Central Characters of $G_{\mathrm{NC}}$, Darboux Normalization, and the Kinematical Inequivalence of NCQM and QM

TL;DR

This paper uses representation theory to show that noncommutative quantum mechanics sectors are generally not equivalent to ordinary quantum mechanics, even after Darboux normalization, clarifying their fundamental differences.

Contribution

It demonstrates that Darboux normalization and Bopp shifts do not establish kinematical equivalence between NCQM and QM sectors at the representation level.

Findings

Nondegenerate NCQM sectors are not unitarily equivalent to QM sectors as G_NC representations.

Darboux normalization does not imply kinematical equivalence of sectors.

The identification of sectors occurs only after passing to a star-product description.

Abstract

We analyze generalized Bopp shifts and Darboux normalization in two-dimensional noncommutative quantum mechanics (NCQM) from the viewpoint of the unitary representation theory of the kinematical symmetry group \(G_{\mathrm{NC}}\). This group is a step-two nilpotent Lie group with three-dimensional center, and the regular part of its unitary dual \(\widehat{G_{\mathrm{NC}}}\) is labelled by central characters \((\hbar,\vartheta,B_{\mathrm{in}})\). Ordinary two-dimensional quantum mechanics (QM) appears inside \(\widehat{G_{\mathrm{NC}}}\) as the family of Weyl-Heisenberg representations inflated along the quotient \(G_{\mathrm{NC}}\rightarrow G_{\mathrm{WH}}\), with central character \((\hbar,0,0)\). We prove that a generic nondegenerate NCQM sector \((\hbar_0,\vartheta_0,B_0)\), with \(\hbar_0,\vartheta_0,B_0\neq 0\) and \(\hbar_0-B_0\vartheta_0\neq 0\), is not unitarily equivalent to the ordinary QM sector \((\hbar_0,0,0)\) as a \(G_{\mathrm{NC}}\)-representation. Consequently, generalized Bopp shifts and Darboux normalizations, although they can produce auxiliary operator quadruples satisfying canonical commutation relations, do not establish kinematical equivalence of the corresponding sectors. We further explain that the apparent identification arises only after passing to a Darboux-normalized or coarse star-product description, where the original \(G_{\mathrm{NC}}\)-central-character label is no longer part of the data. The result clarifies the relation between operator-level Darboux normalization, deformation-quantization equivalence, and representation-theoretic equivalence in NCQM.