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

Chinese Journal of Physics

Stochastic Mechanics in Complex Space

M. S. Wang

Published Dec 1, 2000• 0 citations
OpenAlex

Abstract

We show that a backward stochastic system with an imaginary diffusion coefficient in complex space is equivalent to a real space quantum system if it obeys a Newtonian-like dynamical law and its final spatial distribution is constrained to be in real space. The ensemble averages of the position, momentum, angular momentum and energy of the stochastic processes are the quantum mechanical expectation values of the corresponding operators. The product of the mean-square deviations of the position and momentum of the stochastic process agreees with the corresponding uncertainty relation of quantum mechanics. The equation of motion is invariant under the combined transformations of changing the time t! ¡ t and taking the complex conjugate of the equation. The stochastic motion reduces to the classical motion in the limit where the de Broglie wavelengh of the particle vanishes. PACS. 03.65.Bz – Foundations, theory of measurement, miscellaneous theories. PACS. 02.50.Ga – Markov processes. PACS. 02.50.Cw – Probability theory.

Related papers

Quantum stochastic motion in complex space

M. S. Wang

We show that a quantum system may be associated with a backward stochastic process in complex configuration space when the so-called weak value of the position operator is interpreted as a conditional expectation value. The quantum-mechanical expectation values of the position, momentum, angular momentum, and energy are shown to be the weighted averages of the corresponding quantities for the stochastic process. Moreover, the stochastic trajectory is shown to reduce to the correct classical trajectory in the limit where the de Broglie wavelength vanishes.

Physical Review A•Mar 1, 1998•9 citations

Stochastic Interpretation of Quantum Mechanics in Complex Space

M. S. Wang

We show that a quantum system may be associated to a backward stochastic process in complex configuration space when the so-called weak value of the position operator is interpreted as a conditional expectation value. The ensemble averaged values of such a stochastic trajectory and the corresponding weak values of a quantum particle are shown to be equivalent. Moreover, the stochastic trajectory is shown to reduce to the correct classical trajectory in the limit where the de Broglie wavelength vanishes.

Physical Review Letters•Nov 3, 1997•15 citations

Quantum Mechanics as a Reversible Diffusion Theory

Charalampos Antonakos

This paper proposes an interpretation of quantum mechanics, relying on the time-symmetric stochastic dynamics of quantum particles and on non-classical probability theory. Our main purpose is to demonstrate that the wave function and its complex conjugate can be interpreted as complex probability distributions in two complex diffusion equations related to non-real forward and backward in time stochastic motions respectively. We say non-real because Schroedinger forward and backward diffusions describe both reversible (real trajectories) and irreversible trajectories (non-real trajectories). The reversible trajectories are the only real trajectories and are given by the intersection of those forward and backward processes. It turns out that if we translate this intersection using set-theoretic language, we are led to a reversible diffusion described by Born rule probabilities. This proposal is useful also for explaining more about the role of complex numbers in quantum mechanics that produces this so-called "wave-like" nature of quantum reality. Our perspective also challenges the notion of physical superposition and aims at a derivation of superposition principle not based on the linearity of Schroedinger's equation but relying on pure probability theory. Moreover, it is suggested that, embracing the idea of stochastic processes in quantum theory, explains the reasons for the appearance of classical behavior in large objects, in contrast to the quantum behavior of small ones. In other words, we claim that a combination of a probabilistic and no-ontic view (neither epistemic though) of the wave function with a stochastic hidden-variables approach, may provide some insight into the quantum physical reality and potentially establish the groundwork for a novel interpretation of quantum mechanics.

