Master Dissertation

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Abstract The main purpose of this thesis is to prove that the causality axiom allows a well-defined perturbation theory of quantum electrodynamics. This will be done by using the method of Epstein and Glaser. The thesis introduces the theory of formal power series and regularly varying functions, both tools needed in the main proofs. Basic theory of the scattering matrix, the Poincaré Group and its representations will be introduced. We will use the concept of singular order to investigate the splitting of distributions into an advanced and a retarded part and show when they exist and when they are unique. Applications and the adiabatic limit will be discussed. The thesis is concluded with a consideration of the microlocal approach to the method of Epstein and Glaser and by showing that the translations invariance can be substituted by a condition on the wave front set. Sammenføjning Hovedform˚alet med dette speciale er at vise at man ved hjælp af kausalitetsaxiomet kan indføre en veldefineret pertubationsteori for kvanteelektrodynamik. Dette gøres ved hjælp af Epstein og Glasers metode. Specialet introducerer teorierne for formelle potensrækker og regulært varierende funktioner, som begge skal bruges under hovedbeviserne. Basal teori for spredningsmatrice, Poincaré gruppen og dens repræsentationer vil blive introducerede. Vi vil bruge singulær orden til at undersøge opdelingen af distributioner i en fremtids og fortids del og vise, hvorn˚ar de eksisterer, og hvorn˚ar de er entydige. Anvendelser og den adiabatiske grænse vil blive diskuteret. Specialet afsluttes med at overveje den mikrolokale tilgangsvinkel til Epstein og Glasers metode og med at vise at translations varians kan afløses af en betingelse p˚a bølgefront mængden. Specialet er skrevet p˚a engelsk. 1
Preface Overview This dissertation is about the causal approach to finite quantum electrodynamics (QED). The word causal refers to the main assumption on which the method we will introduce is based. In short it is the statement that what happens at time t > s doesn’t influence what happens at earlier times t < s. The word finite means that there do not arise any ultraviolet divergences. The traditional formalism of quantum field theory (QFT) is known to have problems with ultraviolet divergences. These occur because of misuse of the mathematics. The problem lies in the fact that QFT deals with distributions instead of functions and that the rules of calculus for these are not the same. One cannot simply multiply a distribution by a discontinuous step-function and other simple rules of calculus like multiplication of distributions can be a very complicated matter. In fact, this is the main problem of [4]. In practice the ultraviolet divergencies are dealt with by regularization and renormalization FIXME: explain these. But instead of using these firstaid features on an ill-defined theory one could introduce QFT using the mathematics in the right way from the beginning. This is the basic aim of this dissertation. Structure Chapter 1 In Chapter 1 I introduce the tools needed in our later work. Section 1.2 is about formal power series which is an abstraction of the usual power series used in calculus. A generalization which lets us use most of the wellknown machinery from calculus to settings which have no natural notion of convergence. This will become useful later on when we introduce the scattering matrix as a formal power series. In section 1.3 we will introduce Affine Transformations of distributions. 2
Page 1 and 2: Master Dissertation The Method of E
Page 3: 9 The Microlocal Approach - A Condi
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,
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Let n, m ∈ N then and Hence mf( n
Page 57 and 58:
7.2 Case I: Negative Singular Order
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Therefore φ(x)d(x) also has quasi-
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Corollary 7.10. The limit exists.
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It is obvious for k = 0 and k = 1.
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By compactness of the unit-ball and
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7.3 Case II: Positive Singular Orde
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Chapter 8 Application to QED 8.1 Us
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Finally we can calculate the differ
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Theorem 8.1. Identify ˆg(k) with t
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in the sense that if h ∈ D(M), th
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Note that the definition agrees wit
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Proof. Let xm be a maximal point, t
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[12] Walter Rudin: Functional Analy
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Master Dissertation

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