The Constraints and Spectra of a Deformed Quantum Mechanics

TL;DR

This paper explores a deformed quantum mechanics with momentum-dependent commutators, revealing an intrinsic maximum momentum and its effects on bound state spectra, with implications for experimental distinguishability from string theory models.

Contribution

It introduces a class of modified commutation relations with intrinsic maximum momentum and analyzes their spectral and classical implications, contrasting with string-inspired models.

Findings

Maximum momentum leads to finite energy bounds in harmonic oscillators.

Certain MCRs cause bound state energy shifts with opposite signs to string-inspired models.

Classical behavior emerges in systems with maximum momentum constraints.

Abstract

We examine a deformed quantum mechanics in which the commutator between coordinates and momenta is a function of momenta. The Jacobi identity constraint on a two-parameter class of such modified commutation relations (MCR's) shows that they encode an intrinsic maximum momentum; a sub-class of which also imply a minimum position uncertainty. Maximum momentum causes the bound state spectrum of the one-dimensional harmonic oscillator to terminate at finite energy, whereby classical characteristics are observed for the studied cases. We then use a semi-classical analysis to discuss general concave potentials in one dimension and isotropic power-law potentials in higher dimensions. Among other conclusions, we find that in a subset of the studied MCR's, the leading order energy shifts of bound states are of opposite sign compared to those obtained using string-theory motivated MCR's, and thus these two cases are more easily distinguishable in potential experiments.