This work explores a simple question: why does nature end up using equations that look quantum? Instead of starting from the usual quantum axioms (wavefunctions, operators, measurement rules), we begin with three much weaker and more intuitive requirements: • physical evolution should be reversible • distinguishability between states should have a consistent geometry • classical mechanics must appear as a limiting case From only these conditions, probability flow is forced to acquire a phase degree of freedom, and an information–geometric curvature term inevitably appears. When the two are combined, the resulting dynamics takes the form of the Schrödinger equation. The point of the paper is not to claim that quantum mechanics is fully derived, nor that measurement theory is explained. Rather, it shows that once reversible probabilistic dynamics is required to be geometrically consistent, quantum-type evolution becomes extremely hard to avoid. In this sense, the wavefunction can be interpreted not as a mysterious physical object, but as a natural coordinate system describing consistent probability flow.
This work explores a simple question: why does nature end up using equations that look quantum? Instead of starting from the usual quantum axioms (wavefunctions, operators, measurement rules), we begin with three much weaker and more intuitive requirements: • physical evolution should be reversible • distinguishability between states should have a consistent geometry • classical mechanics must appear as a limiting case From only these conditions, probability flow is forced to acquire a phase degree of freedom, and an information–geometric curvature term inevitably appears. When the two are combined, the resulting dynamics takes the form of the Schrödinger equation. The point of the paper is not to claim that quantum mechanics is fully derived, nor that measurement theory is explained. Rather, it shows that once reversible probabilistic dynamics is required to be geometrically consistent, quantum-type evolution becomes extremely hard to avoid. In this sense, the wavefunction can be interpreted not as a mysterious physical object, but as a natural coordinate system describing consistent probability flow.
This work presents a reconstruction of quantum mechanics in which Schrödinger-type dynamics emerges from reversible probabilistic evolution rather than being postulated. Starting from a normalized probability density, we impose reversible probability dynamics, Fisher information geometry, and classical correspondence. These compatibility conditions yield coupled real equations equivalent to the continuity equation and the Hamilton–Jacobi equation with an additional information curvature (quantum potential) term. By combining density and phase into a complex amplitude, the Schrödinger equation appears as a compact encoding of reversible statistical flow. The result is mathematically equivalent to the Madelung hydrodynamic formulation and structurally connected to Bohmian mechanics, but derived without assuming Hilbert space, operator axioms, or measurement postulates. The work can therefore be viewed as an information-geometric derivation of quantum dynamics: quantum theory arises as the unique consistent representation of reversible probability flow. Keywords: quantum foundations, quantum reconstruction, emergent quantum mechanics, derivation of the Schrödinger equation, information geometry, Fisher information, Hamilton–Jacobi theory, Madelung formulation, Bohmian mechanics, reversible probability dynamics
Quantum mechanics is conventionally formulated in terms of complex Hilbert spaces, linear operators, and the Schrödinger equation. Although geometric structures play an increasingly important role in modern formulations of quantum theory, the wavefunction itself is generally regarded as a fundamental object. In this work, we investigate an alternative formulation in which the fundamental object is a coherence coordinate , where Omega represents radial coherence and Theta represents phase coherence associated with geometric transport. Starting from three assumptions—a coherence coordinate, a probability density generated by radial coherence, and a geometric Hamiltonian—we reconstruct the continuity equation, the quantum Hamilton–Jacobi equation, and ultimately the Schrödinger equation. Within this framework, the wavefunction appears as a derived quantity, while Hilbert spaces emerge naturally as spaces of square-integrable functions on a coherence manifold. The resulting construction establishes a differential-geometric representation of quantum mechanics and provides a common geometric language connecting quantum states, probability flow, geometric phases, holonomies, and gauge transport.
The Schrödinger equation is the fundamental equation of quantum mechanics. Since its discovery in 1926, it has been postulated — accepted as a starting point whose origin remains unexplained. This paper derives it. The derivation begins not with physics but with three conditions that any coherent structure must satisfy in order to exist at all: (i) it must be self-consistent (identity), (ii) it must be capable of persisting through change (temporality), and (iii) it must be distinguishable from its environment (distinction). These three conditions are not physical laws; they are preconditions for anything — a particle, a field, a living organism — to be a definite, persisting, identifiable entity. From these three conditions alone, a unique dynamics on the space of all logically possible configurations can be derived. This dynamics takes the form of a probability flow with exactly three components. The first is a drift that pushes the system toward self-consistent configurations — states in which no internal contradictions remain. The second is a diffusion that prevents the system from collapsing to a single rigid state, maintaining the openness and fluctuation that characterize quantum systems. The third is a circulation — a closed, non-dissipative flow that neither creates nor destroys probability but moves it in loops. This circulation is the origin of time.
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.
The Schrödinger equation has been the central mathematical instrument of quantum mechanics for a century. This paper argues that it is not a quantum mechanical equation at all. It is classical energy conservation applied inside a sphere, with a classical wave form assumed rather than derived. The wavefunction ψ is not a physical object but a lossy geometric encoding. The Born rule—the requirement to square ψ to obtain probabilities—is not a law of nature but a decompression artifact produced by discarding the imaginary phase. The Heisenberg uncertainty principle is a measurement limit, not an ontological statement about particles, and its use to derive the Bohr radius is circular. All distances obtained from this framework are ratios pretending to be absolutes in a universe where only ratios persist. Relativistic Orthogonal Selection Theory (ROST) dissolves each of these confusions by returning to the geometric substrate the equation was always implicitly describing.
Quantum mechanics is traditionally formulated in terms of complex Hilbert spaces, wavefunctions, and linear evolution equations. While extraordinarily successful, this framework leaves open several conceptual questions concerning the geometric origin of the wavefunction, the meaning of quantum phase, and the status of Hilbert space as a fundamental structure. In this work, we investigate a coherence-geometric reformulation in which the fundamental object is a coherence coordinate , where represents radial coherence and represents phase coherence associated with geometric transport. We show that the standard structures of quantum mechanics admit a natural interpretation within a differential-geometric framework. The wavefunction emerges from coherence coordinates, probability density is generated by radial coherence, probability flow is generated by phase coherence, and Hilbert spaces appear as spaces of square-integrable functions defined on a coherence manifold. The resulting formulation provides a unified geometric language connecting quantum mechanics, geometric phases, fiber bundles, holonomies, gauge theory, and differential geometry