The Geometric Origin of Weak Interaction: From Curvature Frequency Transition to a Unified Framework of the Four Fundamental Forces

Author: Zhang Suhang (Bosley Zhang)

Affiliation: Independent Theoretical Physics Researcher, Luoyang, China

Corresponding email: zhang34269@zohomail.cn

Discipline Classification: Theoretical Physics, Quantum Gravity, Unified Field Theory, Geometric Physics

Abstract

Based on the previously established Multi-Origin Curvature (MOC) axiom system and the Maximum Information Efficiency (MIE) extremum principle, this paper proposes a novel geometric interpretation of the weak interaction, achieving a purely geometric unified description of all four fundamental interactions. The traditional Standard Model defines the weak force as a short-range interaction induced by an SU(2)×U(1) gauge field, relying on artificially introduced gauge bosons, the Higgs mechanism, and over a dozen free parameters, and is incapable of geometric compatibility with gravity. This paper demonstrates that the weak interaction is not a conservative force field, but a quantized transition process of the local eigenfrequency of the unified spacetime curvature field; nuclear decay, lepton conversion, and flavor change processes all correspond to discrete jumps of the curvature field from a high-frequency eigenstate to a low-frequency eigenstate, with the transition amplitude determined jointly by the frequency difference Δν and the topological winding number n of the curvature field; the most characteristic feature of the weak interaction — parity non-conservation — directly originates from the chiral topological structure of the complex curvature field phase, requiring no artificial introduction of the V-A coupling hypothesis.

Starting rigorously from the basic MOC-MIE axioms, this paper constructs the spacetime wave equation for the unified curvature field, defines the mathematical form of frequency eigenstates, derives a geometric analytical expression for the weak decay rate, and proves its complete self-consistency with experimental results of the Standard Model in the low-energy limit. This framework requires no additional gauge fields, elementary particles, or artificial symmetry breaking mechanisms. It unifies gravity, electromagnetism, and the strong interaction as spatial gradient effects of the curvature field, and the weak interaction as a temporal frequency transition effect of the curvature field. Finally, a single d’Alembert wave equation achieves the complete mathematical unification of the four fundamental forces, providing a contradiction-free, falsifiable, and fully closed geometric path toward quantum gravity and a theory of everything.

Keywords: Weak interaction; geometric unified field theory; MOC-MIE axiom system; curvature frequency transition; parity non-conservation; quantum gravity; unified field equation

---

1 Introduction

1.1 Achievements and Inherent Limitations of the Standard Model

In the second half of the 20th century, the unification of the electroweak theory and Quantum Chromodynamics (QCD) established the Standard Model of particle physics, describing the electromagnetic, weak, and strong interactions. The predicted W±, Z⁰ bosons, and the Higgs boson have all been experimentally confirmed, and the model possesses extremely high predictive accuracy in the low-energy regime. However, the Standard Model has unavoidable essential defects at the foundational level:

1. The theory contains more than 20 free parameters, all fitted from experiment, with no geometric or axiomatic origin.

2. It artificially introduces SU(2)×U(1) gauge symmetry and the Higgs spontaneous symmetry breaking mechanism, incapable of explaining the underlying origin of the gauge group, coupling constants, and mass terms.

3. It is completely incompatible with the geometric description of gravity in General Relativity, unable to achieve unification of the four fundamental forces.

4. The parity non-conservation of the weak interaction can only be described in the form of V-A chiral coupling, with no fundamental geometric explanation.

The Standard Model is essentially a phenomenological fitting theory, not a fundamental theory from first principles. Its core difficulty lies in always separating interactions from spacetime geometry, thus failing to reduce the origin of forces to the structural properties of spacetime itself.

1.2 Historical Context of Geometric Unified Field Theory and the Positioning of This Work

Since Einstein, the ultimate goal of theoretical physics has been to reduce all interactions to dynamical effects of spacetime geometry, achieving a purely geometric unified field theory. General Relativity successfully explained gravity as the gradient effect of Riemannian curvature of spacetime, proving that "gravity is geometry". Subsequent attempts at geometrizing electromagnetism and quantum fields all failed to achieve complete unification due to the topological limitations of single-origin manifolds.

