On the Groenewold-Van Hove problem for R^{2n}

TL;DR

This paper addresses the Groenewold-Van Hove problem in quantization, providing a complete solution for the case n=1 by identifying obstructions and classifying quantizable subalgebras of polynomials.

Contribution

It rigorously proves the existence of obstructions to quantizing polynomial Poisson algebras on R^{2n} and explicitly characterizes quantizable subalgebras for n=1.

Findings

Obstruction to quantization of polynomial Poisson algebras in R^{2n}

Complete classification of quantizable subalgebras for n=1

Explicit construction of all quantizations for these subalgebras

Abstract

We discuss the Groenewold-Van Hove problem for R^{2n}, and completely solve it when n = 1. We rigorously show that there exists an obstruction to quantizing the Poisson algebra of polynomials on R^{2n}, thereby filling a gap in Groenewold's original proof without introducing extra hypotheses. Moreover, when n = 1 we determine the largest Lie subalgebras of polynomials which can be unambiguously quantized, and explicitly construct all their possible quantizations.