}

is a unitary ray representation of P_{+}^{↑}.

Fortunately, according to Bargmann (Bargmann, 1954) the following holds:^{39}

Theorem 4.2.1 Let U(a, Λ) be a continuous^{40} ray representation of P_{+}^{↑} in H. then there is a continuous unitary representation Û(a, A) of^{41} iSL(2, ℂ) in H with:

U(a, Λ) = {λ Û(a, A) : λ ∈ ℂ} ∀ (a, A) ∈ iSL(2, ℂ) .

Draft, March 26, 2009

39See also (Varadarajan, 2007, Sect. VIII.5) and, for 1 + 2 dimensions, (Grigore, 1993). Ray representations of the Galilei group may always be considered as true representations of some central extension of the Galilei group (Levy-Leblond, 1963).

40Here continuity is to be understood in the sense of Condition (iii) of Definition 1.1.4 .

41As usual, iSL(2, ℂ) denotes the group iSL(2, ℂ) = {(a, A) : A ∈ SL(2, ℂ) , a ∈ ℝ^{4}}

In this sense, iSL(2, ℂ) is more fundamental than P_{+}^{↑} :

If the symmetries which express – in the sense of special relativity

– the equivalence of all inertial systems are Wigner symmetries then they are given by some^{42} continuous unitary representation of iSL(2, ℂ) .

4.2.3 Dirac Particles

Massive Spin-\frac{1}{2} Representations of iSL(2, ℂ)

The simplest non-scalar unitary representations^{43} of iSL(2, ℂ) are of the form^{44}

(Û(a, A)χ) (p) =^{def} e^{(ip^{μ}a_{μ})}A χ(\overrightarrow{Λ_{A^{−1}}p})_{|p^{0}=ω_{p}} ; ω_{p} =^{def} \sqrt{m^{2} + p^{2}} ,  (4.75)

where m is some fixed positive mass and the representation space is the set of all ℂ^{2}-valued wave functions χ(p) with finite Hilbert space norm

∥χ∥ =^{def} \sqrt{∫_{p^{0}=ω_{p}} χ(p)^{∗} \frac{p^{μ}σ_{μ}}{m} χ(p) \frac{dp}{2p^{0}}} , σ_{μ} =^{def} τ^{μ} .  (4.76)

Since

A^{∗}A ≠^{i.a.} 1l_{2} ,

the term p^{μ}σ_{μ} fulfilling

A^{∗}p^{μ}σ_{μ}A = (Λ_{A^{−1}}p)^{μ} σ_{μ}  (4.77)

is needed to make the representation (4.75) unitary.

Sketch of proof for (4.77): With σ^{μ} =^{def} g^{μν}σ_{ν} = τ_{μ} we have

Tr(τ_{λ} (Λ_{A^{−1}}p)^{μ} σ_{μ}) = Tr(τ_{λ}p_{μ}(Λ_{A})_{ν}^{μ}σ^{ν})
=_{(4.60)} 2p_{μ}(Λ_{A})_{λ}^{μ}
=_{(4.64)} Tr(p_{μ}σ^{μ}Aτ_{λ}A^{∗})
= Tr(τ_{λ}A^{∗}p^{μ}σ_{μ}A) for λ = 0, 1, 2, 3 .

Together with (4.60) this implies (4.77).

Draft, March 26, 2009

with multiplication

(a_{1}, A_{1}) ∘ (a_{2}, A_{2}) = (a_{1} + Λ_{A_{1}}a_{2}, A_{1}A_{2}) .

Since SU(2) is simply connected (recall Footnote 33), (4.71) shows that the same is true for iSL(2, ℂ) .

42Of course, not all continuous unitary representations of iSL(2, ℂ) are physically relevant.

43See, e.g. (Streater and Wightman, 1989, p. 15) for a characterization of all irreducible representations of SL(2, ℂ) .

44Note that Λ_{A^{−1}} =_{(4.65)} (Λ_{A})^{−1} .

Exercise 67 Show that^{45} p^{μ}p_{μ} > 0 ⟹ p^{μ}σ_{μ} ≥ 0 and that the representation (4.75) of iSL(2, ℂ) is irreducible.

According to the action of Λ_{U_{φe}} , the components of the the operator Ĵ defined by (4.75) and

e · Ĵ =^{def} i\frac{d}{dφ}Û (0, U_{φe})_{|φ=0} = e · ( \underbrace{ip × ∇_{p}}_{=^{def}L̂} + \underbrace{\frac{1}{2}τ}_{=^{def}Ŝ} )  (4.78)

are interpreted as observables of total angular momentum.

Exercise 68 For Ŝ given by (4.78) prove

A_{φ} (e · Ŝ) Aφ^{−1} = (D̂_{φ} e) · Ŝ ∀ e ∈ ℝ^{3} ,  (4.79)

where D̂_{φ} denotes right-handed rotation by the angle |φ| around an axis oriented along φ, and use this to determine the eigenstates of e · Ŝ for arbitrary e ∈ ℝ^{3} .

(4.78) becomes especially simple for e = \frac{p}{|p|}:

ĥ =^{def} \frac{p}{|p|} · Ĵ = \frac{p}{|p|} · Ŝ helicity operator .  (4.80)

Obviously,

ĥ χ_{±}(p) = ±\frac{1}{2}χ_{±}(p)  (4.81)

holds for

χ_{+}(|p|(cos ϑ cos φ e_{1} + sin ϑ sin φ e_{2} + cos ϑ e_{3})) =^{def} A_{φe_{3}}A_{ϑe_{2}} ( \begin{matrix}1 \\ 0 \end{matrix} )
= ( \begin{matrix}+e^{−i \frac{φ}{2}} cos \frac{ϑ}{2} \\ +e^{+i \frac{φ}{2}} sin \frac{ϑ}{2} \end{matrix} ) ,
χ_{−}(|p|(cos ϑ cos φ e_{1} + sin ϑ sin φ e_{2} + cos ϑ e_{3})) =^{def} A_{φe_{3}}A_{ϑe_{2}} ( \begin{matrix}0 \\ 1 \end{matrix} )
= ( \begin{matrix}−e^{−i \frac{φ}{2}} sin \frac{ϑ}{2} \\ +e^{+i \frac{φ}{2}} cos \frac{ϑ}{2} \end{matrix} )

(4.82)

(compare Exercise 68).

The helicity survives in the limit \frac{m}{|p|} → 0 (to be discussed at the end of this section)

in which, however, the representation of iSL(2, ℂ) becomes reducible (neutrinos!).

Draft, March 26, 2009

45Check det(p^{μ}σ_{μ}) and Tr(p^{μ}σ_{μ}) .

4.2. THE QUANTIZED FREE DIRAC FIELD 161

The wave functions may be written in the form

χ(p) = ∑_{σ=±} b_{σ}(p) χ_{σ}(p) ,  (4.83)

where

χ_{+}(p) =^{def} H_{\frac{p}{ωp}} ( \begin{matrix}1 \\ 0 \end{matrix} ) , χ_{−}(p) =^{def} H_{\frac{p}{ωp}} ( \begin{matrix}0 \\ 1 \end{matrix} ) .  (4.84)

Then, by (4.77), the norm (4.76) becomes

∥χ∥ = \sqrt{∑_{σ=±} ∫ |b_{σ}(p)|^{2} \frac{dp}{2ω_{p}}} .  (4.85)

Moreover, (4.75), (4.83) ⟹ (Û(a, A)χ) (p) = ∑_{σ=±} b′_{σ}(p) χ_{σ}(p) , where:

( \begin{matrix}b′_{+}(p) \\ b′_{−}(p) \end{matrix} ) = e^{ip^{μ}a_{μ}} (H_{\frac{−p}{ωp}} A H_{\frac{p′}{ω_{p′}}} ) ( \begin{matrix}b_{+}(p′) \\ b_{−}(p′) \end{matrix} ) , p′ =^{def} (Λ_{A}^{−1}p)_{|_{p^{0}=ωp} .} (4.86)

Proof of (4.86):

A χ(p′) =_{(4.83)} ∑_{σ} b_{σ}(p′) A χ_{σ}(p′)
=_{(4.84)} ∑_{σ} b_{σ}(p′) H_{\frac{+p}{ωp}} H_{\frac{−p}{ωp}} A H_{\frac{p′}{ω_{p′}}} χ_{σ}(0)
= ∑_{σ} b_{σ}(p′) H_{\frac{+p}{ωp}} ∑_{σ′}(H_{\frac{−p}{ωp}} A H_{\frac{p′}{ω_{p′}}} )_{σ′σ} χ_{σ′}(0)
=_{(4.84)} ∑_{σ′}(∑_{σ}(H_{\frac{−p}{ωp}} A H_{\frac{p′}{ω_{p′}}} )_{σ′σ} b_{σ}(p′))χ_{σ′}(p) .

Exercise 69 Show for arbitrary A ∈ SL(2, ℂ) that^{46}

p′ =^{def} (Λ_{A}^{−1}p)_{|_{p^{0}=ωp}} ⟹ H_{\frac{−p}{ωp}} A H_{\frac{p′}{ω_{p′}}} unitary ,

as to be expected (compare, e.g., Sect. 2.4.3 of (Lücke, rel)).

Draft, March 26, 2009

46Hint: First, show that (4.70) implies

(H_{p/p^{0}})^{2}_{|_{p^{0}=ωp}} = p^{μ}τ_{μ}/m ∀ p ∈ ℝ^{3}

and therefore

H_{\frac{−p′}{ω_{p′}}} = \sqrt{(H_{\frac{−p}{ωp}} A)^{−1}(H_{\frac{−p}{ωp}} A)^{∗}^{−1}} , (see proof of (4.94)). Then recall the polar decomposition (4.72).

Positive Frequency Wave Functions

The transition

χ(p) ⟶ (2π)^{− \frac{3}{2}} ∫_{p^{0}=ω_{p}} χ(p) e^{−ip^{μ}x_{μ}} \frac{dp}{2p^{0}}

would give a usable configuration space version of the above representation (local transformation behavior). However, technically more convenient (for later inclusion of anti-particles) is another, unitarily equivalent, representation in the Hilbert space of 4-component momentum space wave functions

Ψ̂^{(+)}(p) = ∑_{σ=±} b_{σ}(p) ω_{σ}^{(+)}(p)  (4.87)

with norm

∥Ψ̂^{(+)}∥ =^{def} \sqrt{∑_{σ=±} ∫ |b_{σ}(p)|^{2} \frac{dp}{2ω_{p}}}  (4.88)

where

ω_{+}^{(+)}(p) =^{def} \frac{S(H_{\frac{p}{ωpv}} )}{\sqrt{2}} ( \begin{matrix}1 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ω_{−}^{(+)}(p) =^{def} \frac{S(H_{\frac{p}{ωpv}} )}{\sqrt{2}} ( \begin{matrix}0 \\ 1 \\ 0 \\ 1 \end{matrix} ) ;  (4.89)

namely:

(Û(a, A)Ψ̂^{(+)}) (p) =^{def} e^{ip^{μ}a_{μ}} S(A) Ψ̂^{(+)} (\overrightarrow{Λ_{A^{−1}}p})_{|_{p^{0}=ωp}} ,  (4.90)

where

S(A) =^{def} ( \begin{matrix}A & 0 \\ 0 & A^{∗}^{−1} \end{matrix} ) .  (4.91)

Exercise 70 Show that

(4.90), (4.87) ⟹ (Û(a, A)Ψ̂^{(+)}) (p) = ∑_{σ=±} b′_{σ}(p) ω_{σ}^{(+)}(p)

holds with b′_{σ}(p) given by (4.86).

