Exotic deformation quantization

TL;DR

This paper explores formal deformations of Poisson algebras on cotangent bundles, revealing a non-trivial star-product in one-dimensional cases that differs from the standard Moyal product.

Contribution

It introduces a new class of deformation quantizations for Poisson algebras with singularities, specifically in one-dimensional manifolds.

Findings

Existence of a non-trivial star-product in 1D cases

This star-product is not equivalent to the Moyal product

Deformations are Laurent polynomials of fibers

Abstract

We consider formal deformations of the Poisson algebra of functions (with singularities) on $T^*M$ which are Laurent polynomials of fibers. Tn the case: $\dim M=1$ ($M=S^1, {\bf R}$), there exists a non-trivial $\star$-product on this algebra non-equivalent to the standard Moyal product.