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 A

Gaussian quantum fields and stochastic electrodynamics

Lawrence J. Landau

Published Jun 1, 1988• 4 citations
Publisher

Abstract

The relation between quantum electrodynamics and (classical) stochastic electrodynamics is elucidated by means of a general construction which associates with every Gaussian quantum field (for example, vacuum-free fields or coherent states at zero or nonzero temperature) a classical random field, which in the case of quantum electrodynamics yields stochastic electrodynamics. The associated classical and quantum theories yield the same results for all nondisturbing (causally separated) field measurements. This is an additional classical characteristic of coherent quantum states which implies that Bell's inequality is satisfied by quantum field measurements. Furthermore, stochastic electrodynamics is identified as the Wigner distribution of quantum electrodynamics. We point out that in these Gaussian states stochastic electrodynamics is more suitable as a classical model of the quantum field than is the Euclidean field, which results from applying Nelson's stochastic mechanics to each field mode.

DOI
10.1103/physreva.37.4449

Related papers

Classical analogs of quasifree quantum stochastic processes given by stochastic states of the quantized electromagnetic field

C. Hertfelder, Burkhard Kümmerer

The mathematical model describing a light beam prepared in an arbitrary quantum optical state is a quasifree quantum stochastic process on the C* algebra of the canonical commutatation relations. For such quantum stochastic processes the concept of stochastic states is introduced. Stochastic quantum states have a classical analog in the following sense: If the light beam is prepared in a stochastic state, one can construct a generalized classical stochastic process, such that the distributions of the quantum observables and the classical random variables coincide. A sufficient algebraic condition for the stochasticity of a quantum state is formulated. The introduced formalism generalizes the Wigner representation from a single field mode to a continuum of modes. For the special case of a single field mode the stochasticity condition provides a new criterion for the positivity of the Wigner function related to the given state. As an example the quantized eletromagnetic field in empty space at temperature T=0 is discussed. It turns out that the corresponding classical stochastic process is not a white noise but a colored noise with a linearly increasing spectrum.

Journal of Mathematical Physics•Mar 1, 2001•0 citations

Stochastic electrodynamics for the free particle

Luis de la Peña, A. Jáuregui

The theory of stochastic electrodynamics is applied to the free particle and to the particle moving in a homogeneous field, leading to a complete temperature- and time-dependent description in phase space. After a transient time, the marginal description in configuration space coincides entirely with quantum mechanics, while the phase-space description is only mathematically related to the Wigner distribution. The Schrödinger equation appears as a natural—though incomplete—means of describing the statistical behavior of the electron under these conditions.

Journal of Mathematical Physics•Dec 1, 1983•9 citations

The Quantum Electromagnetic Field in the Weyl–Wigner Representation

Emilio Santos

The quantum electromagnetic (EM) field is formulated in the Weyl–Wigner representation (WW), which is equivalent to the standard Hilbert space one (HS). In principle, it is possible to interpret within WW all experiments involving the EM field interacting with macroscopic bodies, the latter treated classically. In the WW formalism, the essential difference between classical electrodynamics and the quantum theory of the EM field is just the assumption that there is a random EM field-filling space, i.e., the existence of a zero-point field with a Gaussian distribution for the field amplitudes. I analyze a typical optical test of a Bell inequality. The model admits an interpretation compatible with local realism, modulo a number of assumptions assumed plausible.

arXiv:2412.13218•Universe•Dec 9, 2024•5 citations

Quantum Entanglement Derived from the Coulomb Field: A Stochastic Electrodynamic Description of Multi-Particle Systems