(4.88) may be written in the form

∥Ψ̂^{(+)}∥ = \sqrt{∫_{p^{0}=ω_{p}} Ψ̂^{(+)}(p)^{∗} \frac{p_{μ}α^{μ}}{m} Ψ̂^{(+)}(p) \frac{dp}{2p^{0}}} ,  (4.92)

where

α^{μ} =^{def} ( \begin{matrix}τ_{μ} & 0 \\ 0 & τ^{μ} \end{matrix} ) ,  (4.93)

since now the generalization

S(A)^{∗} p_{μ}α^{μ}S(A) = (Λ_{A^{−1}}p)_{μ} α^{μ}  (4.94)

of (4.77) holds.

4.2. THE QUANTIZED FREE DIRAC FIELD 163

Sketch of proof for (4.94):

\begin{aligned}Tr (τ_{λ} (Λ_{A^{−1}}p)^{μ} τ_{μ}) & =_{(4.60))} & 2 (Λ_{A^{−1}})_{ν}^{λ} p^{ν} \\ & =_{(4.64))} & Tr (p^{ν}τ_{λ}A^{−1}τ_{ν}A^{∗}^{−1}) . \end{aligned}

⇝_{(4.61)} A^{−1}p^{ν}τ_{ν}A^{∗}^{−1} = (Λ_{A^{−1}}p)^{μ} τ_{μ} .

Together with (4.77) and (4.93) this implies (4.94).

With

γ^{0} =^{def} ( \begin{matrix}0 & 1l_{2} \\ 1l_{2} & 0 \end{matrix} ) , γ^{j} =^{def} γ^{0}α^{j} = ( \begin{matrix}0 & τ^{j} \\ τ_{j} & 0 \end{matrix} )  (4.95)

(4.94) becomes equivalent to

S (A^{−1}) γ^{μ}S(A) = (Λ_{A})_{ν}^{μ} γ^{ν} ,  (4.96)

thanks to

S(A)^{∗}γ^{0} = γ^{0}S(A)^{−1} .  (4.97)

By (4.90), (4.96) is equivalent to

[γ^{μ}p_{μ}, Û(a, A)]_{−} = 0 .  (4.98)

Since Û(a, A) is irreducible and^{47}

det (γ^{μ}p_{μ}) =_{(4.95)} det (α^{μ}p_{μ}) = m^{4} ,

(4.98) implies:^{48}

(γ^{μ}p_{μ} − m) Ψ̂^{(+)}(p)_{|p^{0}=ω_{p}} = 0 .  (4.99)

Conversely, (4.99) implies that Ψ̂^{(+)}(p) is of the form (4.87).

Sketch of proof:

(4.99) ⟺ (S (H_{v})^{−1} (γ^{μ}p_{μ} − m)ψ̂^{(+)}(p) = 0)
⟺_{(4.96)} ((γ^{0} − 1)S (H_{\frac{p}{ωp}} )^{−1} ψ̂^{(+)}(p) = 0)
⟺_{(4.95)} S (H_{\frac{p}{ωp}} )^{−1} ψ̂^{(+)}(p) = ∑_{σ=±} b_{σ}(p)ω_{σ}^{(+)}(0)
⟺_{(4.89)} ψ̂^{(+)}(p) = ∑_{σ=±} b_{σ}(p)ω_{σ}^{(+)}(p) .

Draft, March 26, 2009

47Warning: The matrix γ^{μ}p_{μ} is not selfadjoint (compare (4.109)). 48The correct sign may be easily determined by checking the special case p = 0.

Therefore, the set of admitted configuration space wave functions

Ψ^{(+)}(x) =^{def} \sqrt{2m} (2π)^{−3/2} ∫_{p^{0}=ω_{p}} Ψ̂^{(+)}(p) e^{−ipx} \frac{dp}{2p^{0}}  (4.100)

coincides with the set of all normalizable positive frequency solutions of the Dirac equation

(iγ^{μ}∂_{μ} − m) Ψ(x) = 0 .  (4.101)

Thanks to the factor \sqrt{2m} in (4.100) we have

∥Ψ̂^{(+)}∥^{2} = ∫ Ψ^{(+)}(x)^{∗} Ψ^{(+)}(x) dx ∀ x^{0} ∈ ℝ .  (4.102)

Proof:

∫ Ψ^{(+)}(x)^{∗}Ψ^{(+)}(x) dx

=_{(4.100)} 2m ∫_{p^{0}=ω_{p}} Ψ̂^{(+)}(p)^{∗} Ψ̂^{(+)}(p) \frac{dp}{(2p^{0})^{2}}

=_{(4.99)} ∫_{p^{0}=ω_{p}} Ψ̂^{(+)}(p)^{∗}γ^{μ}p_{μ}Ψ̂^{(+)}(p) \frac{dp}{(2p^{0})^{2}} + ∫_{p^{0}=ω_{p}} (γ^{μ}p_{μ}Ψ̂^{(+)}(p))^{∗} Ψ̂^{(+)}(p) \frac{dp}{(2p^{0})^{2}}

=_{(4.109)} ∫_{p^{0}=ω_{p}} Ψ̂^{(+)}(p)^{∗} γ^{0} Ψ̂^{(+)}(p) \frac{dp}{2p^{0}}

=_{(4.99)} ∫_{p^{0}=ω_{p}} Ψ̂^{(+)}(p)^{∗} \frac{γ^{0}γ^{μ}p_{μ}}{m} Ψ̂^{(+)}(p) \frac{dp}{2p^{0}} .

By (4.95) and (4.92), this implies the statement.

Exercise 71 Show that

S(U_{φ}) = exp (−\frac{i}{4} ∑_{j,k,l=1}^{3} ϵ_{jkl}γ^{j}γ^{k}φ^{l}) , S(H_{\frac{p}{ωp}} ) = \sqrt{\frac{ω_{p} + m}{2m}} γ^{0} (γ^{0} + \frac{γ · p}{ω_{p} + m}) ,

and

γ^{1}γ^{3} \overline{S(A)} = S(A^{∗}^{−1}) γ^{1}γ^{3} ∀ A ∈ SL(2, ℂ) ,

where \overline{S} denotes the matrix resulting from substituting the entries of the matrix S by their complex conjugates (i.e. \overline{S} = S^{∗}^{T}).

Discrete Symmetries

The so-called parity operator 𝔓, describing total spatial reflection of the state, is fixed – up to some irrelevant constant phase factor – by^{49}

𝔓^{−1} Û (a, A) 𝔓 = Û (ℙa, A^{∗}^{−1}) , ℙx =^{def} (x^{0}, −x) ,  (4.103)

Draft, March 26, 2009

49For the necessity of both conditions see, e.g., (Martin and Spearman, 1970, Chapter 5 §1).

4.2. THE QUANTIZED FREE DIRAC FIELD 165 and the requirement of unitarity (positive energy^{50}). Note that

ℙΛ_{A}ℙ = 𝕋Λ_{A}𝕋 = Λ_{A^{∗}}^{−1} ∀ A ∈ SL(2, ℂ) ,

where

𝕋x =^{def} (−x^{0}, x) .

The natural choice is 𝔓^{2} = ˆ1 , i.e.:

(𝔓Ψ^{(+)}) (x) = γ^{0}Ψ^{(+)}(ℙx) ,  (4.104)

by (4.97). Exercise 72 Show that

γ^{0}ω_{σ}^{(+)}(−p) = ω_{σ}^{(+)}(p)

and, therefore,

Ψ^{(+)}(x) = \sqrt{2m} (2π)^{−3/2} ∫_{p^{0}=ω_{p}} ∑_{σ=±} b_{σ}(p) ω_{σ}^{(+)}(p) e^{−ipx} \frac{dp}{2p^{0}}
⟹ (𝔓Ψ^{(+)}) (x) = \sqrt{2m} (2π)^{−3/2} ∫_{p^{0}=ω_{p}} ∑_{σ=±} b_{σ}(−p) ω_{σ}^{(+)}(p) e^{−ipx} \frac{dp}{2p^{0}} .

Similarly the time reversal operator 𝔗 is fixed – up to some irrelevant constant phase factor – by

𝔗^{−1}Û (a, A) 𝔗 = Û (𝕋a, A^{∗}^{−1})

and the condition of anti-unitarity.^{51} The usual choice is:^{52}

(𝔗 Ψ^{(+)}) (x) = iγ^{1}γ^{3} \overline{Ψ^{(+)}(𝕋x)}  (4.105)

(recall Exercise 71). Exercise 73 Show that

γ^{1}γ^{3} \overline{ω_{σ}^{(σ′)}(+p)} = −σσ′ ω_{−σ}^{(+)}(−p)

and, therefore,

Ψ^{(+)}(x) = \sqrt{2m} (2π)^{−3/2} ∫_{p^{0}=ω_{p}} ∑_{σ=±} b_{σ}(p) ω_{σ}^{(+)}(p) e^{−ipx} \frac{dp}{2p^{0}}
⟹ (𝔗Ψ^{(+)}) (x) = \sqrt{2m} (2π)^{−3/2} ∫_{p^{0}=ω_{p}} ∑_{σ=±} iσ \overline{b_{−σ}(−p)} ω_{σ}^{(+)}(p) e^{−ipx} \frac{dp}{2p^{0}} .

Draft, March 26, 2009

50By (4.103), anti-unitarity of 𝔓 would imply 𝔓^{−1}P̂^{0}𝔓 = −P̂^{0} and thus ⟨𝔓Ψ | P̂^{0}𝔓Ψ⟩ =

\overline{⟨Ψ | 𝔓^{−1}P̂^{0}𝔓Ψ⟩} = − ⟨Ψ | P̂^{0}Ψ⟩ for the Hamiltonian P̂^{0} = −i\frac{d}{dx^{0}} Û ((x^{0}, 0, 0, 0), 1l_{4}).

51However, because of anti-linearity of 𝔗, Schur’s lemma is not directly applicable. Note that, for the same reason, 𝔗^{2} = ˆ1 does not depend on the choice of phase factor.

52In the representation (4.95): iγ^{1}γ^{3} = − ( \begin{matrix}τ^{2} & 0 \\ 0 & τ^{2} \end{matrix} ).

