On Weyl calculus in infinitely many variables

TL;DR

This paper develops an abstract framework for Weyl calculus in infinite-dimensional settings, extending previous methods to infinite-dimensional Lie group representations and illustrating with Heisenberg groups.

Contribution

It introduces a novel approach to pseudo-differential Weyl calculus for operators in infinite variables, extending finite-dimensional techniques to infinite-dimensional Lie group representations.

Findings

Extended Weyl calculus to infinite-dimensional Lie groups.

Reproduced classical Weyl-Hörmander calculus in finite dimensions.

Applied the approach to infinite-dimensional Heisenberg groups.

Abstract

We outline an abstract approach to the pseudo-differential Weyl calculus for operators in function spaces in infinitely many variables. Our earlier approach to the Weyl calculus for Lie group representations is extended to the case of representations associated with infinite-dimensional coadjoint orbits. We illustrate the approach by the case of infinite-dimensional Heisenberg groups. The classical Weyl-H\"ormander calculus is recovered for the Schr\"odinger representations of the finite-dimensional Heisenberg groups.