Noncommutative Mapping from the symplectic formalism

TL;DR

This paper generalizes the Bopp's shifts using symplectic formalism to map between commutative and noncommutative phase spaces, reproducing known results without small parameter assumptions and exploring the impact of different NC algebra choices on system symmetries.

Contribution

It introduces a symplectic formalism-based procedure for noncommutative mapping, extending Bopp's shifts, and analyzes the effects of various NC algebra choices on dynamical systems and symmetries.

Findings

Reproduces known $ heta$-deformed results without small parameter approximation

Different NC algebra choices lead to distinct dynamical systems

Some NC choices preserve, break, or restore system symmetries

Abstract

The Bopp's shifts will be generalized through symplectic formalism. A special procedure, like a "diagonalization", which drives the completely deformed symplectic matrix to the standard symplectic form was found as suggested by Faddeev-Jackiw. Consequently, the correspondent transformation matrix guides the mapping from commutative to noncommutative (NC) phase-space coordinates. The Bopp's shifts may be directly generalized from this mapping. In this context, all the NC and scale parameters, introduced into the brackets, will be lifted to the Hamiltonian. Well known results, obtained using $\star$-product, will be reproduced without to consider that the NC parameters are small$(<<1)$. Besides, it will be shown that different choices for NC algebra among the symplectic variables generates distinct dynamical systems, which they may not even connect with each other, and that some of them can preserve, break or restore the symmetry of the system. Further, we will also discuss the charge and mass rescaling in a simple model.