166 CHAPTER 4. QUANTUM ELECTRODYNAMICS Defining

γ^{5} =^{def} iγ^{0}γ^{1}γ^{2}γ^{3} = −iα^{1}α^{2}α^{3} ,  (4.106)

and noting that (4.93) and (4.68) imply

γ^{5} = ( \begin{matrix}−1l_{2} & 0 \\ 0 & +1l_{2} \end{matrix} ) in the representation (4.95) ,

we get the following relations:

S (A^{−1}) γ^{5}S(A) = γ^{5} ,  (4.107)
[γ^{μ}, γ^{ν}]_{+} = \{ \begin{matrix}+2 & if μ = ν ∈ \{0, 5\} , \\ −2 & if μ = ν ∈ \{1, 2, 3\} , \\ 0 & else , \end{matrix}  (4.108)
(γ^{μ})^{∗} = \{ \begin{matrix}+γ^{μ} & for μ ∈ \{0, 5\} , \\ −γ^{μ} & for μ ∈ \{1, 2, 3\} , \end{matrix}  (4.109)
Ŝ · e =^{def} i\frac{d}{dφ}S(U_{φe})_{|φ=0} = \frac{1}{2}γ^{5}γ^{0} γ · e ,  (4.110)
[Ŝ, γ^{5}]_{−} = 0 .  (4.111)

Exercise 74 Show that

V_{+}^{μ}(x) =^{def} Ψ^{(+)}(x)^{∗}γ^{0}γ^{μ}Ψ^{(+)}(x)  (4.112)

is an ordinary current density and

V_{−}^{μ}(x) =^{def} Ψ^{(+)}(x)^{∗}γ^{0}γ^{5}γ^{μ}Ψ^{(+)}(x)  (4.113)

an ordinary axial current density, i.e:^{53}

\begin{matrix}V_{±}^{μ}(x^{0}, x) & ⟶^{Û(a,A)} & (Λ_{A})_{ν}^{μ} V_{±}^{ν} (Λ_{A}^{−1}(x − a)) , \\ V_{±}^{μ}(x^{0}, x) & ⟶^{P̂} & ±g_{μν}V_{±}^{ν}(x^{0}, −x) , \\ V_{±}^{μ}(x^{0}, x) & ⟶^{T̂} & +g_{μν}V_{±}^{ν}(−x^{0}, x) \end{matrix}  (4.114)

Final remark: Physically relevant are only the general relations between the γ-matrices. Transformations of the type

γ^{μ} ⟶ γ^{′μ} = Mγ^{μ}M^{−1} ,

Draft, March 26, 2009

53A more complete listing is given in (Itzykson and Zuber, 1980b, Sect. 3-4-4).

4.2. THE QUANTIZED FREE DIRAC FIELD 167

where M is any unitary 4 × 4-matrix, are always allowed.^{54} E.g., for

M = \frac{1}{\sqrt{2}} ( \begin{matrix}+1l_{2} & +1l_{2} \\ −1l_{2} & +1l_{2} \end{matrix} ) , M^{−1} = \frac{1}{\sqrt{2}} ( \begin{matrix}+1l_{2} & −1l_{2} \\ +1l_{2} & +1l_{2} \end{matrix} )

we get the standard representation

γ^{′0} = −γ^{5} , γ^{′5} = γ^{0} , γ^{′j} = γ^{j} for j = 1, 2, 3  (4.115)

of (Bjorken and Drell, 1964).

The Limit \frac{m}{|p|} → 0 According to (4.110), (4.99) is equivalent to

γ^{5}Ŝ · \frac{p}{|p|} Ψ̂^{(+)}(p) = \frac{1}{2|p|} (ω_{p} − mγ^{0}) Ψ̂^{(+)}(p) .  (4.116)

By (4.108) and (4.111) the latter is equivalent to validity of the following two equations:^{55}

ĥ (1 + γ^{5}) Ψ̂^{(+)}(p) = +\frac{ω_{p}}{2|p|} (1 + γ^{5}) Ψ̂^{(+)}(p) − \frac{m}{2|p|}γ^{0} (1 − γ^{5}) Ψ̂^{(+)}(p) ,

ĥ (1 − γ^{5}) Ψ̂^{(+)}(p) = −\frac{ω_{p}}{2 |p|} (1 − γ^{5}) Ψ̂^{(+)}(p) + \frac{m}{2|p|}γ^{0} (1 + γ^{5}) Ψ̂^{(+)}(p) .

(4.117)

This implies

\begin{matrix}ĥΨ̂_{R}^{(+)}(p) = +\frac{1}{2}Ψ̂_{R}^{(+)}(p) \\ ĥΨ̂_{L}^{(+)}(p) = −\frac{1}{2}Ψ̂_{L}^{(+)}(p) \end{matrix} \} for \frac{m}{|p|} → 0 ,  (4.118)

where

Ψ̂_{R}^{(+)}(p) =^{def} \frac{1}{2}(1 + γ^{5}) Ψ̂^{(+)}(p) ,
Ψ̂_{L}^{(+)}(p) =^{def} \frac{1}{2}(1 − γ^{5}) Ψ̂^{(+)}(p) .

Exercise 75 Show that

γ^{0}γ^{μ} = (\frac{1 + γ^{5}}{2} )^{∗} γ^{0}γ^{μ} \frac{1 + γ^{5}}{2} + (\frac{1 − γ^{5}}{2} )^{∗} γ^{0}γ^{μ} \frac{1 − γ^{5}}{2}

and hence

V_{+}^{μ}(x) = Ψ_{R}^{(+)}(x)^{∗}γ^{0}γ^{μ}Ψ_{R}^{(+)}(x) + Ψ_{L}^{(+)}(x)^{∗}γ^{0}γ^{μ}Ψ_{L}^{(+)}(x) .

Draft, March 26, 2009

54Of course, S(A) (recall Exercise 71) has to be defined accordingly, as well as the ω_{±}^{(+)}(0) :

γ^{0}ω_{±}^{(+)}(0) = ω_{±}^{(+)}(0) , Ŝ^{3}ω_{±}^{(+)}(0) = ±\frac{1}{2}ω_{±}^{(+)}(0) .

55The first of these relations results by adding (4.116) and (4.116) multiplied by γ^{5} (from the left). Recall that the helicity operator is ĥ = \frac{p}{|p|} · Ŝ .

Note that, in the representation (4.95), the equations

Ψ̂_{R}^{(+)}(p) = ( \begin{matrix}0 \\ χ_{R}(p) \end{matrix} ) , Ψ̂_{L}^{(+)}(p) = ( \begin{matrix}χ_{L}(p) \\ 0 \end{matrix} ) ,

with corresponding 2-component spinors χ_{R}(p), χ_{L}(p) , and

ĥ = \frac{1}{2} ( \begin{matrix}τ · \frac{p}{|p|} & 0 \\ 0 & τ · \frac{p}{|p|} \end{matrix} )

hold. Hence, for m = 0 , the equations (4.118) are equivalent to the two so-called Weyl equations

∂_{0}Φ_{R}(x) = −τ · ∇_{x}Φ_{L}(x) , ∂_{0}Φ_{L}(x) = +τ · ∇_{x}Φ_{L}(x) ,
where: Φ_{R(L)}(x) =^{def} (2π)^{− \frac{3}{2}} ∫_{p^{0}=ω_{p}} χ_{R(L)}(p)e^{−ip^{μ}x_{μ}} \frac{dp}{2p^{0}} .

(4.119)

4.2.4 Quantized Dirac Field

Positive Frequency Part of the Dirac Field

Similarly to the electromagnetic field the Dirac field is quantized by replacement of the amplitudes b_{σ}(p) in (4.87) by corresponding annihilation operators b̂_{σ}(p) (of particles with linear momentum p and e_{3}-component σ \frac{1}{2} of the internal angular momentum in the center of mass system of the particle):

Ψ̂^{(+)}(x) = \sqrt{2m} (2π)^{− \frac{3}{2}} ∫_{p^{0}=ω_{p}} \underbrace{∑_{σ=±} b̂_{σ}(p) ω_{σ}^{(+)}(p)}_{=^{def} ˆΨ̂^{(+)}(p)} e^{−ip^{μ}x_{μ}} \frac{dp}{2p^{0}}  (4.120)

(compare (4.100)). Now, however, the b̂_{σ}(p) (respecting the Pauli principle) act in a space

H_{0} =^{def} ⨁_{n=0}^{∞} H^{(n)}

the n-particle components H^{(n)} of which are spanned by totally anti-symmetric wave functions b_{n} :

b_{n}(p_{π1}, σ_{π1}; …; p_{π1}, σ_{π1}) = sign (π) b_{n}(p_{1}, σ_{1}; …; p_{n}, σ_{n}) for all π ∈ S_{n} . (4.121)

The inner product is given by natural generalization of the norm (4.88):

∥b_{n}∥ =^{def} \sqrt{∑_{σ_{1},...σ_{n}=±} ∫ ∥b_{n}(p_{1}, σ_{1}; …; p_{n}, σ_{n}) ∥^{2} ∏_{j} \frac{dp_{j}}{2ω_{p_{j}}}} .  (4.122)

Since, the b̂_{σ}(p) are given by

(b̂_{σ}(p) b)_{0} =^{def} b_{1}(p, σ) ,
(b̂_{σ}(p) b)_{n−1}(p_{1}, σ_{1}; …; p_{n−1}, σ_{n−1}) =^{def} \sqrt{n} b_{n}(p, σ; p_{1}, σ_{1}; …; p_{n}, σ_{n−1})

(4.123)

on their natural domain of definition^{56} we now have the anti-commutation relations

\begin{matrix}[b̂_{σ}(p), b̂_{σ′}(p′)^{∗}]_{+} = 2ω_{p} δ_{σσ′}δ(p − p′) , \\ [b̂_{σ}(p), b̂_{σ′}(p′)]_{+} = 0 . \end{matrix}  (4.124)

All linear relations of the 1-particle theory also hold for the quantized field. The field operator for the current density produced by all particles (not anti-particles) of charge^{57} q is

ĵ_{q}^{(+)}^{μ}(x) =^{def} q Ψ̂^{(+)}(x)^{∗} γ^{0}γ^{μ} Ψ̂^{(+)}(x)  (4.125)

and transforms like a 4-vector field (compare Exercise 74) under the natural extension Û_{0}(a, A) of Û(a, A) to all of H_{0} , fixed – together with the representation of space-time reflections – by the requirements

Û_{0}(a, A)^{−1}Ψ̂^{(+)}(x)Û_{0}(a, A) = S(A)Ψ̂^{(+)} (Λ_{A^{−1}}(x − a)) ,  (4.126)
𝔓^{−1}Ψ̂^{(+)}(x)𝔓 = γ^{0}Ψ̂^{(+)}(x^{0}, −x) ,  (4.127)
𝔗^{−1}Ψ̂^{(+)}(x)𝔗 = iγ^{1}γ^{3}Ψ̂^{(+)}(−x^{0}, x)^{∗}^{T} ,  (4.128)

and

Û_{0}(a, A)Ω_{0} = 𝔓Ω_{0} = 𝔗Ω_{0} = Ω_{0} (=^{def} 1 ∈ H^{(0)} = ℂ) .  (4.129)

consistent with the described 1-particle theory. Due to^{58}

ω_{σ}^{(+)}(p)^{∗}γ^{0}ω_{σ′}^{(+)}(p) = δ_{σσ′}  (4.130)

and

∫ Ψ̂^{(+)}(x)^{∗} γ^{0}γ^{μ} Ψ̂^{(+)}(x) dx = ∫_{p^{0}=ω_{p}} ˆ Ψ̂^{(+)}(p)^{∗} γ^{0} ˆ Ψ̂^{(+)}(p) \frac{dp}{2p^{0}}

(compare proof of (4.102)) the corresponding total charge is

Q̂^{(+)} =^{def} ∫ ĵ_{q}^{(+)}^{0}(x) dx = q ∫_{p^{0}=ω_{p}} ∑_{σ=±} b̂_{σ}^{∗}(p)b̂_{σ}(p) \frac{dp}{2p^{0}} ,  (4.131)

Draft, March 26, 2009

56The natural invariant domain of definition for b̂_{σ}(p) is characterized by the conditions

b_{n}^{σ_{1},...,σ_{n}}(p_{1}, …, p_{n}) =^{def} b_{n}(p_{1}, σ_{1}; …; p_{n}, σ_{n}) ∈ S(ℝ^{3n}) for fixed (σ_{1}, …, σ_{n}) ∈ {+, −}^{n}

and b_{n} = 0 for sufficiently large n .

