Gauged twistor formulation of a massive spinning particle in four dimensions

TL;DR

This paper develops a gauged twistor model for a massive spinning particle in four-dimensional space, deriving constraints and equations for spinor fields that satisfy generalized Dirac-Fierz-Pauli equations, highlighting the role of gauge symmetries.

Contribution

It introduces a novel gauged twistor formulation with gauge invariance and derives the corresponding quantum equations for massive spinning particles.

Findings

Constraints lead to differential equations for twistor functions.

Spinor fields satisfy generalized Dirac-Fierz-Pauli equations.

Clarifies physical roles of $U(1)$ and $SU(2)$ symmetries.

Abstract

We present a gauged twistor model of a free massive spinning particle in four-dimensional Minkowski space. This model is governed by an action, referred to here as the gauged generalized Shirafuji (GGS) action, that consists of twistor variables, auxiliary variables, and $U(1)$ and $SU(2)$ gauge fields on the one-dimensional parameter space of a particle's worldline. The GGS action remains invariant under reparametrization and the local $U(1)$ and $SU(2)$ transformations of the relevant variables, although the $SU(2)$ symmetry is nonlinearly realized. We consider the canonical Hamiltonian formalism based on the GGS action in the unitary gauge by following Dirac's recipe for constrained Hamiltonian systems. It is shown that just sufficient constraints for the twistor variables are consistently derived by virtue of the gauge symmetries of the GGS action. In the subsequent quantization procedure, these constraints turn into simultaneous differential equations for a twistor function. We perform the Penrose transform of this twistor function to define a massive spinor field of arbitrary rank, demonstrating that the spinor field satisfies generalized Dirac-Fierz-Pauli equations with $SU(2)$ indices. We also investigate the rank-one spinor fields in detail to clarify the physical meanings of the $U(1)$ and $SU(2)$ symmetries.