Two-spinor Formulation of First Order Gravity coupled to Dirac Fields

TL;DR

This paper develops a two-spinor formalism for first order gravity coupled with Dirac fields, emphasizing a geometrically well-defined approach that applies to general 4-manifolds and analyzing conserved quantities.

Contribution

It introduces a novel two-spinor formalism for Einstein gravity that is globally well-defined on arbitrary 4-manifolds and includes a detailed treatment of conserved quantities.

Findings

Formalism is applicable to any 4-manifold with arbitrary signature.

A covariant splitting leads to a fundamental first order Lagrangian.

Complete analysis of conserved quantities in the theory.

Abstract

Two-spinor formalism for 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.