Noncommutative spaces, the quantum of time and the Lorentz symmetry

TL;DR

This paper introduces discrete-in-time noncommutative space-times compatible with Lorentz symmetry, deriving their properties and constructing an action using a reparametrized relativistic particle.

Contribution

It presents three new discrete-in-time noncommutative space-times that preserve Lorentz symmetry and provides a realization and action formulation for Snyder-type spaces.

Findings

Discovered three discrete-in-time noncommutative space-times compatible with Lorentz symmetry.

Derived commutation relations similar to Snyder and Yang spaces.

Constructed an action for Snyder-type noncommutative spaces.

Abstract

We introduce three space-times that are discrete in time and compatible with the Lorentz symmetry. We show that these spaces are no commutative, with commutation relations similar to the relations of the Snyder and Yang spaces. Furthermore, using a reparametrized relativistic particle we obtain a realization of the Snyder type spaces and we construct an action for them.