The CPT Group of the Dirac Field

TL;DR

This paper classifies the possible charge conjugation, parity, and time reversal matrices for the Dirac field, revealing two main solution sets and defining a unique CPT group with a geometric structure, applicable across different representations.

Contribution

It identifies only two consistent sets of C, P, T matrices for the Dirac equation and introduces the unique CPT group, linking algebraic properties to geometric symmetries.

Findings

Two sets of consistent C, P, T matrices exist for the Dirac field.

The CPT group is isomorphic to Q x Z_2, with a geometric interpretation.

The matrix groups are consistent across Weyl and Majorana representations.

Abstract

Using the standard representation of the Dirac equation we show that, up to signs, there exist only TWO SETS of consistent solutions for the matrices of charge conjugation (C), parity (P), and time reversal (T). In both cases, P^2=-1, and then two succesive applications of the parity transformation to spin 1/2 fields NECESSARILY amounts to a 2\pi rotation. Each of these sets generates a non abelian group of sixteen elements, G_1 and G_2, which are non isomorphic subgroups of the Dirac algebra, which, being a Clifford algebra, gives a geometric nature to the generators, in particular to C. It turns out that G_1 is isomorphic to D_8 x Z_2, where D_8 is the dihedral group of eight elements (the symmetries of the square) and Z_2 is isomorphic to S^0 (the 0-sphere); while G_2 is isomorphic to a certain semidirect product of D_8 and Z_2. Instead, the corresponding quantum operators for C, P, and T generate a UNIQUE group G, which we call THE CPT GROUP OF THE DIRAC FIELD, and which is isomorphic to Q x Z_2, where Q is the quaternion group. This group, however, is compatible only with the second of the above two matrix solutions, namely with G_2, which is then called THE MATRIX CPT GROUP. It is interesting to remark that G_1, G_2, and G are the only non abelian groups of sixteen elements with three generators. Finally, the matrix groups are also given in the Weyl and Majorana representations, suitable for taking the massless limit and for describing self-conjugate fields.