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Journal of Physics A Mathematical and Theoretical

Classicality of single quantum particles in curved spacetime through the hydrodynamical reformulation of quantum mechanics

Tomer Shushi

Published Aug 9, 2023• 7 citations

Abstract

Abstract Single-particle physics focuses on the behavior and properties of individual particles, providing insight into the building blocks of quantum mechanics. The theory of quantum particles in curved spacetime has been getting attention in recent years for gaining a deeper understanding of the relationship between quantum mechanics and general relativity, the two pillars of modern physics. In this note, we show how single quantum particles can obtain classical behavior. In particular, for a quantum particle that follows the Klein–Gordon equation in curved spacetime in the presence of external potential, we show that when the amplitude of its wavefunction follows the Klein–Gordon equation with an arbitrary effective mass, empty curved spacetime, but with the same curved geometry appearing in the original Klein–Gordon equation of the wavefunction, the quantum force of the particle vanishes, providing a classical description of the quantum particle using a system of coupled classical equations. The result relies on the Madelung hydrodynamical reformulation of quantum mechanics. Understanding how quantum systems transition to a classical behavior is a long-standing challenge in mesoscopic physics, with important implications for a wide range of applications, from quantum computing to condensed matter physics. The result provides a fresh perspective on the relations between quantum and classical effects in curved spacetime.

DOI: 10.1088/1751-8121/aceec7

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Abstract In general relativity, the dynamics of objects is governed by the curvature of spacetime, which is caused by the presence of matter and energy. In contrast, in quantum mechanics, the dynamics is governed by the wavefunction, which completely describes the behavior of the particles. There is an ongoing effort to explore analogs of space and spacetime curvature in the context of quantum mechanics. Such analogies may reveal a deeper structure of quantum reality and its possible relations with Einstein’s theory of gravity. In this note, by coupling the non-relativistic Schrödinger equation with the heat equation and using the hydrodynamical formulation of quantum mechanics, we find that the dynamics of the particle is fully characterized by the normalized curvature of the wavefunction’s amplitude. Such a curvature obtains an analogy to the Ricci curvature of curved space in a Riemannian manifold. The proposed geometric correspondence provides a new pathway to explore quantum dynamics through the lens of differential geometry, the language of general relativity.

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In a recent work \cite{Shaikh:2026nsi}, it was shown that for a non-relativistic particle, the operator equations of motion can be cast in the same differential form as Newton's equation of motion. This establishes a correspondence between quantum operator dynamics and classical mechanics at the level of the equations of motion. In this work, we investigate whether this correspondence extends to quantum field theory and linearized gravity. We explicitly show that the Heisenberg equations of motion for quantum field operators reproduce the corresponding classical equations of motion for a broad class of field theories. The Klein-Gordon equation, Maxwell's equations, Yang-Mills equations, and linearized Einstein equations are shown to arise at the operator level without taking the limit

\hbar\rightarrow0

. We then extend the analysis to curved spacetime and show that the quantum field operators satisfy the corresponding covariant field equations at the operator level. In the weak-field approximation, we show that the Heisenberg evolution of the metric-perturbations reproduces the linearized Einstein equations at the operator level. For a quantum particle in curved spacetime, the Heisenberg equations for position and momentum reproduce classical geodesic motion in the semiclassical limit, thereby extending the operator-level quantum-classical correspondence from fields to particle dynamics.