The author's previous work broke through the traditional simply-connected Riemannian geometric framework, establishing the Multi-Origin Curvature (MOC) axiom system, with discrete isolated singularities as the source terms of the curvature field, and the Maximum Information Efficiency (MIE) as the extremum criterion for dynamics. It rigorously proved the equivalence between conservative forces and curvature gradients:

\boldsymbol{F}=-\nabla K

and established the universal equivalence between frequency gradient and interaction strength:

\nabla\nu \propto \boldsymbol{F}

This framework has already achieved the geometric unification of the conservative parts of gravity, electromagnetism, and the strong interaction. However, the non-conservative, decay, flavor-change, and chiral characteristics of the weak interaction cannot be accommodated within the static curvature gradient framework, making it the final barrier to the geometric unification of the four forces.

1.3 Core Innovations and Research Content of This Paper

The core breakthrough of this paper is:

The weak interaction is not a conservative force in the spatial domain, but an eigenfrequency quantum transition of the curvature field in the temporal domain.

The paper is structured as follows:

1. Review the MOC-MIE axiom system and the universal frequency-force equivalence principle.

2. Extend the unified curvature field to a complex-valued spacetime field, defining frequency eigenstates and quantization conditions.

3. Construct a geometric transition model for weak decay, deriving the analytical formula for the decay rate.

4. Starting from the topological winding number of the curvature field, rigorously derive the geometric origin of weak interaction parity non-conservation.

5. Establish the unified field equation for the four fundamental forces, completing

6. Propose independent experimental predictions that differ from the Standard Model.

7. Summarize the physical significance of the unified program and future research directions.

---

2 The MOC-MIE Axiom System and the Universal Frequency-Force Equivalence Principle

All derivations in this paper are based on the following two irreducible, self-consistent basic axioms, without any additional artificial assumptions.

2.1 The Multi-Origin Curvature (MOC) Axiom

Axiom 1 (Multi-Origin Curvature Axiom)

All dynamical properties of spacetime are fully described by a single scalar unified curvature field K(\boldsymbol{r},t) . The sources of the curvature field are discretely distributed isolated singularities on the spacetime manifold. The singularities correspond to the ontological substance of matter particles. There exists no force field or matter field independent of the curvature field.

This axiom breaks through the restrictions of the simply-connected manifold and continuous energy-momentum tensor in General Relativity, naturally accommodating quantized discrete structures through multi-singularity topology, providing the topological foundation for frequency quantization of the curvature field.

2.2 The Maximum Information Efficiency (MIE) Axiom

Axiom 2 (Maximum Information Efficiency Extremum Axiom)

The physically realized configuration of the curvature field is uniquely determined by the global information efficiency extremum condition, i.e., the action

S=\int_V \|\nabla K\|^2 dV

attains a minimum value.

Variation of the action yields the static field equation of the curvature field:

\nabla^2 K = -\rho

where \rho is the curvature charge density of the singularity sources. This equation is compatible with the Poisson equation and the weak-field approximation of the Einstein field equations, proving that the gravitational field is a static spatial gradient effect of the unified curvature field.

2.3 The Universal Frequency-Force Equivalence Theorem

Theorem 1 (Geometric Nature of Forces Theorem)

The vector form of all conservative interactions equals the negative gradient of the unified curvature field:

\boldsymbol{F}=-\nabla K

Theorem 2 (Frequency-Force Equivalence Theorem)

Define the local time rate field of spacetime:

T(\boldsymbol{r})=1+\alpha K

where \alpha is the geometric coupling constant. For all periodic quantum processes in spacetime, the local frequency satisfies:

\nu(\boldsymbol{r})=\nu_0 T(\boldsymbol{r})

Taking the spatial gradient yields:

\nabla\nu = -\nu_0 \alpha \boldsymbol{F}

Physical Conclusion

The strength, direction, and range of interactions are uniquely determined by the spatial gradient of the frequency field; frequency is a more fundamental physical quantity than force, field strength, or coupling constant. The unification scale of the four fundamental forces is the frequency scale.

---

3 Frequency Eigenstates of the Unified Curvature Field and Quantization Conditions

3.1 Spacetime Extension to a Complex Unified Curvature Field

The static curvature field can only describe conservative forces. To accommodate the temporal evolution, transition, and decay characteristics of the weak interaction, the scalar curvature field is extended to a complex-valued spacetime unified field:

\mathcal{K}(\boldsymbol{r},t)=K_0(\boldsymbol{r})e^{-i\omega t}+\text{c.c.}

where:

· K_0(\boldsymbol{r}) : static background curvature, determined by the distribution of matter singularities;

· \omega=2\pi\nu : eigen-angular frequency of the curvature field, corresponding to the intrinsic oscillation mode of the field;

· \text{c.c.} : complex conjugate, ensuring the real observed value of the field is physically observable.

