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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 \(\hat T^2 = -1\).

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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[1] Aharonov, Y. and Susskind, L. (1967). Observability of the sign change of spinors under 2\(\pi\) rotations, Physical Review 158, 1237–1238. · doi:10.1103/PhysRev.158.1237
[2] de Azcárraga, J. A. (1975). P, C, T, \(\theta\) In Quantum Field Theory, GIFT 7/75, Zaragoza, Spain, pp. 6–7.
[3] de Azcárraga, J. A. and Izquierdo, J. M. (1995). Lie groups, Lie Algebras, Cohomology, and Some Applications in Physics, Cambridge University Press, Cambridge, p. 153. · Zbl 0836.22027
[4] Berestetskii, V. B., Lifshitz, E. M., and Pitaevskii, L. P. (1982). Quantum Electrodynamics, Landau and Lifshitz Course of Theoretical Physics, Vol. 4, 2nd edn., Pergamon Press, Oxford, pp. 69–70.
[5] Capri, A. Z. (2002). Relativistic Quantum Mechanics and Introduction to Quantum Field Theory, World Scientific, New Jersey, pp. 46–51. · Zbl 1025.81001
[6] Feynman, R. P. (1987). The Reason for Antiparticles. In R. P. Feynman and S. Weinberg (Eds.), Elementary Particles and the Laws of Physics. 1986 Dirac Memorial Lectures, Cambridge University Press, Princeton, New Jersey, p. 48.
[7] Heine, V. (1977). Group Theory in Quantum Mechanics, An Introduction to Its Present Usage, Pergamon Press, Oxford, p. 87. · Zbl 0089.18502
[8] Landau, L. D. and Lifshitz, E. M. (1997). Quantum Mechanics, Non-relativistic Theory, Course of Theoretical Physics, Vol. 3, 3rd edn., Butterworth-Heinemann, Newton, MA, p. 393.
[9] MacLane, S. and Birkoff, G. (1979). Algebra, 2nd. edn., Macmillan, New York, p. 409.
[10] Naber, G. L. (1997). Topology, Geometry, and Gauge Fields. Foundations, Springer-Verlag, New York, pp. 367–377. · Zbl 0876.53002
[11] Rauch, H., Zeilinger, A., Badurek, G., and Wilfing, A. (1975). Verification of coherent spinor rotation of fermions. Physics Letters 54A, 425–427.
[12] Sakurai, J. J. (1985). Modern Quantum Mechanics, Benjamin, Menlo Park, California, p. 278.
[13] Silverman, M. P. (1980). The curious problem of spinor rotation, European Journal of Physics 1, 116–123. · doi:10.1088/0143-0807/1/2/009
[14] Socolovsky, M. (2001). On the Geometry of Spin 1/2, Advances in Applied Clifford Algebras 11, 487–494. · Zbl 1042.81013 · doi:10.1007/BF03042041
[15] Socolovsky, M. (2004). The CPT group of the Dirac field, math-ph/0404038, International Journal of Theoretical Physics 43, 1941–1967. · Zbl 1073.81033 · doi:10.1023/B:IJTP.0000049003.90851.60
[16] Sternberg, S. (1997). Group Theory and Physics, Cambridge University Press, Cambridge, p. 160. · Zbl 0829.53001
[17] Werner, S. A., Colella. R., Overhauser, A. W., and Eagen, C. F. (1975). Observation of the phase shift of a neutron due to precession in a magnetic field, Physical Review Letters 35, 1053–1055. · doi:10.1103/PhysRevLett.35.1053
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