Master Dissertation

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Now from equation (5.15) T2(t1, t2) = A2(t1, t2) − A ′ 2(t1, t2) = Θ(t2 − t1)(T1(t1)T1(t2) − T1(t2)T1(t1)) + T1(t1)T1(t2) = Θ(t1 − t2)T1(t1)T1(t2) − Θ(t2 − t1)T1(t2)T1(t1) =: T {T1(t1)T1(t2)}. FIXME: draw parallel to S-matrix with usage of Schrödinger equation. 37
5.3 Splitting of Distributions In our previous example we saw that the splitting of Dn into advanced and retarded parts could be done by use of the Heaviside function. This was because we were dealing with operators in Hilbert space. But in our setting we are considering distributions which cannot simply be multiplied with discontinuous step-functions. In this section we consider the properties of the advanced and retarded distributions. First a technicality. Theorem 5.3. Let Y = P ∪ Q, P = ∅, P ∩ Q = ∅, |Y | = n1 ≤ n − 1 and x /∈ Y . If {Q, x} > P ,|Q| = n2, then we have If {Q, x} < P , then we have R ′ n1+1(Y, x) = −Tn2+1(Q, x)Tn1−n2 (P ). (5.16) A ′ n1+1(Y, x) = −Tn1−n2 (P )Tn2+1(Q, x). (5.17) Proof. Equation (5.16) is proved in [6]. Therefore we only prove equation (5.17). A ′ n1+1(Y, x) = ˜Tn3 (X)Tn−n3 (Y ′ , x), where P2 is the set of all partitions Y = X ∪ Y ′ such that X = ∅. Now let and Now since causality implies P2 Y ′ = Y1 ∪ Y2, where Y1 = Y ′ ∩ P and Y2 = Y ′ ∩ Q X = X1 ∪ X2, where X1 = X ′ ∩ P and X2 = X ∩ Q Q ∪ {x} < P, Y2 < Y1, X2 < X1 and Y1 > Y2 ∪ {x}, A ′ n1+1(Y, x) = ˜Tn3 (X)Tn−n3 (Y ′ , x) P2 = ˜T (X2) ˜ T (X1)T (Y1)T (Y2, x), (5.18) P 0 4 subscripts are omitted for simplicity. P 0 4 form is the set of all partitions of the P 0 4 : P = X1 ∪ Y1, Q = X2 ∪ Y2 and X1 ∪ X2 = ∅. 38
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 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: 5.2 Example - In the Hilbert Space
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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