Second order symmetry operators for the massive Dirac equation

Employing the covariant language of two-spinors, we find what conditions a curved four-dimensional 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.

Automated summary

Employing the covariant language of two-spinors, we determine the conditions for the existence of a second order symmetry operator for the massive Dirac equation in a curved four-dimensional Lorentzian spacetime. These conditions are formulated as the existence of a set of Killing spinors satisfying a set of covariant linear differential equations. Using these Killing spinors, we derive the most general form of such an operator. Partial results for the zeroth and first order are presented and interpreted. Computer algebra tools from the Mathematica package suite xAct were utilized for the calculations.