# Covariant Jacobi Brackets for Test Particles

## TL;DR

This paper introduces a Poincaré-invariant Jacobi structure on the space of test particle observables, extending the Peierls procedure to time-like geodesics in Minkowski space-time, and discusses conditions for Poisson algebra formation.

## Contribution

It presents a natural Jacobi structure on test particle observables invariant under Poincaré transformations, extending the Peierls method to geodesics in Minkowski space.

## Key findings

- Jacobi structure is Poincaré invariant
- Poisson algebras require additional conditions
- Extension of Peierls procedure to geodesics

## Abstract

We show that the space of observables of test particles carries a natural Jacobi structure which is manifestly invariant under the action of the Poincar\'{e} group. Poisson algebras may be obtained by imposing further requirements. A generalization of Peierls procedure is used to extend this Jacobi bracket on the space of time-like geodesics on Minkowski space-time.