We present a strictly local and deterministic geometric foundation for relativistic quantum mechanics by mapping the phenomenological theory of martensitic crystallography onto a four-dimensional pseudo-Riemannian spacetime manifold. By generalizing the classical 3D Hadamard compatibility condition to a 4D covariant tensor field framework, we demonstrate that the fundamental equations of quantum mechanics emerge naturally from the geometric constraints of a continuous lattice undergoing structural phase transformations. Defining a complex field structure from the local volume fraction density of twinned variants and the kinematic phase front normal, the principle of stationary action yields the exact, unapproximated Klein-Gordon and Dirac equations. By doing so, we provide a concrete, solid-state mechanical realization to the historical hydrodynamic formulations of Madelung and the double-solution models of de Broglie. Under this paradigm, abstract probability amplitudes are reinterpreted as real, structural domain distributions, and the Bohm quantum potential is identified as the localized interfacial surface energy density of sub-microscopic twin boundaries. Finally, we resolve the historical problem of wave-function collapse without invoking anthropocentric or non-local interventions; utilizing a framework of macroscopic structural decoherence, we prove that the perception of a single, definitive measurement outcome is an unavoidable consequence of the thermodynamic orthogonality and strict kinematic constraints governing macroscopically bound solid-state detectors embedded within a globally twinned 4D block universe.
We present a strictly local and deterministic geometric foundation for relativistic quantum mechanics by mapping the phenomenological theory of martensitic crystallography onto a four-dimensional pseudo-Riemannian spacetime manifold. By generalizing the classical 3D Hadamard compatibility condition to a 4D covariant tensor field framework, we demonstrate that the fundamental equations of quantum mechanics emerge naturally from the geometric constraints of a continuous lattice undergoing structural phase transformations. Defining a complex field structure from the local volume fraction density of twinned variants and the kinematic phase front normal, the principle of stationary action yields the exact, unapproximated Klein-Gordon and Dirac equations. By doing so, we provide a concrete, solid-state mechanical realization to the historical hydrodynamic formulations of Madelung and the double-solution models of de Broglie. Under this paradigm, abstract probability amplitudes are reinterpreted as real, structural domain distributions, and the Bohm quantum potential is identified as the localized interfacial surface energy density of sub-microscopic twin boundaries. Finally, we resolve the historical problem of wave-function collapse without invoking anthropocentric or non-local interventions; utilizing a framework of macroscopic structural decoherence, we prove that the perception of a single, definitive measurement outcome is an unavoidable consequence of the thermodynamic orthogonality and strict kinematic constraints governing macroscopically bound solid-state detectors embedded within a globally twinned 4D block universe.
We develop a complete geometric framework in which each quantum particle possesses its own private spacetime—a world-block—constructed from Fermi–Walker coordinates. The intrinsic spatial metric on each proper-time slice is a dynamical field governed by an action with a universal stiffness constant A0. A single world-block exhibits kinematic non-locality: the metric perturbation at a point depends on the entire wavefunction through an integral relation, while maintaining dynamic locality via causal wave equations. This duality captures the essential non-locality of quantum mechanics without violating relativistic causality. When two particles interact, their world-blocks are stitched along a common boundary, forming a single compound world-block in ordinary four-dimensional spacetime. The stitching imposes local matching conditions that yield a non-separable strain field, providing a geometric account of quantum entanglement. Measurements correspond to fixing boundary conditions on one part of the compound block; the correlated outcome on the distant part is automatically determined by the shared global geometry, not by any superluminal signal. Thus, the apparent non-locality of EPR correlations is explained as a manifestation of geometric connectivity within a single 4D manifold, consistent with Bell’s theorem because the geometry itself is non-local. In the continuum limit of many overlapping blocks, coarse-graining restores an effective local description, and Newton’s law of universal gravitation emerges exactly, with Newton’s constant given by G = 3c4/(8πA0). The model offers a unified, deterministic, and fully relativistic foundation for quantum mechanics and gravity, without invoking extra dimensions or stochastic elements. Experimental signatures in ultrafast interferometry and possible connections to dark energy are discussed. The framework aligns with recent developments in emergent gravity and provides a concrete geometric realization of spacetime from quantum entanglement.
Background: The unification of General Relativity and Quantum Mechanics represents a fundamental challenge in theoretical physics due to their mutually exclusive conceptual foundations: continuous deterministic geometry versus probabilistic wave functions. While the de Broglie-Bohm pilot-wave theory offers a deterministic, trajectory-based alternative to the standard probabilistic framework, it introduces a non-local, ad-hoc "Quantum Potential" (Q) to account for phenomena such as interference and non-locality. The lack of a geometric origin for Q in standard spacetime remains a significant conceptual barrier to a fully unified theory. Purpose: This paper aims to resolve this interpretational crisis by proposing that the Quantum Potential is not a fundamental or mysterious external force, but rather a derived geometric artifact. We postulate that the physical universe is fundamentally a 4-dimensional Complex Manifold (ℂ4), extending the standard Riemannian spacetime manifold to include imaginary coordinates. Methods: To establish this geometric basis, we treat the quantum wave function not as an abstract probabilistic amplitude, but as a physical, holomorphic map describing a particle's deterministic trajectory through this complex spacetime. Utilizing the Cauchy-Riemann conditions, we demonstrate that the Bohmian amplitude (R) is strictly coupled to the particle's location in the imaginary dimension. We then analyze the dynamics of particles following geodesics within ℂ4. By projecting the complex geodesic equation of motion onto the observable real slice (ℝ4), we separate the real and imaginary components of the complex 4-velocity to observe the energy balance. Conclusions: Our derivation reveals that the Quantum Potential (Q) emerges naturally and is mathematically identical to the kinetic energy component associated with a particle's hidden motion in these imaginary dimensions. This formulation successfully recovers the predictions of the Schrödinger equation while removing the ad-hoc nature of Bohmian mechanics. Furthermore, it interprets quantum non-locality—such as the interference observed in the double-slit experiment—as purely local, deterministic geodesic motion around topological singularities in a curved complex manifold. Ultimately, this framework provides a unified geometric description wherein gravity is the curvature of real coordinates, electromagnetism is the torsion of imaginary coordinates, and quantum mechanics is inertial motion through this complex geometry.
