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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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