Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Geometric derivation of the Schrödinger equation from classical mechanics in curved Weyl spaces

Enrico Santamato

Published Jan 15, 1984• 81 citations
Publisher

Abstract

A theory physically equivalent to traditional nonrelativistic quantum mechanics is presented, in which both dynamical and probabilistic concepts enter in a classical way. Particle trajectories are deterministically governed by classical mechanics, only the initial position being at random. Quantum effects are supposed to arise from a modification of the geometry of space, due to the presence of matter. However, unlike gravitational forces, which are related to the metric of space-time, quantum-mechanical forces are proved to be related to the transplantation law of vectors. The resulting geometry of space, in the nonrelativistic limit, is found to be Weyl's geometry. Both particle motion and geometry of space are obtained from a unique averaged least-action principle.

DOI
10.1103/physrevd.29.216

Related papers

Gauge-invariant statistical mechanics and average action principle for the Klein-Gordon particle in geometric quantum mechanics

Enrico Santamato

A gauge-invariant average action principle is presented that permits the formulation of a completely classical theory which is proved to be equivalent to Klein-Gordon quantum mechanics. In this approach, here called geometric quantum mechanics, the particle motion as well as the space-time geometry are determined simultaneously from the same average action principle. Quantum effects are proved to be related to space-time affine connections rather than to space-time metric tensor components. In this way geometric quantum mechanics is made compatible with axiomatic approaches to both space-time structure and probability calculus.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Nov 15, 1985•32 citations

Contemporary development of Einstein's ideas on space-time and Brownian motion

Yuri A. Rylov

Development of the contemporary theory of physical phenomena in the microcosm is considered to be a result of development of Einstein's ideas on a possibility of the event space modification and on a possibility of stochastic (Brownian) motion of free particles. One considers the space-time modification, based on a new conception of geometry. In the framework of this conception any geometry is obtained as a result of the proper Euclidean geometry deformation. In the framework of this conception it is possible such a space-time geometry, where the free particle motion appears to be stochastic, although the geometry in itself remains to be deterministic. The stochasticity intensity depends on the particle mass. It is small for particle of a large mass, and it is essential for the particle of a small mass. The space-time geometry may be chosen in such a way, that the statistical description of random world lines of particles be equivalent to the quantum description (the Schrödinger equation). At such a description the quantum principles are not used. They can be obtained as a corollaries of the statistical description of nonrelativistic stochastic particles. The nonrelativistic quantum principles may not be used for description of relativistic phenomena.

arXiv:physics/0512069•Dec 8, 2005•0 citations

Nonlinear corrections to the Schrödinger equation from geometric quantum mechanics

Carlos Castro

The nonlinear corrections to the Schrödinger equation based on the results of geometric quantum mechanics (GQM) are derived. Such theory derives quantum mechanics from the underlying Weyl space-time geometry associated with the classical ensemble of particle paths given by the solutions of the Hamilton–Jacobi equation.

Journal of Mathematical Physics•Nov 1, 1990•15 citations

Geometric analogy between quantum dynamics and curved space through quantum hydrodynamics

Tomer Shushi

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.

Classical and Quantum Gravity•Aug 2, 2023•0 citations

The classical Schrödinger equation

K. R. W. Jones

Using a simple geometrical construction based upon the linear action of the Heisenberg--Weyl group we deduce a new nonlinear Schrödinger equation that provides an exact dynamic and energetic model of any classical system whatsoever, be it integrable, nonintegrable or chaotic. Within our model classical phase space points are represented by equivalence classes of wavefunctions that have identical position and momentum expectation values. Transport of these equivalence classes without dispersion leads to a system of wavefunction dynamics such that the expectation values track classical trajectories {\em precisely\/} for arbitrarily long times. Interestingly, the value of $\hbar$ proves immaterial for the purpose of constructing this alternative representation of classical point mechanics. The new feature which $\hbar$ does mediate concerns a simple embedding of the quantum geometric phase within classical mechanics. We discuss problems of physical interpretation and discover a simple route to recover the ordinary linear Schrödinger equation.

