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

arXiv:2112.13996 · Physical review. D/Physical review. D.

Relativistic quantum field theory of stochastic dynamics in the Hilbert space

Pei Wang

Published Jun 27, 2022• 7 citations
arXivHTMLPDF

Abstract

In this paper, we develop an action formulation of stochastic dynamics in the Hilbert space. In this formulation, the quantum theory of random unitary evolution is easily reconciled with special relativity. We generalize the Wiener process into $1+3$-dimensional spacetime, and then define a scalar random field which keeps invariant under Lorentz transformations. By adding to the action of quantum field theory a coupling term between random and quantum fields, we obtain a random-number action which has the statistical spacetime translation and Lorentz symmetries. The canonical quantization of the theory results in a Lorentz-invariant equation of motion for the state vector or density matrix. We derive the path integral formula of $S$ matrix, based on which we develop a diagrammatic technique for doing the calculation. We find the diagrammatic rules for both the stochastic free field theory and stochastic ${\ensuremath{\phi}}^{4}$ theory. The Lorentz invariance of the random $S$ matrix is strictly proved by using the diagrammatic technique. We then develop a diagrammatic technique for calculating the density matrix of final quantum states after scattering. In the absence of interaction, we obtain the exact expressions of both $S$ matrix and density matrix. In the presence of interaction, we prove a simple relation between the density matrices of stochastic and conventional ${\ensuremath{\phi}}^{4}$ theory. Our formalism leads to an ultraviolet divergence that has a similar origin as that in quantum field theory. The divergence can be canceled by renormalizing the coupling strength to random field. We prove that the stochastic quantum field theory is renormalizable even in the presence of interaction. In the models with a linear coupling between random and quantum fields, the random field excites particles out of the vacuum, driving the Universe towards an infinite-temperature state. The number of excited particles follows the Poisson distribution. The collision between particles is not affected by the random field. But the signals of colliding particles are gradually covered by the background excitations caused by random field.

DOI
10.1103/physrevd.105.115037

Related papers

Relativistic model of spontaneous wave-function localization induced by nonHermitian colored noise

Pei Wang

Abstract We develop a quantum field theory based on random nonHermitian actions, which upon quantization lead to stochastic nonlinear Schrödinger dynamics for the state vector. In this framework, Lorentz and spacetime translation symmetries are preserved only in a statistical sense: the probability distribution of the action remains invariant under these transformations. As a result, the theory describes ensembles of quantum-state trajectories whose probability distributions remain invariant under changes of reference frame. As a concrete example, we augment the Dirac action with a purely imaginary term coupling the fermion density operator to a universal colored noise. This noise is constructed by solving the d’Alembert equation with white noise as its source, using a generalized stochastic calculus in 1+3 dimensions. We demonstrate that the colored noise drives stochastic localization of wave packets and derive the localization length analytically. Remarkably, the localization length decreases as the size of the observable universe increases. Our model thus provides a potential framework for relativistic spontaneous wave-function collapse. While establishing consistency with Born’s law remains an open challenge, the present work constitutes a step toward embedding collapse models into a Lorentz-invariant quantum field theory.

arXiv:2501.07050•Physica Scripta•May 5, 2026•1 citation

Schrödinger-picture formulation of a scalar quantum field driven by white noise

Pei Wang

Abstract We develop a Schrödinger-picture formulation for a scalar quantum field driven by a Lorentz-invariant white-noise field. The quantum state of the system is described by a stochastic wave functional that evolves according to a stochastic Schrödinger equation. We show that the Gaussian structure of the wave functional is preserved under the stochastic evolution, allowing the dynamics to be reduced to a set of equations for the corresponding kernel functions. These kernel equations are derived and solved exactly, yielding an explicit time-dependent expression for the wave functional. The exact solution enables a direct analysis of the statistical properties of the quantum field in the space of field configurations. In particular, we show that the expectation value of the field operator obeys the same stochastic equation as the classical field obtained from the Euler–Lagrange equation of the action. We further compute the energy density from the stochastic wave functional and evaluate its ensemble average over noise realizations. The resulting energy production rate coincides with that obtained from the corresponding Lindblad equation. This result indicates that the stochastic quantum state remains well defined even though certain derived observables exhibit ultraviolet divergences associated with the white-noise idealization.