Fusao Ishii

This paper extends the stochastic electrodynamic framework of Paper 1 [3] to multiparticle systems, providing a physical derivation of quantum entanglement. When a highenergy gamma ray creates an electron-positron pair, both particles emerge from the same spacetime point and share a common electromagnetic quantum field. We propose that the electromagnetic vacuum retains a cross-correlation component S(12)E (x1, x2, ω) in their joint electromagnetic quantum field, arising from the common creation event. The spatial dependence of this cross-correlation—a sinc(ωr12/c) factor—is derived from the known two-point vacuum correlation function of the electromagnetic field, not postulated. This cross-correlation is identified as the physical carrier of quantum entanglement. The two particles undergo correlated Brownian motion in six-dimensional configuration space, each with diffusion coefficient D = ℏ/2m derived in [3]. Nelson’s stochastic mechanics in configuration space gives the two-particle Schrödinger equation. The foundational ergodicity assumption underlying Papers 1 and 2—that the coupled electron-vacuum system admits a unique stationary ergodic measure—is proved in the companion Paper 3 [5] via the Caldeira–Leggett model and the Ford–Kac–Mazur theorem. The EPR paradox is resolved: the instantaneous correlation upon measurement is a consequence of conditional probability applied to a joint stochastic process, not a physical signal. Bell inequality violations are derived explicitly: the cross-correlation is nonlocal in configuration space, bypassing Bell’s locality assumption while satisfying the no-signalling theorem. Decoherence arises naturally as disruption of S(12)E by environmental electromagnetic noise. Fermi and Bose statistics emerge as boundary conditions on the joint stochastic process in configuration space.

•Jun 16, 2026•0 citations

General connection between random electrodynamics and quantum electrodynamics for free electromagnetic fields and for dipole oscillator systems

Timothy H. Boyer

A general comparison is presented between random electrodynamics and quantum electrodynamics for the two systems which can be solved exactly in both theories, free electromagnetic fields and point dipole oscillators. The $N$-point correlation functions of the fields are computed in both theories and are found to differ in general because of the dependence upon the order of the quantum operators within products of operators. However, if all products of quantum operators are symmetrized by taking all permutations of the operator order, then the two theories give identical results for the correlation functions. Analogous results hold to all orders in the fine-structure constant for dipole oscllators in quantum and random electrodynamics. The theories agree only if the quantum operator products are symmetrized. In the limit that the oscillator couplings to the radiation fields vanish, th oscllators can be regarded as mechanical oscillators in quantum mechanics and in random mechanics. The theory of random mechanics is defined in terms of this limit which uncouples a mechanical oscillator from the radiation field. The average values of oscillator variables in random mechanics agree with those of symmetrized products in quantum mechanics. The question is then raised as to the physical significance of the many quantum operators which differ only in the order of their factors. It is pointed out that some operator products which are regarded as physically important, such as the square of the angular momentum, indeed involve unsymmetrized products of operators. On this account the average values of the angular momentum squared in the ground state of an isotropic three-dimensional harmonic oscillator differ between the random-mechanical and quantum-mechanical descriptions. However, there seems to be no case in which experiments have shown that the (unsymmetrized) quantum operator value is to be preferred to that provided by random mechanics. The presence of thermal radiation is next treated for free electromagnetic fields and for dipole-oscillator systems. Despite extraordinary differences in the points of view toward thermal radiation taken by the two theories, the conclusion is the same as that found for zero temperature; the two theories agree in their average values if all products of quantum operators are symmetrized. Finally, as a further example of the power of random electrodynamics to give an account of phenomena where Lorentz's classical electron theory failed, we investigate the diamagnetism of a charged three-dimensional isotropic oscillator. The mathematical descriptions at finite temperature are developed in full random electrodynamics and quantum electrodynamics and in second-order perturbation theory in quantum mechanics.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Feb 15, 1975•115 citations

Stochastic Quantization of the Maxwell Field: Some Comments on the Paper of Guerra and Loffredo.

