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arXiv:2512.07616 · Physics Letters A

Sharp values for all dynamical variables via Anti-Wick quantization

Simon Friederich

quant-phSubmitted Dec 8, 2025• Announced Dec 9, 2025• 0 citations

Abstract

This paper proposes an approach to interpreting quantum expectation values that may help address the quantum measurement problem. Quantum expectation values are usually calculated via Hilbert space inner products and, thereby, differently from expectation values in classical mechanics, which are weighted phase-space integrals. It is shown that, by using Anti-Wick quantization to associate dynamical variables with self-adjoint linear operators, quantum expectation values can be interpreted as genuine weighted averages over phase space, paralleling their classical counterparts. This interpretation arises naturally in the Segal-Bargmann space, where creation and annihilation operators act as simple multiplication and differentiation operators. In this setting, the Husimi Q-function - the coherent-state representation of the quantum state - can be seen as a true probability density in phase space. Unlike Bohmian mechanics, the present approach retains the standard correspondence between dynamical variables and self-adjoint operators while paving the way for a classical-like probabilistic interpretation of quantum statistics.

Journal reference: Physics Letters A, Volume 567, 2026, 131226, ISSN 0375-9601, Physics Letters A, 567: 131226 (2026)
DOI: 10.1016/j.physleta.2025.131226.
Comments: 19 pages, no figures

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Given a physical quantum system described by a Hilbert H, for any bounded quantum observable (a bounded self-adjoint operator) T it is possible to define several ''hidden observable'' functions f:H->R associated to T and for any quantum mixed state (a density matrix) D it is possible to define several ''hidden mixed states'' (probability measures) m on H associated to D in such a way that the following equality is verified: Trace[ b(T). D] =integral[b(f(psi)).dm(psi) whatever is the continuous function b:R->R. This formula gives a general way to express any expectation value computable in a quantum theory as a classical statistical mean value.

One of the crucial differences between mathematical models of classical and quantum mechanics (QM) is the use of the tensor product of the state spaces of subsystems as the state space of the corresponding composite system. (To describe an ensemble of classical composite systems, one uses random variables taking values in the Cartesian product of the state spaces of subsystems.) We show that, nevertheless, it is possible to establish a natural correspondence between the classical and the quantum probabilistic descriptions of composite systems. Quantum averages for composite systems (including entangled) can be represented as averages with respect to classical random fields. It is essentially what Albert Einstein dreamed of. QM is represented as classical statistical mechanics with infinite-dimensional phase space. While the mathematical construction is completely rigorous, its physical interpretation is a complicated problem. We present the basic physical interpretation of prequantum classical statistical field theory in Sec. II. However, this is only the first step toward real physical theory.

The approximations of classical mechanics resulting from quantum mechanics are richer than a correspondence of classical dynamical variables with self-adjoint Hilbert space operators. Assertion that classical dynamic variables correspond to self-adjoint Hilbert space operators is disputable and sets unnatural limits on quantum mechanics. Well known examples of classical dynamical variables not associated with self-adjoint Hilbert space operators are discussed as a motivation for the realizations of quantum field theory that lack Hermitian field operators but exhibit interaction.

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In this work, the statement and proof of a fundamental formula in the phase space representation of quantum systems will be carried out for the similitude group, 𝑆𝑖𝑚(2).This formula takes the form∫ 𝒂(𝒀)𝑷(𝒀)𝒅(𝒀) = 〈𝑨 ̂〉, where 𝑌 is the phase space variable and 〈𝐴 ̂〉 is a linear operator on Hilbert space representing a quantum dynamical observable.〈𝐴 ̂〉 is the quantum expected value of the observable in a state of the system.The focus on the similitude group is due to current interest in signal analysis, localization operators and pseudo-differential operators.The fundamental formula states that this may be computed in a classical manner, as an integral against a probability distribution.The formula is intimately related to the quantization-dequantization problem 𝑎(𝑌) ⟷ 𝐴 ̂ which assigns a quantum operator to the classical phase space function 𝑎(𝑌).