Journal of Physics A Mathematical and Theoretical•Aug 6, 2026•0 citations

Schrödinger-picture formulation of a scalar quantum field driven by white noise

Pei Wang

We develop a Schrödinger-picture formulation for a scalar quantum field driven by a Lorentz-invariant white-noise field. The quantum state of the system is described by a stochastic wave functional that evolves according to a stochastic Schrödinger equation. We show that the Gaussian structure of the wave functional is preserved under the stochastic evolution, allowing the dynamics to be reduced to a set of equations for the corresponding kernel functions. These kernel equations are derived and solved exactly, yielding an explicit time-dependent expression for the wave functional. The exact solution enables a direct analysis of the statistical properties of the quantum field in the space of field configurations. In particular, we show that the expectation value of the field operator obeys the same stochastic equation as the classical field obtained from the Euler-Lagrange equation of the action. We further compute the energy density from the stochastic wave functional and evaluate its ensemble average over noise realizations. The resulting energy production rate coincides with that obtained from the corresponding Lindblad equation. This result indicates that the stochastic quantum state remains well defined even though certain derived observables exhibit ultraviolet divergences associated with the white-noise idealization.

quant-pharXiv:2603.15704•Mar 17, 2026•0 citations

Random non-Hermitian action theory for stochastic quantum dynamics: from canonical to path integral quantization

Pei Wang

Abstract We develop a theory of random non-Hermitian action that, after quantization, describes the stochastic nonlinear dynamics of quantum states in Hilbert space. Focusing on fermionic fields, we propose both canonical quantization and path integral quantization, demonstrating that these two approaches are equivalent. Using this formalism, we investigate the evolution of a single-particle Gaussian wave packet under the influence of non-Hermiticity and randomness. Our results show that specific types of non-Hermiticity lead to wave packet localization, while randomness affects the central position of the wave packet, causing the variance of its distribution to increase with the strength of the randomness.

arXiv:2410.10164•Journal of Physics A Mathematical and Theoretical•Mar 19, 2025•3 citations

Quantum Free Particle, Stochastic Action and Its Link to Special Relativity

Francesco R. Ruggeri

In (1), the notion of a stochastic classical action is introduced and a probability is linked to exp( constant * action). The objective of (1) is not to derive quantum mechanics, but to derive stochastic dynamics based on maximum entropy in action space. We have argued in a number of previous notes (2) that one may use the classical action to create a quantum free particle probability. This classical action, however, is not chosen a priori, but rather arises due to other considerations. We suggest that stochasticity is present in Newtonian mechanics at some time scale because dt is finite in dp/dt= Force (Newton’s second law) and so postulate that force delivery at some x,t is probabilistic within some dt and dx. A priori, dt and dx are not known, but if there is stochasticity, there should be a free particle probability, i.e. a probability(x,t) where x,t represent an x, t interaction point. Furthermore, the notion of dx and dt should arise naturally. We note that the form exp(i function(x,t,p,E) may be introduced because free particles have no distributed variable (like energy in the Maxwell-Boltzmann case). exp(i function(x,t)) immediately introduces periodicity, but we have argued that stochasticity can only exist within tiny, fixed dx, dt linked to dp/dt=F. In particular, one classical Newtonian scales this stochasticity is not seen. The periodicity should be directly in x and t space, suggesting exp( i A1 t + i A2 x). This expression, however, should be Lorentz invariant, and A1 and A2 should be constants, at least for a fixed mo and v. The reason that the one must have periodicity is that the dx and dt ranges of stochasticity in dp/dt=Fr must appear naturally. At this point, one is led to exp(-iEt+ipx) with -Et+px being a Lorentz invariant which satisfies the above criteria. This Lorentz invariant happens to be the classical action in both the relativistic and nonrelativistic cases, hence giving rise to action as being a stochastic variable (of (1)) because it is a function of the two scholastic variables x and t. Alternatively, one may choose the exp(iA1x+iA2t) due to its conservation properties. One may then argue that A1 and A2 could also be conserved for x,t held constant, i.e. A1 and A2 could be linked to Newtonian energy and momentum in elastic collisions. This allows one to solve for the special relativistic g(v) =1/sqrt(1-vv/cc). Given the idea of a conserved quantity form for a probability, one may argue that exp(i Action) satisfies a maximization of Shannon’s entropy subject to a conserved constraint (which is again a link with (1).) The question then becomes: Why should the Lorentz invariant A= -Et+px be the classical action.One may note that x,t usually refer to trajectory values and so for a free particle, one has x=vt. This means that one may factor out t, i.e. A= t( -E+pv), but this is automatically the form of the Lagrangian with E being the Hamiltonian. This holds relativistically and non-relativistically with the form: dL/dv partial = p. The remaining question is: Why should the classical action be Lorentz invariant? We consider E(v)-p(v) v = -L such that dE/dv - v dp/dv -p = -dL/dv partial. This means that dE/dv - v dpd/v = 0 ((1)). A Lorentz transformation, however, changes mo in a rest frame into p=mov/sqrt(1-vv/cc) and E=mocc/sqrt(1-vv/cc). One may see that this solution may be obtained from ((1)) showing a direct link between the classical action approach and special relativity. The connection may be carried further as Lt = -Et+px is a Lorentz invariant, but also the classical action (in both the relativistic and nonrelativistic cases). We argue that the use of the action in (1) as a stochastic variable is directly related to special relativity. In summary, from finite dt in dp/dt=F, t and x are stochastic variables within some naturally appearing dx and dt. One may write a linear combination of x and t and to create a stochastic Lorent invariant variable which should appear in exp(i Lorentz invariant) and create the dx and dt units. It may then be shown that this Lorentz invariant in E,p,x,t should be equivalent to the classical action.

