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A ′ 2(x1, x2) = ˜ T1(x1)T1(x2) = −T1(x1)T1(x2) = e 2 γ µ abγν cd : ψa(x1)ψb(x1) :: ψc(x2)ψd(x2) : Aµ(x1)Aν(x2) : ψa(x1)ψb(x1)ψ c(x2)ψd(x2) : = e 2 γ µ ab γν cd = e 2 γ µ ab γν cd + : ψ a(x1) | ψb(x1)ψ c(x2) | ψd(x2) : + : | ψ a(x1)ψb(x1)ψ c(x2)ψd(x2) | : + : | ψ a(x1) | ψb(x1)ψ c(x2) | ψd(x2) | : × : ψa(x1)ψb(x1)ψ c(x2)ψd(x2) : : Aµ(x1)Aν(x2) : + : | Aµ(x1)Aν(x2) | : + 1 i S(+) bc (x1 − x2) : ψa(x1)ψd(x2) : + 1 i S(−) da (x2 − x1) : ψb(x1)ψc(x2) : − S (+) bc (x1 − x2)S (−) da (x2 − x1) × : Aµ(x1)Aν(x2) : +gµνiD (+) 0 (x1 − x2) . (8.1) From (5.8) we see that the retarded distribution is given by (8.1) simply by substituting x1 ↔ x2. For convenience we also interchange the indices µ ↔ ν, a ↔ c and b ↔ d, arriving at R ′ 2(x1, x2) = T1(x2) ˜ T1(x1) = −T1(x2)T1(x1) : ψa(x1)ψb(x1)ψ c(x2)ψd(x2) : = e 2 γ µ ab γν cd − 1 i S(+) da (x2 − x1) : ψb(x1)ψc(x2) : − 1 i S(−) bc (x1 − x2) : ψa(x1)ψd(x2) : − S (+) da (x2 − x1)S (−) bc (x1 − x2) × : Aµ(x1)Aν(x2) : +gµνiD (+) 0 (x2 − x1) 67 . (8.2)
Finally we can calculate the difference D(x1, x2) by (5.11). D(x1, x2) = R ′ n(x1, x2) − A ′ n(x1, x2) = e 2 γ µ abγν cd ψa(x1)ψb(x1)ψ c(x2)ψd(x2) : gµνi(D (+) 0 (x2 − x1) − D (+) 0 (x1 − x2)) − 1 S i (+) da (x2 − x1) + S (−) da (x2 − x1) : ψb(x1)ψc(x2) :: Aµ(x1)Aν(x2) : − gµν S (+) da (x2 − x1)D (+) 0 (x2 − x1) + S (−) da (x2 − x1)D (+) 0 (x1 − x2) : ψb(x1)ψc(x2) : − 1 i (S(−) bc (x1 − x2) + S (+) bc (x1 − x2))ψa(x1)ψd(x2) :: Aµ(x1)Aν(x2) : − gµν S (−) bc (x1 − x2)D (+) 0 (x2 − x1) + S (+) bc (x1 − x2)D (+) 0 (x1 − x2) : ψa(x1)ψd(x2) : + S (+) bc (x1 − x2)S (−) da (x2 − x1) − S (−) bc (x1 − x2)S (+) da (x2 − x1) : Aµ(x1)Aν(x2) : + 1 i gµν S (−) bc (x1 − x2)S (+) da (x2 − x1)D (+) 0 (x2 − x1) . (8.3) − S (+) bc (x1 − x2)S (−) da (x2 − x1)D (+) 0 (x1 − x2) This formula may be considered as a correspondent to the Feynman rules. Each term can be treated separately and represents different scattering scenarios, where the field operators create and annihilate particles. Like the Feynman rules, they may be illustrated by graphs. But notice the huge difference that while all our terms are well-defined distributions, the Feynman rules are ill-defined for closed loops, which here are represented by terms with products of the propagators S and D. Lets us consider the first term. First we note that D + 0 (x1 − x2) − D + 0 (x2 − x1) is in fact the Jordan-Pauli distribution for mass m = 0, where the full Jordan-Pauli distribution is given by 2 D(x) = sgn x0 2π (δ(x2 ) − Θ(x 2 ) m 2 √ x 2 J1(m √ x 2 )). Note that for the 1-dimensional δ-distribution, Therefore 〈δ(δ 2 x 2 ), φ(δ 2 x 2 )〉 = δ −2 〈δ(x 2 ), φ(x 2 )〉. D0(δx) = sgn x0 2π δ(δ2x 2 sgn x0 δ(x ) = 2π 2 ) δ2 Hence if we choose ρ(δ) = δ 2−n we get the quasi-asymptotics 2 [6] page 89. lim δ→0 ρ(δ)δn sgn x0 D0(δx) = 2π δ(x2 ). 68
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Master Dissertation The Method of E
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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
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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 and 26: Chapter 4 The Mathematical Setting
Page 27 and 28: where ψ is the time-dependent Dira
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: Chapter 8 Application to QED 8.1 Us
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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