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On the Groenewold-Van Hove problem for R2n

Mark J. Gotay

math-pharXiv:math-ph/9809015

Abstract

We discuss the Groenewold-Van Hove problem for R2n, and completely solve it when n = 1. We rigorously show that there exists an obstruction to quantizing the Poisson algebra of polynomials on R2n, 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.

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