Bundle Theory of Improper Spin Transformations

{\it We first give a geometrical description of the action of the parity operator (\(\hat{P}\)) on non relativistic spin \({{1}\over{2}}\) Pauli spinors in terms of bundle theory. The relevant bundle, \(SU(2)\odot \Z_2\to O(3)\), is a non trivial extension of the universal covering group \(SU(2)\to SO(3)\). \(\hat{P}\) is the non relativistic limit of the corresponding Dirac matrix operator \({\cal P}=i\gamma_0\) and obeys \(\hat{P}^2=-1\). Then, 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 (\(\hat{T}\)) acting on spinors, and its product with \(\hat{P}\). Both, \(\hat{P}\) and \(\hat{T}\), generate the group \(\Z_4 \times \Z_2\). As in the case of parity, \(\hat{T}\) is the non relativistic limit of the corresponding Dirac matrix operator \({\cal T}=\gamma^3 \gamma^1\), and obeys \(\hat{T}^2=-1\).}