Symmetry operators and separation of variables for Dirac's equation on two-dimensional spin manifolds with external fields

TL;DR

This paper characterizes second order symmetry operators commuting with the Dirac operator on two-dimensional spin manifolds with external fields, linking them to geometric structures like Killing vectors and tensors, and explores their role in variable separation.

Contribution

It provides a comprehensive computation of symmetry operators for the Dirac equation involving external fields on 2D manifolds, connecting them to geometric symmetries and separation schemes.

Findings

Symmetry operators are expressed via Killing vectors and tensors.

Operators from different approaches are shown to coincide.

Examples demonstrate the practical application of the theoretical results.

Abstract

The second order symmetry operators that commute with the Dirac operator with external vector, scalar and pseudo-scalar potentials are computed on a general two-dimensional spin-manifold. It is shown that the operator is defined in terms of Killing vectors, valence two Killing tensors and scalar fields defined on the background manifold. The commuting operator that arises from a non-trivial Killing tensor is determined with respect to the associated system of Liouville coordinates and compared to the the second order operator that arises from that obtained from the unique separation scheme associated with such operators. It shown by the study of several examples that the operators arising from these two approaches coincide.