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Axiomatic Unification of Quantum Mechanics, Thermodynamics, and Geometry via Complex Time, Information Geometry Equivalence, and Gradient Flow

Y. Li

Published Mar 24, 2026• 0 citations
Publisher

Abstract

We propose a self-consistent axiomatic framework that unifies quantum unitary evolution, thermal dissipation, and spacetime geometry based on three fundamental principles: the \emph{complex-time principle}, the \emph{information-geometry equivalence principle}, and the \emph{maximal entropy gradient principle}. Starting from these axioms, we rigorously derive three core equations: a complex-time quantum-thermal master equation, an information-induced gravitational field equation, and a unified gradient-flow dynamics that couples quantum states, thermodynamics, and curvature. The framework naturally recovers standard quantum mechanics, thermodynamics, and general relativity in appropriate limits, while offering new predictions about the relation between quantum entanglement and spacetime structure. All derivations are carried out in full mathematical detail, and the resulting equations are original, closed-form, and internally consistent.

DOI
10.5281/zenodo.19199085

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We propose a self-consistent axiomatic framework that unifies quantum unitary evolution, thermal dissipation, and spacetime geometry based on three fundamental principles: the \emph{complex-time principle}, the \emph{information-geometry equivalence principle}, and the \emph{maximal entropy gradient principle}. Starting from these axioms, we rigorously derive three core equations: a complex-time quantum-thermal master equation, an information-induced gravitational field equation, and a unified gradient-flow dynamics that couples quantum states, thermodynamics, and curvature. From these, we further derive four key derived equations: a generalized Einstein field equation in the low-energy limit, a quantum-thermodynamic uncertainty relation, a relation between entropy production and curvature scalar, and a conservation law for quantum information current. The framework naturally recovers standard quantum mechanics, thermodynamics, and general relativity in appropriate limits, while offering new predictions about the relation between quantum entanglement and spacetime structure. All derivations are carried out in full mathematical detail, and the resulting equations are original, closed-form, and internally consistent.
This paper proposes a fundamental framework based on information geometry that unifies quantum mechanics, thermodynamics, and spacetime geometry into a single foundational theory. Starting from three elementary axioms regarding information as the primary physical ontology, we derive the universal information metric, construct the information action functional, and show that all known physical laws emerge as extremal conditions of this functional. The theory recovers standard quantum mechanics, classical thermodynamics, and general relativity in appropriate limits, while predicting new relations between quantum entanglement, gravitational curvature, and entropy. We conclude that all physical phenomena—matter, energy, spacetime, and interactions—are equivalent to geometric structures on the space of quantum information states.
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We present a fundamental unified framework based on information geometry, in which quantum mechanics, statistical thermodynamics, and general relativity emerge from a single information-theoretic ontology. Starting from three rigorous axioms that establish quantum information as the primary physical substrate, we construct the universal information action functional using the Bures–Fisher–Rao metric on the manifold of quantum density matrices. Full variational derivation yields a unified field equation that governs the evolution of quantum information and emergent spacetime geometry. In appropriate limiting regimes, we recover the Schrödinger equation, equilibrium statistical mechanics, and the Einstein field equations. This work provides the foundational layer of a complete theory of everything based on the principle: geometry is information.
We present a complete, mathematically rigorous formulation of the Information Geometry Unification Principle (IGUP). Starting from three fundamental axioms establishing quantum information as the primary ontological substrate, we construct the Bures–Fisher–Rao metric on the manifold of density matrices, derive covariant derivatives, curvature tensors, and the universal information action. We perform full variational derivations for both the quantum state ρ\rhoρ and the information metric gμνg_{\mu\nu}gμν​, obtaining closed-form unified field equations that govern quantum dynamics, thermodynamics, and gravitational geometry. We show explicitly how the Schrödinger, Liouville, Boltzmann, and Einstein equations arise as limiting cases. This paper constitutes the foundational core of a complete theory of physics based on the identification of geometry with information.
The foundations of modern theoretical physics rest on the interplay between quantum dynamics, thermodynamics, and spacetime geometry. While quantum mechanics describes microscopic dynamics through unitary time evolution, thermodynamic phenomena introduce irreversibility and entropy production. In this work we propose a unified framework in which quantum evolution is formulated on a complex-time manifold that naturally incorporates both dynamical and thermodynamic structures. We introduce a complex-time coordinate combining physical time and thermal flow and derive a generalized evolution equation from a variational principle defined on a complex-time geometric manifold. The resulting equation contains both Hamiltonian dynamics and geometric diffusion terms governed by the Laplace–Beltrami operator. This formulation provides a geometric interpretation of thermodynamic irreversibility within quantum theory and establishes connections with Euclidean quantum field theory, geometric quantum mechanics, and thermal time hypotheses. Possible implications for quantum statistical mechanics and information flow in gravitational systems are discussed.