Master Dissertation

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General notation. Let α = (α1, . . . , αn) ∈ N n 0 be a multi-index and x = (x1, . . . , xn) ∈ R n a point, then |α| = α1 + α2 + . . . + αn. α! = (α1, α2, . . . , αn)! = α1!α2! · · · αn!. x α = x α1 1 · · · xαn n ∂j = ∂/∂xj for j = 1, . . . , n. ∂ α = ∂ α1 1 · · · ∂αn n . Dj = −i∂j for j = 1, . . . n. f (α) (x) = ∂ α f(x). 7
1.2 Formal Power Series The concept of formal power series will be of much importance to this project. They make it possible to employ much of the analytical machinery of usual power series but in settings which have no natural notion of convergence. Let R be a commutative ring with identity. We call the elements of R the scalars. Let R N be the set of all infinite sequences in R. Now we define addition and multiplication of two such sequences in the usual way, that is, and the Cauchy product (an) + (bn) = (an + bn), n (an) × (bn) = k=0 akbn−k = h+k=n ahbk, which in a sense is discrete convolution. Note that addition and multiplication are well-defined operations because each term only depends on a finite number of terms from the original sequences. Theorem 1.1. (R N , +, ×) is a commutative ring with the multiplicative identity 1 := (1, 0, 0, . . .) and additive identity 0 := (0, 0, . . .). Proof. The only non-trivial property is multiplicative associativity. We need to prove that ((a × b) × cn) = (a × (b × c)n). This follows from ((a × b) × cn) = (a × b)hck = and h+k=n (a × (b × c)n) = r+h=n r+s+k=n ar(b × c)h = r+s+k=n arbsck arbsck It is easy to see from the definition of the Cauchy product that if we define X = (0, 1, 0, 0, . . .) then every element of R N on the form (a0, . . . , aN, 0, . . .) can be written on the form N anX n . n≥0 8 .
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: Chapter 1 Introductory Theory 1.1 N
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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