Two-spinor form of first-order gravity coupled to Dirac fields

Godina, M., Matteucci, P., Fatibene, L. and Francaviglia, M. (2000) Two-spinor form of first-order gravity coupled to Dirac fields. General Relativity and Gravitation, 32 (1), 145-160.

Abstract

A two-spinor formalism for the Einstein Lagrangian is developed. The gravitational field is regarded as a composite object derived from soldering forms. Our formalism is geometrically and globally well-defined and may be used in virtually any 4m-dimensional manifold with arbitrary signature as well as without any stringent topological requirement on space-time, such as parallelizability. Interactions and feedbacks between gravity and spinor fields are considered. As is well known, the Hilbert–Einstein Lagrangian is second order also when expressed in terms of soldering forms. A covariant splitting is then analysed leading to a first-order Lagrangian which is recognized to play a fundamental role in the theory of conserved quantities. The splitting and thence the first-order Lagrangian depend on a reference spin connection which is physically interpreted as setting the zero level for conserved quantities. A complete and detailed treatment of conserved quantities is then presented.

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