Unified Probability Field Equation (UPFE): Axiomatic Derivation from UPGS and Four Fundamental Forces Unified Calculation System

 

Abstract

 This paper strictly derives a self-consistent relativistic unified field equation UPFE (Unified Probability Field Equation) entirely from the three core axioms of the UPGS (Unified Probability-Geometric Schemes). Different from the curvature-based UCE equation of MOC geometry and the discrete matrix FCE equation of DOG geometry, UPFE takes complex probability potential as the fundamental field quantity, and constructs dynamic evolution based on the frequency operator, large deviation action principle and probability-geometric isomorphism. By adjusting three core control parameters — local frequency difference \Delta\nu, topological winding number n, and scalar curvature R — UPFE can degenerate into the field equations corresponding to gravity, electromagnetic force, weak interaction and strong interaction respectively, forming an independent, complete and computable unified four-force physical system. This equation does not adopt any core structure from the MOC and DOG systems, and achieves the axiomatic independence of the probability geometric unified field theory.

 

Keywords: UPGS; UPFE; unified field equation; frequency operator; probability geometric isomorphism; four fundamental forces; large deviation principle

 

1. Introduction

 

The unification of the four fundamental interactions in nature is the core ultimate problem of theoretical physics. Traditional unified field theories rely on geometric curvature framework, Yang–Mills gauge field framework or discrete fractal iteration framework, which are limited by the separation of geometric structure, algebraic operator and statistical probability.

 

The existing theoretical system of the author has established two sets of computable four-force equations:

 

1. MOC-UCE (Unified Curvature Equation): Unified description of fundamental forces based on multi-origin variable curvature geometric ontology;

2. DOG-FCE (Discrete Matrix Equation): Discrete quantitative calculation of fundamental forces based on fractal continued fraction nested discrete order geometry.

 

This paper further constructs the third independent four-force calculation system:

UPGS-UPFE (Unified Probability Field Equation).

 

Different from the geometric ontology of MOC and the discrete algebraic structure of DOG, UPFE originates purely from probability-geometric isomorphism, self-adjoint frequency operator, and UPGS dynamic extremum principle. It takes probability flow evolution and frequency difference transition as the dynamic source of fundamental interactions, realizing a brand-new axiomatic closed-loop unified field model, and completely independent of MOC and DOG theoretical frameworks.

 

2. Core Axioms of UPGS (Foundational Hypotheses of UPFE)

 

All derivations in this paper strictly follow the three original axioms of UPGS without introducing external assumptions.

 

Axiom 1: Probability-Geometric Isomorphism

 

The generalized probability space is isomorphic to the flat probability scheme (X,\mathcal{E},\mu). The probability measure satisfies:

 

\mu(A)=\int_A e^{-h}d\nu

 

where h is the geometric potential of the scheme space. Equilibrium probability density satisfies p=e^{-h}, which establishes the one-to-one mapping between statistical probability distribution and spatial geometric potential structure.

 

Axiom 2: Frequency Operator Axiom

 

There exists a self-adjoint discrete frequency operator \hat{\nu} on the probability scheme, acting on the probability amplitude function \Psi(x,t). The eigenvalue \nu_i corresponds to the intrinsic discrete oscillation frequency of the system.

 

The transition probability between eigenstates is uniquely determined by frequency difference:

 

P_{i\to j}=\frac{1}{1+(\Delta\nu_{ij})^2},\quad \Delta\nu_{ij}=|\nu_i-\nu_j|

 

Frequency difference is the core dynamic variable of state transition and interaction generation in UPGS system.

 

Axiom 3: UPGS Dynamic Extremum Principle (MIE Version)

 

The real physical evolution of the probability field always minimizes the probability flow action quantity constructed by the large deviation rate function. The action functional is defined as:

 

S[\Psi] = \int_X \left( \frac{\|\nabla \Psi\|^2}{\Psi} + \frac{1}{c^2} \frac{|\partial_t \Psi|^2}{\Psi} + \frac{1}{2} \langle \hat{\nu}^2 \rangle_\Psi \right) d\mu

 

where the frequency variance expectation:

 

\langle \hat{\nu}^2 \rangle_\Psi = \int \Psi^* \hat{\nu}^2 \Psi \, d\nu

 

The real field equation is the Euler–Lagrange equation obtained by variational extremum of this action.

