Probability concept in physics entered into statistical physics and quantum physics by molecules kinematics; and curvature concept in physics as applying differential geometry to physics, entered into analytical mechanics long ago. Along with introducing space-time curvature concept into general relativity, curvature concept became more important; gauge field theory regards field intensity as curvature of fibre bundles. Curvature concept in quantum mechanics germinated from original derivation of Schrodinger equation; catastrophe scientist Rene Thom advanced curvature interpretations of ψ function and entropy according to differential geometry. Guoqiu Zhao advanced curvature interpretation of quantum mechanics; this new interpretation made relativity theory and quantum mechanics more harmonious, and regarded ψ function as a curvature function. So far Zhao’s quantum curvature interpretation is nearest to Schrodinger’s scientific thought and Einstein’s physics ideal.
Probability concept in physics entered into statistical physics and quantum physics by molecules kinematics; and curvature concept in physics as applying differential geometry to physics, entered into analytical mechanics long ago. Along with introducing space-time curvature concept into general relativity, curvature concept became more important; gauge field theory regards field intensity as curvature of fibre bundles. Curvature concept in quantum mechanics germinated from original derivation of Schrodinger equation; catastrophe scientist Rene Thom advanced curvature interpretations of ψ function and entropy according to differential geometry. Guoqiu Zhao advanced curvature interpretation of quantum mechanics; this new interpretation made relativity theory and quantum mechanics more harmonious, and regarded ψ function as a curvature function. So far Zhao’s quantum curvature interpretation is nearest to Schrodinger’s scientific thought and Einstein’s physics ideal.
Until the introduction of Special Relativity by Hendrik Antoon Lorentz, Herman Minkovski and Albert Einstein in 1905 and General Relativity by Albert Einstein in 1916, Physics has been 3-Dimensional. After this period, Physics became in generally 4-Dimensional with the time as the fourth dimension. For that reason it is remarkable that till now quantum physics has been described by 3-dimensional equations like the Schrodinger wave equation and the relativistic Dirac equation. In this article the 4-Dimensional Relativistic Quantum Mechanical Dirac Equation expressed in the vector probability functions and the complex conjugated vector probability function will be published. To realize this, the classical boundaries of physics has to be changed. It is necessary to go back in time 300 years ago. More than 200 years ago before the Dirac Equation had been published. A Return to the Inception of Physics. The time of Isaac Newton who published in 1687 in the “Philosophiae Naturalis Principia Mathematica” a Universal Fundamental Principle in Physics which was in Harmony with Science and Religion. The Universal Path, the Leitmotiv, the Universal Concept in Physics. Newton found the concept of “Universal Equilibrium” which he formulated in his famous third equation Action = - Reaction. This article presents a New Kind of Physics based on this Universal Fundamental Concept in Physics which results in a New Approach in Quantum Physics and General Relativity. The physical concept of quantum mechanical probability waves has been created during the famous 1927 5th Solvay Conference. During that period there were several circumstances which came together and made it possible to create an unique idea of material waves being complex (partly real and partly imaginary) and describing the probability of the appearance of a physical object (elementary particle). The idea of complex probability waves was new in the beginning of the 20th century. Since then the New Concept has been protected carefully within the Copenhagen Interpretation. When Schrodinger published his famous material wave equation in 1926, he found spherical and elliptical solutions for the presence of the electron within the atom. The first idea of the material waves in Schrodinger’s wave equation was the concept of confined Electromagnetic Waves. But according to Maxwell this was impossible. According to Maxwell’s equations Electromagnetic Waves can only propagate along straight lines and it is impossible that Light (Electromagnetic Waves) could confine with the surface of a sphere or an ellipse. For that reason these material waves in Schrodinger’s wave equation could only be of a different origin than Electromagnetic Waves. Niels Bohr introduced the concept of “Probability Waves” as the origin of the material waves in Schrodinger’s wave equation. And defined the New Concept that the electron was still a particle but the physical presence of the electron in the Atom was equally divided by a spherical probability function. In the New Theory it will be demonstrated that because of a mistake in the Maxwell Equations, in 1927 Confined Electromagnetic waves could not be considered to be the material waves expressed in Schrodinger's wave equation. The New Theory presents a new equation describing electromagnetic field configurations which are also solutions of the Schrodinger's wave equation and the relativistic quantum mechanical Dirac Equation and carry mass, electric charge and magnetic spin at discrete values. Key words : General Relativity; Quantum Physics; Dirac Equation; Gravitational-Electromagnetic Interaction; Black Holes; Gravitational-Electromagnetic Confinement; Electromagnetism; Quantum Optics
Quantum mechanics traditionally focuses on dynamical variables such as energy, phase, and entanglement to characterize and manipulate quantum states. Recent advances in quantum information theory have revealed that the geometry of quantum state space, endowed with the Fubini–Study metric, carries rich structural information that can be exploited for control and resource theories. In this paper we propose a new dynamical framework, the Quantum State Curvature Flow (QSCF), which treats the curvature of a quantum state as an evolving quantity. We define a curvature scalar Kψ associated with a normalized state vector |ψ⟩ in projective Hilbert space, derived from the Riemannian curvature of the Fubini–Study metric. We then postulate a partial differential equation for the time evolution of Kψ, analogous to curvature flow equations in differential geometry, and examine its properties. The QSCF formalism uncovers novel invariants, links curvature dynamics to entanglement evolution, and suggests new strategies for continuous quantum control. While our work is purely theoretical, it opens a pathway toward a deeper understanding of quantum state geometry and its role in quantum technologies.
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 In general relativity, the dynamics of objects is governed by the curvature of spacetime, which is caused by the presence of matter and energy. In contrast, in quantum mechanics, the dynamics is governed by the wavefunction, which completely describes the behavior of the particles. There is an ongoing effort to explore analogs of space and spacetime curvature in the context of quantum mechanics. Such analogies may reveal a deeper structure of quantum reality and its possible relations with Einstein’s theory of gravity. In this note, by coupling the non-relativistic Schrödinger equation with the heat equation and using the hydrodynamical formulation of quantum mechanics, we find that the dynamics of the particle is fully characterized by the normalized curvature of the wavefunction’s amplitude. Such a curvature obtains an analogy to the Ricci curvature of curved space in a Riemannian manifold. The proposed geometric correspondence provides a new pathway to explore quantum dynamics through the lens of differential geometry, the language of general relativity.
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.
Abstract This paper derives a quantitative relation between the probability amplitude of a wavefield and the local curvature produced by its phase congruence. Starting from the covariant continuity equation for the probability current, we show that the amplitude A(x) of a complex field Ψ = A e^{iΦ} evolves according to the expansion scalar of its phase flow, which obeys the Raychaudhuri equation. The resulting expression, R_{\mu\nu} u^\mu u^\nu ≈ 2 · (1/A) · D²A / Dτ², demonstrates that curvature acts as the differential measure of amplitude concavity along the phase trajectories. Therefore, probability is not an independent quantity but the visible manifestation of phase-induced geometric curvature. The framework provides a natural geometric interpretation of quantum probability densities as curvature projections within analytic phase geometry.