We propose a minimal effective field theory extension of Bohmian mechanics via a weak real Lorentz-scalar field φ coupled to the Dirac bilinear ψ-barψ through a strictly dimensionless Yukawa coupling g. In the non-relativistic limit this adds a classical term gφ(x) directly to the Bohmian quantum potential, Q_eff = Q + gφ(x). The model is fully relativistic, Lagrangian-based, and observationally equivalent to standard quantum mechanics plus a classical Yukawa fifth force. The paper derives the field equations, exact sourced scalar field, conservation laws, and realistic laboratory predictions for atom/neutron interferometry (μeV benchmark) and Casimir forces (meV benchmark). Current bounds already constrain |g| ≲ 10^{-12}–10^{-15}; next-generation experiments can test the predictions. This construction — giving the quantum potential an explicit hidden-sector scalar origin while remaining fully testable by existing fifth-force searches — is original and has not appeared in the literature.
We propose a minimal effective field theory extension of Bohmian mechanics via a weak real Lorentz-scalar field φ coupled to the Dirac bilinear ψ-barψ through a strictly dimensionless Yukawa coupling g. In the non-relativistic limit this adds a classical term gφ(x) directly to the Bohmian quantum potential, Q_eff = Q + gφ(x). The model is fully relativistic, Lagrangian-based, and observationally equivalent to standard quantum mechanics plus a classical Yukawa fifth force. The paper derives the field equations, exact sourced scalar field, conservation laws, and realistic laboratory predictions for atom/neutron interferometry (μeV benchmark) and Casimir forces (meV benchmark). Current bounds already constrain |g| ≲ 10^{-12}–10^{-15}; next-generation experiments can test the predictions. This construction — giving the quantum potential an explicit hidden-sector scalar origin while remaining fully testable by existing fifth-force searches — is original and has not appeared in the literature.
We propose a minimal effective field theory extension of Bohmian mechanics via a weak real Lorentz-scalar field φ coupled to the Dirac bilinear ψ-barψ through a strictly dimensionless Yukawa coupling g. In the non-relativistic limit this adds a classical term gφ(x) directly to the Bohmian quantum potential, Q_eff = Q + gφ(x). The model is fully relativistic, Lagrangian-based, and observationally equivalent to standard quantum mechanics plus a classical Yukawa fifth force. The paper derives the field equations, exact sourced scalar field, conservation laws, and realistic laboratory predictions for atom/neutron interferometry (μeV benchmark) and Casimir forces (meV benchmark). Current bounds already constrain |g| ≲ 10^{-12}–10^{-15}; next-generation experiments can test the predictions. This construction — giving the quantum potential an explicit hidden-sector scalar origin while remaining fully testable by existing fifth-force searches — is original and has not appeared in the literature.
We propose an exploratory framework for a Bohmian model of quantum matter propagating on a classical curved spacetime background. The gravitational sector is governed by classical Einstein field equations throughout; no quantisation of spacetime is attempted. The wave function emerges as the scalar contraction Ψ=ψνψν∈C of a complex-valued tensorial field ψμ, encoding quantum dynamics in a geometric object. The wave tensor interacts with spacetime via the stress–energy tensor Tμν, mediated by a real scalar field a of dimension volume, so that aTμνψμψν yields the correct potential energy. We derive a covariant Adapted Schrödinger Equation as the unique minimal covariant lift of the standard equation, justify it from four guiding principles, and verify three internal consistency checks. Under seven explicit approximations the framework reproduces the Schrödinger equation with Coulomb potential for the hydrogen atom. We also derive a dynamical equation for ψμ that entails the Adapted Schrödinger Equation by contraction. Two open problems are then resolved. First, a complete Lagrangian formulation is provided: a real-valued action for Ψ yields the Adapted Schrödinger Equation via the Euler–Lagrange equations; a separate action for ψμ, extended by a non-polynomial term, yields the full dynamical equation variationally. Second, two experimental predictions are derived. Expanding to first post-Newtonian order, the perturbation Hamiltonian has coefficients (3, 1) on the kinetic and potential operators; via the virial theorem these produce a coordinate-time blueshift, which after photon propagation yields the universal Einstein gravitational redshift δν/ν=Φ/c2, confirming consistency with the equivalence principle. The same kinetic coefficient independently predicts that free quantum wave packets spread more slowly by the fractional amount 3|Φ|/c2, a correction absent in standard non-relativistic quantum mechanics.