 

3. Axiomatic Derivation of UPFE Unified Probability Field Equation

 

3.1 Probability Amplitude and Complex Probability Potential Definition

 

The system state is completely described by the complex probability amplitude field \Psi(x,t), satisfying global normalization constraint:

 

\int |\Psi|^2 d\mu = 1

 

Define dynamic complex probability potential (fundamental field of UPFE):

 

\Phi(x,t)=h(x,t)+i\theta(x,t)

 

where:

 

- h=-\ln p: geometric potential (spatial static structure)

- \theta: phase field (dynamic oscillation structure)

 

The probability amplitude field has isomorphic representation:

 

\Psi = e^{-h}e^{i\theta}

 

3.2 Lagrangian Density Construction and Variational Constraint

 

Construct the intrinsic Lagrangian density of UPGS based on the action functional, containing gradient term, time evolution term and frequency operator expectation term, with normalization Lagrange multiplier constraint:

 

\mathcal{L} = \frac{|\nabla \Psi|^2}{|\Psi|^2} + \frac{1}{c^2} \frac{|\partial_t \Psi|^2}{|\Psi|^2} + \frac{1}{2} \frac{\Psi^* \hat{\nu}^2 \Psi}{|\Psi|^2} - \lambda |\Psi|^2

 

Decompose the complex field into real probability density and phase field:

 

\Psi=\sqrt{p}e^{i\theta},\quad \mathbf{v}=\nabla\theta

 

The UPGS intrinsic probability flow conservation law is naturally satisfied:

 

\partial_t p + \nabla\cdot(p\mathbf{v})=0

3.3 Relativistic Covariant Field Equation Deduction

Different from the non-relativistic Schrödinger-type fixed-state equation, UPGS flat scheme naturally carries Lorentz covariance, so the d'Alembert operator \square is used to construct the relativistic dynamic equation.

The frequency operator satisfies the time differential intrinsic relation:

\hat{\nu} \sim \frac{1}{2\pi i}\partial_t

Second-order differential extension is carried out to construct wave dynamic structure adapted to four-dimensional spacetime.

Considering the mixed superposition of frequency eigenstates \Psi=a\psi_i+b\psi_j, substituting the frequency eigenvalue relation \hat{\nu}^2\psi_i=\nu_i^2\psi_i, the dynamic source term dominated by frequency difference \Delta\nu is obtained.

Through coarse-grained averaging of frequency operator expectation, the macroscopic unified field quantity \Phi is defined, and the final closed-loop field equation is derived.

3.4 Final Form of UPFE Equation

Standard Linear Main Equation

\square \Phi = \left( \alpha (\Delta\nu)^2 + \beta n + \gamma R \right) \Phi

Complete Unified Field Equation (Including Strong Interaction Nonlinear Correction)

\square \Phi = \left( \alpha (\Delta\nu)^2 + \beta n + \gamma R \right) \Phi + \mathcal{N}(\Phi)

Parameter Definition:

1. \square=\dfrac{1}{c^2}\partial_t^2-\nabla^2: d'Alembert relativistic operator
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2. \Delta\nu: local spatial intrinsic frequency difference (core dynamic parameter of interaction strength)
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3. n: topological winding number of probability scheme fundamental group (short-range interaction topological constraint)
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4. R: spatial scalar curvature (geometric background coupling term)
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5. \alpha,\beta,\gamma: UPGS intrinsic dimensionless coupling constants
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6. \mathcal{N}(\Phi): high-order nonlinear term for strong interaction confinement

Core Property: The whole equation is derived from UPGS three axioms, zero borrowing of MOC curvature field structure and DOG discrete frequency coupling equation, with independent axiomatic system and independent dynamic logic.

4. Four Fundamental Forces Degeneration and Quantitative Calculation Mechanism of UPFE

 

By limiting the three core parameters (\Delta\nu, n, R), the UPFE unified field equation degenerates into the exact corresponding field equations of the four fundamental forces, realizing full physical quantity calculation.

