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Relativistic Bohmian Physics, with a Microscopic Interpretation
Stephane H. Maes
Abstract
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.
____
1. Introduction
This paper is presenting a consistent relativistic QFT theory, reviewing and compiling the status of research, and view in the Physics community. The hybrid model, combining the different proposed ways for a covariant theory, is guided by QFT, SM and results of the Multi-fold Theory [1-199,258,259], where the latter is also a non-local theory [1,62,131,137,140,152,251] that seems to explain some open issues with the Standard Model (SM [2]) and the Standard Cosmological Model, and often recovers interesting insights, recovering different conventional results (See [8,252] and Appendix A).
With this paper we want to see what can be said about relativistic Bohmian Physics. This paper is not necessarily arguing that Bohmian Physics characterizes well our real universe..
The extension of Bohmian mechanics to the relativistic domain, and to QFT presents specific theoretical difficulties [200]. The non-local nature of the pilot-wave guidance equations inherently requires a temporal ordering of spacelike separated events [201]. Historically, this requirement was satisfied by introducing an arbitrary preferred foliation of spacetime. Furthermore, the standard Bohmian particle model struggles to accommodate the creation and annihilation events that are characteristic of quantum field theory [202].
In section2, this paper constructs a theoretical framework that is Lorentz-invariant and compatible with quantum field theory. The proposed synthesis couples the Tomonaga-Schwinger equation for the relativistic wave functional with a hypersurface Bohm-Dirac model [201].
To ensure Lorentz invariance without adding absolute spatiotemporal structure, the necessary foliation is covariantly determined by the universal wave function [204]. To address particle non-conservation, i.e., particle creation an annihilation events, 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 [202].
A hybrid model is adopted, assigning localized particle beables to fermions and continuous field beables to bosons [206]. 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 quantum field theory [207].
The theoretical framework proposed in the paper is a synthesis of established proposals developed within the specialized community researching Bohmian mechanics[3], complemented with a perspective to complete the model. While it is not part of standard, mainstream QFT, which generally relies on operational, non-realist interpretations, every component of the proposed framework (sections 2 and 3) represents the state-of-the-art in relativistic pilot-wave research, developed primarily between 1999 and 2014; all put together into one consistent framework[4] in this paper.
The core mathematical and conceptual components originate from the following well-known research efforts in foundational physics:
- The Hypersurface Bohm-Dirac Model (HBDM): The mechanism for generalizing Bohmian trajectories across arbitrary curved spacelike surfaces without relying on flat, equal-time hyperplanes was developed in [200].
- Bell-Type Quantum Field Theories (Stochastic Jumps): The stochastic Markov jump process used to model discrete particle creation and annihilation events via “minimal jump rates” on a Fock space was formulated for continuum theories in [201].
- Covariant Foliation: The critical idea of eliminating absolute spacetime structures by extracting the required foliation dynamically from the universal wave function itself was detailed in [202].
- Tomonaga-Schwinger Integration: The approach of replacing the standard functional Schrödinger equation with the covariant, many-fingered time Tomonaga-Schwinger equation to support Bohmian dynamics has been explored in the literature as a method to avoid preferred foliations [201].
- Hybrid Model (Fields and Particles): The assignment of discrete particle positions to fermions and continuous functional fields to bosons is a recognized and debated conceptual approach within the Bohmian QFT literature [203].
The paper brings together these highly specialized, disparate theoretical patches into a single, cohesive model or framework. The synthesis provides a unified solution to the historical problems of non-locality and particle creation in pilot-wave theories, relying entirely on established proposals.
In section 3, we discuss the experimental indistinguishability of this approach, with non-Bohmian relativistic quantum mechanics, or QFT, to see if indistinguishability is preserved as it is between Bohmian and non-Bohmian non-relativistic quantum mechanics [221].
Finally, in section 4, this framework is microscopically interpreted with the multi-fold theory [1-199,251,258,259]. Linking relativistic Bohmian (field) theory and the multifold theory was motivated by the observation that both, especially if a relativistic Bohmian model exists, are non-local theory [1,62,131,137,140,152,251]. In this context, the covariantly determined foliation emerges physically from a discrete, non-commutative spacetime concretized by 2D random walks of massless Higgs bosons[5] [1,8-10,22,32,62,72,131,137,152,170,176,181,251]. The multi-fold mechanism and the E/G conjecture (entanglement is gravity) provide a local, physical model for the non-local Bohmian guidance equations [1,22,24,131,137,152,251], 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 [1,23,33-35,63,68,72,116,150,131,137,152,177,251]. It turns out that the two theories are compatible, and multi-fold configurations can provide subtle microscopic interpretations to Bohmian models.
As such this paper is part of an upcoming set of studies that will focus on explaining conventional Physics behaviors, relying, in particular, on the notion of 2D random walks of massless Higgs bosons [1,8-10,22,32,62,72,131,137,152,170,176,181,251], as we already started in [199]. More will be published and can be tracked at [8].
Appendices A through C discuss respectively: overview of the multi-fold theory, the notion of quantum non-equilibrium, and the modeling[6] of neutrinos in relativistic quantum Bohmian QFT.
2. Covariant Foliations and Stochastic Jump Processes: A Unified Framework for Relativistic Bohmian Quantum Field Theory
2.1. Introduction
Bohmian mechanics, also known as the de Broglie-Bohm pilot-wave theory, provides a deterministic formulation of quantum phenomena, wherein the statistical uncertainties of standard quantum mechanics are epistemic rather than fundamental [208].
In the non-relativistic regime, the theory proposes an objective configuration of particle positions
(1)
evolving in time according to a guidance equation determined by the wave function
[202]. The wave function itself evolves according to the standard Schrödinger equation.
This framework resolves the quantum measurement problem by ensuring that particles always possess definite configurations, independent of observation. The standard textbook collapse rule becomes a consequence of the deterministic dynamics when considering the interaction between a measured subsystem and the macroscopic environment [209].
Despite its conceptual clarity, the generalization of Bohmian mechanics to relativistic quantum field theory encounters two primary obstacles.
First, the non-relativistic guidance equation is explicitly non-local. The velocity of a given particle depends instantaneously on the positions of all other particles in the system. In a relativistic spacetime, the concept of instantaneous dependence requires a global definition of simultaneity, appearing to necessitate a preferred spacetime foliation [210]. The introduction of such an absolute structure violates the principles of fundamental relativity. This leads to models that are phenomenologically adequate but fundamentally non-relativistic [211].
Second, standard Bohmian mechanics is formulated for a fixed number of point particles. Relativistic QFT is fundamentally characterized by processes of particle creation and annihilation, which are phenomena that defy the continuous, unbroken trajectories of the traditional de Broglie-Bohm interpretation [202].
To circumvent these limitations, we propose a unified theoretical architecture for a relativistic Bohmian QFT that synthesizes several theoretical developments.
- To address the foliation problem, the Hypersurface Bohm-Dirac model is utilized, wherein the arbitrary fixed foliation is replaced by a foliation covariantly determined by the quantum state itself [201].
- To maintain covariance at the level of quantum state evolution, the standard functional Schrödinger equation is replaced by the Tomonaga-Schwinger equation, which propagates the state vector over arbitrary spacelike hypersurfaces [203].
- To account for variable particle numbers, the deterministic continuous dynamics are augmented by a stochastic Markov jump process defined on Fock space, following the Bell-type quantum field theories [202].
- Finally, in Section 4, the multi-fold theory [1-199,251,258,259] is introduced to provide a fundamental, discrete physical mechanism, and microscopic interpretation, for these phenomena/model choices [1,6,8-10,22,23,27,32-34,62,63,72,116,124,131,137,152,170,176,181].
2.2. Standard Relativistic Extensions and Their Limitations
The most direct attempt to construct a relativistic Bohmian theory involves adapting the relativistic wave equations, such as the Dirac equation for fermions or the Klein-Gordon equation for bosons, to the pilot-wave formalism [213].
2.2.1 The Single-Particle Dirac Equation
For a single relativistic spin-1/2 particle, the evolution of the wave function is governed by the Dirac equation. Setting the reduced Planck constant ℏ, and the speed of light to unity, the equation reads
(2)
[202]. The wave function
takes values in the four-dimensional spinor space
.
A natural candidate for the deterministic guidance equation utilizes the conserved Dirac probability current.
The velocity field for the particle is given by
, (3)
where the probability current four-vector is
(4)
[200]. This single-particle equation is strictly covariant and successfully defines a sub-luminal, deterministic trajectory for a single electron. The probability density
(5)
is conserved, satisfying the continuity equation
, (6)
However, the fundamental incompatibility with relativity emerges immediately when the system is expanded to multiple interacting particles.
2.2.2 The Multi-Particle Dirac Equation, and Non-Locality
For an N-particle system, the wave function
takes values in the tensor product space
.
The corresponding multi-particle Dirac equation must account for the spatial coordinates of all N particles. The guiding equation for the k-th particle in this configuration space is formulated as:
(7)
Where ak are the Dirac matrices operating on the spinor index of the k-th particle. The critical issue lies in the evaluation of the right-hand side of this equation. The velocity of the k-th particle at a specific time t depends on the value of the many-body wave function evaluated at the exact positions of all N particles at that same time t.
In a relativistic context, the notion of “same time” for spatially separated particles is frame dependent. If the guidance equation is evaluated in one specific Lorentz frame, the trajectories will generally differ from those calculated in another Lorentz frame [203]. To achieve a unique set of trajectories, it is necessary to select a single, privileged foliation of spacetime, i.e., a sequence of equal-time hyperplanes, and declare that the trajectories defined by this specific foliation are the true physical trajectories [210].
2.2.3 The Problem of Absolute Structure
The mathematical introduction of a preferred foliation solves the ambiguity of the trajectories but introduces a conceptual problem. According to the classification of physical structures in [211], a theory is seriously Lorentz-invariant only if it does not contain any absolute structures beyond the Lorentz metric itself.
An absolute structure is a geometric object that influences the dynamics of the system, but it is not influenced by the system in return [211].
If a preferred foliation is added to a theory simply to evaluate the non-local guidance equations, it acts as a fixed background structure. While the statistical predictions of the theory remain Lorentz invariant due to the equivariance of the probability distribution, the underlying model relies on an undetectable, absolute Newtonian time. Such a formulation means that the theory is not relativistic in a fundamental sense. To resolve this, the foliation must be transformed from an absolute geometric background into a dynamical entity.
2.3. The Hypersurface Bohm-Dirac Formulation
To construct a framework that supports non-local interactions without relying on flat equal-time hyperplanes, the Hypersurface Bohm-Dirac model generalizes the Bohmian dynamics to arbitrary curved foliations of spacetime [201].
2.3.1 Mathematical Definition of the Model
The Hypersurface Bohm-Dirac model assumes a foliation
of spacetime into a one-parameter family of smooth, spacelike hypersurfaces
,where s is a scalar parameter [205]. Each point x on a hypersurface
has a future-oriented, timelike unit normal vector field denoted by n(x).
The state of the system is given by the N-particle Dirac wave function, which solves the multi-time Dirac equations, or a suitable constraint equation ensuring consistency across the foliation. The physical configuration is represented by the N-path
, (8)
consisting of the worldlines of the N particles [214].
The law of motion for the configuration is defined by constructing a multi-particle current tensor. The current for the k-th particle, generalizing the single-particle Dirac current, is defined as:
, (9)
where
(10)
is the unit normal vector evaluated at the position of the i-th particle on the specific hypersurface
. The actual particle trajectories are defined by the Hypersurface Bohm-Dirac tangency condition. The tangent vector to the worldline of the k-th particle, denoted
, must be proportional to the current jk evaluated at the intersection points of all N worldlines with the hypersurface S [201]:
(11)
2.3.2 Positivity and Determinism
For the Hypersurface Bohm-Dirac model to be physically viable, the trajectories must be strictly future-directed, i.e., timelike or null, meaning that the time component of the velocity must be positive. This requirement is satisfied by the mathematical structure of the current jk.
By considering the action of a suitable local Lorentz transformation on the operator
, for an arbitrary timelike unit vector n, one can transform n into the standard rest frame vector (1,0,0,0) [201]. In this frame, the operator reduces to
, which is the identity matrix. Consequently, the operator
is strictly positive definite [201].
This mathematical property guarantees that the currents jk are always future-directed timelike or null vectors, ensuring that the Bohmian particles never travel backward in time, or never exceed the speed of light relative to the local geometry.
2.4. Covariant Determination of the Time Foliation
The Hypersurface Bohm-Dirac model provides the mathematical approach to define non-local trajectories across arbitrary spacelike surfaces [214]. However, if the foliation
is specified externally as an absolute structure, the theory remains fundamentally non-relativistic.
The solution to this conceptual problem is to derive the foliation directly from the wave function itself.
2.4.1 Extracting Geometry from the Wave Function
Instead of positing the foliation
as an independent, fixed background, it can be defined as a functional of the universal wave function,
(12)
Because the wave function evolves according to a Lorentz-covariant physical law, any geometric structure mathematically derived from it will automatically transform covariantly under Lorentz transformations. This strategy ensures that the theory does not contain any absolute structures other than the spacetime metric.
The foliation becomes a dynamical variable, inextricably linked to the quantum state of the matter fields. The underlying physical microscopic interpretation is at this stage a postulate. It will be addressed and potentially explained in section 4.
2.4.2 Constructing the Covariant Vector Field
To derive a foliation from the wave function, one must first construct a covariant timelike vector field. A natural candidate is the expectation value of the total four-momentum density or the total probability current. Given a generic many-body wave function
, the total current vector field
can be defined as the expectation value of the local current operator
:
(13)
For a system of Dirac particles, the local current operator ensures that the expected vector field
is strictly timelike. Once the time like vector field
is established, it can be normalized to produce a future-directed unit vector field [204]:
(14)
2.4.3 Defining the Leaves of the Foliation
The normalized vector field
specifies a unique temporal direction at every point in spacetime, dictated entirely by the local properties of the quantum state. The required foliation
is then constructed such that its leaves (the spacelike hypersurfaces
) are everywhere orthogonal to
.
If the vector field
has zero vorticity, it is Frobenius integrable, meaning it admits a family of exact orthogonal hypersurfaces. If the vector field has non-zero vorticity, and is not strictly hypersurface-orthogonal, one can still extract a unique foliation by utilizing the integral curves of
to define the level sets of a scalar phase function, or by imposing a minimal-surface condition mapped to the boundary values of the universe.
Figure 1: Covariant Spacetime Foliation and Hypersurface Bohm-Dirac Trajectories Description. The figure shows a visual representation of a (2+1)-dimensional spacetime section illustrating the covariantly determined foliation
. The spacelike hypersurfaces
(colored blue) flex according to the dynamically changing total probability current
. The deterministic worldlines of two fermions (red and green curves) are shown intersecting the hypersurfaces, with their local tangent vectors aligned with the Hypersurface Bohm-Dirac current jk defined by the normal vectors n(xi) of the foliation.
By substituting the covariantly derived normal vector
into the Hypersurface Bohm-Dirac tangency condition, the complete set of equations governing the particle trajectories becomes strictly Lorentz invariant [201]. The dynamics of the particles are determined by the wave function, and the spatial non-locality, required to evaluate the current jk is mediated along the dynamically generated surfaces of
.
2.5. Functional Evolution via the Tomonaga-Schwinger Equation
In standard non-relativistic quantum field theory, the state is represented in the functional Schrödinger picture by a wave functional
. This functional encodes the probability amplitude for a specific field configuration across all space at a fixed time t. The evolution of this functional is governed by the functional Schrödinger equation:
(15)
This evolution equation is fundamentally noncovariant. It relies on a global time parameter t, implying an integration over a flat, equal-time hyperplane. If the Bohmian particles are to be guided by a wave function evaluated on the curved, covariantly determined hypersurfaces of
, the wave function itself must be definable on those surfaces.
2.5.1 The Generalization to Arbitrary Hypersurfaces
To align the quantum state evolution with the covariant foliation, the functional Schrödinger equation must be replaced by the Tomonaga-Schwinger equation [203]. The Tomonaga-Schwinger framework operates in the interaction picture, or a suitably defined multi-time Heisenberg picture. It replaces the global time coordinate t with a functional dependence on an arbitrary spacelike hypersurface
. The quantum state is designated as a functional vector
.
The dynamical evolution of the state under an infinitesimal local deformation of the hypersurface in the normal direction at a point x is governed by the following functional differential equation:
(16)
where
denotes the local deformation of the hypersurface at spacetime point x, and
is the Hamiltonian density operator.
2.5.2 Micro causality and Integrability
For the Tomonaga-Schwinger functional evolution to be physically consistent and mathematically integrable, the order in which local deformations are applied must not affect the final quantum state, provided the deformations occur at spacelike separated points. Foliation independence requires that the second functional derivatives commute:
(17)
This mathematical requirement dictates that the Hamiltonian density operators must commute at spacelike separations. This is the standard micro causality condition of quantum field theory:
(18)
The utilization of the Tomonaga-Schwinger equation fulfills a dual purpose within the proposed framework. First, it ensures that the wave functional can be evaluated consistently and covariantly on any arbitrary leaf of the dynamically determined foliation
[203]. Second, it provides the necessary mathematical foundation to calculate the local probability currents and stochastic jump rates required for the Bohmian configuration dynamics.
2.6. Primitive Model: Beables for Bosons and Fermions
To generate a complete Bohmian quantum field theory, one must specify the primitive model [206]. The primitive model consists of the local beables, which are the fundamental mathematical entities that represent actual physical configurations in spacetime. A choice must be made regarding whether these fundamental entities are discrete point particles or continuous fields.
Our choices here are guided by considerations of scales and behavior differences between fermions and bosons in a multi-fold universe (In section 4 and appendices, we present the consistency of the framework as recovering these guiding principles). However, these can also be seen just a selection of what works out of the arsenal of options developed in the literature [200-249].
2.6.1 Field Beables for Bosonic Degrees of Freedom
Bosonic degrees of freedom, such as the electromagnetic field, or the Higgs field [201], are most naturally represented by continuous field configurations rather than discrete particles. For a scalar boson field[7] governed by the Klein-Gordon equation, the fundamental local beable is the scalar field configuration
. The guidance equation for this field is derived from the functional gradient of the phase of the wave functional. Expressing the wave functional in polar form as
, (19)
where
is the generalized parameter labeling the leaves of the foliation, the deterministic evolution of the scalar field is given by:
(20)
For gauge bosons, such as the electromagnetic vector potential
, a similar functional guidance equation applies. Following minimalist models, the beables for quantum electrodynamics are restricted to the transverse degrees of freedom of the vector potential,
, to maintain strict gauge invariance [207]. The wave functional
satisfies the functional Tomonaga-Schwinger equation involving the Hamiltonian of the specific gauge theory.
2.6.2 Particle Beables for Fermionic Degrees of Freedom
Model TypeBoson RepresentationFermion RepresentationDynamicsPhenomenological ChallengesDirac Sea ModelField BeablesInfinite Particle SeaPurely DeterministicInfinite unobservable mass/charge densities.Minimalist ModelField BeablesNone (Emergent from Bosons)Purely DeterministicMatter is not fundamental.Bell-Type Model (Proposed)Field BeablesFinite Particle ConfigurationsDeterministic Drift + Stochastic JumpsRequires rigorous stochastic measure theory on Fock space.Table 1: Comparing model options to model particles. As a proposal, we recommend the Bell-type approach.
For fermionic degrees of freedom, point particles remain the conventional choice [201]. Formulating a continuous field model for fermions requires utilizing anti-commuting Grassmann variables, which lack a clear physical interpretation as classical fields in ordinary spacetime [215]. Two dominant theoretical approaches exist to model fermion particles in a relativistic quantum field theory:
- The first option is the Dirac Sea model. This deterministic model interprets the quantum vacuum as an infinitely dense sea of actual particles occupying negative energy states [216]. Particle trajectories, including those of the unobservable particles in the Dirac sea, evolve deterministically according to the generalized Hypersurface Bohm-Dirac guidance equations [218]. While this preserves the continuity of trajectories without requiring stochastic jumps, the postulation of an unobservable, infinite density of actual particles raises physical difficulties, particularly regarding mass and charge regularization [218].
- The second option, adopted in this framework, relies on Bell-type quantum field theories [202]. Originating from lattice models, and extended to the continuum, this approach incorporates explicit creation and annihilation events [202]. Fermions are modeled as point particles that trace continuous trajectories guided by the Hypersurface Bohm-Dirac equation, but these deterministic segments are interrupted by discontinuous stochastic jumps [202]
The proposed framework utilizes a hybrid model, defining continuous functional fields for bosons and stochastic point particles for fermions[8]. This combination aligns with standard QFT, where fermions interact via the exchange of gauge bosons, generating the creation and annihilation processes observed in high-energy physics. Neutrinos are handled in Appendix C.
2.7. Stochastic Dynamics on Fock Space
To accommodate the non-conservation of particle number implied by the interaction terms in the Hamiltonian density, the deterministic motion of the configuration must be supplemented by discontinuous jumps [202]. The configuration space is no longer a fixed 3N-dimensional space, but rather the disjoint union of configuration sectors corresponding to different particle numbers, representing the spatial equivalent of the Fock space [202].
A generic configuration q consists of a specific number of particles located at specific spatial coordinates on the hypersurface
, alongside the continuous field configuration for the bosons. The total evolution of the physical state is defined as a piecewise deterministic Markov process, combining the continuous flow (the Hypersurface Bohm-Dirac drift), and discrete jumps representing creation or annihilation events [203].
2.7.1 The Master Equation
The probability distribution
for the configuration q across the entire Fock space is governed by a master equation. This equation, often referred to as a Kolmogorov forward equation, must account for both the deterministic divergence of the probability current within a specific sector and the stochastic transition rates between different sectors.
Let
denote the transition rate, defined as the probability density per unit parameter time to jump from configuration q’ to configuration q. The temporal evolution of the probability density
with respect to the foliation parameter t is described by the integro-differential equation:
(21)
In this equation,
represents the continuous flux associated with the deterministic guidance equations generated by the free Hamiltonian [202]. The integral term on the right-hand side represents the net gain and loss of probability density due to stochastic jumps into and out of the configuration state q.
2.7.2 Derivation of the Minimal Jump Rates
The transition rates
chosen as free phenomenological parameters. They must be constrained by the fundamental interaction Hamiltonian
to ensure that the stochastic process strictly reproduces the exact quantum statistics of the underlying field theory. To derive these rates, the time derivative of the target quantum probability density
is evaluated using the full Hamiltonian [202]:
. (22)
The free Hamiltonian H0 generates the continuous deterministic velocity field v(q) via the standard continuity equation. The interaction Hamiltonian HI, which contains the creation and annihilation operators coupling the fermionic and bosonic sectors, generates the discrete jump dynamics. According to the Schrödinger evolution, the variation of the probability density due exclusively to the interaction term is given by:
(23)
To map this quantum probability variation to the classical stochastic master equation, the gain-loss integral of the master equation must equal the interaction derivative. A sufficient condition for solving this integral equation is to equate the integrands directly. However, mathematical consistency dictates that transition rates must be non-negative real numbers
. (24)
Defining
(25)
as the positive part of a real number x, the minimal jump rate formula is derived as [202]:
(26)
This prescription is termed “minimal” because it mathematically ensures that for any given pair of configurations q and q’, stochastic jumps only occur in one direction at any given instant. If the imaginary term is positive, probability flows from q’ to q, and the rate is non-zero. If the term is negative, the required jump is in the reverse direction q to q’, and the rate
correctly evaluates to zero, leaving the reverse rate
to handle the transition. This formulation completely fixes the stochastic dynamics of the creation and annihilation events at the vertices determined by the interaction Hamiltonian.
Figure 2: Bell-type Stochastic Jump Process in Spacetime. It shows A schematic of a particle creation and annihilation event as modeled by a Bell-type stochastic Markov process. A single fermion worldline (blue) propagates deterministically until a stochastic jump occurs on a specific hypersurface
. At this interaction vertex, governed by the minimal jump rate
, the particle annihilates, and a fermion-antifermion pair (red and green worldlines) is created, representing a discontinuous jump from the 1-particle sector to the 3-particle sector in Fock space.
2.8. The Equivariance Theorem
For the proposed Bohmian QFT to be empirically equivalent to standard QFT, it must reproduce the Born rule statistics across all times and across all leaves of the foliation [207].
This statistical compatibility is known as equivariance. If the probability distribution of the particle and field configurations
equals the modulus squared of the wave functional
on an initial spacelike hypersurface
, the property of equivariance guarantees that
(27)
on any subsequent hypersurface evaluated by the Tomonaga-Schwinger equation. The proof of this theorem requires demonstrating consistency for both the continuous and stochastic components of the dynamics.
2.8.1 Continuity and Deterministic Equivariance
For the purely deterministic segments of the particle trajectories, which are governed by the free Hamiltonian H0, equivariance requires that the probability density satisfies a continuity equation compatible with the continuous unitary evolution of the wave functional. Using the Hypersurface Bohm-Dirac model, the multi-particle probability current defined in Section 2.3.1 is strictly divergence-free in the absence of interactions. This follows directly from the application of the free Dirac equation and its adjoint:
(28)
To relate this four-divergence to the evolution of the probability density across the foliation, the equation must be split into components normal to and tangent to the hypersurface Ss , where s is the scalar parameter labeling the leaves. The continuity equation along the foliation parameter s is expressed as:
(29)
Where gk is the geometric area-expansion factor arising from the normal flow of the metric between adjacent hypersurfaces, and
is the projection of the velocity field onto the local tangent space of the hypersurface
. The mathematical integration of this equation proves that the crossing probability density
(30)
is strictly preserved along the deterministic trajectories bridging the leaves of the foliation. Therefore, the deterministic drift maintains the Born rule [201].
2.8.2 Stochastic Equivariance on Fock Space
When the interaction terms HI are involved, the state vector is subjected to the Markov jump processes [202]. To prove equivariance in the presence of particle creation and annihilation, it must be shown that the master equation governing the jumps exactly reproduces the non-unitary disruption of the wave functional’s modulus squared within a given sector. The master equation defined in Section 2.7.1 is explicitly structured to satisfy this requirement.
By assuming the initial equilibrium condition
(31)
and substituting this into the minimal jump rate equation, one obtains:
(32)
The notion of equilibrium in Bohmian Physics is further discussed in Appendix B.
To evaluate the net gain and loss integral in the master equation, one substitutes this expression and its reverse counterpart. Utilizing the fundamental algebraic identity for the positive parts of real numbers,
, (33)
the right-hand side of the master equation simplifies algebraically. The resulting expression matches exactly the temporal variation of
induced by the interaction Hamiltonian HI , as derived from the Schrödinger equation.
Because the total divergence of the probability distribution maps identically to the unitary evolution of the wave functional, the ensemble equivariance theorem holds over the covariantly defined foliation. The minimal jump rates algorithmically mediate transitions across Fock space, replicating the creation and annihilation events mandated by quantum gauge interactions without violating the statistical predictions of the standard regularized theory [202].
2.9. Phenomenological Implications and “Serious” Relativity
The theoretical construction presented in this paper satisfies several rigorous physical criteria. By elevating the Tomonaga-Schwinger equation to evaluate states on arbitrary spacelike surfaces, and by allowing the wave functional itself to define the unique covariant foliation
governing the Hypersurface Bohm-Dirac tangency conditions, the framework escapes the usual criticism that pilot-wave theories require a phenomenologically absolute Newtonian time. Because the structure
is a dynamical entity dependent entirely on the Lorentz-invariant wavefunction and the invariant spacetime metric
, the theory lacks any absolute structures [211]. It is therefore fundamentally covariant. No preferred inertial frame exists a priori. The effective preferred frame is an emergent property of the specific initial conditions of the universe’s wave function, analogous to how the distribution of matter in general relativity defines an effective cosmic rest frame without violating fundamental diffeomorphism invariance. Section 4, justify and interpret the physicality of
.
Furthermore, the introduction of macroscopic contextuality provides a law-like, top-down stochastic mechanism to explain quantum decoherence [209]. In these extended models, such as the Contextual Bohmian Quantum Field Theory (CBQFT) [209], macroscopic context variables, such as detector configurations, or symmetry-breaking thermal sectors, act as a dynamical background, that modulates the Bell-type jump rates on the single Fock space [209]. Such integrations securely anchor the macroscopic arrow of time in the fundamental non-equilibrium stochastic transitions across the configuration space, employing a hylomorphic loop that links quantum events directly to contextual structure [209].
Finally, because the jump rates rely on a carefully regulated stochastic projection defined locally upon the hypersurfaces, the model avoids generating ultraviolet singularities beyond those natively present in the regularized QFT. The combined model allows fermions to exist as identifiable local beables (stochastic point particles) while mediating forces via explicit continuous boson field configurations [206]. This eliminates the necessity of postulating an unobservable, infinitely dense Dirac sea [216], replacing it with localized creation and annihilation dynamics governed by rigorously derived transition probabilities.
3. The Foundations and Limits of Observational Equivalence: Bohmian Mechanics, Relativistic Foliations, and Field Models
3.1 Observations and The Model of de Broglie-Bohm Theory
The mathematical formalism of orthodox, aka standard or conventional, quantum mechanics, as presented in standard textbooks, functions primarily as an operational recipe for calculating the probabilities of measurement outcomes [221]. It deliberately refrains from providing a direct, objective description of physical processes occurring in the absence of an observer [221]. This conceptual gap is bridged by Bohmian mechanics, also known as the de Broglie-Bohm pilot-wave theory, or the causal interpretation, which offers a realistic, deterministic, and observer-independent account of microscopic reality [221].
As we saw, within this framework, a physical system is not described solely by its wave function; rather, its physical state is represented by a dual structure: the wave function Y(q,t), and the actual configuration
(34)
of the system, where each Qi(t) represents the precise spatial position of a point-like particle [221]. The wave function, postulated to belong to a standard square-integrable Hilbert space, evolves continuously according to the linear Schrödinger equation [221]:
(35)
Simultaneously, the actual positions of the particles evolve continuously according to a first-order differential guiding equation, which represents the simplest Galilean-covariant law of motion compatible with the Schrödinger dynamics [221]:
(36)
In this manner, the pilot wave guides the configuration of the universe through a deterministic choreography [223]. The particles are directed by the local phase gradient of the wave, ensuring that they avoid regions of destructive interference and accumulate in regions of constructive interference [225]. By integrating actual configurations into the fundamental model, the theory operates as a “quantum theory without observers” [221] [9].
It successfully resolves the measurement problem, eliminates the need for wave function collapse as a physical postulate, and bypasses orthodox concepts such as wave-particle complementarity, unsharp physical values, or the necessity of human consciousness to register physical events [221].
3.2 The Mechanism of Observational Equivalence in Non-Relativistic Space
The observational indistinguishability of Bohmian mechanics from standard quantum mechanics in non-relativistic regimes is not an accidental coincidence. It is a consequence of the Quantum Equilibrium Hypothesis (QEH) [222].
In standard quantum mechanics, the Born rule is treated as an axiomatic postulate that maps wave functions directly to probabilities [225]. In de Broglie-Bohm theory, the connection between probability density and the wave function has the status of a theorem, emerging from the underlying deterministic dynamics of the system [221].
The basis for this equivalence is the property of equivariance [222]. If the initial configuration Q(t0) of an ensemble of systems is chosen at random with a probability density
that matches the squared wave function
, then the configuration
at any subsequent time t will remain distributed according to
[222]. This dynamical compatibility is proved by the continuity equation, which is derived directly from the Schrödinger equation and the guiding equation [222]:
(37)
(38)
, where
(39)
is the Bohmian velocity field [224]. Since the physical configuration density
and the probability density
satisfy the exact same continuity equation and evolve under the same velocity field, any initial statistical alignment is preserved for all times [224].
To analyze localized subsystems, Bohmian mechanics utilizes the conditional wave function [222]. If the actual configuration of the universe is split into
, (40)
where X is the configuration of the subsystem under study and Y is the configuration of its environment, the conditional wave function y(x) of the subsystem is defined by inserting the actual environmental configuration Y(t) into the universal wave function
[222]:
(41)
As the particles in the environment evolve, the actual configuration Y(t) selects an effective, localized branch of the universal wave function [228]. When the subsystem interacts with a measurement apparatus, the actual configuration of the joint system-apparatus enters one of the non-overlapping wave packets of the superposition [224]. The empty wave packets, although mathematically present, become dynamically irrelevant because the actual configuration has moved far away from them in configuration space, preventing any future interference [225]. This branch selection physically derives the “collapse of the wave function” without needing to introduce collapse as an independent dynamical law [222].
This construction explains why macroscopic observations are identical across both interpretations [222]. Every experimental record, whether a pointer orientation on a dial, an ink pattern on paper, or a localized spot on a detector screen, is ultimately a spatial configuration of matter [222].Because Bohmian mechanics tracks actual configurations, and because these configurations are distributed according to the
quantum equilibrium, the predictions of Bohmian mechanics for any experiment are identical to those of the orthodox formalism [222].
Under the quantum equilibrium hypothesis, any measurement of a generalized observable can be mathematically represented by a positive-operator-valued measure (POVM) O(dz) acting on the wave function y, such that the probability distribution of the result Z is given by [223]:
(42)
This formulation reveals that properties like spin or momentum are not intrinsic localized variables carried by the particles [224]. Instead, they are contextual properties of the wave function in relation to the specific experimental configuration, emerging deterministically during the measurement process itself [223].
3.3 Extension of Equivalence to Relativistic Spacetime
Extending Bohmian mechanics to relativistic spacetime requires addressing a fundamental tension: the “spirit” of relativity is local, four-dimensional spacetime geometry, whereas the “spirit” of quantum mechanics is nonlocal entanglement [223]. Because the velocity of any single Bohmian particle depends on the positions of all other particles in the universe, the guiding equation requires a mechanism to determine which points on different particle world lines correspond to the same instant of time [232]. In non-relativistic space, this simultaneity is provided by absolute time, but in relativistic space, there is no absolute temporal coordinator [232].
To resolve this, relativistic Bohmian models introduce a preferred, spacelike foliation of spacetime into spacelike hypersurfaces [231]
. (43)
This foliation provides a physical coordinate system for calculating nonlocal interactions [233]. A priori, introducing a preferred reference frame appears to violate Lorentz invariance, suggesting a conflict with fundamental relativity [233]. However, this frame remains completely unobservable at the statistical level [227]. If the particles are distributed in accordance with the quantum equilibrium distribution, the statistical predictions of the relativistic Bohmian model are identical to those of standard relativistic quantum mechanics, which are Lorentz-invariant [227]. Thus, the preferred foliation cannot be detected experimentally, preserving observational equivalence even though fundamental Lorentz invariance is broken at the subquantum level [227].
To achieve Lorentz invariance at a fundamental level, the required foliation can be extracted covariantly from the wave function itself (
(44)
) [227] [10].
If the foliation is defined by a covariant vector field generated by the wave function, such as a conserved, timelike probability current
, the foliation ceases to exist as an independent, absolute spacetime structure [233]. This approach is realized in the Hypersurface Bohm-Dirac Model (HBDM), which describes a system of N Dirac particles using a multi-time wave function
[233]. The particles’ actual world lines Xk(s) are guided by the multi-particle Dirac current evaluated at the intersections of the world lines with a common leaf
of the foliation [233]:
(45)
This dynamical law is covariant under the Lorentz group [233].