•Jan 26, 2026•0 citations

Time evolution with symmetric stochastic action

P. D. Drummond

The time evolution of quantum fields is shown to be equivalent to a time-symmetric Fokker-Planck equation.Results are obtained using a Q-function representation, including fermion-fermion, boson-boson, and fermion-boson interactions with linear, quadratic, cubic, and quartic Hamiltonians, typical of QED and many other cases.For local boson-boson coupling, the resulting probability distribution is proved to have a positive, time-symmetric path integral and action principle, leading to a forward-backward stochastic process in both time directions.The solution corresponds to a c-number field equilibrating in an additional dimension.Paths are stochastic trajectories of fields in space-time, which are samples of a statistical mechanical steady state in a higher-dimensional space.We derive numerical methods and examples of solutions to the resulting stochastic partial differential equations in a higher time dimension, giving agreement with examples of simple bosonic quantum dynamics.This approach may lead to useful computational techniques for quantum field theory, as well as to new ontological models of physical reality.

arXiv:1910.00001•Physical Review Research•Mar 15, 2021•14 citations
In this paper, we develop an action formulation of stochastic dynamics in the Hilbert space. In this formulation, the quantum theory of random unitary evolution is easily reconciled with special relativity. We generalize the Wiener process into 1+31+31+3-dimensional spacetime, and then define a scalar random field which keeps invariant under Lorentz transformations. By adding to the action of quantum field theory a coupling term between random and quantum fields, we obtain a random-number action which has the statistical spacetime translation and Lorentz symmetries. The canonical quantization of the theory results in a Lorentz-invariant equation of motion for the state vector or density matrix. We derive the path integral formula of SSS matrix, based on which we develop a diagrammatic technique for doing the calculation. We find the diagrammatic rules for both the stochastic free field theory and stochastic \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 theory. The Lorentz invariance of the random SSS matrix is strictly proved by using the diagrammatic technique. We then develop a diagrammatic technique for calculating the density matrix of final quantum states after scattering. In the absence of interaction, we obtain the exact expressions of both SSS matrix and density matrix. In the presence of interaction, we prove a simple relation between the density matrices of stochastic and conventional \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 theory. Our formalism leads to an ultraviolet divergence that has a similar origin as that in quantum field theory. The divergence can be canceled by renormalizing the coupling strength to random field. We prove that the stochastic quantum field theory is renormalizable even in the presence of interaction. In the models with a linear coupling between random and quantum fields, the random field excites particles out of the vacuum, driving the Universe towards an infinite-temperature state. The number of excited particles follows the Poisson distribution. The collision between particles is not affected by the random field. But the signals of colliding particles are gradually covered by the background excitations caused by random field.
Abstract We develop a quantum field theory based on random nonHermitian actions, which upon quantization lead to stochastic nonlinear Schrödinger dynamics for the state vector. In this framework, Lorentz and spacetime translation symmetries are preserved only in a statistical sense: the probability distribution of the action remains invariant under these transformations. As a result, the theory describes ensembles of quantum-state trajectories whose probability distributions remain invariant under changes of reference frame. As a concrete example, we augment the Dirac action with a purely imaginary term coupling the fermion density operator to a universal colored noise. This noise is constructed by solving the d’Alembert equation with white noise as its source, using a generalized stochastic calculus in 1+3 dimensions. We demonstrate that the colored noise drives stochastic localization of wave packets and derive the localization length analytically. Remarkably, the localization length decreases as the size of the observable universe increases. Our model thus provides a potential framework for relativistic spontaneous wave-function collapse. While establishing consistency with Born’s law remains an open challenge, the present work constitutes a step toward embedding collapse models into a Lorentz-invariant quantum field theory.
