Star product algebras of test functions

TL;DR

This paper characterizes when Gelfand-Shilov spaces form topological algebras under the Moyal star product, showing the conditions on parameters for algebraic structure and convergence, relevant for quantum field theory on noncommutative spacetime.

Contribution

It establishes the precise conditions on Gelfand-Shilov spaces for the Moyal star product to be continuous and convergent, linking functional analysis with quantum field theory applications.

Findings

Gelfand-Shilov spaces are topological algebras under the Moyal star product if and only if α ≥ β.

Series expansion of the Moyal product converges absolutely in these spaces if and only if β < 1/2.

The star product depends continuously on the noncommutativity parameter in the topology of these spaces.

Abstract

We prove that the Gelfand-Shilov spaces $S^\beta_\alpha$ are topological algebras under the Moyal star product if and only if $\alpha\ge\beta$. These spaces of test functions can be used in quantum field theory on noncommutative spacetime. The star product depends on the noncommutativity parameter continuously in their topology. We also prove that the series expansion of the Moyal product converges absolutely in $S^\beta_\alpha$ if and only if $\beta<1/2$.