A deterministic quantum-gravity framework is proposed that embeds Bohmian mechanics within the spin–torsion geometry of Einstein–Cartan–Kibble–Sciama theory. Starting from the Palatini form of the action for a Dirac field, a Gordon decomposition followed by a Foldy–Wouthuysen expansion shows that the familiar Bohm quantum potential, namely the term proportional to the Laplacian of the wave-function amplitude divided by that amplitude, coincides exactly with the axial-torsion contribution to the Ricci scalar. A covariant, spin-augmented guidance law then follows, and the algebraically determined torsion field, sourced by the global spin density of an entangled state, mediates non-local correlations without superluminal signalling. The same spin–torsion couplings produce an effective dark stress–energy tensor, for which observational tests are outlined across a wide range of scales: neutron-star magnetospheres, black-hole jets, gravitational-wave inspirals and spin-polarized interferometry. This geometric mechanism extends to spinless particles via torsion’s universal back-reaction on the connection, yielding a deterministic reformulation of quantum mechanics without metric quantization. A simple one-dimensional toy model reproduces the Tsirelson bound for Bell-inequality violations, underscoring the empirical and conceptual viability of this geometric route to quantum gravity, which dispenses with the need to quantize the metric.
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.
We present a geometric framework for quantum entanglement within the Cosmic Energy Inversion Theory wherein space-time torsion, dynamically sourced by primordial energy field gradients, physically mediates non-local quantum correlations. Unlike standard quantum mechanics treating entanglement as axiomatic, CEIT attributes correlations to torsion-induced phase coupling propagating at light speed through energy field variations, preserving relativistic causality while explaining Bell violations geometrically. The modified von Neumann entropy incorporates geometric contributions scaling as λℰ⁻¹∇ₐTᵅμν∂μΦₑₙₜ∂νΦₑₙₜ, where Φₑₙₜ quantifies how space-time twisting modulates correlation strength. Integration with Loop Quantum Gravity establishes holographic entropy encoding on torsion defined minimal surfaces, resolving black hole information paradoxes through geometric mechanisms. Numerical validation against gravitational gradient measurements yields 98.7% agreement with observed fidelity ratios, while pulsar coherence data constrains electromagnetic coupling parameters within 2.7% precision. The framework predicts Bell parameter modifications ΔB=0.182±0.026 in particle accelerator environments testable via CERN MATISSE interferometry, squeezed-light gravitational wave correlations accessible through LIGO observations, and enhanced cosmological structure formation signatures in JWST high-redshift spectroscopy. Experimental verification would establish entanglement as emergent space-time geometry rather than fundamental quantum axiom, unifying quantum mechanics with general relativity through six independently calibrated parameters.
Understanding quantum theory in terms of a geometric picture sounds great. There are different approaches to this idea. Here we shall present a geometric picture of quantum theory using the de-Broglie--Bohm causal interpretation of quantum mechanics. We shall show that it is possible to understand the key character of de-Broglie--Bohm theory, the quantum potential, as the conformal degree of freedom of the space--time metric. In this way, gravity should give the causal structure of the space--time, while quantum phenomena determines the scale. Some toy models in terms of tensor and scalar--tensor theories will be presented. Then a few essential physical aspects of the idea including the effect on the black holes, the initial Big--Bang singularity and non locality are investigated. We shall formulate a quantum equivalence principle according to which gravitational effects can be removed by going to a freely falling frame while quantum effects can be eliminated by choosing an appropriate scale. And we shall see that the best framework for both quantum and gravity is Weyl geometry. Then we shall show how one can get the de-Broglie--Bohm quantum theory out of a Weyl covariant theory. Extension to the case of many particle systems and spinning particles is discussed at the end.
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.
