Fundamental Relativistic Rotator. Hessian singularity and the issue of the minimal interaction with electromagnetic field

TL;DR

This paper investigates the dynamics of relativistic rotators, revealing that their Hessian singularity leads to indeterminate motion and unphysical constraints, which cannot be resolved by minimal electromagnetic interaction.

Contribution

It demonstrates that the Hessian singularity in relativistic rotators persists even with minimal electromagnetic interaction, causing unphysical constraints on motion.

Findings

Hessian singularity causes indeterminate evolution.

Minimal electromagnetic interaction does not remove the singularity.

External fields impose unphysical constraints on initial conditions.

Abstract

There are two relativistic rotators with Casimir invariants of the Poincar\'{e} group being fixed parameters. The particular models of spinning particles were studied in the past both at the classical and quantum level. Recently, a minimal interaction with electromagnetic field has been considered. We show that the dynamical systems can be uniquely singled out from among other relativistic rotators by the unphysical requirement that the Hessian referring to the physical degrees of freedom should be singular. Closely related is the fact that the equations of free motion are not independent, making the evolution indeterminate. We show that the Hessian singularity cannot be removed by the minimal interaction with the electromagnetic field. By making use of a nontrivial Hessian null space, we show that a single constraint appears in the external field for consistency of the equations of motion with the Hessian singularity. The constraint imposes unphysical limitation on the initial conditions and admissible motions. We discuss the mechanism of appearance of unique solutions in external fields on an example of motion in the uniform magnetic field. We give a simple model to illustrate that similarly constrained evolution cannot be determinate in arbitrary fields.