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

TL;DR

The paper develops a complete finite-dimensional representation theory for the full Lorentz group $O(1,3)$ by showing it decomposes as a semidirect product $O(1,3) \cong SO^+(1,3) \rtimes K_4$, where $K_4$ is the discrete reflection group generated by parity and time reversal. By tensoring every $SO^+(1,3)$ representation with the four one-dimensional irreps of $K_4$ (labeled $1, P, T, PT$), there are exactly four corresponding $O(1,3)$ representations for each $SO^+(1,3)$ rep, making the action of $\mathcal{P}$ and $\mathcal{T}$ on tensors transparent. The work provides explicit classifications and tables for scalars, vectors, and higher tensors, with key examples such as the metric, angular momentum, the Faraday tensor, and the Levi-Civita symbol, illustrating how Maxwell’s equations remain reflection-covariant within this framework. This representation-theoretic bookkeeping clarifies how spacetime reflections affect Lorentzian quantities, offering a practical tool for analyzing parity and time-reversal properties and pointing toward extensions to spinor/Pin groups and CPT structure.

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.

Figures (2)

• Figure 1: In solid red we have a trajectory $x(t) = \operatorname{sech}^2(t-1)$. In the top panel we see the trajectory plotted with its velocity (dashed blue). In the middle panel we see the naive reflection of both functions. This naive reflection does not send the velocity function to the velocity function of the reflected trajectory. Finally on the bottom we see the negation ensures the velocity is mapped correctly under reflection.
• Figure 2: The four connected components of the group $\operatorname{O}(1,3)$, along with the reflections mapping between these components: $\mathcal{R}$ = $\operatorname{diag}\{1,-1,-1,-1\}$.