[Paper Review] Classical Representations of Quantum Mechanics Related to Statistically Complete Observables

Werner Stulpe|ArXiv.org|Oct 16, 2006

Quantum Mechanics and Applications54 references18 citations

TL;DR

This paper proposes a classical phase-space representation of quantum mechanics using statistically complete observables, enabling a map from density operators to non-negative probability densities on phase space. It establishes that while exact satisfaction of all classical postulates (including correct marginals) is impossible due to Wigner's theorem, approximate satisfaction is achievable, and quantum dynamics can be reformulated as a Liouville-type equation on phase space via dequantization maps.

ABSTRACT

We present a reformulation of quantum mechanics in terms of probability measures and functions on a general classical sample space and in particular in terms of probability densities and functions on phase space. The basis of our proceeding is the existence of so-called statistically complete observables and the duality between the state spaces and the spaces of the observables, the latter holding in the quantum as well as in the classical case. In the phase-space context, we further discuss joint position-momentum observables, Hilbert spaces of infinitely differentiable functions on phase space, and dequantizations. Finally, the relation of quantum dynamics to the classical Liouville dynamics is investigated.

Motivation & Objective

• To develop a classical representation of quantum mechanics on phase space using statistically complete observables.
• To address the fundamental challenge of representing quantum states as non-negative probability densities on phase space.
• To investigate the conditions under which quantum expectation values can be expressed as integrals over phase-space functions.
• To explore the possibility of reformulating quantum dynamics as a classical Liouville equation on phase space.
• To clarify the limitations imposed by Wigner's theorem on exact classical representations with non-negative densities.

Proposed method

• Constructs a map $W \mapsto \rho_W$ from density operators to non-negative phase-space probability densities.
• Imposes postulates: affine and injective mapping, expectation value representation via $\mathrm{tr}(WA) = \int \rho_W f_A \, dq\,dp$, and correct position/momentum marginals.
• Uses statistically complete observables to ensure injectivity and enable dequantization maps $A \mapsto f_A$.
• Applies group representations and coherent states to generate continuous resolutions of identity on phase space.
• Derives a Liouville-type equation for time evolution by mapping the von Neumann equation to phase space via the transform $\hat{T}$.
• Demonstrates that for the harmonic oscillator, the phase-space dynamics satisfies $\frac{\partial \rho_t}{\partial t} = -\{H, \rho_t\}$ under specific conditions.

Experimental results

Research questions

• RQ1 Can quantum mechanics be consistently represented on a classical phase space using non-negative probability densities?
• RQ2 To what extent can the quantum expectation values be represented as integrals over phase-space functions?
• RQ3 Why is it impossible to satisfy both the correct marginal distributions and non-negative densities simultaneously?
• RQ4 How can quantum dynamics be reformulated as a classical Liouville equation on phase space?
• RQ5 What is the role of statistically complete observables in enabling a classical representation of quantum states?

Key findings

• A map $W \mapsto \rho_W$ exists that satisfies postulates (i) and (ii) exactly: it is affine and injective.
• Postulate (iii), representing quantum expectation values via phase-space functions, holds essentially for bounded operators and approximately for all bounded operators.
• Postulate (iv), requiring correct position and momentum marginals, cannot be satisfied exactly due to Wigner's theorem.
• The Wigner function $\rho^W_W$ satisfies (i), (ii), and (iv) exactly, but is not a true probability density as it can be negative.
• For the harmonic oscillator with $\sigma = 1/\sqrt{2m\omega}$, the time evolution of $\rho_t = \hat{T} \tau_t \hat{T}^{-1} \rho$ satisfies the classical Liouville equation $\frac{\partial \rho_t}{\partial t} = -\{H, \rho_t\}$.
• The time derivative $\dot{\rho}_t$ from the Liouville operator $\hat{L}$ and the partial derivative $\frac{\partial \rho_t}{\partial t}$ coincide only under smoothness and domain conditions, highlighting a subtle distinction in dynamics.