Space Inversion, Time Reversal, and Other Discrete Symmetries in Local Field Theories

The general algebraic relations between space inversion, time reversal, and the internal symmetry group are analyzed within the framework of a Lorentz-invariant local field theory. The problem of unitary representations of the full Poincar\'e group including the space and time reflection operators has been studied by Wigner, and the representations are classified into 4 cases. It is shown that, with the added assumption of the local field theory, Wigner's cases 2,3, and 4 either do not not occur or can be reduced to his case 1. The concept of minimal group extension is introduced and the related mathematical analysis is given. The symmetry properties under space inversion, time reversal, and other discrete operators such as charge conjugation are analyzed separately for each of the three known interactions: strong, electromagnetic, and weak.