Strict quantization of polynomial Poisson structures

TL;DR

This paper develops a method using combinatorial star products to achieve strict deformation quantizations of polynomial Poisson structures on , extending previous results to more complex nonlinear cases with explicit examples and functional analysis considerations.

Contribution

It introduces a systematic approach for strict quantization of polynomial Poisson structures of arbitrary degree, including explicit constructions and analysis of properties like positivity and continuity.

Findings

Constructed explicit strict quantizations for nonlinear polynomial Poisson structures.

Demonstrated the existence of *-algebra representations on pre-Hilbert spaces.

Analyzed properties such as classical limit continuity and *-involution compatibility.

Abstract

We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on $\mathbb R^d$, generalizing known results for constant and linear Poisson structures to polynomial Poisson structures of arbitrary degree. We give several examples of nonlinear Poisson structures and construct explicit formal star products whose deformation parameter can be evaluated to any real value of $\hbar$, giving strict quantizations on the space of analytic functions on $\mathbb R^d$ with infinite radius of convergence. We also address further questions such as continuity of the classical limit $\hbar \to 0$, compatibility with *-involutions, and the existence of positive linear functionals. The latter can be used to realize the strict quantizations as *-algebras of operators on a pre-Hilbert space which we demonstrate in a concrete example.