Parity Oddness of $\operatorname{Spin}(1,4)$ Dirac Mass Terms

Abstract

In the representation theory of Lorentzian orthogonal groups there are well known arguments as to why the parity operator $\mathcal{P}$ and the time reversal operator $\mathcal{T}$ should be realized as linear and anti-linear operators respectively (Wigner 1932). In this paper it is shown that the only operators satisfying the requisite properties for the spinor representation of the de Sitter group $\operatorname{SO}(1,4)$ lead to fermion self-couplings which are necessarily parity odd, ruling out standard Dirac mass terms for theories with fermions obeying $\operatorname{Spin}(1,4)$ symmetry.