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.
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.
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.
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 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.
This work develops a Lorentz-invariant variational framework in which Fisher-information geometry appears as an intrinsic structural contribution to quantum dynamics. Motivated by longstanding attempts to connect quantum mechanics with information-theoretic principles, we introduce an action functional depending on the density and phase fields in the Madelung representation. Variation of this action yields a modified Klein–Gordon equation containing a single nonlinear term proportional to the four-dimensional Fisher-information curvature of the probability density. The standard Klein–Gordon equation is recovered when the structural parameter vanishes, ensuring full compatibility with established relativistic dynamics. Taking the nonrelativistic limit, we obtain a uniquely determined nonlinear Schrödinger equation in which the correction term is the functional derivative of the Fisher information. The resulting dynamics preserve probability, maintain the Hamilton–Jacobi correspondence, and contain the linear Schrödinger equation as a special case. Analytical expressions for Gaussian and superposed states demonstrate how the structural modification scales with spatial localization and interference structure, providing clear qualitative signatures that distinguish the model from previous nonlinear extensions and offer a theoretical basis for future experimental verification. The results establish a mathematically transparent link between information geometry and quantum dynamics and provide a foundation for future extensions to fermionic, gauge, and many-body systems.
Bohmian mechanics, also referred to as the de Broglie-Bohm pilot-wave theory, represents a deterministic and nonlocal interpretation of quantum mechanics. Central to this framework is a description of quantum motion in terms of particle trajectories, whose reconciliation with relativity remains a fundamental challenge. Although relativistic guidance laws have been proposed, their predicted dynamics has remained experimentally untested. Here we use weak measurements in a single-photon interferometer to jointly reconstruct the photon energy and momentum weak values and the associated relativistic velocity field. The reconstructed average trajectories agree with theoretical predictions. Near destructive interference, the flow exhibits local reconstructed velocities exceeding the speed of light and negative values of the effective squared-mass parameter. Our results establish an experimental basis for investigating the physical meaning of quantum trajectories and their role in describing motion in relativistic spacetime.
A mathematically explicit extension of quantum hydrodynamics is developed in which a scalar coherence field ΦG modifies phase transport and is driven by the information-geometric structure of the probability density together with probability transport itself. The construction starts from the Madelung representation Ψ = √ρ eiS/ℏ, augments the phase gradient by the covariant shift ∇S ↦ ∇S − αℏ∇ΦG, and defines the dynamics through a variational (ρ, S) sector together with a dissipative Onsager-type relaxation law for ΦG. This yields a closed PDE system consisting of a continuity equation, a modified Hamilton–Jacobi equation, and a coherence-field equation that reduces in the overdamped limit to an elliptic closure. The source of ΦG is neither bare mass density nor bare probability density: it is a local Fisher-information density together with probability transport, so that ΦG measures coherence demand. The field equations are written explicitly; the coupling sign is fixed by an energy-stability argument, the transport-forcing sign by a dissipation argument, and the induced coherence-pressure term is computed exactly. A structural no-signalling condition is obtained by restricting the nonlinearity to Fisher-marginal source functionals that are blind to the entanglement Fisher excess under marginalisation. A coarse-grained Yukawa sector is shown to admit a conditional Newtonian limit under explicitly stated screening and additivity assumptions; this remains a conjectural macroscopic regime rather than a completed derivation of gravity. The paper closes with a falsifiable numerical programme—ablation studies, dispersion diagnostics, Fisher-information monitoring, and an explicit pilot solver scaffold—together with a claim-status ledger, a testability map, and a development roadmap that separate what is defined, what is derived, what is heuristic, and what remains unproven. Keywords: quantum hydrodynamics; Fisher–Rao geometry; Madelung dynamics; coherence field; no-signalling; information geometry.