TY - JOUR

T1 - Nonlocal, non-commutative picture in quantum mechanics and distinguished continuous canonical maps

AU - Hakioglu, T.

PY - 2002/11

Y1 - 2002/11

N2 - 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 h-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 [Proc. Camb. Phil. Soc., 60, 581 (1964)] and Curtright, Fairlie and Zachos [Phys. Rev., D 58, 025002 (1998)]. 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.

AB - 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 h-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 [Proc. Camb. Phil. Soc., 60, 581 (1964)] and Curtright, Fairlie and Zachos [Phys. Rev., D 58, 025002 (1998)]. 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.

UR - http://www.scopus.com/inward/record.url?scp=0036848719&partnerID=8YFLogxK

U2 - 10.1238/Physica.Regular.066a00345

DO - 10.1238/Physica.Regular.066a00345

M3 - Article

AN - SCOPUS:0036848719

SN - 0031-8949

VL - 66

SP - 345

EP - 353

JO - Physica Scripta

JF - Physica Scripta

IS - 5

ER -