The mathematical consistency of the HBDM is robust under complex geometric conditions [b20]. When the foliation is determined by the law:
(46)
, where
(47)
is the normal unit one-form of the foliation (representing equal timelike distance from a given initial hypersurface), the spacelike leaves generically develop geometric kinks [236]. Even with these kinks, or under degenerate foliations where multiple leaves overlap in a localized region, the particle trajectories remain mathematically well-defined, and the
probability distribution remains equivariant [236]. This geometric stability ensures that the HBDM remains empirically equivalent to standard relativistic quantum mechanics across all physical scenarios [236].
Alternative relativistic approaches have also been investigated to reconcile nonlocality with spacetime geometry [232]. One notable example is the “Opposite Arrows of Time Model,” which attempts to resolve the tension by introducing bidirectional causal influences, though it remains less mathematically developed than the HBDM [232].
3.4 Empirical Equivalence in Quantum Field Theory (QFT)
As we saw previously, the extension of Bohmian mechanics to the domain of Quantum Field Theory (QFT) is essential to account for relativistic phenomena such as particle creation, particle annihilation, and infinite degrees of freedom [237]. To maintain observational equivalence, Bohmian QFTs are formulated by defining a primitive model (local beables) in physical space, and deriving a guidance law, that preserves the
distribution of the corresponding regularized QFT [202]. There are two primary approaches to constructing these field-theoretic models [237]. They are discussed in the next subsections.
3.4.1 Fock Space and Particle Model (Bell-Type QFTs)
Bell-type QFTs, originally proposed by John S. Bell on a spatial lattice and subsequently extended to the continuum[11], take the concept of point-like particles seriously in the QFT regime [202]. At any given time t, the system has a definite, actual number of particles N(t) at precise spatial positions, meaning the actual configuration Qt resides in a single n-particle sector of a multi-sector configuration space [202]:
(48)
The universal wave function
is represented by a state vector in a Fock space and can exist in a superposition of different particle numbers [202]. The dynamics of the actual configuration Qt are governed by two distinct laws of motion [202]:
- Continuous Bohmian Motion: Between creation and annihilation events, the particles move smoothly within their current sector
, guided by a deterministic guiding equation derived from the spatial current of the free Hamiltonian [230].
- Stochastic Jumps: To account for the non-conservation of particle numbers, the configuration jumps stochastically to a different sector of [230]
.
These jumps occur at random times and are governed by transition rates Tnm (from configuration m to n) designed to be the minimal jump rates necessary to preserve the equivariance of the |Y|2 Fock-space distribution [202]. On a lattice, the minimal transition rate is formulated as [227]:
(49)
where the probability current Jnm is defined by [227]:
(50)
And
(51)
is the probability of being in configuration m [227]. In the continuum, these rates correspond to stochastic creation and annihilation events where physical particle world lines begin and end [230]. Because the jump rates are constructed to preserve the
distribution, a law of large numbers ensures that the empirical frequencies of measurement outcomes in a typical Bohmian universe are identical to the statistical predictions of standard QFT [230].
3.4.2 Functional Wave Representation and Field Model
An alternative Bohmian approach to QFT utilizes a wave functional representation, which suggests a model of classical field configurations rather than point particles [237].
For bosonic systems, such as scalar or electromagnetic fields, the state of the universe is described by a wave functional
evolving via a functional Schrödinger equation [202]. The local beable is an actual classical field configuration
, that evolves deterministically over physical space, guided by the phase S of the wave functional [227]:
(52)
Fermionic degrees of freedom can be integrated into this field model using the Minimalist Model developed by Ward Struyve and Antony Westman [227]. In this model, there is no physical particle model for fermions [227]. Instead, the fermionic degrees of freedom are represented solely by the wave function, which is mapped to a bosonic field (such as the electromagnetic vector potential A(x,t), and a continuous charge density
defined on physical space [227]. The wave function is written as a function of the bosonic field
, (53)
where f labels the fermionic states, and the equilibrium distribution is defined as [227]:
(54)
The actual charge density is calculated dynamically from the wave function and the actual bosonic field configuration [227]:
(55)
Where
is the charge density operator [227]. This formulation provides a continuous, field-based model that remains fully indistinguishable from standard QFT predictions, demonstrating that empirical equivalence can be maintained without postulating classical particle trajectories for fermions [227].
3.5 Subquantum Physics: Violating the Boundaries of Indistinguishability
While Bohmian mechanics is observationally equivalent to standard quantum mechanics under the assumption of quantum equilibrium, the two interpretations are not mathematically identical [226].
The pilot-wave framework permits the existence of quantum non-equilibrium states, where the actual distribution of particle configurations deviates from the Born rule [226]:
. (56)
The existence of quantum non-equilibrium would render Bohmian mechanics empirically distinguishable from, and falsifiable against, standard quantum mechanics [226].
In non-equilibrium states, reviewed in Appendix B, several key principles of orthodox quantum mechanics are violated, as detailed in the following table:
Physical PrincipleQuantum Equilibrium
(ρ=∣ψ| 2)Quantum Non-Equilibrium
(not equal)Signal LocalityStrictly respected; faster-than-light signaling is impossible [241]Violated; enables instantaneous “signal nonlocality” at the statistical level [226]Uncertainty PrincipleAdhered to; precise joint measurements of conjugate variables are forbidden [224]Violated; allows subquantum measurements that bypass uncertainty limits [226]Quantum EntanglementRequires a secondary classical channel to decode nonlocal correlations [243]Functions as an independent, instantaneous communication channel [243]Born Rule ValidityUniversally confirmed across all laboratory measurements [226]Violated; yields anomalous statistical distributions in experiments [226]Table 2: Quantum Equilibrium vs. Quantum Non-equilibrium Bohmian Physics (See also appendix B).
To explain why our current universe appears to be in a state of strict quantum equilibrium, physicists have developed a subquantum statistical mechanics [245]. By constructing a coarse-grained subquantum H-function analogous to the Boltzmann H-function of classical thermodynamics, one can demonstrate that non-equilibrium distributions naturally relax toward equilibrium over time [226]:
(57)
Under a subquantum H-theorem, this quantity is non-increasing (
(58)
), showing that the Born rule represents a stable, maximum-entropy statistical attractor [240]. Consequently, standard quantum mechanics can be understood as an equilibrium phenomenology, and its adherence to relativistic signal-locality is a direct consequence of this equilibrium state masking the underlying subquantum nonlocality [241].
This thermodynamic view suggests that the indistinguishability of Bohmian mechanics is historically contingent [240]. If the early universe began in a state of quantum non-equilibrium, physical signatures of Born rule violations might have survived quantum relaxation [240]. Prominent cosmological and astrophysical regimes proposed for detecting these relic deviations include (This is revisited in Appendix B):
- The Cosmic Microwave Background (CMB): Non-equilibrium fluctuations of the primordial scalar inflaton field prior to inflation could have bypassed complete relaxation [240]. This would imprint anomalous power-spectrum signatures on the CMB anisotropies, which could be observed in cosmological surveys [240].
- Primordial Reheating Phase: During the transition from inflation to the radiation-dominated era, the rapid coupling and energy transfer between fields (modeled by coupled harmonic oscillators) could freeze or slow down quantum relaxation, preserving non-equilibrium signatures in relic particle populations [240].
- Bouncing Cosmological Models: In models where the Big Bang is replaced by a quantum gravitational contraction-to-expansion bounce, quantum gravitational processes could actively generate fresh non-equilibrium states, propagating Born rule violations into the early expanding universe [240].
- Black Hole Evaporation: Extremely strong gravitational fields and high-energy processes near black hole horizons, particularly during evaporation, could generate quantum non-equilibrium in the emitted Hawking radiation [240].
3.6 Comparative Analysis of Interpretational Models
To systematically compare the modeling, and dynamical characteristics of these different formulations, Table 3 maps the mathematical and physical profiles of orthodox quantum mechanics against the various Bohmian extensions.
4. Interpretation with the Multi-Fold Universe Framework
The covariant foliation and stochastic jump processes, described in sections 2.3 through 2.8, represent a mathematical formalism for a relativistic Bohmian quantum field theory.
To construct a complete physical model, one must address the microscopic origin of the generated foliation
, the physical mechanism executing the non-local pilot-wave guidance, and the fundamental nature of the chosen field-particle beables.
The multi-fold theory provides a possible microscopic spacetime geometry, or macroscopic interpretation, that satisfies these requirements seamlessly [1-199, 251,258,259]. An overview of the multi-fold theory is presented in Appendix A, with pointers to references.
4.1 Spacetime Concretization and the Covariant Foliation
Interpretation / FormulationPrimitive Model (Local Beables)Dynamical EquationsWave Function RoleMechanism of Empirical EquivalenceTreatment of MeasurementCopenhagen (Orthodox)None; physical properties are indeterminate until measured [221]Schrödinger equation plus instantaneous projection postulate [223] Complete description of physical state; probability amplitude carrier [224]Defined axiomatically via the Born rule [225]Discontinuous collapse triggered by external classical apparatus [223]Non-Relativistic Bohmian MechanicsPoint-like particles Qi(t) in physical space 3 [222]Schrödinger equation plus deterministic guiding equation [222]Physical/nomological field guiding actual configurations [222] Quantum Equilibrium Hypothesis and equivariance [222]Continuous, observer-free branch selection via conditional wave function [228] Relativistic Hypersurface Bohm-Dirac ModelParticle world lines Xk(s) in Minkowski spacetime [233] Dirac equation plus HBDM guidance law coupled to foliation F [233] Multi-time pilot wave defining spacetime probability currents [233]Equilibrium statistics matching standard covariant QMC predictions [227]Branch selection along the spacelike leaves of the foliation [233] Bell-Type QFT (Particle Model)Point-like particles with variable number N(t) [230] Schrödinger equation plus continuous guidance and stochastic jumps [202]Fock-space state vector guiding configuration jumps between sectors [202]Minimal jump rates preserving equivariance of the probability distribution [202] Observer-free branch selection via stochastic particle configuration jumps and the conditional wave function [202]Field-Theoretic QFT (Struyve-Westman)Classical fields F(x,t) and charge density r(x,t) [227]Functional Schrödinger equation plus deterministic field guidance [227]Functional pilot wave guiding field evolution on configuration space [227] Equivariance of the wave functional probability density [227] Continuous field evolution selecting stable macroscopic charge concentrations [227]Table 3: Mapping of maps the mathematical and physical profiles of orthodox quantum mechanics against the various Bohmian extensions
In the standard Bohmian model, the foliation
is determined functionally by the global wave function.
The multi-fold theory can offer a physical mechanism for this geometrical structure: spacetime itself emerges as a discrete, non-commutative, fractal geometry concretized by the 2D random walks of massless Higgs bosons [1,8-10,22,32,62,72,131,137,152,170,176,181,251], and their location or location history. Because these random walks’ locations should follow random sprinkled Poisson distributions [1,23,72,131,137,152,170,181,251], the resulting discrete structure successfully preserves continuous macroscopic Lorentz invariance while remaining non-commutative at the smallest scales [1,23,72,131,137,152,170,181,251]. Therefore, the covariantly determined spacelike hypersurfaces S are not abstract mathematical structures mapped onto a continuous background; rather, they correspond directly to the physical locations, and historical locations, concretized by the underlying massless Higgs random walks. This way, the background depends on the matter distribution of the massless Higgs bosons (the quantum state), satisfying the conditions for a dynamically determined foliation without problematically positing absolute external structures.
4.2 The E/G Conjecture and the Quantum Potential
A persistent conceptual challenge in any pilot-wave formulation is the non-local nature of the guidance equation: how does a particle instantaneously “know” the configuration of all other entangled particles along the spacelike leaf S?
The multi-fold theory resolves this by proposing that (EPR (Einstein-Podolsky-Rosen)) entanglement generates direct physical pathways, the multi-folds, between entangled systems [1,5,7,8-10,71,131,137,152,140,18,251]. According to the E/G (Entanglement is Gravity) conjecture of the multi-fold theory, entanglement physically generates gravity-like attractive effective potentials, or effective curvature, within the 4D spacetime region spanning the entangled entities [1,5,7-10,22,24,131,137,152,188,251]. These multi-fold mechanisms permit “spooky actions at a distance” to result from strictly local interactions operating within the folds [1,9,10,140]. Consequently, the non-local Bohmian current correlations and the resulting quantum potential are physically mediated through the multi-fold connections. Local physics and hidden variables are thus fully reconcilable with Bell experiments within this multi-fold discrete spacetime [1,8-10,62,131,137,152,251].
Figure 3: Multi-Fold Mechanism and the E/G Conjecture Description. It is a conceptual diagram illustrating the integration of the multi-fold theory with the non-local Bohmian pilot-wave. Two entangled particle beables (represented as Kerr-Newman soliton Q-balls, or random walk patterns) exist at spacelike separated positions on a discrete hypersurface S (constructed from 2D massless Higgs random walks). A multi-fold mechanism is depicted as an extra-dimensional mapping (a fold) connecting the two particles, transmitting the non-local guidance potential while simultaneously generating gravity-like effective curvature (the E/G conjecture [24]) in the intervening 4D spacetime.
4.3 Multi-fold Space Time Matter Induction and Scattering, and Beables
4.3.1 Multi-fold Particles
The relativistic Bohmian model proposed in Section 2.6 requires a hybrid model: continuous field beables for bosons and discrete point-particle beables for fermions.
The multi-fold space-time matter induction and scattering processes, operating in an unconstrained 7D (or 5D) embedding space, explain the emergence of these specific beables [1,23,33-35,63,68,72,116,150,131,137,152,177,251]. In the multi-fold geometry, the particles of the Standard Model particles with gravity effects non-negligible at its scales (SMG) are modeled as distinct structural manifestations of the underlying massless Higgs boson random walks [1,8-10,22,32,62,72,131,137,152,170,176,181,251].
At spatial scales small enough below the multi-fold gravity electroweak symmetry breaking, inspired by figure 1 in [72], massless Higgs bosons propagate freely as conformal field theories (CFTs) [68,70,177].
Field Beables: Bosonic degrees of freedom map directly to the continuous, massless patterns of these random walks [68,70,150,170,177].
At higher spatial scales [4,8-10,29,68,70,124,150,170,177], still below the multi-fold gravity electroweak symmetry breaking, we have, in addition to bosonic field beables:
- Particle Beables: Fermionic degrees of freedom map to the pattern of these random walks into massless fermions. [4,8-10,29,68,70,72,124,150,170,177].
At still higher spatial scales, above the multi-fold gravity electroweak symmetry breaking, we have, in addition to bosonic field beables:
- Particle Beables: Fermionic degrees of freedom map to the condensation of these random walks into massive and charged Dirac Kerr-Newman soliton Q-balls (microscopic black holes) [4,8-10,29,68,70,72,124,150,170,177].
This provides a direct physical justification for the hybrid choice of primitives. (Massless) Bosons behave as extended fields generated by the random walk configurations in agreement with [124] and references therein, whereas fermions act as highly localized, particle-like condensates, in agreement with [124] for fermionic fields (particles hence modeled in the multi-fold universe as extended microscopic black hole like patterns of random walks or condensates of preons / massless Higgs bosons) and with [1,8-10,22,131,137,150,152,170,251] and references therein, in terms of scales required to have non-commutativity of spacetime required to support fermions, their zitterbewegung, and spin statistics, and Quantum Physics in general. It is also related to [199].
4.3.2 Further Multi-fold Corroboration
In the multi-fold theory, massless bosons, and fermions occupy different scales (as do massive Bosons) [170]. We showed that this is essential to the notion of zitterbewegung and spin statistics theorem for fermions and the anti-commutativity of spacetime at the core of the existence of quantum mechanics and its uncertainty principle.
The Non-commutativity of spacetime is also key to the expansion of a dS (asymptotically de Sitter) universe [6,181], and random walks of the 2D massless bosons [1,8-10,22,32,62,72,131,137,152,170,176,181,251], which we [1,6-10,124,131,137,140,152,176,251,258], and others [124,170,253-256], have shown to recover, in a fractal spacetime, Schrödinger’s equations / quantum Physics and General relativity (GR) [253,257]. So, in an expanding universe, non-commutativity implies 2D random walks [1,6,72,137,181][12], which imply fractals (and QFT [6,62,124,175,181,253-256]), GR and SMG [1,6-10,22,29,62,72,131,137,152,181,251]. The important message is that:
- Spacetime is predicted non-commutative, a result also consistent with all the coherent theories of gravity [1,6,8-10,22,62,112,131,137,152,155,167,181,251],
- Therefore, spacetime does not have to be continuous [1,8-10,22,29,36,62]. Limits to the continuum do not even have to be taken in the fermion as particle beables.
4.4 Microscopic Origin of Stochastic Jumps
4.4.1 Particle Creation and Annihilation
Finally, the multi-fold framework provides a physical mechanism for the Bell-type stochastic Markov jump processes described in Section 2.7.
The creation and annihilation events governed by the minimal jump rates s(q/q’) can represent the physical condensation and dissociation of the Kerr-Newman soliton Q-balls [4,35,162,177,181], or the random walk patterns [1,8-10,22,29,32,35,62,72,131,137,152,162,170,176,177,181,251].
When an interaction vertex is reached along the foliation
a given particle beable (a soliton condensate, or random walk pattern) may break apart back into the constituent massless Higgs random walk patterns, corresponding to a particle annihilation event. Conversely, background random walks may stochastically condensate into a localized microscopic black hole condensate or random walk pattern, representing a particle creation event [4].
The transition rates for these structural phase changes are governed strictly by the interaction Hamiltonian HI, preserving the exact mathematical equivariance of the master equation across the Fock space while anchoring the jumps in a specific, physical microscopic process.
4.4.2 A Bit More
Beyond [1,8-10,22,29,32,62,72,131,137,152,162,170,176,177,181,251], we are in the process of showing that key characteristics of quantum mechanics are a direct result of the 2D random walks [199]. More papers will appear. Please watch [8].
5. Conclusions
The de Broglie-Bohm interpretation of quantum mechanics provides a mathematically rigorous, observer-independent representation of physical reality that is empirically indistinguishable from orthodox quantum mechanics [221]. This observational equivalence is maintained in non-relativistic regimes by the Quantum Equilibrium Hypothesis, which establishes the Born rule as an emergent statistical distribution that is dynamically preserved by the equivariance of the guiding equation [222].
This equivalence successfully extends to relativistic spacetime and quantum field theories [230]. In the relativistic domain, the tension between quantum nonlocality and Lorentz covariance is resolved by the Hypersurface Bohm-Dirac Model (HBDM), which can extract a preferred spacelike foliation directly from the covariant structure of the wave function [233]. In the context of QFT, the pilot-wave model accommodates particle creation and annihilation either through stochastic jumps in a Fock-space particle model (Bell-type QFTs), or via deterministic functional evolution in a field model [227].
However, the indistinguishability of these models is contingent upon the universe having reached a state of statistical quantum equilibrium [226]. If the universe was prepared in a state of primordial quantum non-equilibrium, the underlying subquantum mechanics of the pilot-wave theory would become manifest, resulting in observable violations of standard quantum limits [226]. The search for these relic signatures in the Cosmic Microwave Background, primordial reheating phases, bouncing cosmologies, or Hawking radiation represents a potentially viable pathway toward empirically distinguishing Bohmian mechanics from standard quantum mechanics and testing the ultimate boundaries of quantum foundations [240].
The framework derived in this paper establishes a cohesive, explicitly formulated relativistic Bohmian quantum field theory a priori mapped onto a fundamental discrete spacetime, even if in the limit it might be continuum. This synthesis formally resolves the historical incompatibilities between pilot-wave theory, special relativity, and QFT.
The deterministic continuity equations ensure that particle and field beables track the free Hamiltonian dynamics along the leaves of a dynamically generated, covariant spacetime foliation, which physically originates from the 2D random walks of massless Higgs bosons. The non-local guidance required by the pilot-wave model can be microscopically interpreted as supported by the multi-fold mechanisms [1,137,140] and the E/G conjecture [1,22,24,137], ensuring that entanglement-driven gravity-like potentials facilitate communication across the spacelike leaves. Furthermore, the minimal jump rates algorithmically mediate transitions across Fock space, replicating the creation and annihilation events as the physical condensation and dissociation of Kerr-Newman soliton Q-balls, or formation and dissociation of random walk patterns. Interestingly the scale differences and methods required to model relativistic Bohmian bosons and Fermions recover multi-fold results in terms of spacetime scales for massless bosons, and fermions, and massive particles per [35,177,181], in ways that directly corroborates the relativistic Bohmian models of (charged) fermions vs. bosons[13].
The mathematical proof of equivariance guarantees that the statistical predictions of the theory remain isomorphic to standard regularized QFT, validating the empirical adequacy of the model. This synthesis provides a robust picture of the relativistic quantum domain, restoring localized physical realism options to QFT without compromising Lorentz invariance or the principles of QFT.
This paper demonstrates a consistent way to evolve non-relativistic Bohmian mechanics into a relativistic Bohmian QFT model; and, that the resulting hybrid model could be microscopically justified with multi-fold considerations, as has often been the case for explaining conventional physics behaviors[14] (See Appendix A). We expect more such examples to come, as part of our current work to infer some of this, in particular, using 2D random walks of massless Higgs bosons e.g., [199].
Appendix A. The Multi-fold Theory
For more details on the latest developments, updates to papers , discussion and the full story only succinctly summarized here, please consider the complete list of papers compiled at [8]. In particular, while some more recent references are provided, the focus in this appendix is not to provide our latest findings. There are rather tracked on that web site [8].
In a multi-fold universe [1,8-10,22,131,137,152,251], gravity emerges from entanglement through the multi-fold mechanisms. As a result, gravity-like effects appear in between entangled particles [1,24,25], whether they are real or virtual. Long range, massless gravity results from entanglement of massless virtual particles [1,25]. Entanglement of massive virtual particles leads to massive gravity contributions at very smalls scales [1,26]. It is at the base of the E/G Conjecture [24], and the main characteristics of the multi-fold theory [22]. Multi-folds mechanisms [140] also result in a spacetime that is discrete, with a random walk fractal structure and non-commutative geometry that is Lorentz invariant and where spacetime nodes and particles can be modeled with microscopic black holes [1,4,16,27-31,68,72,121,131,137,150,152,170,177,181,251]. All these recover General Relativity (GR) at large scales, and semi-classical model remain valid till smaller scale than usually expected [1,6,131,137,152,181,251]. Gravity can therefore be added to the Standard Model (SM) resulting into what we define as SMG: the SM with gravity effects non-negligible at its scales. This can contribute to resolving several open issues with the Standard Model without new Physics[15] other than gravity. These considerations hint at an even stronger relationship between (multi-fold) gravity and the Standard Model, as finally shown in [23].
Among the multi-fold SMG discoveries, the apparition of an always in-flight, and hence non-interacting, right-handed neutrinos, coupled to the Higgs boson is quite notable. It is supposedly always around right-handed neutrinos, due to chirality flips by gravity of the massless Weyl fermions, induced by 7D space time matter induction and scattering models [1,23,33-35,63,68,72,116,150,131,137,152,177,251], and hidden behind the Higgs boson or field at the entry points and exit points of the multi-folds. Massless Higgs bosons modeled as minimal microscopic black holes mark concretized spacetime locations. They can condensate into Dirac Kerr-Newman soliton Qballs to produce massive and charged particles [1,4,161], thereby providing a microscopic explanation for a Higgs driven inflation [27], the electroweak symmetry breaking [29,35], the Higgs mechanism, the mass acquisition [29,35,36,72,170,177,181] and the chirality of fermions and spacetime [29,35,36,72,177]; all resulting from the multi-fold gravity electroweak symmetry breaking [35,177]. The multi-fold theory has also concrete implications on New Physics like supersymmetry, superstrings, M-theory and Loop Quantum Gravity (LQG) [1,8-21,112,131,137,152,182].
The multi-fold paper [1,137] proposes contributions to several open problems in physics, like the reconciliation of General Relativity (GR) with Quantum Physics, explaining the origin of gravity proposed as emerging from quantum (EPR- Einstein Podolsky Rosen) entanglement between particles, detailing contributions to dark matter and dark energy, and explaining other Standard Model mysteries without requiring New Physics beyond the Standard Model other than the addition of gravity to the Standard Model Lagrangian, and the 2D massless Higgs boson random walks. All this is achieved in a multi-fold universe that may well model our real universe, which remains to be validated.
With the proposed model of [1,137], spacetime and Physics are modeled from Planck scales to quantum and macroscopic scales, and semi-classical approaches appear valid till very small scales. In [1,137], it is argued that spacetime is discrete, with a random walk-based fractal structure, fractional and noncommutative at, and above Planck scales (with a 2-D behavior and Lorentz invariance preserved by random walks till the early moments of the universe) [1,8-10,22,29,31,62,131,137,148,152,169,170,181,182,251]. Spacetime results from past random walks of particles. Spacetime locations and particles can be modeled as microscopic black holes (Schwarzschild for photons and concretized spacetime coordinates, and metrics between Reissner Nordström [2], and Kerr Newman [3] for massive, and possibly charged, particles – the latter being possibly extremal) [4,162]. Although possibly surprising, [1,137] recovers results which are consistent with others (see [4], and its references), while also being able to justify the initial assumptions of black holes from the models of gravity or entanglement in a multi-fold universe. The resulting gravity model recovers General Relativity at larger scale, as a 4D process, with massless gravity, but also with massive gravity components at very small scales, which make gravity non-negligible at these scales. Semi-classical models also turn out to work well till way smaller scales than usually expected.
Multi-folds are encountered in GR at Planck scales [5,6] and in Quantum Mechanics[16] (QM) if different suitable quantum reference frames (QRFs) are to be equivalent relatively to entangled, coherent or correlated systems [7,58]. This shows that GR and QM are different facets of something that they cannot well model: multi-folds.
Considering results as in [6,52,181], and our answers to so many open issues with the SM and the ΛCDM can be qualitatively explained with the SMG and multi-fold mechanisms, as discussed in [1-199,251,258,259], we can then argue that these conclusions can apply to our real universe, especially considering how the multi-fold mechanisms recover GR [1,137], and can be encountered in GR at Planck scales [6,181], with the spacetime reconstruction [1,62], and with the top-down-up-and-upper derivation of the multi-fold theory [6,181].
Appendix B. About Quantum Equilibrium and Bohmian Physics
B.1 In Equilibrium or Out of Equilibrium?
Here we expand, with some repetition on the quantum equilibrium discussed in the body of the paper. Equations are renumbered to be self-contained.
In the de Broglie–Bohm, or pilot-wave, interpretation of quantum mechanics, a system is said to be in quantum equilibrium when the actual spatial distribution of its physical particles, denoted by ,
exactly matches the probability density predicted by standard quantum mechanics, which is the square of the wave function’s amplitude, . This equality,
(B.1)
is known as the quantum equilibrium hypothesis (QEH).
Because the wave function is treated as an objective, physical pilot wave rather than a mere representation of observer probability, the Born rule is not a fundamental axiom in this theory. Consequently, a system can theoretically be out of equilibrium (a state known as quantum non-equilibrium, where
) (B.2)
through several physical mechanisms, like arbitrary Initial Conditions at the Beginning of the Universe. In pilot-wave theory, the actual distribution of particles and the wave-intensity are governed by separate mathematical identities. While both satisfy the same continuity equation, i.e., meaning that if a system starts in equilibrium at
t = 0 , (B.3)it will remain in equilibrium for all future times (a property called equivariance as discussed earlier in this paper), there is no fundamental law which dictates that they must be equal initially. At the birth of the universe, e.g., the Big Bang, the particles could have been distributed in an arbitrary, non-standard configuration
(B.4)
placing the early universe in a state of quantum non-equilibrium.
In general, it would them eventually evolve towards equilibrium by Dynamical Relaxation. According to Valentini’s “sub-quantum H-theorem”, i.e., the pilot-wave analog to Boltzmann’s H-theorem in classical statistical mechanics, chaotic interactions and the complex “mixing” of Bohmian trajectories typically force an initial non-equilibrium state to rapidly relax to equilibrium on a coarse-grained level [241,245]. This rapid relaxation explains why we only observe standard quantum probabilities in laboratory experiments today.
However, there are cases where this may not happen.
- This relaxation could be suppressed or “frozen” under extreme conditions (Cosmological Freezing):
- Cosmic Expansion: In the very early, rapidly expanding universe (such as during the inflationary epoch), the expansion of space can stretch physical wavelengths faster than the particles can traverse them. This limits the displacement of the trajectories, effectively “freezing” the relaxation process and leaving primordial field modes in a state of non-equilibrium. This frozen non-equilibrium could theoretically leave detectable, non-standard statistical signatures in the Cosmic Microwave Background (CMB) or survive in relic primordial particles today.
- Another counter example could be a case of Incomplete Relaxation and Lack of Chaos. For a system to relax to equilibrium, its pilot-wave trajectories must be sufficiently chaotic. Chaos in Bohmian mechanics is primarily generated by trajectories scattering off the moving nodal points, i.e., where
(B.5)
of the wave function.
Therefore, if a quantum system has a highly coherent, non-chaotic wave function, such as certain simple Gaussian superpositions or systems with few degrees of freedom, the quantum flow behaves as a laminar flow. In these cases, the sub-quantum
-function does not decrease to its minimum value but instead saturates, preventing complete relaxation and keeping the system permanently out of equilibrium.
- Gravitational Collapse and Black Hole Evaporation could also prevent equilibrium. Strong gravitational fields represent another frontier where quantum equilibrium might break down. It has been theorized that the extreme physics of gravitational collapse and subsequent black hole evaporation via Hawking radiation can generate quantum non-equilibrium. If so, the particles and radiation emitted by an evaporating black hole would violate the Born rule, carrying distinct non-equilibrium statistics into the surrounding space.
Then, we should also note that the case of Velocity/Momentum Non-Equilibrium from (Bohm’s 1952 Formulation) [147,148]. In Louis de Broglie’s original 1927 pilot-wave theory (See [b5,aa3]), the first-order guidance equation (
(B.6)
Where p is momentum and S is the phase of the wave function) is a fundamental law of motion.
However, in David Bohm’s 1952 second-order Newtonian reformulation, the guidance equation is treated merely as a constraint on the initial momenta [247,248]. If this constraint is relaxed, particles can start with momenta
(B.7)
This “extended” non-equilibrium is highly unstable [248]. For instance, in a ground-state system like a hydrogen atom, where standard pilot-wave theory dictates the velocity is zero, dropping the momentum constraint causes a rapid growth of spatial non-equilibrium, ultimately causing stable bound states to fly apart.B.2 A Multi-fold Perspective
B.2.1. From the Big Bang
Based on multi-fold considerations as discussed in [1-199,251,258,259], even the earliest steps are modeled with (2D random walk Physics). In such cases, we would not have initial conditions out of equilibrium. In fact, in the multi-fold fluctuation case [1,137], probability and wave necessarily have the same norm. Distributing the fluctuations along a large region, still rely on the same phenomena and equations, and should result the same.
Similarly, in the context of the total collision model [107] or other cyclic expansions after a big crunch, we would expect to come from an equilibrium, and, therefore, to have the equilibrium preserved.
In either case, there are no situations where the boundary conditions would be following (B.2).
Could an extended non-equilibrium occur? Per [248], this would imply a (global) rapidly expanding chaotic region, unrelated to inflationary effects with no clear mechanism to ever recover equilibrium. In such a universe, no bound structure would exist[17].
So, in a multi-fold universe, the initial moment of the universe would necessarily be in equilibrium, and relativistic Bohmian QFT / Physics always remain undistinguishable from conventional QFT, even when looking in traces of the big bang. In Appendix A, we mentioned that a multi-fold universe often models well our real universe. Maybe it is the case again, and quantum equilibrium is never broken in our real universe.
B.2.2 Relativistic Bohmian Physics And Beable Wave Function
As discussed in the main paper, with [4,71] and the multi-fold spin model [1,8-10,22,131,137,150,152,161,251], we saw that the wavefunction seems to be a beable.
The multi-fold SM (or rather SMG) elementary particles as random walk patterns or condensates of massless Higgs bosons, aka multi-fold preons) [4,35,170,176,177,181] also imply that all particles are beables. With such a model, the wavefunction is expected to always satisfy (B.1), and therefore never to be out of equilibrium satisfying (B.2).
Appendix C. Treatment of Neutral Fermions (Neutrinos) in Relativistic Bohmian QFT
The evaluation of fermionic behavior in relativistic Bohmian mechanics poses specific modeling challenges when dealing with neutral particles such as neutrinos. Because certain formulations rely heavily on macroscopic charge densities to map physical reality, addressing neutral fermions requires a deliberate selection of the underlying primitive model.
Per [157], and references therein, especially [250], neutrinos can’t carry any charge no matter how small it would be, or our current Physics would collapse, e.g., QED and Electroweak theories would have to be reworked.
C.1 The Limitation of the Field-Theoretic (Struyve-Westman) Model
In the Minimalist Model developed by Struyve and Westman [227], fermionic degrees of freedom are not treated as localized physical particles. Instead, they are represented entirely by the wave function, which is mapped to a macroscopic continuous field beable, such as the electromagnetic vector potential ,
and an associated continuous charge density
defined on physical space.
Because neutrinos are electrically neutral, they do not couple to the electromagnetic vector potential and do not generate an electromagnetic charge density.
Consequently, applying this specific minimalist field-theoretic model to neutrinos requires extending the continuous functional guidance equations beyond electromagnetism. The model must substitute the electromagnetic vector potential with the gauge bosons of the weak interaction and evaluate the corresponding weak-isospin density.
C.2 Resolution via the Hybrid Bell-Type Particle Model
To avoid the phenomenological challenges of relying on macroscopic charge density fields to represent matter, the primary theoretical framework adopted in this paper relies on a hybrid model utilizing Bell-type quantum field theories.
In this adopted model, as we saw in the main body of the paper, bosonic degrees of freedom (which mediate forces) are represented by continuous functional fields, while all fermions (charged or neutral) are modeled as distinct, localized point-particle beables.
Neutrinos are treated as actual point particles with variable configurations N(t) across a Fock space rather than as a continuous charge density. This way, the hybrid model naturally copes with them.
The deterministic drift of these neutral point particles is supplemented by discrete, stochastic Markov jumps representing creation and annihilation events. These events are governed by minimal jump rates derived directly from the fundamental interaction Hamiltonian, ensuring that weak-force interactions involving neutrinos are successfully modeled without violating standard quantum statistics.
C.3 Neutrinos in the Multi-Fold Universe Framework
When integrating this framework into the multi-fold theory, where spacetime is concretized by 2D random walks of massless Higgs bosons, and entanglement creates multi-folds, neutrinos exhibit highly specific behaviors that resolve broader cosmological and standard model issues:
- Chirality Flips and Right-Handed Neutrinos: The multi-fold theory accounts for the apparition of always in-flight, non-interacting right-handed neutrinos coupled to the Higgs boson. This mechanism is driven by gravity-induced chirality flips of massless Weyl fermions via 7D spacetime matter induction and scattering models [1,8-10,42,47,131,137,152,159,165,251].
