We propose a unified framework in which quantum dynamics, gauge interactions, and gravitational behavior arise from a common operator-based microscopic structure. The approach is based on a complex operator composed of two Hermitian components whose non-commutativity generates interaction effects. A positive density operator is constructed, and entropy is used as a central organizing principle connecting microscopic and macroscopic physics. Under general assumptions, the formalism reproduces unitary quantum evolution, provides a mechanism for the emergence of gauge structure, and connects entropy variation to gravitational dynamics. The framework is illustrated through an explicit solvable model demonstrating how interaction strength influences thermodynamic behavior. This work establishes a concrete mathematical basis for exploring a unified origin of fundamental interactions.
We propose a unified framework in which quantum dynamics, gauge interactions, and gravitational behavior arise from a common operator-based microscopic structure. The approach is based on a complex operator composed of two Hermitian components whose non-commutativity generates interaction effects. A positive density operator is constructed, and entropy is used as a central organizing principle connecting microscopic and macroscopic physics. Under general assumptions, the formalism reproduces unitary quantum evolution, provides a mechanism for the emergence of gauge structure, and connects entropy variation to gravitational dynamics. The framework is illustrated through an explicit solvable model demonstrating how interaction strength influences thermodynamic behavior. This work establishes a concrete mathematical basis for exploring a unified origin of fundamental interactions.
We develop an operator-based framework in which quantum dynamics, interaction structure, and geometric features arise from the modular properties of a density operator constructed from a non-commuting pair of Hermitian operators. A positive state is defined via exponential coarse-graining, and time evolution is derived from the associated modular Hamiltonian rather than postulated. Using a systematic Baker–Campbell–Hausdorff expansion, interaction terms are shown to be organized by nested commutators, yielding a natural algebraic hierarchy analogous to effective field theory. Interpreting the coarse-graining parameter as a renormalization scale, we derive flow equations for operator-valued couplings, establishing a renormalization group structure in operator space. A continuum field-theoretic extension and a lattice regularization suitable for numerical investigation are constructed. Local automorphisms of the operator algebra give rise to gauge connections, while the quantum Fisher information metric defines an emergent notion of geometry. Under additional assumptions of locality and approximate equilibrium, the framework reproduces the structure of entropic approaches to gravitational dynamics. These results provide a unified algebraic perspective in which dynamics, interactions, and geometry are encoded in the modular structure of quantum states.
Entropic Dynamics (ED) provides a statistical-inferential foundation for physical laws, deriving motion and field equations from principles of entropy maximization rather than quantization postulates. ED reconstructs quantum mechanics by treating the evolution of probability distributions on configuration space as driven by information constraints, yielding the Schrodinger equation as a non-dissipative diffusion process. Building on this foundation, the present work extends the ED framework into a Unified Entropic Dynamics (UED) formulation that encompasses classical, quantum, relativistic, thermodynamic, and gravitational phenomena within a single information geometric principle. By maximizing entropy subject to constraints on diffusion, drift, and gauge covariance over a manifold endowed with a supermetric H_ab, we derive a universal field equation that merges the Fokker-Planck and Hamilton-Jacobi structures into one covariant form. When specialized to different dynamical variables, this equation reproduces the harmonic oscillator, Schrodinger, Maxwell, Klein-Gordon, and gravitational wave equations, thereby revealing a deep equivalence between probabilistic inference and dynamical law. The UED framework demonstrates that spacetime geometry, quantum coherence and thermodynamic diffusion emerge as complementary expressions of the same entropic process, establishing a unified inferential foundation for both microscopic and macroscopic physics. In this formulation, energy, probability, and entropy are intertwined aspects of information geometry, providing a consistent inferential foundation for understanding classical, quantum, and gravitational dynamics as complementary expressions of a single entropic law.
The Entropy-Originated Topological Framework (EOTF) extends conventional quantum mechanics by embedding its algebraic, wave, and interaction representations within an entropic–topological manifold. While the Heisenberg, Schrödinger, and Dirac pictures traditionally provide unitarily equivalent formulations of quantum dynamics, the EOTF generalizes their equivalence to a deeper invariance of informational and geometric flow. Within this framework, operator evolution, wave function propagation, and interaction dynamics emerge as distinct projections of a unified entropic operator field defined on a diagram-Hilbert manifold. The correspondence preserves the predictive structure of quantum mechanics while revealing its geometric and thermodynamic origin—thereby linking quantum pictures to the emergent structure of spacetime, gauge interactions, and matter.
The Entropy-Originated Topological Framework (EOTF) extends conventional quantum mechanics by embedding its algebraic, wave, and interaction representations within an entropic–topological manifold. While the Heisenberg, Schrödinger, and Dirac pictures traditionally provide unitarily equivalent formulations of quantum dynamics, the EOTF generalizes their equivalence to a deeper invariance of informational and geometric flow. Within this framework, operator evolution, wave function propagation, and interaction dynamics emerge as distinct projections of a unified entropic operator field defined on a diagram-Hilbert manifold. The correspondence preserves the predictive structure of quantum mechanics while revealing its geometric and thermodynamic origin—thereby linking quantum pictures to the emergent structure of spacetime, gauge interactions, and matter.
The Entropy-Originated Topological Framework (EOTF) extends conventional quantum mechanics by embedding its algebraic, wave, and interaction representations within an entropic-topological manifold. While the Heisenberg, Schrödinger, and Dirac pictures traditionally provide unitarily equivalent formulations of quantum dynamics, the EOTF generalizes their equivalence to a deeper invariance of informational and geometric flow. Within this framework, operator evolution, wave function propagation, and interaction dynamics emerge as distinct projections of a unified entropic operator field defined on a diagram-Hilbert manifold. The correspondence preserves the predictive structure of quantum mechanics while revealing its geometric and thermodynamic origin-thereby linking quantum pictures to the emergent structure of spacetime, gauge interactions, and matter.
Modern theoretical physics employs several fundamental mathematical structures, including Green functions, Hilbert spaces, operator algebras, Gibbs ensembles, gauge fields, and spacetime geometry. These structures are traditionally introduced independently and serve as separate foundations for quantum mechanics, quantum field theory, statistical mechanics, and general relativity. In the present work, we propose an alternative mathematical organization based on a single Reconstruction Principle. Beginning with one primitive mathematical architecture, successive reconstruction layers generate transport operators, Green functions, Diagram Hilbert Spaces, complex quantum operators, Gibbs ensembles, entropic projections, and finally macroscopic spacetime together with gauge fields. Each layer is reconstructed uniquely from its predecessor without introducing additional primitive mathematical structures. Consequently, Hilbert space, Green functions, and spacetime are interpreted as successive mathematical representations of one underlying architecture rather than independent foundations. The resulting reconstruction hierarchy provides a unified conceptual framework connecting quantum mechanics, statistical mechanics, gauge theory, and emergent gravity while substantially reducing the number of independent assumptions.