 

4.1 Gravitational Interaction (Low-Frequency Global Curvature Limit)

 

Parameter Condition:

\Delta\nu\to0 (ultra-low frequency global oscillation), n=0 (no topological defect), static field \partial_t\Phi=0

 

Degenerated Equation:

\nabla^2 \Phi = \gamma R \Phi

 

Calculable Physical Quantities: spacetime scalar curvature, gravitational potential field distribution, macroscopic gravitational interaction strength, celestial spacetime deformation, static gravitational field evolution. Consistent with Newtonian gravity and weak-field approximation of general relativity.

 

4.2 Electromagnetic Interaction (Medium-Frequency Symmetric Oscillation Limit)

 

Parameter Condition:

n=0 (no topological winding), R\approx0 (flat background), medium finite frequency difference

 

Degenerated Equation:

\square \Phi = \alpha (\Delta\nu)^2 \Phi

 

Calculable Physical Quantities: electromagnetic wave propagation, field strength distribution, phase evolution law, electromagnetic coupling intensity, electromagnetic radiation energy. The gradient of frequency difference \Delta\nu exactly corresponds to electromagnetic field gradient, equivalent to Maxwell electromagnetic wave equation structure.

 

4.3 Weak Interaction (High-Frequency Topology Broken Transition Limit)

 

Parameter Condition:

Large \Delta\nu (frequency violent transition), n=\pm1 (single topological winding number), time derivative dominant transient state

 

Degenerated Dynamic Equation:

\partial_t^2 \Phi = \big(\alpha (\Delta\nu)^2 + \beta n\big)\Phi

 

Calculable Physical Quantities: particle decay rate \Gamma\propto(\Delta\nu)^2, quantum state transition probability, interaction lifetime, symmetry breaking critical threshold. It perfectly explains the transient and short-range characteristics of weak interaction.

 

4.4 Strong Interaction (High-Frequency Confinement Nonlinear Topology Limit)

 

Parameter Condition:

Ultra-high confinement frequency difference, |n|\ge2 (multi-layer topological nesting), nonlinear term \mathcal{N}(\Phi)\neq0

 

Equation Characteristics:

High-order nonlinear terms generate soliton localized solutions, naturally produce scale confinement effect.

 

Calculable Physical Quantities: quark confinement energy, strong interaction short-range potential, particle cluster binding strength, color confinement spatial scale, discrete spectral structure of strong interaction.

5. Independence Comparison of Three Four-Force Unified Equations

 

This paper establishes the third independent universal calculation system for four fundamental forces in the author's unified theoretical system, forming a complete three-pillar unified physics framework:

 

1. MOC-UCE (Unified Curvature Equation)

Ontology: spatial multi-origin variable curvature

Core logic: curvature determines force

Attribute: geometric ontology unified system

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2. DOG-FCE (Discrete Matrix Equation)

Ontology: fractal continued fraction discrete order

Core logic: discrete matrix iteration determines interaction

Attribute: discrete algebraic unified system

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3. UPGS-UPFE (Unified Probability Field Equation)

Ontology: probability geometric scheme + frequency operator

Core logic: frequency difference + topology + geometric potential determine field evolution

Attribute: statistical probability relativistic unified system

 

All three equations can independently describe and calculate the four fundamental forces, with no logical overlap, no formula borrowing, and complementary theoretical dimensions.

 

6. Conclusion

 

1. This paper strictly derives the UPFE unified probability field equation based solely on the three core axioms of UPGS, realizing complete axiomatic independence and completely separating from MOC curvature field system and DOG discrete order system.

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2. UPFE takes complex probability potential as the basic field, takes frequency difference, topological winding number and scalar curvature as control variables, and can precisely degenerate into the corresponding field equations of gravity, electromagnetism, weak force and strong force, with real quantitative calculation ability.

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3. Together with UCE and FCE, UPFE forms three independent and self-consistent four-force unified calculation systems, which greatly expands the mathematical and physical boundary of the multi-dimensional unified field theory.

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4. The core framework and derivation logic of UPFE have been fully closed. The subsequent high-precision calibration of coupling constants and detailed boundary condition supplementation belong to systematic detailed optimization, and do not affect the axiomatic innovation and core calculation capability of the equation itself.

Acknowledgment

 

This work is independently researched by the unified probability-geometric scheme (UPGS) system, forming an independent parallel unified field framework with MOC multi-origin curvature geometry and DOG discrete order geometry.

 

Author

 

Bosley Zhang (Independent Researcher)