Compositions of states and observables in Fock spaces

Abstract

This article is concerned with compositions in the context of three standard quantizations in the Fock spaces framework, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for time dynamics issues (see [15], [7]). Anti-Wick quantization and Segal-Bargmann transforms are implied here for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick symbol of the composition of two operators, we obtain an absolutely converging series and for the Weyl symbol, the remainder term of the expansion is absolutely converging, still in the Fock spaces framework.

Positivity results for Weyl's pseudodifferential calculus on the Wiener space

“…The preceding articles contain different constructions of the calculus, L 2 -boundedness properties, the definition of two different symbol classes. Tools and results analogous to the finite dimensional results exist, such as a Beals characterization [5,3], composition results [4], in the shape of semiclassical asymptotic expansions in powers of a small parameter h. One of the constructions relies on the notion of Wigner function, as in the finite dimensional case. Parallel calculus are available, like the Anti-Wick calculus (associating an operator with a symbol) and the Wick calculus (associating a function with an operator).…”

“…Parallel calculus are available, like the Anti-Wick calculus (associating an operator with a symbol) and the Wick calculus (associating a function with an operator). A special kind of heat operators links these calculi together [20,4,3]. The Wick calculus is defined thanks to a family of coherent states, which are elements of a space of square summable functions defiened on the Wiener space and have a counterpart in the Fock space.…”

Integral formulas for the Weyl and anti-Wick symbols

“…In the article [9], Hübner and Spohn have introduced the so-called scattering identification operator (unbounded operator) and considered the wave operator in the context of the infinite dimension, concerning some physical model. With L. Jager, we have proved in [1] that the first operator (scattering identification) can also be defined with the anti-Wick quantization. That is, this operator is defined by an integral formula and it would be interesting to know if the second operator (wave) shares that property.…”