Abstract We develop a Schrödinger-picture formulation for a scalar quantum field driven by a Lorentz-invariant white-noise field. The quantum state of the system is described by a stochastic wave functional that evolves according to a stochastic Schrödinger equation. We show that the Gaussian structure of the wave functional is preserved under the stochastic evolution, allowing the dynamics to be reduced to a set of equations for the corresponding kernel functions. These kernel equations are derived and solved exactly, yielding an explicit time-dependent expression for the wave functional. The exact solution enables a direct analysis of the statistical properties of the quantum field in the space of field configurations. In particular, we show that the expectation value of the field operator obeys the same stochastic equation as the classical field obtained from the Euler–Lagrange equation of the action. We further compute the energy density from the stochastic wave functional and evaluate its ensemble average over noise realizations. The resulting energy production rate coincides with that obtained from the corresponding Lindblad equation. This result indicates that the stochastic quantum state remains well defined even though certain derived observables exhibit ultraviolet divergences associated with the white-noise idealization.
We develop a Schrödinger-picture formulation for a scalar quantum field driven by a Lorentz-invariant white-noise field. The quantum state of the system is described by a stochastic wave functional that evolves according to a stochastic Schrödinger equation. We show that the Gaussian structure of the wave functional is preserved under the stochastic evolution, allowing the dynamics to be reduced to a set of equations for the corresponding kernel functions. These kernel equations are derived and solved exactly, yielding an explicit time-dependent expression for the wave functional. The exact solution enables a direct analysis of the statistical properties of the quantum field in the space of field configurations. In particular, we show that the expectation value of the field operator obeys the same stochastic equation as the classical field obtained from the Euler-Lagrange equation of the action. We further compute the energy density from the stochastic wave functional and evaluate its ensemble average over noise realizations. The resulting energy production rate coincides with that obtained from the corresponding Lindblad equation. This result indicates that the stochastic quantum state remains well defined even though certain derived observables exhibit ultraviolet divergences associated with the white-noise idealization.
Abstract We develop a theory of random non-Hermitian action that, after quantization, describes the stochastic nonlinear dynamics of quantum states in Hilbert space. Focusing on fermionic fields, we propose both canonical quantization and path integral quantization, demonstrating that these two approaches are equivalent. Using this formalism, we investigate the evolution of a single-particle Gaussian wave packet under the influence of non-Hermiticity and randomness. Our results show that specific types of non-Hermiticity lead to wave packet localization, while randomness affects the central position of the wave packet, causing the variance of its distribution to increase with the strength of the randomness.
In (1), the notion of a stochastic classical action is introduced and a probability is linked to exp( constant * action). The objective of (1) is not to derive quantum mechanics, but to derive stochastic dynamics based on maximum entropy in action space. We have argued in a number of previous notes (2) that one may use the classical action to create a quantum free particle probability. This classical action, however, is not chosen a priori, but rather arises due to other considerations. We suggest that stochasticity is present in Newtonian mechanics at some time scale because dt is finite in dp/dt= Force (Newton’s second law) and so postulate that force delivery at some x,t is probabilistic within some dt and dx. A priori, dt and dx are not