Generalized Bopp shift, Darboux Canonicalization, and the Kinematical Inequivalence of NCQM and QM

Abstract: Two-dimensional noncommutative quantum mechanics (NCQM) is often formulated through linear transformations of represented position and momentum operators and through Darboux-type canonicalizations. We clarify the representation-theoretic meaning of such constructions at the level of kinematical symmetry groups and irreducible unitary representations. The standard NCQM commutators are naturally encoded by a step-two nilpotent Lie group $G_{\hbox{\tiny{NC}}}$ with three-dimensional center, whose irreducible sectors are labeled by central characters (equivalently, coadjoint-orbit labels), parametrized on the regular stratum by $(\hbar,\vartheta,B_{\mathrm{in}})$. In this language, ordinary two-dimensional quantum mechanics (QM) is the quotient (equivalently, inflation) sector $(\hbar,0,0)\subset \widehat{G_{\hbox{\tiny{NC}}}}$, the unitary dual of $G_{\hbox{\tiny{NC}}}$; i.e., it consists of those representations that factor through the central quotient $G_{\hbox{\tiny{NC}}}\twoheadrightarrow G_{\hbox{\tiny{WH}}}$, where $G_{\hbox{\tiny{WH}}}$ denotes the Weyl--Heisenberg group. We show that generalized Bopp-shift and Seiberg--Witten-type linear recombinations of represented operators, and the existence of an auxiliary quadruple satisfying the canonical commutation relations obtained by Darboux canonicalization, do not imply unitary equivalence between a fixed generic NCQM sector $(\hbar_{0},\vartheta_{0},B_{0})$ and the ordinary-quantum-mechanics sector $(\hbar_{0},0,0)$ of $\widehat{G_{\hbox{\tiny{NC}}}}$.