Unitary Quantum Physics with Time-Space Noncommutativity

TL;DR

This paper develops quantum physics in noncommutative spacetime, focusing on time-space noncommutativity, and explores its implications for symmetries, causality, and spectral properties of Hamiltonians.

Contribution

It provides a formal construction of noncommutative quantum mechanics with time-space noncommutativity, analyzing symmetries, causality, and spectral equivalence of Hamiltonians.

Findings

Spectral equivalence of commutative and noncommutative Hamiltonians under certain conditions

Detailed treatment of the Moyal plane in noncommutative quantum mechanics

Insights into causality and observables in noncommutative spacetime

Abstract

In this work quantum physics in noncommutative spacetime is developed. It is based on the work of Doplicher et al. which allows for time-space noncommutativity. The Moyal plane is treated in detail. In the context of noncommutative quantum mechanics, some important points are explored, such as the formal construction of the theory, symmetries, causality, simultaneity and observables. The dynamics generated by a noncommutative Schrodinger equation is studied. We prove in particular the following: suppose the Hamiltonian of a quantum mechanical particle on spacetime has no explicit time dependence, and the spatial coordinates commute in its noncommutative form (the only noncommutativity being between time and a space coordinate). Then the commutative and noncommutative versions of the Hamiltonian have identical spectra.