We present a deterministic extension of de Broglie-Bohm pilot-wave theory that resolves the fundamental action-reaction asymmetry of the standard formulation. By introducing a reciprocal coupling term sourced by the imaginary part of the momentum weak value, Im(pw) = -(ħ/2)∇ ln ρ, we enable the particle configuration to influence the guiding wavefunction locally. To prevent the anti-diffusive instabilities inherent in such back-reaction models, we introduce a parameter-free "Fluctuation Clamp" mechanism. This mechanism budgets allowed gradient sharpening against a numerical vacuum surrogate scale, ensuring stability over 10^5 simulation steps while preserving time-symmetry and normalization. The model recovers standard quantum mechanics in equilibrium (ρ = |ψ|²) and in the weak-coupling limit (λ → 0), but predicts falsifiable deviations in non-equilibrium regimes, specifically in neutron interferometry (velocity shifts Δv ~ 5 × 10^-3 m/s) and tunneling time measurements (dwell time shifts δτ/τ ~ 0.1).
We present a deterministic extension of de Broglie-Bohm pilot-wave theory that resolves the fundamental action-reaction asymmetry of the standard formulation. By introducing a reciprocal coupling term sourced by the imaginary part of the momentum weak value, Im(pw) = -(ħ/2)∇ ln ρ, we enable the particle configuration to influence the guiding wavefunction locally. To prevent the anti-diffusive instabilities inherent in such back-reaction models, we introduce a parameter-free "Fluctuation Clamp" mechanism. This mechanism budgets allowed gradient sharpening against a numerical vacuum surrogate scale, ensuring stability over 10^5 simulation steps while preserving time-symmetry and normalization. The model recovers standard quantum mechanics in equilibrium (ρ = |ψ|²) and in the weak-coupling limit (λ → 0), but predicts falsifiable deviations in non-equilibrium regimes, specifically in neutron interferometry (velocity shifts Δv ~ 5 × 10^-3 m/s) and tunneling time measurements (dwell time shifts δτ/τ ~ 0.1).
Abstract Standard pilot-wave theory contains a structural asymmetry: the wave guides the particle, but the particle exerts no reciprocal influence on the wave. This paper develops a geometrically complete extension in which action–reaction symmetry is restored. Particle deviation from guidance is sourced by the imaginary part of the momentum weak value, a quantity intrinsic to quantum theory that vanishes in equilibrium and activates precisely where amplitude structure exists. Unregulated back-reaction generates deterministic instability through positive feedback between deviation and density gradients. Stability is restored through geometric phase accumulation along deviating trajectories, which feeds back into the wave’s phase structure and reshapes subsequent guidance. The resulting framework remains deterministic and time-symmetric and yields three independent falsifiable predictions: the weak-value source of deviation, trajectory-dependent phase reshaping, and the existence of a critical feedback strength separating unstable and bounded dynamical regimes.
This paper closes the action–reaction post-quantum series by proposing thermodynamic fluctuations as an intrinsic stabilizer for Sutherland’s back-reaction dynamics. Building on prior work in self-organization, weak values, geometric phase, and torsion bounds, we introduce a deterministic, audit-friendly entropy budget (a “clamp”) that caps probability-flow deviations without requiring an external thermal bath. The cap is set by the model’s own vacuum-fluctuation surrogate already present in the numerical scheme, avoiding a new temperature-like tuning parameter. We clarify connections to fluctuation theorems (Jarzynski/Crooks/Sagawa–Ueda) without claiming new equalities. AI-assisted large-scale simulations (1D, up to 10^6 iterations) show stable localization in harmonic and free-particle cases at scales beyond manual computation, with transparent logs indicating when and how the clamp acted. Standard quantum mechanics emerges in the zero-back-reaction limit (and, when applicable, the zero-noise limit). We outline five falsifiable predictions for trapped-ion and interferometry platforms and give explicit criteria under which the mechanism is constrained or ruled out.
We formulate a regularized reciprocal extension of quantum hydrodynamics motivated by action-reaction symmetry in pilot-wave theory and by Sutherland-type post-quantum back-reaction schemes. The reciprocal anomaly is sourced by the imaginary part of the local momentum weak value. In the formal unregularized limit this gives the minimal closure ∆p proportional to Im(pw ). Since that reduced closure is singular at nodes and generically behaves as an effectively open dynamics, we construct a bounded completion on the torus by introducing: (i) a nodal regularization Aε =−(ℏ/2m)∇ln(ρ+ ε), (ii) a smooth information-limited mobility controlled by a nonnegative global admissibility functional built from relative entropy and regularized Fisher information, and (iii) a transported Eulerian memory field encoding cumulative reciprocal activity. The resulting model is a globally coupled nonlinear PDE for (ρ,S,M) with Hamiltonian advection, bounded reciprocal transport, capillarity-regularized quantum pressure, and local memory loading. We prove exact identities for weak-value decomposition, mass conservation, reciprocal-current entropy- gradient form, and memory-energy balance. We then state an energy-controlled local classical well-posedness theorem for fixed ε>0, based on a high-regularity Sobolev–capillarity energy. A modal asymptotic result shows that if the mobility exponent satisfies n>1, ultraviolet amplification is suppressed in a Fisher-dominated fixed-amplitude scaling regime, with an extended condition for amplitude-decaying spectra. In reciprocal equilibrium the model reduces to a regularized Madelung-type quantum hydrodynamics, recovering the standard unregularized system formally as ε →0 away from nodal singularities. The framework remains a candidate closure rather than an established physical theory, but its mathematical content is explicit enough to be analyzed, simulated, and falsified.
