A Nuclear Weyl Algebra

TL;DR

This paper constructs a topologically complete Weyl algebra with a convergent star product for locally convex vector spaces, connecting algebraic and analytical frameworks, and illustrating applications in classical field theory.

Contribution

It introduces a locally convex topology on the Weyl algebra ensuring convergence of the star product, extending algebraic models to analytic settings, and explores its properties and applications.

Findings

Star product converges absolutely in the completed algebra.

The completed algebra contains exponential functions.

The construction is functorial and compatible with symmetries.

Abstract

A bilinear form on a possibly graded vector space $V$ defines a graded Poisson structure on its graded symmetric algebra together with a star product quantizing it. This gives a model for the Weyl algebra in an algebraic framework, only requiring a field of characteristic zero. When passing to $\mathbb{R}$ or $\mathbb{C}$ one wants to add more: the convergence of the star product should be controlled for a large completion of the symmetric algebra. Assuming that the underlying vector space carries a locally convex topology and the bilinear form is continuous, we establish a locally convex topology on the Weyl algebra such that the star product becomes continuous. We show that the completion contains many interesting functions like exponentials. The star product is shown to converge absolutely and provides an entire deformation. We show that the completion has an absolute Schauder basis whenever $V$ has an absolute Schauder basis. Moreover, the Weyl algebra is nuclear iff $V$ is nuclear. We discuss functoriality, translational symmetries, and equivalences of the construction. As an example, we show how the Peierls bracket in classical field theory on a globally hyperbolic spacetime can be used to obtain a local net of Weyl algebras.