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arXiv:2608.13894

de Broglie-Bohm Dynamics with Schrödinger Source Fields: A Framework for Subquantum Theory

Said Mikki

quant-phmath-phSubmitted Aug 14, 2026• Announced Aug 17, 2026• 0 citations
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Abstract

We extend de Broglie--Bohm (dBB) pilot-wave theory by introducing a complex source field into the Schrödinger equation and examining its effects on quantum equilibrium and nonequilibrium dynamics. In dBB theory, the physical particle distribution P, rather than the Born density $|ψ|^2$, carries the ensemble probability, so a source term may modify the pilot wave without violating conservation of total particle probability. We derive the source-modified Hamilton--Jacobi and continuity equations, the transport equation for the nonequilibrium ratio $f=P/|ψ|^2$, and an exact entropy-production formula. Three applications follow. First, suitably designed sources can drive exponential relaxation toward quantum equilibrium. Second, the entropy-production rate admits a Prigogine-type bilinear form, providing a basis for a subquantum thermodynamics with entropy-producing and entropy-extracting regimes. Third, a tuned source can exactly cancel the Bohmian quantum potential, yielding classical particle trajectories while the guiding wave retains nontrivial structure and the nonequilibrium ratio remains conserved. The resulting framework provides a unified setting for studying source-driven quantum nonequilibrium, entropy exchange, and classicalization, and motivates further investigation of the physical and ontological status of the source field.

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Pilot Wave Theory and Quantum Fields

Matthias Lienert

Pilot wave theory (PWT), also called de Broglie-Bohm theory or Bohmian Mechanics, is a deterministic nonlocal `hidden variables' quantum theory without fundamental uncertainty. It is in agreement with all experimental facts about nonrelativistic quantum mechanics (QM) and furthermore explains its mathematical structure. But in general, PWT describes a nonequilibrium state, admitting new physics beyond standard QM. This essay is concerned with the problem how to generalise the PWT approach to quantum field theories (QFTs). First, we briefy state the formulation of nonrelativistic PWT and review its major results. We work out the parts of its structure that it shares with the QFT case. Next, we come to the main part: We show how PW QFTs can be constructed both for field and particle ontologies. In this context, we discuss some of the existing models as well as general issues: most importantly the status of Lorentz invariance in the context of quantum nonlocality. The essay concludes with a more speculative outlook in which the potential of PWT for open QFT questions as well as quantum nonequilibrium physics are considered.

