Nonlocal, noncommutative picture in quantum mechanics and distinguished canonical maps

TL;DR

This paper explores how classical nonlinear canonical maps induce nonlocal effects in quantum mechanics through noncommutative phase space, linking entanglement, nonlocality, and applications in noncommutative scalar field theories.

Contribution

It demonstrates the nonlocal quantum effects of classical canonical maps in noncommutative phase space and connects entanglement with nonlocality using the Weyl map.

Findings

Classical canonical maps induce nonlocality in quantum phase space.

Entanglement can be generated via noncommutative phase space transformations.

Applications include vacuum soliton configurations in noncommutative scalar field theories.

Abstract

Classical nonlinear canonical (Poisson) maps have a distinguished role in quantum mechanics. They act unitarily on the quantum phase space and generate $\hbar$-independent quantum canonical maps. It is shown that such maps act in the noncommutative phase space as dictated by the classical covariance. A crucial observation made is that under the classical covariance the local quantum mechanical picture can become nonlocal in the Hilbert space. This nonlocal picture is made equivalent by the Weyl map to a noncommutative picture in the phase space formulation of the theory. The connection between the entanglement and nonlocality of the representation is explored and specific examples of the generation of entanglement are provided by using such concepts as the generalized Bell states. That the results have direct application in generating vacuum soliton configurations in the recently popular scalar field theories of noncommutative coordinates is also demonstrated.