This review paper comprehensively examines the influence of spatial torsion on quantum fluctuations from the perspectives of Metric-Affine Gravity (MAG) and the Stochastic Variational Method (SVM). We first outline the fundamental framework of MAG, a generalized theory that includes both torsion and non-metricity, and discuss the geometrical significance of torsion within this context. Subsequently, we summarize SVM, a powerful technique that facilitates quantization while effectively incorporating geometrical effects. By integrating these frameworks, we evaluate how the geometrical structures originating from torsion affect quantum fluctuations, demonstrating that they induce non-linearity in quantum mechanics. Notably, torsion, traditionally believed to influence only spin degrees of freedom, can also affect spinless degrees of freedom via quantum fluctuations. Furthermore, extending beyond the results of previous work [Koide and van de Venn, Phys. Rev. A112, 052217 (2025)], we investigate the competitive interplay between the Levi-Civita curvature and torsion within the non-linearity of the Schrödinger equation. Finally, we discuss the structural parallelism between SVM and information geometry, highlighting that the splitting of time derivatives in stochastic processes corresponds to the dual connections in statistical manifolds. These insights pave the way for future extensions to gravity theories involving non-metricity and are expected to deepen our understanding of unresolved cosmological problems.
Previous works of this series have realized the classical unification of gravitation and electromagnetic fields based on the spacetime density tensor, and the intrinsically generated high-order couplings consistently explain various observational anomalies of compact celestial objects. As a realistic quantum framework, existing schemes for unifying Bohmian theory and gravity generally adopt a dual structure that separates spacetime from the pilot wave. The nonlocality of quantum entanglement in multi-particle systems leads to fundamental conflicts with the local causality of relativity. Meanwhile, torsion-free Riemannian geometry cannot accommodate spin-
1/2 fermion spinors, nor can it intrinsically generate the short-range
SU(2) weak and
SU(3) strong gauge interactions, making it difficult to fully embed the Standard Model of quantum field theory. On the basis of maintaining the monistic ontological postulate of the spacetime density tensor, this paper carries out theoretical expansion in two stages. First, the Bohmian pilot wave, quantum potential and multi-particle entanglement are defined as global coherent perturbations of the spacetime density tensor. This work strictly distinguishes the field structures, dynamic equations and propagation mechanisms of separable quantum states and entangled quantum states, and reconstructs the covariant quantum guiding equation. It resolves the paradox between quantum nonlocality and relativistic causality from an ontological perspective, achieving the unification of classical field theory and Bohmian quantum mechanics. Second, Einstein–Cartan (EC) geometry is introduced with the additional degree of freedom of spacetime torsion. Two types of quantum correlations, namely orbital entanglement and spin entanglement, are differentiated. Fermion spinor fields, as well as
U(1) electromagnetic,
SU(2) weak and
SU(3) strong gauge interactions, are all mapped to local chiral twists and color topological torsional perturbation modes of spacetime. Four core innovations are presented in this paper: 1. Quantum effects are attributed to global coherent fluctuations of spacetime density. The essential geometric differences between ordinary separable quantum states and entangled states are clarified. Entanglement originates from the global coherent deformation of spacetime rather than superluminal action, thoroughly eliminating the dual separation between quantum fields and spacetime. 2. With the aid of EC torsion geometry, fermion spinors are no longer background-independent fields but geometric projections of local spin torsion of spacetime, realizing the geometric ontology of intrinsic spin. 3. The complete set of
U(1)/SU(2)/SU(3) gauge symmetries arise intrinsically from multi-scale topological perturbations of the spacetime density tensor, removing the artificial introduction of gauge groups as implemented in the Standard Model. 4. A complete covariant unified field system covering gravitation, electromagnetism, Bohmian quantum effects, weak interactions and strong interactions is constructed, alongside a quantitative calculation system for orbital entanglement and spin entanglement. Under corresponding limiting conditions, the system rigorously reduces to General Relativity, Maxwell's equations, covariant Bohmian dynamics, electroweak Lagrangian and standard QCD Lagrangian. Key physical phenomena including parity violation of the weak interaction, color confinement, asymptotic freedom and quantum nonlocality are consistently explained via geometric structures. This study ultimately establishes a self-consistent unified field theory with a single ontology, which is compatible with classical field theory, realistic Bohmian quantum mechanics and the Standard Model of quantum field theory.