arXiv:1212.6786•Dec 30, 2012•1 citation

Geometrization of the Schrödinger equation: Application of the Maupertuis Principle to quantum mechanics

Antonia Karamatskou, Hagen M. Kleinert

In its geometric form, the Maupertuis Principle states that the movement of a classical particle in an external potential V(x) can be understood as a free movement in a curved space with the metric gμν(x) = 2M[V(x) - E]δμν. We extend this principle to the quantum regime by showing that the wavefunction of the particle is governed by a Schrödinger equation of a free particle moving through curved space. The kinetic operator is the Weyl-invariant Laplace–Beltrami operator. On the basis of this observation, we calculate the semiclassical expansion of the particle density.

International Journal of Geometric Methods in Modern Physics•May 23, 2014•8 citations
A theory physically equivalent to traditional nonrelativistic quantum mechanics is presented, in which both dynamical and probabilistic concepts enter in a classical way. Particle trajectories are deterministically governed by classical mechanics, only the initial position being at random. Quantum effects are supposed to arise from a modification of the geometry of space, due to the presence of matter. However, unlike gravitational forces, which are related to the metric of space-time, quantum-mechanical forces are proved to be related to the transplantation law of vectors. The resulting geometry of space, in the nonrelativistic limit, is found to be Weyl's geometry. Both particle motion and geometry of space are obtained from a unique averaged least-action principle.
A gauge-invariant average action principle is presented that permits the formulation of a completely classical theory which is proved to be equivalent to Klein-Gordon quantum mechanics. In this approach, here called geometric quantum mechanics, the particle motion as well as the space-time geometry are determined simultaneously from the same average action principle. Quantum effects are proved to be related to space-time affine connections rather than to space-time metric tensor components. In this way geometric quantum mechanics is made compatible with axiomatic approaches to both space-time structure and probability calculus.
Development of the contemporary theory of physical phenomena in the microcosm is considered to be a result of development of Einstein's ideas on a possibility of the event space modification and on a possibility of stochastic (Brownian) motion of free particles. One considers the space-time modification, based on a new conception of geometry. In the framework of this conception any geometry is obtained as a result of the proper Euclidean geometry deformation. In the framework of this conception it is possible such a space-time geometry, where the free particle motion appears to be stochastic, although the geometry in itself remains to be deterministic. The stochasticity intensity depends on the particle mass. It is small for particle of a large mass, and it is essential for the particle of a small mass. The space-time geometry may be chosen in such a way, that the statistical description of random world lines of particles be equivalent to the quantum description (the Schrödinger equation). At such a description the quantum principles are not used. They can be obtained as a corollaries of the statistical description of nonrelativistic stochastic particles. The nonrelativistic quantum principles may not be used for description of relativistic phenomena.
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.
Using a simple geometrical construction based upon the linear action of the Heisenberg--Weyl group we deduce a new nonlinear Schrödinger equation that provides an exact dynamic and energetic model of any classical system whatsoever, be it integrable, nonintegrable or chaotic. Within our model classical phase space points are represented by equivalence classes of wavefunctions that have identical position and momentum expectation values. Transport of these equivalence classes without dispersion leads to a system of wavefunction dynamics such that the expectation values track classical trajectories {\em precisely\/} for arbitrarily long times. Interestingly, the value of ℏ\hbarℏ proves immaterial for the purpose of constructing this alternative representation of classical point mechanics. The new feature which ℏ\hbarℏ does mediate concerns a simple embedding of the quantum geometric phase within classical mechanics. We discuss problems of physical interpretation and discover a simple route to recover the ordinary linear Schrödinger equation.
In its geometric form, the Maupertuis Principle states that the movement of a classical particle in an external potential V(x) can be understood as a free movement in a curved space with the metric gμν(x) = 2M[V(x) - E]δμν. We extend this principle to the quantum regime by showing that the wavefunction of the particle is governed by a Schrödinger equation of a free particle moving through curved space. The kinetic operator is the Weyl-invariant Laplace–Beltrami operator. On the basis of this observation, we calculate the semiclassical expansion of the particle density.