57q need not be the electric charge!

58(4.130) is trivial for p = 0 and therefore, by (4.89) and (4.97), also valid for p ≠ 0.

independent of x^{0} ∈ ℝ.

The theory with only positive frequencies has (among others) the following disadvantages:

(P1): The current density (4.125) violates Einstein’s causality principle, i.e:

x×y ≠⇒ [ĵ_{q}^{(+)}^{α}(x), ĵ_{q}^{(+)}^{β}(y)]_{−} = 0 .

(P2): In general, minimal coupling with an exterior field does not allow a solution for which both the incoming and the outgoing free Dirac field have vanishing negative frequency parts.^{59}

Local Operator Field

Obviously problem (P2) requires the additional introduction of negative frequency solutions Ψ̂^{(−)}(x) of the Dirac field operator equation. These can be constructed using the 4-spinors

ω_{σ}^{(−)}(p) =^{def} γ^{5}ω_{−σ}^{(+)}(−p) .  (4.132)

Exercise 76 Show the following:^{60}

1.

ω_{+}^{(−)}(p) = S(H_{\frac{p}{ωp}} )^{−1} ( \begin{matrix}0 \\ −1 \\ 0 \\ +1 \end{matrix} ) = S(H_{\frac{p}{ωp}} )^{−1} ( \begin{matrix}−1 \\ 0 \\ +1 \\ 0 \end{matrix} ) , ω_{−}^{(−)}(p) (4.133)

in the representation (4.95).

2.

S(H_{\frac{p}{ωp}} ) S(H_{\frac{p}{ωp}} ) = \frac{p_{μ}γ^{μ}}{m} γ^{0} .

3.

∑_{σ,σ′=±} (ω_{σ}^{σ′}(σ′p))_{r} \overline{(ω_{σ}^{σ′}(σ′p))_{r′}} = (\frac{p_{μ}γ^{μ}}{m} γ^{0})_{r,r′} , with p^{0} = ω_{p} .

4.

∑_{σ,σ′=±} σ′ (ω_{σ}^{σ′}(σ′p))_{r} \overline{(ω_{σ}^{σ′}(σ′p))_{r′}} = (γ^{0})_{r,r′} .

Draft, March 26, 2009

59This is related to Klein’s paradox (see (Telegdi, 1995) and references given there).

60Hint: To prove the second statement, exploit (4.94) for the special case p = 0 , A = H_{\frac{p}{ωp}} .

Then

Ψ̂^{(−)}(x) = \sqrt{2m} (2π)^{− \frac{3}{2}} ∫_{p^{0}=−ω_{p}} \overbrace{∑_{σ=±} d̂_{σ}^{∗}(−p)ω_{σ}^{(−)}(p)}^{ˆΨ̂^{(−)}(p) =^{def}} e^{−ip^{μ}x_{μ}} \frac{dp}{2ω_{p}}  (4.134)

is indeed a solution of the Dirac equation:

(γ^{μ}p_{μ} − m) ˆ Ψ̂^{(−)}(p)_{|p^{0}=−ω_{p}} = 0 .

Sketch of proof:

S(H_{\frac{p}{ωp}} ) (γ^{μ}p_{μ} − m)_{|p^{0}=−ω_{p}} ω_{σ}^{(−)}(p)

=_{(4.133)} S(H_{\frac{p}{ωp}} ) (γ^{μ}p_{μ} − m)_{|p^{0}=−ω_{p}} S(H_{\frac{−p}{ωp}} )ω_{σ}^{(−)}(0)

=_{(4.96)} ((Λ_{H_{\frac{−p}{ωp}}} )_{ν}^{μ}γ^{ν}p_{μ} − m)_{|p^{0}=−ω_{p}} ω_{σ}^{(−)}(0)

= − (γ^{μ}(Λ_{H_{\frac{p}{ωp}}} (ω_{p}, −p))_{μ} + m) ω_{σ}^{(−)}(0)

= −m (γ^{0} + 1)ω_{σ}^{(−)}(0)

= 0 .

With (4.132) and (4.132) one can show that the equations (4.126)–(4.128) hold also for

Ψ̂(x) =^{def} Ψ̂^{(+)}(x) + Ψ̂^{(−)}(x)  (4.135)

instead of Ψ̂^{(+)}(x) if the

˘b̂_{σ}(p) =^{def} σd̂_{σ}(p)

transform the same way as the b̂_{σ}(p) do.

Hence, in principle, one could identify the operators ˘ b̂_{σ}(p) with the operators b̂_{σ}(p) . However, a solution of the problems mentioned above will be achieved^{61} only if the d̂_{σ}(p) the are interpreted as annihilation operators for anti-particles, with invariant domain D̆_{0} ⊂ H̆_{0} (compare Section 2.4.1). Then the operators

b̂_{σ}(p) = b̂_{σ}(p) ⊗ ˆ1 , d̂_{σ}(p) = (−1)^{Q̂^{(+)}/q} ˆ1 ⊗ d̂_{σ}(p) ,

well-defined on

D_{D} =^{def} D_{0} ⊗ D̆_{0} ⊂ H_{D} =^{def} H_{0} ⊗ H̆_{0} ,

fulfill the anti-commutation relations

[d̂_{σ}(p), b̂_{σ′}(p′)]_{+} = [d̂_{σ}(p)^{∗}, b̂_{σ′}(p′)]_{+} = 0  (4.136)

Draft, March 26, 2009

61See, e.g., (Seiler, 1978).

in addition to (4.124) and the corresponding relations for the d̂_{σ}(p) . From this, using the results of Exercise 76, one easily derives the anti-commutation relations

\begin{aligned}[(Ψ̂^{(σ)}(x))_{r} , (Ψ̂^{(σ′)}(x′))_{r′}]_{+} & = & 0 , \\ [(Ψ̂^{(σ)}(x))_{r} , (Ψ̂^{(σ′)}(x′))_{r′}^{∗}]_{+} & = & δ_{σσ′} ((iγ^{μ}∂_{μ} + m)γ^{0})_{rr′} iΔ_{m}^{(σ)}(x − x′) . \end{aligned}

(4.137)

Thus the solution (4.135) of the Dirac equation

(iγ^{μ}∂_{μ} − m) Ψ̂(x) = 0 ,  (4.138)

is a local Fermi field, i.e.:

x×x′ ⟹ [(Ψ̂(x))_{r} , (Ψ̂(x′))_{r′}^{(∗)}]_{+} = 0 .  (4.139)

Moreover, the anti- commutation relations (4.137) imply that

ĵ_{q}^{μ}(x) =^{def} q:Ψ̂(x)^{∗}γ^{0}γ^{μ}Ψ̂(x):  (4.140)

is a local Bose field, i.e.:

x×x′ ⟹ [ĵ_{q}^{α}(x), ĵ_{q}^{β}(x′)]_{−} = 0 .  (4.141)

Normal ordering :.: in (4.140) means that Fermi creation and annihilation operators have to be anti-commute, if necessary, irrespective of the actual anticommutation relations until no creation operator is on the right of any annihilation operator.^{62}

ĵ_{q}^{μ}(x) is interpreted as observable of the total current density since, e.g., the following relations hold:

∫ ĵ_{q}^{0}(x) dx = Q̂_{q} =^{def} q ∫_{p^{0}=ω_{p}} ∑_{σ=±} (b̂_{σ}^{∗}(p)b̂_{σ}(p) − d̂_{σ}^{∗}(p)d̂_{σ}(p)) \frac{dp}{2p^{0}} , (4.142)
∂_{μ}ĵ_{q}^{μ}(x) = 0 ,  (4.143)
ĵ_{q}^{μ}(x) = ĵ_{q}^{μ}(x)^{∗} .  (4.144)

Since

Δ_{m}(0, x) = 0 , (∂_{0}Δ(x))_{|_{x^{0}=0}} = −δ(x) ,  (4.145)

the anti-commutation relations (4.137) imply the canonical anti-commutation relations

[(Ψ̂(x^{0}, x))_{r} , (Ψ̂(x^{0}, x′))_{r′}]_{+} = 0 ,
[(Ψ̂(x^{0}, x))_{r} , (Ψ̂(x^{0}, x′))_{r′}^{∗}]_{+} = δ_{rr′}δ(x − x′) .

(4.146)

Draft, March 26, 2009

62Another effect of normal ordering, besides making ĵ_{q}^{μ}(x) well-defined (as operator-valued distribution, not just a quadratic form), is that the quantized current density – contrary to the classical one – is no longer positive (compare remark on Corollary 2.2.14).

Therefore,^{63}

P̂_{D}^{μ} =^{def} ∫ :Ψ̂(x)^{∗}i∂^{μ}Ψ̂(x): dx  (4.147)

is the generator of space-time translations, i.e.

i [P̂_{D}^{μ}, Ψ̂(x)]_{−} = ∂^{μ}Ψ̂(x) .  (4.148)

Final remark: Note that the local Dirac formalism is physically consistent only since anti-particles actually exist and are different from the corresponding particles (e.g. having charge of different sign). Contrary to this, local Bose fields may well describe particles (e.g. photons) that are identical with their anti-particles.

The Limit \frac{m}{|p|} → 0

Particle or anti-particles in states which can be created by applying (smeared versions of)

Ψ̂_{R}(x) = \frac{1}{2}(1 + γ^{5})Ψ̂(x)  (4.149)

or Ψ̂_{R}(x)^{∗} to the vacuum vector are called right handed. Similarly, particle or anti-particles in states which can be created by applying (smeared versions of)

Ψ̂_{L}(x) = \frac{1}{2}(1 − γ^{5})Ψ̂(x)  (4.150)

or Ψ̂_{R}(x)^{∗} to the vacuum vector are called left handed.

The transition \frac{m}{|p|} → 0 may be performed as discussed at the end of 4.2.3:

for m = 0 : [Ψ̂_{R}(x), ĥ]_{+} = +\frac{1}{2} Ψ̂_{R}(x) [Ψ̂_{L}(x), ĥ]_{+} = −\frac{1}{2} Ψ̂_{L}(x)

where now: ĥ =^{def} ∫ dx:Ψ̂(x)^{∗} (\frac{1}{2}γ^{5}γ^{0}γ^{j}) \frac{i∂^{j}}{|∇_{x}|}Ψ̂(x): .

(4.151)

(compare (4.110) and (4.117)). This implies:

helicity of right handed massless particles:  +1/2
helicity of left handed massless particles:  −1/2
helicity of left handed massless anti- particles: +1/2
helicity of right handed massless anti- particles: −1/2

(4.152)

Draft, March 26, 2009

63Note that [ÂB̂, Ĉ]_{−} = Â[B̂, Ĉ]_{+} − [Â, Ĉ]_{+}B̂.

