On the group generated by C, P and T: I 2 = T 2 = P 2 = ITP = −1, with applications to pseudo-scalar mesons

Abstract

We study faithful representations of the discrete Lorentz symmetry operations of parity P and time reversal T , which involve complex phases when acting on fermions. If the phase of P is a rational multiple of π then P 2n = 1 for some positive integer n and it is shown that, when this is the case, P and T generate a discrete group, a dicyclic group (also known as a generalised quaternion group) which are generalisations of the dihedral groups familiar from crystallography. Charge conjugation C introduces another complex phase and, again assuming rational multiples of π for complex phases, TC generates a cyclic group of order 2m for some positive integer m. There is thus a doubly infinite series of possible finite groups labelled by n and m. Demanding that C commutes with P and T forces n = m = 2 and the group generated by P and T is uniquely determined to be the quaternion group. Neutral pseudo-scalar mesons can be simultaneous C and P eigenstates, T commutes with P and C when acting on fermion bi-linears so neutral pseudo-scalar mesons can also be T eigenstates. The T -parity should therefore be experimentally observable and the CPT theorem dictates that T = CP.