The extension of Bohmian mechanics to the relativistic domain, and to quantum field theory presents specific theoretical difficulties. The non-local nature of the pilot-wave guidance equations inherently requires a temporal ordering of spacelike separated events. Historically, this requirement has been satisfied by introducing an arbitrary preferred foliation of spacetime, which leads to covariance concerns. Furthermore, the standard Bohmian particle model struggles to accommodate particle creation and annihilation events, which are characteristic of quantum field theory (QFT). This paper constructs a theoretical framework that is rigorously Lorentz-invariant, and compatible with QFT. The proposed synthesis couples the Tomonaga-Schwinger equation for the relativistic wave functional with a hypersurface Bohm-Dirac model (HBDM). To ensure Lorentz invariance without adding absolute spatiotemporal structure, the necessary foliation is covariantly determined by the universal wave function. To address particle non-conservation, the deterministic trajectories are supplemented by a Bell-type stochastic Markov jump process defined on Fock space, utilizing minimal jump rates derived directly from the interaction Hamiltonian. A hybrid model is adopted, assigning localized particle beables to fermions, and continuous field beables to bosons. The exact equivariance of the probability distributions across the covariantly defined leaves of the foliation is proven, demonstrating that the model reproduces the empirical predictions of standard regularized QFT. In an appendix, we discuss the modeling of neutrinos. Finally, this framework is interpreted with the multi-fold theory. In this context, the covariantly determined foliation emerges physically from a discrete, non-commutative spacetime concretized by 2D random walks of massless Higgs bosons. The multi-fold mechanism and the E/G conjecture (entanglement is gravity) provide a local, physical model for the non-local Bohmian guidance equations, while space-time matter induction and scattering explain the origin of the proposed hybrid field-particle beables as patterns and condensates of these random walks. Sizes or scales also align with multi-fold modeling of QFT with 2D random walks, and spacetime non-commutativity to support Quantum Physics and Fermions. These add to many examples where the multi-fold preons as massless Higgs bosons can microscopically interpretate Quantum Physics behaviors.
We derive the Schr¨odinger, Dirac, and Klein-Gordon equations from the scalar-time continuity law of Time-Scalar Field Theory (TSFT). By promoting evolution to the scalar-time parameter Θ and introducing a local time-rate field α(x), classical Hamiltonian and quantum operator dynamics unify under the relation dOdΘ = α{O,H} (classical) and dO dΘ = i¯hα[ ˆH ,O] (quantum). Schr¨odinger’s equation follows from ∂Θ = α∂t; the Dirac equation emerges from a Θ-weighted tetrad with a spin-connection term ∝ ∂μ ln α; and the Klein-Gordon equation arises as the relativistic scalar field equation in the effective metric induced by Θ. Standardquantum mechanics is recovered when ∇α = 0, while TSFT predicts falsifiable gradient corrections accessible to precision matter-wave and spinor interferometry. We supply quantitative estimates (e.g. δϕ ∼ 10−3 rad for Q = 109 cavities) and cross-link TSFT’s microphysics to its cosmological continuity, emphasizing multi-scale coherence and experiment-first falsifiability.
We present a three-dimensional field ontology for Bohmian mechanics in which the single-particle wave functions defined in ordinary physical space are taken as ontological elements alongside particle positions. In this view, each particle is associated with a wave function in three-dimensional space conditioned on the actual positions of the remaining particles. The universal wave function on configuration space is retained within the formalism, but its role is interpreted as nomological: it determines the coupled evolution of the three-dimensional fields without itself being treated as a fundamental physical field. The corresponding evolution equations for the conditioned three-dimensional wave functions follow from the standard Bohmian formalism and, together with the guidance equation, yield dynamics fully equivalent to those of the conventional theory. The empirical and predictive content of Bohmian mechanics is thereby preserved, while its ontological structure is expressed in explicitly three-dimensional terms. Within a ψ-ontic perspective, this proposal makes explicit an ontological organization of Bohmian mechanics in which the dynamical fields are defined on physical space. While fully equivalent to the conventional formulation, expressing the theory in these terms may facilitate interpretation, visualization, and pedagogical presentation.
In this paper we generalize the ideas of de Broglie and Bohm to the relativistic case which is based on the relativistic Schrödinger equation. In this regard, the relativistic forms of the guidance equation and quantum potential are derived. In our formulation of Relativistic Bohmian Mechanic, the quantum equilibrium hypothesis (ρ=∣ψ∣2) and the probabilistic interpretation of the wave function are not an intrinsic feature of the theory as well as expected from the theoretical structure of Bohmian mechanics but still we can extract the statistical predictions of this theory. In fact with assuming the quantum equilibrium hypothesis in non-relativistic case and then acceleration to particles by an external field we go to the relativistic regime. So in this case the quantum equilibrium would not be established and theory will have testable predictions that can be experienced, or this will be compared to the result of relativistic quantum mechanics. In some simple cases such calculations have been done.