Master Dissertation

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Lemma 5.2. The product (x1 + x2 + . . . + xn) m has n−1 m m − k1 m − i ki · · · · , k1 terms on the form x k1 1 · · · xkn n . k2 Proof. We note that the number of ways to pick x1, k1 times from m possibilities is given by the binomial coefficient. That is, there must be m k1 terms on the form x k1 1 (x2 + · · · + xn) m−k1 . Now there are m − k1 multiplications left and thus m−k1 ways to pick x2, k2 times. Hence there k2 are m m−k1 k1 terms on the form x k1 k2 1 xk2 2 (x3 + · · · xn) m−k1−k2 . This way we end up with n−1 m m − k1 m − i ki · · · · , k1 terms on the form x k1 1 · · · xkn n . Proof of Theorem 5.1. First note that by the lemma k2 〈Tm, (f1 + f2 + · · · + fm) ⊗ · · · ⊗ (f1 + f2 + · · · + fm)〉 = n−1 m m − k1 m − i ki · · · · k1+···+km=m k1 since Tm is symmetric. Hence 〈Tm, f1 + f2 + · · · + fm〉 k2 kn kn × 〈Tm, f k1 1 kn ⊗ · · · ⊗ f km m 〉, = 〈Tm, (f1 + f2 + · · · + fm) ⊗ . . . ⊗ (f1 + f2 + · · · + fm)〉 − n−1 m m − k1 m − i ki · · · · 〈Tm, f k1 1 k1+···+km=m (k1,...,km)=(1,...,1) Hence by 5.5 k1 〈Tm, f1 ⊗ · · · ⊗ fm〉〈Tn−m, fm+1 ⊗ · · · ⊗ fn〉 k2 kn ⊗ · · · ⊗ f km m 〉. = 〈Tn, (f1 + f2 + · · · + fm) ⊗ . . . ⊗ (f1 + f2 + · · · + fm) ⊗ (fm+1 ⊗ · · · ⊗ fn)〉 − n−1 m m − k1 m − i ki · · · · k1+···+km=m (k1,...,km)=(1,...,1) × 〈Tn, f k1 1 k1 k2 ⊗ f km m ⊗ (fm+1 ⊗ · · · ⊗ fn)〉 = 〈Tn, f1 ⊗ f2 ⊗ · · · ⊗ fm ⊗ fm+1 ⊗ · · · ⊗ fn〉 as wanted. 33 kn
We see that the Tn’s are time-ordered products (therefore the T ). Similarly the causality condition for S −1 (g) is that if supp g1 < supp g2, then S(g1 + g2) −1 = S(g1) −1 S(g2) −1 . This again implies that ˜Tn(x1, . . . , xn) = ˜ Tm(x1, . . . , xm) ˜ Tn−m(xm+1, . . . , xn), if {x1, . . . , xm} < {xm+1, . . . , xn}. We are now ready to sketch the game plan for the inductive construction of the time ordered products Tn. 34
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: equal powers must have equal coeffi
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

Master Dissertation

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