Master Dissertation

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Thus the condition of Definition 1.14 is satisfied by ˆ d0. Further from (7.3) and hence by (7.7) a m 〈d0(ax), φ(x)〉 = 〈d0(x), φ( x a )〉 = 〈 ˆ d0( p a ), ˇ φ(p)〉. 〈 ˆ d0(p), ˇ φ(p)〉 = 〈d0(x), φ(p)〉 〈d0(ax), φ(x)〉 = a −(m+ω) 〈d0(x), φ(x)〉. The condition of Definition 1.14 is satisfied by d0. The singular order now gives us a way to distinguish two different situations in the splitting process. We investigate the splitting first in the case where ω < 0 and then for ω ≥ 0. 53
7.2 Case I: Negative Singular Order 7.2.1 Existence In this subsection we will show that there exists a decomposition of d into an advanced part a and a retarded part r. Choose ɛ > 0 such that ω + ɛ < 0 then t ω+ɛ → ∞ for t → 0 − hence by (6.11) ρ(t) → ∞ for t → 0. Hence by (7.1) lim δ→0 〈d(x), φ(x δ )〉 = lim δ→0 δm 〈d0(x), φ(x)〉 〈d(δx), φ(x)〉 = lim = 0. (7.8) δ→0 ρ(δ) Let v = (v1, . . . , vn−1) ∈ Γ + n−1 (0) be an arbitrary but fixed vector and define a hyperplane by PH := {x ∈ M n−1 n−1 |v · x := 〈vj, xj〉 = 0}. Lemma 7.6. All products satisfy and j=1 〈vj, xj〉 ≥ 0 if x ∈ Γ + n−1 (0) (7.9) 〈vj, xj〉 ≤ 0 if x ∈ Γ − n−1 (0). (7.10) Proof. For all j, (v0 j )2 − 3 i=1 (vi j )2 ≥ 0 hence v0 j ≥ 3 i=1 (vi j )2 since v0 j ≥ 0. Now if x ∈ Γ+ n−1 (0), then for all j 〈vj, xj〉 = v 0 j x 0 j − 3 i=1 v i jx i j ≥ v 0 j x 0 j − 3 i=1 If on the other hand x ∈ Γ − n−1 (0), then x0 j 〈vj, xj〉 = v 0 j x 0 j − 3 i=1 (v i j )2 v i jx i j ≤ v 0 j x 0 j − x 0 j ≤ x 0 j FIXME: read this through carefully. 3 Thus PH splits causal support (5.20) of d. 54 i=1 3 i=1 (x i j )2 ≥ v 0 j x 0 j − v 0 j x 0 j = 0 ≤ 0 for all j, hence 3 i=0 (v i j )2 − x 0 j (v i j )2 3 i=0 (v i j )2 = 0
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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 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: Let n, m ∈ N then and Hence mf( n
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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