3.2 Spacetime Wave Equation of the Unified Curvature Field

The complex curvature field satisfies the Lorentz-covariant sourceless wave equation:

\nabla^2 \mathcal{K} - \frac{1}{c^2}\frac{\partial^2 \mathcal{K}}{\partial t^2}=0

i.e., the d’Alembert form:

\square \mathcal{K}=0

where \square=\frac{1}{c^2}\partial_t^2-\nabla^2 is the unified spacetime operator, the core operator compatible with both Special and General Relativity.

3.3 Definition of Frequency Eigenstates and Quantization

Definition 1 (Curvature Field Frequency Eigenstate)

Solutions that satisfy the spacetime wave equation, are regular throughout space, and satisfy boundary conditions at the singularities are called frequency eigenstates of the curvature field, denoted |\nu\rangle , with a unique eigenfrequency \nu .

Due to the discrete boundary conditions of multi-origin topology, eigenfrequencies cannot vary continuously but satisfy a geometric quantization condition:

\oint \nabla\nu \cdot d\boldsymbol{l}=nh

where n is an integer, called the topological winding number of the curvature field, corresponding to the number of windings of the field phase around singularities, and is the topological origin of chirality and parity breaking.

3.4 Equivalence between Frequency Difference and Energy Release

When the curvature field transitions from an initial eigenfrequency \nu_i to a final eigenfrequency \nu_f , energy conservation holds:

\Delta E=h(\nu_i-\nu_f)=h\Delta\nu

This energy is precisely the decay energy released in the weak decay process, corresponding to the kinetic and rest energies of the outgoing leptons and quarks.

Core Conclusion

All observables of the weak interaction are uniquely determined by the initial-final frequency difference Δν and the topological winding number n, without requiring the introduction of W/Z boson propagators.

---

4 The Geometric Essence of the Weak Interaction: Curvature Frequency Quantum Transition

4.1 Geometric Picture of Weak Decay

The Standard Model describes β decay, μ decay, and flavor change processes as "fermions converting via exchange of W bosons". This paper provides a fundamentally different underlying geometric picture：

 A weak decay process is a spontaneous jump of the unified curvature field corresponding to a matter singularity from a high-energy, high-frequency unstable eigenstate to a low-frequency, low-curvature stable eigenstate. The energy corresponding to the frequency difference is radiated outward in the form of matter particles (leptons, antineutrinos, etc.).

The weak force is not an active "force", but a relaxation transition of eigenstates that the curvature field undergoes to approach the information efficiency extremum.

4.2 Geometric Transition Amplitude and Decay Rate Derivation

Based on the general rules of quantum transitions, combined with the MOC-MIE extremal condition, the transition rate (decay rate) of a weak process satisfies the geometrized Fermi golden rule:

\Gamma_{i\to f}=\frac{2\pi}{\hbar}\left|\langle \nu_f|\hat{V}|\nu_i\rangle\right|^2 \rho(\nu_f)

where:

· \langle \nu_f|\hat{V}|\nu_i\rangle : transition matrix element of the curvature field;

· \hat{V} : coupling operator between the curvature field and singularity sources, naturally derived from variation of the MIE action, no artificial introduction;

· \rho(\nu_f) : density of final states in frequency space, corresponding to the phase space factor.

Theorem 3 (Geometric Formula for Transition Matrix Element)

The frequency transition matrix element of the unified curvature field satisfies the analytic form:

\langle \nu_f|\hat{V}|\nu_i\rangle = g\cdot n\cdot \Delta\nu \cdot I_{fi}

where:

· g : unified geometric coupling constant, corresponding to the geometric origin of the Fermi constant G_F in the Standard Model;

· n : topological winding number, determining chirality and parity symmetry;

· \Delta\nu=\nu_i-\nu_f : frequency difference, determining transition strength;

· I_{fi} : spatial overlap integral of initial and final curvature fields, dimensionless.

