Two-Component Spinor Calculus
Spinor calculus is presented by relying on spin-space formalism. Given the existence of unprimed and primed spin-space, one has the isomorphism between such vector spaces and their duals, realized by a symplectic form. Moreover, for Lorentzian metrics, complex conjugation is the anti-isomorphism between unprimed and primed spin-space. Finally, for any space-time point, its tangent space is isomorphic to the tensor product of unprimed and primed spin-spaces via the Infeld-van der Waerden symbols. Hence the correspondence between tensor fields and spinor fields. Euclidean conjugation in Riemannian geometries is also discussed in detail. The Maxwell field strength is written in this language, and many useful identities are given. The curvature spinors of general relativity are then constructed explicitly, and the Petrov classification of space-times is obtained in terms of the Weyl spinor for conformal gravity
KeywordsSymplectic Form Spinor Field Curvature Spinor Spinor Index Riemann Curvature Tensor
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