- Spacetime Location: These right-handed neutrinos are theorized to be hidden behind the Higgs boson (or field) precisely at the entry and exit points of the multi-folds [35,42,47,67,119,159,177,x].
The multi-fold mechanisms suggest that neutrinos are strictly governed by these chiral and gravitational dynamics. Specifically, the framework explicitly notes that neutrinos are probably not Majorana fermions [1,8-10,42,47,119,131,137,152,159,165,251].
The multi-fold Standard Model with Gravity (SMG) operates without the need for New Physics beyond the SM, as SMG [1,22,24], explicitly predicting the absence of conventional sterile neutrinos [1,8-10,22,42,47,67,131,137,152,159,165,251].
____
Cite as: Stephane H. Maes, (2026), “Relativistic Bohmian Physics, with a Microscopic Interpretation”, https://shmaesphysics.wordpress.com/2026/08/12/relativistic-bohmian-physics-with-a-microscopic-interpretation/, August 12, 2026.
____
References
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[2]: Wikipedia, “Reissner–Nordström metric”, https://en.wikipedia.org/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric. Retrieved on March 21, 2020.
[3]: Wikipedia, “Kerr–Newman metric”, https://en.wikipedia.org/wiki/Kerr-Newman_metric. Retrieved on March 21, 2020.
[4]: Stephane H Maes, (2021), “More on Multi-fold Particles as Microscopic Black Holes with Higgs Regularizing Extremality and Singularities”, viXra:2210.0004v1, https://shmaesphysics.wordpress.com/2021/02/28/more-on-multi-fold-particles-as-microscopic-black-holes-with-higgs-regularizing-extremality-and-singularities/, February 25, 2021.
[5]: Stephane H Maes, (2020), “Multi-folds, The Fruit From The Loops? Fixing “Oops for The Loops” May Encounter Multi-folds in General Relativity And The E/G Conjecture”, viXra:2212.0206v1, https://shmaesphysics.wordpress.com/2021/12/31/multi-folds-the-fruit-from-the-loops-fixing-oops-for-loops-encounters-multi-folds-and-the-e-g-conjecturein-general-relativity/, January 1, 2022.
[6]: Stephane H Maes, (2022), “Deriving the Multi-fold Theory from General Relativity at Planck scale”, viXra:2302.0129v1, https://shmaesphysics.wordpress.com/2022/02/22/deriving-the-multi-fold-theory-from-general-relativity-at-planck-scale/, February 22, 2022.
[7]: Stephane H Maes, (2022), “From Quantum Relational Equivalence to Multi-folds Encounter in the Real Universe and Confirmation of the E/G conjecture”, viXra:2302.0108v1, https://shmaesphysics.wordpress.com/2022/02/12/from-quantum-relational-equivalence-to-multi-folds-encounter-in-the-real-universe-and-confirmation-of-the-e-g-conjecture/, February 7, 2022.
[8]: Stephane Maes, (2020-25), “Web Site Tracking all Publications around the Multi-fold universe”, Navigation page listing all papers, https://shmaesphysics.wordpress.com/shmaes-physics-site-navigation/.
[9]: Stephane H Maes, (2021), ”The Multi-fold Theory: A synopsis”, viXra:2112.0144v1, https://shmaesphysics.wordpress.com/2021/12/24/the-multi-fold-theory-a-synopsis-so-far-v2-end-of-2021/, December 24, 2021. Note that additional links will always be available at https://shmaesphysics.wordpress.com/2021/05/03/the-multi-fold-theory-a-synopsis-so-far/ to track the latest and interim versions of the synopsis, as they may be published under different tittle or URL/publication numbers.
[10]: Stephane H Maes, (2022), “Understanding the Multi-fold theory principles and the SM_G”, osf.io/xc74t, https://shmaesphysics.wordpress.com/2022/03/11/understanding-the-multi-fold-theory-principles-and-the-sm_g/, March 11, 2022. Also, as Stephane H Maes, (2022), “A tutorial on the Multi-fold theory principles and the SM_G”, viXra:2303.0154v1, https://shmaesphysics.wordpress.com/blog-2/a-tutorial-on-the-multi-fold-theory-principles-and-the-sm_g/, March11, 2022.
[11]: Stephane H. Maes, (2022), “Comment on LQG, Superstrings, Supersymmetry and most GUTs/TOEs, all have big problems exposed by the Multi-fold Theory”, https://shmaesphysics.wordpress.com/2021/12/27/the-multi-fold-theory-a-synopsis/#comment-3293. Published on January 9, 2022.
[12]: Stephane H. Maes, (2020), “Comment on why no supersymmetry”, https://shmaesphysics.wordpress.com/2020/10/11/circular-arguments-in-string-and-superstring-theory-from-a-multi-fold-universe-perspective/#comment-934. Published on October 12, 2020.
[13]: Stephane H Maes, (2020), “Renormalization and Asymptotic Safety of Gravity in a Multi-Fold Universe: More Tracking of the Standard Model at the Cost of Supersymmetries, GUTs and Superstrings”, viXra:2102.0137v1, https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/, September 18, 2020.
[14]: Stephane H Maes, (2020), “Circular Arguments in String and Superstring Theory from a Multi-fold Universe Perspective”, viXra:2103.0195v1, https://shmaesphysics.wordpress.com/2020/10/11/circular-arguments-in-string-and-superstring-theory-from-a-multi-fold-universe-perspective/, October 5, 2020.
[15]: Stephane H Maes, (2021), “The String Swampland and de Sitter Vacua: A Consistent Perspective for Superstrings and Multi-fold Universes”, viXra:2208.0078v1, https://shmaesphysics.wordpress.com/2021/01/12/the-string-swampland-and-de-sitter-vacua-a-consistent-perspective-for-superstrings-and-multi-fold-universes/, January 9, 2021.
[16]: Stephane H Maes, (2021), “Quantum Gravity Asymptotic Safety from 2D Universal Regime and Smooth Transition to Dual Superstrings”, viXra:2208.0151v1, https://shmaesphysics.wordpress.com/2021/02/07/quantum-gravity-asymptotic-safety-from-2d-universal-regime-and-smooth-transition-to-dual-superstrings/, January 29, 2021.
[17]: Stephane H Maes, (2020), “A Non-perturbative Proof of the Asymptotic Safety of 4D Einstein Gravity, With or Without Matter”, https://doi.org/10.5281/zenodo.7953796, https://shmaesphysics.wordpress.com/2022/05/04/a-non-perturbative-proof-of-the-asymptotic-safety-of-4d-einstein-gravity-with-or-without-matter/, May 4, 2022, viXra:2305.0138.
[18]: Stephane H Maes, (2020), “Dualities or Analogies between Superstrings and Multi-fold Universe”, viXra:2006.0178v1, https://shmaesphysics.wordpress.com/2020/06/14/dualities-or-analogies-between-superstrings-and-multi-fold-universes/, June 14, 2020.
[19]: Stephane H Maes, (2020), “Alignments and Gaps Between Multi-fold Universes And Loop Quantum Gravity”, viXra:2006.0229v1, https://shmaesphysics.wordpress.com/2020/06/19/multi-fold-universes-analysis-of-loop-quantum-gravity/, June 18, 2020.
[20]: Stephane H Maes, (2020), ”Superstrings Encounter of the Second, Third or Fourth Types?”, viXra:2010.0140v1, https://shmaesphysics.wordpress.com/2020/07/19/superstrings-encounter-of-the-second-third-or-fourth-types/, July 5, 2020.
[21]: Stephane H Maes, (2022), “Oops For The Loops II: Real Oops; LQG Does Not Optimize the Hilbert Einstein Action”, viXra:2301.0036v1, https://shmaesphysics.wordpress.com/2022/01/05/oops-for-the-loops-ii-real-oops-lqg-does-not-optimize-the-hilbert-einstein-action/, January 5, 2022.
[22]: Stephane H. Maes, (2022), “What is the Multi-fold Theory? Its Main Characteristics in a Few Words”, vixra:2207.0172v1, https://shmaesphysics.wordpress.com/2022/07/28/what-is-the-multi-fold-theory-its-main-characteristics-in-a-few-words/, July 28, 2022.
[23]: Stephane H. Maes, (2022), “Justifying the Standard Model U(1) x SU(2) x SU(3) Symmetry in a Multi-fold Universe”, https://doi.org/10.5281/zenodo.8422911, https://shmaesphysics.wordpress.com/2022/08/08/justifying-the-standard-model-u1-x-su2-x-su3-symmetry-in-a-multi-fold-universe/, August 8, 2022, (viXra:2310.0040v1).
[24]: Stephane H Maes, (2020), “The E/G conjecture: entanglement is gravity and gravity is entanglement”, viXra:2010.0139v1, https://shmaesphysics.wordpress.com/2020/10/15/the-e-g-conjecture-entanglement-is-gravity-and-gravity-is-entanglement/, October 15, 2020.
[25]: Stephane H Maes, (2020), “Gravity-like Attractions and Fluctuations between Entangled Systems?”, viXra:2010.0010v1, https://shmaesphysics.wordpress.com/2020/06/25/gravity-like-attractions-and-fluctuations-between-entangled-systems/, June 24, 2020.
[26]: Stephane H Maes, (2020), ”Massless and Massive Multi-Gravity in a Multi-fold Universe”, viXra:2010.0095v1, https://shmaesphysics.wordpress.com/2020/06/30/massless-and-massive-multi-gravity-in-a-multi-fold-universe/, June 19, 2020.
[27]: Stephane H Maes, (2020), ”Explaining Dark Energy, Small Cosmological Constant and Inflation Without New Physics?”, viXra:2006.0261v1, https://shmaesphysics.wordpress.com/2020/06/19/explaining-dark-energy-small-cosmological-constant-and-inflation-without-new-physics/, June 19, 2020.
[28]: Stephane H Maes, (2020), ”Ultimate Unification: Gravity-led Democracy vs. Uber-Symmetries”, viXra:2006.0211v1, https://shmaesphysics.wordpress.com/2020/06/16/ultimate-unification-gravity-led-democracy-vs-uber-symmetries/, June 16, 2020.
[29]: Stephane H. Maes, (2022), “Invalidation and Proof of the Mass Gap, and Viability of The Standard Model on a Discrete Spacetime”, https://doi.org/10.5281/zenodo.8237456, https://shmaesphysics.wordpress.com/2022/07/15/invalidation-and-proof-of-the-mass-gap-and-viability-of-the-standard-model-on-a-discrete-spacetime/, July 15, 2022. (viXra:2308.0059).
[30]: Stephane H. Maes, (2022), Stephane H. Maes, (2022), “A Conjecture: No Dark Matter will be discovered at LHC, or elsewhere”, (v2), https://doi.org/10.5281/zenodo.8175806, https://shmaesphysics.wordpress.com/2022/07/08/a-prediction-no-dark-matter-will-be-discovered-at-lhc-or-elsewhere/, July 8, 2022, viXra:2307.0119.
[31]: Stephane H Maes, (2022), “Unruh effects, Hawking Black Hole Evaporation, Quantum Corrected Larmor Formula, Numbers of Particles in Curved Spacetime: “Same-Same, but Just A Bit Different””, https://doi.org/10.5281/zenodo.8306942, https://shmaesphysics.wordpress.com/2022/07/25/unruh-effects-hawking-black-hole-evaporation-quantum-corrected-larmor-formula-numbers-of-particles-in-curved-spacetime-same-same-but-just-a-bit-different/, July 25, 2022, (viXra:2309.0005).
[32]: Stephane H Maes, (2020), “Multi-fold Higgs Fields and Bosons”, viXra:2204.0146v1, https://shmaesphysics.wordpress.com/2020/11/10/multi-fold-higgs-fields-and-bosons/, November 6, 2020.
[33]: Stephane H Maes, (2020), “Tracking Down The Standard Model With Gravity In Multi-Fold Universes”, viXra:2011.0208v1, https://shmaesphysics.wordpress.com/2020/08/30/tracking-down-the-standard-model-with-gravity-in-multi-fold-universes/, August 20, 2020.
[34]: Stephane H. Maes, (2020), “Particles of The Standard Model In Multi-Fold Universes”, viXra:2111.0071v1, https://shmaesphysics.wordpress.com/2020/11/05/particles-of-the-standard-model-in-multi-fold-universes/, November 4, 2020.
[35]: Stephane H Maes, (2021), “Multi-fold Gravity-Electroweak Theory and Symmetry Breaking”, viXra:2211.0100, https://shmaesphysics.wordpress.com/2021/03/28/multi-fold-gravity-electroweak-theory-and-symmetry-breaking/, March 16, 2021.
[36]: Stephane H Maes, (2020), “Viable Lattice Spacetime and Absence of Quantum Gravitational Anomalies in a Multi-fold Universe”, viXra:2205.0143v1, https://shmaesphysics.wordpress.com/2020/12/13/viable-lattice-spacetime-and-absence-of-quantum-gravitational-anomalies-in-a-multi-fold-universe/, December 4, 2020.
[37]: Stephane H Maes, (2022), “Can Chirality Flips Occur in a Multi-Fold Universe? What About Conservation Laws? II”, viXra:2204.0152v2, https://shmaesphysics.wordpress.com/2022/08/20/can-chirality-flips-occur-in-a-multi-fold-universe-what-about-conservation-laws-ii/, August 20, 2022, and Stephane H Maes, (2020), “Can Chirality Flips Occur in a Multi-Fold Universe? What About Conservation Laws?”, viXra:2204.0152, https://shmaesphysics.wordpress.com/2020/12/07/can-chirality-flips-occur-in-a-multi-fold-universe-what-about-conservation-laws/, December 6, 2020.
[38]: Stephane H Maes, (2020), ”Derivation of the Equivalence Principle in a Multi-fold Universe”, viXra:2010.0090v1, https://shmaesphysics.wordpress.com/2020/06/29/derivation-of-the-equivalence-principle-in-a-multi-fold-universe/, June 19, 2020.
[39]: Stephane H Maes, (2020), “Progress on Proving the Mass gap for Yang Mills and Gravity (maybe it’s already proved…)”, viXra:2006.0155v1, https://shmaesphysics.wordpress.com/2020/06/12/progresses-on-proving-the-mass-gap-for-yang-mills-and-gravity-maybe-its-already-proven/, June 12, 2020.
[40]: Stephane H Maes, (2020), “Gravity Induced Anomalies Smearing in Standard Model so that Protons May Never Decay, Except in Black holes“, viXra:2006.0128v1, https://shmaesphysics.wordpress.com/2020/06/13/gravity-induced-anomalies-smearing-in-standard-model-so-that-protons-may-never-decay-except-in-black-holes/, June 13, 2020.
[41]: Stephane H Maes, (2022), ”Gravity or Magnetic Monopoles? You Cannot Have Both! II“, viXra:2006.0190v2, https://shmaesphysics.wordpress.com/2022/08/20/gravity-or-magnetic-monopoles-you-cannot-have-both-2/, August 20, 2022; Stephane H Maes, (2020), ”Gravity or Magnetic Monopoles? You Cannot Have Both!“, viXra:2006.0190, https://shmaesphysics.wordpress.com/2020/06/15/gravity-or-magnetic-monopoles-you-cannot-have-both/, June 15, 2020.
[42]: Stephane H Maes, (2020), ”Right-handed neutrinos? Mass? Ask Gravity”, viXra:2007.0018v1, https://shmaesphysics.wordpress.com/2020/06/21/right-handed-neutrinos-ask-gravity/, June 23, 2020.
[43]: Stephane H Maes, (2020), ”Strong CP Violation Tamed in The Presence of Gravity”, viXra:2007.0025v1, https://shmaesphysics.wordpress.com/2020/06/23/strong-cp-violation-tamed-in-the-presence-of-gravity/ , June 21, 2020.
[44]: Stephane H Maes, (2020), “Gravity Dictates the Number of Fermion Generations: 3”, viXra:2007.0068v1, https://shmaesphysics.wordpress.com/2020/06/24/gravity-dictates-the-number-of-fermion-generations-3/, June 24, 2020.
[45]: Stephane H Maes, (2020), “Gravity Stabilizes Electroweak Vacuum – No Bubble of Nothing to Worry About!”, viXra:2007.0173v1, https://shmaesphysics.wordpress.com/2020/06/24/gravity-stabilizes-electroweak-vacuum-no-bubble-of-nothing-to-worry-about/, June 24, 2020.
[46]: Stephane H Maes, (2020), ”More Matter Than Antimatter, All Falling Down”, viXra:2010.0121v2, https://shmaesphysics.wordpress.com/2020/07/05/more-matter-than-antimatter-all-falling-down/, July 5, 2020. (V2: April 8, 2021)
[47]: Stephane H Maes, (2020), “No Conventional Sterile Neutrinos In a Multi-fold Universe: just SMG business as usual”, viXra:2103.0202v1, https://shmaesphysics.wordpress.com/2020/10/02/no-conventional-sterile-neutrinos-in-a-multi-fold-universe-just-smg-business-as-usual/, October 1, 2020.
[48]: Stephane H Maes, (2021), “New Physics with LHCb to explain loss of lepton universality, or just gravity?”, viXra:2103.0191v1, https://shmaesphysics.wordpress.com/2021/03/29/new-physics-with-lhcb-to-explain-loss-of-lepton-universality-or-just-gravity/, March 29, 2021.
[49]: Stephane H. Maes, “A bold prediction on the muon anomalous magnetic moment, and expected results to be published on April 7, 2021 by the Fermilab Muon g-2, and its explanation”, viXra:2104.0030v1, https://shmaesphysics.wordpress.com/2021/04/01/a-bold-prediction-on-the-muon-anomalous-magnetic-moment-and-expected-resulted-to-be-published-on-april-7-2021-by-the-fermilab-muon-g-2-and-its-explanation/, April 1, 2021.
[50]: Stephane H Maes, (2021), “New Physics is often not so new”, osf.io/z3sj6, https://shmaesphysics.wordpress.com/2021/04/27/new-physics-is-often-not-so-new/, April 27, 2021, https://zenodo.org/records/7791704.
[51]: Stephane H Maes, (2022), “Direction of Possible Multi-folds Corrections to the W Boson Mass”, osf.io/qvewa, https://shmaesphysics.wordpress.com/2022/04/08/direction-of-possible-multi-folds-corrections-to-the-w-boson-mass/, April 8, 2022, viXra:2304.0020.
[52]: Stephane H Maes, (2022), “Multi-folds in Yang Mills Feynman Diagrams”, osf.io/y8fpd, https://shmaesphysics.wordpress.com/2022/04/05/multi-folds-in-yang-mills-feynman-diagrams/, April 5, 2022, viXra:2303.0161.
[53]: Stephane H. Maes, (2022), “Time-Varying Multi-fold Dark Energy Effects and Implications for the Hubble Tension”, https://doi.org/10.5281/zenodo.10396357, https://shmaesphysics.wordpress.com/2022/11/13/time-varying-multi-fold-dark-energy-effects-and-implications-for-the-hubble-tension/, November 13, 2022, osf.io/g2vzy/, viXra:2312.0083v1. Also, as Stephane H. Maes, (2022), “The Possibility of a Multi-fold Time-Varying Hubble Constant”, viXra:2312.0083v1.
[54]: Stephane H Maes, (2020), ”Explaining Dark Matter Without New Physics?”, viXra:2007.0006, https://shmaesphysics.wordpress.com/2020/06/21/explaining-dark-matter-without-new-physics/, June 21, 2020.
[55]: Stephane H Maes, (2020), “Multi-Fold Universe Dark Matter Successful Explanation and the “Too Thin Universe” but “Too Strong Gravity Lensing by Galaxy Clusters””, viXra:2102.0079v1, https://shmaesphysics.wordpress.com/2020/09/15/multi-fold-universe-dark-matter-successful-explanation-and-the-too-thin-universe-but-too-strong-gravity-lensing-by-galaxy-clusters/, September 14, 2020.
[56]: Stephane H Maes, (2020), ”Multi-Fold Universe Dark Matter Effects Survive Low-Mass Galaxies with Dark Matter Deficits and Excesses”, viXra:2105.0042v1, https://shmaesphysics.wordpress.com/2020/10/14/multi-fold-universe-dark-matter-effects-survive-low-mass-galaxies-with-dark-matter-deficits-and-excesses/, October 14, 2020.
[57]: Stephane H Maes, (2020), ”Multi-Fold Dark Matter Effects and Early Supermassive Black Holes”, viXra:2105.0041v1, https://shmaesphysics.wordpress.com/2020/10/15/multi-fold-dark-matter-effects-and-early-supermassive-black-holes/, October 15, 2020.
[58]: Stephane H Maes, (2022), “Hints of Multi-fold Dark Matter Effects in the Universe”, osf.io/krw7g, https://shmaesphysics.wordpress.com/2022/03/14/hints-of-multi-fold-dark-matter-effects-in-the-universe/, March 14, 2022, https://zenodo.org/record/7791678.
[59]: Stephane H Maes, (2022), “Multi-fold Dark Matter and Energy Effects Fit The Ratios to Normal Matter in the Universe”, https://zenodo.org/doi/10.5281/zenodo.10071554, https://shmaesphysics.wordpress.com/2022/08/14/multi-fold-dark-matter-and-energy-effects-fit-the-ratios-to-normal-matter-in-the-universe/, August 14, 2022, (https://osf.io/mahsu, viXra:2311.0018v1).
[60]: Stephane H. Maes, (2022), “Explaining Imbalance of Tidally Ejected Stars from Open Stars Clusters Without MOND”, https://doi.org/10.5281/zenodo.10421124, https://shmaesphysics.wordpress.com/2022/11/19/explaining-imbalance-of-tidally-ejected-stars-from-open-stars-clusters-without-mond/, November 19, 2022, https://osf.io/bp64c.
[61]: Stephane H. Maes, {2022), “Black holes effects outside the black holes do not mean that Hawking radiation is not occurring at its horizon”, https://shmaesphysics.wordpress.com/different-approaches-to-compute-hawking-black-holes-decay/#comment-5027, November 23, 2022.
[62]: Stephane H Maes, (2022), “Multi-fold Discrete Fractal Spacetime, and the Viability of Local vs. Non-Local Hidden Variables”, https://doi.org/10.5281/zenodo.10344634, https://shmaesphysics.wordpress.com/2022/10/30/multi-fold-discrete-fractal-spacetime-and-the-viability-of-local-vs-non-local-hidden-variable-viability/, October 30, 2022, osf.io/qevys, viXra:2312.0065v1.
[63]: Stephane H Maes, (2021), “Multi-fold Embeddings, Space Time Matter Induction or Gravity Asymptotically Safe and The AdS/CFT Correspondence Conjecture, they all can recover the Standard Model”, viXra:2212.0120v1, https://shmaesphysics.wordpress.com/2021/12/20/multi-fold-embeddings-space-time-matter-induction-or-gravity-asymptotically-safe-and-the-ads-cft-correspondence-conjecture-they-all-can-recover-the-standard-model-or-smg/, December 20, 2021.
[64]: Stephane H. Maes, (2022), “A Better Quantum Extremal Surface and Island Interpretation that explains the Associated Massive Gravity”, https://doi.org/10.5281/zenodo.10437116, https://shmaesphysics.wordpress.com/2022/12/03/a-better-quantum-extremal-surface-and-island-interpretation-that-explains-the-associated-massive-gravity/, December 3, 2022, (https://osf.io/dn3kh).
[65]: Stephane H Maes, (2021), “Pointers to Nowhere with Geometric Unity Theory, or Some Ways Forward in Multi-fold Universes?”, viXra:2210.0081v1, https://shmaesphysics.wordpress.com/2021/03/07/pointers-to-nowhere-with-geometric-unity-theory-or-some-ways-forward-in-multi-fold-universes/, March 7, 2021.
[66]: Stephane H Maes, (2022), “The Replica Trick, Wormholes, Island formula, and Quantum Extremal Surfaces, and How the AdS/CFT Correspondence Conjecture, and Hence the M-theory, Encounters Multi-folds”, https://doi.org/10.5281/zenodo.10207057, https://shmaesphysics.wordpress.com/2022/09/20/the-replica-trick-its-wormholes-islands-and-quantum-extremal-surfaces-and-how-the-ads-cft-correspondence-conjecture-and-hence-the-m-theory-encounters-multi-folds/, September 26, 2022, (osf.io/xwf6q/). Also published as: Stephane H Maes, (2022), “The Replica Trick, Wormholes, Island formula, and Quantum Extremal Surfaces”, September 26, 2022 (viXra:2311.0154v1).
[67]: Stephane H Maes, (2021), “Right-handed Neutrinos and Traversable Wormholes: the key to entanglement, gravity and multi-folds extensions to ER=EPR?”, viXra:2211.0173v1, https://shmaesphysics.wordpress.com/2021/04/03/right-handed-neutrinos-and-traversable-wormholes-the-key-to-entanglement-gravity-and-multi-folds-extensions-to-erepr/, April 3, 2021.
[68]: Stephane H Maes, (2021), “Multi-fold Non-Commutative Spacetime, Higgs and The Standard Model with Gravity”, viXra:2212.0037v1, https://shmaesphysics.wordpress.com/2021/04/18/multi-fold-non-commutative-spacetime-higgs-and-the-standard-model-with-gravity/, April 11, 2021.
[69]: Stephane H Maes, (2022), “Trans-Planckian Censorship Conjecture: Factual in Multi-fold Universes as well as GR Universes”, viXra:2303.0025v1, https://shmaesphysics.wordpress.com/2022/03/13/trans-planckian-censorship-conjecture-factual-in-multi-fold-universes-as-well-as-gr-universes/, March 12, 2022.
[70]: Stephane H Maes, (2021), “Spacetime and Gravity are 2D around Planck Scales: A Universal Property of Consistent Quantum Gravity”, viXra:2211.0001v1, https://shmaesphysics.wordpress.com/2021/03/23/spacetime-and-gravity-are-2d-around-planck-scales-a-universal-property-of-consistent-quantum-gravity/, March 20, 2021.
[71]: Stephane H Maes, (2020), “The W-type Multi-Fold Hypothesis and Quantum Physics Interpretation of wave Functions and QFT”, viXra:2207.0118v1, https://shmaesphysics.wordpress.com/2020/12/24/the-w-type-multi-fold-hypothesis-and-quantum-physics-interpretation-of-wave-functions-and-qft/, December 20, 2020.
[72]: Stephane H Maes, (2022), “2D Random Walks of Massless Higgs Bosons as Microscopic Interpretation of the Asymptotic Safety of Gravity, and of the Standard Model”, https://doi.org/10.5281/zenodo.10467452, https://shmaesphysics.wordpress.com/2022/12/28/2d-random-walks-of-massless-higgs-bosons-as-microscopic-interpretation-of-the-asymptotic-safety-of-gravity-and-of-the-standard-model/, December 28, 2022, (osf.io/udhbf). Also published as: Stephane H. Maes, (2022), “2D Random Walks of Massless Higgs Bosons”, viXra:2401.0073v1, https://shmaesphysics.wordpress.com/2022/12/28/2d-random-walks-of-massless-higgs-bosons-as-microscopic-interpretation-of-the-asymptotic-safety-of-gravity-and-of-the-standard-model/, December 28, 2022.
[73]: Stephane H Maes, (2020), “No Gravity Induced Wave Function Collapse in a Multi-fold Universe”, viXra:2012.0152v1, https://shmaesphysics.wordpress.com/2020/09/11/no-gravity-induced-wave-function-collapse-in-a-multi-fold-universe/, September 11, 2020. Also as: Stephane H Maes, (2020), “No Gravity Superposition Induced Wave Function Collapse in a Multi-fold Universe”, viXra:2012.0152v1, https://shmaesphysics.wordpress.com/2020/09/11/no-gravity-induced-wave-function-collapse-in-a-multi-fold-universe/, September 11, 2020.
[74]: Stephane H. Maes, (2022), “Multi-fold Gravity can Violate P-Symmetry. It is Aligned With Observations of Asymmetry of the Orientation of Tetrahedra of Galaxies”, https://doi.org/10.5281/zenodo.10443847, https://shmaesphysics.wordpress.com/2022/12/10/multi-fold-gravity-can-violate-p-symmetry-it-is-aligned-with-observations-of-asymmetry-of-the-orientation-of-tetrahedra-of-galaxies/, December 10, 2022. Also published as Stephane H. Maes, (2022), “Multi-fold Gravity can Violate Parity Symmetry”, https://shmaesphysics.wordpress.com/2022/12/10/multi-fold-gravity-can-violate-p-symmetry-it-is-aligned-with-observations-of-asymmetry-of-the-orientation-of-tetrahedra-of-galaxies/, December 10, 2022.
[75]: Stephane H Maes, (2021), ““Quantum Gravity Emergence from Entanglement in a Multi-Fold Universe”: 2D or 2+1D spacetime at small scales”, viXra:2103.0142, https://shmaesphysics.wordpress.com/2021/03/20/quantum-gravity-emergence-from-entanglement-in-a-multi-fold-universe-2d-or-21d-spacetime-at-small-scales/, March 20, 2021.
[76]: Stephane H Maes, (2022), “Comments on Multi-fold mechanisms as Hermitian vs. Unitary processes”, https://shmaesphysics.wordpress.com/2020/06/25/gravity-like-attractions-and-fluctuations-between-entangled-systems/#comment-4359, July 27, 2022.
[77]: Stephane H. Maes, (2021), “Comment on 4D spacetime and follow-up comments”, https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/#comment-1416, January 16, 2021. Retrieved on February 21, 2021.
[78]: Stephane H. Maes, (2021), “Comment on 4D spacetime and follow-up comments: https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/#comment-1579, February 18, 2021. Retrieved on February 21, 2021.
[79]: Stephane H. Maes, (2021), “Comment on 2D spacetime and follow-up comments”, https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/#comment-1695, March 3, 2021. Retrieved on March 31, 2021.
[80]: Stephane H. Maes, (2021), “Comment on 4D spacetime and follow-up comments”, https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/#comment-1891, March 30, 2021. Retrieved on March 31, 2021.
[81]: Stephane H. Maes, (2021), “Comment on 4D spacetime and follow-up comments”, https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/#comment-1906, April 1, 2021. Retrieved on December 27, 2022.
[82]: Stephane H. Maes, (2022), “Additional arguments for 4D spacetime for our real universe”, https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/#comment-4679, September 21, 2022. Retrieved on December 27, 2022.
[83]: Stephane H Maes, (2020), “Call for Collaboration”, https://shmaesphysics.wordpress.com/2020/09/07/do-you-want-a-phd-or-who-knows-a-nobel-price-in-physics/, September 6, 2020.
[84]: Stephane H Maes, (2020), “Multi-fold Gravitons In-N-Out Spacetime”, viXra:2010.0155v1, https://shmaesphysics.wordpress.com/2020/07/27/multi-fold-gravitons-in-n-out-spacetime/, July 27, 2020, (posted September 6, 2020)
[85]: Stephane H Maes, (2022), “Gravitational Bootstrap, S-matrix, Superstrings, and The Plausible Unphysicality of Gravitons”, viXra:2301.0155v1, https://shmaesphysics.wordpress.com/2022/02/06/gravitational-bootstrap-s-matrix-superstrings-and-the-plausible-unphysicality-of-gravitons/, February 6, 2022.
[86]: Stephane H Maes, (2020), “Particles, Especially Virtual Particles, in a Multi-fold Universe vs. QFT”, viXra:2010.0133v1, https://shmaesphysics.wordpress.com/2020/07/11/particles-especially-virtual-particles-in-a-multi-fold-universe-vs-qft/ , July 10, 2020.
[87]: Stephane H Maes, (2020), “Comments to “Yes, Stephen Hawking Lied To Us All About How Black Holes Decay””, https://osf.io/v7thb/, https://shmaesphysics.wordpress.com/2020/07/11/comments-to-yes-stephen-hawking-lied-to-us-all-about-how-black-holes-decay/, July 11, 2020.
[88]: Stephane H Maes, (2020), “Different approaches to compute Hawking Black Holes Decay”, viXra:2208.0009v1, https://shmaesphysics.wordpress.com/different-approaches-to-compute-hawking-black-holes-decay/, August 1, 2022. (Originally published July 11, 2020).
[89]: Stephane H Maes, (2020), “Multi-Fold Black Holes: Entropy, Evolution and Quantum Extrema”, viXra:2105.0136v1, https://shmaesphysics.wordpress.com/2020/11/01/multi-fold-black-holes-entropy-evolution-and-quantum-extrema/, October 31, 2020.
[90]: Stephane H Maes, (2020), “No Gravity Shield in Multi-folds Universes”, viXra:2010.0032v1, https://shmaesphysics.wordpress.com/2020/06/26/no-gravity-shields-in-multi-folds-universes/ , June 26, 2020.
[91]: Stephane H Maes, (2020), “Area Laws Between Multi-Fold Universes and AdS”, viXra:2010.0207v1, https://shmaesphysics.wordpress.com/2020/08/10/area-laws-between-multi-fold-universes-and-ads/, August 10, 2020.
[92]: Stephane H. Maes, (2022), “Comments on radiation black hole simulation on a lattice”, https://shmaesphysics.wordpress.com/2022/07/25/unruh-effects-hawking-black-hole-evaporation-quantum-corrected-larmor-formula-numbers-of-particles-in-curved-spacetime-same-same-but-just-a-bit-different/#comment-5099, December 2, 2022.
[93]: Stephane H. Maes, (2021), “Neutrons are forming an external skin in Nuclei and Neutron Stars”, https://zenodo.org/doi/10.5281/zenodo.14582585, https://shmaes.wordpress.com/2021/05/08/neutrons-are-forming-an-external-skin-in-nuclei-and-neutron-stars/, May 8, 2021. (V1) (V3 is January 7, 2024). (osf.io/zdy4s/, viXra:2501.0029).
[94]: Stephane H Maes, (2022), “The Yang Mills Double Copy leads to New AdS/CFT + Gravity Correspondences, or How the M-theory encounters Multi-fold Universes”, v1.1, https://doi.org/10.5281/zenodo.7827248, https://shmaesphysics.wordpress.com/2022/04/22/the-yang-mills-double-copy-leads-to-new-ads-cft-gravity-correspondences-or-how-the-m-theory-encounters-multi-fold-universes/, April 22, 2022. (v1: at zenodo.7827249).
[95]: Stephane H. Maes, (2022), “Schwinger effect and charged black holes”, https://shmaesphysics.wordpress.com/2020/11/01/multi-fold-black-holes-entropy-evolution-and-quantum-extrema/#comment-4686, September 25, 2022.
[96]: Stephane H. Maes, “Comments on asymmetry of distributions of ejected star from gas clusters”, https://shmaesphysics.wordpress.com/2020/06/21/explaining-dark-matter-without-new-physics/#comment-4813 and subsequent comments, October 27, 2022
[97]: Stephane H Maes, (2020), “Implicit Multi-Fold Mechanisms in a Neural Network Model of the Universe”, viXra:2012.0191v1, https://shmaesphysics.wordpress.com/2020/09/12/implicit-multi-fold-mechanisms-in-a-neural-network-model-of-the-universe/, September 12, 2020.
[98]: Stephane H Maes, (2020), “Interpretation of “Neural Network as the World””, viXra:2012.0197v1, https://shmaesphysics.wordpress.com/2020/09/14/interpretation-of-neural-network-as-the-world/, September 14, 2020.
