Quantum Analytical Mechanics: Quantum Mechanics with Hidden Variables

TL;DR

The paper addresses the quantum measurement problem and the role of hidden variables by proposing quantum analytical mechanics (QAM), a completion of standard quantum mechanics based on time-reversal invariant diffusion and stochastic trajectories in configuration space. It derives a Hamilton-Jacobi–Madelung formulation from forward-backward stochastic dynamics, linking observables to stochastic variables on configuration space and reproducing Schrödinger dynamics via $\psi=\sqrt{\rho}\,e^{iS/\hbar}$. The approach enables a dynamical account of measurement, illustrated through a levitated mesoscopic oscillator where trajectories reproduce observed coherence and spectra, and a Stern-Gerlach-type setup where spin emerges as a dynamical internal variable on $\mathbb{R}^3\times SO(3)$, connecting to the Pauli equation. The author argues that QAM is not a rival to Hilbert-space QM but a physically richer extension that preserves existing results while providing real trajectories and duration estimates for quantum processes, potentially reshaping the understanding of measurement and hidden-variable questions.

Abstract

The question about the existence of so-called ``hidden'' variables in quantum mechanics and the perception of the completeness of quantum mechanics are two sides of the same coin. Quantum analytical mechanics constitutes a completion of standard quantum mechanics based on the concept of stochastic trajectories in the configuration space of a quantum system. For particle systems, configuration space is made up out of their coordinates and, if relevant, their orientation. Quantum analytical mechanics derives equations of motion for these variables which allow a description of the measurement process as a dynamical physical process. After all, it is exactly these variables experiments are designed to interact with. The theory is not a replacement of Hilbert space quantum mechanics but a mathematical completion enriching our toolset for the description of quantum phenomena.

Figures (2)

• Figure 1: Left: sample paths for the ground state and a coherent state with energy $3\hbar\omega$ in the classical motion. Right: probability distributions for the positions of the particle in these states.
• Figure 2: Sample paths of particles bearing an intrinsic magnetic moment in a Stern-Gerlach experiment. The first 3 cm, which are enlarged on the left, are within the magnet, the screen is put at 9 cm distance to the beginning of the magnet. These dimensions replicate the original experimental setup (taken from Michael-Diss with permission).