Weyl calculus in Wiener spaces and in QED

Abstract

The concern of this article is a semiclassical Weyl calculus on an infinite dimensional Hilbert space $H$. If $(i, H, B)$ is a Wiener triplet associated to $H$, the quantum state space will be the space of $L^2$ functions on $B$ with respect to a Gaussian measure with $h/2$ variance, where $h$ is the semiclassical parameter. We prove the boundedness of our pseudodifferential operators (PDO) in the spirit of Calder\'on-Vaillancourt with an explicit bound, a Beals type characterization, and metaplectic covariance. An application to a model of quantum electrodynamics (QED) is added in the last section, for fixed spin $1/2$ particles interacting with the quantized electromagnetic field (photons). We prove that some observable time evolutions, the spin evolutions, the magnetic and electric evolutions when subtracting their free evolutions, are PDO in our class.