Discrete Lorentz symmetry and discrete time translational symmetry

TL;DR

This paper proposes that Lorentz symmetry and time translational symmetry can be discretized similarly to spatial translational symmetry in crystals, leading to new discrete symmetry frameworks in 1+1-dimensional spacetime.

Contribution

It introduces the concept of discrete Lorentz and time translational symmetries, extending the idea of broken continuous symmetries from space to spacetime in a systematic way.

Findings

Classifies all possible discrete Lorentz and time translational symmetries in 1+1D

Provides methods to construct field theories with these discrete symmetries

Suggests implications for symmetry breaking in fundamental physics

Abstract

The Lorentz symmetry and the space and time translational symmetry are fundamental symmetries of nature. Crystals are the manifestation of the continuous space translational symmetry being spontaneously broken into a discrete one. We argue that, following the space translational symmetry, the continuous Lorentz symmetry should also be broken into a discrete one, which further implies that the continuous time translational symmetry is broken into a discrete one. We deduce all the possible discrete Lorentz and discrete time translational symmetries in 1+1-dimensional spacetime, and show how to build a field theory or a lattice field theory that has these symmetries.