2026/02/28 by S. Hasibul Hassan Chowdhury
Physics and Astronomy · Mathematics · #Noncommutative and Quantum Gravity Theories #Advanced Operator Algebra Research #Quantum Mechanics and Applications
paper · pdf · doi:10.1016/j.geomphys.2026.105942
We study Bopp shifts in two-dimensional noncommutative quantum mechanics (NCQM) through a functorial lens. A nondegenerate NCQM sector with central character (ℏ,ϑ,B\rm in) determines a self-adjoint infinitesimal representation of the NCQM Lie algebra \mathfrakg\rm NC. A Darboux normalization of its represented phase-space operators produces a self-adjoint infinitesimal representation of the Weyl-Heisenberg Lie algebra \mathfrakg\rm WH with central parameter ℏ, and hence defines a Bopp-shift functor collapsing (ℏ,ϑ,B\rm in) ↦ ℏ. In particular, a generic NCQM sector is not equivalent, as a \mathfrakg\rm NC-sector, to the ordinary QM sector (ℏ,0,0), even though their Bopp-shift images have the same Weyl-Heisenberg parameter. To measure what this collapse forgets, we construct a toroidal shadow functor assigning to each periodic datum L=(ax,ay) and each NCQM sector χ a phase-space noncommutative four-torus A4_Θχ,L. Its K0-trace pairing yields sector-sensitive gap labels whose top-degree coefficient is (ϑ B\rm in-ℏ)/((2π)2ℏ). This coefficient is independent of the spatial cell area axay and equals the Pfaffian \rm Pf(Θχ,L), the top-degree generator of the K0-trace range. Since strong Morita equivalence preserves this trace range up to positive scaling, non-proportionality of the trace ranges obstructs Morita equivalence of the shadows, separating equal-ℏ sectors that the Bopp-shift functor identifies. In the arithmetic subfamily where ϑ/A and B\rm inA/ℏ are algebraic, the trace-range scale is forced to be trivial and |\rm Pf| becomes a computable separation criterion: |ϑ B\rm in-ℏ| ≠ |ϑ' B'\rm in-ℏ| already implies inequivalence.