Master Dissertation

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Contents 1 Introductory Theory 6 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Formal Power Series . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Affine Transformations of Distributions . . . . . . . . . . . . 12 2 The Poincaré Group 14 2.1 The Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 The Poincaré Group . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Spinor Representations of the Lorentz Group . . . . . . . . . 17 3 The Scattering Matrix 19 4 The Mathematical Setting of QFT 22 4.1 The Fock Space . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 The Wightman Axioms . . . . . . . . . . . . . . . . . . . . . 25 5 The Method of Epstein and Glaser 27 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.2 Example - In the Hilbert Space Setting of Quantum Mechanics 36 5.3 Splitting of Distributions . . . . . . . . . . . . . . . . . . . . . 38 6 Regularly Varying Functions 41 7 Splitting of Numerical Distributions 49 7.1 The Singular Order of a Distribution . . . . . . . . . . . . . . 49 7.2 Case I: Negative Singular Order . . . . . . . . . . . . . . . . . 54 7.2.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . 54 7.2.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . 59 7.3 Case II: Positive Singular Order . . . . . . . . . . . . . . . . . 64 8 Application to QED 66 8.1 Using the Game Plan . . . . . . . . . . . . . . . . . . . . . . 66 8.2 The Adiabatic Limit . . . . . . . . . . . . . . . . . . . . . . . 69 ii
9 The Microlocal Approach - A Condition on the Wave Front Set. 71 iii
Page 1: Master Dissertation The Method of E
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 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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Master Dissertation

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