Nonlocal, Non-Commutative Picture in Quantum Mechanics and Distinguished Continuous Canonical Maps
Tugrul Hakioglu
It is shown that continuous classical nonlinear canonical (Poisson) maps have a distinguished role in quantum mechanics. They act unitarily on the quantum phase space and generate ℏ-independent quantum nonlinear canonical maps. It is also shown that such maps act in the non-commutative phase space under the classical covariance. A crucial result of the work is that under the action of Poisson maps a local quantum mechanical picture is converted onto a non-local picture which is then represented in a non-local Hilbert space. On the other hand, it is known that a non-local picture is equivalent by the Weyl map to a non-commutative picture which, in the context of this work, corresponds to a phase space formulation of the theory. As a result of this equivalence, a phase space Schrödinger picture can be formulated. In particular, we obtain the ⋆-genvalue equation of Fairlie (1964 Proc. Camb. Phil. Soc. 60 581) and Curtright, Fairlie and Zachos (1998 Phys. Rev. D 58 025002). In a non-local picture entanglement becomes a crucial concept. The connection between the entanglement and non-locality is explored in the context of Poisson maps and specific examples of the generation of entanglement from a local wavefunction are provided by using the concept of generalized Bell states. The results obtained are also relevant for the non-commutative soliton picture in the non-commutative field theories. We elaborate on this in the context of the scalar non-commutative field theory.
arXiv:hep-th/0108125•Physica Scripta•Jan 1, 2002•1 citation