arXiv:2502.10523•Feb 14, 2025•0 citations

Real-space diffusion theory from quantum mechanics using analytic continuation

Leonardo Di G. Sigalotti, Otto Rendón, José-Rubén Luévano

We show that a physical correspondence between Brownian motion and quantum mechanics can be established by formal analytic continuation if Wick rotation ( t \→ i t ) is replaced by the mirror symmetric, complex conjugate time transformation t \→ ± i t . Invariance of the square modulus of the wave function under this transformation reveals a tight connection between Born's interpretation of the probability density and the proper time t 2 = \( − i t \) \( + i t \) , as being both the result of a time symmetry breaking in the informational content of the microscopic (quantum) world, which takes place as we move to a macroscopic level. We find that under the transformation t \→ ± i t , the Schrödinger equation and its complex conjugate conforms with a stochastic-mechanical model of the Brownian motion of a particle. From the present analysis, Nelson's osmotic velocity arises naturally and the quantum phase function admits a classical interpretation in terms of a continuous (scalar) potential field that influences the motion of the particle. The maximum phase variation defines the direction of the osmotic and current velocity. Whereas the paper focuses on the correlation between quantum mechanics and Ficks's second law for a constant diffusion coefficient, a discussion is also provided on the possibility of a quantum mechanical formulation for anomalous diffusion, where the diffusion coefficient is a function of space and/or time.

Heliyon•Oct 1, 2024•1 citation

Stochastic mechanics of reciprocal diffusions

Bernard C. Levy, Arthur J. Krener

The dynamics and kinematics of reciprocal diffusions were examined in a previous paper [J. Math. Phys. 34, 1846 (1993)], where it was shown that reciprocal diffusions admit a chain of conservation laws, which close after the first two laws for two disjoint subclasses of reciprocal diffusions, the Markov and quantum diffusions. For the case of quantum diffusions, the conservation laws are equivalent to Schrödinger’s equation. The Markov diffusions were employed by Schrödinger [Sitzungsber. Preuss. Akad. Wiss. Phys. Math Kl. 144 (1931); Ann. Inst. H. Poincaré 2, 269 (1932)], Nelson [Dynamical Theories of Brownian Motion (Princeton University, Princeton, NJ, 1967); Quantum Fluctuations (Princeton University, Princeton, NJ, 1985)], and other researchers to develop stochastic formulations of quantum mechanics, called stochastic mechanics. We propose here an alternative version of stochastic mechanics based on quantum diffusions. A procedure is presented for constructing the quantum diffusion associated to a given wave function. It is shown that quantum diffusions satisfy the uncertainty principle, and have a locality property, whereby given two dynamically uncoupled but statistically correlated particles, the marginal statistics of each particle depend only on the local fields to which the particle is subjected. However, like Wigner’s joint probability distribution for the position and momentum of a particle, the finite joint probability densities of quantum diffusions may take negative values.

Journal of Mathematical Physics•Feb 1, 1996•24 citations

Stochastic mechanics and Feynman path integrals

M. S. Wang

A relation between the kernel of a quantum system in Feynman's path-integral theory and the transition probability of the corresponding Markov process, with the diffusion coefficient rotated to the imaginary axis, is obtained in the framework of stochastic mechanics. If a stationary state is considered, the relation reduces to a form relating a Euclidean (i.e., imaginary time) quantum theory to a Markov process.

