Bundle Theory of Improper Spin Transformations

Abstract

{\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 $\hat{T}^2=-1$.}