Constrained dynamics of two interacting relativistic particles in the Faddeev-Jackiw symplectic framework

TL;DR

This paper applies the Faddeev-Jackiw symplectic formalism to analyze the constraints, gauge symmetries, and degrees of freedom of a model with two interacting relativistic particles, providing insights into bigravity theories.

Contribution

It demonstrates the use of the Faddeev-Jackiw formalism to systematically identify constraints and gauge structures in a relativistic particle model with interactions, including gauge fixing and quantization.

Findings

Identified the complete set of physical constraints and zero-modes.

Derived transformation laws for dynamical variables related to gauge symmetries.

Computed the physical degrees of freedom and Faddeev-Jackiw brackets.

Abstract

The Faddeev-Jackiw symplectic formalism for constrained systems is applied to analyze the dynamical content of a model describing two massive relativistic particles with interaction, which can also be interpreted as a bigravity model in one dimension. We systematically investigate the nature of the physical constraints, for which we also determine the zero-modes structure of the corresponding symplectic matrix. After identifying the whole set of constraints, we find out the transformation laws for all the set of dynamical variables corresponding to gauge symmetries, encoded in the remaining zero-modes. In addition, we use an appropriate gauge-fixing procedure, the conformal gauge, to compute the quantization brackets (Faddeev-Jackiw brackets) and also obtain the number of physical degree of freedom. Finally, we argue that this symplectic approach can be helpful for assessing physical constraints and understanding the gauge structure of theories of interacting spin-2 fields.