Master Dissertation

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Of these properties, causality plays a key role in the method of Epstein and Glaser. Equation (5.4) leads to the condition stated in the following theorem. Theorem 5.1. If {x1, . . . , xm} > {xm+1, . . . , xn}, then Tn(x1, . . . , xn) = Tm(x1, . . . , xm)Tn−m(xm+1, . . . , xn), Before proving this result we first note that if (5.4) is satisfied, S(g1 + g2) = = ∞ n=0 n 1 λ n! d 4 x1 · · · d 4 xnTn(x1, . . . , xn) × (g1(x1) + g2(x1)) · · · (g1(xn) + g2(xn)) ∞ n λ n 1 m!(n − m)! n=0 m=0 d 4 x1 · · · d 4 xnTn(x1, . . . , xn) × (g2(x1) · · · g2(xm)) · · · (g1(xm+1) · · · g1(xn)), as there are 2n terms from the product of the test-functions and ways to pick g2 m times. On the other hand S(g2)S(g1) = = = ∞ m=0 m 1 λ m! d 4 x1 · · · d 4 xmTm(x1, . . . , xm) × g2(x1) · · · g2(xm) ∞ k 1 × λ k! k=0 ∞ m=0 k=0 ∞ n=0 m=0 d 4 ˜x1 · · · d 4 ˜xkTk(˜x1, . . . , ˜xk) × g1(˜x1) · · · g1(˜xk) ∞ m+k 1 λ m!k! d 4 x1 · · · d 4 xmd 4 ˜x1 · · · d 4 ˜xk × Tm(x1, . . . , xm)Tk(˜x1, . . . , ˜xk) n! m!(n−m)! × g2(x1) · · · g2(x1) · · · g2(xm)g1(˜x1) · · · g1(˜xk) n λ n 1 dx1 · · · dxnTm(x1, . . . , xm) m!(n − m)! × Tn−m(xm+1, . . . , xn)g2(x1) · · · g2(xm) × g1(xm+1) · · · g1(xn). From the definition of the metric on formal power series we know that 31
equal powers must have equal coefficients. We conclude that for every m, n = d 4 x1 · · · d 4 xnTn(x1, . . . , xn)g2(x1) · · · g2(xm)g1(xm+1) · · · g1(xn) d 4 x1 · · · d 4 xnTm(x1, . . . , xm)Tn−m(xm+1, . . . , xn) × g2(x1) · · · g2(xm)g1(xm+1) · · · g1(xn) (5.5) Further note that from the calculus of the tensor product 〈T2, (f1 + f2) ⊗ (f1 + f2)〉 = 〈T2, f1 ⊗ f1〉 + 〈T2, f1 ⊗ f2〉 + 〈T2, f2 ⊗ f1〉 + 〈T2, f2 ⊗ f2〉, FIXME: maybe better to write it out in n dimensions and since Tm is symmetric 〈Tm, f1 ⊗ f2〉 = 1 2 (〈Tm, (f1 + f2) ⊗ (f1 + f2)〉 − 〈Tm, f1 ⊗ f1〉 − 〈Tm, f2 ⊗ f2〉). (5.6) Now for transparency and to get a good idea of how to prove Theorem 5.1 let us prove it for n = 3 and m = 2. That is, Claim. 〈T3, f1 ⊗ f2 ⊗ f3〉 = 〈T2, f1 ⊗ f2〉〈T1, f3〉. Proof. By 5.6 Hence by 5.5 〈T2, f1 ⊗ f2〉 = 1 〈T2, (f1 + f2) ⊗ (f1 + f2)〉 − 〈T2, f1 ⊗ f1〉 2 − 〈T2, f2 ⊗ f2〉 〈T2, f1 ⊗ f2〉〈T1, f3〉 = 1 〈T3, (f1 + f2) ⊗ (f1 + f2) ⊗ f3〉 − 〈T3, f1 ⊗ f1 ⊗ f3〉 2 − 〈T3, f2 ⊗ f2 ⊗ f3〉 = 1 〈T3, f1 ⊗ f1 ⊗ f3〉 + 〈T3, f2 ⊗ f2 ⊗ f3〉 + 2〈T3, f1 ⊗ f2 ⊗ f3〉 2 − 〈T3, f1 ⊗ f1 ⊗ f3〉 − 〈T3, f2 ⊗ f2 ⊗ f3〉 = 〈T3, f1 ⊗ f2 ⊗ f3〉. Which proves the claim. Now before proving this for arbitrary n and m we need the following lemma. 32
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: Expected properties of the S-matrix
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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