Master Dissertation

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We site Corollary 1.3 of [6] here 2 . Corollary 4.3. Any (bounded) operator A ∈ F can be expressed as A = A (−) + A (+) where A (−) contain annihilation and A (+) creation operators, only. Definition 4.4. We say that an operator A ∈ F is normally ordered if all annihilation operators are placed to the right of the creation operators. The normal ordering of A is expressed by : A :. In the light of Corollary 4.3 we may define. Definition 4.5. A contraction between two field operators A and B is defined by | AB | := [A (−) , B (+) ]±, where [·, ·]± is the commutator in the case of Bose fields and anti-commutator in the case of Fermi fields. The following theorem tells us how to order field operators normally. We state it without its proof. 3 Theorem 4.6 (Theorem of Wick). A product of n field operators can be normally ordered as follows: A1A2 · · · An =: A1A2 · · · An : + | A1A1 | · · · An + permutations + : | A1 | A2 · · · Aj | . . .An : + · · · + : | A1A2 | | A3A4 | · · · : +permutations, where the sum contains all normal products with all possible pairings of contractions. FIXME: think about proving it We assume the contractions are complex numbers, which therefore can be taken out of the normal products. Though we have to remember to take the sign into account in the case of Fermi operators. In QED there exists only three contractions. | ψa(x)ψb(y) | := {ψ (−) a (x), ψ (+) b (y)} =: 1 i S(+) ab | ψa(x)ψb(y) | := {ψ (−) a (x), ψ (+) b (y)} =: 1 i S(−) ba (x − y) (4.1) (y − x) (4.2) | Aµ(x)Aν(y) | := [A (−) µ (x), A (+) ν (y)] = gµνiD (+) 0 (x − y), (4.3) 2 I have taken the liberty of reformulating the corollary such that its meaning will be more apparent for my application. I have not altered the meaning of it, though. 3 See [6] 23
where ψ is the time-dependent Dirac field 4 and ψ = ψ (+) γ 0 is the Dirac adjoint. The gamma matrices are γ µ 0 σµ = σµ 0 with σµ given by (2.6) and (2.7). We here regard (4.1), (4.2) and (4.3) as definitions of the operators S = S (−) + S (+) and D (+) 0 , with Aµ given by [6] page 148. 4 Se [6] page 83 24 ,
Page 1 and 2: Master Dissertation The Method of E
Page 3 and 4: 9 The Microlocal Approach - A Condi
Page 5 and 6: Preface Overview This dissertation
Page 7 and 8: Chapter 8 In Chapter 8 we will turn
Page 9 and 10: Chapter 1 Introductory Theory 1.1 N
Page 11 and 12: 1.2 Formal Power Series The concept
Page 13 and 14: Theorem 1.6. Let anX n be a formal
Page 15 and 16: 1.3 Affine Transformations of Distr
Page 17 and 18: Chapter 2 The Poincaré Group 2.1 T
Page 19 and 20: Definition 2.2. Given some point x
Page 21 and 22: 〈y, y〉 = det σ(y) = det A det
Page 23 and 24: Note that 〈Woutφ, ψ〉 = 〈 li
Page 25: Chapter 4 The Mathematical Setting
Page 29 and 30: The Wightman Axioms. Axiom 1 (quant
Page 31 and 32: Axiom 0 (Well-definedness) Let g =
Page 33 and 34: Expected properties of the S-matrix
Page 35 and 36: equal powers must have equal coeffi
Page 37 and 38: We see that the Tn’s are time-ord
Page 39 and 40: 5.2 Example - In the Hilbert Space
Page 41 and 42: 5.3 Splitting of Distributions In o
Page 43 and 44: A proof of this is found in [6] pag
Page 45 and 46: Lemma 6.3. Let f : [x0, ∞) → R
Page 47 and 48: We want to show by induction that f
Page 49 and 50: substituting t := s − x. We may w
Page 51 and 52: Proof. Reduce ρ to be slowly varyi
Page 53 and 54: In our setting this is not defined,
Page 55 and 56: Let n, m ∈ N then and Hence mf( n
Page 57 and 58: 7.2 Case I: Negative Singular Order
Page 59 and 60: Therefore φ(x)d(x) also has quasi-
Page 61 and 62: Corollary 7.10. The limit exists.
Page 63 and 64: It is obvious for k = 0 and k = 1.
Page 65 and 66: By compactness of the unit-ball and
Page 67 and 68: 7.3 Case II: Positive Singular Orde
Page 69 and 70: Chapter 8 Application to QED 8.1 Us
Page 71 and 72: Finally we can calculate the differ
Page 73 and 74: Theorem 8.1. Identify ˆg(k) with t
Page 75 and 76: in the sense that if h ∈ D(M), th
Page 77 and 78:
Note that the definition agrees wit
Page 79 and 80:
Proof. Let xm be a maximal point, t
Page 81:
[12] Walter Rudin: Functional Analy
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Master Dissertation

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