A Unified Operator-Flow Framework on Hilbert Spaces:Curvature Trichotomy and the Emergence of Schrödinger, Heisenberg, and Dirac Dynamics
Lâu, Thiat-uí
Description This work develops a unified operator-flow framework on Hilbert spaces in which the standard dynamical formulations of quantum theory—the Schrödinger equation, the Heisenberg operator evolution, and Dirac-type spinor dynamics—arise as curvature-selected representations of a single abstract unitary flow. Starting from an axiomatic assignment of an admissible field tensor FFF to an anti-Hermitian generator G(F)G(F)G(F), the framework establishes the existence of a strongly continuous unitary flow acting on vectors, operators, and Clifford modules. From the same field tensor, an effective metric geff(F)g_{\mathrm{eff}}(F)geff(F) and its scalar curvature κ(F)\kappa(F)κ(F) are constructed, providing a geometric structure underlying the operator evolution. The central result, the Curvature Trichotomy Theorem, shows that the sign of κ(F)\kappa(F)κ(F) uniquely determines the curvature-adapted representation of the flow: positive curvature selects the Schrödinger picture, zero curvature selects the Heisenberg picture, and negative curvature selects a Dirac-type Clifford-module representation. In this way, the three classical formulations of quantum dynamics are interpreted not merely as unitarily equivalent pictures, but as distinct geometric realizations of a single operator-flow governed by curvature. Analytically, the work relies on standard results from the theory of generators of unitary groups, non-autonomous evolution equations, operator algebras, and spin geometry. Conceptually, it provides a geometric classification principle for quantum dynamics, linking representation theory, operator flows, and curvature within a unified mathematical structure. Preprint notice This manuscript is a preprint made publicly available for the purpose of early dissemination and scholarly discussion. It has not yet undergone peer review. The content, structure, or presentation may be revised in a subsequent peer-reviewed publication, and the final published version—if accepted—may differ from this preprint.
•Dec 12, 2025•0 citations