[99]: Stephane H Maes, (2020), “Entangled Neural Networks from Multi-fold Universes to Biology”, viXra:2207.0174v1, https://shmaesphysics.wordpress.com/2020/12/31/entangled-neural-networks-from-multi-fold-universes-to-biology/, December 25, 2020.
[100]: Wikipedia, “Lambda-CDM model”, https://en.wikipedia.org/wiki/Lambda-CDM_model. Retrieved on August 14, 2022.
[101]: Stephane H. Maes, (2022), “CO2 and CH4 absorption powered by nuclear fusion, via fission, is the only way to manage climate change and the Planet’s trigger points”, viXra:2211.0154v1, https://shmaes.wordpress.com/2022/04/09/co2-and-ch4-absorption-powered-fission-is-the-only-way-to-manage-climate-change-and-the-planets-trigger-points/, April 9, 2022.
[102]: Stephane H Maes, (2021), “Oops For The Loops: Mounting LQG Woes And A Challenge To The LQG Community”, viXra:2212.0168, https://shmaesphysics.wordpress.com/2021/12/30/oops-for-loops-mounting-lqg-woes-and-a-challenge-to-the-lgq-community/, December 29, 2021.
[103]: Stephane H. Maes, (2021-2022), “Our universe is 4D”, Comments and following comments at https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/#comment-1416. January 16, 2021, and after.
[104]: Stephane H Maes, (2021), “Multi-fold gravity and double copy of gauge theory”, osf.io/xun82, https://shmaesphysics.wordpress.com/2021/05/04/multi-fold-gravity-and-double-copy-of-gauge-theory/, May 4, 2021, viXra:2303.0114.
[105]: Stephane H Maes, (2020), “Entanglement Concretizes Time in a Multi-fold Universe”, viXra:2010.0083v1, https://shmaesphysics.wordpress.com/2020/06/28/entanglement-concretizes-time-in-a-multi-fold-universe/, June 28, 2020. Also published as: Stephane H Maes, (2020), “Entanglement and Random Walks Concretize Time in a Multi-fold Universe”, viXra:2010.0083v1, https://shmaesphysics.wordpress.com/2020/06/28/entanglement-concretizes-time-in-a-multi-fold-universe/, June 28, 2020.
[106]: Stephane H Maes, (2021), “How the ER = EPR, GR = QM and AdS/CFT correspondence conjectures, can be explained in multi-fold theory, along with the E/G conjecture. A call to the Physics Community!”, viXra:2111.0144v2, https://shmaesphysics.wordpress.com/2021/11/28/how-the-er-epr-gr-qm-and-ads-cft-correspondence-conjectures-can-be-explained-in-multi-fold-theory-and-the-e-g-conjecture-explains-and-realize-in-a-multi-fold-universe-a-call-to-the-physics-comm/, December 28, 2021.
[107]: Stephane H Maes, (2020), “A Multi-fold Universe Genesis Inspired By Explosive Total Collision: The Source Of The Big Bang?”, viXra:2208.0082v1, https://shmaesphysics.wordpress.com/2021/01/17/a-multi-fold-universe-genesis-inspired-by-total-explosion-collision-the-source-of-the-big-bang/, January 12, 2021.
[108]: Stephane H. Maes, (2022), “JWST and the Big Bang invalidation”, https://shmaesphysics.wordpress.com/2021/01/17/a-multi-fold-universe-genesis-inspired-by-total-explosion-collision-the-source-of-the-big-bang/#comment-4577, and following comments. August 21, 2022.
[109]: Stephane H. Maes, (2022), “Schwinger effect dominates near the horizon of charged black holes near extremality and reduces the charge”, https://shmaesphysics.wordpress.com/2022/07/25/unruh-effects-hawking-black-hole-evaporation-quantum-corrected-larmor-formula-numbers-of-particles-in-curved-spacetime-same-same-but-just-a-bit-different/#comment-4687, September 23, 2022.
[110] Stephane H Maes, (2022), “Charm of the proton”, https://shmaesphysics.wordpress.com/2021/03/29/new-physics-with-lhcb-to-explain-loss-of-lepton-universality-or-just-gravity/#comment-3791, March 27, 2022.
[111]: Stephane H. Maes, (2022-2023), “Confusing mathematical duality to predict quantum computing algorithm, with building a wormhole”, https://shmaesphysics.wordpress.com/2020/10/11/circular-arguments-in-string-and-superstring-theory-from-a-multi-fold-universe-perspective/comment-page-1/#comment-5093, and following related comments, November 30, 2022.
[112]: Stephane H. Maes, (2023), “No supersymmetry”, https://shmaesphysics.wordpress.com/2023/11/21/no-supersymmetry/, November 21,2023.
[113]: Stephane H. Maes, (2023), “Justification for the multi-fold mappings, and dynamic multi-fold mechanism”, https://shmaesphysics.wordpress.com/2020/12/24/the-w-type-multi-fold-hypothesis-and-quantum-physics-interpretation-of-wave-functions-and-qft/comment-page-1/#comment-8092, October 29, 2023.
[114]: Stephane H. Maes, (2023), “Yeah or Nay on Black Holes as Explanation for Dark Energy?”, osf.io/369pd, https://shmaesphysics.wordpress.com/2023/03/01/yeah-or-nay-on-black-holes-as-explanation-for-dark-energy/, V3, March 26, 2023. (V2: March 12, 2023, V1: Stephane H. Maes, (2023), “Yeah or Nay on Black Holes as Explanation for Dark Energy?”, viXra:2303.0031, https://shmaesphysics.wordpress.com/2023/03/01/yeah-or-nay-on-black-holes-as-explanation-for-dark-energy/, March 1, 2023).
[115]: Stephane H. Maes, (2023), “Dynamic sources, Dynamic Multi-folds, and General Relativity Lense-Thirring and Frame Dragging Effects”, https://doi.org/10.5281/zenodo.14737010, https://shmaesphysics.wordpress.com/2023/03/12/dynamic-sources-dynamic-multi-folds-and-general-relativity-lens-thirring-and-frame-dragging-effects/, March 12, 2023, https://osf.io/ytmw6/download/
[116]: Stephane H. Maes, (2023), “The Multi-fold Least Action Principle, a Quasi Theory Of Everything”, https://doi.org/10.5281/zenodo.14542569, https://shmaesphysics.wordpress.com/2023/02/19/the-multi-fold-least-action-principle-a-quasi-theory-of-everything/, February 19, 2023. (osf.io/2ncqf/, viXra:2412.0145v1).
[117]: Stephane H. Maes, (2023), “Maybe, black holes do not systematically decohere quantum states”, https://shmaesphysics.wordpress.com/2020/11/01/multi-fold-black-holes-entropy-evolution-and-quantum-extrema/#comment-6315, March 7, 2023.
[118]: Stephane H. Maes, (2023), “No electroweak / Higgs mass hierarchy problem in multi-fold theory”, https://shmaesphysics.wordpress.com/2021/03/28/multi-fold-gravity-electroweak-theory-and-symmetry-breaking/#comment-6794, March 30, 2023.
[119]: Stephane H. Maes, “Right-handed neutrinos in the multi-fold stabilize the multi-fold unconstrained KK space time matter induction and scattering”, https://shmaesphysics.wordpress.com/2021/04/03/right-handed-neutrinos-and-traversable-wormholes-the-key-to-entanglement-gravity-and-multi-folds-extensions-to-erepr/comment-page-1/#comment-6875, April 8, 2023.
[120]: Stephane H. Maes, (2023), “No lack of clumpiness, just as needed”, https://shmaesphysics.wordpress.com/2020/06/21/explaining-dark-matter-without-new-physics/comment-page-1/#comment-6974. April 12, 2023.
[121]: Stephane H. Maes, (2023), “Multi-fold Universes, Multiverses and Many Worlds”, https://doi.org/10.5281/zenodo.15339413, https://shmaesphysics.wordpress.com/2023/04/08/multi-fold-universes-multi-folds-and-many-worlds/, April 8, 2023.
[122]: Stephane H. Maes, (2023), ‘Comment on black hole decoherence”, https://shmaesphysics.wordpress.com/2020/11/01/multi-fold-black-holes-entropy-evolution-and-quantum-extrema/#comment-6315, March 7, 2023.
[123]: Stephane H Maes, (2023), “Comments of the universe is too smooth”, https://shmaesphysics.wordpress.com/2020/06/21/explaining-dark-matter-without-new-physics/#comment-6086, February 9, 2023, and https://shmaesphysics.wordpress.com/2020/06/21/explaining-dark-matter-without-new-physics/#comment-6974, April 12, 2023.
[124]: Stephane H Maes, (2023), “Our real universe is macroscopically 4D. Hints come from every direction & show that it had to be so”, https://doi.org/10.5281/zenodo.17575694, https://shmaesphysics.wordpress.com/2023/04/23/our-real-universe-is-macroscopically-4d-hints-come-from-every-directions-show-that-it-had-to-be-so/, April 23, 2023 (viXra:2511.0054v1).
[125]: Stephane H Maes, (2023), “No Gravitational Evaporation of Everything à la Schwinger, only for Black Holes”, https://shmaesphysics.wordpress.com/2023/07/15/no-gravitational-evaporation-of-everything-a-la-schwinger-only-for-black-holes/, July 15, 2023.
[126]: Stephane H Maes, (2023), “Unstable QFT and SM with Gravity except in a Multi-fold Universe”, https://shmaesphysics.wordpress.com/2023/07/19/unstable-qft-and-sm-with-gravity-except-in-a-multi-fold-universe/, July 19, 2023.
[127]: Stephane H. Maes, (2023), “Comments about massive galaxies without dark matter”, https://shmaesphysics.wordpress.com/2020/10/14/multi-fold-universe-dark-matter-effects-survive-low-mass-galaxies-with-dark-matter-deficits-and-excesses/#comment-7430, July 20, 2023.
[128]: Stephane H Maes, (2023), “Less Cracks in the Standard Cosmology in a Multi-fold Universe with its Quantum Random walks”, https://shmaesphysics.wordpress.com/2023/06/20/less-cracks-in-the-standard-cosmology-in-a-multi-fold-universe-with-its-quantum-random-walks/, June 19, 2023.
[129]: Stephane H. Maes, (2023), “Ad Astra With Warp Drives? Probably Not”, https://shmaesphysics.wordpress.com/2023/12/09/ad-astra-longe-with-warp-drives-probably-not/, December 9, 2023.
[130]: Stephane H Maes, (2023), “2D gravity and 2D Yang Mills Physics is all what matters”, https://shmaesphysics.wordpress.com/2023/04/23/our-real-universe-is-macroscopically-4d-hints-come-from-every-directions-show-that-it-had-to-be-so/comment-page-1/#comment-7588, August 5, 2023.
[131]: Stephane H Maes (2023), “The Multi-fold Theory – Draft Raw Compendium of Research Papers (till August, 2023)”, https://doi.org/10.5281/zenodo.8242021, https://shmaesphysics.wordpress.com/2023/08/12/the-multi-fold-theory-draft-raw-compendium-of-research-papers-till-august-2023/, August 12, 2023, (https://osf.io/swqmb).
[133]: Stephane H. Maes, (2023), “Persisting on No Decoherence due to Gravity, Black Holes, or Spacetime Curvature Superpositions”, https://shmaesphysics.wordpress.com/2023/08/18/persisting-on-no-decoherence-due-to-gravity-black-holes-or-spacetime-curvature-superpositions/, August 18, 2023.
[132]: Stephane H Maes, (2023), “Barnett’s resolution of the Minkowski – Abraham dilemma holds, no 4-vector issue”, https://zenodo.org/records/10071847, https://shmaes.wordpress.com/2023/08/11/barnetts-resolution-of-the-minkowski-abraham-dilemma-holds-no-4-vector-issue/ August 13, 2023, (https://osf.io/bd8ju, viXra:2311.0023v1).
[134]: Stephane H. Maes, (2023), “No supersymmetry at D<=4 with a positive cosmological constant”, https://shmaesphysics.wordpress.com/2022/07/08/a-prediction-no-dark-matter-will-be-discovered-at-lhc-or-elsewhere/#comment-7563. July 23, 2023.
[135]: Stephane H. Maes, (2023), “The universe is exactly the only thing that it could be if it is a 4D multi-fold universe! No fine-tuning problem, no invocation of God or multiverses”, https://shmaesphysics.wordpress.com/2023/04/08/multi-fold-universes-multi-folds-and-many-worlds/comment-page-1/#comment-8037, October 7, 2023.
[136]: Stephane H. Maes, (2020-2023), “Quantum Gravity Emergence from Entanglement in a Multi-Fold Universe – V3: Update to section 4.1 – Multi-folds for Entanglement and EPR”, https://shmaesphysics.wordpress.com/quantum-gravity-emergence-from-entanglement-in-a-multi-fold-universe-v3-update-to-section-4-1-multi-folds-for-entanglement-and-epr/.
[137]: Stephane H. Maes, (2020-2023), “Quantum Gravity Emergence from Entanglement in a Multi-Fold Universe”, V3, https://zenodo.org/doi/10.5281/zenodo.7792911, October 29, 2023.
[138]: Stephane H. Maes, (2023), “Path integrals and wormholes impact on the cosmological constant”, https://shmaesphysics.wordpress.com/2022/09/20/the-replica-trick-its-wormholes-islands-and-quantum-extremal-surfaces-and-how-the-ads-cft-correspondence-conjecture-and-hence-the-m-theory-encounters-multi-folds/comment-page-1/#comment-7978, September 3, 2023.
[139]: Stephane H. Maes, (2023), “Microscopic interpretation of mass acquisition from massless Higgs bosons”, https://shmaesphysics.wordpress.com/2021/02/28/more-on-multi-fold-particles-as-microscopic-black-holes-with-higgs-regularizing-extremality-and-singularities/#comment-8412, November 5, 2023.
[140]: Stephane H. Maes, (2023), “Justifying the Multi-folds Mechanisms, Mapping, Tenancy and More”, https://shmaesphysics.wordpress.com/2023/11/10/justifying-the-multi-folds-mechanisms-mapping-tenancy-and-more/, November 10, 2023.
[141]: Stephane H. Maes, (2023), “In multi-fold theory, the expansion of the universe is not a mirage”, https://shmaesphysics.wordpress.com/2020/06/19/explaining-dark-energy-small-cosmological-constant-and-inflation-without-new-physics/#comment-7242, June 20, 2023.
[142]: Stephane H. Maes, “Multi-fold dark energy is also a fluctuation of quantum vacuum fluctuations”, https://shmaesphysics.wordpress.com/2020/06/19/explaining-dark-energy-small-cosmological-constant-and-inflation-without-new-physics/#comment-6111, February 18, 2023.
[143]: Stephane H. Maes, (2023), “It’s experimentally validated: antimatter falls down, no antigravity”, https://shmaesphysics.wordpress.com/2020/07/05/more-matter-than-antimatter-all-falling-down/#comment-8018 and subsequent comments, September 27, 2023.
[144]: Stephane H. Maes, (2023), “Particle internal symmetries and anti-particles when modeled as microscopic black holes or random walk patterns”, https://shmaesphysics.wordpress.com/2021/02/28/more-on-multi-fold-particles-as-microscopic-black-holes-with-higgs-regularizing-extremality-and-singularities/#comment-8634, November 28, 2023.
[145]: Stephane H. Maes, (2023), “Information has no mass”, https://shmaesphysics.wordpress.com/2023/12/14/information-has-no-mass/, December 14, 2023
[146]: Stephane H. Maes, (2023), “Gravity is Quantum”, https://shmaesphysics.wordpress.com/2023/12/19/gravity-is-quantum/, December 19, 2023.
[147]: Stephane H. Maes, (2023), “Multi-folds for Entanglement and EPR”, https://zenodo.org/doi/10.5281/zenodo.10059877, https://shmaesphysics.wordpress.com/2023/11/01/multi-folds-for-entanglement-and-epr/, October 29, 2023, (viXra:2311.0001v1), also as “Update to section 4.1 of “Quantum Gravity Emergence from Entanglement in a Multi-Fold Universe” – Multi-folds for Entanglement and EPR”, https://shmaesphysics.wordpress.com/2023/10/29/update-to-section-4-1-of-quantum-gravity-emergence-from-entanglement-in-a-multi-fold-universe-multi-folds-for-entanglement-and-epr/, (https://osf.io/54ycm/).
[148]: Stephane H. Maes, (2023), “A Fractal spacetime just leads to rescaled cosmological constant. Yet that may provide a time varying effect”, https://shmaesphysics.wordpress.com/2022/10/30/multi-fold-discrete-fractal-spacetime-and-the-viability-of-local-vs-non-local-hidden-variable-viability/comment-page-1/#comment-8896. December 14, 2023.
[149]: Stephane H. Maes, (2023), “Multi-Fold dark Matter effects & Rotation Curve Differences in Galaxies in Clusters, Yet Respect of the Strong Equivalence Principle”, https://doi.org/10.5281/zenodo.13766006, https://shmaesphysics.wordpress.com/2023/01/29/multi-fold-dark-matter-effects-rotation-curve-differences-in-galaxies-in-custers-yet-respect-of-the-strong-equivalence-principle/, January 29, 2023, (osf.io/texvj, vixra:2409.0088v1).
[150]: Stephane H. Maes, (2022), “Multi-folds, Non-Commutative Spacetime, Spin, and All That”, https://doi.org/10.5281/zenodo.11114501, https://shmaesphysics.wordpress.com/2022/12/31/the-principles-of-quantum-mechanics/, December 31, 2022. Also, as https://shmaesphysics.wordpress.com/2022/12/31/multi-folds-non-commutative-spacetime-spin-and-all-that/, (osf.io/au7wc, viXra:2405.0022v1).
[151]: Stephane H. Maes, (2024), “Collapses, Singularities, Censorship, Conjectures, and More”, https://shmaesphysics.wordpress.com/2024/07/16/collapses-singularities-censorship-conjectures-and-more/, July 16, 2024.
[152]: Stephane H. Maes, (2024), “June 2024 Status of the Multi-fold Theory”, https://doi.org/10.5281/zenodo.13345964, https://shmaesphysics.wordpress.com/2024/08/19/june-2024-status-of-the-multi-fold-theory/, June 22, 2024.
[153]: Stephane H. Maes, (2024), “About binaries alleged gravity anomalies”, https://shmaesphysics.wordpress.com/2023/01/29/multi-fold-dark-matter-effects-rotation-curve-differences-in-galaxies-in-custers-yet-respect-of-the-strong-equivalence-principle/#comment-9726, September 10,
[154]: Stephane H Maes, (2024), “A different and more generic proof based on the multi-fold theory, that confinement implies mass effects and chiral symmetry breaking”, https://shmaesphysics.wordpress.com/2021/02/28/more-on-multi-fold-particles-as-microscopic-black-holes-with-higgs-regularizing-extremality-and-singularities/comment-page-1/#comment-9952, November 3, 2024.
[155]: Stephane H. Maes, (2024), “A discrete multi-fold spacetime realized by random walk, implies a non-commutative spacetime”, https://shmaesphysics.wordpress.com/2022/12/31/multi-folds-non-commutative-spacetime-spin-and-all-that/#comment-9820, September 23, 2024.
[156]: Stephane H Maes, (2021), “Comments on “No issue of unnaturalness and mass hierarchy with the Higgs mass””, https://shmaesphysics.wordpress.com/2021/04/27/new-physics-is-often-not-so-new/comment-page-1/#comment-3026, December 24, 2021.
[157]: Stephane H. Maes, (2024), “Physics Salvages The Third Law Of Black Hole Thermodynamics”, https://shmaesphysics.wordpress.com/2024/11/24/physics-salvages-the-third-law-of-black-hole-thermodynamics/, November 24, 2024.
[158]: Stephane H. Maes, “No Issue with the Quantization of Electrostatic Fields”, https://shmaesphysics.wordpress.com/2024/12/31/no-issue-with-the-quantization-of-electrostatic-fields/, December 31, 2024.
[159]: Stephane H. Maes, (2025), “Looks like we were right. Neutrinos are probably not Majorana Fermions”, https://shmaesphysics.wordpress.com/2020/06/21/right-handed-neutrinos-ask-gravity/#comment-10501, March 23, 2025.
[160]: Stephane H. Maes, (2024), “Gödel’s incompleteness theorems imply no full theory is possible”, https://shmaesphysics.wordpress.com/2023/02/19/the-multi-fold-least-action-principle-a-quasi-theory-of-everything/#comment-10066, November 30, 2024. Based on: Stephane H. Maes, (1989), “Feynman Path Integrals”, and communication to Prof. J. Weyers, Quantum Physics Seminar as part of BS in Physics, FYMA, UC Louvain.
[161]: Stephane H. Maes, (2024), “Spin as angular momentum of massless Higgs bosons in Higgs condensate or random walks”, https://shmaesphysics.wordpress.com/2022/12/31/multi-folds-non-commutative-spacetime-spin-and-all-that/#comment-10027, November 26, 2024.
[162]: Stephane H. Maes, (2025), “No Extremality or Singularity for Particles as Multi-fold Black Holes”, https://shmaesphysics.wordpress.com/2025/03/27/no-extremality-or-singularity-for-particles-as-multi-fold-black-holes/, March 27, 2025.
[163]: Stephane H. Maes, (2024), “Explaining the mass anisotropy of certain semi-Dirac Fermions in semimetals”, https://shmaesphysics.wordpress.com/2021/02/28/more-on-multi-fold-particles-as-microscopic-black-holes-with-higgs-regularizing-extremality-and-singularities/comment-page-1/#comment-9951, November 3, 2024.
[164]: Stephane H. Maes, (2024), “Gravity in SMG contributes in the right direction to the discrepancies of CP violation in B-mesons with the SM”, https://shmaesphysics.wordpress.com/2022/07/08/a-prediction-no-dark-matter-will-be-discovered-at-lhc-or-elsewhere/#comment-10078, December 6, 2024.
[165]: Stephane H. Maes, (2024), “As we predicted, still no sterile neutrino”, https://shmaesphysics.wordpress.com/2020/10/02/no-conventional-sterile-neutrinos-in-a-multi-fold-universe-just-smg-business-as-usual/#comment-10039, November 26, 2024.
[166]: David Tong, (2018), “Gauge Theory”, Cambridge University, https://www.damtp.cam.ac.uk/user/tong/gaugetheory/gt.pdf.
[167]: Stephane H. Maes, (2025), “Early inflation dS instabilities, require a mechanisms like the multi-fold spacetime reconstruction with 2D random walks of massless bosons”, https://shmaesphysics.wordpress.com/2024/12/12/consistencies-and-implications-of-2d-massless-random-walks-discrete-non-commutative-spacetime-and-no-flat-supersymmetry/comment-page-1/#comment-11204, November 4, 2025.
[168]: Stephane H. Maes, (2024), “Continuous mathematics to model Physics are just degenerate approximations of finite / discrete mathematics”, https://shmaesphysics.wordpress.com/2023/11/21/no-supersymmetry/#comment-9457, July 13, 2024.
[169]: Stephane H. Maes, (2022), “Background independence implies discreteness”, https://shmaesphysics.wordpress.com/2021/04/18/multi-fold-non-commutative-spacetime-higgs-and-the-standard-model-with-gravity/comment-page-1/#comment-3304, January 11, 2022.
[170]: Stephane H. Maes, “Consistencies and Implications of 2D Massless Random Walks: Discrete Non-commutative Spacetime, and No Flat Supersymmetry”, https://shmaesphysics.wordpress.com/2024/12/12/consistencies-and-implications-of-2d-massless-random-walks-discrete-non-commutative-spacetime-and-no-flat-supersymmetry/, December 12, 2024.
[171]: nLab, “Skyrmions”, https://ncatlab.org/nlab/show/skyrmion. Retrieved for this paper on December 10, 2024.
[172] : Wikipedia, “Skyrmion”, https://en.wikipedia.org/wiki/Skyrmion. Retrieved for this paper on December 10, 2024.
[173]: Stephane H. Maes, (2024), “QFT on discrete spacetime is OK”, https://shmaesphysics.wordpress.com/2020/12/13/viable-lattice-spacetime-and-absence-of-quantum-gravitational-anomalies-in-a-multi-fold-universe/comment-page-1/#comment-10322, December 30, 2024.
[174]: Stephane H. Maes, (2025), “Mathematics, Physics / QFT on discrete spacetime is more general than Mathematics and Physics / QFT on continuous spacetime”, https://shmaesphysics.wordpress.com/2022/10/30/multi-fold-discrete-fractal-spacetime-and-the-viability-of-local-vs-non-local-hidden-variable-viability/#comment-9455, July 13, 2024.
[175]: Stephane H. Maes, (2025), “QFT on discrete and / or non-commutative spacetime”, https://shmaesphysics.wordpress.com/2024/12/12/consistencies-and-implications-of-2d-massless-random-walks-discrete-non-commutative-spacetime-and-no-flat-supersymmetry/#comment-10326, January 7, 2025.
[176]: Stephane H. Maes, (2025), “Preons as Massless Higgs Bosons”, https://shmaesphysics.wordpress.com/2025/01/09/preons-as-massless-higgs-bosons/, January 9, 2025.
[177]: Stephane H. Maes, (2025), “Microscopic Interpretation of The Gravity Electroweak Symmetry Breaking”, https://doi.org/10.5281/zenodo.15099220, https://shmaesphysics.wordpress.com/2025/03/27/microscopic-interpretation-of-the-gravity-electroweak-symmetry-breaking/, March 27, 2025. (https://osf.io/q5erz).
[178]: Stephane H. Maes, (2025), “Gravity From Relative Entropic Action Is Not Necessarily Entropic Gravity”, https://shmaesphysics.wordpress.com/2025/04/06/gravity-from-relative-entropic-action-is-not-necessarily-entropic-gravity/, April 6, 2025.
[179]: Stephane H Maes, (2025), “Time-Varying Dark Energy Effect, How to Beat a Dead Horse. No, It Not Provide A First Evidence of String Theory”, https://shmaesphysics.wordpress.com/2020/10/11/circular-arguments-in-string-and-superstring-theory-from-a-multi-fold-universe-perspective/#comment-10609, April 5, 2025.
[180]: Stephane H. Maes, (2021), “Comments on CPT / anti-universe”, https://shmaesphysics.wordpress.com/2020/10/02/no-conventional-sterile-neutrinos-in-a-multi-fold-universe-just-smg-business-as-usual/#comment-1489 and comments after, January 30, 2021.
[181]: Stephane H. Maes, (2025), “2D Random Walks Imply a Strictly Positive Cosmological Constant, Possibly Variable in Time”, https://shmaesphysics.wordpress.com/2025/04/24/2d-random-walks-imply-a-strictly-positive-cosmological-constant-possibly-variable-in-time/, April 24, 2025.
[182]: Stephane H. Maes, (2025), “No Hilbert Einstein Action With Positive Dark Energy / Cosmological Constant From A String Action”, https://zenodo.org/doi/10.5281/zenodo.15385159, https://shmaesphysics.wordpress.com/2025/05/08/no-hilbert-einstein-action-with-positive-dark-energy-cosmological-constant-from-a-strings-action/, May8, 2025, (https://osf.io/7nh45/download/).
[183]: Stephane H. Maes, (2025), “Hopefully somebody will tell Vopson that he misunderstands Landauer”, https://shmaesphysics.wordpress.com/2023/12/14/information-has-no-mass/#comment-10716, April 26, 2025.
[184]: Stephane H Maes, (2022), “Why is The Multi-fold Theory on viXra, and not peer-reviewed (yet)?”, https://shmaesphysics.wordpress.com/why-is-the-multi-fold-theory-on-vixra-and-not-peer-reviewed-yet/, July 8, 2022.
[185]: Stephane H. Maes, (2025), “Adaptive Co-Design of Quantum Machine Learning Algorithms and Error Correction Protocols using Reinforcement Learning”, https://zenodo.org/doi/10.5281/zenodo.15428357, https://shmaes.wordpress.com/2025/05/15/adaptive-co-design-of-quantum-machine-learning-algorithms-and-error-correction-protocols-using-reinforcement-learning/, May 15, 2025, (https://osf.io/5jte9/download/).
[186]: Stephane H. Maes, (2025), “Preserving the Power of Preprints: Why Minimal Oversight is Key to Scientific Progress”, https://zenodo.org/doi/10.5281/zenodo.15617193, https://shmaes.wordpress.com/2025/06/05/preserving-the-power-of-preprints-why-minimal-oversight-is-key-to-scientific-progress/, June 5, 2025.
[187]: Stephane H Maes, (2025), “Rotating Black holes based on charged black holes”, https://shmaesphysics.wordpress.com/2024/11/24/physics-salvages-the-third-law-of-black-hole-thermodynamics/#comment-11016, June 28, 20265.
[188]: Stephane H. Maes, (2025), “Information Energy Momentum Tensor, E/G Conjecture vs. Alleged Massive Information”, https://shmaesphysics.wordpress.com/2025/06/11/information-energy-momentum-tensor-e-g-conjecture-vs-alleged-massive-information/, June 11, 2025.
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[2] Or SMG, which described the SM with gravity effects non-negligible at its scales [1,131,137,152,251].
[3] In general, Bohmian Physics is presented as a quantum mechanics theory and interpretation (undistinguishable from the other interpretations [252]), but rarely do we encounter a relativistic or QFT version.
[4] The consistent framework is in our view a differentiator vs. straight reviews.
[5] The 2D random walks of massless Higgs bosons, aka multi-fold preons, ensure random sprinkled Poisson distribution of the discrete spacetime locations, which leads to Lorentz invariance behaviors [260] at higher scales [1,23,72,131,137,152,170,181,251].
[6] At least for our proposed framework.
[7] When this is a massless scalar boson remember the massless Higgs boson of the multi-fold theory, or multi-fold preon [176], which in a multi-fold theory plays also the role of dilaton [23,29,32,64,66,72,170,176,181,230], inflaton [15,27,32,251], double copy gauge scalar boson [94,104], Higgs boson condensing into Higgs and massive particles [1,23,33-35,63,68,72,116,150,131,137,152,177,251], or in random walk patters that forms the massless ones [1,8-10,22,32,62,72,131,137,152,170,176,181,251], or eventually as the source of concretized and historical locations, that form the spacetime [1,6,8-10,22,23,32,62,63,72,116,131,137,152,170,176,181].
[8] This provides a direct physical justification for the hybrid choice of primitives. (Massless) Bosons behave as extended fields generated by the random walk configurations in agreement with [124] and references therein, whereas fermions act as highly localized, particle-like condensates, in agreement with [124] for fermionic fields (particles hence modeled in the multi-fold universe as extended microscopic black hole like patterns of random walks or condensates of preons / massless Higgs bosons) and with [1,8-10,22,131,137,150,152,170,251] and references therein, in terms of scales required to have non-commutativity of spacetime required to support fermions, their zitterbewegung, and spin statistics, and Quantum Physics in general. It is also related to [199].
[9] The multi-fold theory also prefers such an interpretation, i.e., that we are dealing with beables even in the absence of observers [1-199,251,258,259].
[10] Here,
are equivalent notations.
[11] Interestingly, in the case of a multi-fold universe, the spacetime is discrete [1,6,8-10,22,29,36,62,68,72,112,124,131,137,152,155,167-170,173-176,181,190], and there is no need to go back to the continuum.
[12] That statement may be dispute in general. In the context of a universe compatible with GR [1,6,62,112,137,169,170,253], and in expansion [112,167,170,181], it holds. That covers multi-fold universes. Furthermore, if Physical Actions and Path Integral apply, something that always does in a complete physical description of a physical system [116,159], then we have [1,137,155].
[13] Even the special models for (relativist, in between parentheses because they are always relativist) neutrinos also capture their special relationship to the multi-fold Higgs bosons / massless Higgs boson models as in [1,8-10,22,42,47,67,119,131,137,152,159,251].
[14] Relativistic Bohmian Physics may or may not characterize well our real universe. That has not been determined in this paper.
[15] Conventional physicists may argue it is New Physics. We consider that it isn’t because no new particles or interactions are introduced. We just add gravity, as we know it should be, and multi-fold mechanism and let conventional Physics unfold with the considerations. It does change conventional results or explanations, usually with same observables, and it does live in a discrete spacetime etc., because of conventional analysis of these consequences. We also do not cover stable field effects like Skyrmions [166], that we prefer to see as a collective effect for the theory. Beside the SM particles, there are other collective solutions/solitons in gauge theories, they have behaviors as particles, but they are quasi-particles composed of collective effects of a large set of particles. We see them as Qballs or patterns that can appear by multi-fold space time matter induction and scattering, under specific circumstances, nothing more. They are topological solitons and can appear in BEC, as expected with massless Higgs boson condensates [171,172].
[16] Standing in for Quantum Physics in general.
[17] It is granted that another mechanism, not yet discovered or thought about, could always be discovered in the future.
____
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Relativistic Bohmian Physics, with a Microscopic Interpretation
Stephane H. Maes
Abstract
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.
____
1. Introduction
This paper is presenting a consistent relativistic QFT theory, reviewing and compiling the status of research, and view in the Physics community. The hybrid model, combining the different proposed ways for a covariant theory, is guided by QFT, SM and results of the Multi-fold Theory [1-199,258,259], where the latter is also a non-local theory [1,62,131,137,140,152,251] that seems to explain some open issues with the Standard Model (SM [2]) and the Standard Cosmological Model, and often recovers interesting insights, recovering different conventional results (See [8,252] and Appendix A).
With this paper we want to see what can be said about relativistic Bohmian Physics. This paper is not necessarily arguing that Bohmian Physics characterizes well our real universe..
The extension of Bohmian mechanics to the relativistic domain, and to QFT presents specific theoretical difficulties [200]. The non-local nature of the pilot-wave guidance equations inherently requires a temporal ordering of spacelike separated events [201]. Historically, this requirement was satisfied by introducing an arbitrary preferred foliation of spacetime. Furthermore, the standard Bohmian particle model struggles to accommodate the creation and annihilation events that are characteristic of quantum field theory [202].
In section2, this paper constructs a theoretical framework that is Lorentz-invariant and compatible with quantum field theory. The proposed synthesis couples the Tomonaga-Schwinger equation for the relativistic wave functional with a hypersurface Bohm-Dirac model [201].
To ensure Lorentz invariance without adding absolute spatiotemporal structure, the necessary foliation is covariantly determined by the universal wave function [204]. To address particle non-conservation, i.e., particle creation an annihilation events, 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 [202].
A hybrid model is adopted, assigning localized particle beables to fermions and continuous field beables to bosons [206]. 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 quantum field theory [207].
The theoretical framework proposed in the paper is a synthesis of established proposals developed within the specialized community researching Bohmian mechanics[3], complemented with a perspective to complete the model. While it is not part of standard, mainstream QFT, which generally relies on operational, non-realist interpretations, every component of the proposed framework (sections 2 and 3) represents the state-of-the-art in relativistic pilot-wave research, developed primarily between 1999 and 2014; all put together into one consistent framework[4] in this paper.
The core mathematical and conceptual components originate from the following well-known research efforts in foundational physics:
- The Hypersurface Bohm-Dirac Model (HBDM): The mechanism for generalizing Bohmian trajectories across arbitrary curved spacelike surfaces without relying on flat, equal-time hyperplanes was developed in [200].
