Sharp values for all dynamical variables via Anti-Wick quantization

TL;DR

The paper addresses the quantum measurement problem by proposing Anti-Wick quantization within Segal–Bargmann space to reinterpret quantum expectation values as genuine phase-space averages. It shows that, for phase-space densities of the form $P(\mathbf{z})=|F(\mathbf{z})|^2\mu_\hbar(\mathbf{z})$, the quantum expectation $\langle A \rangle = \langle \psi|\hat{A}|\psi\rangle$ equals the classical-like integral $\int A(\mathbf{z})P(\mathbf{z})\,d\mathbf{z}$, with $\hat{A}$ given by a Toeplitz/Anti-Wick construction and $|F\rangle$ the Segal–Bargmann transform of $|\psi\rangle$. The Husimi Q-function then emerges as a bona fide phase-space probability density, enabling a classical-like probabilistic interpretation of quantum statistics while preserving the operator correspondence. The work highlights a coherent path toward solving the measurement problem by embedding phase-space sharpness at the level of the dynamical variables through Anti-Wick quantization and motivates further development of the underlying microdynamics and generalizations to broader quantization frameworks. It also discusses implications for contextuality, nonlocality, and the time evolution of Q-functions, outlining future research directions in microdynamics and complex-geometry generalizations such as Berezin quantization.

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.