Discrete Time Evolution and Energy Nonconservation in Noncommutative Physics

TL;DR

This paper explores how noncommutative spacetime leads to discrete time evolution and energy nonconservation, developing scattering theory to analyze transition probabilities in such unconventional quantum systems.

Contribution

It introduces a scattering framework for discrete time translations in noncommutative physics, enabling analysis of energy nonconservation phenomena.

Findings

Energy is conserved only modulo a fixed unit in noncommutative theories.

Developed a formal scattering theory for discrete time translations.

Implications for phenomenology in theories with compact extra dimensions.

Abstract

Time-space noncommutativity leads to quantisation of time and energy nonconservation when time is conjugate to a compact spatial direction like a circle. In this context energy is conserved only modulo some fixed unit. Such a possibility arises for example in theories with a compact extra dimension with which time does not commute. The above results suggest striking phenomenological consequences in extra dimensional theories and elsewhere. In this paper we develop scattering theory for discrete time translations. It enables the calculation of transition probabilities for energy nonconserving processes and has a central role both in formal theory and phenomenology. We can also consider space-space noncommutativity where one of the spatial directions is a circle. That leads to the quantisation of the remaining spatial direction and conservation of momentum in that direction only modulo some fixed unit, as a simple adaptation of the results in this paper shows.