- Bell-Type Quantum Field Theories (Stochastic Jumps): The stochastic Markov jump process used to model discrete particle creation and annihilation events via “minimal jump rates” on a Fock space was formulated for continuum theories in [201].
- Covariant Foliation: The critical idea of eliminating absolute spacetime structures by extracting the required foliation dynamically from the universal wave function itself was detailed in [202].
- Tomonaga-Schwinger Integration: The approach of replacing the standard functional Schrödinger equation with the covariant, many-fingered time Tomonaga-Schwinger equation to support Bohmian dynamics has been explored in the literature as a method to avoid preferred foliations [201].
- Hybrid Model (Fields and Particles): The assignment of discrete particle positions to fermions and continuous functional fields to bosons is a recognized and debated conceptual approach within the Bohmian QFT literature [203].
The paper brings together these highly specialized, disparate theoretical patches into a single, cohesive model or framework. The synthesis provides a unified solution to the historical problems of non-locality and particle creation in pilot-wave theories, relying entirely on established proposals.
In section 3, we discuss the experimental indistinguishability of this approach, with non-Bohmian relativistic quantum mechanics, or QFT, to see if indistinguishability is preserved as it is between Bohmian and non-Bohmian non-relativistic quantum mechanics [221].
Finally, in section 4, this framework is microscopically interpreted with the multi-fold theory [1-199,251,258,259]. Linking relativistic Bohmian (field) theory and the multifold theory was motivated by the observation that both, especially if a relativistic Bohmian model exists, are non-local theory [1,62,131,137,140,152,251]. In this context, the covariantly determined foliation emerges physically from a discrete, non-commutative spacetime concretized by 2D random walks of massless Higgs bosons[5] [1,8-10,22,32,62,72,131,137,152,170,176,181,251]. The multi-fold mechanism and the E/G conjecture (entanglement is gravity) provide a local, physical model for the non-local Bohmian guidance equations [1,22,24,131,137,152,251], 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 [1,23,33-35,63,68,72,116,150,131,137,152,177,251]. It turns out that the two theories are compatible, and multi-fold configurations can provide subtle microscopic interpretations to Bohmian models.
As such this paper is part of an upcoming set of studies that will focus on explaining conventional Physics behaviors, relying, in particular, on the notion of 2D random walks of massless Higgs bosons [1,8-10,22,32,62,72,131,137,152,170,176,181,251], as we already started in [199]. More will be published and can be tracked at [8].
Appendices A through C discuss respectively: overview of the multi-fold theory, the notion of quantum non-equilibrium, and the modeling[6] of neutrinos in relativistic quantum Bohmian QFT.
2. Covariant Foliations and Stochastic Jump Processes: A Unified Framework for Relativistic Bohmian Quantum Field Theory
2.1. Introduction
Bohmian mechanics, also known as the de Broglie-Bohm pilot-wave theory, provides a deterministic formulation of quantum phenomena, wherein the statistical uncertainties of standard quantum mechanics are epistemic rather than fundamental [208].
In the non-relativistic regime, the theory proposes an objective configuration of particle positions
(1)
evolving in time according to a guidance equation determined by the wave function
[202]. The wave function itself evolves according to the standard Schrödinger equation.
This framework resolves the quantum measurement problem by ensuring that particles always possess definite configurations, independent of observation. The standard textbook collapse rule becomes a consequence of the deterministic dynamics when considering the interaction between a measured subsystem and the macroscopic environment [209].
Despite its conceptual clarity, the generalization of Bohmian mechanics to relativistic quantum field theory encounters two primary obstacles.
First, the non-relativistic guidance equation is explicitly non-local. The velocity of a given particle depends instantaneously on the positions of all other particles in the system. In a relativistic spacetime, the concept of instantaneous dependence requires a global definition of simultaneity, appearing to necessitate a preferred spacetime foliation [210]. The introduction of such an absolute structure violates the principles of fundamental relativity. This leads to models that are phenomenologically adequate but fundamentally non-relativistic [211].
Second, standard Bohmian mechanics is formulated for a fixed number of point particles. Relativistic QFT is fundamentally characterized by processes of particle creation and annihilation, which are phenomena that defy the continuous, unbroken trajectories of the traditional de Broglie-Bohm interpretation [202].
To circumvent these limitations, we propose a unified theoretical architecture for a relativistic Bohmian QFT that synthesizes several theoretical developments.
- To address the foliation problem, the Hypersurface Bohm-Dirac model is utilized, wherein the arbitrary fixed foliation is replaced by a foliation covariantly determined by the quantum state itself [201].
- To maintain covariance at the level of quantum state evolution, the standard functional Schrödinger equation is replaced by the Tomonaga-Schwinger equation, which propagates the state vector over arbitrary spacelike hypersurfaces [203].
- To account for variable particle numbers, the deterministic continuous dynamics are augmented by a stochastic Markov jump process defined on Fock space, following the Bell-type quantum field theories [202].
- Finally, in Section 4, the multi-fold theory [1-199,251,258,259] is introduced to provide a fundamental, discrete physical mechanism, and microscopic interpretation, for these phenomena/model choices [1,6,8-10,22,23,27,32-34,62,63,72,116,124,131,137,152,170,176,181].
2.2. Standard Relativistic Extensions and Their Limitations
The most direct attempt to construct a relativistic Bohmian theory involves adapting the relativistic wave equations, such as the Dirac equation for fermions or the Klein-Gordon equation for bosons, to the pilot-wave formalism [213].
2.2.1 The Single-Particle Dirac Equation
For a single relativistic spin-1/2 particle, the evolution of the wave function is governed by the Dirac equation. Setting the reduced Planck constant ℏ, and the speed of light to unity, the equation reads
(2)
[202]. The wave function
takes values in the four-dimensional spinor space
.
A natural candidate for the deterministic guidance equation utilizes the conserved Dirac probability current.
The velocity field for the particle is given by
, (3)
where the probability current four-vector is
(4)
[200]. This single-particle equation is strictly covariant and successfully defines a sub-luminal, deterministic trajectory for a single electron. The probability density
(5)
is conserved, satisfying the continuity equation
, (6)
However, the fundamental incompatibility with relativity emerges immediately when the system is expanded to multiple interacting particles.
2.2.2 The Multi-Particle Dirac Equation, and Non-Locality
For an N-particle system, the wave function
takes values in the tensor product space
.
The corresponding multi-particle Dirac equation must account for the spatial coordinates of all N particles. The guiding equation for the k-th particle in this configuration space is formulated as:
(7)
Where ak are the Dirac matrices operating on the spinor index of the k-th particle. The critical issue lies in the evaluation of the right-hand side of this equation. The velocity of the k-th particle at a specific time t depends on the value of the many-body wave function evaluated at the exact positions of all N particles at that same time t.
In a relativistic context, the notion of “same time” for spatially separated particles is frame dependent. If the guidance equation is evaluated in one specific Lorentz frame, the trajectories will generally differ from those calculated in another Lorentz frame [203]. To achieve a unique set of trajectories, it is necessary to select a single, privileged foliation of spacetime, i.e., a sequence of equal-time hyperplanes, and declare that the trajectories defined by this specific foliation are the true physical trajectories [210].
2.2.3 The Problem of Absolute Structure
The mathematical introduction of a preferred foliation solves the ambiguity of the trajectories but introduces a conceptual problem. According to the classification of physical structures in [211], a theory is seriously Lorentz-invariant only if it does not contain any absolute structures beyond the Lorentz metric itself.
An absolute structure is a geometric object that influences the dynamics of the system, but it is not influenced by the system in return [211].
If a preferred foliation is added to a theory simply to evaluate the non-local guidance equations, it acts as a fixed background structure. While the statistical predictions of the theory remain Lorentz invariant due to the equivariance of the probability distribution, the underlying model relies on an undetectable, absolute Newtonian time. Such a formulation means that the theory is not relativistic in a fundamental sense. To resolve this, the foliation must be transformed from an absolute geometric background into a dynamical entity.
2.3. The Hypersurface Bohm-Dirac Formulation
To construct a framework that supports non-local interactions without relying on flat equal-time hyperplanes, the Hypersurface Bohm-Dirac model generalizes the Bohmian dynamics to arbitrary curved foliations of spacetime [201].
2.3.1 Mathematical Definition of the Model
The Hypersurface Bohm-Dirac model assumes a foliation
of spacetime into a one-parameter family of smooth, spacelike hypersurfaces
,where s is a scalar parameter [205]. Each point x on a hypersurface
has a future-oriented, timelike unit normal vector field denoted by n(x).
The state of the system is given by the N-particle Dirac wave function, which solves the multi-time Dirac equations, or a suitable constraint equation ensuring consistency across the foliation. The physical configuration is represented by the N-path
, (8)
consisting of the worldlines of the N particles [214].
The law of motion for the configuration is defined by constructing a multi-particle current tensor. The current for the k-th particle, generalizing the single-particle Dirac current, is defined as:
, (9)
where
(10)
is the unit normal vector evaluated at the position of the i-th particle on the specific hypersurface
. The actual particle trajectories are defined by the Hypersurface Bohm-Dirac tangency condition. The tangent vector to the worldline of the k-th particle, denoted
, must be proportional to the current jk evaluated at the intersection points of all N worldlines with the hypersurface S [201]:
(11)
2.3.2 Positivity and Determinism
For the Hypersurface Bohm-Dirac model to be physically viable, the trajectories must be strictly future-directed, i.e., timelike or null, meaning that the time component of the velocity must be positive. This requirement is satisfied by the mathematical structure of the current jk.
By considering the action of a suitable local Lorentz transformation on the operator
, for an arbitrary timelike unit vector n, one can transform n into the standard rest frame vector (1,0,0,0) [201]. In this frame, the operator reduces to
, which is the identity matrix. Consequently, the operator
is strictly positive definite [201].
This mathematical property guarantees that the currents jk are always future-directed timelike or null vectors, ensuring that the Bohmian particles never travel backward in time, or never exceed the speed of light relative to the local geometry.
2.4. Covariant Determination of the Time Foliation
The Hypersurface Bohm-Dirac model provides the mathematical approach to define non-local trajectories across arbitrary spacelike surfaces [214]. However, if the foliation
is specified externally as an absolute structure, the theory remains fundamentally non-relativistic.
The solution to this conceptual problem is to derive the foliation directly from the wave function itself.
2.4.1 Extracting Geometry from the Wave Function
Instead of positing the foliation
as an independent, fixed background, it can be defined as a functional of the universal wave function,
(12)
Because the wave function evolves according to a Lorentz-covariant physical law, any geometric structure mathematically derived from it will automatically transform covariantly under Lorentz transformations. This strategy ensures that the theory does not contain any absolute structures other than the spacetime metric.
The foliation becomes a dynamical variable, inextricably linked to the quantum state of the matter fields. The underlying physical microscopic interpretation is at this stage a postulate. It will be addressed and potentially explained in section 4.
2.4.2 Constructing the Covariant Vector Field
To derive a foliation from the wave function, one must first construct a covariant timelike vector field. A natural candidate is the expectation value of the total four-momentum density or the total probability current. Given a generic many-body wave function
, the total current vector field
can be defined as the expectation value of the local current operator
:
(13)
For a system of Dirac particles, the local current operator ensures that the expected vector field
is strictly timelike. Once the time like vector field
is established, it can be normalized to produce a future-directed unit vector field [204]:
(14)
2.4.3 Defining the Leaves of the Foliation
The normalized vector field
specifies a unique temporal direction at every point in spacetime, dictated entirely by the local properties of the quantum state. The required foliation
is then constructed such that its leaves (the spacelike hypersurfaces
) are everywhere orthogonal to
.
If the vector field
has zero vorticity, it is Frobenius integrable, meaning it admits a family of exact orthogonal hypersurfaces. If the vector field has non-zero vorticity, and is not strictly hypersurface-orthogonal, one can still extract a unique foliation by utilizing the integral curves of
to define the level sets of a scalar phase function, or by imposing a minimal-surface condition mapped to the boundary values of the universe.
Figure 1: Covariant Spacetime Foliation and Hypersurface Bohm-Dirac Trajectories Description. The figure shows a visual representation of a (2+1)-dimensional spacetime section illustrating the covariantly determined foliation
. The spacelike hypersurfaces
(colored blue) flex according to the dynamically changing total probability current
. The deterministic worldlines of two fermions (red and green curves) are shown intersecting the hypersurfaces, with their local tangent vectors aligned with the Hypersurface Bohm-Dirac current jk defined by the normal vectors n(xi) of the foliation.
By substituting the covariantly derived normal vector
into the Hypersurface Bohm-Dirac tangency condition, the complete set of equations governing the particle trajectories becomes strictly Lorentz invariant [201]. The dynamics of the particles are determined by the wave function, and the spatial non-locality, required to evaluate the current jk is mediated along the dynamically generated surfaces of
.
2.5. Functional Evolution via the Tomonaga-Schwinger Equation
In standard non-relativistic quantum field theory, the state is represented in the functional Schrödinger picture by a wave functional
. This functional encodes the probability amplitude for a specific field configuration across all space at a fixed time t. The evolution of this functional is governed by the functional Schrödinger equation:
(15)
This evolution equation is fundamentally noncovariant. It relies on a global time parameter t, implying an integration over a flat, equal-time hyperplane. If the Bohmian particles are to be guided by a wave function evaluated on the curved, covariantly determined hypersurfaces of
, the wave function itself must be definable on those surfaces.
2.5.1 The Generalization to Arbitrary Hypersurfaces
To align the quantum state evolution with the covariant foliation, the functional Schrödinger equation must be replaced by the Tomonaga-Schwinger equation [203]. The Tomonaga-Schwinger framework operates in the interaction picture, or a suitably defined multi-time Heisenberg picture. It replaces the global time coordinate t with a functional dependence on an arbitrary spacelike hypersurface
. The quantum state is designated as a functional vector
.
The dynamical evolution of the state under an infinitesimal local deformation of the hypersurface in the normal direction at a point x is governed by the following functional differential equation:
(16)
where
denotes the local deformation of the hypersurface at spacetime point x, and
is the Hamiltonian density operator.
2.5.2 Micro causality and Integrability
For the Tomonaga-Schwinger functional evolution to be physically consistent and mathematically integrable, the order in which local deformations are applied must not affect the final quantum state, provided the deformations occur at spacelike separated points. Foliation independence requires that the second functional derivatives commute:
(17)
This mathematical requirement dictates that the Hamiltonian density operators must commute at spacelike separations. This is the standard micro causality condition of quantum field theory:
(18)
The utilization of the Tomonaga-Schwinger equation fulfills a dual purpose within the proposed framework. First, it ensures that the wave functional can be evaluated consistently and covariantly on any arbitrary leaf of the dynamically determined foliation
[203]. Second, it provides the necessary mathematical foundation to calculate the local probability currents and stochastic jump rates required for the Bohmian configuration dynamics.
2.6. Primitive Model: Beables for Bosons and Fermions
To generate a complete Bohmian quantum field theory, one must specify the primitive model [206]. The primitive model consists of the local beables, which are the fundamental mathematical entities that represent actual physical configurations in spacetime. A choice must be made regarding whether these fundamental entities are discrete point particles or continuous fields.
Our choices here are guided by considerations of scales and behavior differences between fermions and bosons in a multi-fold universe (In section 4 and appendices, we present the consistency of the framework as recovering these guiding principles). However, these can also be seen just a selection of what works out of the arsenal of options developed in the literature [200-249].
2.6.1 Field Beables for Bosonic Degrees of Freedom
Bosonic degrees of freedom, such as the electromagnetic field, or the Higgs field [201], are most naturally represented by continuous field configurations rather than discrete particles. For a scalar boson field[7] governed by the Klein-Gordon equation, the fundamental local beable is the scalar field configuration
. The guidance equation for this field is derived from the functional gradient of the phase of the wave functional. Expressing the wave functional in polar form as
, (19)
where
is the generalized parameter labeling the leaves of the foliation, the deterministic evolution of the scalar field is given by:
(20)
For gauge bosons, such as the electromagnetic vector potential
, a similar functional guidance equation applies. Following minimalist models, the beables for quantum electrodynamics are restricted to the transverse degrees of freedom of the vector potential,
, to maintain strict gauge invariance [207]. The wave functional
satisfies the functional Tomonaga-Schwinger equation involving the Hamiltonian of the specific gauge theory.
2.6.2 Particle Beables for Fermionic Degrees of Freedom
Model TypeBoson RepresentationFermion RepresentationDynamicsPhenomenological ChallengesDirac Sea ModelField BeablesInfinite Particle SeaPurely DeterministicInfinite unobservable mass/charge densities.Minimalist ModelField BeablesNone (Emergent from Bosons)Purely DeterministicMatter is not fundamental.Bell-Type Model (Proposed)Field BeablesFinite Particle ConfigurationsDeterministic Drift + Stochastic JumpsRequires rigorous stochastic measure theory on Fock space.Table 1: Comparing model options to model particles. As a proposal, we recommend the Bell-type approach.
For fermionic degrees of freedom, point particles remain the conventional choice [201]. Formulating a continuous field model for fermions requires utilizing anti-commuting Grassmann variables, which lack a clear physical interpretation as classical fields in ordinary spacetime [215]. Two dominant theoretical approaches exist to model fermion particles in a relativistic quantum field theory:
- The first option is the Dirac Sea model. This deterministic model interprets the quantum vacuum as an infinitely dense sea of actual particles occupying negative energy states [216]. Particle trajectories, including those of the unobservable particles in the Dirac sea, evolve deterministically according to the generalized Hypersurface Bohm-Dirac guidance equations [218]. While this preserves the continuity of trajectories without requiring stochastic jumps, the postulation of an unobservable, infinite density of actual particles raises physical difficulties, particularly regarding mass and charge regularization [218].
- The second option, adopted in this framework, relies on Bell-type quantum field theories [202]. Originating from lattice models, and extended to the continuum, this approach incorporates explicit creation and annihilation events [202]. Fermions are modeled as point particles that trace continuous trajectories guided by the Hypersurface Bohm-Dirac equation, but these deterministic segments are interrupted by discontinuous stochastic jumps [202]
The proposed framework utilizes a hybrid model, defining continuous functional fields for bosons and stochastic point particles for fermions[8]. This combination aligns with standard QFT, where fermions interact via the exchange of gauge bosons, generating the creation and annihilation processes observed in high-energy physics. Neutrinos are handled in Appendix C.
2.7. Stochastic Dynamics on Fock Space
To accommodate the non-conservation of particle number implied by the interaction terms in the Hamiltonian density, the deterministic motion of the configuration must be supplemented by discontinuous jumps [202]. The configuration space is no longer a fixed 3N-dimensional space, but rather the disjoint union of configuration sectors corresponding to different particle numbers, representing the spatial equivalent of the Fock space [202].
A generic configuration q consists of a specific number of particles located at specific spatial coordinates on the hypersurface
, alongside the continuous field configuration for the bosons. The total evolution of the physical state is defined as a piecewise deterministic Markov process, combining the continuous flow (the Hypersurface Bohm-Dirac drift), and discrete jumps representing creation or annihilation events [203].
2.7.1 The Master Equation
The probability distribution
for the configuration q across the entire Fock space is governed by a master equation. This equation, often referred to as a Kolmogorov forward equation, must account for both the deterministic divergence of the probability current within a specific sector and the stochastic transition rates between different sectors.
Let
denote the transition rate, defined as the probability density per unit parameter time to jump from configuration q’ to configuration q. The temporal evolution of the probability density
with respect to the foliation parameter t is described by the integro-differential equation:
(21)
In this equation,
represents the continuous flux associated with the deterministic guidance equations generated by the free Hamiltonian [202]. The integral term on the right-hand side represents the net gain and loss of probability density due to stochastic jumps into and out of the configuration state q.
2.7.2 Derivation of the Minimal Jump Rates
The transition rates
chosen as free phenomenological parameters. They must be constrained by the fundamental interaction Hamiltonian
to ensure that the stochastic process strictly reproduces the exact quantum statistics of the underlying field theory. To derive these rates, the time derivative of the target quantum probability density
is evaluated using the full Hamiltonian [202]:
. (22)
The free Hamiltonian H0 generates the continuous deterministic velocity field v(q) via the standard continuity equation. The interaction Hamiltonian HI, which contains the creation and annihilation operators coupling the fermionic and bosonic sectors, generates the discrete jump dynamics. According to the Schrödinger evolution, the variation of the probability density due exclusively to the interaction term is given by:
(23)
To map this quantum probability variation to the classical stochastic master equation, the gain-loss integral of the master equation must equal the interaction derivative. A sufficient condition for solving this integral equation is to equate the integrands directly. However, mathematical consistency dictates that transition rates must be non-negative real numbers
. (24)
Defining
(25)
as the positive part of a real number x, the minimal jump rate formula is derived as [202]:
(26)
This prescription is termed “minimal” because it mathematically ensures that for any given pair of configurations q and q’, stochastic jumps only occur in one direction at any given instant. If the imaginary term is positive, probability flows from q’ to q, and the rate is non-zero. If the term is negative, the required jump is in the reverse direction q to q’, and the rate
correctly evaluates to zero, leaving the reverse rate
to handle the transition. This formulation completely fixes the stochastic dynamics of the creation and annihilation events at the vertices determined by the interaction Hamiltonian.
Figure 2: Bell-type Stochastic Jump Process in Spacetime. It shows A schematic of a particle creation and annihilation event as modeled by a Bell-type stochastic Markov process. A single fermion worldline (blue) propagates deterministically until a stochastic jump occurs on a specific hypersurface
. At this interaction vertex, governed by the minimal jump rate
, the particle annihilates, and a fermion-antifermion pair (red and green worldlines) is created, representing a discontinuous jump from the 1-particle sector to the 3-particle sector in Fock space.
2.8. The Equivariance Theorem
For the proposed Bohmian QFT to be empirically equivalent to standard QFT, it must reproduce the Born rule statistics across all times and across all leaves of the foliation [207].
This statistical compatibility is known as equivariance. If the probability distribution of the particle and field configurations
equals the modulus squared of the wave functional
on an initial spacelike hypersurface
, the property of equivariance guarantees that
(27)
on any subsequent hypersurface evaluated by the Tomonaga-Schwinger equation. The proof of this theorem requires demonstrating consistency for both the continuous and stochastic components of the dynamics.
2.8.1 Continuity and Deterministic Equivariance
For the purely deterministic segments of the particle trajectories, which are governed by the free Hamiltonian H0, equivariance requires that the probability density satisfies a continuity equation compatible with the continuous unitary evolution of the wave functional. Using the Hypersurface Bohm-Dirac model, the multi-particle probability current defined in Section 2.3.1 is strictly divergence-free in the absence of interactions. This follows directly from the application of the free Dirac equation and its adjoint:
(28)
To relate this four-divergence to the evolution of the probability density across the foliation, the equation must be split into components normal to and tangent to the hypersurface Ss , where s is the scalar parameter labeling the leaves. The continuity equation along the foliation parameter s is expressed as:
(29)
Where gk is the geometric area-expansion factor arising from the normal flow of the metric between adjacent hypersurfaces, and
is the projection of the velocity field onto the local tangent space of the hypersurface
. The mathematical integration of this equation proves that the crossing probability density
(30)
is strictly preserved along the deterministic trajectories bridging the leaves of the foliation. Therefore, the deterministic drift maintains the Born rule [201].
2.8.2 Stochastic Equivariance on Fock Space
When the interaction terms HI are involved, the state vector is subjected to the Markov jump processes [202]. To prove equivariance in the presence of particle creation and annihilation, it must be shown that the master equation governing the jumps exactly reproduces the non-unitary disruption of the wave functional’s modulus squared within a given sector. The master equation defined in Section 2.7.1 is explicitly structured to satisfy this requirement.
By assuming the initial equilibrium condition
(31)
and substituting this into the minimal jump rate equation, one obtains:
(32)
The notion of equilibrium in Bohmian Physics is further discussed in Appendix B.
To evaluate the net gain and loss integral in the master equation, one substitutes this expression and its reverse counterpart. Utilizing the fundamental algebraic identity for the positive parts of real numbers,
, (33)
the right-hand side of the master equation simplifies algebraically. The resulting expression matches exactly the temporal variation of
induced by the interaction Hamiltonian HI , as derived from the Schrödinger equation.
Because the total divergence of the probability distribution maps identically to the unitary evolution of the wave functional, the ensemble equivariance theorem holds over the covariantly defined foliation. The minimal jump rates algorithmically mediate transitions across Fock space, replicating the creation and annihilation events mandated by quantum gauge interactions without violating the statistical predictions of the standard regularized theory [202].
2.9. Phenomenological Implications and “Serious” Relativity
The theoretical construction presented in this paper satisfies several rigorous physical criteria. By elevating the Tomonaga-Schwinger equation to evaluate states on arbitrary spacelike surfaces, and by allowing the wave functional itself to define the unique covariant foliation
governing the Hypersurface Bohm-Dirac tangency conditions, the framework escapes the usual criticism that pilot-wave theories require a phenomenologically absolute Newtonian time. Because the structure
is a dynamical entity dependent entirely on the Lorentz-invariant wavefunction and the invariant spacetime metric
, the theory lacks any absolute structures [211]. It is therefore fundamentally covariant. No preferred inertial frame exists a priori. The effective preferred frame is an emergent property of the specific initial conditions of the universe’s wave function, analogous to how the distribution of matter in general relativity defines an effective cosmic rest frame without violating fundamental diffeomorphism invariance. Section 4, justify and interpret the physicality of
.
Furthermore, the introduction of macroscopic contextuality provides a law-like, top-down stochastic mechanism to explain quantum decoherence [209]. In these extended models, such as the Contextual Bohmian Quantum Field Theory (CBQFT) [209], macroscopic context variables, such as detector configurations, or symmetry-breaking thermal sectors, act as a dynamical background, that modulates the Bell-type jump rates on the single Fock space [209]. Such integrations securely anchor the macroscopic arrow of time in the fundamental non-equilibrium stochastic transitions across the configuration space, employing a hylomorphic loop that links quantum events directly to contextual structure [209].
Finally, because the jump rates rely on a carefully regulated stochastic projection defined locally upon the hypersurfaces, the model avoids generating ultraviolet singularities beyond those natively present in the regularized QFT. The combined model allows fermions to exist as identifiable local beables (stochastic point particles) while mediating forces via explicit continuous boson field configurations [206]. This eliminates the necessity of postulating an unobservable, infinitely dense Dirac sea [216], replacing it with localized creation and annihilation dynamics governed by rigorously derived transition probabilities.
3. The Foundations and Limits of Observational Equivalence: Bohmian Mechanics, Relativistic Foliations, and Field Models
3.1 Observations and The Model of de Broglie-Bohm Theory
The mathematical formalism of orthodox, aka standard or conventional, quantum mechanics, as presented in standard textbooks, functions primarily as an operational recipe for calculating the probabilities of measurement outcomes [221]. It deliberately refrains from providing a direct, objective description of physical processes occurring in the absence of an observer [221]. This conceptual gap is bridged by Bohmian mechanics, also known as the de Broglie-Bohm pilot-wave theory, or the causal interpretation, which offers a realistic, deterministic, and observer-independent account of microscopic reality [221].
As we saw, within this framework, a physical system is not described solely by its wave function; rather, its physical state is represented by a dual structure: the wave function Y(q,t), and the actual configuration
(34)
of the system, where each Qi(t) represents the precise spatial position of a point-like particle [221]. The wave function, postulated to belong to a standard square-integrable Hilbert space, evolves continuously according to the linear Schrödinger equation [221]:
(35)
Simultaneously, the actual positions of the particles evolve continuously according to a first-order differential guiding equation, which represents the simplest Galilean-covariant law of motion compatible with the Schrödinger dynamics [221]:
(36)
In this manner, the pilot wave guides the configuration of the universe through a deterministic choreography [223]. The particles are directed by the local phase gradient of the wave, ensuring that they avoid regions of destructive interference and accumulate in regions of constructive interference [225]. By integrating actual configurations into the fundamental model, the theory operates as a “quantum theory without observers” [221] [9].
It successfully resolves the measurement problem, eliminates the need for wave function collapse as a physical postulate, and bypasses orthodox concepts such as wave-particle complementarity, unsharp physical values, or the necessity of human consciousness to register physical events [221].
3.2 The Mechanism of Observational Equivalence in Non-Relativistic Space
The observational indistinguishability of Bohmian mechanics from standard quantum mechanics in non-relativistic regimes is not an accidental coincidence. It is a consequence of the Quantum Equilibrium Hypothesis (QEH) [222].
In standard quantum mechanics, the Born rule is treated as an axiomatic postulate that maps wave functions directly to probabilities [225]. In de Broglie-Bohm theory, the connection between probability density and the wave function has the status of a theorem, emerging from the underlying deterministic dynamics of the system [221].
The basis for this equivalence is the property of equivariance [222]. If the initial configuration Q(t0) of an ensemble of systems is chosen at random with a probability density
that matches the squared wave function
, then the configuration
at any subsequent time t will remain distributed according to
[222]. This dynamical compatibility is proved by the continuity equation, which is derived directly from the Schrödinger equation and the guiding equation [222]:
(37)
(38)
, where
(39)
is the Bohmian velocity field [224]. Since the physical configuration density
and the probability density
satisfy the exact same continuity equation and evolve under the same velocity field, any initial statistical alignment is preserved for all times [224].
To analyze localized subsystems, Bohmian mechanics utilizes the conditional wave function [222]. If the actual configuration of the universe is split into
, (40)
where X is the configuration of the subsystem under study and Y is the configuration of its environment, the conditional wave function y(x) of the subsystem is defined by inserting the actual environmental configuration Y(t) into the universal wave function
[222]:
(41)
As the particles in the environment evolve, the actual configuration Y(t) selects an effective, localized branch of the universal wave function [228]. When the subsystem interacts with a measurement apparatus, the actual configuration of the joint system-apparatus enters one of the non-overlapping wave packets of the superposition [224]. The empty wave packets, although mathematically present, become dynamically irrelevant because the actual configuration has moved far away from them in configuration space, preventing any future interference [225]. This branch selection physically derives the “collapse of the wave function” without needing to introduce collapse as an independent dynamical law [222].
This construction explains why macroscopic observations are identical across both interpretations [222]. Every experimental record, whether a pointer orientation on a dial, an ink pattern on paper, or a localized spot on a detector screen, is ultimately a spatial configuration of matter [222].Because Bohmian mechanics tracks actual configurations, and because these configurations are distributed according to the
quantum equilibrium, the predictions of Bohmian mechanics for any experiment are identical to those of the orthodox formalism [222].
Under the quantum equilibrium hypothesis, any measurement of a generalized observable can be mathematically represented by a positive-operator-valued measure (POVM) O(dz) acting on the wave function y, such that the probability distribution of the result Z is given by [223]:
(42)
This formulation reveals that properties like spin or momentum are not intrinsic localized variables carried by the particles [224]. Instead, they are contextual properties of the wave function in relation to the specific experimental configuration, emerging deterministically during the measurement process itself [223].
3.3 Extension of Equivalence to Relativistic Spacetime
Extending Bohmian mechanics to relativistic spacetime requires addressing a fundamental tension: the “spirit” of relativity is local, four-dimensional spacetime geometry, whereas the “spirit” of quantum mechanics is nonlocal entanglement [223]. Because the velocity of any single Bohmian particle depends on the positions of all other particles in the universe, the guiding equation requires a mechanism to determine which points on different particle world lines correspond to the same instant of time [232]. In non-relativistic space, this simultaneity is provided by absolute time, but in relativistic space, there is no absolute temporal coordinator [232].
To resolve this, relativistic Bohmian models introduce a preferred, spacelike foliation of spacetime into spacelike hypersurfaces [231]
. (43)
This foliation provides a physical coordinate system for calculating nonlocal interactions [233]. A priori, introducing a preferred reference frame appears to violate Lorentz invariance, suggesting a conflict with fundamental relativity [233]. However, this frame remains completely unobservable at the statistical level [227]. If the particles are distributed in accordance with the quantum equilibrium distribution, the statistical predictions of the relativistic Bohmian model are identical to those of standard relativistic quantum mechanics, which are Lorentz-invariant [227]. Thus, the preferred foliation cannot be detected experimentally, preserving observational equivalence even though fundamental Lorentz invariance is broken at the subquantum level [227].
To achieve Lorentz invariance at a fundamental level, the required foliation can be extracted covariantly from the wave function itself (
(44)
) [227] [10].
If the foliation is defined by a covariant vector field generated by the wave function, such as a conserved, timelike probability current
, the foliation ceases to exist as an independent, absolute spacetime structure [233]. This approach is realized in the Hypersurface Bohm-Dirac Model (HBDM), which describes a system of N Dirac particles using a multi-time wave function
[233]. The particles’ actual world lines Xk(s) are guided by the multi-particle Dirac current evaluated at the intersections of the world lines with a common leaf
of the foliation [233]:
(45)
This dynamical law is covariant under the Lorentz group [233].
The mathematical consistency of the HBDM is robust under complex geometric conditions [b20]. When the foliation is determined by the law:
(46)
, where
(47)
is the normal unit one-form of the foliation (representing equal timelike distance from a given initial hypersurface), the spacelike leaves generically develop geometric kinks [236]. Even with these kinks, or under degenerate foliations where multiple leaves overlap in a localized region, the particle trajectories remain mathematically well-defined, and the
probability distribution remains equivariant [236]. This geometric stability ensures that the HBDM remains empirically equivalent to standard relativistic quantum mechanics across all physical scenarios [236].
Alternative relativistic approaches have also been investigated to reconcile nonlocality with spacetime geometry [232]. One notable example is the “Opposite Arrows of Time Model,” which attempts to resolve the tension by introducing bidirectional causal influences, though it remains less mathematically developed than the HBDM [232].
3.4 Empirical Equivalence in Quantum Field Theory (QFT)
As we saw previously, the extension of Bohmian mechanics to the domain of Quantum Field Theory (QFT) is essential to account for relativistic phenomena such as particle creation, particle annihilation, and infinite degrees of freedom [237]. To maintain observational equivalence, Bohmian QFTs are formulated by defining a primitive model (local beables) in physical space, and deriving a guidance law, that preserves the
distribution of the corresponding regularized QFT [202]. There are two primary approaches to constructing these field-theoretic models [237]. They are discussed in the next subsections.
3.4.1 Fock Space and Particle Model (Bell-Type QFTs)
Bell-type QFTs, originally proposed by John S. Bell on a spatial lattice and subsequently extended to the continuum[11], take the concept of point-like particles seriously in the QFT regime [202]. At any given time t, the system has a definite, actual number of particles N(t) at precise spatial positions, meaning the actual configuration Qt resides in a single n-particle sector of a multi-sector configuration space [202]:
(48)
The universal wave function
is represented by a state vector in a Fock space and can exist in a superposition of different particle numbers [202]. The dynamics of the actual configuration Qt are governed by two distinct laws of motion [202]:
- Continuous Bohmian Motion: Between creation and annihilation events, the particles move smoothly within their current sector
, guided by a deterministic guiding equation derived from the spatial current of the free Hamiltonian [230].
