# Spatio-temporal Symmetry - Point Groups with Time Translations

## TL;DR

This paper introduces spatio-temporal point groups combining spatial symmetries with time translations, providing their derivation, classification, and visualization, with applications to physical systems exhibiting combined space-time symmetry.

## Contribution

It systematically derives and catalogs spatio-temporal point groups from conventional point groups, including their irreducible representations and visualizations, for the first time.

## Key findings

- Complete listing of spatio-temporal point groups
- Visualization methods using stereographic projections
- Examples of physical systems with combined space-time symmetry

## Abstract

Spatial symmetries occur in combination with temporal symmetries in a wide range of physical systems in nature, including time-periodic quantum systems typically described by the Floquet formalism. In this context, groups formed by three-dimensional point group symmetry operations in combination with time translation operations are discussed in this work. The derivation of these 'spatio-temporal' groups from conventional point groups and their irreducible representations is outlined, followed by a complete listing. The groups are presented in a template similar to space group operations, and are visualized using a modified version of conventional stereographic projections. Simple examples of physical processes that simultaneously exhibit symmetry in space and time are identified and used to illustrate the application of spatio-temporal groups.