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 define a class of Lorentz-invariant Bohmian quantum models for N entangled but noninteracting Dirac particles. Lorentz invariance is achieved for these models through the incorporation of an additional dynamical space-time structure provided by a foliation of space time. These models can be regarded as the extension of Bohm's model for N Dirac particles, corresponding to the foliation into the equal-time hyperplanes for a distinguished Lorentz frame, to more general foliations. As with Bohm's model, there exists for these models an equivariant measure on the leaves of the foliation. This makes possible a simple statistical analysis of position correlations analogous to the equilibrium analysis for (the nonrelativistic) Bohmian mechanics.
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 present a novel covariant relativistic guidance formula à la de Broglie-Bohm derived from conserved quantities associated with the Lagrangian of the Dirac equation in a 1+1-dimensional spacetime. In line with the standard guidance formula of Bohmian mechanics, this generalized guidance formula incorporates space-time derivatives of the Dirac spinor phases and consistently reproduces relativistic features such as mass-dependent trajectories and zitterbewegung. We formally show that this formulation explicitly reveals how the relativistic quantum potential affects the trajectories. Using numerical simulations, we compare these trajectories with those presented in the usual approaches. Finally, we show how the guidance formula naturally transitions to the standard de Broglie-Bohm theory in the non-relativistic limit and gives additional contributions in the expansions giving new insight into the zitterbewegung in pilot-wave theories.
In relativistic space-time, Bohmian theories can be formulated by introducing a privileged foliation of space-time. The introduction of such a foliation-as extra absolute space-time structure-would seem to imply a clear violation of Lorentz invariance, and thus a conflict with fundamental relativity. Here, we consider the possibility that, instead of positing it as extra structure, the required foliation could be covariantly determined by the wave function. We argue that this allows for the formulation of Bohmian theories that seem to qualify as fundamentally Lorentz invariant. We conclude with some discussion of whether or not they might also qualify as fundamentally relativistic.