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Aspects of Non-commutative Geometry: Deformation Quantum Mechanics

Basil J. Hiley Related
Physics 26/10/2017

In a little known 1931 paper, von Neumann shows how it is possible to discuss quantum mechanics, not in terms of Hilbert space, but in terms of a non-commutative quantum phase space. This structure turns out to be identical to the algebra introduced in 1949 by Moyal in a different context. The resulting non-commutative symplectic algebra contains the classical Poisson algebra as a sub-structure resulting in deformation quantum mechanics. The importance of this algebra for the subject of this conference is that it shows exactly how the Bohm approach emerges from this non-commutative structure as a projection onto a commutative phase space. It also shows how weak values arise in this non-commutative structure. I will explain how this general structure faces similar problems to that raised in Penrose’s palatial twistor theory which involves combining a symplectic structure with an orthogonal Clifford algebra in a non-trivial way.