General Covariant Equations for Fields of Arbitrary Spin

TL;DR

This paper derives general covariant wave equations for particles of arbitrary spin in curved spacetime, extending previous special relativity results and introducing a gauge covariant derivative for a gauge theory of gravitation.

Contribution

It generalizes the Gel'fand-Iaglom wave equation to curved space and introduces a gauge covariant derivative with local Lorentz invariance.

Findings

Derived covariant field equations for arbitrary spin in curved spacetime.

Introduced a gauge covariant derivative based on local Lorentz transformations.

Provided a foundation for a gauge theory of gravitation.

Abstract

To obtain a generalized wave equation for a field in general covariant form, one must define covariant differentiation of a generalized wave function describing particles with arbitrary spin. Gel'fand and Iaglom, Dirac, and Fierz and Pauli have studied the generalized wave equation in the special theory of relativity. In this paper, the Gel'fand-Iaglom field equation was generalized to the curved space based on the veilbein framework. The general covariant field equations for arbitrary spin were obtained. More importantly, in this work, we introduced the gauge covariant derivative with the local Lorentz transformation, which is the foundation for a gauge theory of gravitation.