4.3 The S-Matrix of Quantum Electrodynamics (QED)

Let us choose Heaviside units (in addition to ℏ = c = 1). Then, by (4.51) and Footnote 25,

ζ = 1 .  (4.153)

4.3.1 Naive Interaction Picture of QED

Asymptotic Description

For simplicity we consider the interaction of the electromagnetic field with the electron-positron field, only. Then, in the Gupta-Bleuler formalism, the interacting system of quantum electrodynamics is asymptotically identified (in the sense of 2.3.1) with the following ‘free’ system:

The basic Hilbert space (containing unphysical degrees of freedom) is

H = H_{D} ⊗ H_{GB} ,  (4.154)

where H_{D} is the Hilbert space of the free electron-positron system as described in 4.2.4 and H_{GB} is the Hilbert space of the Gupta-Bleuler description of the quantized free electromagnetic field given in 4.1.3. The Dirac field Ψ̂(x) will now be identified with Ψ̂(x)⊗ˆ1 and the Gupta-Bleuler field Â_{GB}^{μ}(x) with ˆ1⊗Â_{GB}^{μ}(x) . Then the vacuum state vector

Ω = Ω_{D} ⊗ Ω_{GB}

of the total system is cyclic w.r.t. the fields Â_{GB}^{μ}(x) , Ψ̂(x)_{r} and Ψ̂(x)_{l}^{†}, well-defined as tempered field operators with invariant domain^{64}

D = D_{D} ⊗ D_{A} .

Here, of course, Ω_{D} =^{def} Ω_{0} ⊗Ω̆_{0} denotes the vacuum state vector of the Dirac theory and Ω_{GB} the vacuum state vector of the Gupta-Bleuler formalism (also denoted by Ω in 4.1.3). The new vacuum state vector is invariant under the representation

Û(a, A) = Û_{D}(a, A) ⊗ V̂ (a, Λ_{A})

of iSL(2, ℂ) (compare 4.2.2), where Û_{D}(a, A) =^{def} Û_{0}(a, A) × ˘ Û_{0}(a, A) denotes the representation of iSL(2, ℂ) for the Dirac theory and V̂ (a, Λ) the representation of P_{+}^{↑} given in 4.1.3 . It is unitary w.r.t. the indefinite inner product

(Φ_{1} | Φ_{2}) =^{def} ⟨Φ_{1} | ˆ1 ⊗ η̂Φ_{2}⟩ .  (4.155)

Draft, March 26, 2009

64Here ⊗ denotes the algebraic tensor product whereas in (4.154), of course, the topological tensor product of Hilbert spaces has to be taken.

4.3. THE S-MATRIX OF QUANTUM ELECTRODYNAMICS (QED) 175

with η̂ defined by (4.29).

The subset of D describing physical states is

D_{F} = \{Φ ∈ D : \overline{∂_{μ}Â_{GB}^{μ (+)}(x)}Φ = 0\}

(compare (4.33)). Elements of D_{F} describe the same physical state if their difference is in

D_{00} = {Φ ∈ D_{F} : (Φ | Φ) = 0}

(compare (4.17)). For physical states the expectation values^{65} of the quantized free electromagnetic field

F̂_{GB}^{μν}(x) = ∂^{μ}Â_{GB}^{ν}(x) − ∂^{ν}Â_{GB}^{μ}(x)

(compare (4.13)) fulfill the free Maxwell equations

∂_{μ} (Φ | F̂_{GB}^{μν}(x) Φ) = 0 ∀ Φ ∈ D_{F} .

(compare final remark of 4.1.3) although the current density (4.27) of the free Dirac field does not vanish on D_{F} . This means that the free electromagnetic field operators describe only the radiative part – not the field dragged along by the asymptotic charged particles (and contributing to their physical mass).

The Hamiltonian of the FS is

Ĥ_{0} = P̂_{D}^{0} + P̂_{GB}^{0}  (4.156)

with P̂_{D}^{0} resp. P̂_{GB}^{0} given by (4.147) resp. (4.48).

Formal Minimal Coupling

By (4.153) and (4.145), the commutation relations (4.27) imply the canonical commutation relations

[Â_{GB}^{μ}(x), (\frac{∂}{∂x′^{0}} Â_{GB}^{μ′}(x′))_{|_{x′^{0}=x^{0}}}]_{−} = −i g^{μμ′} δ(x − x′)

[Â_{GB}^{μ}(x), Â_{GB}^{μ}(x^{0}, x′)]_{−} = [\frac{∂}{∂x′^{0}} Â_{GB}^{μ′}(x), (\frac{∂}{∂x′^{0}} Â_{GB}^{μ′}(x′))_{|_{x′^{0}=x^{0}}}]_{−} = 0 .

(4.157)

These and the canonical anti-commutation relations (4.146) for the Dirac field (together with (4.27) and (4.137)) imply^{66} that a formal solution of the fundamental differential equations

□ Â_{int}^{μ}(x) = ĵ_{int}^{μ}(x) =^{‘def′} e^{iĤx^{0}} ĵ_{−e}^{μ}(0, x) e^{−iĤx^{0}} ,

(iγ^{μ}(∂_{μ} − ie g_{μν}Â_{int}^{ν}(x)) − m)Ψ̂_{int}(x) = 0 ,

(4.158)

Draft, March 26, 2009

65Recall that expectation values are given via (. | .) , in the Gupta-Bleuler formalism 66Recall Section 3.1.2

of QED is given by

Â_{int}^{μ}(x) =^{def} e^{iĤx^{0}} Â_{GB}^{μ}(0, x) e^{−iĤx^{0}} ,
Ψ̂_{int}(x) =^{def} e^{iĤx^{0}} Ψ̂(0, x) e^{−iĤx^{0}} ,

(4.159)

with

Ĥ = P̂_{D}^{0} + P̂_{GB}^{0} + Ĥ_{int} ,  (4.160)

where

Ĥ_{int} = ∫ g_{μν} :ĵ_{−e}^{μ}(0, x)Â_{GB}^{ν}(0, x): dx .  (4.161)

Remark: Here, e is the modulus of the electron charge in Heaviside units (see Appendix A.3.4 von (Lücke, edyn)). Note that the definition of ĵ_{int}(x) is only a formal one. ĵ_{q}^{μ}(x) was defined in (4.140), but now ^{∗} has to be replaced by ^{†}, of course. The transition from (4.138)/(4.9) to (4.158) is called minimal coupling .

Exercise 77 Show that

[ĵ_{−e}^{μ}(0, x′), Ψ̂(0, x)]_{−} = +e δ(x − x′) γ^{0}γ^{μ} Ψ̂(0, x) .

By (4.156), the operator (3.9) is

Ĥ_{I}(x^{0}) = ∫ g_{μν} :ĵ_{−e}^{μ}(x)Â_{GB}^{ν}(x): dx ,

i.e. we have to set^{67}

Ŝ_{1}(x) = ie g_{μν} :Ψ̂(x)^{†}γ^{0}γ^{μ}Ψ̂(x)Â_{GB}^{ν}(x):  (4.162)

in (3.26) (and let g → 1) if the time ordering is suitably defined.

Transition Probabilities

If the actual state of the IS looks for t → −∞ like the state of the FS described by Φ then the probability for a positive outcome of an ideal test whether the IS is in a state looking for t → +∞ like the state of the FS described by Φ′ is the transition probability

p(Φ → Φ′) =^{def} |(Φ′ | Ŝ_{0} Φ)|^{2}  (4.163)

Draft, March 26, 2009

67Note that, by Lemma 4.1.3,

Φ ∈ D_{00} ⟹ ∫ Ŝ_{1}(x) g(x) dx Φ ∈ \overline{D_{00}}^{⟨.|.⟩}

holds for all g ∈ S(ℝ^{4}) .

(recall Section 2.3.1). In practice one is only interested in states Φ, Φ′ of the form

∫ ĉ_{k_{1}}(p_{1})^{†} ⋯ ĉ_{k_{N}} (p_{N})^{†} φ(p_{1}, …, p_{N}) dp_{1} ⋯ dp_{N} Ω , φ ∈ S (ℝ^{3N}) ,

where the ĉ_{k}(p)^{†} are suitable creation operators. Then the essential task is to calculate the scattering amplitudes

(ĉ_{j_{1}}(p′_{1})^{†} ⋯ ĉ_{j_{N′}}(p′_{N′})^{†} Ω | (Ŝ_{0} − ˆ1) ĉ_{k_{1}}(p_{1})^{†} ⋯ ĉ_{k_{N}} (p_{N})^{†} Ω) .  (4.164)

4.3.2 General Perturbation Theory

Generalization of Wick’s Theorem

To every type of field appearing in Ŝ_{1}(x) and its adjoint (if not identical to the field itself) one assign a characteristic type of line; e.g. wavy lines for photons, simple lines with upward orientation for electrons etc.

The transition matrix

Ŝ_{0} − ˆ1 = ∑_{n=1}^{∞} ∫ T (Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{n})) dx_{1} ⋯ dx_{n}  (4.165)

(compare (3.26)) is evaluated by first writing the expressions

T (Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{n})) − ⟨Ω| T (Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{n})) Ω⟩

as linear combinations of normally ordered products. This involves so-called internal contractions of pairs of field operators appearing in Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{n}) and depending on different variables x_{ν} . These contractions will be characterized by joining typical lines attached the operators to be contracted. For instance, the (dashed) line in

:Φ̂_{11}(x_{1}) ⋯ Φ̂_{j1}(x_{1}) ⋯: :Φ̂_{12}(x_{2}) ⋯: :Φ̂_{1n}(x_{n}) ⋯ Φ̂_{kn}(x_{n}) ⋯:

means internal contraction of the pair of field operators Φ̂_{j_{1}}(x_{1}), Φ̂_{k_{n}}(x_{n}) (corresponding to the line type), i.e. this pair has to be replaced by its propagator^{68}

(Ω | T (Φ̂_{j_{1}}(x_{1})Φ̂_{k_{n}}(x_{n}))Ω) .

Draft, March 26, 2009

68The definition of the propagators implies that only contractions of pairs with fitting line types can be different from zero.

If Φ̂_{j_{1}}(x_{1}) and Φ̂_{k_{n}}(x_{n}) are both Fermi fields with an odd number of Fermi fields in between, a factor -1 has to be applied in addition. Here the time ordering T is defined as the least singular covariant operation fulfilling the condition

T (Φ̂(x)Φ̂′(x′)) =^{def} \{ \begin{matrix}Φ̂(x)Φ̂′(x′) & for x^{0} > x^{′0} , \\ σΦ̂′(x′)Φ̂(x) & for x^{0} < x^{′0} , \end{matrix}
where: σ =^{def} \{ \begin{matrix}−1 & if both Φ̂ and Φ̂′ are Fermi fields , \\ +1 & else . \end{matrix}

(4.166)

For simplicity we assume that Ŝ_{1}(x) , which should be a Bose field,^{69} is a monomial^{70} of field operators. Then one may prove^{71}

T (Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{n})) − ⟨Ω| T (Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{n})) Ω⟩

= :Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{n}): +:Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{j}) · · · · · · Ŝ_{1}(x_{k}) ⋯ Ŝ_{1}(x_{n}): + all other 1-fold contractions +:⋯ Ŝ_{1}(x_{j_{1}}) ⋯ Ŝ_{1}(x_{j_{2}}) · · · · · · Ŝ_{1}(x_{j_{3}}) ⋯ Ŝ_{1}(x_{j_{4}}) · · · · · ·: + all other 2-fold contractions +:⋯ Ŝ_{1}(x_{j_{1}}) ⋯ Ŝ_{1}(x_{j_{2}}) · · · · · · Ŝ_{1}(x_{j_{3}}) · · · · · ·: + all other incomplete multiple contractions .

for pairwise different x_{ν}^{0} by straightforward generalization of the techniques used in Section 3.2.1.