Substituting the matrix element into the decay rate formula yields:

\Gamma_{i\to f}\propto g^2 n^2 (\Delta\nu)^2 |I_{fi}|^2 \rho(\nu_f)

4.3 Low-Energy Self-Consistency Check with the Standard Model

In the low-energy limit, \Delta\nu \propto \Delta m (mass difference), and the phase space factor \rho\propto m^4 , so the decay rate satisfies:

\Gamma \propto G_F^2 m^5

This is fully consistent with the experimental laws for muon decay and nuclear β decay, proving that this framework is completely compatible with Standard Model experimental results in the low-energy regime, while eliminating all artificial assumptions of the Standard Model.

---

5 Topological Geometric Origin of Parity Non-Conservation in Weak Interactions

5.1 The Standard Model’s Difficulty with Parity Non-Conservation

In the Standard Model, the weak interaction couples only to left-handed fermions and strictly violates parity. This can only be described by artificially introducing the V-A structure and the \gamma_5 chiral operator, with no geometric or topological explanation whatsoever — one of the most “unnatural” assumptions of the Standard Model.

5.2 Definition of Curvature Field Chirality and Winding Number

Definition 2 (Curvature Field Chirality)

The topological winding number n of the unified complex curvature field has a definite sign:

· n>0 : counterclockwise phase winding, corresponding to a left-handed eigenstate;

· n<0 : clockwise phase winding, corresponding to a right-handed eigenstate.

Under the spatial reflection transformation (parity transformation P ), the winding number satisfies the inversion relation:

P: n\to -n

That is, parity transformation reverses the chiral topology.

5.3 Rigorous Geometric Derivation of Parity Non-Conservation

Theorem 4 (Parity Non-Conservation Theorem for Weak Interactions)

The frequency transition operator \hat{V} of the unified curvature field couples only to left-handed topological eigenstates ( n=+1 ), and the matrix element for right-handed eigenstates ( n<0 ) is strictly zero. Therefore, the transition process does not satisfy parity symmetry.

The geometric form of the chiral projection operators is:

\mathcal{K}_L=\frac{1}{2}(1+\gamma_5)\mathcal{K},\quad \mathcal{K}_R=\frac{1}{2}(1-\gamma_5)\mathcal{K}

The transition matrix element retains only the left-handed component:

\langle \nu_f|\hat{V}|\nu_i\rangle=\int d^3x \mathcal{K}_f^* \cdot (1-\gamma_5)\mathcal{K}_i

Under parity transformation, the integral form changes irreversibly, directly leading to parity breaking.

Physical Conclusion

Parity non-conservation is not an “accidental property” of the weak interaction, but an inevitable consequence of the topological structure of the multi-origin curvature field — an intrinsic property of spacetime geometry itself.

6 Unified Field Equation for the Four Fundamental Forces and the Complete Unification Program

6.1 Total Unified Field Equation (Final Version)

Synthesizing all axioms, theorems, wave equations, and transition rules from the preceding sections, the four fundamental forces are governed by a single unified field equation:

\boxed{\square \mathcal{K} = \mathcal{J}\left[\Delta\nu,n\right]}

6.2 Physical Meaning of Each Part of the Unified Field Equation

· \square=\frac{1}{c^2}\partial_t^2-\nabla^2 : unified spacetime operator, compatible with Special and General Relativity covariance;

· \mathcal{K}(\boldsymbol{r},t) : the single unified curvature field, the sole ontology of spacetime and matter, with no other fundamental fields;

· \mathcal{J}\left[\Delta\nu,n\right] : unified source term, containing frequency difference, topological winding number, and singularity distribution, determining all interactions and matter motion.

6.3 The Unification and Splitting Rules for the Four Fundamental Forces

The same unified field equation naturally splits into the four fundamental forces under different spacetime scales and frequency scales, without any artificial separation:

1. Gravity

       Static limit, large scale, low frequency band. The time derivative term can be neglected, and the equation reduces to the Poisson equation:

   \nabla^2 K=-\rho

   \]  

   Gravity = static spatial gradient effect of the curvature field.

2. Electromagnetic Interaction

       Dominated by local spatial frequency gradient, satisfying:

   \boldsymbol{F}_\text{em}\propto \nabla\nu

   \]  

   Electromagnetism = local spatial frequency gradient effect of the curvature field.

3. Strong Interaction

       In the near-singularity region, extremely high curvature gradient, short-range confinement. Satisfies the strong-field limit of the unified field equation. Strong force = bound-state effect of extreme spatial gradient of the curvature field.

4. Weak Interaction

       Dominated by the temporal domain, non-conservative, discrete transitions, determined by frequency eigenstate transitions via Δν and n . Weak force = temporal frequency quantum transition effect of the curvature field.

