Nonlocal, noncommutative picture in quantum mechanics and distinguished canonical maps

Abstract

Classical nonlinear canonical (Poisson) maps have a distinguished role in quantum mechanics. They act unitarily on the quantum phase space and generate $\hslash$-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.

14 pages, one figure

References in corpus (4)

• The Quantum Hall Fluid and Non-Commutative Chern Simons Theory
• Quantum Theories on Noncommutative Spaces with Nontrivial Topology: Aharonov-Bohm and Casimir Effects
• The Moyal-Lie Theory of Phase Space Quantum Mechanics
• Quantum Deformed Canonical Transformations, W_{\infty}-algebras and Unitary Transformations

Cited by in corpus (1)

• Lagrangian and Noncommutativity