From Bopp Shifts to Toroidal Shadows: K-Theoretic Gap Labels in Noncommutative Quantum Mechanics

Abstract

We study Bopp shifts in two-dimensional noncommutative quantum mechanics (NCQM) through a functorial lens. A nondegenerate NCQM sector with central character $(\hslash ,\vartheta ,B_{\mathrm{in}})$ determines a self-adjoint infinitesimal representation of the NCQM Lie algebra ${\mathfrak{g}}_{\mathrm{NC}}$. A Darboux normalization of its represented phase-space operators produces a self-adjoint infinitesimal representation of the Weyl-Heisenberg Lie algebra ${\mathfrak{g}}_{\mathrm{WH}}$ with central parameter $\hslash$, and hence defines a Bopp-shift functor collapsing $(\hslash ,\vartheta ,B_{\mathrm{in}})\mapsto \hslash$. In particular, a generic NCQM sector is not equivalent, as a ${\mathfrak{g}}_{\mathrm{NC}}$-sector, to the ordinary QM sector $(\hslash ,0,0)$, 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 $L=(a_x,a_y)$ and each NCQM sector $\chi$ a phase-space noncommutative four-torus $A_{{\Theta }_{\chi ,L}}^4$. Its $K_0$-trace pairing yields sector-sensitive gap labels whose top-degree coefficient is $(\vartheta B_{\mathrm{in}}-\hslash )/((2\pi {)}^2\hslash )$. This coefficient is independent of the spatial cell area $a_x{a}_y$ and equals the Pfaffian $\mathrm{Pf}({\Theta }_{\chi ,L})$, the top-degree generator of the $K_0$-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-$\hslash$ sectors that the Bopp-shift functor identifies. In the arithmetic subfamily where $\vartheta /A$ and $B_{\mathrm{in}}A/\hslash$ are algebraic, the trace-range scale is forced to be trivial and $|\mathrm{Pf}|$ becomes a computable separation criterion: $|\vartheta B_{\mathrm{in}}-\hslash |\ne |{\vartheta }^{\prime }{B}_{\mathrm{in}}^{\prime }-\hslash |$ already implies inequivalence.

arXiv commentUpdated to match the published version of record, including the revised concluding outlook and corresponding bibliography changes

References in corpus (11)

• Topological Field Theory of Time-Reversal Invariant Insulators
• The Non-Commutative Geometry of the Quantum Hall Effect
• A Four Dimensional Generalization of the Quantum Hall Effect
• Exploring 4D Quantum Hall Physics with a 2D Topological Charge Pump
• Non-Commutative Quantum Mechanics
• Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
• Four-Dimensional Quantum Hall Effect with Ultracold Atoms
• On Quantum Mechanics on Noncommutative Quantum Phase Space
• On the Plethora of Representations Arising in Noncommutative Quantum Mechanics and An Explicit Construction of Noncommutative 4-tori
• Triply Extended Group of Translations of ${\mathbb{R}}^4$ as Defining Group of NCQM: relation to various gauges
• Gauge invariant energy spectra in 2-dimensional noncommutative quantum mechanics