Standard pilot-wave theory exhibits a fundamental dynamical asymmetry: the wavefunc- tion guides the particle trajectory through the Bohmian guidance equation, yet the par- ticle exerts no reciprocal influence on the wave. This lack of action–reaction symmetry is conceptually problematic if the wavefunction is regarded as physically real. This work develops Geometric Weak Value Dynamics (GWVD), a deterministic and time-symmetric extension of pilot-wave theory that restores reciprocal coupling between particle and wave. The key mechanism is the identification of the imaginary part of the quantum momentum weak value as a deviation source proportional to the score function ∇ln ϱ of the prob- ability density. Because the score function defines the Fisher information geometry of statistical manifolds, the resulting dynamics can be interpreted as a geometric flow on the manifold of probability distributions. Unregulated deviation produces anti-diffusive am- plification of density gradients, corresponding to runaway growth of Fisher information. Stability is achieved through trajectory-dependent geometric phase accumulation, which feeds back locally into the wavefunction phase and introduces curvature-dependent reg- ulation of particle motion. Crucially, a Lagrangian formulation is presented from which both the modified Schrödinger equation and the modified guidance equation are derived simultaneously via a single variational principle, establishing equivariance by construc- tion and reducing the free parameters of the theory to a single consistency condition. The theory predicts three experimentally testable signatures: deviation velocity proportional to Im(𝑝𝑤), trajectory-dependent interference modulation via geometric phase, and a criti- cal feedback threshold 𝛼crit = λ/2 separating unstable and bounded regimes. Extension to higher spatial dimensions reveals emergent vorticity, vortex lattices, and mode-selective pattern formation. GWVD maps onto a minimal Turing activator–inhibitor system, suggest- ing a deep connection between quantum dynamics, information geometry, and nonlinear pattern formation.
This manuscript gives a strengthened first-principles closure of a weak-osmotic action-reaction pilot-wave dynamics. The aim is not to claim a final field theory, but to close the nonrelativistic finite-resolution source-active dynamics on a declared mathematical domain. The primitive wave ontology is , so that the local momentum weak value has the standard decomposition ; the imaginary part is precisely Nelson's osmotic velocity , well known from stochastic quantum mechanics. The saturation length is physically identified with the thermal de Broglie wavelength . The raw source-per-density field is projected in onto the additive subspace generated by operationally protected marginals; for any finite protected family and this subspace is closed and the projection is well defined. The empirical reaction current is the closest current to the bare weak-osmotic current subject to exact Born equivariance . The finite- resolution theory is locally well-posed. Continuum well-posedness, relativistic hypersurface integrability, and empirical calibration remain stated maturation tasks. A concrete bipartite benchmark density is specified in Appendix B for independent numerical reproduction.
Abstract Bohmian mechanics supplements the quantum wavefunction with deterministic particle trajectories, offering an alternate, dynamical language for quantum theory. However, the Bohmian wavefunction evolves independently of these trajectories, and is thus unaffected by the observable properties of the system. While this property is widely assumed necessary to ensure agreement with quantum mechanics, much work has recently been dedicated to understanding classical pilot-wave systems, which feature a two-way coupling between particle and wave. These systems—including the “walking droplet” system of Couder and Fort (Couder and Fort (2006) Phys. Rev. Lett. 97:154101) and its various abstractions (Dagan and Bush (2020) CR Mecanique 348:555–571; Durey and Bush (2020) Front. Phys. 8:300; (2021) Chaos 31:033136; Darrow and Bush (2024) Symmetry 16:149)—allow us to investigate the limits of classical systems and offer a touchstone between quantum and classical dynamics. In this work, we present a general result that bridges Bohmian mechanics with this classical pilot-wave theory. Namely, Darrow and Bush ((2024) Symmetry 16:149) recently introduced a Lagrangian pilot-wave framework to study quantum-like behaviours in classical systems; with a particular choice of particle-wave coupling, they recover key dynamics hypothesised in de Broglie’s early double-solution theory (de Broglie (1970) Foundations Phys. 1:5–15). We here show that, with a different choice of coupling, their de Broglie-like system reduces exactly to single-particle Bohmian mechanics in the non-relativistic limit. Our result clarifies that, while multi-particle entanglement is impossible to replicate in general with local, classical theories, no such restriction exists for single-particle quantum mechanics. Moreover, connecting with the previous work of Darrow and Bush, our work demonstrates that de Broglie’s and Bohm’s theories can be connected naturally within a single Lagrangian framework. Finally, we present an application of the present work in developing a single-particle analogue for position measurement in a de Broglie-like setting.