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arXiv:2606.25085

Note About Koopman-von Neumann Theory and Density Matrix

J. Kluson

quant-phSubmitted Jun 23, 2026• Announced Jun 25, 2026• 0 citations
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Abstract

In this short note we study Koopman-von Neumann theory for N-particle system. We argue that it is natural to identify classical N-particle distribution function as diagonal form of density matrix operator in coordinate representation. We also determine generalized BBGKY hierarchy for reduced density matrix in coordinate representation.
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13 pages

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Generalized master equations determining the time dependence of both diagonal and off-diagonal density matrix elements are derived. The equations are valid for systems of arbitrary size and initial conditions, and have all the generality of the von Neumann equation for the density matrix. They are general enough for use in determining all higher-order contributions to transport coefficients. Their advantage over the von Neumann equation is that they are in forms suited to reduction by time-independent many-body approximation techniques.
Hilbert-space techniques are widely used not only for quantum theory, but also for classical physics. Two important examples are the Koopman-von Neumann (KvN) formulation and the method of ``classical'' wave functions. As this paper explains, these two approaches are conceptually distinct. In particular, the method of classical wave functions was not due to Bernard Koopman and John von Neumann, but was developed independently by a number of later researchers, perhaps first by Mario Schönberg, with key contributions from Angelo Loinger, Giacomo Della Riccia, Norbert Wiener, and E. C. George Sudarshan. The primary goals of this paper are to explain these two approaches, describe the relevant history in detail, and give credit where credit is due.
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