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

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 \(hbar,vartheta,B_in\). 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 \(hbar₀,vartheta₀,B₀\) and the ordinary-quantum-mechanics sector \(hbar₀,0,0\) of widehat{G_{hbox{tiny{NC}}}}.