S. C. Lb

Recently GUER~A and LOFF~]~D0 (:) have constructed a stochastice elctromagnetic field by applying the stochastic quantization method of Nelson (2) to the classical electromagnetic field. According to their results the (( space part ~) of the Euclidean electromagnetic field F~j (with i, ] = 1, 2, 3) can be regarded as the ground-state stochastic electromagnetic field defined in the real physical time, in complete analogy with the case of the scalar massive boson field (3). In this note we shall comment on a basic assumption contained in the paper of Guerra and Loffredo. The main feature in their construction is that the magnetic field B(~, t) is assumed to be promoted to a random ~iarkov process by the stochastic quantization procedure. The electric field E(~, t) is then split into a couple of functions /~( ~, t) expressible in terms of B(~, t). By considering the classical Maxwell field in terms of normal canonical co-ordinates and employing the stochastic differential equation of Ito type, they obtained the ground-state stochastic magnetic field with zero mean and covarianee given by

•Jan 1, 1981•0 citations
The relation between quantum electrodynamics and (classical) stochastic electrodynamics is elucidated by means of a general construction which associates with every Gaussian quantum field (for example, vacuum-free fields or coherent states at zero or nonzero temperature) a classical random field, which in the case of quantum electrodynamics yields stochastic electrodynamics. The associated classical and quantum theories yield the same results for all nondisturbing (causally separated) field measurements. This is an additional classical characteristic of coherent quantum states which implies that Bell's inequality is satisfied by quantum field measurements. Furthermore, stochastic electrodynamics is identified as the Wigner distribution of quantum electrodynamics. We point out that in these Gaussian states stochastic electrodynamics is more suitable as a classical model of the quantum field than is the Euclidean field, which results from applying Nelson's stochastic mechanics to each field mode.
The mathematical model describing a light beam prepared in an arbitrary quantum optical state is a quasifree quantum stochastic process on the C* algebra of the canonical commutatation relations. For such quantum stochastic processes the concept of stochastic states is introduced. Stochastic quantum states have a classical analog in the following sense: If the light beam is prepared in a stochastic state, one can construct a generalized classical stochastic process, such that the distributions of the quantum observables and the classical random variables coincide. A sufficient algebraic condition for the stochasticity of a quantum state is formulated. The introduced formalism generalizes the Wigner representation from a single field mode to a continuum of modes. For the special case of a single field mode the stochasticity condition provides a new criterion for the positivity of the Wigner function related to the given state. As an example the quantized eletromagnetic field in empty space at temperature T=0 is discussed. It turns out that the corresponding classical stochastic process is not a white noise but a colored noise with a linearly increasing spectrum.
The theory of stochastic electrodynamics is applied to the free particle and to the particle moving in a homogeneous field, leading to a complete temperature- and time-dependent description in phase space. After a transient time, the marginal description in configuration space coincides entirely with quantum mechanics, while the phase-space description is only mathematically related to the Wigner distribution. The Schrödinger equation appears as a natural—though incomplete—means of describing the statistical behavior of the electron under these conditions.
The quantum electromagnetic (EM) field is formulated in the Weyl–Wigner representation (WW), which is equivalent to the standard Hilbert space one (HS). In principle, it is possible to interpret within WW all experiments involving the EM field interacting with macroscopic bodies, the latter treated classically. In the WW formalism, the essential difference between classical electrodynamics and the quantum theory of the EM field is just the assumption that there is a random EM field-filling space, i.e., the existence of a zero-point field with a Gaussian distribution for the field amplitudes. I analyze a typical optical test of a Bell inequality. The model admits an interpretation compatible with local realism, modulo a number of assumptions assumed plausible.
