Solution of the Dirac equation in a curved space with static metric

TL;DR

This paper derives exact solutions to the Dirac equation in a 1+1 dimensional curved space with static metric, exploring its algebraic structure and interactions with potentials.

Contribution

It introduces a specific representation of spin connections compatible with covariance and provides exact solutions and interaction formulations in a curved spacetime setting.

Findings

Exact solutions for the Dirac equation in 1+1 static curved space.

Operator algebra involving covariant derivatives and Riemann-Christoffel connections.

Formulation and solution of interacting Dirac models with potentials.

Abstract

Compatibility of symmetric quantization of the Dirac equation in a curved space with general covariance gives a special representation of the spin connections in which their dot product with the Dirac gamma matrices becomes equal to the "covariant divergence" of the latter. Requiring that the square of the equation gives the conventional Klein-Gordon equation in a curved space results in an operator algebra for the Dirac gamma matrices that involves the "covariant derivative" connections and the Riemann-Christoffel connections. In 1+1 space-time with static metric, we obtain exact solutions of this Dirac equation model for some examples. We also formulate the interacting theory of the model with various coupling modes and solve it in the same space for a given potential configuration.