We explore whether quantum field theory can be understood as the statistical mechanics of a time-reversal-invariant stochastic generalization of Hamiltonian dynamics. The motivation for this project, started with this paper, is to assign sharp values to all observables and thereby avoid the quantum measurement problem. In classical mechanics, motion is deterministic and corresponds to an evolution of the phase space probability density according to Liouville's equation that is governed by first derivatives of the Hamiltonian in phase space. We derive a generalization of the Liouville equation with natural constraints -- namely, reduction to classical Hamiltonian dynamics as the stochasticity parameter
ℏ↦0, Fokker-Planck form for the probability density evolution, local Hamiltonian dependence, time-reversal invariance, energy conservation, and minimality -- which turns out to be a Fokker-Planck equation with a generalized diffusion matrix that is symmetric, traceless, and constructed from the Hessian of the Hamiltonian. We then show that the Schrödinger equation in the coherent-state phase-space formulation of certain bosonic QFTs has precisely this form, with the Husimi function playing the role of the phase space probability density. The question to what extent this equation can be interpreted in terms of objective stochastic field theories is discussed in a companion paper.
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.
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.
A recently proposed stochastic hidden variable model for quantum mechanics has been claimed to involve "retrocausality" due to the appearance of equations of motion with future-time boundary conditions. We formulate an equivalent system of forward-time equations of motion that gives rise to the same trajectories as solutions, but involves only initial-time boundary conditions. The forward-time dynamics involves a guidance term for the dynamical variables, determined by the phase-space distribution corresponding to a quantum wavefunction. We show, however, that this particular guidance term can be recovered as the mean-field limit of averaged pairwise interactions among an ensemble of finitely many particles.
de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function
ψ. Given a Schrodinger wave function
ψ, we use its value
ψ0 at some fixed time (say,
t=0) to define the probability measure
∣ψ0∣2dx on the system configuration space
M (
=Rn). On the resulting probability space
M0, we introduce a stochastic process
ξ(t) corresponding to the Heisenberg position operator
XH(t) such that, in the Heisenberg state
∣ψh⟩ corresponding to
ψ0, the expectation value of
XH(t) equals that of
ξ(t) in
M0. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process
ξ(t) which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the
ψ0-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.
Standard quantum mechanics relies on two distinct dynamical principles: unitary evolution and collapse. A mathematically self-contained variational framework is presented that replaces this dualism with a single principle, in which nonrelativistic Schrödinger dynamics are not postulated but emerge as an admissible optimality condition of a primal-dual boundary-value problem. By expressing the state in terms of hydrodynamic variables
(ρ,j) subject to a continuity constraint, it is shown that Fisher-information regularization yields the linear Schrödinger equation within the admissible single-valued variational class. Rather than evolving an initial state forward in time, the dynamics arise from minimizing a global action that connects the initial and final boundary constraints, with the selected solution corresponding to a specific hydrodynamic flow within an ensemble of admissible histories. A von Neumann pointer model illustrates how Born-rule statistics for recorded outcomes arise without introducing a separate collapse law. Within this formulation, quantum uncertainty is interpreted as effective randomness over boundary-compatible histories rather than as a fundamental stochastic postulate. The resulting framework provides a nonrelativistic proof of concept for how a single time-symmetric variational reformulation can recover key features of quantum theory.