Second order symmetry operators for the massive Dirac equation

TL;DR

This paper characterizes the geometric conditions under which second order symmetry operators exist for the massive Dirac equation in curved spacetimes, using Killing spinors and computer algebra tools.

Contribution

It derives the necessary conditions for second order symmetry operators for the Dirac equation in curved spacetime using a spinor approach and explicit differential equations.

Findings

Conditions for existence of second order symmetry operators are formulated.

The general form of such operators is derived using Killing spinors.

Partial results for zeroth and first order operators are provided.

Abstract

Employing the covariant language of two-spinors, we find what conditions a curved Lorentzian spacetime must satisfy for existence of a second order symmetry operator for the massive Dirac equation. The conditions are formulated as existence of a set of Killing spinors satisfying a set of covariant linear differential equations. Using these Killing spinors, we then state the most general form of such an operator. Partial results for the zeroth and first order are presented and interpreted as well. Computer algebra tools from the Mathematica package suite xAct were used for the calculations.