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Foundations of Physics

On the Relationship Between the Wigner-Moyal and Bohm Approaches to Quantum Mechanics: A Step to a More General Theory?

Basil J. Hiley

Published Jun 11, 2009• 27 citations
Publisher
DOI
10.1007/s10701-009-9320-y

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Physical Review A•Dec 1, 1994•79 citations
Discussed are some geometric aspects of the phase space formalism in quantum mechanics in the sense of Weyl, Wigner, Moyal, and Ville. We analyze the relationship between this formalism and geometry of the Galilei group, classical momentum mapping, theory of unitary projective representations of groups, and theory of groups algebras. Later on, we present some generalization to quantum mechanics on locally compact Abelian groups. It is based on Pontryagin duality. Indicated are certain physical aspects in quantum dynamics of crystal lattices, including the phenomenon of ‘Umklapp–Prozessen’. Copyright © 2011 John Wiley & Sons, Ltd.
The displaced parity operators are shown to have properties that bear a deep relationship with those of the Wigner functions. By exploiting these properties we show that these operators play an important role in linking together many of the aspects of the various exact phase-space mappings of quantum mechanics. These include the Wigner and Weyl representations, coherent states and the Bargmann representation, the P and Q representations, the Weyl correspondence, and the Moyal star product formalism. We also introduce corresponding displaced Fourier operators and show that their squares are just the displaced parity operators. The formalism is extended to squeezed and displaced parity operators, and their corresponding central role in the theory of squeezed coherent states and general squeezing is explained. We also elucidate the part played by the displaced parity operators in the Moyal star product and its extensions, as a first step towards a potential application of these operators in such modern developments as deformation theory and quantum groups. Finally, we indicate how the apparatus developed might also find applications in other recent exact classical mappings of many-particle quantum mechanics or quantum field theory, which are not special cases of deformation theory. Prime examples here include the powerful so-called independent-cluster method techniques, which incorporate the coupled-cluster method formalism with its inbuilt supercoherent states. Throughout the work we stress the central and unifying role played by the displaced parity operator and its generalizations.