Bundle theory of improper spin transformations

Abstract

We give a geometrical description of the action of the parity operator ( (P) over cap) P) on non-relativistic spin 1/ 2 Pauli spinors in terms of bundle theory. The relevant bundle, SU(2) circle dot Z(2) -> O( 3), is a non-trivial extension of the universal covering group SU( 2) -> SO( 3). (P) over cap is the non-relativistic limit of the corresponding Dirac matrix operator P = i gamma(0) and obeys (P) over cap (2) = -1 From the direct product of O( 3) by Z(2), naturally induced by the structure of the Galilean group, we identify, in its double cover, the time-reversal operator ((T) over cap) T) acting on spinors, and its product with (P) over cap. (P) over cap (T) over cap generate the group Z(4) x Z(2). As in the case of parity, (T) over cap T is the non-relativistic limit of the corresponding Dirac matrix operator T = gamma(3) gamma(1), and obeys (T) over cap (2) = - 1.