Weyl calculus in Wiener spaces and in QED

TL;DR

This paper develops a semiclassical Weyl calculus for infinite-dimensional Hilbert spaces, establishing boundedness and covariance properties, and applies it to analyze quantum electrodynamics models involving spin and electromagnetic fields.

Contribution

It introduces a novel semiclassical Weyl calculus on infinite-dimensional spaces and demonstrates its application to QED, including boundedness results and evolution analysis.

Findings

Boundedness of pseudodifferential operators with explicit bounds

Metaplectic covariance of the calculus

Identification of certain QED evolutions as pseudodifferential operators

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.