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Bundle theory of improper spin transformations. (English) Zbl 1077.81046
Summary: 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, $$SU(2) \odot \mathbb Z_2 \to O(3)$$, 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 $$\mathcal P = i\gamma_0$$ and obeys $${\hat P}^2 = -1$$. From the direct product of $$O(3)$$ by $$\mathbb Z_2$$, 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 $$\mathbb Z_4 \times \mathbb Z_2$$. As in the case of parity, $$\hat T$$ is the non-relativistic limit of the corresponding Dirac matrix operator $$\mathcal T = \gamma^3 \gamma^1$$, and obeys $$\hat T^2 = -1$$.

##### MSC:
 81R05 Finite-dimensional groups and algebras motivated by physics and their representations 22E70 Applications of Lie groups to the sciences; explicit representations 53C27 Spin and Spin$${}^c$$ geometry 53C80 Applications of global differential geometry to the sciences 81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics
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