Twistor formulation of a massive particle with rigidity

TL;DR

This paper reformulates a massive rigid particle model in four dimensions using twistors, develops a gauge-fixed Hamiltonian formalism, and demonstrates that the particle's spin quantum number is a non-negative integer, with mass depending on spin.

Contribution

It introduces a novel twistor-based formulation for a massive particle with rigidity, including gauge fixing, Hamiltonian analysis, and quantization, revealing spin quantization and mass dependence.

Findings

Spin quantum number is a non-negative integer.

Mass of the spinor field depends on the spin.

Twistor formulation maintains local U(1)×U(1) invariance.

Abstract

A massive rigid particle model in $(3+1)$ dimensions is reformulated in terms of twistors. Beginning with a first-order Lagrangian, we establish a twistor representation of the Lagrangian for a massive particle with rigidity. The twistorial Lagrangian derived in this way remains invariant under a local $U(1) \times U(1)$ transformation of the twistor and other relevant variables. Considering this fact, we carry out a partial gauge-fixing so as to make our analysis simple and clear. We develop the canonical Hamiltonian formalism based on the gauge-fixed Lagrangian and perform the canonical quantization procedure of the Hamiltonian system. Also, we obtain an arbitrary-rank massive spinor field in $(3+1)$ dimensions via the Penrose transform of a twistor function defined in the quantization procedure. Then we prove, in a twistorial fashion, that the spin quantum number of a massive particle with rigidity can take only non-negative integer values, which result is in agreement with the one shown earlier by Plyushchay. Interestingly, the mass of the spinor field is determined depending on the spin quantum number.