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.
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.
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
Quantum mechanics is traditionally formulated on complex Hilbert spaces, while geometry is usually introduced only after the quantum formalism has been established. In this work, we investigate an alternative viewpoint in which a coherence manifold constitutes the fundamental object and Hilbert space emerges as a derived structure. Local quantum dynamics are described by a complex coherence coordinate W= Omega + iTheta, where Omega represents radial coherence and Theta represents holonomy phase. The wavefunction arises naturally as phi=exp(Omega+iTheta), and the continuity equation, quantum Hamilton–Jacobi equation, and Schrödinger equation emerge from the differential geometry of the coherence manifold. We further propose that the discrete quadruples (a,b,c,d), originally introduced in the Triple-Coherence framework, should be interpreted as coherence quantum numbers rather than phenomenological fitting parameters. This leads naturally to a coherence operator whose spectrum is determined by a rank-four coherence lattice. The resulting framework combines ideas from geometric quantum mechanics, geometric quantization, spectral geometry, fiber-bundle theory, and representation theory within a common geometric language. We argue that elementary particles may be interpreted as stable eigenstates of coherence geometry and that the observed particle spectrum may represent a subset of a larger coherence spectrum.
Quantum mechanics is traditionally formulated on complex Hilbert spaces, whereas gauge theories are formulated on fiber bundles equipped with connections and curvature. Despite the central role of geometry in modern physics, the Hilbert-space structure of quantum theory is usually postulated rather than derived. In this work, we investigate a differential-geometric formulation in which quantum states are represented as points on a coherence manifold characterized by a radial coherence coordinate and a holonomy-induced phase coordinate. The resulting geometry naturally admits the structure of a fiber bundle whose base space is associated with a coherence lattice and whose fiber corresponds to a phase circle. Within this framework, Hilbert spaces emerge as function spaces defined on the coherence manifold, while Schrödinger dynamics arise from the Laplace–Beltrami operator associated with the underlying geometry. The construction unifies concepts from geometric quantum mechanics, fiber-bundle gauge theory, holonomy, and coherence geometry within a common differential-geometric framework.
Quantum mechanics is traditionally formulated on complex Hilbert spaces, while geometric structures are introduced only after the quantum formalism has been established. In this work, we investigate a coherence-geometric reformulation in which geometry is fundamental and Hilbert space emerges as a derived structure. The framework is based on a coherence manifold with local coordinates , where denotes radial coherence and denotes holonomy phase. The wavefunction arises naturally as , and the continuity equation, quantum Hamilton–Jacobi equation, and Schrödinger equation emerge from the differential geometry of the coherence manifold. A second ingredient is a discrete coherence spectrum characterized by quadruples . These integers are interpreted as coherence quantum numbers rather than phenomenological fitting parameters. The resulting framework introduces a coherence operator whose spectrum is determined by a rank-four coherence lattice. We further propose a Coherence Resonance Condition in which stable particle states correspond to rational phase locking between local coherence oscillations and global holonomy cycles. The framework combines concepts from geometric quantum mechanics, geometric quantization, Berry holonomies, Yang–Mills gauge theory, spectral geometry, and representation theory within a common geometric language.
Quantum mechanics is conventionally formulated on complex Hilbert spaces, whereas gauge theories are formulated through connections, holonomies, and curvature. Despite their common use of complex structures, the geometric relationship between these frameworks remains incompletely understood. In this work, we develop a geometric interpretation of the Triple-Coherence Framework based on the complex coherence coordinate Z=M+iQ . The empirical Triple-Coherence relation is reformulated as an exponential generator structure, naturally leading to a discrete coherence lattice. Combining this radial structure with an angular phase coordinate yields the representation Z=R exp i(phi). We investigate the possibility that the MQ plane constitutes a more fundamental geometric arena than the conventional Hilbert-space description of quantum mechanics. Within this framework, quantum states correspond to points on an MQ manifold, Hilbert spaces emerge as function spaces on this geometry, and Yang–Mills holonomies generate the phase structure of quantum states. The resulting formulation provides a common language connecting Triple-Coherence coordinates, Schrödinger dynamics, Wilson loops, gauge transport, and particle stability.
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