Bypassing the Groenewold-van Hove Obstruction: A New Argument in Favor of Born-Jordan Quantization

TL;DR

This paper proposes a modified quantization approach that bypasses the Groenewold-van Hove obstruction by only requiring commutator-Poisson bracket correspondence for specific Hamiltonian functions, aligning with Born-Jordan quantization.

Contribution

It introduces a weakened quantization condition that allows for a consistent quantization procedure avoiding known obstructions, connecting to the historic Born-Jordan rule.

Findings

Constructs a well-defined quantization method under the weakened condition.

Shows the method aligns with Born-Jordan quantization for polynomial observables.

Demonstrates the approach bypasses the Groenewold-van Hove no-go theorem.

Abstract

There are known obstructions to a full quantization in the spirit of Dirac's approach, the most known being the Groenewold-van Hove no-go result. We show, following a suggestion of S. K. Kauffmann, that it is possible to construct a well-defined quantization procedure by weakening the usual requirement that commutators should correspond to Poisson brackets. The weaker requirement consists in demanding that this correspondence should only hold for Hamiltonian functions of the type T(p)+V(q). This reformulation leads to a non-injective quantization of all observables which, when restricted to polynomials, is the rule proposed by Born and Jordan in the early days of quantum mechanics.