A Convenient Representation Theory of Lorentzian Pseudo-Tensors: $\mathcal{P}$ and $\mathcal{T}$ in $\operatorname{O}(1,3)$

TL;DR

This paper introduces a new, simplified framework for understanding the finite-dimensional representation theory of the Lorentz group, emphasizing the role of spacetime reflections and providing a classification scheme for Lorentzian tensors.

Contribution

It presents a novel approach to Lorentz group representations by decomposing it into a semi-direct product, enabling straightforward classification of tensors under spacetime reflections.

Findings

Classifies Lorentzian tensors using the semi-direct product structure.

Shows how to track spacetime reflection properties of tensors.

Provides examples illustrating the representation theory and reflection properties.

Abstract

A novel approach to the finite dimensional representation theory of the entire Lorentz group $\operatorname{O}(1,3)$ is presented. It is shown how the entire Lorentz group may be understood as a semi-direct product between its identity component and the Klein four group of spacetime reflections: $\operatorname{O}(1,3) = \operatorname{SO}^+(1,3) \rtimes \operatorname{K}_4$. This gives way to a convenient classification of tensors transforming under $\operatorname{O}(1,3)$, namely that there are four representations of $\operatorname{O}(1,3)$ for each representation of $\operatorname{SO}^+(1,3)$, and it is shown how the representation theory of the Klein group $\operatorname{K}_4$ allows for simple book keeping of the spacetime reflection properties of general Lorentzian tensors, and combinations thereof, with several examples given. There is a brief discussion of the time reversal of the electromagnetic field, concluding in agreement with standard texts such as Jackson, and works by Malament.