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.
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.
We propose a new way to unify General Relativity (GR) with Bohmian Mechanics (BM) that remains faithful to GR’s tensorial and deterministic philosophy. In our approach, the conventional quantum wave function emerges as the scalar contraction of a tensorial field ψμ, namely Ψ=ψνψν, thereby encoding quantum dynamics in a geometric object. Crucially, the tensor ψν interacts directly with spacetime via the stress-energy tensor, with an auxiliary vector field aμ serving as the mediator between the quantum and gravitational sectors.We analyze the equations for one single particle without accounting for spin. As an illustrative example, we show that under suitable approximations our formulation reproduces the non-relativistic Schr\"odinger equation with a Coulomb potential\textemdash as encountered in the hydrogen atom. We further establish a dynamical equation for ψν and a corresponding volume-like equation for aμ.
Previous works of this series have realized the classical unification of gravitation and electromagnetic fields based on the spacetime density tensor, and the intrinsically generated high-order couplings consistently explain various observational anomalies of compact celestial objects. As a realistic quantum framework, existing schemes for unifying Bohmian theory and gravity generally adopt a dual structure that separates spacetime from the pilot wave. The nonlocality of quantum entanglement in multi-particle systems leads to fundamental conflicts with the local causality of relativity. Meanwhile, torsion-free Riemannian geometry cannot accommodate spin-1/2 fermion spinors, nor can it intrinsically generate the short-range SU(2) weak and SU(3) strong gauge interactions, making it difficult to fully embed the Standard Model of quantum field theory. On the basis of maintaining the monistic ontological postulate of the spacetime density tensor, this paper carries out theoretical expansion in two stages. First, the Bohmian pilot wave, quantum potential and multi-particle entanglement are defined as global coherent perturbations of the spacetime density tensor. This work strictly distinguishes the field structures, dynamic equations and propagation mechanisms of separable quantum states and entangled quantum states, and reconstructs the covariant quantum guiding equation. It resolves the paradox between quantum nonlocality and relativistic causality from an ontological perspective, achieving the unification of classical field theory and Bohmian quantum mechanics. Second, Einstein–Cartan (EC) geometry is introduced with the additional degree of freedom of spacetime torsion. Two types of quantum correlations, namely orbital entanglement and spin entanglement, are differentiated. Fermion spinor fields, as well as U(1) electromagnetic, SU(2) weak and SU(3) strong gauge interactions, are all mapped to local chiral twists and color topological torsional perturbation modes of spacetime. Four core innovations are presented in this paper: 1. Quantum effects are attributed to global coherent fluctuations of spacetime density. The essential geometric differences between ordinary separable quantum states and entangled states are clarified. Entanglement originates from the global coherent deformation of spacetime rather than superluminal action, thoroughly eliminating the dual separation between quantum fields and spacetime. 2. With the aid of EC torsion geometry, fermion spinors are no longer background-independent fields but geometric projections of local spin torsion of spacetime, realizing the geometric ontology of intrinsic spin. 3. The complete set of U(1)/SU(2)/SU(3) gauge symmetries arise intrinsically from multi-scale topological perturbations of the spacetime density tensor, removing the artificial introduction of gauge groups as implemented in the Standard Model. 4. A complete covariant unified field system covering gravitation, electromagnetism, Bohmian quantum effects, weak interactions and strong interactions is constructed, alongside a quantitative calculation system for orbital entanglement and spin entanglement. Under corresponding limiting conditions, the system rigorously reduces to General Relativity, Maxwell's equations, covariant Bohmian dynamics, electroweak Lagrangian and standard QCD Lagrangian. Key physical phenomena including parity violation of the weak interaction, color confinement, asymptotic freedom and quantum nonlocality are consistently explained via geometric structures. This study ultimately establishes a self-consistent unified field theory with a single ontology, which is compatible with classical field theory, realistic Bohmian quantum mechanics and the Standard Model of quantum field theory.
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.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
Background: The unification of General Relativity and Quantum Mechanics represents a fundamental challenge in theoretical physics due to their mutually exclusive conceptual foundations: continuous deterministic geometry versus probabilistic wave functions. While the de Broglie-Bohm pilot-wave theory offers a deterministic, trajectory-based alternative to the standard probabilistic framework, it introduces a non-local, ad-hoc "Quantum Potential" (Q) to account for phenomena such as interference and non-locality. The lack of a geometric origin for Q in standard spacetime remains a significant conceptual barrier to a fully unified theory. Purpose: This paper aims to resolve this interpretational crisis by proposing that the Quantum Potential is not a fundamental or mysterious external force, but rather a derived geometric artifact. We postulate that the physical universe is fundamentally a 4-dimensional Complex Manifold (ℂ4), extending the standard Riemannian spacetime manifold to include imaginary coordinates. Methods: To establish this geometric basis, we treat the quantum wave function not as an abstract probabilistic amplitude, but as a physical, holomorphic map describing a particle's deterministic trajectory through this complex spacetime. Utilizing the Cauchy-Riemann conditions, we demonstrate that the Bohmian amplitude (R) is strictly coupled to the particle's location in the imaginary dimension. We then analyze the dynamics of particles following geodesics within ℂ4. By projecting the complex geodesic equation of motion onto the observable real slice (ℝ4), we separate the real and imaginary components of the complex 4-velocity to observe the energy balance. Conclusions: Our derivation reveals that the Quantum Potential (Q) emerges naturally and is mathematically identical to the kinetic energy component associated with a particle's hidden motion in these imaginary dimensions. This formulation successfully recovers the predictions of the Schrödinger equation while removing the ad-hoc nature of Bohmian mechanics. Furthermore, it interprets quantum non-locality—such as the interference observed in the double-slit experiment—as purely local, deterministic geodesic motion around topological singularities in a curved complex manifold. Ultimately, this framework provides a unified geometric description wherein gravity is the curvature of real coordinates, electromagnetism is the torsion of imaginary coordinates, and quantum mechanics is inertial motion through this complex geometry.