Bundle Theory of Improper Spin Transformations

Abstract

We give a geometrical description of the action of the parity operator (\(\hat{P}\)) on non-relativistic spin 1/2 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 \(\hbox{\it SU}(2)\to \hbox{\it SO}(3). \hat{P}\) is the non-relativistic limit of the corresponding Dirac matrix operator \({\cal P}=i\gamma_0\) and obeys \(\hat{P}^2=-1\). 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 \(\hat{P}. \hat{P}\) 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 \(\hat{T}^2=-1\).