Let K̂(x_{1}, …, x_{n}) be any of the incompletely contracted :Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{n}):-terms (maybe :Ŝ_{1}(x_{1}) ⋯ Ŝ_{1}(x_{n}): itself) and assume that products of propagators are suitably defined, if necessary.^{72} Then its contribution to (4.164) is evaluated by rewriting

ĉ_{j_{N′}} (p′_{N′}) ⋯ ĉ_{j_{1}} (p′_{1}) K̂(x_{1}, …, x_{n}) ĉ_{k_{1}}^{†}(p_{1}) ⋯ ĉ_{k_{N}}^{†} (p_{N})  (4.167)

as the K(x_{1}, …, x_{n}; p_{1}, …, p_{N}; p′_{1}, …, p′_{N′})-fold of ˆ1 plus some linear combination of normally ordered field products. Then, by (4.165), the contribution of K̂(x_{1}, …, x_{n}) to (4.164) is given by

\frac{1}{n!} ∫ K(x_{1}, …, x_{n}; p_{1}, …, p_{N}; p′_{1}, …, p′_{N′}) dx_{1} ⋯ dx_{n} ,

where, by straightforward generalization of Wick’s theorem,

Draft, March 26, 2009

69Note that the time ordering of the Ŝ_{1}(x_{ν})-factors used in (3.26) resp. (4.165) was that for Bose fields.

70The necessary modifications for polynomials are obvious.

71As in 3.2.1, all field operators have to be formally considered as different in the contraction schemes. Recall ⟨Ω | Ŝ_{0} Ω⟩ = ⟨Ω | Ω⟩.

72Actually, as discussed in Section 3.2.2 for the λΦ̂^{4}-theory, physically correct definition of these products is the task of renormalization theory.

K(x_{1}, …, x_{n}; p_{1}, …, p_{N}; p′_{1}, …, p′_{N′})

= ĉ_{j_{N′}}(p′_{N′}) ⋯ ĉ_{j_{1}} (p′_{1}) :K̂(x_{1}, . . . . . . . . . , x_{n}): ĉ_{k_{1}}^{†}(p_{1}) ⋯ ĉ_{k_{N}}^{†} (p_{N}) + all other complete external contractions . (4.168)

External contractions are those involving at least one of the operators ĉ_{k}(p)^{†} or ĉ_{j}(p′) and for which the corresponding 2-point function (without time-ordering) is used instead of the propagator.

Feynman Diagrams with External Momenta

The nontrivial terms contribution to (4.168) can be represented by Feynman diagrams G of the following type:

1. G consists vertices (corresponding to the variables x_{ν}) numbered 1, …, V_{G} > 0 and lines representing internal or external contractions, the free ends of the latter being provided with a unique characterization of the operator (ĉ_{k}(p)^{†} or ĉ_{j}(p′)) to be contracted.

2. Every vertex is connected with a family of contraction lines corresponding to the family of field factors building Ŝ_{1}(x) .

3. Every line representing an internal contraction forms a direct link between two vertices.

4. Every line representing an external contraction has at least one free end.

5. A free end of a line representing an external contraction has to be the lowest point of this line if it corresponds to a creations operator.

6. A free end of a line representing an external contraction has to be the highest point of this line if it corresponds to an annihilation operator.

As demonstrated in 3.2.3 for the λΦ̂^{4}-theory one can show that the terms in (4.168) corresponding to diagrams with vacuum subdiagrams may be skipped.

The admitted diagrams are heuristically interpreted as follows:

• Every lower end of an exterior line represents an incoming particle corresponding to the attached information.

• Every higher end of an exterior line represents an outgoing particle corresponding to the attached information.

• Every internal line represent a virtual particle corresponding to the line type.

[figure: ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; ^{p′^{1}}; p_{1}; ^{p′^{2}}; p_{2}; 2; 1; ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; ^{p′^{2}}; p_{1}; ^{p′^{1}}; p_{2}; 2; 1]

[figure: ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; ^{p′^{1}}; p_{1}; ^{p′^{2}}; p_{2}; 1; 2; ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; ^{p′^{2}}; p_{1}; ^{p′^{1}}; p_{2}; 1; 2]

Figure 4.1: Møller scattering^{73}

• Every vertex represents an event where the particles corresponding to the connected lines interact with each other being annihilated or created respectively.

The four diagrams of QED sketched in Figure 4.1, e.g., describe scattering of two electrons with incoming momenta p_{1}, p_{2} and outgoing momenta p′_{1}, p′_{2}.

Similarly, the four diagrams of Figure 4.2 describe scattering of an electron having initial momentum p_{1} and final momentum p′_{1} with a positron having initial momentum p_{2} and final momentum p′_{2} .

4.3.3 The Feynman Rules of QED

In QED, i.e. if Ŝ_{1}(x) is given by (4.162), every vertex is connected with exactly three lines:^{74} a wavy photon line corresponding to Â_{GB} , an incoming solid fermion line corresponding to Ψ̂ , and an outgoing solid fermion line corresponding to \overline{Ψ̂} = Ψ̂^{†}γ^{0} . This is sketched in Figure 4.3.

The propagators corresponding to internal lines connecting the indices j and k > j are those given by Figure 4.4. If one is interested only in linearly po-

Draft, March 26, 2009

73Actually, the spin states should also be indicated at the ends of the external lines.

74Actually, since Ŝ_{1}(x) is a sum of monomials (recall Footnote 70) should be considered as a family of lines with indices to be summed over according to the Feynman rules formulated below.

75The orientation is relevant only for external lines.

76Die notation r_{j}, l_{k} for the indices is to indicate the original position (left/right) of the contracted field operators relative to each other. For the definition of the bosonic propagator recall (3.28)/(3.30) and (4.26). For the fermionic propagator recall (4.137).

[figure: ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; p_{2}; p_{1}; ^{p′^{2}}; ^{p′^{1}}; 2; 1; ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; ^{p′^{1}}; p_{1}; ^{p′^{2}}; p_{2}; 2; 1]

[figure: ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; p_{2}; p_{1}; ^{p′^{2}}; ^{p′^{1}}; 1; 2; ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; ⌢ ⌢ ⌢ ⌣ ⌣ ⌣; ^{p′^{1}}; p_{1}; ^{p′^{2}}; p_{2}; 1; 2]

Figure 4.2: Bhabba scattering

[figure: ν; ^{(())}; Â(x_{ν}); Ψ̂(x_{ν}); \overline{Ψ̂}(x_{ν})]

Figure 4.3: A typical^{75} vertex of QED

[figure: j; k] ˆ= ⟨Ω | T ((Ψ̂(x_{j}))_{r_{j}} (Ψ̂(x_{k})^{†}γ^{0})_{l_{k}}) Ω⟩

=^{def} +i(2π)^{−4} lim_{ϵ→+0} ∫ dq \frac{(γ^{μ}q_{μ} + m)_{r_{j}l_{k}}}{q^{2} − m^{2} + iϵ} e^{−iq(x_{j}−x_{k})}

[figure: j; k] ˆ= ⟨Ω | T ((Ψ̂(x_{j})^{†}γ^{0})_{l_{j}} (Ψ̂(x_{k}))_{r_{k}}) Ω⟩

=^{def} −i(2π)^{−4} lim_{ϵ→+0} ∫ dq \frac{(γ^{μ}q_{μ} + m)_{r_{k}l_{j}}}{q^{2} − m^{2} + iϵ} e^{−iq(x_{k}−x_{j})}

[figure: j; k; ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣] ˆ= ⟨Ω | T (Â^{μ_{j}}(x_{j})Â^{μ_{k}}(x_{k})) Ω⟩

=^{def} −g^{μ_{j}μ_{k}} i(2π)^{−4} lim_{ϵ→+0} ∫ \frac{dk}{k^{ν}k_{ν} + iϵ}e^{±ik(x_{j}−x_{k})}

Figure 4.4: Internal Lines^{76} of QED (j < k) larized asymptotic ‘particles’ then the ĉ_{j}(p) used in (4.164) are Dirac operators b̂_{σ}(p), d̂_{σ}(p) (compare (4.2.4)) or transversal photon operators^{77}

â_{ϵ}(p) =^{def} ϵ(p) · ∑_{j=1}^{3} e_{j} â^{j}(p)

with

ϵ^{0}(p) = 0 , ϵ(p) = ϵ(p)^{∗} , |ϵ(p)| = 1 , p · ϵ(p) = 0 .  (4.169)

Then the 2-point functions corresponding to external lines are those of Figure 4.5.

As in 3.2.3, any two diagrams G_{1} , G_{2} are considered as equal if they differ only by their diagramatical realization resp. are called equivalent (G_{1} ≅ G_{2}) if they differ only by the distribution of their vertex indices.

If all integrals over internal momenta exist^{78} (absence of ultraviolet divergences) then Â_{G} can be evaluated by the following naive Feynman rules of QED:

1. Assign suitable momenta to every line of G and then replace it by a corresponding factor:

−i(2π)^{−4}ϵ(k − j) lim_{ϵ→+0} \frac{(γ^{ν}q_{ν} + m)_{|_{r_{k}l_{j}}}}{q^{2} − m^{2} + iϵ} ˆ= [figure: k; j; q; −q]

−g^{μ_{j}μ_{k}} i(2π)^{−4} lim_{ϵ→+0} \frac{1}{k^{2} + iϵ} ˆ= [figure: j; k; k; −k; ⌢ ⌢ ⌢ ⌢ ⌣ ⌣ ⌣]

(2π)^{− \frac{3}{2}} ϵ′^{μ_{j}}(k′) ˆ=  ^{(}^{(}^{(}^{)}^{)} k′ ϵ′ −k′ _{j}_{}

\sqrt{2m}(2π)^{− \frac{3}{2}} (ω_{σ}^{(−)}(−p)^{∗}γ^{0})_{l_{j}} ˆ=  [figure: p; j; p; σ]

etc.

2. For j = 1, …, V_{G} replace vertex j by the factor^{79}

−ie (γ_{μ_{j}})_{l_{j}r_{j}} (2π)^{4}δ(P)

Draft, March 26, 2009

77Here e_{j} and â^{j}(p) are to be understood in the sense of (4.57).

78This is the case if G is a so-called tree diagram, i.e. if it does not contain closed loops.

79The δ-function results from integration over x_{1}, …, x_{V_{G}} .

