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On the relationship between the Wigner–Moyal approach and the quantum operator algebra of von Neumann

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Abstract

In this paper we discuss the close relationship between the Wigner–Moyal algebra and the original non-commutative quantum algebra introduced by von Neumann in 1931. We show that the “distribution function”, F(P, X, t) is simply the quantum mechanical density matrix for a single particle where the coordinates, X and P, are not the coordinates of a point particle, but the mean co-ordinate of a cell structure (a ‘blob’) in phase space. This provides an intrinsically non-local and non-commutative description of an individual, which only becomes a point particle in the commutative limit. In this general structure, the Wigner function appears as a transition probability amplitude which accounts for the appearance of negative values. The Moyal and Baker brackets play a significant role in the time evolution, producing the quantum Hamilton–Jacobi equation used in the Bohm approach. It is the non-commutative structure based on a symplectic geometry that generates a generalised phase space for quantum processes.

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Notes

  1. Here we are concerned with conceptual questions so the mathematical formalism will be kept as simple as possible using a two-dimensional phase space. The generalisation to a higher dimensional phase space is straightforward.

  2. We will put \(\hbar =1\) except in Sect. 2.7 where it helps the discussion to show its presence explicitly.

  3. We will follow convention and revert to lower case X and P.

  4. The eigenvalues of an idempotent operator are 1 or 0, to exist or not to exist.

  5. The distinction between left and right multiplication is necessary even in conventional quantum field theory when one deals with the Pauli and Dirac particles. The double arrow symbol (23) is used for the energy term, for example, in the Lagrangian for the Dirac field [33]. It is therefore not surprising that Eq. (22) involves energy.

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Acknowledgments

I would like to thank Fabio Frescura for many fruitful discussions and for drawing my attention to the von Neumann paper in the first place. I would also like to thank Maurice de Gosson for his help in my struggle to understand the deep topological structures that lie in symplectic geometry. I would also like to thank the members of the TPRU for their many helpful comments as this work unfolded.

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  1. University College, Gower Street, London, WC1E 6BT, UK

    B. J. Hiley

  2. TPRU, Birkbeck, University of London, Malet Street, London, WC1E 7HX, UK

    B. J. Hiley

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Hiley, B.J. On the relationship between the Wigner–Moyal approach and the quantum operator algebra of von Neumann. J Comput Electron 14, 869–878 (2015). https://doi.org/10.1007/s10825-015-0728-7

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