The Panvitalistic Theory (PVT) replaces the external time parameter of conventional quantum mechanics with internal angular curvature \(\pi \equiv T/L\). The dynamics of any physical system are governed solely by the volume-invariance constraint \(\delta V = 0\) in a discrete, rational 12-dimensional spacetime. We derive the complete set of 12 volume operators and their geometric commutator algebra, show that the time-dependent Schrödinger equation is an artefact of an ontologically inconsistent external-time postulate, and re-derive the Bohr model of the hydrogen atom—including radii \(r_n\), energy levels \(E_n \propto -1/n^2\), and the fine-structure constant \(\alpha\)—directly from the volume constraint. A macroscopic Earth-satellite analogue confirms the geometric origin of quantization. The framework eliminates the measurement problem, the problem of time, and the QT–GRT incompatibility at the axiomatic level while reproducing all spectroscopic successes of standard quantum theory.
The Panvitalistic Theory (PVT) replaces the external time parameter of conventional quantum mechanics with internal angular curvature (\pi \equiv T/L). The dynamics of any physical system are governed solely by the volume-invariance constraint (\delta V = 0) in a discrete, rational 12-dimensional spacetime. We derive the complete set of 12 volume operators and their geometric commutator algebra, show that the time-dependent Schrödinger equation is an artefact of an ontologically inconsistent external-time postulate, and re-derive the Bohr model of the hydrogen atom—including radii (r_n), energy levels (E_n \propto -1/n^2), and the fine-structure constant (\alpha)—directly from the volume constraint. A macroscopic Earth-satellite analogue confirms the geometric origin of quantization. The framework eliminates the measurement problem, the problem of time, and the QT–GRT incompatibility at the axiomatic level while reproducing all spectroscopic successes of standard quantum theory.
Despite its successes, quantum mechanics (QM) has remained mysterious to all who have encountered it. Starting with Bohr and progressing into the present, the departure from intuitive, physical reality has widened. The connection between QM and reality is more than just a “philosophical” issue. It reveals that QM is not a correct or complete theory of the physical world and that inescapable internal inconsistencies and incongruities arise when attempts are made to treat it as physical as opposed to a purely mathematical “tool.” Some of these issues are discussed in a review by F. Laloe [Am. J. Phys. 69, 655 (2001)]. In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of e moving in the Coulombic field of the proton and the wave equation as modified by Schrodinger, a classical approach is explored that yields a remarkably accurate model and provides insight into physics on the atomic level. The proverbial view, deeply seated in the wave-particle duality notion, that there is no large-scale physical counterpart to the nature of the electron may not be correct. Physical laws and intuition may be restored when dealing with the wave equation and quantummechanical problems. Specifically, a theory of classical quantum mechanics (CQM) is derived from first principles that successfully applies physical laws on all scales. Rather than using the postulated Schrodinger boundary condition “Ψ → 0 as r → ∞,” which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell’s equations, the classical wave equation is solved with the constraint that the bound (n = 1)state electron cannot radiate energy. By further application of Maxwell’s equations to electromagnetic and gravitational fields at particle production, the Schwarzschild metric is derived from the classical wave equation, which modifies general relativity to include conservation of spacetime in addition to momentum and matter/energy. The result gives a natural relationship among Maxwell’s equations, special relativity, and general relativity. CQM holds over a scale of spacetime of 85 orders of magnitude — it correctly predicts the nature of the universe from the scale of the quarks to that of the cosmos. A review is given by G. Landvogt [Internat. J. Hydrogen Energy 28, 1155 (2003)].
This paper proposes and validates a revolutionary foundational physics framework, demonstrating for the first time that quantum mechanics can naturally emerge from pure principles of geometric constraints. We establish a two-level unified theory: at the top level is the Relation Matrix R_{ij} describing the abstract relational network of the universe, and at the bottom level is the Minimal Phase Closed Triangle (MPCT) network serving as the physical implementation unit. By introducing a curvature constraint action, we rigorously derive the Schrödinger equation, quantization conditions, and Planck's constant, and for the first time directly calculate the fine-structure constant from geometric principles. Core Innovations: Proposed the new paradigm "Existence is relation, relation is geometry," defining fundamental physical entities as relations rather than particles. Established an exact mathematical mapping between the Relation Matrix R and the MPCT network. Derived quantum mechanics from the curvature constraint action \mathcal{I} = \int [L_c + \frac{\lambda}{2}(\kappa - \kappa_R)^2]dt. Discovered the geometric compression factor \eta(n,l,m) and its scaling law \eta \propto n^{0.67}. Implemented strict curvature calculations based on the relation matrix, verifying \eta \propto (\text{Var}(\kappa_R))^{0.15}. Numerical simulations of the hydrogen atom system confirmed the quantized distribution of MPCT areas, discovering the new scaling law \Delta A \propto n^{3.55}. Theoretical predictions agree with numerical results within 3.4% accuracy, with the fine-structure constant calculation error at 15.2%. This work provides a novel geometric perspective for understanding the essence of quantum phenomena and indicates a possible path toward unifying quantum mechanics and gravity. Keywords: Foundations of quantum mechanics, Geometric constraints, Relation matrix, Minimal Phase Closed Triangle, Fine-structure constant, Hydrogen atom, Scaling laws, Emergent phenomena