(^{)} k′ ϵ′ j_{}\begin{aligned}ˆ= & ⟨Ω| â_{ϵ′}(k′)Â^{μ_{j}}(x_{j})Ω⟩ \\ & = (2π)^{− \frac{3}{2}} ϵ^{′μ_{j}}(p′) e^{+ik′x_{j}}_{|_{k^{′0}=|k′|}} \end{aligned}

(^{)} j k ϵ^{}\begin{aligned}ˆ= & ⟨Ω| Â^{μ_{j}}(x_{j})â_{ϵ}^{†}(k)Ω⟩ \\ & = (2π)^{− \frac{3}{2}} ϵ^{μ_{j}}(p) e^{−ikx_{j}}_{|_{k^{0}=|k|}} \end{aligned}

[figure: p′; σ′; j] \begin{aligned}ˆ= & ⟨Ω| b̂_{σ′}(p′) (Ψ̂(x_{j})^{†}γ^{0})_{l} Ω⟩ \\ & = \sqrt{2m} (2π)^{− \frac{3}{2}} (ω_{σ′}^{(+)}(p′)^{∗}γ^{0})_{l} e^{+ip′x_{j}}_{|_{p^{′0}=ω_{_{p′}′}}} \end{aligned}

[figure: p; σ; j] \begin{aligned}ˆ= & ⟨Ω| (Ψ̂(x_{j}))_{r} b̂_{σ}(p)^{†}Ω⟩ \\ & = \sqrt{2m} (2π)^{− \frac{3}{2}} (ω_{σ}^{(+)}(p))_{r} e^{−ipx_{j}}_{|_{p^{0}=ωp}} \end{aligned}

[figure: p′; σ′; j] \begin{aligned}ˆ= & ⟨Ω| d̂_{σ′}(p′) (Ψ̂(x_{j}))_{r} Ω⟩ \\ & = \sqrt{2m} (2π)^{− \frac{3}{2}} (ω_{σ′}^{(−)}(−p′))_{r} e^{+ip′x_{j}}_{|_{p^{′0}=ω_{p′}}} \end{aligned}

[figure: p; j; σ] \begin{aligned}ˆ= & ⟨Ω| (Ψ̂(x_{j})^{†}γ^{0})_{l} d̂_{σ}^{†}(p)Ω⟩ \\ & = \sqrt{2m} (2π)^{− \frac{3}{2}} (ω_{σ}^{(−)}(−p)^{∗}γ^{0})_{l} e^{−ipx_{j}}_{|_{p^{0}=ωp}} \end{aligned}

(^{)} k′ ϵ′ k ϵ ˆ= ⟨Ω| â_{ϵ′}(k′)â_{ϵ}(k)^{†}Ω⟩ = 2|k| ϵ′(k′) · ϵ(k′) δ(k − k′)

p′ σ′ p σ ˆ= ⟨Ω| b̂_{σ′}(p′)b̂_{σ}(p)^{†}Ω⟩ = 2ω_{p} δ_{σσ′}δ(p − p′)

p′ σ′ p σ ˆ= ⟨Ω| d̂_{σ′}(p′)d̂_{σ}(p)^{†}Ω⟩ = 2ω_{p} δ_{σσ′}δ(p − p′)

Figure 4.5: External lines of QED where

P =^{def} \{ \begin{matrix}sum of all 4-momenta^{80} assigned to line ends \\ connected with vertex j. \end{matrix}

3. Take the product of all factors and sum over all indices r_{j}, l_{j}, μ_{j} and integrate over all momenta assigned to internal lines (only one integration per internal line).

4. Finally multiply by σ_{G} ∈ {+1, −1} to be determined as follows:

Write down the corresponding contraction scheme, e.g.

ĉ_{j_{N′}} (p′_{N′}) ⋯ ĉ_{j_{1}} (p′_{1}) :Ŝ_{1}(x_{1}) · · · · · · Ŝ (x_{V_{G}}): ĉ_{k_{1}}^{†}(p_{1}) ⋯ ĉ_{k_{N}}^{†} (p_{N}) ,

and rearrange the operators such that all contracted pairs become direct neighbors without changing the relative order of the operators forming any such pair. Then σ_{G} is the signum of the overall permutation resulting this way.

4.3.4 Example: Compton Scattering

For Compton scattering, i.e. for electron-photon scattering in second order^{81} of perturbation theory, only graphs equivalent to

[figure: 2; 1; k, ϵ; p, σ; p′, σ′; k′, ϵ′; or; 2; 1; k′, ϵ′; p, σ; p′, σ′; k, ϵ]

are relevant. The factors corresponding to the graph

[figure: −q; p; k; −k′; −p′; 2; 1; G_{1} =; k, ϵ; p, σ; p′, σ′; k′, ϵ′; +q]

Draft, March 26, 2009

‘particle’ type.

80Here, the 3-momenta assigned to external lines have to lifted to the mass shell of the asymptotic 81This means that only contributions with V_{G} ≤ 2 are considered.

4.3. THE S-MATRIX OF QUANTUM ELECTRODYNAMICS (QED) 185 are the following:

\sqrt{2m} (2π)^{−3/2} (ω_{σ}^{(+)}(p))_{r_{1}}

(2π)^{−3/2}ϵ^{μ_{1}}(k) ,

−ie (γ_{μ_{1}})_{l_{1}r_{1}} (2π)^{4} δ(p + k − q) ,

−i (2π)^{−4} lim_{ϵ→+0} \frac{(γ^{ν}q_{ν} + m)_{r_{2}l_{1}}}{q^{2} − m^{2} + iϵ}

−ie (γ_{μ_{2}})_{l_{2}r_{2}} (2π)^{4} δ(q − p′ − k′) ,

(2π)^{−3/2}ϵ′^{μ_{2}}(k′) ,

\sqrt{2m} (2π)^{−3/2} (ω_{σ′}^{(+)}(p′)^{∗}γ^{0})_{l_{2}} .

The signum of the permutation mapping

b̂_{σ′}(p′)Ψ̂(x_{1})^{†}Ψ̂(x_{1})Ψ̂(x_{2})^{†}Ψ̂(x_{2})b̂_{σ}^{†}(p)

onto

(Ψ̂(x_{1})b̂_{σ}^{†}(p)) (Ψ̂(x_{1})^{†}Ψ̂(x_{2})) (b̂_{σ′}(p′)Ψ̂(x_{2})^{†})

is σ_{G_{1}} = −1 . Therefore

Â_{G_{1}} = \frac{me^{2}}{2iπ^{2}} lim_{ϵ→+0} ∫ δ(p + k − q)δ(q − p′ − k′)ω_{σ′}^{(+)}(p′)^{∗}γ^{0} \begin{matrix}ϵ′ \\ / \end{matrix}(k′) ·

·\frac{q/+ m}{q^{2} − m^{2} + iϵ} ϵ/(k)ω_{σ}^{(+)}(p) dq

= \frac{me^{2}}{2iπ^{2}} δ(p + k − p′ − k′)ω_{σ′}^{(+)}(p′)^{∗}γ^{0} \begin{matrix}ϵ′ \\ / \end{matrix}(k′) \frac{p/+ k/+ m}{2p · k} ϵ/(k)ω_{σ}^{(+)}(p) ,

where we use the usual conventions

x/ =^{def} γ^{μ}x_{μ} , p/ =^{def} γ^{μ}p_{μ} etc.

Similarly we get

Â_{G_{2}} = \frac{me^{2}}{2iπ^{2}} δ(p + k − p′ − k′)ω_{σ′}^{(+)}(p′)^{∗}γ^{0} ϵ/(k) \frac{p/− \begin{matrix}k′ \\ / \end{matrix}+ m}{−2p · k′} \begin{matrix}ϵ′ \\ / \end{matrix}(k′)ω_{σ}^{(+)}(p)

for

[figure: 2; q; 1; −q; G_{2} =; k′, ϵ′; −k′; p, σ; p; p′, σ′; −p′; k, ϵ; k]

and hence

⟨b̂_{σ′}(p′)â_{ϵ′}(k′)Ω | (Ŝ_{0} − ˆ1)_{2. order} b̂_{σ}(p)^{†}â_{ϵ}(k)^{†}Ω⟩

= \frac{me^{2}}{2iπ^{2}} δ(p + k − p′ − k′) ω_{σ′}^{(+)}(p′)^{∗}Γ(p, k, k′)ω_{σ}^{(+)}(p) ; where:

Γ(p, k, k′) =^{def} γ^{0} (\begin{matrix}ϵ′ \\ / \end{matrix}(k′) \frac{p/+ k/+ m}{2p · k} ϵ/(k) + ϵ/(k) \frac{p/− \begin{matrix}k′ \\ / \end{matrix}+ m}{−2p · k′} \begin{matrix}ϵ′ \\ / \end{matrix}(k′)) .

(4.170)

The cross section for unpolarized incoming and outcoming electrons^{82} is

σ(p, k, ϵ, ϵ′)

= \frac{(2π)^{2}}{8p · k} ∑_{σ,σ′=±} ∫ |\frac{⟨b̂_{σ′}(p′)â_{ϵ′}(k′)Ω | (Ŝ_{0} − ˆ1)_{2. order} b̂_{σ}(p)^{†}â_{ϵ}(k)^{†}Ω⟩}{δ(p′ + k′ − p − k)}|^{2} ×

× δ(p′ + k′ − p − k) \frac{dk′}{2k^{′0}} \frac{dp′}{2p^{′0}} ,

(4.171)

where p resp. k is the momentum of the incoming electron resp. photon and ϵ resp. ϵ′ is the polarization of the incoming resp. outgoing photon. Note that all 4-momenta have to be on the corresponding mass shell:

k^{0} = |k| , k′^{0} = |k′| , p^{0} = ω_{p} = \sqrt{p^{2} + m^{2}} , p′^{0} = ω_{p′} .

In order to evaluate (4.171) we have to calculate

I =^{def} ∫ ∑_{σ,σ′=±} |ω_{σ′}^{(+)}(p′)^{∗} Γ(p, k, k′) ω_{σ}^{(+)}(p)|^{2} δ(p′ + k′ − p − k) \frac{dp′}{2ω_{p′}} \frac{dk′}{2|k′|} . (4.172)

By p = 0 and ϵ^{0}(k) = ϵ^{′0}(k′) = 0 , (4.108) gives

( p/+ m) ϵ/^{(′)}(p)ω_{σ}^{(+)}(p) = − ϵ/^{(′)}(p) ( p/− m) ω_{σ}^{(+)}(p) = 0 .

With

(ω^{′∗}Γ̂ω)^{∗}^{T} = (ω^{′∗})^{∗} \binom{}{Γ̂^{∗}}^{T} ω^{∗}^{T} = (ω^{∗}Γ̂^{∗}ω′)^{T}

this implies

|ω^{′∗}Γ̂ω^{∗}|^{2} = (ω^{′∗}Γ̂ω) (ω^{∗}Γ̂ω′) .

In the standard representation (4.115) the latter together with

∑_{σ=±} (ω_{σ}^{(+)}(p))_{r} (ω_{σ}^{(+)}(p)^{∗}γ^{0})_{l} = ( \frac{p/+ m}{2m})_{rl}  (4.173)

Draft, March 26, 2009 82This means summation of the polarizations of the outgoing electron and averaging over the the polarizations of the incoming electron. Therefore, we need an additional factor \frac{1}{2} compared to the formula of Exercise 53.