This paper extends the stochastic electrodynamic framework of Paper 1 [3] to multiparticle systems, providing a physical derivation of quantum entanglement. When a highenergy gamma ray creates an electron-positron pair, both particles emerge from the same spacetime point and share a common electromagnetic quantum field. We propose that the electromagnetic vacuum retains a cross-correlation component S(12)E (x1, x2, ω) in their joint electromagnetic quantum field, arising from the common creation event. The spatial dependence of this cross-correlation—a sinc(ωr12/c) factor—is derived from the known two-point vacuum correlation function of the electromagnetic field, not postulated. This cross-correlation is identified as the physical carrier of quantum entanglement. The two particles undergo correlated Brownian motion in six-dimensional configuration space, each with diffusion coefficient D = ℏ/2m derived in [3]. Nelson’s stochastic mechanics in configuration space gives the two-particle Schrödinger equation. The foundational ergodicity assumption underlying Papers 1 and 2—that the coupled electron-vacuum system admits a unique stationary ergodic measure—is proved in the companion Paper 3 [5] via the Caldeira–Leggett model and the Ford–Kac–Mazur theorem. The EPR paradox is resolved: the instantaneous correlation upon measurement is a consequence of conditional probability applied to a joint stochastic process, not a physical signal. Bell inequality violations are derived explicitly: the cross-correlation is nonlocal in configuration space, bypassing Bell’s locality assumption while satisfying the no-signalling theorem. Decoherence arises naturally as disruption of S(12)E by environmental electromagnetic noise. Fermi and Bose statistics emerge as boundary conditions on the joint stochastic process in configuration space.
A general comparison is presented between random electrodynamics and quantum electrodynamics for the two systems which can be solved exactly in both theories, free electromagnetic fields and point dipole oscillators. The NNN-point correlation functions of the fields are computed in both theories and are found to differ in general because of the dependence upon the order of the quantum operators within products of operators. However, if all products of quantum operators are symmetrized by taking all permutations of the operator order, then the two theories give identical results for the correlation functions. Analogous results hold to all orders in the fine-structure constant for dipole oscllators in quantum and random electrodynamics. The theories agree only if the quantum operator products are symmetrized. In the limit that the oscillator couplings to the radiation fields vanish, th oscllators can be regarded as mechanical oscillators in quantum mechanics and in random mechanics. The theory of random mechanics is defined in terms of this limit which uncouples a mechanical oscillator from the radiation field. The average values of oscillator variables in random mechanics agree with those of symmetrized products in quantum mechanics. The question is then raised as to the physical significance of the many quantum operators which differ only in the order of their factors. It is pointed out that some operator products which are regarded as physically important, such as the square of the angular momentum, indeed involve unsymmetrized products of operators. On this account the average values of the angular momentum squared in the ground state of an isotropic three-dimensional harmonic oscillator differ between the random-mechanical and quantum-mechanical descriptions. However, there seems to be no case in which experiments have shown that the (unsymmetrized) quantum operator value is to be preferred to that provided by random mechanics. The presence of thermal radiation is next treated for free electromagnetic fields and for dipole-oscillator systems. Despite extraordinary differences in the points of view toward thermal radiation taken by the two theories, the conclusion is the same as that found for zero temperature; the two theories agree in their average values if all products of quantum operators are symmetrized. Finally, as a further example of the power of random electrodynamics to give an account of phenomena where Lorentz's classical electron theory failed, we investigate the diamagnetism of a charged three-dimensional isotropic oscillator. The mathematical descriptions at finite temperature are developed in full random electrodynamics and quantum electrodynamics and in second-order perturbation theory in quantum mechanics.
Recently GUER~A and LOFF~]~D0 (:) have constructed a stochastice elctromagnetic field by applying the stochastic quantization method of Nelson (2) to the classical electromagnetic field. According to their results the (( space part ~) of the Euclidean electromagnetic field F~j (with i, ] = 1, 2, 3) can be regarded as the ground-state stochastic electromagnetic field defined in the real physical time, in complete analogy with the case of the scalar massive boson field (3). In this note we shall comment on a basic assumption contained in the paper of Guerra and Loffredo. The main feature in their construction is that the magnetic field B(~, t) is assumed to be promoted to a random ~iarkov process by the stochastic quantization procedure. The electric field E(~, t) is then split into a couple of functions /~( ~, t) expressible in terms of B(~, t). By considering the classical Maxwell field in terms of normal canonical co-ordinates and employing the stochastic differential equation of Ito type, they obtained the ground-state stochastic magnetic field with zero mean and covarianee given by