- Stochastic Jumps: To account for the non-conservation of particle numbers, the configuration jumps stochastically to a different sector of [230]
.
These jumps occur at random times and are governed by transition rates Tnm (from configuration m to n) designed to be the minimal jump rates necessary to preserve the equivariance of the |Y|2 Fock-space distribution [202]. On a lattice, the minimal transition rate is formulated as [227]:
(49)
where the probability current Jnm is defined by [227]:
(50)
And
(51)
is the probability of being in configuration m [227]. In the continuum, these rates correspond to stochastic creation and annihilation events where physical particle world lines begin and end [230]. Because the jump rates are constructed to preserve the
distribution, a law of large numbers ensures that the empirical frequencies of measurement outcomes in a typical Bohmian universe are identical to the statistical predictions of standard QFT [230].
3.4.2 Functional Wave Representation and Field Model
An alternative Bohmian approach to QFT utilizes a wave functional representation, which suggests a model of classical field configurations rather than point particles [237].
For bosonic systems, such as scalar or electromagnetic fields, the state of the universe is described by a wave functional
evolving via a functional Schrödinger equation [202]. The local beable is an actual classical field configuration
, that evolves deterministically over physical space, guided by the phase S of the wave functional [227]:
(52)
Fermionic degrees of freedom can be integrated into this field model using the Minimalist Model developed by Ward Struyve and Antony Westman [227]. In this model, there is no physical particle model for fermions [227]. Instead, the fermionic degrees of freedom are represented solely by the wave function, which is mapped to a bosonic field (such as the electromagnetic vector potential A(x,t), and a continuous charge density
defined on physical space [227]. The wave function is written as a function of the bosonic field
, (53)
where f labels the fermionic states, and the equilibrium distribution is defined as [227]:
(54)
The actual charge density is calculated dynamically from the wave function and the actual bosonic field configuration [227]:
(55)
Where
is the charge density operator [227]. This formulation provides a continuous, field-based model that remains fully indistinguishable from standard QFT predictions, demonstrating that empirical equivalence can be maintained without postulating classical particle trajectories for fermions [227].
3.5 Subquantum Physics: Violating the Boundaries of Indistinguishability
While Bohmian mechanics is observationally equivalent to standard quantum mechanics under the assumption of quantum equilibrium, the two interpretations are not mathematically identical [226].
The pilot-wave framework permits the existence of quantum non-equilibrium states, where the actual distribution of particle configurations deviates from the Born rule [226]:
. (56)
The existence of quantum non-equilibrium would render Bohmian mechanics empirically distinguishable from, and falsifiable against, standard quantum mechanics [226].
In non-equilibrium states, reviewed in Appendix B, several key principles of orthodox quantum mechanics are violated, as detailed in the following table:
Physical PrincipleQuantum Equilibrium
(ρ=∣ψ| 2)Quantum Non-Equilibrium
(not equal)Signal LocalityStrictly respected; faster-than-light signaling is impossible [241]Violated; enables instantaneous “signal nonlocality” at the statistical level [226]Uncertainty PrincipleAdhered to; precise joint measurements of conjugate variables are forbidden [224]Violated; allows subquantum measurements that bypass uncertainty limits [226]Quantum EntanglementRequires a secondary classical channel to decode nonlocal correlations [243]Functions as an independent, instantaneous communication channel [243]Born Rule ValidityUniversally confirmed across all laboratory measurements [226]Violated; yields anomalous statistical distributions in experiments [226]Table 2: Quantum Equilibrium vs. Quantum Non-equilibrium Bohmian Physics (See also appendix B).
To explain why our current universe appears to be in a state of strict quantum equilibrium, physicists have developed a subquantum statistical mechanics [245]. By constructing a coarse-grained subquantum H-function analogous to the Boltzmann H-function of classical thermodynamics, one can demonstrate that non-equilibrium distributions naturally relax toward equilibrium over time [226]:
(57)
Under a subquantum H-theorem, this quantity is non-increasing (
(58)
), showing that the Born rule represents a stable, maximum-entropy statistical attractor [240]. Consequently, standard quantum mechanics can be understood as an equilibrium phenomenology, and its adherence to relativistic signal-locality is a direct consequence of this equilibrium state masking the underlying subquantum nonlocality [241].
This thermodynamic view suggests that the indistinguishability of Bohmian mechanics is historically contingent [240]. If the early universe began in a state of quantum non-equilibrium, physical signatures of Born rule violations might have survived quantum relaxation [240]. Prominent cosmological and astrophysical regimes proposed for detecting these relic deviations include (This is revisited in Appendix B):
- The Cosmic Microwave Background (CMB): Non-equilibrium fluctuations of the primordial scalar inflaton field prior to inflation could have bypassed complete relaxation [240]. This would imprint anomalous power-spectrum signatures on the CMB anisotropies, which could be observed in cosmological surveys [240].
- Primordial Reheating Phase: During the transition from inflation to the radiation-dominated era, the rapid coupling and energy transfer between fields (modeled by coupled harmonic oscillators) could freeze or slow down quantum relaxation, preserving non-equilibrium signatures in relic particle populations [240].
- Bouncing Cosmological Models: In models where the Big Bang is replaced by a quantum gravitational contraction-to-expansion bounce, quantum gravitational processes could actively generate fresh non-equilibrium states, propagating Born rule violations into the early expanding universe [240].
- Black Hole Evaporation: Extremely strong gravitational fields and high-energy processes near black hole horizons, particularly during evaporation, could generate quantum non-equilibrium in the emitted Hawking radiation [240].
3.6 Comparative Analysis of Interpretational Models
To systematically compare the modeling, and dynamical characteristics of these different formulations, Table 3 maps the mathematical and physical profiles of orthodox quantum mechanics against the various Bohmian extensions.
4. Interpretation with the Multi-Fold Universe Framework
The covariant foliation and stochastic jump processes, described in sections 2.3 through 2.8, represent a mathematical formalism for a relativistic Bohmian quantum field theory.
To construct a complete physical model, one must address the microscopic origin of the generated foliation
, the physical mechanism executing the non-local pilot-wave guidance, and the fundamental nature of the chosen field-particle beables.
The multi-fold theory provides a possible microscopic spacetime geometry, or macroscopic interpretation, that satisfies these requirements seamlessly [1-199, 251,258,259]. An overview of the multi-fold theory is presented in Appendix A, with pointers to references.
4.1 Spacetime Concretization and the Covariant Foliation
Interpretation / FormulationPrimitive Model (Local Beables)Dynamical EquationsWave Function RoleMechanism of Empirical EquivalenceTreatment of MeasurementCopenhagen (Orthodox)None; physical properties are indeterminate until measured [221]Schrödinger equation plus instantaneous projection postulate [223] Complete description of physical state; probability amplitude carrier [224]Defined axiomatically via the Born rule [225]Discontinuous collapse triggered by external classical apparatus [223]Non-Relativistic Bohmian MechanicsPoint-like particles Qi(t) in physical space 3 [222]Schrödinger equation plus deterministic guiding equation [222]Physical/nomological field guiding actual configurations [222] Quantum Equilibrium Hypothesis and equivariance [222]Continuous, observer-free branch selection via conditional wave function [228] Relativistic Hypersurface Bohm-Dirac ModelParticle world lines Xk(s) in Minkowski spacetime [233] Dirac equation plus HBDM guidance law coupled to foliation F [233] Multi-time pilot wave defining spacetime probability currents [233]Equilibrium statistics matching standard covariant QMC predictions [227]Branch selection along the spacelike leaves of the foliation [233] Bell-Type QFT (Particle Model)Point-like particles with variable number N(t) [230] Schrödinger equation plus continuous guidance and stochastic jumps [202]Fock-space state vector guiding configuration jumps between sectors [202]Minimal jump rates preserving equivariance of the probability distribution [202] Observer-free branch selection via stochastic particle configuration jumps and the conditional wave function [202]Field-Theoretic QFT (Struyve-Westman)Classical fields F(x,t) and charge density r(x,t) [227]Functional Schrödinger equation plus deterministic field guidance [227]Functional pilot wave guiding field evolution on configuration space [227] Equivariance of the wave functional probability density [227] Continuous field evolution selecting stable macroscopic charge concentrations [227]Table 3: Mapping of maps the mathematical and physical profiles of orthodox quantum mechanics against the various Bohmian extensions
In the standard Bohmian model, the foliation
is determined functionally by the global wave function.
The multi-fold theory can offer a physical mechanism for this geometrical structure: spacetime itself emerges as a discrete, non-commutative, fractal geometry concretized by the 2D random walks of massless Higgs bosons [1,8-10,22,32,62,72,131,137,152,170,176,181,251], and their location or location history. Because these random walks’ locations should follow random sprinkled Poisson distributions [1,23,72,131,137,152,170,181,251], the resulting discrete structure successfully preserves continuous macroscopic Lorentz invariance while remaining non-commutative at the smallest scales [1,23,72,131,137,152,170,181,251]. Therefore, the covariantly determined spacelike hypersurfaces S are not abstract mathematical structures mapped onto a continuous background; rather, they correspond directly to the physical locations, and historical locations, concretized by the underlying massless Higgs random walks. This way, the background depends on the matter distribution of the massless Higgs bosons (the quantum state), satisfying the conditions for a dynamically determined foliation without problematically positing absolute external structures.
4.2 The E/G Conjecture and the Quantum Potential
A persistent conceptual challenge in any pilot-wave formulation is the non-local nature of the guidance equation: how does a particle instantaneously “know” the configuration of all other entangled particles along the spacelike leaf S?
The multi-fold theory resolves this by proposing that (EPR (Einstein-Podolsky-Rosen)) entanglement generates direct physical pathways, the multi-folds, between entangled systems [1,5,7,8-10,71,131,137,152,140,18,251]. According to the E/G (Entanglement is Gravity) conjecture of the multi-fold theory, entanglement physically generates gravity-like attractive effective potentials, or effective curvature, within the 4D spacetime region spanning the entangled entities [1,5,7-10,22,24,131,137,152,188,251]. These multi-fold mechanisms permit “spooky actions at a distance” to result from strictly local interactions operating within the folds [1,9,10,140]. Consequently, the non-local Bohmian current correlations and the resulting quantum potential are physically mediated through the multi-fold connections. Local physics and hidden variables are thus fully reconcilable with Bell experiments within this multi-fold discrete spacetime [1,8-10,62,131,137,152,251].
Figure 3: Multi-Fold Mechanism and the E/G Conjecture Description. It is a conceptual diagram illustrating the integration of the multi-fold theory with the non-local Bohmian pilot-wave. Two entangled particle beables (represented as Kerr-Newman soliton Q-balls, or random walk patterns) exist at spacelike separated positions on a discrete hypersurface S (constructed from 2D massless Higgs random walks). A multi-fold mechanism is depicted as an extra-dimensional mapping (a fold) connecting the two particles, transmitting the non-local guidance potential while simultaneously generating gravity-like effective curvature (the E/G conjecture [24]) in the intervening 4D spacetime.
4.3 Multi-fold Space Time Matter Induction and Scattering, and Beables
4.3.1 Multi-fold Particles
The relativistic Bohmian model proposed in Section 2.6 requires a hybrid model: continuous field beables for bosons and discrete point-particle beables for fermions.
The multi-fold space-time matter induction and scattering processes, operating in an unconstrained 7D (or 5D) embedding space, explain the emergence of these specific beables [1,23,33-35,63,68,72,116,150,131,137,152,177,251]. In the multi-fold geometry, the particles of the Standard Model particles with gravity effects non-negligible at its scales (SMG) are modeled as distinct structural manifestations of the underlying massless Higgs boson random walks [1,8-10,22,32,62,72,131,137,152,170,176,181,251].
At spatial scales small enough below the multi-fold gravity electroweak symmetry breaking, inspired by figure 1 in [72], massless Higgs bosons propagate freely as conformal field theories (CFTs) [68,70,177].
Field Beables: Bosonic degrees of freedom map directly to the continuous, massless patterns of these random walks [68,70,150,170,177].
At higher spatial scales [4,8-10,29,68,70,124,150,170,177], still below the multi-fold gravity electroweak symmetry breaking, we have, in addition to bosonic field beables:
- Particle Beables: Fermionic degrees of freedom map to the pattern of these random walks into massless fermions. [4,8-10,29,68,70,72,124,150,170,177].
At still higher spatial scales, above the multi-fold gravity electroweak symmetry breaking, we have, in addition to bosonic field beables:
- Particle Beables: Fermionic degrees of freedom map to the condensation of these random walks into massive and charged Dirac Kerr-Newman soliton Q-balls (microscopic black holes) [4,8-10,29,68,70,72,124,150,170,177].
This provides a direct physical justification for the hybrid choice of primitives. (Massless) Bosons behave as extended fields generated by the random walk configurations in agreement with [124] and references therein, whereas fermions act as highly localized, particle-like condensates, in agreement with [124] for fermionic fields (particles hence modeled in the multi-fold universe as extended microscopic black hole like patterns of random walks or condensates of preons / massless Higgs bosons) and with [1,8-10,22,131,137,150,152,170,251] and references therein, in terms of scales required to have non-commutativity of spacetime required to support fermions, their zitterbewegung, and spin statistics, and Quantum Physics in general. It is also related to [199].
4.3.2 Further Multi-fold Corroboration
In the multi-fold theory, massless bosons, and fermions occupy different scales (as do massive Bosons) [170]. We showed that this is essential to the notion of zitterbewegung and spin statistics theorem for fermions and the anti-commutativity of spacetime at the core of the existence of quantum mechanics and its uncertainty principle.
The Non-commutativity of spacetime is also key to the expansion of a dS (asymptotically de Sitter) universe [6,181], and random walks of the 2D massless bosons [1,8-10,22,32,62,72,131,137,152,170,176,181,251], which we [1,6-10,124,131,137,140,152,176,251,258], and others [124,170,253-256], have shown to recover, in a fractal spacetime, Schrödinger’s equations / quantum Physics and General relativity (GR) [253,257]. So, in an expanding universe, non-commutativity implies 2D random walks [1,6,72,137,181][12], which imply fractals (and QFT [6,62,124,175,181,253-256]), GR and SMG [1,6-10,22,29,62,72,131,137,152,181,251]. The important message is that:
- Spacetime is predicted non-commutative, a result also consistent with all the coherent theories of gravity [1,6,8-10,22,62,112,131,137,152,155,167,181,251],
- Therefore, spacetime does not have to be continuous [1,8-10,22,29,36,62]. Limits to the continuum do not even have to be taken in the fermion as particle beables.
4.4 Microscopic Origin of Stochastic Jumps
4.4.1 Particle Creation and Annihilation
Finally, the multi-fold framework provides a physical mechanism for the Bell-type stochastic Markov jump processes described in Section 2.7.
The creation and annihilation events governed by the minimal jump rates s(q/q’) can represent the physical condensation and dissociation of the Kerr-Newman soliton Q-balls [4,35,162,177,181], or the random walk patterns [1,8-10,22,29,32,35,62,72,131,137,152,162,170,176,177,181,251].
When an interaction vertex is reached along the foliation
a given particle beable (a soliton condensate, or random walk pattern) may break apart back into the constituent massless Higgs random walk patterns, corresponding to a particle annihilation event. Conversely, background random walks may stochastically condensate into a localized microscopic black hole condensate or random walk pattern, representing a particle creation event [4].
The transition rates for these structural phase changes are governed strictly by the interaction Hamiltonian HI, preserving the exact mathematical equivariance of the master equation across the Fock space while anchoring the jumps in a specific, physical microscopic process.
4.4.2 A Bit More
Beyond [1,8-10,22,29,32,62,72,131,137,152,162,170,176,177,181,251], we are in the process of showing that key characteristics of quantum mechanics are a direct result of the 2D random walks [199]. More papers will appear. Please watch [8].
5. Conclusions
The de Broglie-Bohm interpretation of quantum mechanics provides a mathematically rigorous, observer-independent representation of physical reality that is empirically indistinguishable from orthodox quantum mechanics [221]. This observational equivalence is maintained in non-relativistic regimes by the Quantum Equilibrium Hypothesis, which establishes the Born rule as an emergent statistical distribution that is dynamically preserved by the equivariance of the guiding equation [222].
This equivalence successfully extends to relativistic spacetime and quantum field theories [230]. In the relativistic domain, the tension between quantum nonlocality and Lorentz covariance is resolved by the Hypersurface Bohm-Dirac Model (HBDM), which can extract a preferred spacelike foliation directly from the covariant structure of the wave function [233]. In the context of QFT, the pilot-wave model accommodates particle creation and annihilation either through stochastic jumps in a Fock-space particle model (Bell-type QFTs), or via deterministic functional evolution in a field model [227].
However, the indistinguishability of these models is contingent upon the universe having reached a state of statistical quantum equilibrium [226]. If the universe was prepared in a state of primordial quantum non-equilibrium, the underlying subquantum mechanics of the pilot-wave theory would become manifest, resulting in observable violations of standard quantum limits [226]. The search for these relic signatures in the Cosmic Microwave Background, primordial reheating phases, bouncing cosmologies, or Hawking radiation represents a potentially viable pathway toward empirically distinguishing Bohmian mechanics from standard quantum mechanics and testing the ultimate boundaries of quantum foundations [240].
The framework derived in this paper establishes a cohesive, explicitly formulated relativistic Bohmian quantum field theory a priori mapped onto a fundamental discrete spacetime, even if in the limit it might be continuum. This synthesis formally resolves the historical incompatibilities between pilot-wave theory, special relativity, and QFT.
The deterministic continuity equations ensure that particle and field beables track the free Hamiltonian dynamics along the leaves of a dynamically generated, covariant spacetime foliation, which physically originates from the 2D random walks of massless Higgs bosons. The non-local guidance required by the pilot-wave model can be microscopically interpreted as supported by the multi-fold mechanisms [1,137,140] and the E/G conjecture [1,22,24,137], ensuring that entanglement-driven gravity-like potentials facilitate communication across the spacelike leaves. Furthermore, the minimal jump rates algorithmically mediate transitions across Fock space, replicating the creation and annihilation events as the physical condensation and dissociation of Kerr-Newman soliton Q-balls, or formation and dissociation of random walk patterns. Interestingly the scale differences and methods required to model relativistic Bohmian bosons and Fermions recover multi-fold results in terms of spacetime scales for massless bosons, and fermions, and massive particles per [35,177,181], in ways that directly corroborates the relativistic Bohmian models of (charged) fermions vs. bosons[13].
The mathematical proof of equivariance guarantees that the statistical predictions of the theory remain isomorphic to standard regularized QFT, validating the empirical adequacy of the model. This synthesis provides a robust picture of the relativistic quantum domain, restoring localized physical realism options to QFT without compromising Lorentz invariance or the principles of QFT.
This paper demonstrates a consistent way to evolve non-relativistic Bohmian mechanics into a relativistic Bohmian QFT model; and, that the resulting hybrid model could be microscopically justified with multi-fold considerations, as has often been the case for explaining conventional physics behaviors[14] (See Appendix A). We expect more such examples to come, as part of our current work to infer some of this, in particular, using 2D random walks of massless Higgs bosons e.g., [199].
Appendix A. The Multi-fold Theory
For more details on the latest developments, updates to papers , discussion and the full story only succinctly summarized here, please consider the complete list of papers compiled at [8]. In particular, while some more recent references are provided, the focus in this appendix is not to provide our latest findings. There are rather tracked on that web site [8].
In a multi-fold universe [1,8-10,22,131,137,152,251], gravity emerges from entanglement through the multi-fold mechanisms. As a result, gravity-like effects appear in between entangled particles [1,24,25], whether they are real or virtual. Long range, massless gravity results from entanglement of massless virtual particles [1,25]. Entanglement of massive virtual particles leads to massive gravity contributions at very smalls scales [1,26]. It is at the base of the E/G Conjecture [24], and the main characteristics of the multi-fold theory [22]. Multi-folds mechanisms [140] also result in a spacetime that is discrete, with a random walk fractal structure and non-commutative geometry that is Lorentz invariant and where spacetime nodes and particles can be modeled with microscopic black holes [1,4,16,27-31,68,72,121,131,137,150,152,170,177,181,251]. All these recover General Relativity (GR) at large scales, and semi-classical model remain valid till smaller scale than usually expected [1,6,131,137,152,181,251]. Gravity can therefore be added to the Standard Model (SM) resulting into what we define as SMG: the SM with gravity effects non-negligible at its scales. This can contribute to resolving several open issues with the Standard Model without new Physics[15] other than gravity. These considerations hint at an even stronger relationship between (multi-fold) gravity and the Standard Model, as finally shown in [23].
Among the multi-fold SMG discoveries, the apparition of an always in-flight, and hence non-interacting, right-handed neutrinos, coupled to the Higgs boson is quite notable. It is supposedly always around right-handed neutrinos, due to chirality flips by gravity of the massless Weyl fermions, induced by 7D space time matter induction and scattering models [1,23,33-35,63,68,72,116,150,131,137,152,177,251], and hidden behind the Higgs boson or field at the entry points and exit points of the multi-folds. Massless Higgs bosons modeled as minimal microscopic black holes mark concretized spacetime locations. They can condensate into Dirac Kerr-Newman soliton Qballs to produce massive and charged particles [1,4,161], thereby providing a microscopic explanation for a Higgs driven inflation [27], the electroweak symmetry breaking [29,35], the Higgs mechanism, the mass acquisition [29,35,36,72,170,177,181] and the chirality of fermions and spacetime [29,35,36,72,177]; all resulting from the multi-fold gravity electroweak symmetry breaking [35,177]. The multi-fold theory has also concrete implications on New Physics like supersymmetry, superstrings, M-theory and Loop Quantum Gravity (LQG) [1,8-21,112,131,137,152,182].
The multi-fold paper [1,137] proposes contributions to several open problems in physics, like the reconciliation of General Relativity (GR) with Quantum Physics, explaining the origin of gravity proposed as emerging from quantum (EPR- Einstein Podolsky Rosen) entanglement between particles, detailing contributions to dark matter and dark energy, and explaining other Standard Model mysteries without requiring New Physics beyond the Standard Model other than the addition of gravity to the Standard Model Lagrangian, and the 2D massless Higgs boson random walks. All this is achieved in a multi-fold universe that may well model our real universe, which remains to be validated.
With the proposed model of [1,137], spacetime and Physics are modeled from Planck scales to quantum and macroscopic scales, and semi-classical approaches appear valid till very small scales. In [1,137], it is argued that spacetime is discrete, with a random walk-based fractal structure, fractional and noncommutative at, and above Planck scales (with a 2-D behavior and Lorentz invariance preserved by random walks till the early moments of the universe) [1,8-10,22,29,31,62,131,137,148,152,169,170,181,182,251]. Spacetime results from past random walks of particles. Spacetime locations and particles can be modeled as microscopic black holes (Schwarzschild for photons and concretized spacetime coordinates, and metrics between Reissner Nordström [2], and Kerr Newman [3] for massive, and possibly charged, particles – the latter being possibly extremal) [4,162]. Although possibly surprising, [1,137] recovers results which are consistent with others (see [4], and its references), while also being able to justify the initial assumptions of black holes from the models of gravity or entanglement in a multi-fold universe. The resulting gravity model recovers General Relativity at larger scale, as a 4D process, with massless gravity, but also with massive gravity components at very small scales, which make gravity non-negligible at these scales. Semi-classical models also turn out to work well till way smaller scales than usually expected.
Multi-folds are encountered in GR at Planck scales [5,6] and in Quantum Mechanics[16] (QM) if different suitable quantum reference frames (QRFs) are to be equivalent relatively to entangled, coherent or correlated systems [7,58]. This shows that GR and QM are different facets of something that they cannot well model: multi-folds.
Considering results as in [6,52,181], and our answers to so many open issues with the SM and the ΛCDM can be qualitatively explained with the SMG and multi-fold mechanisms, as discussed in [1-199,251,258,259], we can then argue that these conclusions can apply to our real universe, especially considering how the multi-fold mechanisms recover GR [1,137], and can be encountered in GR at Planck scales [6,181], with the spacetime reconstruction [1,62], and with the top-down-up-and-upper derivation of the multi-fold theory [6,181].
Appendix B. About Quantum Equilibrium and Bohmian Physics
B.1 In Equilibrium or Out of Equilibrium?
Here we expand, with some repetition on the quantum equilibrium discussed in the body of the paper. Equations are renumbered to be self-contained.
In the de Broglie–Bohm, or pilot-wave, interpretation of quantum mechanics, a system is said to be in quantum equilibrium when the actual spatial distribution of its physical particles, denoted by ,
exactly matches the probability density predicted by standard quantum mechanics, which is the square of the wave function’s amplitude, . This equality,
(B.1)
is known as the quantum equilibrium hypothesis (QEH).
Because the wave function is treated as an objective, physical pilot wave rather than a mere representation of observer probability, the Born rule is not a fundamental axiom in this theory. Consequently, a system can theoretically be out of equilibrium (a state known as quantum non-equilibrium, where
) (B.2)
through several physical mechanisms, like arbitrary Initial Conditions at the Beginning of the Universe. In pilot-wave theory, the actual distribution of particles and the wave-intensity are governed by separate mathematical identities. While both satisfy the same continuity equation, i.e., meaning that if a system starts in equilibrium at
t = 0 , (B.3)it will remain in equilibrium for all future times (a property called equivariance as discussed earlier in this paper), there is no fundamental law which dictates that they must be equal initially. At the birth of the universe, e.g., the Big Bang, the particles could have been distributed in an arbitrary, non-standard configuration
(B.4)
placing the early universe in a state of quantum non-equilibrium.
In general, it would them eventually evolve towards equilibrium by Dynamical Relaxation. According to Valentini’s “sub-quantum H-theorem”, i.e., the pilot-wave analog to Boltzmann’s H-theorem in classical statistical mechanics, chaotic interactions and the complex “mixing” of Bohmian trajectories typically force an initial non-equilibrium state to rapidly relax to equilibrium on a coarse-grained level [241,245]. This rapid relaxation explains why we only observe standard quantum probabilities in laboratory experiments today.
However, there are cases where this may not happen.
- This relaxation could be suppressed or “frozen” under extreme conditions (Cosmological Freezing):
- Cosmic Expansion: In the very early, rapidly expanding universe (such as during the inflationary epoch), the expansion of space can stretch physical wavelengths faster than the particles can traverse them. This limits the displacement of the trajectories, effectively “freezing” the relaxation process and leaving primordial field modes in a state of non-equilibrium. This frozen non-equilibrium could theoretically leave detectable, non-standard statistical signatures in the Cosmic Microwave Background (CMB) or survive in relic primordial particles today.
- Another counter example could be a case of Incomplete Relaxation and Lack of Chaos. For a system to relax to equilibrium, its pilot-wave trajectories must be sufficiently chaotic. Chaos in Bohmian mechanics is primarily generated by trajectories scattering off the moving nodal points, i.e., where
(B.5)
of the wave function.
Therefore, if a quantum system has a highly coherent, non-chaotic wave function, such as certain simple Gaussian superpositions or systems with few degrees of freedom, the quantum flow behaves as a laminar flow. In these cases, the sub-quantum
-function does not decrease to its minimum value but instead saturates, preventing complete relaxation and keeping the system permanently out of equilibrium.
- Gravitational Collapse and Black Hole Evaporation could also prevent equilibrium. Strong gravitational fields represent another frontier where quantum equilibrium might break down. It has been theorized that the extreme physics of gravitational collapse and subsequent black hole evaporation via Hawking radiation can generate quantum non-equilibrium. If so, the particles and radiation emitted by an evaporating black hole would violate the Born rule, carrying distinct non-equilibrium statistics into the surrounding space.
Then, we should also note that the case of Velocity/Momentum Non-Equilibrium from (Bohm’s 1952 Formulation) [147,148]. In Louis de Broglie’s original 1927 pilot-wave theory (See [b5,aa3]), the first-order guidance equation (
(B.6)
Where p is momentum and S is the phase of the wave function) is a fundamental law of motion.
However, in David Bohm’s 1952 second-order Newtonian reformulation, the guidance equation is treated merely as a constraint on the initial momenta [247,248]. If this constraint is relaxed, particles can start with momenta
(B.7)
This “extended” non-equilibrium is highly unstable [248]. For instance, in a ground-state system like a hydrogen atom, where standard pilot-wave theory dictates the velocity is zero, dropping the momentum constraint causes a rapid growth of spatial non-equilibrium, ultimately causing stable bound states to fly apart.B.2 A Multi-fold Perspective
B.2.1. From the Big Bang
Based on multi-fold considerations as discussed in [1-199,251,258,259], even the earliest steps are modeled with (2D random walk Physics). In such cases, we would not have initial conditions out of equilibrium. In fact, in the multi-fold fluctuation case [1,137], probability and wave necessarily have the same norm. Distributing the fluctuations along a large region, still rely on the same phenomena and equations, and should result the same.
Similarly, in the context of the total collision model [107] or other cyclic expansions after a big crunch, we would expect to come from an equilibrium, and, therefore, to have the equilibrium preserved.
In either case, there are no situations where the boundary conditions would be following (B.2).
Could an extended non-equilibrium occur? Per [248], this would imply a (global) rapidly expanding chaotic region, unrelated to inflationary effects with no clear mechanism to ever recover equilibrium. In such a universe, no bound structure would exist[17].
So, in a multi-fold universe, the initial moment of the universe would necessarily be in equilibrium, and relativistic Bohmian QFT / Physics always remain undistinguishable from conventional QFT, even when looking in traces of the big bang. In Appendix A, we mentioned that a multi-fold universe often models well our real universe. Maybe it is the case again, and quantum equilibrium is never broken in our real universe.
B.2.2 Relativistic Bohmian Physics And Beable Wave Function
As discussed in the main paper, with [4,71] and the multi-fold spin model [1,8-10,22,131,137,150,152,161,251], we saw that the wavefunction seems to be a beable.
The multi-fold SM (or rather SMG) elementary particles as random walk patterns or condensates of massless Higgs bosons, aka multi-fold preons) [4,35,170,176,177,181] also imply that all particles are beables. With such a model, the wavefunction is expected to always satisfy (B.1), and therefore never to be out of equilibrium satisfying (B.2).
Appendix C. Treatment of Neutral Fermions (Neutrinos) in Relativistic Bohmian QFT
The evaluation of fermionic behavior in relativistic Bohmian mechanics poses specific modeling challenges when dealing with neutral particles such as neutrinos. Because certain formulations rely heavily on macroscopic charge densities to map physical reality, addressing neutral fermions requires a deliberate selection of the underlying primitive model.
Per [157], and references therein, especially [250], neutrinos can’t carry any charge no matter how small it would be, or our current Physics would collapse, e.g., QED and Electroweak theories would have to be reworked.
C.1 The Limitation of the Field-Theoretic (Struyve-Westman) Model
In the Minimalist Model developed by Struyve and Westman [227], fermionic degrees of freedom are not treated as localized physical particles. Instead, they are represented entirely by the wave function, which is mapped to a macroscopic continuous field beable, such as the electromagnetic vector potential ,
and an associated continuous charge density
defined on physical space.
Because neutrinos are electrically neutral, they do not couple to the electromagnetic vector potential and do not generate an electromagnetic charge density.
Consequently, applying this specific minimalist field-theoretic model to neutrinos requires extending the continuous functional guidance equations beyond electromagnetism. The model must substitute the electromagnetic vector potential with the gauge bosons of the weak interaction and evaluate the corresponding weak-isospin density.
C.2 Resolution via the Hybrid Bell-Type Particle Model
To avoid the phenomenological challenges of relying on macroscopic charge density fields to represent matter, the primary theoretical framework adopted in this paper relies on a hybrid model utilizing Bell-type quantum field theories.
In this adopted model, as we saw in the main body of the paper, bosonic degrees of freedom (which mediate forces) are represented by continuous functional fields, while all fermions (charged or neutral) are modeled as distinct, localized point-particle beables.
Neutrinos are treated as actual point particles with variable configurations N(t) across a Fock space rather than as a continuous charge density. This way, the hybrid model naturally copes with them.
The deterministic drift of these neutral point particles is supplemented by discrete, stochastic Markov jumps representing creation and annihilation events. These events are governed by minimal jump rates derived directly from the fundamental interaction Hamiltonian, ensuring that weak-force interactions involving neutrinos are successfully modeled without violating standard quantum statistics.
C.3 Neutrinos in the Multi-Fold Universe Framework
When integrating this framework into the multi-fold theory, where spacetime is concretized by 2D random walks of massless Higgs bosons, and entanglement creates multi-folds, neutrinos exhibit highly specific behaviors that resolve broader cosmological and standard model issues:
- Chirality Flips and Right-Handed Neutrinos: The multi-fold theory accounts for the apparition of always in-flight, non-interacting right-handed neutrinos coupled to the Higgs boson. This mechanism is driven by gravity-induced chirality flips of massless Weyl fermions via 7D spacetime matter induction and scattering models [1,8-10,42,47,131,137,152,159,165,251].
- Spacetime Location: These right-handed neutrinos are theorized to be hidden behind the Higgs boson (or field) precisely at the entry and exit points of the multi-folds [35,42,47,67,119,159,177,x].
The multi-fold mechanisms suggest that neutrinos are strictly governed by these chiral and gravitational dynamics. Specifically, the framework explicitly notes that neutrinos are probably not Majorana fermions [1,8-10,42,47,119,131,137,152,159,165,251].
The multi-fold Standard Model with Gravity (SMG) operates without the need for New Physics beyond the SM, as SMG [1,22,24], explicitly predicting the absence of conventional sterile neutrinos [1,8-10,22,42,47,67,131,137,152,159,165,251].
____
Cite as: Stephane H. Maes, (2026), “Relativistic Bohmian Physics, with a Microscopic Interpretation”, https://shmaesphysics.wordpress.com/2026/08/12/relativistic-bohmian-physics-with-a-microscopic-interpretation/, August 12, 2026.
____
References
[1]: Stephane H. Maes, (2020-2022) “Quantum Gravity Emergence from Entanglement in a Multi-Fold Universe”, HIJ, Vol 2, No 4, pp 136-219, Dec 2022, https://doi.org/10.55672/hij2022pp136-219, https://shmaesphysics.wordpress.com/2020/06/09/paper-published-as-preprint-quantum-gravity-emergence-from-entanglement-in-a-multi-fold-universe/, https://shmaesphysics.wordpress.com/2022/11/09/quantum-gravity-emergence-from-entanglement-in-a-multi-fold-universe-2/, and viXra:2006.0088, (June 9, 2020). Errata/improvements/latest updates at https://zenodo.org/doi/10.5281/zenodo.7792911.
[2]: Wikipedia, “Reissner–Nordström metric”, https://en.wikipedia.org/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric. Retrieved on March 21, 2020.
[3]: Wikipedia, “Kerr–Newman metric”, https://en.wikipedia.org/wiki/Kerr-Newman_metric. Retrieved on March 21, 2020.
[4]: Stephane H Maes, (2021), “More on Multi-fold Particles as Microscopic Black Holes with Higgs Regularizing Extremality and Singularities”, viXra:2210.0004v1, https://shmaesphysics.wordpress.com/2021/02/28/more-on-multi-fold-particles-as-microscopic-black-holes-with-higgs-regularizing-extremality-and-singularities/, February 25, 2021.
