The Anti-Unitarity of Time Reversal & Co-representations of Lorentzian Pin Groups

TL;DR

This paper explores the representation theory of Lorentzian groups, introducing co-representations to naturally incorporate anti-linearity of time reversal operators, and examines implications for spinors and de Sitter symmetry.

Contribution

It presents a new framework using co-representations to naturally include anti-linearity in double covers of Lorentzian groups, clarifying the structure of time reversal operators.

Findings

Standard double covers can be extended to include anti-linearity naturally.

A mapping between Majorana and Weyl spinors is constructed.

A co-representation for the de Sitter group is developed, showing no scalar fermion mass terms.

Abstract

In the representation theory of Lorentzian orthogonal groups, there are well known arguments as to why the parity inversion operator $\mathcal{P}$ and the time reversal operator $\mathcal{T}$, should be realized as linear and anti-linear operators respectively (Wigner 1932). Despite this, standard constructions of double covers of the Lorentzian orthogonal groups naturally build time reversal operators in such a manner that they are linear, and the anti-linearity is put in ad-hoc after the fact. This article introduces a viewpoint naturally incorporating the anti-linearity into the construction of these double covers, through what Wigner called co-representations, a kind of semi-linear representation. It is shown how the standard spinoral double covers of the Lorentz group -- $\operatorname{Pin}(1,3)$ and $\operatorname{Pin}(3,1)$ -- may be naturally centrally extended for this purpose, and the relationship between the $\mathcal{C}$, $\mathcal{P}$, and $\mathcal{T}$ operators is discussed. Additionally a mapping is constructed demonstrating an interesting equivalence between Majorana and Weyl spinors. Finally a co-representation is built for the de Sitter group $\operatorname{Pin}(1,4)$, and it is demonstrated how a theory with this symmetry has no truly scalar fermion mass terms.