Abstract
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.
Highlights
In many evolutions taking place in a flat or a curved background, isometries of the underlying spacetime metric lead to conservation laws that have clear geometrical meanings expressed by means of their local generators, Killing vector fields
An important place where some or all of these tensorial objects enter the analysis is the study of symmetry operators for the Dirac-type equations describing the motion in curved background with or without additional interactions. It has become a well-established fact that while conformal KY (CKY) forms take part in symmetry operators, via the R-commuting argument for the massless Dirac equation, KY forms are indispensable in constructing first-order symmetries of the massless as well as massive Dirac equation in a curved spacetime
In searching for the most general symmetry operator commuting with the Dirac equation in the presence of an electromagnetic field, they found that the symmetry operator can be constructed from KY forms of the underlying background, provided that they separately fulfil some conditions involving the field itself
Summary
In many evolutions taking place in a flat or a curved background, isometries of the underlying spacetime metric lead to conservation laws that have clear geometrical meanings expressed by means of their local generators, Killing vector fields. An important place where some or all of these tensorial objects enter the analysis is the study of symmetry operators for the Dirac-type equations describing the motion in curved background with or without additional interactions It has become a well-established fact that while CKY forms take part in symmetry operators, via the R-commuting argument for the massless Dirac equation, KY forms are indispensable in constructing first-order symmetries of the massless as well as massive Dirac equation in a curved spacetime. In searching for the most general symmetry operator commuting with the Dirac equation in the presence of an electromagnetic field, they found that the symmetry operator can be constructed from KY forms of the underlying background, provided that they separately fulfil some conditions involving the field itself These included some conditions found before by Carter and McLenaghan [12].
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