Time-Space Noncommutativity: Quantised Evolutions

TL;DR

This paper explores quantum physics on noncommutative spaces with quantized time, revealing that in such models, time translations are discrete and conservation laws are modified, impacting scattering processes and quantum field theory formulation.

Contribution

The work extends quantum physics on the Moyal plane to noncommutative cylinders and spheres, demonstrating discrete time translations and their effects on conservation laws.

Findings

Time translations are discrete in noncommutative models.

Hamiltonian conservation is modified in scattering processes.

Framework for scattering theory and quantum field theory on noncommutative spaces.

Abstract

In previous work, we developed quantum physics on the Moyal plane with time-space noncommutativity, basing ourselves on the work of Doplicher et al.. Here we extend it to certain noncommutative versions of the cylinder, $\mathbb{R}^{3}$ and $\mathbb{R}\times S^{3}$. In all these models, only discrete time translations are possible, a result known before in the first two cases. One striking consequence of quantised time translations is that even though a time independent Hamiltonian is an observable, in scattering processes, it is conserved only modulo $\frac{2\pi}{\theta}$, where $\theta$ is the noncommutative parameter. (In contrast, on a one-dimensional periodic lattice of lattice spacing $a$ and length $L=Na$, only momentum mod $\frac{2\pi}{L}$ is observable (and can be conserved).) Suggestions for further study of this effect are made. Scattering theory is formulated and an approach to quantum field theory is outlined.