Interference, noncommutativity, and determinateness in quantum mechanics

Jos Uffink, Jan Hilgevoord

S. Hasibul Hassan Chowdhury

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

G. D. Barbosa, Nelson Pinto-Neto

We carry out an investigation into the possibility of developing a Bohmian interpretation based on the continuous motion of point particles for noncommutative quantum mechanics. The conditions for such an interpretation to be consistent are determined, and the implications of its adoption for noncommutativity are discussed. A Bohmian analysis of the noncommutative harmonic oscillator is carried out in detail. By studying the particle motion in the oscillator orbits, we show that small-scale physics can have influence at large scales, something similar to the IR-UV mixing.

Urbasi Sinha, Debadrita Ghosh

Shidong Liang

K. H. C. Castello-Branco, Andrey Gomes Martins

We study the free-fall of a quantum particle in the context of noncommutative quantum mechanics (NCQM). Assuming noncommutativity of the canonical type between the coordinates of a two-dimensional configuration space, we consider a neutral particle trapped in a gravitational well and exactly solve the energy eigenvalue problem. By resorting to experimental data from the GRANIT experiment, in which the first energy levels of freely falling quantum ultracold neutrons were determined, we impose an upper-bound on the noncommutativity parameter. We also investigate the time of flight of a quantum particle moving in a uniform gravitational field in NCQM. This is related to the weak equivalence principle. As we consider stationary, energy eigenstates, i.e., delocalized states, the time of flight must be measured by a quantum clock, suitably coupled to the particle. By considering the clock as a small perturbation, we solve the (stationary) scattering problem associated and show that the time of flight is equal to the classical result, when the measurement is made far from the turning point. This result is interpreted as an extension of the equivalence principle to the realm of NCQM.