known, but if there is stochasticity, there should be a free particle probability, i.e. a probability(x,t) where x,t represent an x, t interaction point. Furthermore, the notion of dx and dt should arise naturally. We note that the form exp(i function(x,t,p,E) may be introduced because free particles have no distributed variable (like energy in the Maxwell-Boltzmann case). exp(i function(x,t)) immediately introduces periodicity, but we have argued that stochasticity can only exist within tiny, fixed dx, dt linked to dp/dt=F. In particular, one classical Newtonian scales this stochasticity is not seen. The periodicity should be directly in x and t space, suggesting exp( i A1 t + i A2 x). This expression, however, should be Lorentz invariant, and A1 and A2 should be constants, at least for a fixed mo and v. The reason that the one must have periodicity is that the dx and dt ranges of stochasticity in dp/dt=Fr must appear naturally. At this point, one is led to exp(-iEt+ipx) with -Et+px being a Lorentz invariant which satisfies the above criteria. This Lorentz invariant happens to be the classical action in both the relativistic and nonrelativistic cases, hence giving rise to action as being a stochastic variable (of (1)) because it is a function of the two scholastic variables x and t. Alternatively, one may choose the exp(iA1x+iA2t) due to its conservation properties. One may then argue that A1 and A2 could also be conserved for x,t held constant, i.e. A1 and A2 could be linked to Newtonian energy and momentum in elastic collisions. This allows one to solve for the special relativistic g(v) =1/sqrt(1-vv/cc). Given the idea of a conserved quantity form for a probability, one may argue that exp(i Action) satisfies a maximization of Shannon’s entropy subject to a conserved constraint (which is again a link with (1).) The question then becomes: Why should the Lorentz invariant A= -Et+px be the classical action.One may note that x,t usually refer to trajectory values and so for a free particle, one has x=vt. This means that one may factor out t, i.e. A= t( -E+pv), but this is automatically the form of the Lagrangian with E being the Hamiltonian. This holds relativistically and non-relativistically with the form: dL/dv partial = p. The remaining question is: Why should the classical action be Lorentz invariant? We consider E(v)-p(v) v = -L such that dE/dv - v dp/dv -p = -dL/dv partial. This means that dE/dv - v dpd/v = 0 ((1)). A Lorentz transformation, however, changes mo in a rest frame into p=mov/sqrt(1-vv/cc) and E=mocc/sqrt(1-vv/cc). One may see that this solution may be obtained from ((1)) showing a direct link between the classical action approach and special relativity. The connection may be carried further as Lt = -Et+px is a Lorentz invariant, but also the classical action (in both the relativistic and nonrelativistic cases). We argue that the use of the action in (1) as a stochastic variable is directly related to special relativity. In summary, from finite dt in dp/dt=F, t and x are stochastic variables within some naturally appearing dx and dt. One may write a linear combination of x and t and to create a stochastic Lorent invariant variable which should appear in exp(i Lorentz invariant) and create the dx and dt units. It may then be shown that this Lorentz invariant in E,p,x,t should be equivalent to the classical action.
The time evolution of quantum fields is shown to be equivalent to a time-symmetric Fokker-Planck equation.Results are obtained using a Q-function representation, including fermion-fermion, boson-boson, and fermion-boson interactions with linear, quadratic, cubic, and quartic Hamiltonians, typical of QED and many other cases.For local boson-boson coupling, the resulting probability distribution is proved to have a positive, time-symmetric path integral and action principle, leading to a forward-backward stochastic process in both time directions.The solution corresponds to a c-number field equilibrating in an additional dimension.Paths are stochastic trajectories of fields in space-time, which are samples of a statistical mechanical steady state in a higher-dimensional space.We derive numerical methods and examples of solutions to the resulting stochastic partial differential equations in a higher time dimension, giving agreement with examples of simple bosonic quantum dynamics.This approach may lead to useful computational techniques for quantum field theory, as well as to new ontological models of physical reality.