Abstract This paper develops a formal correspondence between quantum probability and geometric curvature, showing that the amplitude of the wavefunction represents the observable manifestation of microscopic curvature generated by the phase field. Within the generalized phase–metric framework, the gradient of the phase Φ(x) deforms the background geometry, producing a curvature tensor that governs the local modulation of probability density. The analysis demonstrates that the conservation of the phase current acquires curvature corrections, yielding a Raychaudhuri-like equation linking the second covariant derivative of the amplitude to the Ricci curvature. Consequently, quantum indeterminacy can be reinterpreted as an epistemic projection of phase-induced curvature rather than intrinsic randomness. This approach unites the probabilistic and geometric aspects of quantum theory and integrates naturally into the Temporal Potential (ZPAT) framework, where temporal curvature governs amplitude evolution. The resulting picture provides an analytic–geometric synthesis bridging phase dynamics, probability flow, and spacetime structure.
Abstract This paper develops a formal correspondence between quantum probability and geometric curvature, showing that the amplitude of the wavefunction represents the observable manifestation of microscopic curvature generated by the phase field. Within the generalized phase–metric framework, the gradient of the phase Φ(x) deforms the background geometry, producing a curvature tensor that governs the local modulation of probability density. The analysis demonstrates that the conservation of the phase current acquires curvature corrections, yielding a Raychaudhuri-like equation linking the second covariant derivative of the amplitude to the Ricci curvature. Consequently, quantum indeterminacy can be reinterpreted as an epistemic projection of phase-induced curvature rather than intrinsic randomness. This approach unites the probabilistic and geometric aspects of quantum theory and integrates naturally into the Temporal Potential (ZPAT) framework, where temporal curvature governs amplitude evolution. The resulting picture provides an analytic–geometric synthesis bridging phase dynamics, probability flow, and spacetime structure.
Abstract The present work develops a covariant Schrödinger formalism in which phase, curvature, and probability flow emerge as interdependent aspects of a single field. Starting from a phase-induced metric deformation gμν(Φ)g_{\mu\nu}(\Phi)gμν(Φ), we construct a geometric Schrödinger equation that naturally couples to an Einstein-like tensor Gμν[g]G_{\mu\nu}[g]Gμν[g]. This yields a unified system linking the quantum amplitude AAA, the local curvature R[g]R[g]R[g], and the analytic hierarchy of transcendental constants (π, ζ(3), ζ(5), …). Within this framework, the classical probability interpretation is reinterpreted as the observable projection of phase–curvature dynamics. The resulting equations provide a self-consistent description of how quantum interference, analytic number structure, and spacetime geometry co-evolve within a single differential system.
We develop a geometric formulation of stochastic dynamics in which noise, diffusion, path probabilities, fluctuation theorems, and entropy production arise from the intrinsic geometry of an evolving manifold rather than from externally imposed randomness. Within the theory of moving manifolds, we establish a curvature-noise correspondence: fluctuations are governed by the inverse curvature tensor, while entropy production is controlled by curvature deformation. The invariant continuity law on a moving hypersurface yields a geometric Fokker-Planck equation, and curvature-velocity coupling generates a quadratic Onsager-Machlup functional determining path weights. The resulting entropy functional satisfies a curvature-driven monotonicity law, providing a geometric derivation of the Second Law. In two dimensions, the curvature invariant reduces to Gaussian curvature and encodes topology, so topological transitions produce discrete entropy jumps. When the ambient space carries a Minkowskian signature, the same curvature-kinetic quadratic form that generates dissipative thermal weights produces oscillatory phase weights, and the Laplace-Beltrami operator governing entropy evolution acquires a Schrödinger-type structure. This provides a geometric resolution of the apparent distinction between classical stochastic behaviour and quantum dynamics. These results show that stochastic behaviour, thermodynamic irreversibility, and quantum transition amplitudes are unified within the moving manifold framework. Geometry does not merely accommodate stochasticity; stochastic behaviour arises as a consequence of deterministic geometric evolution. The theory predicts curvature-controlled anisotropic diffusion, entropy jumps at topology-changing events, and a geometric thermal-quantum crossover in which classical stochastic weights and quantum amplitudes are generated by the same curvature-kinetic action.