Physical Review A•Feb 1, 1988•6 citations
We show that a backward stochastic system with an imaginary diffusion coefficient in complex space is equivalent to a real space quantum system if it obeys a Newtonian-like dynamical law and its final spatial distribution is constrained to be in real space. The ensemble averages of the position, momentum, angular momentum and energy of the stochastic processes are the quantum mechanical expectation values of the corresponding operators. The product of the mean-square deviations of the position and momentum of the stochastic process agreees with the corresponding uncertainty relation of quantum mechanics. The equation of motion is invariant under the combined transformations of changing the time t! ¡ t and taking the complex conjugate of the equation. The stochastic motion reduces to the classical motion in the limit where the de Broglie wavelengh of the particle vanishes. PACS. 03.65.Bz – Foundations, theory of measurement, miscellaneous theories. PACS. 02.50.Ga – Markov processes. PACS. 02.50.Cw – Probability theory.
We show that a quantum system may be associated with a backward stochastic process in complex configuration space when the so-called weak value of the position operator is interpreted as a conditional expectation value. The quantum-mechanical expectation values of the position, momentum, angular momentum, and energy are shown to be the weighted averages of the corresponding quantities for the stochastic process. Moreover, the stochastic trajectory is shown to reduce to the correct classical trajectory in the limit where the de Broglie wavelength vanishes.
We show that a quantum system may be associated to a backward stochastic process in complex configuration space when the so-called weak value of the position operator is interpreted as a conditional expectation value. The ensemble averaged values of such a stochastic trajectory and the corresponding weak values of a quantum particle are shown to be equivalent. Moreover, the stochastic trajectory is shown to reduce to the correct classical trajectory in the limit where the de Broglie wavelength vanishes.
This paper proposes an interpretation of quantum mechanics, relying on the time-symmetric stochastic dynamics of quantum particles and on non-classical probability theory. Our main purpose is to demonstrate that the wave function and its complex conjugate can be interpreted as complex probability distributions in two complex diffusion equations related to non-real forward and backward in time stochastic motions respectively. We say non-real because Schroedinger forward and backward diffusions describe both reversible (real trajectories) and irreversible trajectories (non-real trajectories). The reversible trajectories are the only real trajectories and are given by the intersection of those forward and backward processes. It turns out that if we translate this intersection using set-theoretic language, we are led to a reversible diffusion described by Born rule probabilities. This proposal is useful also for explaining more about the role of complex numbers in quantum mechanics that produces this so-called "wave-like" nature of quantum reality. Our perspective also challenges the notion of physical superposition and aims at a derivation of superposition principle not based on the linearity of Schroedinger's equation but relying on pure probability theory. Moreover, it is suggested that, embracing the idea of stochastic processes in quantum theory, explains the reasons for the appearance of classical behavior in large objects, in contrast to the quantum behavior of small ones. In other words, we claim that a combination of a probabilistic and no-ontic view (neither epistemic though) of the wave function with a stochastic hidden-variables approach, may provide some insight into the quantum physical reality and potentially establish the groundwork for a novel interpretation of quantum mechanics.
We show that a physical correspondence between Brownian motion and quantum mechanics can be established by formal analytic continuation if Wick rotation ( t \→ i t ) is replaced by the mirror symmetric, complex conjugate time transformation t \→ ± i t . Invariance of the square modulus of the wave function under this transformation reveals a tight connection between Born's interpretation of the probability density and the proper time t 2 = −it − i t −it +it + i t +it , as being both the result of a time symmetry breaking in the informational content of the microscopic (quantum) world, which takes place as we move to a macroscopic level. We find that under the transformation t \→ ± i t , the Schrödinger equation and its complex conjugate conforms with a stochastic-mechanical model of the Brownian motion of a particle. From the present analysis, Nelson's osmotic velocity arises naturally and the quantum phase function admits a classical interpretation in terms of a continuous (scalar) potential field that influences the motion of the particle. The maximum phase variation defines the direction of the osmotic and current velocity. Whereas the paper focuses on the correlation between quantum mechanics and Ficks's second law for a constant diffusion coefficient, a discussion is also provided on the possibility of a quantum mechanical formulation for anomalous diffusion, where the diffusion coefficient is a function of space and/or time.
The dynamics and kinematics of reciprocal diffusions were examined in a previous paper [J. Math. Phys. 34, 1846 (1993)], where it was shown that reciprocal diffusions admit a chain of conservation laws, which close after the first two laws for two disjoint subclasses of reciprocal diffusions, the Markov and quantum diffusions. For the case of quantum diffusions, the conservation laws are equivalent to Schrödinger’s equation. The Markov diffusions were employed by Schrödinger [Sitzungsber. Preuss. Akad. Wiss. Phys. Math Kl. 144 (1931); Ann. Inst. H. Poincaré 2, 269 (1932)], Nelson [Dynamical Theories of Brownian Motion (Princeton University, Princeton, NJ, 1967); Quantum Fluctuations (Princeton University, Princeton, NJ, 1985)], and other researchers to develop stochastic formulations of quantum mechanics, called stochastic mechanics. We propose here an alternative version of stochastic mechanics based on quantum diffusions. A procedure is presented for constructing the quantum diffusion associated to a given wave function. It is shown that quantum diffusions satisfy the uncertainty principle, and have a locality property, whereby given two dynamically uncoupled but statistically correlated particles, the marginal statistics of each particle depend only on the local fields to which the particle is subjected. However, like Wigner’s joint probability distribution for the position and momentum of a particle, the finite joint probability densities of quantum diffusions may take negative values.