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By M. Aizenman (Chief Editor)

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Where gt := We prove the existence of the asymptotic fields and show that they realize a CCR representation satisfying the Fock property. This result was first proven in [HK]. Up to technical details due to a more singular character of the interaction, the proof of the existence of asymptotic fields follows the proof of the analogous result in [DG1]. The proof of the Fock property is based on the general theory of CCR representations. Its main ingredient is the concept of the number operator associated to the regular CCR representation described in Sect.

Dv w = θ(∇v w). (34) Now we have to prove that it is a HKT-connection. We claim that the connection D preserves the hypercomplex structure. This claim is equivalent to Dv (Ia w) = Ia Dv w. Lifting to U , it is equivalent to θ(∇v Ia w) = Ia θ(∇v w). Since the direct sum decomposition is invariant of the hypercomplex structure, the projection map θ is hypercomplex. Therefore, it commutes with the complex structures. Then the above identity is equivalent to θ(∇v Ia w) = θ(Ia ∇v w). This identity holds because ∇ is hypercomplex.

42 J. Derezi´nski, C. Gérard The second part of our paper is devoted to the study of spatially cutoff P (ϕ)2 Hamiltonians. In Sect. g. K]. One of the most difficult results about such Hamiltonians are the so-called higher order estimates due to Rosen. They are described with some of their consequences in Sect. 7. Strictly speaking, their proof contained in [Ro2] does not cover the class of Hamiltonians that we consider. Therefore, we indicate how to modify the arguments of [Ro2] to cover our class of coupling functions g.

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Communications in Mathematical Physics - Volume 213 by M. Aizenman (Chief Editor)


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