4.3. THE S-MATRIX OF QUANTUM ELECTRODYNAMICS (QED) 187 implies

I = ∫ \frac{dp′}{2ω_{p′}} \frac{dk′}{2 |k′|} δ(p′ + k′ − p − k) Tr(γ^{0} Γ̂(p, k, k′)^{∗} \frac{\begin{matrix}p′ \\ / \end{matrix}+ m}{2m}) ,

where

Γ̂(p, k, k′) =^{def} γ^{0} ( \frac{\begin{matrix}ϵ′ \\ / \end{matrix}(k′) k/ ϵ/(k)}{2 |k| m} + \frac{ϵ/(k) \begin{matrix}k′ \\ / \end{matrix} \begin{matrix}ϵ′ \\ / \end{matrix}(k′)}{2 |k′| m})  (4.174)

and hence

Γ̂(p, k, k′)^{∗} = γ^{0} ( \frac{ϵ/(k) k/ \begin{matrix}ϵ′ \\ / \end{matrix}(k′)}{2 |k| m} + \frac{\begin{matrix}ϵ′ \\ / \end{matrix}(k′) \begin{matrix}k′ \\ / \end{matrix} ϵ/(k)}{2 |k′| m})  (4.175)

(because of γ = −γ^{∗} and γ^{0} = γ^{0}^{∗}).

In the laboratory system, i.e. for

p = 0 , p^{0} = m ,

the conditions

p′ + k′ = p + k , p^{′2} = m^{2} , k^{′2} = 0

are well known to imply the so-called Compton condition

k^{′0} = |k′| = \frac{|k|}{1 + \frac{|k|}{m}(1 − cos ϑ)} , cos ϑ =^{def} \frac{k}{|k|} · \frac{k′}{|k′|} . (4.176)

Therefore p′ and k′ are uniquely fixed, in the laboratory system, by k and the direction of k′. Hence, there is a function f(|k| , (ϑ, φ)) with

f(|k| , (ϑ, φ)) = Tr(γ^{0}γ̂(p, k, k′)^{∗} \frac{\begin{matrix}p′ \\ / \end{matrix}+ m}{2m}) .  (4.177)

(ϑ, φ) are polar angles of k′ (with ϑ = 0 for k ∥ k′). For p = 0 , therefore,

I = ∫ \frac{|k′|^{2} d |k′| dω}{2 |k′|} ∫ dp′ θ(p^{′0})δ(p^{′2} − m^{2})δ(p′ + k′ − p − k)f(k, ϑ, φ)
= dω d|k′| \frac{|k′|}{2}θ(p^{0} + k^{0} − k^{′0})δ ((p + k − k′)^{2} − m^{2}) f(k, ϑ, φ)
= ∫ dω ∫_{0}^{p^{0}+k^{0}} d|k′| \frac{|k′|}{2}δ(2m |k| − |k′| (2m + 2 |k| (1 − cos ϑ))) f(k, ϑ, φ)

(upper boundary redundant due to (4.176))

= ∫ dω \frac{m |k|}{4 (m + |k| (1 − cos ϑ))^{2}} f(k, ϑ, φ) .

By (4.176), this implies

\frac{dI}{dΩ} = \frac{|k′|^{2}}{4m |k|}f(k, ϑ, φ) .  (4.178)

(4.177) and (4.172)/(4.175), on the other hand, imply

\begin{aligned}f(k, ϑ, φ) & = & Tr( \frac{\begin{matrix}p′ \\ / \end{matrix}+ m}{2m} ( \frac{\begin{matrix}ϵ′ \\ / \end{matrix}(k′) k/ ϵ/(k)}{2 |k| m} + \frac{ϵ/(k) \begin{matrix}k′ \\ / \end{matrix} \begin{matrix}ϵ′ \\ / \end{matrix}(k′)}{2 |k′| m}) × \\ & & × \frac{p/+ m}{2m} ( \frac{ϵ/(k) k/ \begin{matrix}ϵ′ \\ / \end{matrix}(k′)}{2 |k| m} + \frac{\begin{matrix}ϵ′ \\ / \end{matrix}(k′) \begin{matrix}k′ \\ / \end{matrix} ϵ/(k)}{2 |k′| m})) . \end{aligned}  (4.179)

Calculation of this trace is easily done by computer algebra. In Section 17.7 of the REDUCE^{83} manual there is already a listing of the corresponding program:

  ON DIV;
  MASS K= 0, KP= 0, P= MC, PP= MC; VECTOR EP,E;
  MSHELL K,KP,P,PP;
  LET P.EP= 0, P.E= 0, P.PP= MC**2+K.KP, P.K= MC*NK,P.KP=
      MC*NKP, PP.EP= -KP.EP, PP.E= K.E, PP.K= MC*NKP, PP.KP=
      MC*NK, K.EP= 0, K.KP= MC*(NK-NKP), KP.E= 0, EP.EP= -1, E.E=-1;
  (G(L,PP) + MC)/(2*MC)*(G(L,E,EP,K)/(2*K.P) + G(L,EP,E,KP)/(2*KP.P))
    * (G(L,P) + MC)/(2*MC)*(G(L,K,EP,E)/(2*K.P) + G(L,KP,E,EP)/(2*KP.P))$
  WRITE "1/4 Trace = ",WS;

This gives the following result for (4.179):

1/4 Trace = MC^{(−2)} ∗ (1/2∗EP.E^{2} + 1/8∗NKP∗NK^{(−1)} + 1/8∗NKP^{(−1)}∗NK - 1/4)

The explicit meaning of this REDUCE message is:

\frac{1}{4}f(k, ϑ, φ) = (mc)^{−2} (\frac{1}{2}(ϵ′ · ϵ)^{2} + \frac{1}{8} \frac{|k′|}{|k|} + \frac{1}{8} \frac{|k|}{|k′|} − \frac{1}{4}) .

Since we use natural units this, together with (4.178), proves the so-called Klein-Nishina formula:

\frac{d}{dΩ}σ(p, k, ϵ, ϵ′)_{|_{p=0}} = \frac{1}{4}(r_{0})^{2} (\frac{|k′|}{|k|})^{2} (\frac{|k′|}{|k|} + \frac{|k|}{|k′|} − 2 + 4 (ϵ′(k′) · ϵ(k))^{2}) ,

where:

r_{0} =^{def} classical radius of the electron
= \frac{e^{2}}{4πm} in natural units in the Heaviside system
ˆ= 2, 82 . . . · 10^{−13}cm .

Draft, March 26, 2009

83For MATHEMATICA the package TRACER by M. Jamin and M.E. Lautenbacher (Jamin and Lautenbacher, 1993) is useful in this context.

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Index

1-particle mass shell, 40

adiabatic limit, 131 almost local, 83 annihilation

operator, 48 part, 51

anti-symmetric, 7 asymptotic

completeness, 79 condition, 75, 76

asymptotically localized, 89 automorphism, 26 axial current, 166

Banach algebra, 20 Bogoliubov-Shirkov

causality, 124

generalized, 134 conjecture, 131

Bose field

local, 172 bounded

states, 76

canonical commutation relations, 32, 105 field theory, 106 quantization, 110

Cauchy Schwarz inequality, 23 causality

Bogoliubov-Shirkov, 124

generalized, 134 formal, 126

charge operator, 99 charged scalar field, 93 chronological

product, 111 classical, 8 commutant, 21 compatible, 8 complex Lorentz group, 72 Compton

condition, 187 scattering, 184

contraction, 114

external, 179 counter term, 117, 131 coupling constant, 104, 119 covariance

formal, 125 covering

group

universal, 156 mapping, 156

creation

part, 51 cyclic vector, 53

decoherent histories, 14 diagram

tree-, 182 diagrams

admitted, 120 connected, 122 equal, 121, 182 equivalent, 121, 182

Dirac equation, 164 distributive, 8 dynamical semi-group, 15 dynamics, 15 Dyson series, 104

effect, 23 energy, 39 essentially local, 133 extended tube, 73

faithful, 27 Fermi field

local, 172 Feynman

parameterization, 137 rules, 120

field operators, 51 Fock

functional, 33 ground state, 33 property, 50 representation, 33, 35 space, 47 vacuum, 35

folium, 37 four-momentum, 40 Friedrichs’ theorem, 67

gauge transformations of second kind,

94 generalized

functions, 50 wave operator, 77

generating functional, 33

Haag’s theorem, 109 Haag-Ruelle-Kastler

sequence, 81 Heisenberg

group, 32 picture, 75

homogeneous history, 14

inertial mass, 39 infinite tensor product, 35 interaction

-picture, 101 self-, 104

involution, 21 irreducibility, 99 isomorphism, 26 isotony, 99

Jaffe space, 56 Jauch-Piron condition, 10

Klein’s paradox, 170 Klein-Gordon

equation, 43 solution

smooth, 44 Klein-Nishina formula, 188

Lüders postulate, 13 Laplace transform, 61 lattice, 7 left handed, 173 line

external, 120 internal, 120

local almost, 83 commutativity, 52, 55, 99 essentially, 133 field algebras, 99 observables

algebra of, 79 localized

asymptotically, 89 logic, 8, 10 Lorentz group, 72

measurement of first kind, 13 microcausality, 52 minimal coupling, 176 momentum amplitude, 43 morphism, 26

negative frequency part, 51 nonlocalizable, 56 nonoverlapping, 79 normal ordering, 172

observable, 12 order

of an entire function, 136 orthocomplementation, 8

Paley-Wiener theorem, 44 partial state, 25 Pauli matrices, 155 principle, 168

Pauli-Villars regularization, 117 perturbation, 104 Poincaré covariance, 99 Poisson bracket, 108 polar decomposition, 157 position amplitude, 43 positive

frequency part, 51 semi definite, 59

probability measure, 8 product

chronological, 111 inequality, 20

projection valued measure, 18 propagator, 177 proper complex Lorentz group, 72 propositions, 9

quantization, 34 quasi-support, 133 quasilocal operator, 128 questions, 9

reflexive, 7 renormalization, 119, 131 retarded commutator, 135 right handed, 173

scattering amplitudes, 177 Bhabba, 180 Compton, 184 Møller, 180 state, 76

Schrödinger representation, 33 semi-ordering, 7 simple

quantum reasoning, 12 tests, 9

spectral measure, 18 theorem, 18

spectrum condition, 54 standard quantum logic, 15 ∗-antiautomorphism, 37 ∗-morphism, 26 state, 9, 23

instantaneous, 101 symmetry, 15

tadpole, 114

three-momentum, 39 time-translations, 15 transition

matrix, 177

probability, 17, 176 transitive, 7

ultraviolet divergences, 117 unitarily equivalent, 29 unitarity

formal, 125 universal

covering group, 156 lower bound, 7 upper bound, 8

vacuum diagram, 121 Fock, 109 physical, 109 state, 54 vector, 48

variation

of a measure, 135 von Neumann algebra, 21

weak asymptotic completeness, 77 local commutativity, 74, 98

weakly continuous, 15 modular, 8

Weyl equations, 168 relations, 32

Wightman axioms, 139 theory, 54, 94