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Renormalisation Circumvents Haag's Theorem Some no-go results & canonical quantum fields Axiomatics & Haag's theorem Renormalisation bypasses Haag's theorem Renormalisation circumvents Haag's theorem Lutz Klaczynski, HU Berlin Theory seminar DESY Zeuthen, 31.03.2016 Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Some no-go results & canonical quantum fields Axiomatics & Haag's theorem Renormalisation bypasses Haag's theorem Outline 1 Some no-go results & canonical quantum fields Sharp-spacetime quantum fields Interaction picture Theorems by Powers & Baumann (CCR/CAR) Schrader's result 2 Axiomatics & Haag's theorem Wightman axioms Haag vs. Gell-Mann & Low Haag's theorem for nontrivial spin 3 Renormalisation bypasses Haag's theorem Renormalisation Mass shift wrecks unitary equivalence Conclusion Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Sharp-spacetime quantum fields Some no-go results & canonical quantum fields Interaction picture Axiomatics & Haag's theorem Theorems by Powers & Baumann (CCR/CAR) Renormalisation bypasses Haag's theorem Schrader's result The quest for understanding canonical quantum fields 1 1 Constructive approaches by Glimm & Jaffe : d ≤ 3 2 2 Axiomatic quantum field theory by Wightman & G˚arding : axioms to construe quantum fields in terms of operator theory hard work, some results (eg PCT, spin-statistics theorem) interesting: triviality results (no-go theorems) 3 other axiomatic approaches, eg algebraic quantum field theory General form of no-go theorems φ quantum field with properties so-and-so ) φ is trivial. 3 forms of triviality (1) φ free, (2) φ = cI or (3) φ = 0 1J.Glimm, A.Jaffe, Springer (1981) 2A. S. Wightman, L.G˚arding,Arkiv f¨orFysik, 28, 129 - 184 (1964) Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Sharp-spacetime quantum fields Some no-go results & canonical quantum fields Interaction picture Axiomatics & Haag's theorem Theorems by Powers & Baumann (CCR/CAR) Renormalisation bypasses Haag's theorem Schrader's result Triviality of sharp-spacetime fields Theorem (Wightman, 1964) Let '(x) be a Poincar´e-covariant Hermitian scalar field, that is, U(a; Λ)'(x)U(a; Λ)y = '(Λx + a) and suppose it is a well-defined operator with vacuum Ω0 in its domain. Then there is a constant c 2 C such that p n=2 '(x)Ω0 = cΩ0 thus hΩ0j'(x1):::'(xn)Ω0i = c : Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Sharp-spacetime quantum fields Some no-go results & canonical quantum fields Interaction picture Axiomatics & Haag's theorem Theorems by Powers & Baumann (CCR/CAR) Renormalisation bypasses Haag's theorem Schrader's result Canonical computation Free canonical field Z 3 d p 1 −ip·x ip·x y '(x) = 3 p [e a(p) + e a (p)]: (2π) 2Ep Question: jj'(x)Ω0jj = ? Answer: unphysical question, result: Z 3 2 1 d p 1 jj'(x)Ω0jj = hΩ0j'(x)'(x)Ω0i = = 1 2 (2π)3 pp2 + m2 Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Sharp-spacetime quantum fields Some no-go results & canonical quantum fields Interaction picture Axiomatics & Haag's theorem Theorems by Powers & Baumann (CCR/CAR) Renormalisation bypasses Haag's theorem Schrader's result What to make of it? 1 Implement Heisenberg's uncertainty: Z '(t; f ) = d3x f (x)'(t; x); free fields smoothed out in space, then Z 3 2 2 1 d p jf (p)j jj'(t; f )Ω0jj = < 1: 2 (2π)3 pp2 + m2 2 Too strong an assumption: x 7! hΨjU(x; I)Φi continuous for all state vectors Ψ; Φ 2 H, must be weakened. Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Sharp-spacetime quantum fields Some no-go results & canonical quantum fields Interaction picture Axiomatics & Haag's theorem Theorems by Powers & Baumann (CCR/CAR) Renormalisation bypasses Haag's theorem Schrader's result Expedient: operator-valued distributions 1 Smearing in both space and time such that f 7! hΨj'(f )Φi Ψ; Φ 2 H; is distribution on Schwartz space S(M). 2 Poincar´ecovariance: for all f Schwartz U(a; Λ)'(f )U(a; Λ)y = '(fa; Λgf ); where (fa; Λgf )(x) = f (Λ−1(x − a)). 3 Sharp-time field '(t; f ) := lim '(δt ⊗ f ) !0 if existent, fδt g>0 Dirac family centred at t 2 R. Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Sharp-spacetime quantum fields Some no-go results & canonical quantum fields Interaction picture Axiomatics & Haag's theorem Theorems by Powers & Baumann (CCR/CAR) Renormalisation bypasses Haag's theorem Schrader's result Interaction picture Hamiltonian Wightman, 1967 R 3 Hamiltonian H = H0 + Hint, where Hint = d x Hint(x). Then Z iH0t −iH0t 3 iH0t −iH0t HI (t) = e Hinte = d x e Hint(x)e entails jjHI (t)Ω0jj = 0 for all t, thus HI (t)Ω0 = 0. −iH t Proof. By e 0 Ω0 = Ω0 and translational invariance one has Z Z 2 3 3 jjHI (t)Ω0jj = d x d y hΩ0jHint(x)Hint(y)Ω0i Z Z 3 3 = d x d y hΩ0jHint(0)Hint(y)Ω0i Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Sharp-spacetime quantum fields Some no-go results & canonical quantum fields Interaction picture Axiomatics & Haag's theorem Theorems by Powers & Baumann (CCR/CAR) Renormalisation bypasses Haag's theorem Schrader's result Response: simply vacuum diagrams jjHI (t)Ω0jj not of interest, yet in Vacuum expectation of Dyson's series R −i dt HI (t) hΩ0jSΩ0i = hΩ0jT fe gΩ0i X (−1)n Z Z = dt ::: dt hΩ jTfH (t ) ::: H (t )gΩ i n! 1 n 0 I 1 I n 0 n≥0 there one finds Z Z dt ds hΩ0jTfHI (t)HI (s)gΩ0i = vacuum diagrams but cancel out in Gell-Mann-Low formula. Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Sharp-spacetime quantum fields Some no-go results & canonical quantum fields Interaction picture Axiomatics & Haag's theorem Theorems by Powers & Baumann (CCR/CAR) Renormalisation bypasses Haag's theorem Schrader's result Canonical (anti)commutation relations (CCR/CAR) Triviality of fields obeying CCR/CAR Theorem (Baumann): scalar field '(t; f ) in d ≥ 5 Nontrivial commutator given by ['(t; f );@t '(t; g)] = i(f ; g) ) '(t; f ) free Weyl form of CCR: d ≥ 4 (Sinha). Theorem (Powers): Dirac field (t; f ) in d ≥ 3 Nontrivial anticommutator given by f (t; f ); y(t; g)g = i(f ; g) ) (t; f ) free: Response: so be it. Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Sharp-spacetime quantum fields Some no-go results & canonical quantum fields Interaction picture Axiomatics & Haag's theorem Theorems by Powers & Baumann (CCR/CAR) Renormalisation bypasses Haag's theorem Schrader's result Euclidean version of Haag's theorem Theorem (Schrader, 1974) 0 2 dµ0(') Gaussian measure on distributions D (R ), interaction term Z 2 V`(λ, ') = λ d x : P('(x)) : B` 2 P bounded normalised polynomial, B` = [−`=2; `=2] and −[V`(λ,ϕ)−E`(λ)] dµ`,λ(') = e dµ0(') −E (λ) R −V (λ,ϕ) with e ` = e ` dµ0('). Then dµ1,λ exists but dµ0 and dµ1,λ have mutually disjoint support: Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Some no-go results & canonical quantum fields Wightman axioms Axiomatics & Haag's theorem Haag vs. Gell-Mann & Low Renormalisation bypasses Haag's theorem Haag's theorem for nontrivial spin Wightman framework: axioms (scalar field) Axiom 0 (Relativistic Hilbert space) Hilbert space H, strongly continuous unitary representation (a; Λ) 7! U(a; Λ) of the connected Poincar´egroup, unique vacuum Ω0 2 H U(a; Λ)Ω0 = Ω0 8(a; Λ): Axiom I (Spectral condition) generator of translations Pµ has spectrum in forward lightcone σ(P) ⊂ V +. Axiom II (Quantum field) Poincar´e-covariant operator family f'(f )gf 2S(M), common dense and Poincar´e-stabledomain D ⊂ H, generate dense subspace D0 = C['(f ): f 2 S(M)]Ω0 ⊂ H. Axiom III (Locality) ['(f );'(g)] = 0 if f ; g have spacelike-separated support. Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Some no-go results & canonical quantum fields Wightman axioms Axiomatics & Haag's theorem Haag vs. Gell-Mann & Low Renormalisation bypasses Haag's theorem Haag's theorem for nontrivial spin Haag's theorem Haag's theorem (Wightman, Jost, Schroer, Reeh, Schlieder) 1 '(t; f ) and '0(t; f ) sharp-time Wightman quantum fields of mass m > 0 with Hilbert spaces H and H0, respectively. In any spacetime dimension d ≥ 2. 2 Suppose at time t there is a unitary intertwiner V : H0 ! H such that −1 '(t; f ) = V '0(t; f )V : Then if '0(t; f ) is free, so is '(t; f ). Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Some no-go results & canonical quantum fields Wightman axioms Axiomatics & Haag's theorem Haag vs. Gell-Mann & Low Renormalisation bypasses Haag's theorem Haag's theorem for nontrivial spin Proof of Haag's theorem part I (sketch) 1. First prove intertwiner relates vacua: V Ω0 = Ω. Then follows −1 −1 hΩ0j'0(t; f )'0(t; h)Ω0i = hΩ0jV '(t; f )VV '(t; h)V Ω0i = hΩj'(t; f )'(t; h)Ωi: 2. This implies hΩ0j'0(t; x)'0(t; y)Ω0i = hΩj'(t; x)'(t; y)Ωi in the sense of distributions, ie 1 Z d3p eip·(x−y) hΩj'(t; x)'(t; y)Ωi = : 2 (2π)3 pp2 + m2 Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Some no-go results & canonical quantum fields Wightman axioms Axiomatics & Haag's theorem Haag vs. Gell-Mann & Low Renormalisation bypasses Haag's theorem Haag's theorem for nontrivial spin Proof of Haag's theorem part II (sketch) 3. Use Poincar´ecovariance to show 1 Z d3p eip·(x−y) hΩj'(x)'(y)Ωi = : 2 (2π)3 pp2 + m2 for spacelike distances (x − y)2 < 0.
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