Application of covariant analytic mechanics with differential forms to gravity with Dirac field

TL;DR

This paper extends covariant analytic mechanics using differential forms to include the Dirac field and gravity, demonstrating its covariant properties and applicability to various fundamental fields, while noting limitations with higher order curvature theories.

Contribution

It applies covariant analytic mechanics to Dirac fields and gravity, showing its general covariance and exploring first and second order formalisms in gravitational theories.

Findings

Covariant analytic mechanics is applicable to Dirac fields and gravity.

The formalism maintains general coordinate and gauge covariance.

Higher order curvature gravitation theories are incompatible with the second order formalism.

Abstract

We apply the covariant analytic mechanics with the differential forms to the Dirac field and the gravity with the Dirac field. The covariant analytic mechanics treats space and time on an equal footing regarding the differential forms as the basic variables. A significant feature of the covariant analytic mechanics is that the canonical equations, in addition to the Euler-Lagrange equation, are not only manifestly general coordinate covariant but also gauge covariant. Combining our study and the previous works (the scalar field, the abelian and non-abelian gauge fields and the gravity without the Dirac field), the applicability of the covariant analytic mechanics is checked for all fundamental fields. We study both the first and second order formalism of the gravitational field coupled with matters including the Dirac field. It is suggested that gravitation theories including higher order curvatures cannot be treated by the second order formalism in the covariant analytic mechanics.