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

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 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.

Spin raising and lowering operators for Rarita-Schwinger fields

“…Du . Now, we have to prove that the spin- 3 2 field defined in (24) satisfies both of the Rarita-Schwinger field equations in (21) and (22).…”

“…This resembles a condition on Killing-Yano forms that can be used in the construction of symmetry operators of a massive Dirac equation with an electromagnetic minimal coupling term [21]. Those symmetry operators are constructed from the Killing-Yano forms ω that satisfy the condition…”

“…This can be seen as follows. The Rarita-Schwinger equation given in (21) and (22) can be written in a more compact form. Let us consider a Clifford-valued 1-form e = e a ⊗ e a and define the action of the covariant exterior derivative D on Ψ = ψ a ⊗ e a as…”

“…Let us consider a massless Rarita-Schwinger field Ψ = ψ a ⊗ e a in four dimensions which satisfies (21) and (22) with an extra condition…”

On quantum numbers for Rarita–Schwinger fields

“…5) where the overall sign in the above depends on the signature of the background metric. Using this we can obtain the dual expression of O (n+3) µν .…”