[5]: Stephane H Maes, (2020), “Multi-folds, The Fruit From The Loops? Fixing “Oops for The Loops” May Encounter Multi-folds in General Relativity And The E/G Conjecture”, viXra:2212.0206v1, https://shmaesphysics.wordpress.com/2021/12/31/multi-folds-the-fruit-from-the-loops-fixing-oops-for-loops-encounters-multi-folds-and-the-e-g-conjecturein-general-relativity/, January 1, 2022.
[6]: Stephane H Maes, (2022), “Deriving the Multi-fold Theory from General Relativity at Planck scale”, viXra:2302.0129v1, https://shmaesphysics.wordpress.com/2022/02/22/deriving-the-multi-fold-theory-from-general-relativity-at-planck-scale/, February 22, 2022.
[7]: Stephane H Maes, (2022), “From Quantum Relational Equivalence to Multi-folds Encounter in the Real Universe and Confirmation of the E/G conjecture”, viXra:2302.0108v1, https://shmaesphysics.wordpress.com/2022/02/12/from-quantum-relational-equivalence-to-multi-folds-encounter-in-the-real-universe-and-confirmation-of-the-e-g-conjecture/, February 7, 2022.
[8]: Stephane Maes, (2020-25), “Web Site Tracking all Publications around the Multi-fold universe”, Navigation page listing all papers, https://shmaesphysics.wordpress.com/shmaes-physics-site-navigation/.
[9]: Stephane H Maes, (2021), ”The Multi-fold Theory: A synopsis”, viXra:2112.0144v1, https://shmaesphysics.wordpress.com/2021/12/24/the-multi-fold-theory-a-synopsis-so-far-v2-end-of-2021/, December 24, 2021. Note that additional links will always be available at https://shmaesphysics.wordpress.com/2021/05/03/the-multi-fold-theory-a-synopsis-so-far/ to track the latest and interim versions of the synopsis, as they may be published under different tittle or URL/publication numbers.
[10]: Stephane H Maes, (2022), “Understanding the Multi-fold theory principles and the SM_G”, osf.io/xc74t, https://shmaesphysics.wordpress.com/2022/03/11/understanding-the-multi-fold-theory-principles-and-the-sm_g/, March 11, 2022. Also, as Stephane H Maes, (2022), “A tutorial on the Multi-fold theory principles and the SM_G”, viXra:2303.0154v1, https://shmaesphysics.wordpress.com/blog-2/a-tutorial-on-the-multi-fold-theory-principles-and-the-sm_g/, March11, 2022.
[11]: Stephane H. Maes, (2022), “Comment on LQG, Superstrings, Supersymmetry and most GUTs/TOEs, all have big problems exposed by the Multi-fold Theory”, https://shmaesphysics.wordpress.com/2021/12/27/the-multi-fold-theory-a-synopsis/#comment-3293. Published on January 9, 2022.
[12]: Stephane H. Maes, (2020), “Comment on why no supersymmetry”, https://shmaesphysics.wordpress.com/2020/10/11/circular-arguments-in-string-and-superstring-theory-from-a-multi-fold-universe-perspective/#comment-934. Published on October 12, 2020.
[13]: Stephane H Maes, (2020), “Renormalization and Asymptotic Safety of Gravity in a Multi-Fold Universe: More Tracking of the Standard Model at the Cost of Supersymmetries, GUTs and Superstrings”, viXra:2102.0137v1, https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/, September 18, 2020.
[14]: Stephane H Maes, (2020), “Circular Arguments in String and Superstring Theory from a Multi-fold Universe Perspective”, viXra:2103.0195v1, https://shmaesphysics.wordpress.com/2020/10/11/circular-arguments-in-string-and-superstring-theory-from-a-multi-fold-universe-perspective/, October 5, 2020.
[15]: Stephane H Maes, (2021), “The String Swampland and de Sitter Vacua: A Consistent Perspective for Superstrings and Multi-fold Universes”, viXra:2208.0078v1, https://shmaesphysics.wordpress.com/2021/01/12/the-string-swampland-and-de-sitter-vacua-a-consistent-perspective-for-superstrings-and-multi-fold-universes/, January 9, 2021.
[16]: Stephane H Maes, (2021), “Quantum Gravity Asymptotic Safety from 2D Universal Regime and Smooth Transition to Dual Superstrings”, viXra:2208.0151v1, https://shmaesphysics.wordpress.com/2021/02/07/quantum-gravity-asymptotic-safety-from-2d-universal-regime-and-smooth-transition-to-dual-superstrings/, January 29, 2021.
[17]: Stephane H Maes, (2020), “A Non-perturbative Proof of the Asymptotic Safety of 4D Einstein Gravity, With or Without Matter”, https://doi.org/10.5281/zenodo.7953796, https://shmaesphysics.wordpress.com/2022/05/04/a-non-perturbative-proof-of-the-asymptotic-safety-of-4d-einstein-gravity-with-or-without-matter/, May 4, 2022, viXra:2305.0138.
[18]: Stephane H Maes, (2020), “Dualities or Analogies between Superstrings and Multi-fold Universe”, viXra:2006.0178v1, https://shmaesphysics.wordpress.com/2020/06/14/dualities-or-analogies-between-superstrings-and-multi-fold-universes/, June 14, 2020.
[19]: Stephane H Maes, (2020), “Alignments and Gaps Between Multi-fold Universes And Loop Quantum Gravity”, viXra:2006.0229v1, https://shmaesphysics.wordpress.com/2020/06/19/multi-fold-universes-analysis-of-loop-quantum-gravity/, June 18, 2020.
[20]: Stephane H Maes, (2020), ”Superstrings Encounter of the Second, Third or Fourth Types?”, viXra:2010.0140v1, https://shmaesphysics.wordpress.com/2020/07/19/superstrings-encounter-of-the-second-third-or-fourth-types/, July 5, 2020.
[21]: Stephane H Maes, (2022), “Oops For The Loops II: Real Oops; LQG Does Not Optimize the Hilbert Einstein Action”, viXra:2301.0036v1, https://shmaesphysics.wordpress.com/2022/01/05/oops-for-the-loops-ii-real-oops-lqg-does-not-optimize-the-hilbert-einstein-action/, January 5, 2022.
[22]: Stephane H. Maes, (2022), “What is the Multi-fold Theory? Its Main Characteristics in a Few Words”, vixra:2207.0172v1, https://shmaesphysics.wordpress.com/2022/07/28/what-is-the-multi-fold-theory-its-main-characteristics-in-a-few-words/, July 28, 2022.
[23]: Stephane H. Maes, (2022), “Justifying the Standard Model U(1) x SU(2) x SU(3) Symmetry in a Multi-fold Universe”, https://doi.org/10.5281/zenodo.8422911, https://shmaesphysics.wordpress.com/2022/08/08/justifying-the-standard-model-u1-x-su2-x-su3-symmetry-in-a-multi-fold-universe/, August 8, 2022, (viXra:2310.0040v1).
[24]: Stephane H Maes, (2020), “The E/G conjecture: entanglement is gravity and gravity is entanglement”, viXra:2010.0139v1, https://shmaesphysics.wordpress.com/2020/10/15/the-e-g-conjecture-entanglement-is-gravity-and-gravity-is-entanglement/, October 15, 2020.
[25]: Stephane H Maes, (2020), “Gravity-like Attractions and Fluctuations between Entangled Systems?”, viXra:2010.0010v1, https://shmaesphysics.wordpress.com/2020/06/25/gravity-like-attractions-and-fluctuations-between-entangled-systems/, June 24, 2020.
[26]: Stephane H Maes, (2020), ”Massless and Massive Multi-Gravity in a Multi-fold Universe”, viXra:2010.0095v1, https://shmaesphysics.wordpress.com/2020/06/30/massless-and-massive-multi-gravity-in-a-multi-fold-universe/, June 19, 2020.
[27]: Stephane H Maes, (2020), ”Explaining Dark Energy, Small Cosmological Constant and Inflation Without New Physics?”, viXra:2006.0261v1, https://shmaesphysics.wordpress.com/2020/06/19/explaining-dark-energy-small-cosmological-constant-and-inflation-without-new-physics/, June 19, 2020.
[28]: Stephane H Maes, (2020), ”Ultimate Unification: Gravity-led Democracy vs. Uber-Symmetries”, viXra:2006.0211v1, https://shmaesphysics.wordpress.com/2020/06/16/ultimate-unification-gravity-led-democracy-vs-uber-symmetries/, June 16, 2020.
[29]: Stephane H. Maes, (2022), “Invalidation and Proof of the Mass Gap, and Viability of The Standard Model on a Discrete Spacetime”, https://doi.org/10.5281/zenodo.8237456, https://shmaesphysics.wordpress.com/2022/07/15/invalidation-and-proof-of-the-mass-gap-and-viability-of-the-standard-model-on-a-discrete-spacetime/, July 15, 2022. (viXra:2308.0059).
[30]: Stephane H. Maes, (2022), Stephane H. Maes, (2022), “A Conjecture: No Dark Matter will be discovered at LHC, or elsewhere”, (v2), https://doi.org/10.5281/zenodo.8175806, https://shmaesphysics.wordpress.com/2022/07/08/a-prediction-no-dark-matter-will-be-discovered-at-lhc-or-elsewhere/, July 8, 2022, viXra:2307.0119.
[31]: Stephane H Maes, (2022), “Unruh effects, Hawking Black Hole Evaporation, Quantum Corrected Larmor Formula, Numbers of Particles in Curved Spacetime: “Same-Same, but Just A Bit Different””, https://doi.org/10.5281/zenodo.8306942, https://shmaesphysics.wordpress.com/2022/07/25/unruh-effects-hawking-black-hole-evaporation-quantum-corrected-larmor-formula-numbers-of-particles-in-curved-spacetime-same-same-but-just-a-bit-different/, July 25, 2022, (viXra:2309.0005).
[32]: Stephane H Maes, (2020), “Multi-fold Higgs Fields and Bosons”, viXra:2204.0146v1, https://shmaesphysics.wordpress.com/2020/11/10/multi-fold-higgs-fields-and-bosons/, November 6, 2020.
[33]: Stephane H Maes, (2020), “Tracking Down The Standard Model With Gravity In Multi-Fold Universes”, viXra:2011.0208v1, https://shmaesphysics.wordpress.com/2020/08/30/tracking-down-the-standard-model-with-gravity-in-multi-fold-universes/, August 20, 2020.
[34]: Stephane H. Maes, (2020), “Particles of The Standard Model In Multi-Fold Universes”, viXra:2111.0071v1, https://shmaesphysics.wordpress.com/2020/11/05/particles-of-the-standard-model-in-multi-fold-universes/, November 4, 2020.
[35]: Stephane H Maes, (2021), “Multi-fold Gravity-Electroweak Theory and Symmetry Breaking”, viXra:2211.0100, https://shmaesphysics.wordpress.com/2021/03/28/multi-fold-gravity-electroweak-theory-and-symmetry-breaking/, March 16, 2021.
[36]: Stephane H Maes, (2020), “Viable Lattice Spacetime and Absence of Quantum Gravitational Anomalies in a Multi-fold Universe”, viXra:2205.0143v1, https://shmaesphysics.wordpress.com/2020/12/13/viable-lattice-spacetime-and-absence-of-quantum-gravitational-anomalies-in-a-multi-fold-universe/, December 4, 2020.
[37]: Stephane H Maes, (2022), “Can Chirality Flips Occur in a Multi-Fold Universe? What About Conservation Laws? II”, viXra:2204.0152v2, https://shmaesphysics.wordpress.com/2022/08/20/can-chirality-flips-occur-in-a-multi-fold-universe-what-about-conservation-laws-ii/, August 20, 2022, and Stephane H Maes, (2020), “Can Chirality Flips Occur in a Multi-Fold Universe? What About Conservation Laws?”, viXra:2204.0152, https://shmaesphysics.wordpress.com/2020/12/07/can-chirality-flips-occur-in-a-multi-fold-universe-what-about-conservation-laws/, December 6, 2020.
[38]: Stephane H Maes, (2020), ”Derivation of the Equivalence Principle in a Multi-fold Universe”, viXra:2010.0090v1, https://shmaesphysics.wordpress.com/2020/06/29/derivation-of-the-equivalence-principle-in-a-multi-fold-universe/, June 19, 2020.
[39]: Stephane H Maes, (2020), “Progress on Proving the Mass gap for Yang Mills and Gravity (maybe it’s already proved…)”, viXra:2006.0155v1, https://shmaesphysics.wordpress.com/2020/06/12/progresses-on-proving-the-mass-gap-for-yang-mills-and-gravity-maybe-its-already-proven/, June 12, 2020.
[40]: Stephane H Maes, (2020), “Gravity Induced Anomalies Smearing in Standard Model so that Protons May Never Decay, Except in Black holes“, viXra:2006.0128v1, https://shmaesphysics.wordpress.com/2020/06/13/gravity-induced-anomalies-smearing-in-standard-model-so-that-protons-may-never-decay-except-in-black-holes/, June 13, 2020.
[41]: Stephane H Maes, (2022), ”Gravity or Magnetic Monopoles? You Cannot Have Both! II“, viXra:2006.0190v2, https://shmaesphysics.wordpress.com/2022/08/20/gravity-or-magnetic-monopoles-you-cannot-have-both-2/, August 20, 2022; Stephane H Maes, (2020), ”Gravity or Magnetic Monopoles? You Cannot Have Both!“, viXra:2006.0190, https://shmaesphysics.wordpress.com/2020/06/15/gravity-or-magnetic-monopoles-you-cannot-have-both/, June 15, 2020.
[42]: Stephane H Maes, (2020), ”Right-handed neutrinos? Mass? Ask Gravity”, viXra:2007.0018v1, https://shmaesphysics.wordpress.com/2020/06/21/right-handed-neutrinos-ask-gravity/, June 23, 2020.
[43]: Stephane H Maes, (2020), ”Strong CP Violation Tamed in The Presence of Gravity”, viXra:2007.0025v1, https://shmaesphysics.wordpress.com/2020/06/23/strong-cp-violation-tamed-in-the-presence-of-gravity/ , June 21, 2020.
[44]: Stephane H Maes, (2020), “Gravity Dictates the Number of Fermion Generations: 3”, viXra:2007.0068v1, https://shmaesphysics.wordpress.com/2020/06/24/gravity-dictates-the-number-of-fermion-generations-3/, June 24, 2020.
[45]: Stephane H Maes, (2020), “Gravity Stabilizes Electroweak Vacuum – No Bubble of Nothing to Worry About!”, viXra:2007.0173v1, https://shmaesphysics.wordpress.com/2020/06/24/gravity-stabilizes-electroweak-vacuum-no-bubble-of-nothing-to-worry-about/, June 24, 2020.
[46]: Stephane H Maes, (2020), ”More Matter Than Antimatter, All Falling Down”, viXra:2010.0121v2, https://shmaesphysics.wordpress.com/2020/07/05/more-matter-than-antimatter-all-falling-down/, July 5, 2020. (V2: April 8, 2021)
[47]: Stephane H Maes, (2020), “No Conventional Sterile Neutrinos In a Multi-fold Universe: just SMG business as usual”, viXra:2103.0202v1, https://shmaesphysics.wordpress.com/2020/10/02/no-conventional-sterile-neutrinos-in-a-multi-fold-universe-just-smg-business-as-usual/, October 1, 2020.
[48]: Stephane H Maes, (2021), “New Physics with LHCb to explain loss of lepton universality, or just gravity?”, viXra:2103.0191v1, https://shmaesphysics.wordpress.com/2021/03/29/new-physics-with-lhcb-to-explain-loss-of-lepton-universality-or-just-gravity/, March 29, 2021.
[49]: Stephane H. Maes, “A bold prediction on the muon anomalous magnetic moment, and expected results to be published on April 7, 2021 by the Fermilab Muon g-2, and its explanation”, viXra:2104.0030v1, https://shmaesphysics.wordpress.com/2021/04/01/a-bold-prediction-on-the-muon-anomalous-magnetic-moment-and-expected-resulted-to-be-published-on-april-7-2021-by-the-fermilab-muon-g-2-and-its-explanation/, April 1, 2021.
[50]: Stephane H Maes, (2021), “New Physics is often not so new”, osf.io/z3sj6, https://shmaesphysics.wordpress.com/2021/04/27/new-physics-is-often-not-so-new/, April 27, 2021, https://zenodo.org/records/7791704.
[51]: Stephane H Maes, (2022), “Direction of Possible Multi-folds Corrections to the W Boson Mass”, osf.io/qvewa, https://shmaesphysics.wordpress.com/2022/04/08/direction-of-possible-multi-folds-corrections-to-the-w-boson-mass/, April 8, 2022, viXra:2304.0020.
[52]: Stephane H Maes, (2022), “Multi-folds in Yang Mills Feynman Diagrams”, osf.io/y8fpd, https://shmaesphysics.wordpress.com/2022/04/05/multi-folds-in-yang-mills-feynman-diagrams/, April 5, 2022, viXra:2303.0161.
[53]: Stephane H. Maes, (2022), “Time-Varying Multi-fold Dark Energy Effects and Implications for the Hubble Tension”, https://doi.org/10.5281/zenodo.10396357, https://shmaesphysics.wordpress.com/2022/11/13/time-varying-multi-fold-dark-energy-effects-and-implications-for-the-hubble-tension/, November 13, 2022, osf.io/g2vzy/, viXra:2312.0083v1. Also, as Stephane H. Maes, (2022), “The Possibility of a Multi-fold Time-Varying Hubble Constant”, viXra:2312.0083v1.
[54]: Stephane H Maes, (2020), ”Explaining Dark Matter Without New Physics?”, viXra:2007.0006, https://shmaesphysics.wordpress.com/2020/06/21/explaining-dark-matter-without-new-physics/, June 21, 2020.
[55]: Stephane H Maes, (2020), “Multi-Fold Universe Dark Matter Successful Explanation and the “Too Thin Universe” but “Too Strong Gravity Lensing by Galaxy Clusters””, viXra:2102.0079v1, https://shmaesphysics.wordpress.com/2020/09/15/multi-fold-universe-dark-matter-successful-explanation-and-the-too-thin-universe-but-too-strong-gravity-lensing-by-galaxy-clusters/, September 14, 2020.
[56]: Stephane H Maes, (2020), ”Multi-Fold Universe Dark Matter Effects Survive Low-Mass Galaxies with Dark Matter Deficits and Excesses”, viXra:2105.0042v1, https://shmaesphysics.wordpress.com/2020/10/14/multi-fold-universe-dark-matter-effects-survive-low-mass-galaxies-with-dark-matter-deficits-and-excesses/, October 14, 2020.
[57]: Stephane H Maes, (2020), ”Multi-Fold Dark Matter Effects and Early Supermassive Black Holes”, viXra:2105.0041v1, https://shmaesphysics.wordpress.com/2020/10/15/multi-fold-dark-matter-effects-and-early-supermassive-black-holes/, October 15, 2020.
[58]: Stephane H Maes, (2022), “Hints of Multi-fold Dark Matter Effects in the Universe”, osf.io/krw7g, https://shmaesphysics.wordpress.com/2022/03/14/hints-of-multi-fold-dark-matter-effects-in-the-universe/, March 14, 2022, https://zenodo.org/record/7791678.
[59]: Stephane H Maes, (2022), “Multi-fold Dark Matter and Energy Effects Fit The Ratios to Normal Matter in the Universe”, https://zenodo.org/doi/10.5281/zenodo.10071554, https://shmaesphysics.wordpress.com/2022/08/14/multi-fold-dark-matter-and-energy-effects-fit-the-ratios-to-normal-matter-in-the-universe/, August 14, 2022, (https://osf.io/mahsu, viXra:2311.0018v1).
[60]: Stephane H. Maes, (2022), “Explaining Imbalance of Tidally Ejected Stars from Open Stars Clusters Without MOND”, https://doi.org/10.5281/zenodo.10421124, https://shmaesphysics.wordpress.com/2022/11/19/explaining-imbalance-of-tidally-ejected-stars-from-open-stars-clusters-without-mond/, November 19, 2022, https://osf.io/bp64c.
[61]: Stephane H. Maes, {2022), “Black holes effects outside the black holes do not mean that Hawking radiation is not occurring at its horizon”, https://shmaesphysics.wordpress.com/different-approaches-to-compute-hawking-black-holes-decay/#comment-5027, November 23, 2022.
[62]: Stephane H Maes, (2022), “Multi-fold Discrete Fractal Spacetime, and the Viability of Local vs. Non-Local Hidden Variables”, https://doi.org/10.5281/zenodo.10344634, https://shmaesphysics.wordpress.com/2022/10/30/multi-fold-discrete-fractal-spacetime-and-the-viability-of-local-vs-non-local-hidden-variable-viability/, October 30, 2022, osf.io/qevys, viXra:2312.0065v1.
[63]: Stephane H Maes, (2021), “Multi-fold Embeddings, Space Time Matter Induction or Gravity Asymptotically Safe and The AdS/CFT Correspondence Conjecture, they all can recover the Standard Model”, viXra:2212.0120v1, https://shmaesphysics.wordpress.com/2021/12/20/multi-fold-embeddings-space-time-matter-induction-or-gravity-asymptotically-safe-and-the-ads-cft-correspondence-conjecture-they-all-can-recover-the-standard-model-or-smg/, December 20, 2021.
[64]: Stephane H. Maes, (2022), “A Better Quantum Extremal Surface and Island Interpretation that explains the Associated Massive Gravity”, https://doi.org/10.5281/zenodo.10437116, https://shmaesphysics.wordpress.com/2022/12/03/a-better-quantum-extremal-surface-and-island-interpretation-that-explains-the-associated-massive-gravity/, December 3, 2022, (https://osf.io/dn3kh).
[65]: Stephane H Maes, (2021), “Pointers to Nowhere with Geometric Unity Theory, or Some Ways Forward in Multi-fold Universes?”, viXra:2210.0081v1, https://shmaesphysics.wordpress.com/2021/03/07/pointers-to-nowhere-with-geometric-unity-theory-or-some-ways-forward-in-multi-fold-universes/, March 7, 2021.
[66]: Stephane H Maes, (2022), “The Replica Trick, Wormholes, Island formula, and Quantum Extremal Surfaces, and How the AdS/CFT Correspondence Conjecture, and Hence the M-theory, Encounters Multi-folds”, https://doi.org/10.5281/zenodo.10207057, https://shmaesphysics.wordpress.com/2022/09/20/the-replica-trick-its-wormholes-islands-and-quantum-extremal-surfaces-and-how-the-ads-cft-correspondence-conjecture-and-hence-the-m-theory-encounters-multi-folds/, September 26, 2022, (osf.io/xwf6q/). Also published as: Stephane H Maes, (2022), “The Replica Trick, Wormholes, Island formula, and Quantum Extremal Surfaces”, September 26, 2022 (viXra:2311.0154v1).
[67]: Stephane H Maes, (2021), “Right-handed Neutrinos and Traversable Wormholes: the key to entanglement, gravity and multi-folds extensions to ER=EPR?”, viXra:2211.0173v1, https://shmaesphysics.wordpress.com/2021/04/03/right-handed-neutrinos-and-traversable-wormholes-the-key-to-entanglement-gravity-and-multi-folds-extensions-to-erepr/, April 3, 2021.
[68]: Stephane H Maes, (2021), “Multi-fold Non-Commutative Spacetime, Higgs and The Standard Model with Gravity”, viXra:2212.0037v1, https://shmaesphysics.wordpress.com/2021/04/18/multi-fold-non-commutative-spacetime-higgs-and-the-standard-model-with-gravity/, April 11, 2021.
[69]: Stephane H Maes, (2022), “Trans-Planckian Censorship Conjecture: Factual in Multi-fold Universes as well as GR Universes”, viXra:2303.0025v1, https://shmaesphysics.wordpress.com/2022/03/13/trans-planckian-censorship-conjecture-factual-in-multi-fold-universes-as-well-as-gr-universes/, March 12, 2022.
[70]: Stephane H Maes, (2021), “Spacetime and Gravity are 2D around Planck Scales: A Universal Property of Consistent Quantum Gravity”, viXra:2211.0001v1, https://shmaesphysics.wordpress.com/2021/03/23/spacetime-and-gravity-are-2d-around-planck-scales-a-universal-property-of-consistent-quantum-gravity/, March 20, 2021.
[71]: Stephane H Maes, (2020), “The W-type Multi-Fold Hypothesis and Quantum Physics Interpretation of wave Functions and QFT”, viXra:2207.0118v1, https://shmaesphysics.wordpress.com/2020/12/24/the-w-type-multi-fold-hypothesis-and-quantum-physics-interpretation-of-wave-functions-and-qft/, December 20, 2020.
[72]: Stephane H Maes, (2022), “2D Random Walks of Massless Higgs Bosons as Microscopic Interpretation of the Asymptotic Safety of Gravity, and of the Standard Model”, https://doi.org/10.5281/zenodo.10467452, https://shmaesphysics.wordpress.com/2022/12/28/2d-random-walks-of-massless-higgs-bosons-as-microscopic-interpretation-of-the-asymptotic-safety-of-gravity-and-of-the-standard-model/, December 28, 2022, (osf.io/udhbf). Also published as: Stephane H. Maes, (2022), “2D Random Walks of Massless Higgs Bosons”, viXra:2401.0073v1, https://shmaesphysics.wordpress.com/2022/12/28/2d-random-walks-of-massless-higgs-bosons-as-microscopic-interpretation-of-the-asymptotic-safety-of-gravity-and-of-the-standard-model/, December 28, 2022.
[73]: Stephane H Maes, (2020), “No Gravity Induced Wave Function Collapse in a Multi-fold Universe”, viXra:2012.0152v1, https://shmaesphysics.wordpress.com/2020/09/11/no-gravity-induced-wave-function-collapse-in-a-multi-fold-universe/, September 11, 2020. Also as: Stephane H Maes, (2020), “No Gravity Superposition Induced Wave Function Collapse in a Multi-fold Universe”, viXra:2012.0152v1, https://shmaesphysics.wordpress.com/2020/09/11/no-gravity-induced-wave-function-collapse-in-a-multi-fold-universe/, September 11, 2020.
[74]: Stephane H. Maes, (2022), “Multi-fold Gravity can Violate P-Symmetry. It is Aligned With Observations of Asymmetry of the Orientation of Tetrahedra of Galaxies”, https://doi.org/10.5281/zenodo.10443847, https://shmaesphysics.wordpress.com/2022/12/10/multi-fold-gravity-can-violate-p-symmetry-it-is-aligned-with-observations-of-asymmetry-of-the-orientation-of-tetrahedra-of-galaxies/, December 10, 2022. Also published as Stephane H. Maes, (2022), “Multi-fold Gravity can Violate Parity Symmetry”, https://shmaesphysics.wordpress.com/2022/12/10/multi-fold-gravity-can-violate-p-symmetry-it-is-aligned-with-observations-of-asymmetry-of-the-orientation-of-tetrahedra-of-galaxies/, December 10, 2022.
[75]: Stephane H Maes, (2021), ““Quantum Gravity Emergence from Entanglement in a Multi-Fold Universe”: 2D or 2+1D spacetime at small scales”, viXra:2103.0142, https://shmaesphysics.wordpress.com/2021/03/20/quantum-gravity-emergence-from-entanglement-in-a-multi-fold-universe-2d-or-21d-spacetime-at-small-scales/, March 20, 2021.
[76]: Stephane H Maes, (2022), “Comments on Multi-fold mechanisms as Hermitian vs. Unitary processes”, https://shmaesphysics.wordpress.com/2020/06/25/gravity-like-attractions-and-fluctuations-between-entangled-systems/#comment-4359, July 27, 2022.
[77]: Stephane H. Maes, (2021), “Comment on 4D spacetime and follow-up comments”, https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/#comment-1416, January 16, 2021. Retrieved on February 21, 2021.
[78]: Stephane H. Maes, (2021), “Comment on 4D spacetime and follow-up comments: https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/#comment-1579, February 18, 2021. Retrieved on February 21, 2021.
[79]: Stephane H. Maes, (2021), “Comment on 2D spacetime and follow-up comments”, https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/#comment-1695, March 3, 2021. Retrieved on March 31, 2021.
[80]: Stephane H. Maes, (2021), “Comment on 4D spacetime and follow-up comments”, https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/#comment-1891, March 30, 2021. Retrieved on March 31, 2021.
[81]: Stephane H. Maes, (2021), “Comment on 4D spacetime and follow-up comments”, https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/#comment-1906, April 1, 2021. Retrieved on December 27, 2022.
[82]: Stephane H. Maes, (2022), “Additional arguments for 4D spacetime for our real universe”, https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/#comment-4679, September 21, 2022. Retrieved on December 27, 2022.
[83]: Stephane H Maes, (2020), “Call for Collaboration”, https://shmaesphysics.wordpress.com/2020/09/07/do-you-want-a-phd-or-who-knows-a-nobel-price-in-physics/, September 6, 2020.
[84]: Stephane H Maes, (2020), “Multi-fold Gravitons In-N-Out Spacetime”, viXra:2010.0155v1, https://shmaesphysics.wordpress.com/2020/07/27/multi-fold-gravitons-in-n-out-spacetime/, July 27, 2020, (posted September 6, 2020)
[85]: Stephane H Maes, (2022), “Gravitational Bootstrap, S-matrix, Superstrings, and The Plausible Unphysicality of Gravitons”, viXra:2301.0155v1, https://shmaesphysics.wordpress.com/2022/02/06/gravitational-bootstrap-s-matrix-superstrings-and-the-plausible-unphysicality-of-gravitons/, February 6, 2022.
[86]: Stephane H Maes, (2020), “Particles, Especially Virtual Particles, in a Multi-fold Universe vs. QFT”, viXra:2010.0133v1, https://shmaesphysics.wordpress.com/2020/07/11/particles-especially-virtual-particles-in-a-multi-fold-universe-vs-qft/ , July 10, 2020.
[87]: Stephane H Maes, (2020), “Comments to “Yes, Stephen Hawking Lied To Us All About How Black Holes Decay””, https://osf.io/v7thb/, https://shmaesphysics.wordpress.com/2020/07/11/comments-to-yes-stephen-hawking-lied-to-us-all-about-how-black-holes-decay/, July 11, 2020.
[88]: Stephane H Maes, (2020), “Different approaches to compute Hawking Black Holes Decay”, viXra:2208.0009v1, https://shmaesphysics.wordpress.com/different-approaches-to-compute-hawking-black-holes-decay/, August 1, 2022. (Originally published July 11, 2020).
[89]: Stephane H Maes, (2020), “Multi-Fold Black Holes: Entropy, Evolution and Quantum Extrema”, viXra:2105.0136v1, https://shmaesphysics.wordpress.com/2020/11/01/multi-fold-black-holes-entropy-evolution-and-quantum-extrema/, October 31, 2020.
[90]: Stephane H Maes, (2020), “No Gravity Shield in Multi-folds Universes”, viXra:2010.0032v1, https://shmaesphysics.wordpress.com/2020/06/26/no-gravity-shields-in-multi-folds-universes/ , June 26, 2020.
[91]: Stephane H Maes, (2020), “Area Laws Between Multi-Fold Universes and AdS”, viXra:2010.0207v1, https://shmaesphysics.wordpress.com/2020/08/10/area-laws-between-multi-fold-universes-and-ads/, August 10, 2020.
[92]: Stephane H. Maes, (2022), “Comments on radiation black hole simulation on a lattice”, https://shmaesphysics.wordpress.com/2022/07/25/unruh-effects-hawking-black-hole-evaporation-quantum-corrected-larmor-formula-numbers-of-particles-in-curved-spacetime-same-same-but-just-a-bit-different/#comment-5099, December 2, 2022.
[93]: Stephane H. Maes, (2021), “Neutrons are forming an external skin in Nuclei and Neutron Stars”, https://zenodo.org/doi/10.5281/zenodo.14582585, https://shmaes.wordpress.com/2021/05/08/neutrons-are-forming-an-external-skin-in-nuclei-and-neutron-stars/, May 8, 2021. (V1) (V3 is January 7, 2024). (osf.io/zdy4s/, viXra:2501.0029).
[94]: Stephane H Maes, (2022), “The Yang Mills Double Copy leads to New AdS/CFT + Gravity Correspondences, or How the M-theory encounters Multi-fold Universes”, v1.1, https://doi.org/10.5281/zenodo.7827248, https://shmaesphysics.wordpress.com/2022/04/22/the-yang-mills-double-copy-leads-to-new-ads-cft-gravity-correspondences-or-how-the-m-theory-encounters-multi-fold-universes/, April 22, 2022. (v1: at zenodo.7827249).
[95]: Stephane H. Maes, (2022), “Schwinger effect and charged black holes”, https://shmaesphysics.wordpress.com/2020/11/01/multi-fold-black-holes-entropy-evolution-and-quantum-extrema/#comment-4686, September 25, 2022.
[96]: Stephane H. Maes, “Comments on asymmetry of distributions of ejected star from gas clusters”, https://shmaesphysics.wordpress.com/2020/06/21/explaining-dark-matter-without-new-physics/#comment-4813 and subsequent comments, October 27, 2022
[97]: Stephane H Maes, (2020), “Implicit Multi-Fold Mechanisms in a Neural Network Model of the Universe”, viXra:2012.0191v1, https://shmaesphysics.wordpress.com/2020/09/12/implicit-multi-fold-mechanisms-in-a-neural-network-model-of-the-universe/, September 12, 2020.
[98]: Stephane H Maes, (2020), “Interpretation of “Neural Network as the World””, viXra:2012.0197v1, https://shmaesphysics.wordpress.com/2020/09/14/interpretation-of-neural-network-as-the-world/, September 14, 2020.
[99]: Stephane H Maes, (2020), “Entangled Neural Networks from Multi-fold Universes to Biology”, viXra:2207.0174v1, https://shmaesphysics.wordpress.com/2020/12/31/entangled-neural-networks-from-multi-fold-universes-to-biology/, December 25, 2020.
[100]: Wikipedia, “Lambda-CDM model”, https://en.wikipedia.org/wiki/Lambda-CDM_model. Retrieved on August 14, 2022.
[101]: Stephane H. Maes, (2022), “CO2 and CH4 absorption powered by nuclear fusion, via fission, is the only way to manage climate change and the Planet’s trigger points”, viXra:2211.0154v1, https://shmaes.wordpress.com/2022/04/09/co2-and-ch4-absorption-powered-fission-is-the-only-way-to-manage-climate-change-and-the-planets-trigger-points/, April 9, 2022.
[102]: Stephane H Maes, (2021), “Oops For The Loops: Mounting LQG Woes And A Challenge To The LQG Community”, viXra:2212.0168, https://shmaesphysics.wordpress.com/2021/12/30/oops-for-loops-mounting-lqg-woes-and-a-challenge-to-the-lgq-community/, December 29, 2021.
[103]: Stephane H. Maes, (2021-2022), “Our universe is 4D”, Comments and following comments at https://shmaesphysics.wordpress.com/2020/09/19/renormalization-and-asymptotic-safety-of-gravity-in-a-multi-fold-universe-more-tracking-of-the-standard-model-at-the-cost-of-supersymmetries-guts-and-superstrings/#comment-1416. January 16, 2021, and after.