This paper presents a comprehensive, alternative non-linear geometric framework aimed at unifying classical general relativity with quantum mechanics under a cohesive paradigm termed Probabilistic Determinism. We propose that the smooth spacetime metric manifold constitutes the sole fundamental physical reality, and that subatomic fermions are localized, topologically stable metric solitons possessing spin-1/2 connection profile invariants. By introducing a novel covariant field relation—the Anbi Metric Deformation Equation—we demonstrate that localized energy-momentum density traces inherently compress the coordinate network itself at the sub-microscopic scale. This localized spatial contraction provides an asymptotic geometric buffer that regularizes field interactions, natively resolving the ultraviolet divergence catastrophes of quantum field theories without requiring arbitrary mathematical renormalization. Furthermore, utilizing a relativistic de Broglie-Bohm phase connection, we demonstrate how deterministic trajectories traversing this dynamically compressed, fluctuating background manifold scale up macroscopically to recreate observed quantum statistical probability distributions as a strict geometric continuity fluid law.
While the foundational architecture of quantum mechanics has remained structurally bound to the probabilistic interpretations of the Copenhagen school due to the historical inability to resolve high-frequency ultraviolet boundary singularities, we present a definitive, constructive mathematical framework that establishes the absolute global regularity and determinism of fermion fields within native four-dimensional spacetime. By embedding the four-component Dirac spinor into a density-weighted non-linear state vector, the coupled system is processed through a gauge-covariant intrinsic Friedrichs mollifier J_λ integrated with an anti-symmetric Riesz transform kernel R_m. At the contact interface, the singular Dirac-delta loop divergences undergo an instantaneous, exact cross-term cancellation, giving spontaneous birth to a fourth-order biharmonic quantum-chromodynamic dissipative spectral cage - λ^2 * G_vac * 𝔐^4. At the vanishing scaffold limit (λ → 0), the continuous regularization parameter collapses uniformly (λ^2 / λ^2 = 1), freezing the anomalous energy dissipation ledger into a strictly positive local invariant. This structural closure uncovers the long-sought geometric origin of the particle angular momentum, proving analytically that the quantum spin (𝔖 = 1/2) is the unique, necessary topological reflex of the continuous spacetime fabric twisting to preserve gauge invariance under the mass gap constraint of non-Abelian fields. Furthermore, this non-local functional integration permanently eradicates the probabilistic wave-function collapse paradox, explaining quantum entanglement (EPR anomalies) as a continuous, deterministic redistribution of field density across a structurally un-cuttable topological domain. The predictive sovereignty of this framework is rigorously verified against empirical accelerator data, offering an ironclad mathematical blueprint that anchors quantum causality from within continuous four-dimensional engineering realities.
A novel model for the problem of Quantum Gravity. The formal topologies of continuous General Relativity and discrete Quantum Mechanics are fundamentally irreconcilable without an intervening boundary constraint. We propose the dual-manifold formalism: a deterministic framework that reclassifies wave-function collapse as a localised thermodynamic phase transition. By strictly bounding quantum state reduction at the Bekenstein limit, we demonstrate that macroscopic spacetime curvature emerges natively as the mandatory metric deformation tethered to localised mass-energy. Mapped via standard thermodynamic equations of state, this localised thermal flux mechanically recovers continuous gravity without invoking unobserved dimensions or novel gauge bosons.
This paper develops a theoretical framework that seeks to unify the foundational principles of quantum mechanics, general relativity, and thermodynamics by extending spatial geometry to include a compactified fourth spatial dimension. The central proposal is that many counterintuitive features of quantum systems—such as wave function collapse, superposition, entanglement, tunneling, and decoherence—emerge as phenomena from the projection of four-dimensional quantum states into observable three-dimensional space. Within this formulation, apparent quantum randomness is reinterpreted as a consequence of observing only a lower-dimensional slice of a higher-dimensional structure. A key innovation of the framework is the introduction of a scalar degree of freedom, termed the entropion field, which governs decoherence and encodes the thermodynamic arrow of time. The entropion field couples directly to both quantum fields and spacetime curvature, leading to modified Einstein field equations and an entropic extension of quantum field dynamics. This mechanism provides a consistent account of classicality emerging from quantum substrates, links entropy growth to projection dynamics, and clarifies how spacetime geometry constrains quantum evolution. The theory is presented through a structured progression of conceptual foundations, mathematical formalism, physical interpretation, and predictive consequences. Particular emphasis is placed on testable deviations from standard theories, including altered interference signatures in 4D-projected systems, entropion-modulated decoherence rates, and revised mechanisms of tunneling and entanglement. Broader implications are explored for quantum information theory, cosmology, condensed matter, and computational theory. Collectively, these results establish a geometrically grounded unification scheme that integrates the three major pillars of modern physics into a consistent and falsifiable framework.