Recursion operator in a noncommutative Minkowski phase space

TL;DR

This paper develops a recursion operator for geodesic flow in a noncommutative Minkowski phase space, introducing a Hamiltonian framework, canonical transformations, and conserved quantities within this noncommutative geometric setting.

Contribution

It constructs a recursion operator in a noncommutative Minkowski phase space and relates it to Hamiltonian dynamics and conserved quantities, advancing noncommutative geometric methods.

Findings

Recursion operator formulated for noncommutative Minkowski phase space.

Derived constants of motion from the recursion operator.

Re-expressed physical quantities in initial noncommutative coordinates.

Abstract

A recursion operator for a geodesic flow, in a noncommutative (NC) phase space endowed with a Minkowski metric, is constructed and discussed in this work. A NC Hamiltonian function $\mathcal{H}_{nc}$ describing the dynamics of a free particle system in such a phase space, equipped with a noncommutative symplectic form $\omega_{nc}$ is defined. A related NC Poisson bracket is obtained. This permits to construct the NC Hamiltonian vector field, also called NC geodesic flow. Further, using a canonical transformation induced by a generating function from the Hamilton-Jacobi equation, we obtain a relationship between old and new coordinates, and their conjugate momenta. These new coordinates are used to re-write the NC recursion operator in a simpler form, and to deduce the corresponding constants of motion. Finally, all obtained physical quantities are re-expressed and analyzed in the initial NC canonical coordinates.