Abstract We propose a geometric reinterpretation of the Schrödinger equation in which the phase gradient functions as a U(1) connection and its curvature defines a measurable field strength. This framework transforms quantum probability flow into a covariant gauge theory, revealing a direct equivalence between phase curvature, spatial geometry, and analytic constants such as π and ζ(3). The result suggests that the geometric structure underlying quantum interference may itself be a manifestation of curvature in a higher-dimensional phase manifold.
Abstract We develop a novel framework connecting quantum mechanics with gravitational geometry through a deformation of probability amplitudes. Inspired by Rohit Dhormare’s geometric formulation of probability under gravitational fields, we reinterpret the influence of curvature on probability not within classical settings, but in the domain of quantum amplitudes. We propose a quantum probabilistic manifold where space-time curvature induces topological and metric deformations on Hilbert space projections, affecting interference, measurement outcomes, and entropy. This formulation offers new pathways for understanding quantum gravity, decoherence, and information localization.
Abstract This paper reformulates the Schrödinger dynamics in a fully geometric and gauge-theoretic language, where the phase of the quantum wavefunction acts as a U(1) connection and its curvature manifests as measurable interference and geometric deformation. By coupling the connection Aμ=ℏ−1∂μSA_\mu = \hbar^{-1}\partial_\mu SAμ=ℏ−1∂μS to a curved phase metric gμν(n)=gμν+χnAμAν+ηnA2Πμνg^{(n)}_{\mu\nu} = g_{\mu\nu} + \chi_n A_\mu A_\nu + \eta_n A^2 \Pi_{\mu\nu}gμν(n)=gμν+χnAμAν+ηnA2Πμν, the Schrödinger equation acquires a covariant form that unites probability flow, curvature, and analytic hierarchy. The corresponding field strength Fμν=∂μAν−∂νAμF_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\muFμν=∂μAν−∂νAμ defines the geometric curvature of phase space, leading to a Maxwell-like equation for the phase field and an emergent Lorentz-type force on the probability current. The framework provides a direct correspondence between phase holonomy, Cn-carrier parity, and the analytic ζ-layer structure, suggesting a measurable bridge between gauge geometry and transcendental constants in multi-path interferometry.
The absence of a common mathematical structure underlying quantum theory and relativistic physics remains a central obstacle to unification. While relativity is formulated in terms of spacetime geometry and covariance, quantum theory is intrinsically probabilistic, relying on amplitudes, uncertainty, and statistical interpretation. Existing unification programs typically attempt to quantize geometry or to geometrize quantum fields, but the probabilistic structure itself is rarely treated as a fundamental geometric object. In this paper, we propose a unified description of relativistic and quantum dynamics based on probabilistic geometry. Probability densities are taken as primary variables and are endowed with an intrinsic information–geometric structure defined by the Fisher–Rao metric. Within this framework, physical motion is described as the covariant evolution of probability distributions on a statistical manifold rather than as trajectories in a preassigned phase space or Hilbert space. The central result is a unified probabilistic–geometric field equation in which the relativistic d’Alembertian acting on the probability density is coupled to the scalar curvature of the associated information manifold. In an appropriate nonrelativistic limit, this equation reproduces the quantum potential as a curvature term, establishing a direct correspondence between quantum effects and informational geometry. Relativistic covariance is preserved throughout, allowing quantum and relativistic structures to appear as different regimes of a single probabilistic equation. The proposed framework does not introduce new particles, interactions, or quantization postulates. Instead, it provides a structural unification in which quantum and relativistic features emerge from the geometry of probability itself.