First-order symmetries of the Dirac equation in a curved background: a unified dynamical symmetry condition

It has been shown that, for all dimensions and signatures, the most general first-order linear symmetry operators for the Dirac equation including interaction with Maxwell field in a curved background are given in terms of Killing-Yano (KY) forms. As a general gauge invariant condition it is found that among all KY forms of the underlying (pseudo) Riemannian manifold, only those which Clifford commute with the Maxwell field take part in the symmetry operator. It is also proved that associated with each KY form taking part in the symmetry operator, one can define a quadratic function of velocities which is a geodesic invariant as well as a constant of motion for the classical trajectory. Some geometrical and physical implications of the existence of KY forms are also elucidated.