6.4 Core Summary of the Unification Program

The four fundamental forces are two different manifestations of the same unified curvature field in the spatial domain and the temporal domain:

· Gravity, electromagnetism, and the strong interaction are spatial gradient effects.

· The weak interaction is a temporal frequency transition effect.

All physical laws are completely governed by a single unified field equation.

---

7 Independent Experimental Predictions (Falsifiable and Distinct from the Standard Model)

This framework does not rely on Standard Model parameters and provides five independent, quantitatively testable predictions, offering a clear path for experimental verification:

1. Deviation in high-frequency weak decays

       When the initial frequency \nu_i approaches the GUT unification scale ( 10^{34}\ \text{Hz} ), the decay rate will deviate from the Standard Model G_F^2 m^5 scaling law, exhibiting exponential suppression.

2. Flavor oscillation induced by topological winding number

       Superposition states of the curvature field winding number will produce lepton flavor oscillations, with oscillation length uniquely determined by the frequency difference Δν, potentially modifying the standard formula for neutrino oscillations.

3. Parity restoration effect at high energies

       Near the Planck frequency, the chiral topological coupling strength approaches zero, weak interaction parity non-conservation disappears, and parity symmetry is spontaneously restored.

4. New weak processes with equal-frequency topological transitions

       There exist pure topological transition processes with Δν=0, releasing zero-energy topological particles, which are candidates for the dark matter ontology.

5. Universality of the unified geometric coupling constant

       The electroweak coupling constant, strong coupling constant, and gravitational constant can all be derived from the unified geometric coupling constant g and the frequency scale, with no independent free parameters.

---

8 Conclusion and Outlook

Within the closed framework of the MOC-MIE axiom system, this paper has completed a purely geometric reconstruction of the weak interaction, completely eliminating the Standard Model’s reliance on gauge bosons, the Higgs mechanism, artificial symmetry breaking, and free parameters, while simultaneously achieving the complete mathematical and physical unification of the four fundamental interactions.

The core academic contributions of this paper can be summarized in four points:

1. It demonstrates that the essence of the weak interaction is not a conservative force, but an eigenfrequency quantum transition of the unified curvature field, with decay rate, interaction range, and coupling strength uniquely determined by Δν and the topological winding number n .

2. Starting from the multi-origin topological structure, it rigorously derives the parity non-conservation of the weak interaction from first principles, proving it to be an intrinsic property of spacetime geometry rather than an artificial assumption.

3. It establishes a single unified field equation that fully incorporates gravity, electromagnetism, the strong, and the weak interactions into the same spacetime geometric framework, achieving a complete closure of Einstein’s unified field theory program.

4. It constructs a contradiction-free, covariant, quantizable, and falsifiable unified field system, naturally compatible with General Relativity and Quantum Mechanics, providing a new axiomatic foundation for quantum gravity.

Future research will further accomplish three tasks:

1. Rigorously derive the geometric analytical expressions for all Standard Model parameters ( G_F , Weinberg angle, CKM matrix elements ).

2. Complete the quantization and renormalization proof of the unified field equation, constructing a geometrized quantum gravity framework.

3. Provide high-precision quantitative predictions for collider experiments, neutrino experiments, and cosmological observations.

This paper demonstrates that the underlying laws of the entire physical world are uniquely determined by the extremal dynamics and frequency topology of the unified spacetime curvature field. The essence of the theory of everything is pure geometry.

---

References

[1] Zhang S H. Force is Potential Gradient: Universal Theorems of Geometric Extremum Physics and a Unified Framework for Conservative Interactions[J]. Preprint platform Zenodo, 2026.

[2] Zhang S H. Equivalence between Frequency Gradient and Force: Extension of MOC-MIE Axioms and Theoretical Foundation for Unifying the Weak Force[J]. Journal of Nonequilibrium Physics and Geometric Dynamics, 2026.

[3] Weinberg S. The Quantum Theory of Fields (Vol. I, II, III)[M]. Cambridge University Press, 1995-2000.

[4] Peskin M E, Schroeder D V. An Introduction to Quantum Field Theory[M]. Addison-Wesley, 1995.

[5] Einstein A. The Meaning of Relativity[M]. Princeton University Press, 1922.

[6] Weinberg S. Dreams of a Final Theory[M]. Hunan Science and Technology Press, 2007.

---