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

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 gives 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.