Noncommutative quantum mechanics as a gauge theory

TL;DR

This paper demonstrates how noncommutative quantum mechanics can be reformulated as a gauge theory using the BFT embedding, enabling multiple gauge choices and explicit quantization, thus providing new insights into its structure.

Contribution

It applies the BFT embedding to transform noncommutative quantum mechanics into a gauge theory, revealing a one-to-one correspondence and exploring different gauge quantizations.

Findings

Existence of a one-to-one mapping between second class system and gauge-invariant sector.

Explicit quantization performed for two different gauges.

Multiple equivalent descriptions of noncommutative quantum mechanics established.

Abstract

The classical counterpart of noncommutative quantum mechanics is a constrained system containing only second class constraints. The embedding procedure formulated by Batalin, Fradkin and Tyutin (BFT) enables one to transform this system into an Abelian gauge theory exhibiting only first class constraints. The appropriateness of the BFT embedding, as implemented in this work, is verified by showing that there exists a one to one mapping linking the second class model with the gauge invariant sector of the gauge theory. As is known, the functional quantization of a gauge theory calls for the elimination of its gauge freedom. Then, we have at our disposal an infinite set of alternative descriptions for noncommutative quantum mechanics, one for each gauge. We study the relevant features of this infinite set of correspondences. The functional quantization of the gauge theory is explicitly performed for two gauges and the results compared with that corresponding to the second class system. Within the operator framework the gauge theory is quantized by using Dirac's method.