This Volume III develops a complete geometric formulation of quantum mechanics, demonstrating that quantum phenomena emerge from the very structure of spacetime. We demonstrate that Einstein's field equations—when interpreted in the spectral domain and taking phase coherence into account—already contain the entire structure of quantization. Quantum mechanics appears not as an axiom, but as a geometric consequence. Based on the experimental and mathematical results consolidated in Volumes~0, I, and II, the present work shows that quantization need not be postulated: it arises inevitably from three fundamental geometric elements—proper frequency, parallel transport, and phase coherence. The action appears as the phase accumulated along the worldline, and its discretization into coherence windows, derived without external hypotheses, naturally leads to the condition Sn=nh and to all associated quantum phenomenology. We derive the Schrödinger equation as the limit of short windows, obtain the commutator [x,p]=iℏ from the incompatibility between spatiotemporal synchronizations, and reinterpret interference, propagators, and decoherence as spectral properties of parallel transport. The Hilbert space arises as the spectral space of the metric, observables correspond to generators of isometric deformations, while the collapse is understood as the geometric selection of coherent subspaces. The formalism extends to relativistic particles, spin, quantum statistics, the Dirac equation, harmonic oscillators, and field quantization, where spectral modes of the metric assume the role of particles: photons as harmonic 1-forms, weak modes as connections of 2-forms, gluons as 3-forms, and the Higgs as the fundamental scalar curvature mode. The reinterpretation of the Standard Model thus becomes a direct consequence of spectral geometry. The Volume culminates in the Central Theorem —Geometry = Quantization— showing that the entire quantum structure is a manifestation of the spectral granularity of spacetime. We conclude with experimental predictions and falsification criteria, as well as the conceptual framework necessary for Volume~IV, dedicated to the complete quantization of the metric within the same paradigm.
Abstract—This paper completely deconstructs the calculus-centered mathematical redundancy and probabilistic uncertainty faced by modern physics, and re-establishes the fundamental mechanism of the universe into a clear system of elementary algebraic equations. Conventional academia has treated "time (t)" as a physical variable, interpreting the behavior of the microscopic world through complex partial differential equations and probability clouds. This Foundation establishes a unique grand premise: "Time (t) is not a physical reality but merely a unit for establishing a reference, and a Field is not an applied Force but the geometric structure of space (the thickness of a water pipe)." Based on this premise, the three pillars of quantum mechanics—Louis de Broglie's matter wave theory, Werner Heisenberg's uncertainty principle, and Erwin Schrödinger's wave equation—are substituted into closed algebraic structures. Cross-verification through six empirical cases from both macroscopic and microscopic realms—including multidimensional orbits of hydrogen atoms, particle constraints in quantum dots, actual nuclear proton behavior, and deep learning data-driven energy levels—proves that this algebraic formula system precisely converges with the high-level calculus predictions of conventional academia down to the decimal points with 100% accuracy.
Standard quantum mechanics offers remarkably preciseempirical predictions but often frames microscopicmechanisms through complex differential formalismsand probabilistic clouds. This paperpresents an algebraic-geometric reconstruction of selectedstationary-state quantum-mechanical relationsby introducing two foundational framework axioms:treating time (t) as a reference measure for indexingstate transitions rather than a dynamic variable, anddefining a physical Field as a local geometric constraintstructure F(x). By coupling an independent geometricwave-matching postulate (L = nλ) with the establishedde Broglie relation (p = h/λ), we algebraically reconstructBohr angular momentum quantization and thediscrete hydrogenic energy spectrum without directlysolving the Schr¨odinger eigenvalue equation. Furthermore,we evaluate Heisenberg’s uncertainty principle atits saturated lower bound (∆x∆p = K), proposing amechanical spatial compression model hypothesis formomentum spread. Cross-verifications across benchmarkcases demonstrate numerical agreement withinstated rounding precision.
While the complex formalism of quantum mechanics has been extraordinarily successful, the fundamental geometric origin of its core elements—the imaginary unit, operator non-commutativity, quantum phase, and the intrinsic geometry with quantum spin—has remained unresolved for more than a century. Here we resolve these foundational questions by deriving an exact and physically equivalent formulation of the Schrödinger equation as a Real Geometric Flow—a moving-frame evolution on a 2𝑁-dimensional real state space. In this restored framework, the global imaginary unit 𝑖 fully decomposes into a set of local skew-symmetric generators that drive planar rotations in real space, revealing that the traditional complex formalism is a compressed representation of an intrinsic rotational geometry.Within the real geometric flow, energy eigenvalues appear as the Curvatures of the state trajectory, energy gaps correspond to Torsions that couple internal planes—naturally generating the geometry of quantum spin—and quantum phase emerges naturally as the Holonomy of the real moving frame. This formulation preserves the full empirical content of quantum mechanics while exposing the geometric mechanisms hidden beneath its notation, providing the first coherent explanation of curvature, torsion, non-commutativity, spin behavior, and quantum phase within a single framework.By demonstrating that the algebraic postulates of quantum mechanics arise from a deeper geometric order—one that stands alongside curvature-based theories such as General Relativity—this work establishes a unified differential-geometric foundation for quantum dynamics.It opens new directions for quantum control, quantum information, and fundamental physics.