Rigorous Quantum Field Theory A Festschrift for Jacques Bros
Jacques Bros has greatly advanced our present understanding of rigorous quantum field theory through numerous fundamental contributions. This book arose from an international symposium held in honour of Jacques Bros on the occasion of his 70th birthd
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Series Editors H. Bass J. Oesterlé A. Weinstein
Rigorous Quantum Field Theory A Festschrift for Jacques Bros
Anne Boutet de Monvel Detlev Buchholz Daniel Iagolnitzer Ugo Moschella Editors
Birkhäuser Verlag Basel xBoston xBerlin
Editors: Anne Boutet de Monvel ,QVWLWXWGH0DWKpPDWLTXHVGH-XVVLHX 8QLYHUVLWp3DULV 175 rue du Chevaleret 3DULV )UDQFH HPDLO DERXWHW#PDWKMXVVLHXIU
Daniel Iagolnitzer 6HUYLFHGH3K\VLTXH7KpRULTXH CEA/Saclay Orme des Merisiers *LIVXU a (with respect to the order of N ) for all a ∈ A. The set α(A) = (α(a1 ), . . . , α(a|A| )) inherits its order from A. In what follows, the letter U is reserved for the ordered set N \ (A ∪ α(A)) (of course, U is empty in the contractions considered above where |A| = n2 ). For any given contraction, the corresponding momenta kA and kU are called the internal and external momenta, respectively. The situation is very similar on the noncommutative Minkowski space, where by application of (2.3) we find φ(q + x1 ) · · · φ(q + xn ) ≡ 0 for n odd and for n even, i φnf (q) = dxN f (xN ) dμ(kA ) e−ikA (xA −xα(A) ) e− 2 ki Qkj
kα(A) =−kA
C∈C(N ) |A|= n 2
=
C∈C(N ) |A|= n 2
i . (3.34) =: − 4πR2 This, together with the fact that φ is a free field, proves the statement.
In the homogeneous case the equation φ = 0 is clearly invariant under the transformation φ → φ+λ1. This invariance is still present as the following lemma shows Lemma 2. The equation of motion for the quantum field φ is invariant under the gauge transformation γ λ (φ) := φ + λ, λ ∈ C. Proof. The identity operator 1, as an operator-valued distribution, associates to a test function its total integral. In the equation of motion we then have γ λ (φ) = φ = −
1 1 φ(v0 ) = − γ λ (φ)(v0 ) , 4πR2 4πR2
where the last equality follows from the fact that the state v0 = [4πR2 h] has zero total integral.
3 Massless Scalar Field in a Two-dimensional de Sitter Universe
35
We can then define a gauge transformation as the automorphism of the field algebra generated by γ λ : φ → φ + λ, λ ∈ C. (3.35) Let us now introduce the operator Q = i φ+ (v0 ) − φ− (v0 ) ,
(3.36)
where the operators φ± (f ) are defined as creators–annihilators of states v0 in the Fock-Krein space K. This is a continuous operator from the n-particle space K(n) to K(n±1) . Moreover it is easily verified that Q satisfies the following commutation relations: [Q, φ(f )] = −i dσ(x)f (x). (3.37) One can compute directly that — exactly as in the flat case — 1 dσ(x)f (x) ; [φ+ (v0 ), φ− (v0 )] = 0 . [φ± (v0 ), φ(f )] = ∓ 2
(3.38)
The natural question now arises as to whether this charge Q is the integral of a local expression. The classical charge of any solution of the wave equation in d + 1 dimensions ϕ = 0 is defined by integrating along any space-like dsurface the “time-like” derivative along the direction orthogonal to the surface dvΣ ∂nˆ ϕ. (3.39) Σ
Such an expression is “conserved” (i.e. independent of the spacelike surface Σ) because of the equation of motion. The integrand in eq.
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