James P Crawford 2003 Class. Quantum Grav. 20 2945 doi:10.1088/0264-9381/20/13/337
Spinor metrics, spin connection compatibility and spacetime geometry from spin geometry
James P Crawford
Show affiliationsWe show first that it is possible to consider the charge conjugation matrix as a metric (inner product) on the spin space. This metric is complementary to the usual Dirac spinor metric in that the Dirac metric defines the inner product of a spinor with a conjugate spinor, whereas the charge conjugation metric defines the inner product of a spinor with another spinor. The invariance group of the Dirac metric, U(2, 2), is distinct from that of the charge metric, Sp(4; 
), but their joint subgroup, Sp(4; 
), contains the cover of the Lorentz group, S
(2; ). It is possible to find a spin connection that is metric compatible with both spin metrics, and also compatible with covariant constancy of the Dirac matrices, and this condition also then determines the spacetime curvature as the spin curvature. However, we show that if the condition of covariant constancy of the Dirac matrices is relaxed, it is possible to maintain metricity for both spin metrics, and to obtain both spacetime curvature and torsion from the spin curvature.
- PACS
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04.20.Gz Spacetime topology, causal structure, spinor structure
11.30.Cp Lorentz and Poincare invariance
11.30.Er Charge conjugation, parity, time reversal, and other discrete symmetries
- MSC
- Subjects
- Dates
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Issue 13 (7 July 2003)
Received 1 November 2002
Published 13 June 2003
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Spinor metrics, spin connection compatibility and spacetime geometry from spin geometry
James P Crawford 2003 Class. Quantum Grav. 20 2945
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