Bundle Theory of Improper Spin Transformations

We first give a geometrical description of the action of the parity operator (P̂ ) on non relativistic spin 12 Pauli spinors in terms of bundle theory. The relevant bundle, SU(2)⊙Z2 → O(3), is a non trivial extension of the universal covering group SU(2) → SO(3). P̂ is the non relativistic limit of the corresponding Dirac matrix operator P = iγ0 and obeys P̂ 2 = −1. Then, from the direct product of O(3) by Z2, naturally induced by the structure of the galilean group, we identify, in its double cover, the time reversal operator (T̂ ) acting on spinors, and its product with P̂ . Both, P̂ and T̂ , generate the group Z4 × Z2. As in the case of parity, T̂ is the non relativistic limit of the corresponding Dirac matrix operator T = γγ, and obeys T̂ 2 = −1.