Symmetry Operators and Separation of Variables for Dirac's Equation on Two-Dimensional Spin Manifolds

TL;DR

This paper develops a formalism to identify symmetry operators for Dirac's equation on 2D spin manifolds, enabling separation of variables and solving the equation in specific geometries like Minkowski space.

Contribution

It introduces a signature-independent formalism for second-order symmetry operators, facilitating separation of variables for Dirac's equation on 2D Lorentzian manifolds.

Findings

Characterization of orthonormal frames allowing separation

Application to Minkowski space with complex variables

General second-order symmetry operators identified

Abstract

A signature independent formalism is created and utilized to determine the general second-order symmetry operators for Dirac's equation on two-dimensional Lorentzian spin manifolds. The formalism is used to characterize the orthonormal frames and metrics that permit the solution of Dirac's equation by separation of variables in the case where a second-order symmetry operator underlies the separation. Separation of variables in complex variables on two-dimensional Minkowski space is also considered.