[104]: Stephane H Maes, (2021), “Multi-fold gravity and double copy of gauge theory”, osf.io/xun82, https://shmaesphysics.wordpress.com/2021/05/04/multi-fold-gravity-and-double-copy-of-gauge-theory/, May 4, 2021, viXra:2303.0114.
[105]: Stephane H Maes, (2020), “Entanglement Concretizes Time in a Multi-fold Universe”, viXra:2010.0083v1, https://shmaesphysics.wordpress.com/2020/06/28/entanglement-concretizes-time-in-a-multi-fold-universe/, June 28, 2020. Also published as: Stephane H Maes, (2020), “Entanglement and Random Walks Concretize Time in a Multi-fold Universe”, viXra:2010.0083v1, https://shmaesphysics.wordpress.com/2020/06/28/entanglement-concretizes-time-in-a-multi-fold-universe/, June 28, 2020.
[106]: Stephane H Maes, (2021), “How the ER = EPR, GR = QM and AdS/CFT correspondence conjectures, can be explained in multi-fold theory, along with the E/G conjecture. A call to the Physics Community!”, viXra:2111.0144v2, https://shmaesphysics.wordpress.com/2021/11/28/how-the-er-epr-gr-qm-and-ads-cft-correspondence-conjectures-can-be-explained-in-multi-fold-theory-and-the-e-g-conjecture-explains-and-realize-in-a-multi-fold-universe-a-call-to-the-physics-comm/, December 28, 2021.
[107]: Stephane H Maes, (2020), “A Multi-fold Universe Genesis Inspired By Explosive Total Collision: The Source Of The Big Bang?”, viXra:2208.0082v1, https://shmaesphysics.wordpress.com/2021/01/17/a-multi-fold-universe-genesis-inspired-by-total-explosion-collision-the-source-of-the-big-bang/, January 12, 2021.
[108]: Stephane H. Maes, (2022), “JWST and the Big Bang invalidation”, https://shmaesphysics.wordpress.com/2021/01/17/a-multi-fold-universe-genesis-inspired-by-total-explosion-collision-the-source-of-the-big-bang/#comment-4577, and following comments. August 21, 2022.
[109]: Stephane H. Maes, (2022), “Schwinger effect dominates near the horizon of charged black holes near extremality and reduces the charge”, https://shmaesphysics.wordpress.com/2022/07/25/unruh-effects-hawking-black-hole-evaporation-quantum-corrected-larmor-formula-numbers-of-particles-in-curved-spacetime-same-same-but-just-a-bit-different/#comment-4687, September 23, 2022.
[110] Stephane H Maes, (2022), “Charm of the proton”, https://shmaesphysics.wordpress.com/2021/03/29/new-physics-with-lhcb-to-explain-loss-of-lepton-universality-or-just-gravity/#comment-3791, March 27, 2022.
[111]: Stephane H. Maes, (2022-2023), “Confusing mathematical duality to predict quantum computing algorithm, with building a wormhole”, https://shmaesphysics.wordpress.com/2020/10/11/circular-arguments-in-string-and-superstring-theory-from-a-multi-fold-universe-perspective/comment-page-1/#comment-5093, and following related comments, November 30, 2022.
[112]: Stephane H. Maes, (2023), “No supersymmetry”, https://shmaesphysics.wordpress.com/2023/11/21/no-supersymmetry/, November 21,2023.
[113]: Stephane H. Maes, (2023), “Justification for the multi-fold mappings, and dynamic multi-fold mechanism”, https://shmaesphysics.wordpress.com/2020/12/24/the-w-type-multi-fold-hypothesis-and-quantum-physics-interpretation-of-wave-functions-and-qft/comment-page-1/#comment-8092, October 29, 2023.
[114]: Stephane H. Maes, (2023), “Yeah or Nay on Black Holes as Explanation for Dark Energy?”, osf.io/369pd, https://shmaesphysics.wordpress.com/2023/03/01/yeah-or-nay-on-black-holes-as-explanation-for-dark-energy/, V3, March 26, 2023. (V2: March 12, 2023, V1: Stephane H. Maes, (2023), “Yeah or Nay on Black Holes as Explanation for Dark Energy?”, viXra:2303.0031, https://shmaesphysics.wordpress.com/2023/03/01/yeah-or-nay-on-black-holes-as-explanation-for-dark-energy/, March 1, 2023).
[115]: Stephane H. Maes, (2023), “Dynamic sources, Dynamic Multi-folds, and General Relativity Lense-Thirring and Frame Dragging Effects”, https://doi.org/10.5281/zenodo.14737010, https://shmaesphysics.wordpress.com/2023/03/12/dynamic-sources-dynamic-multi-folds-and-general-relativity-lens-thirring-and-frame-dragging-effects/, March 12, 2023, https://osf.io/ytmw6/download/
[116]: Stephane H. Maes, (2023), “The Multi-fold Least Action Principle, a Quasi Theory Of Everything”, https://doi.org/10.5281/zenodo.14542569, https://shmaesphysics.wordpress.com/2023/02/19/the-multi-fold-least-action-principle-a-quasi-theory-of-everything/, February 19, 2023. (osf.io/2ncqf/, viXra:2412.0145v1).
[117]: Stephane H. Maes, (2023), “Maybe, black holes do not systematically decohere quantum states”, https://shmaesphysics.wordpress.com/2020/11/01/multi-fold-black-holes-entropy-evolution-and-quantum-extrema/#comment-6315, March 7, 2023.
[118]: Stephane H. Maes, (2023), “No electroweak / Higgs mass hierarchy problem in multi-fold theory”, https://shmaesphysics.wordpress.com/2021/03/28/multi-fold-gravity-electroweak-theory-and-symmetry-breaking/#comment-6794, March 30, 2023.
[119]: Stephane H. Maes, “Right-handed neutrinos in the multi-fold stabilize the multi-fold unconstrained KK space time matter induction and scattering”, https://shmaesphysics.wordpress.com/2021/04/03/right-handed-neutrinos-and-traversable-wormholes-the-key-to-entanglement-gravity-and-multi-folds-extensions-to-erepr/comment-page-1/#comment-6875, April 8, 2023.
[120]: Stephane H. Maes, (2023), “No lack of clumpiness, just as needed”, https://shmaesphysics.wordpress.com/2020/06/21/explaining-dark-matter-without-new-physics/comment-page-1/#comment-6974. April 12, 2023.
[121]: Stephane H. Maes, (2023), “Multi-fold Universes, Multiverses and Many Worlds”, https://doi.org/10.5281/zenodo.15339413, https://shmaesphysics.wordpress.com/2023/04/08/multi-fold-universes-multi-folds-and-many-worlds/, April 8, 2023.
[122]: Stephane H. Maes, (2023), ‘Comment on black hole decoherence”, https://shmaesphysics.wordpress.com/2020/11/01/multi-fold-black-holes-entropy-evolution-and-quantum-extrema/#comment-6315, March 7, 2023.
[123]: Stephane H Maes, (2023), “Comments of the universe is too smooth”, https://shmaesphysics.wordpress.com/2020/06/21/explaining-dark-matter-without-new-physics/#comment-6086, February 9, 2023, and https://shmaesphysics.wordpress.com/2020/06/21/explaining-dark-matter-without-new-physics/#comment-6974, April 12, 2023.
[124]: Stephane H Maes, (2023), “Our real universe is macroscopically 4D. Hints come from every direction & show that it had to be so”, https://doi.org/10.5281/zenodo.17575694, https://shmaesphysics.wordpress.com/2023/04/23/our-real-universe-is-macroscopically-4d-hints-come-from-every-directions-show-that-it-had-to-be-so/, April 23, 2023 (viXra:2511.0054v1).
[125]: Stephane H Maes, (2023), “No Gravitational Evaporation of Everything à la Schwinger, only for Black Holes”, https://shmaesphysics.wordpress.com/2023/07/15/no-gravitational-evaporation-of-everything-a-la-schwinger-only-for-black-holes/, July 15, 2023.
[126]: Stephane H Maes, (2023), “Unstable QFT and SM with Gravity except in a Multi-fold Universe”, https://shmaesphysics.wordpress.com/2023/07/19/unstable-qft-and-sm-with-gravity-except-in-a-multi-fold-universe/, July 19, 2023.
[127]: Stephane H. Maes, (2023), “Comments about massive galaxies without dark matter”, https://shmaesphysics.wordpress.com/2020/10/14/multi-fold-universe-dark-matter-effects-survive-low-mass-galaxies-with-dark-matter-deficits-and-excesses/#comment-7430, July 20, 2023.
[128]: Stephane H Maes, (2023), “Less Cracks in the Standard Cosmology in a Multi-fold Universe with its Quantum Random walks”, https://shmaesphysics.wordpress.com/2023/06/20/less-cracks-in-the-standard-cosmology-in-a-multi-fold-universe-with-its-quantum-random-walks/, June 19, 2023.
[129]: Stephane H. Maes, (2023), “Ad Astra With Warp Drives? Probably Not”, https://shmaesphysics.wordpress.com/2023/12/09/ad-astra-longe-with-warp-drives-probably-not/, December 9, 2023.
[130]: Stephane H Maes, (2023), “2D gravity and 2D Yang Mills Physics is all what matters”, https://shmaesphysics.wordpress.com/2023/04/23/our-real-universe-is-macroscopically-4d-hints-come-from-every-directions-show-that-it-had-to-be-so/comment-page-1/#comment-7588, August 5, 2023.
[131]: Stephane H Maes (2023), “The Multi-fold Theory – Draft Raw Compendium of Research Papers (till August, 2023)”, https://doi.org/10.5281/zenodo.8242021, https://shmaesphysics.wordpress.com/2023/08/12/the-multi-fold-theory-draft-raw-compendium-of-research-papers-till-august-2023/, August 12, 2023, (https://osf.io/swqmb).
[133]: Stephane H. Maes, (2023), “Persisting on No Decoherence due to Gravity, Black Holes, or Spacetime Curvature Superpositions”, https://shmaesphysics.wordpress.com/2023/08/18/persisting-on-no-decoherence-due-to-gravity-black-holes-or-spacetime-curvature-superpositions/, August 18, 2023.
[132]: Stephane H Maes, (2023), “Barnett’s resolution of the Minkowski – Abraham dilemma holds, no 4-vector issue”, https://zenodo.org/records/10071847, https://shmaes.wordpress.com/2023/08/11/barnetts-resolution-of-the-minkowski-abraham-dilemma-holds-no-4-vector-issue/ August 13, 2023, (https://osf.io/bd8ju, viXra:2311.0023v1).
[134]: Stephane H. Maes, (2023), “No supersymmetry at D<=4 with a positive cosmological constant”, https://shmaesphysics.wordpress.com/2022/07/08/a-prediction-no-dark-matter-will-be-discovered-at-lhc-or-elsewhere/#comment-7563. July 23, 2023.
[135]: Stephane H. Maes, (2023), “The universe is exactly the only thing that it could be if it is a 4D multi-fold universe! No fine-tuning problem, no invocation of God or multiverses”, https://shmaesphysics.wordpress.com/2023/04/08/multi-fold-universes-multi-folds-and-many-worlds/comment-page-1/#comment-8037, October 7, 2023.
[136]: Stephane H. Maes, (2020-2023), “Quantum Gravity Emergence from Entanglement in a Multi-Fold Universe – V3: Update to section 4.1 – Multi-folds for Entanglement and EPR”, https://shmaesphysics.wordpress.com/quantum-gravity-emergence-from-entanglement-in-a-multi-fold-universe-v3-update-to-section-4-1-multi-folds-for-entanglement-and-epr/.
[137]: Stephane H. Maes, (2020-2023), “Quantum Gravity Emergence from Entanglement in a Multi-Fold Universe”, V3, https://zenodo.org/doi/10.5281/zenodo.7792911, October 29, 2023.
[138]: Stephane H. Maes, (2023), “Path integrals and wormholes impact on the cosmological constant”, https://shmaesphysics.wordpress.com/2022/09/20/the-replica-trick-its-wormholes-islands-and-quantum-extremal-surfaces-and-how-the-ads-cft-correspondence-conjecture-and-hence-the-m-theory-encounters-multi-folds/comment-page-1/#comment-7978, September 3, 2023.
[139]: Stephane H. Maes, (2023), “Microscopic interpretation of mass acquisition from massless Higgs bosons”, https://shmaesphysics.wordpress.com/2021/02/28/more-on-multi-fold-particles-as-microscopic-black-holes-with-higgs-regularizing-extremality-and-singularities/#comment-8412, November 5, 2023.
[140]: Stephane H. Maes, (2023), “Justifying the Multi-folds Mechanisms, Mapping, Tenancy and More”, https://shmaesphysics.wordpress.com/2023/11/10/justifying-the-multi-folds-mechanisms-mapping-tenancy-and-more/, November 10, 2023.
[141]: Stephane H. Maes, (2023), “In multi-fold theory, the expansion of the universe is not a mirage”, https://shmaesphysics.wordpress.com/2020/06/19/explaining-dark-energy-small-cosmological-constant-and-inflation-without-new-physics/#comment-7242, June 20, 2023.
[142]: Stephane H. Maes, “Multi-fold dark energy is also a fluctuation of quantum vacuum fluctuations”, https://shmaesphysics.wordpress.com/2020/06/19/explaining-dark-energy-small-cosmological-constant-and-inflation-without-new-physics/#comment-6111, February 18, 2023.
[143]: Stephane H. Maes, (2023), “It’s experimentally validated: antimatter falls down, no antigravity”, https://shmaesphysics.wordpress.com/2020/07/05/more-matter-than-antimatter-all-falling-down/#comment-8018 and subsequent comments, September 27, 2023.
[144]: Stephane H. Maes, (2023), “Particle internal symmetries and anti-particles when modeled as microscopic black holes or random walk patterns”, https://shmaesphysics.wordpress.com/2021/02/28/more-on-multi-fold-particles-as-microscopic-black-holes-with-higgs-regularizing-extremality-and-singularities/#comment-8634, November 28, 2023.
[145]: Stephane H. Maes, (2023), “Information has no mass”, https://shmaesphysics.wordpress.com/2023/12/14/information-has-no-mass/, December 14, 2023
[146]: Stephane H. Maes, (2023), “Gravity is Quantum”, https://shmaesphysics.wordpress.com/2023/12/19/gravity-is-quantum/, December 19, 2023.
[147]: Stephane H. Maes, (2023), “Multi-folds for Entanglement and EPR”, https://zenodo.org/doi/10.5281/zenodo.10059877, https://shmaesphysics.wordpress.com/2023/11/01/multi-folds-for-entanglement-and-epr/, October 29, 2023, (viXra:2311.0001v1), also as “Update to section 4.1 of “Quantum Gravity Emergence from Entanglement in a Multi-Fold Universe” – Multi-folds for Entanglement and EPR”, https://shmaesphysics.wordpress.com/2023/10/29/update-to-section-4-1-of-quantum-gravity-emergence-from-entanglement-in-a-multi-fold-universe-multi-folds-for-entanglement-and-epr/, (https://osf.io/54ycm/).
[148]: Stephane H. Maes, (2023), “A Fractal spacetime just leads to rescaled cosmological constant. Yet that may provide a time varying effect”, https://shmaesphysics.wordpress.com/2022/10/30/multi-fold-discrete-fractal-spacetime-and-the-viability-of-local-vs-non-local-hidden-variable-viability/comment-page-1/#comment-8896. December 14, 2023.
[149]: Stephane H. Maes, (2023), “Multi-Fold dark Matter effects & Rotation Curve Differences in Galaxies in Clusters, Yet Respect of the Strong Equivalence Principle”, https://doi.org/10.5281/zenodo.13766006, https://shmaesphysics.wordpress.com/2023/01/29/multi-fold-dark-matter-effects-rotation-curve-differences-in-galaxies-in-custers-yet-respect-of-the-strong-equivalence-principle/, January 29, 2023, (osf.io/texvj, vixra:2409.0088v1).
[150]: Stephane H. Maes, (2022), “Multi-folds, Non-Commutative Spacetime, Spin, and All That”, https://doi.org/10.5281/zenodo.11114501, https://shmaesphysics.wordpress.com/2022/12/31/the-principles-of-quantum-mechanics/, December 31, 2022. Also, as https://shmaesphysics.wordpress.com/2022/12/31/multi-folds-non-commutative-spacetime-spin-and-all-that/, (osf.io/au7wc, viXra:2405.0022v1).
[151]: Stephane H. Maes, (2024), “Collapses, Singularities, Censorship, Conjectures, and More”, https://shmaesphysics.wordpress.com/2024/07/16/collapses-singularities-censorship-conjectures-and-more/, July 16, 2024.
[152]: Stephane H. Maes, (2024), “June 2024 Status of the Multi-fold Theory”, https://doi.org/10.5281/zenodo.13345964, https://shmaesphysics.wordpress.com/2024/08/19/june-2024-status-of-the-multi-fold-theory/, June 22, 2024.
[153]: Stephane H. Maes, (2024), “About binaries alleged gravity anomalies”, https://shmaesphysics.wordpress.com/2023/01/29/multi-fold-dark-matter-effects-rotation-curve-differences-in-galaxies-in-custers-yet-respect-of-the-strong-equivalence-principle/#comment-9726, September 10,
[154]: Stephane H Maes, (2024), “A different and more generic proof based on the multi-fold theory, that confinement implies mass effects and chiral symmetry breaking”, https://shmaesphysics.wordpress.com/2021/02/28/more-on-multi-fold-particles-as-microscopic-black-holes-with-higgs-regularizing-extremality-and-singularities/comment-page-1/#comment-9952, November 3, 2024.
[155]: Stephane H. Maes, (2024), “A discrete multi-fold spacetime realized by random walk, implies a non-commutative spacetime”, https://shmaesphysics.wordpress.com/2022/12/31/multi-folds-non-commutative-spacetime-spin-and-all-that/#comment-9820, September 23, 2024.
[156]: Stephane H Maes, (2021), “Comments on “No issue of unnaturalness and mass hierarchy with the Higgs mass””, https://shmaesphysics.wordpress.com/2021/04/27/new-physics-is-often-not-so-new/comment-page-1/#comment-3026, December 24, 2021.
[157]: Stephane H. Maes, (2024), “Physics Salvages The Third Law Of Black Hole Thermodynamics”, https://shmaesphysics.wordpress.com/2024/11/24/physics-salvages-the-third-law-of-black-hole-thermodynamics/, November 24, 2024.
[158]: Stephane H. Maes, “No Issue with the Quantization of Electrostatic Fields”, https://shmaesphysics.wordpress.com/2024/12/31/no-issue-with-the-quantization-of-electrostatic-fields/, December 31, 2024.
[159]: Stephane H. Maes, (2025), “Looks like we were right. Neutrinos are probably not Majorana Fermions”, https://shmaesphysics.wordpress.com/2020/06/21/right-handed-neutrinos-ask-gravity/#comment-10501, March 23, 2025.
[160]: Stephane H. Maes, (2024), “Gödel’s incompleteness theorems imply no full theory is possible”, https://shmaesphysics.wordpress.com/2023/02/19/the-multi-fold-least-action-principle-a-quasi-theory-of-everything/#comment-10066, November 30, 2024. Based on: Stephane H. Maes, (1989), “Feynman Path Integrals”, and communication to Prof. J. Weyers, Quantum Physics Seminar as part of BS in Physics, FYMA, UC Louvain.
[161]: Stephane H. Maes, (2024), “Spin as angular momentum of massless Higgs bosons in Higgs condensate or random walks”, https://shmaesphysics.wordpress.com/2022/12/31/multi-folds-non-commutative-spacetime-spin-and-all-that/#comment-10027, November 26, 2024.
[162]: Stephane H. Maes, (2025), “No Extremality or Singularity for Particles as Multi-fold Black Holes”, https://shmaesphysics.wordpress.com/2025/03/27/no-extremality-or-singularity-for-particles-as-multi-fold-black-holes/, March 27, 2025.
[163]: Stephane H. Maes, (2024), “Explaining the mass anisotropy of certain semi-Dirac Fermions in semimetals”, https://shmaesphysics.wordpress.com/2021/02/28/more-on-multi-fold-particles-as-microscopic-black-holes-with-higgs-regularizing-extremality-and-singularities/comment-page-1/#comment-9951, November 3, 2024.
[164]: Stephane H. Maes, (2024), “Gravity in SMG contributes in the right direction to the discrepancies of CP violation in B-mesons with the SM”, https://shmaesphysics.wordpress.com/2022/07/08/a-prediction-no-dark-matter-will-be-discovered-at-lhc-or-elsewhere/#comment-10078, December 6, 2024.
[165]: Stephane H. Maes, (2024), “As we predicted, still no sterile neutrino”, https://shmaesphysics.wordpress.com/2020/10/02/no-conventional-sterile-neutrinos-in-a-multi-fold-universe-just-smg-business-as-usual/#comment-10039, November 26, 2024.
[166]: David Tong, (2018), “Gauge Theory”, Cambridge University, https://www.damtp.cam.ac.uk/user/tong/gaugetheory/gt.pdf.
[167]: Stephane H. Maes, (2025), “Early inflation dS instabilities, require a mechanisms like the multi-fold spacetime reconstruction with 2D random walks of massless bosons”, https://shmaesphysics.wordpress.com/2024/12/12/consistencies-and-implications-of-2d-massless-random-walks-discrete-non-commutative-spacetime-and-no-flat-supersymmetry/comment-page-1/#comment-11204, November 4, 2025.
[168]: Stephane H. Maes, (2024), “Continuous mathematics to model Physics are just degenerate approximations of finite / discrete mathematics”, https://shmaesphysics.wordpress.com/2023/11/21/no-supersymmetry/#comment-9457, July 13, 2024.
[169]: Stephane H. Maes, (2022), “Background independence implies discreteness”, https://shmaesphysics.wordpress.com/2021/04/18/multi-fold-non-commutative-spacetime-higgs-and-the-standard-model-with-gravity/comment-page-1/#comment-3304, January 11, 2022.
[170]: Stephane H. Maes, “Consistencies and Implications of 2D Massless Random Walks: Discrete Non-commutative Spacetime, and No Flat Supersymmetry”, https://shmaesphysics.wordpress.com/2024/12/12/consistencies-and-implications-of-2d-massless-random-walks-discrete-non-commutative-spacetime-and-no-flat-supersymmetry/, December 12, 2024.
[171]: nLab, “Skyrmions”, https://ncatlab.org/nlab/show/skyrmion. Retrieved for this paper on December 10, 2024.
[172] : Wikipedia, “Skyrmion”, https://en.wikipedia.org/wiki/Skyrmion. Retrieved for this paper on December 10, 2024.
[173]: Stephane H. Maes, (2024), “QFT on discrete spacetime is OK”, https://shmaesphysics.wordpress.com/2020/12/13/viable-lattice-spacetime-and-absence-of-quantum-gravitational-anomalies-in-a-multi-fold-universe/comment-page-1/#comment-10322, December 30, 2024.
[174]: Stephane H. Maes, (2025), “Mathematics, Physics / QFT on discrete spacetime is more general than Mathematics and Physics / QFT on continuous spacetime”, https://shmaesphysics.wordpress.com/2022/10/30/multi-fold-discrete-fractal-spacetime-and-the-viability-of-local-vs-non-local-hidden-variable-viability/#comment-9455, July 13, 2024.
[175]: Stephane H. Maes, (2025), “QFT on discrete and / or non-commutative spacetime”, https://shmaesphysics.wordpress.com/2024/12/12/consistencies-and-implications-of-2d-massless-random-walks-discrete-non-commutative-spacetime-and-no-flat-supersymmetry/#comment-10326, January 7, 2025.
[176]: Stephane H. Maes, (2025), “Preons as Massless Higgs Bosons”, https://shmaesphysics.wordpress.com/2025/01/09/preons-as-massless-higgs-bosons/, January 9, 2025.
[177]: Stephane H. Maes, (2025), “Microscopic Interpretation of The Gravity Electroweak Symmetry Breaking”, https://doi.org/10.5281/zenodo.15099220, https://shmaesphysics.wordpress.com/2025/03/27/microscopic-interpretation-of-the-gravity-electroweak-symmetry-breaking/, March 27, 2025. (https://osf.io/q5erz).
[178]: Stephane H. Maes, (2025), “Gravity From Relative Entropic Action Is Not Necessarily Entropic Gravity”, https://shmaesphysics.wordpress.com/2025/04/06/gravity-from-relative-entropic-action-is-not-necessarily-entropic-gravity/, April 6, 2025.
[179]: Stephane H Maes, (2025), “Time-Varying Dark Energy Effect, How to Beat a Dead Horse. No, It Not Provide A First Evidence of String Theory”, https://shmaesphysics.wordpress.com/2020/10/11/circular-arguments-in-string-and-superstring-theory-from-a-multi-fold-universe-perspective/#comment-10609, April 5, 2025.
[180]: Stephane H. Maes, (2021), “Comments on CPT / anti-universe”, https://shmaesphysics.wordpress.com/2020/10/02/no-conventional-sterile-neutrinos-in-a-multi-fold-universe-just-smg-business-as-usual/#comment-1489 and comments after, January 30, 2021.
[181]: Stephane H. Maes, (2025), “2D Random Walks Imply a Strictly Positive Cosmological Constant, Possibly Variable in Time”, https://shmaesphysics.wordpress.com/2025/04/24/2d-random-walks-imply-a-strictly-positive-cosmological-constant-possibly-variable-in-time/, April 24, 2025.
[182]: Stephane H. Maes, (2025), “No Hilbert Einstein Action With Positive Dark Energy / Cosmological Constant From A String Action”, https://zenodo.org/doi/10.5281/zenodo.15385159, https://shmaesphysics.wordpress.com/2025/05/08/no-hilbert-einstein-action-with-positive-dark-energy-cosmological-constant-from-a-strings-action/, May8, 2025, (https://osf.io/7nh45/download/).
[183]: Stephane H. Maes, (2025), “Hopefully somebody will tell Vopson that he misunderstands Landauer”, https://shmaesphysics.wordpress.com/2023/12/14/information-has-no-mass/#comment-10716, April 26, 2025.
[184]: Stephane H Maes, (2022), “Why is The Multi-fold Theory on viXra, and not peer-reviewed (yet)?”, https://shmaesphysics.wordpress.com/why-is-the-multi-fold-theory-on-vixra-and-not-peer-reviewed-yet/, July 8, 2022.
[185]: Stephane H. Maes, (2025), “Adaptive Co-Design of Quantum Machine Learning Algorithms and Error Correction Protocols using Reinforcement Learning”, https://zenodo.org/doi/10.5281/zenodo.15428357, https://shmaes.wordpress.com/2025/05/15/adaptive-co-design-of-quantum-machine-learning-algorithms-and-error-correction-protocols-using-reinforcement-learning/, May 15, 2025, (https://osf.io/5jte9/download/).
[186]: Stephane H. Maes, (2025), “Preserving the Power of Preprints: Why Minimal Oversight is Key to Scientific Progress”, https://zenodo.org/doi/10.5281/zenodo.15617193, https://shmaes.wordpress.com/2025/06/05/preserving-the-power-of-preprints-why-minimal-oversight-is-key-to-scientific-progress/, June 5, 2025.
[187]: Stephane H Maes, (2025), “Rotating Black holes based on charged black holes”, https://shmaesphysics.wordpress.com/2024/11/24/physics-salvages-the-third-law-of-black-hole-thermodynamics/#comment-11016, June 28, 20265.
[188]: Stephane H. Maes, (2025), “Information Energy Momentum Tensor, E/G Conjecture vs. Alleged Massive Information”, https://shmaesphysics.wordpress.com/2025/06/11/information-energy-momentum-tensor-e-g-conjecture-vs-alleged-massive-information/, June 11, 2025.
[189] Stephane H. Maes, (2025), “No Naked Singularity, Whatever The Physical Collapse”, https://doi.org/10.5281/zenodo.16181569, https://shmaesphysics.wordpress.com/2025/07/19/no-naked-singularity-whatever-the-physical-collapse/, July 19, 2025
[190]: Stephane H. Maes, (2024), “Discrete spacetime, undecidability, mass gap and the possibilities of TOEs”, https://shmaesphysics.wordpress.com/2022/07/15/invalidation-and-proof-of-the-mass-gap-and-viability-of-the-standard-model-on-a-discrete-spacetime/#comment-10067, November 30, 2024.
[191]: Stephane H Maes, (2024), “Discovering non paradoxal CTCs / time travel with some 2D random walks”, https://shmaesphysics.wordpress.com/2024/12/12/consistencies-and-implications-of-2d-massless-random-walks-discrete-non-commutative-spacetime-and-no-flat-supersymmetry/#comment-10217, December 26, 2024.
[192]: Stephane H. Maes, (2023), “About reviews of Black Holes at LHC/CERN”, https://shmaesphysics.wordpress.com/2023/04/23/our-real-universe-is-macroscopically-4d-hints-come-from-every-directions-show-that-it-had-to-be-so/#comment-7389, July 11, 2023.
[193]: Stephane H. Maes, (2023), “Making contact with Parallel universe. Nonsense!”, https://shmaesphysics.wordpress.com/2023/04/23/our-real-universe-is-macroscopically-4d-hints-come-from-every-directions-show-that-it-had-to-be-so/#comment-7181, June 12, 2023.
[194] : Stephane H. Maes, (2025), “About New theory proposes time has three dimensions, with space as a secondary effect. Non sense!”, https://shmaesphysics.wordpress.com/2020/06/28/entanglement-concretizes-time-in-a-multi-fold-universe/#comment-11009, June 21, 2025, &. https://shmaesphysics.wordpress.com/2020/06/28/entanglement-concretizes-time-in-a-multi-fold-universe/#comment-11018, June 29, 2025.
[195]: Stephane H. Maes, (2023), “2D / JT gravity are well behaved, and without graviton. A sign of Physicality, 2D random walks and gravity by entanglement”, https://shmaesphysics.wordpress.com/2023/04/23/our-real-universe-is-macroscopically-4d-hints-come-from-every-directions-show-that-it-had-to-be-so/#comment-7588, August 5, 2023.
[196]: Stephane H. Maes, (2023), “Equivalency between 2D random walks and (relativistic) QM/ QFT” , https://shmaesphysics.wordpress.com/2022/11/09/quantum-gravity-emergence-from-entanglement-in-a-multi-fold-universe-2/#comment-8083, October 21, 2023.
[197]: Stephane H Maes, (2025), “Wick Rotation and Spacetime Orientation in the Multi-fold Gravity Electroweak Theory”, https://shmaesphysics.wordpress.com/2021/03/28/multi-fold-gravity-electroweak-theory-and-symmetry-breaking/#comment-11222, November 12, 2025.
[198]: Stephane H Maes, (2022), “Multi-fold Theory and The Linear Growth of The Complexity of a Quantum Circuit”, https://shmaesphysics.wordpress.com/2021/11/28/how-the-er-epr-gr-qm-and-ads-cft-correspondence-conjectures-can-be-explained-in-multi-fold-theory-and-the-e-g-conjecture-explains-and-realize-in-a-multi-fold-universe-a-call-to-the-physics-comm/#comment-3797, March 29, 2022.
[199]: Stephane H. Maes, (2026), “Food for thought: the multi-fold massless Higgs bosons / preons in 2D random walks + creation/annihilation are the post quantum theory”, https://shmaesphysics.wordpress.com/2025/01/09/preons-as-massless-higgs-bosons/#comment-11535, April 24, 2026 and sub-sequent comments after.
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[2] Or SMG, which described the SM with gravity effects non-negligible at its scales [1,131,137,152,251].
[3] In general, Bohmian Physics is presented as a quantum mechanics theory and interpretation (undistinguishable from the other interpretations [252]), but rarely do we encounter a relativistic or QFT version.
[4] The consistent framework is in our view a differentiator vs. straight reviews.
[5] The 2D random walks of massless Higgs bosons, aka multi-fold preons, ensure random sprinkled Poisson distribution of the discrete spacetime locations, which leads to Lorentz invariance behaviors [260] at higher scales [1,23,72,131,137,152,170,181,251].
[6] At least for our proposed framework.
[7] When this is a massless scalar boson remember the massless Higgs boson of the multi-fold theory, or multi-fold preon [176], which in a multi-fold theory plays also the role of dilaton [23,29,32,64,66,72,170,176,181,230], inflaton [15,27,32,251], double copy gauge scalar boson [94,104], Higgs boson condensing into Higgs and massive particles [1,23,33-35,63,68,72,116,150,131,137,152,177,251], or in random walk patters that forms the massless ones [1,8-10,22,32,62,72,131,137,152,170,176,181,251], or eventually as the source of concretized and historical locations, that form the spacetime [1,6,8-10,22,23,32,62,63,72,116,131,137,152,170,176,181].
[8] This provides a direct physical justification for the hybrid choice of primitives. (Massless) Bosons behave as extended fields generated by the random walk configurations in agreement with [124] and references therein, whereas fermions act as highly localized, particle-like condensates, in agreement with [124] for fermionic fields (particles hence modeled in the multi-fold universe as extended microscopic black hole like patterns of random walks or condensates of preons / massless Higgs bosons) and with [1,8-10,22,131,137,150,152,170,251] and references therein, in terms of scales required to have non-commutativity of spacetime required to support fermions, their zitterbewegung, and spin statistics, and Quantum Physics in general. It is also related to [199].
[9] The multi-fold theory also prefers such an interpretation, i.e., that we are dealing with beables even in the absence of observers [1-199,251,258,259].
[10] Here,
are equivalent notations.
[11] Interestingly, in the case of a multi-fold universe, the spacetime is discrete [1,6,8-10,22,29,36,62,68,72,112,124,131,137,152,155,167-170,173-176,181,190], and there is no need to go back to the continuum.
[12] That statement may be dispute in general. In the context of a universe compatible with GR [1,6,62,112,137,169,170,253], and in expansion [112,167,170,181], it holds. That covers multi-fold universes. Furthermore, if Physical Actions and Path Integral apply, something that always does in a complete physical description of a physical system [116,159], then we have [1,137,155].
[13] Even the special models for (relativist, in between parentheses because they are always relativist) neutrinos also capture their special relationship to the multi-fold Higgs bosons / massless Higgs boson models as in [1,8-10,22,42,47,67,119,131,137,152,159,251].
[14] Relativistic Bohmian Physics may or may not characterize well our real universe. That has not been determined in this paper.
[15] Conventional physicists may argue it is New Physics. We consider that it isn’t because no new particles or interactions are introduced. We just add gravity, as we know it should be, and multi-fold mechanism and let conventional Physics unfold with the considerations. It does change conventional results or explanations, usually with same observables, and it does live in a discrete spacetime etc., because of conventional analysis of these consequences. We also do not cover stable field effects like Skyrmions [166], that we prefer to see as a collective effect for the theory. Beside the SM particles, there are other collective solutions/solitons in gauge theories, they have behaviors as particles, but they are quasi-particles composed of collective effects of a large set of particles. We see them as Qballs or patterns that can appear by multi-fold space time matter induction and scattering, under specific circumstances, nothing more. They are topological solitons and can appear in BEC, as expected with massless Higgs boson condensates [171,172].
[16] Standing in for Quantum Physics in general.
[17] It is granted that another mechanism, not yet discovered or thought about, could always be discovered in the future.
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