•May 1, 2011•2 citations
We extend de Broglie--Bohm (dBB) pilot-wave theory by introducing a complex source field into the Schrödinger equation and examining its effects on quantum equilibrium and nonequilibrium dynamics. In dBB theory, the physical particle distribution P, rather than the Born density ∣ψ∣2|ψ|^2∣ψ∣2, carries the ensemble probability, so a source term may modify the pilot wave without violating conservation of total particle probability. We derive the source-modified Hamilton--Jacobi and continuity equations, the transport equation for the nonequilibrium ratio f=P/∣ψ∣2f=P/|ψ|^2f=P/∣ψ∣2, and an exact entropy-production formula. Three applications follow. First, suitably designed sources can drive exponential relaxation toward quantum equilibrium. Second, the entropy-production rate admits a Prigogine-type bilinear form, providing a basis for a subquantum thermodynamics with entropy-producing and entropy-extracting regimes. Third, a tuned source can exactly cancel the Bohmian quantum potential, yielding classical particle trajectories while the guiding wave retains nontrivial structure and the nonequilibrium ratio remains conserved. The resulting framework provides a unified setting for studying source-driven quantum nonequilibrium, entropy exchange, and classicalization, and motivates further investigation of the physical and ontological status of the source field.
We illustrate through explicit numerical calculations how the Born rule probability densities of non-relativistic quantum mechanics emerge naturally from the particle dynamics of de Broglie–Bohm pilot-wave theory. The time evolution of a particle distribution initially not equal to the absolute square of the wave function is calculated for a particle in a two-dimensional infinite potential square well. Under the de Broglie–Bohm ontology, the box contains an objectively existing ‘pilot wave’ which guides the electron trajectory, and this is represented mathematically by a Schrödinger wave function composed of a finite out-of-phase superposition of M energy eigenstates (with M ranging from 4 to 64). The electron density distributions are found to evolve naturally into the Born rule ones and stay there; in analogy with the classical case this represents a decay to ‘quantum equilibrium’. The proximity to equilibrium is characterized by the coarse-grained subquantum H -function which is found to decrease roughly exponentially towards zero over the course of time. The time scale τ for this relaxation is calculated for various values of M and the coarse-graining length ε . Its dependence on M is found to disagree with an earlier theoretical prediction. A power law, τ ∝ M −1 , is found to be fairly robust for all coarse-graining lengths and, although a weak dependence of τ on ε is observed, it does not appear to follow any straightforward scaling. A theoretical analysis is presented to explain these results. This improvement in our understanding of time scales for relaxation to quantum equilibrium is likely to be of use in the development of models of relaxation in the early Universe, with a view to constraining possible violations of the Born rule in inflationary cosmology.
Numerical simulations indicate that the Born rule does not need to be postulated in the de Broglie–Bohm pilot-wave theory, but tends to arise dynamically (relaxation to quantum equilibrium). These simulations were done for a particle in a two-dimensional box whose wave function obeys the non-relativistic Schrödinger equation and is therefore scalar. The chaotic nature of the de Broglie–Bohm trajectories, thanks to the nodes of the wave function which yield to vortices, is crucial for a fast relaxation to quantum equilibrium. For spinors, we typically do not expect any node. However, in the case of the Dirac equation, the de Broglie–Bohm velocity field has vorticity even in the absence of nodes. This observation raises the question of the origin of relaxation to quantum equilibrium for fermions. In this article, we provide numerical evidence to show that Dirac particles also undergo relaxation, by simulating the evolution of various non-equilibrium distributions for two-dimensional systems (the two-dimensional Dirac oscillator and the Dirac particle in a two-dimensional spherical step potential).
Abstract We investigate the dynamical stability and relaxation of subquantum non-equilibrium states within the framework of De Broglie Bohm field theory during primordial inflation. By evaluating the quantum Hamilton-Jacobi formalism on a covariant Friedmann-Lemaître Robertson-Walker (FLRW) background, I demonstrate that non-linearities intrinsic to the field-theoretic quantum potential generate microscopic phase-space chaos. I solve the Jacobi deviation equation for coupled Fourier modes and extract the maximal Lyapunov exponent spectrum. I find that sub-horizon modes exhibit rapid exponential trajectory divergence driving coarse-grained subquantum entropy toward zero via Valentini's subquantum H-theorem. This dynamic phase-mixing enforces the Born rule well before horizon crossing, rendering anti-relaxation statistically prohibited for generic inflationary potentials. Finally, I map residual non-equilibrium corrections onto the Cosmic Microwave Background (CMB) power spectrum at low multipoles establishing current observational bounds from Planck 2018 data and providing forecasts for B-mode polarization measurements from LiteBIRD and CMB-S4.
We reply to claims (by Deutsch, Zeh, Brown and Wallace) that the pilot-wave theory of de Broglie and Bohm is really a many-worlds theory with a superfluous configuration appended to one of the worlds. Assuming that pilot-wave theory does contain an ontological pilot wave (a complex-valued field in configuration space), we show that such claims arise from not interpreting pilot-wave theory on its own terms. Specifically, the theory has its own ('subquantum') theory of measurement, and in general describes a 'nonequilibrium' state that violates the Born rule. Furthermore, in realistic models of the classical limit, one does not obtain localised pieces of an ontological pilot wave following alternative macroscopic trajectories: from a de Broglie-Bohm viewpoint, alternative trajectories are merely mathematical and not ontological. Thus, from the perspective of pilot-wave theory itself, many worlds are an illusion. It is further argued that, even leaving pilot-wave theory aside, the theory of many worlds is rooted in the intrinsically unlikely assumption that quantum measurements should be modelled on classical measurements, and is therefore unlikely to be true.
In this work, we derive Born’s rule from the pilot-wave theory of de Broglie and Bohm. Based on a toy model involving a particle coupled to an environment made of “qubits” (i.e., Bohmian pointers), we show that entanglement together with deterministic chaos leads to a fast relaxation from any statistical distribution ρ(x) of finding a particle at point x to the Born probability law |Ψ(x)|2. Our model is discussed in the context of Boltzmann’s kinetic theory, and we demonstrate a kind of H theorem for the relaxation to the quantum equilibrium regime.
Pilot wave theory (PWT), also called de Broglie-Bohm theory or Bohmian Mechanics, is a deterministic nonlocal `hidden variables' quantum theory without fundamental uncertainty. It is in agreement with all experimental facts about nonrelativistic quantum mechanics (QM) and furthermore explains its mathematical structure. But in general, PWT describes a nonequilibrium state, admitting new physics beyond standard QM. This essay is concerned with the problem how to generalise the PWT approach to quantum field theories (QFTs). First, we briefy state the formulation of nonrelativistic PWT and review its major results. We work out the parts of its structure that it shares with the QFT case. Next, we come to the main part: We show how PW QFTs can be constructed both for field and particle ontologies. In this context, we discuss some of the existing models as well as general issues: most importantly the status of Lorentz invariance in the context of quantum nonlocality. The essay concludes with a more speculative outlook in which the potential of PWT for open QFT questions as well as quantum nonequilibrium physics are considered.