Master Dissertation

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Thus we note that the scattering matrix is in fact just the limit of the time evolution in the interaction picture. That is S = lim lim Ũ(t, s). t→∞ s→−∞ If the interaction V (t) is a bounded operator we may write the unitary operator in terms of the Dyson series 3 Ũ(t, s) = 1 + ∞ (−i) n n=1 Claim. t s s s t s dt1 t1 s tn−1 dt2 · · · dtn s ˜ V (t1) · · · ˜ V (tn). (3.2) tn−1 dt1 . . . dtn = s 1 (t − s)n n! Proof. It is obvious for n = 1. Assume it is right for n = k − 1 then t tk−1 t t1 1 dt1 · · · dtk = dt1 (k − 1)! = This proves the claim by induction. s 1 (k − 1)! t 1 = (k − 1)! = 1 (t − s)k k! s t−s 0 s k−1 dτ (t1 − s) k−1 dt1 h k−1 dt1 It follows that the sum (3.2) converges in the operator-norm since each term is bounded by t s dt1 . . . tn−1 s dtn ˜ V (t1) . . . ˜ V (tn) = 1 n! (t − s)n ˜ V ) n If the the interaction decreases for large times such that then the S-matrix is given by S = ∞ (−i) n n=0 +∞ −∞ +∞ −∞ dt1 which is also norm-convergent. 3 See [8] page 326. t1 dsV (s) < ∞, dt2 · · · −∞ 21 tn−1 −∞ dtn ˜ V (t1) · · · ˜ V (tn),
Chapter 4 The Mathematical Setting of QFT 4.1 The Fock Space Let H n = H ⊗ · · · ⊗ H be the tensor product space of n single-particle Hilbert spaces H. Remembering that identical particles must obey either Bose or Fermi statistics 1 , we symmetrize the states φn ∈ H n S + n φn = 1 n! π φn(xπ1 , . . . , xπn) in the case of bosons and antisymmetrize (−1) π (xπ1 , . . . , xπn) S − n φn = 1 n! π in the case of fermions. We then form a Hilbert space from the direct sum of tensor products. Definition 4.1. The Fock space F is defined as F ± = ⊕ ∞ n=0S ± n H n with H0 := {λ|0〉|λ ∈ C} where |0〉 is called the vacuum. It is easy to prove that F is indeed a Hilbert space. It follows from the continuity of the inner products. We simply state it here. Proposition 4.2. The Fock space F is a Hilbert space with the inner product ∞ 〈Φ, Ψ〉 = 〈Φn, Ψn〉n, n=0 where 〈·, ·〉n is the inner product on Hn. 1 See [8] chapter 6 22
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: Note that 〈Woutφ, ψ〉 = 〈 li
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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