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.
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.
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.
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 paper folds a minimal full closure into a previously closure-neutral source-term framework for Sutherland-type action-reaction pilot-wave dynamics. The aim is not to claim an accepted extension of quantum mechanics, but to define a mathematically closed effective post-quantum candidate whose assumptions, conservation properties, and falsification conditions are explicit. The central object is the guidance mismatch field Delta p(x,t). In the closed model this mismatch is no longer free: it is fixed by a regularized weak-osmotic closure, Delta p = -lambda R_delta[Im(p_w)], where Im(p_w) is the imaginary component of the local momentum weak value. The raw Sutherland source s = (1/m) grad rho * Delta p is then globally compensated by sigma = s - Lambda rho, with Lambda chosen so that int sigma d^3x = 0. This produces local source-active redistribution without global norm creation. The corresponding wave equation contains a trace-preserving non-Hermitian effective term, i hbar (sigma/2rho) psi, while the particle-sector velocity is v_B + Delta p/m. The result is a closed dynamical loop psi -> rho,S -> Delta p -> s -> sigma -> psi, together with a particle-sector transport law. Downstream derivations are revised accordingly: branch drift becomes S_a - Lambda N_a, tunnelling corrections become S_B - Lambda N_B, local amplification requires s/rho - Lambda > div v_B, and linear onset is governed by deviation from the global source-per-mass average. The paper also preserves the frameworks epistemic boundary: no-signalling, relativistic extension, and derivation of lambda, delta, and the regularization length remain open problems.
Decoherence is a central obstacle in the control and interpretation of open quantum systems, traditionally treated as an effectively irreversible process driven by environmental coupling. In this work, we introduce a framework for symmetry-driven coherence restoration (SDCR), demonstrating that coherence loss in open systems need not be strictly monotonic when specific temporal, geometric, or algebraic symmetries are aligned within the reduced dynamics. Crucially, the framework operates entirely within standard open-quantum-system theory and does not invoke new particles, collapse mechanisms, or modifications of fundamental quantum laws. SDCR is formulated as a symmetry-selection mechanism acting on the reduced evolution of quantum systems, where dominant decohering channels are parametrically suppressed through structured symmetry alignment. This leads to partial recovery of phase coherence and to small, systematic phase corrections that preserve established scaling relations. The framework is developed consistently across Lindblad, Redfield, and influence-functional formalisms, highlighting its compatibility with both Markovian and non-Markovian dynamics. To organize coupled phase and memory channels, the theory employs octonionic and quaternionic embeddings strictly as internal algebraic tools, without attributing physical reality to additional dimensions or degrees of freedom. The phenomenological consequences of SDCR are explored in contexts where phase stability is experimentally accessible, including neutrino oscillations, matter-wave and photonic interferometry, and precision gravimetry. In these settings, SDCR predicts controlled, symmetry-correlated phase offsets that remain below existing experimental bounds yet lie within the sensitivity of current and near-future experiments. Importantly, all predicted effects vanish in the recovery limit when symmetry alignment is removed, rendering the framework explicitly falsifiable. Overall, this work presents SDCR as a conservative, testable geometric perspective on coherence control in open quantum systems. By reframing decoherence suppression as a problem of symmetry alignment rather than active feedback or new dynamics, the framework provides a unified lens through which coherence preservation and phase stability can be investigated across disparate quantum platforms.
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.