On the Moyal Star Product of Resurgent Series

TL;DR

This paper explores the Moyal star product within deformation quantization using resurgence theory, defining a new class of algebro-resurgent series that remains stable under this product.

Contribution

It introduces the concept of algebro-resurgent series and demonstrates their stability under the Moyal star product, linking resurgence theory with deformation quantization.

Findings

Defined the space of algebro-resurgent series.

Proved stability of this space under Moyal star product.

Connected resurgence theory with algebraic structures in quantization.

Abstract

We analyze the Moyal star product in deformation quantization from the resurgence theory perspective. By putting algebraic conditions on Borel transforms, one can define the space of ``algebro-resurgent series'' (a subspace of $1$-Gevrey formal series in $i\hbar/2$ with coefficients in $C\{q,p\}$), which we show is stable under Moyal star product.