Compatibility of symmetric quantization with general covariance in the Dirac equation and spin connections

TL;DR

This paper explores how symmetric quantization can be compatible with general covariance in the Dirac equation, deriving specific spin connection representations and solving the equations exactly in 1+1 dimensions.

Contribution

It introduces a special representation of spin connections consistent with symmetric quantization and derives exact solutions in a curved 1+1 dimensional spacetime.

Findings

Derived a matrix operator algebra involving a universal curvature constant.

Obtained exact solutions of Dirac and Klein-Gordon equations in 1+1 static spacetime.

Established compatibility conditions between symmetric quantization and general covariance.

Abstract

By requiring unambiguous symmetric quantization leading to the Dirac equation in a curved space, we obtain a special representation of the spin connections in terms of the Dirac gamma matrices and their space-time derivatives. We also require that squaring the equation give the Klein-Gordon equation in a curved space in its canonical from (without spinor components coupling and with no first order derivatives). These requirements result in matrix operator algebra for the Dirac gamma matrices that involves a universal curvature constant. We obtain exact solutions of the Dirac and Klein-Gordon equations in 1+1 space-time for a given static metric.