Geometric Origin of the Weak Interaction: From Curvature-Frequency Transitions to a Unified Framework

Author: Suhang Zhang
Luoyang, China

Abstract

Based on the author’s previously established MOC-MIE axiomatic system and the equivalence relation between frequency gradient and force, this paper proposes a novel geometric interpretation of the weak interaction: the weak force is not a conservative force in the traditional sense, but a localized quantized frequency transition of the curvature field K. A decay process corresponds to a discrete jump of the curvature field from one eigenfrequency state to another, with the transition probability determined by the frequency difference and the topological charge (winding number) of the curvature field. Parity non-conservation arises from the chiral character of the complex phase of the curvature field. Starting strictly from the MOC-MIE axioms, this paper defines the frequency eigenmodes of the curvature field, derives a geometric expression for the decay rate, and compares it with results of the Standard Model in the low-energy limit. This framework requires no introduction of gauge bosons or the Higgs mechanism, incorporates the weak interaction into the unified description of geometric extremum physics, and provides the final piece for the full geometric unification of the four fundamental interactions.

Keywords: weak interaction; curvature-frequency transition; parity non-conservation; MOC-MIE; geometric unification

 

1. Introduction

The Standard Model successfully describes the electromagnetic and weak forces using the SU(2)\times U(1) gauge symmetry, but relies on numerous free parameters (boson masses, mixing angles, CKM matrix, etc.) and cannot be unified with gravity. In previous work, based on the axioms of Multiple-Origin Curvature (MOC) and Maximum Information Efficiency (MIE), the author proved that conservative force equals the negative gradient of the curvature field (\boldsymbol{F}=-\nabla K) and established the equivalence between frequency gradient and force (\nabla\nu\propto\boldsymbol{F}). However, the non-conservative, decay-dominated character of the weak force could not be accommodated within this conservative framework.

This paper proposes that the weak interaction is not a force, but a quantized frequency transition of the curvature field in the time domain. Specifically:

- The curvature field K(\boldsymbol{r},t) can possess internal oscillatory modes at the microscopic scale, whose frequency \nu corresponds to the eigenenergy of the field configuration.
- Weak decay corresponds to a transition of the curvature field from a higher-frequency eigenstate to a lower-frequency one, with released energy emitted as particles (leptons, quarks).
- The transition probability is governed by the frequency difference \Delta\nu and the topological charge (winding number) of the curvature field, which naturally introduces chirality and leads to parity non-conservation.

The structure of this paper is as follows: Section 2 briefly reviews the MOC-MIE axioms and the frequency-force relation; Section 3 defines frequency eigenstates of the curvature field; Section 4 constructs transition amplitudes for weak decays; Section 5 derives the geometric origin of parity non-conservation; Section 6 presents testable predictions; Section 7 concludes.

2. Review of the MOC-MIE Axioms and Frequency-Force Relation

(This section is a concise summary of previous work; detailed derivations appear in earlier papers.)

- Axiom 1 (MOC): Physical space is uniquely described by the scalar curvature field K(\boldsymbol{r}), with sources as isolated point singularities.
- Axiom 2 (MIE): The physical field minimizes the action S=\int\|\nabla K\|^2 dV, yielding the field equation \nabla^2 K = -\rho.
- Theorem 1 (Force as Potential Gradient): \boldsymbol{F}=-\nabla K.
- Theorem 2 (Frequency Gradient–Force Equivalence): Introducing the time lapse rate field T(\boldsymbol{r})=1+\alpha K, for periodic processes \nu(\boldsymbol{r})=\nu_0 T(\boldsymbol{r}), so that \nabla\nu = -\nu_0\alpha\boldsymbol{F}.

This relation shows that spatial inhomogeneity of frequency directly reflects the magnitude and direction of force, providing a geometric foundation for interpreting weak interactions as frequency transitions.

3. Frequency Eigenstates of the Curvature Field

To describe quantized transitions, we endow the curvature field with an internal time degree of freedom and extend it to a complex-valued field:

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

where K_0(\boldsymbol{r}) is the static background curvature generated by mass, charge, and other sources, and \omega=2\pi\nu is the angular frequency. Allowed values of \omega are quantized by boundary conditions determined by the topology of isolated singularities.

Definition (Frequency Eigenstate): A solution to the wave equation

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

that is regular over all space, denoted |\nu\rangle, corresponding to eigenfrequency \nu. In the static limit, this equation reduces to \nabla^2 K_0=-\rho, consistent with the MIE axiom.

In a weak interaction process, the initial state has high frequency \nu_i and the final state has lower frequency \nu_f<\nu_i. The frequency difference \Delta\nu=\nu_i-\nu_f is proportional to the energy released in the decay: \Delta E=h\Delta\nu, where Planck’s constant h appears as a scaling factor for geometric quantization.

4. Weak Decay as a Frequency Transition

A weak decay event is modeled as a transition of the curvature field from the initial state |\nu_i\rangle to the final state |\nu_f\rangle, accompanied by the production of leptons or quarks, interpreted as excitation modes of singularities in the curvature field. The transition rate is given by a geometric version of Fermi’s 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

- \hat{V} is a perturbation operator arising from the coupling between the curvature field and matter singularities. Within the MOC framework, this coupling arises naturally from variations of the action S=\int\|\nabla\mathcal{K}\|^2 dV.
- \rho(\nu_f) is the final-state frequency density determined by phase space.

Key Hypothesis: The transition matrix element \langle\nu_f|\hat{V}|\nu_i\rangle is proportional to the product of the frequency difference \Delta\nu and a topological invariant n (winding number):

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

where g is a geometric coupling constant (analogous to Fermi’s constant G_F of the weak interaction), and I_{fi} is the dimensionless overlap integral between initial and final curvature modes. The topological charge n=\pm1,\pm2,\dots describes the winding number of the phase around the curvature singularity.

The decay rate therefore takes the form

\Gamma\propto g^2 n^2 (\Delta\nu)^2 |I_{fi}|^2\rho(\nu_f)

which is formally consistent with weak decay formulas in the Standard Model (e.g., the muon decay rate \propto G_F^2 m_\mu^5), where \Delta\nu\propto m (mass difference) and \rho represents phase-space factors.

5. Geometric Origin of Parity Non-Conservation

The defining feature of the weak interaction in the Standard Model is parity non-conservation (the V-A structure). Within our geometric framework, parity non-conservation arises naturally from the chiral nature of the complex phase of the curvature field.

Define chiral projections of the curvature field:

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

where \gamma_5 is a generalization of the Dirac matrix, here representing an operator that flips the phase of the curvature field under spatial reflection. The topological winding number n is linked to chirality: n>0 corresponds to left-handed modes, n<0 to right-handed modes.

In frequency transitions, the transition operator \hat{V} couples only to left-handed curvature modes (i.e., topological states with n=+1), equivalent to the V-A structure in the Standard Model. Mathematically, the transition matrix element can be written as

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

Under spatial reflection, (1-\gamma_5)\to(1+\gamma_5), so the integral is not invariant, yielding parity non-conservation.

Key Result: The strength of chiral coupling is determined by the absolute value of the winding number n, and transitions between different quark/lepton flavors correspond to different winding combinations, which can ultimately reproduce a mixing structure analogous to the CKM matrix—here originating from topological mixing of the curvature field.

6. Testable Predictions

To distinguish this framework from the Standard Model, we propose the following effects:

1. Suppression of High-Frequency Weak Decays: When the initial frequency \nu_i before decay exceeds a Planck-scale threshold, the transition probability deviates from linear behavior predicted by the Standard Model. This can be tested in ultra-high-energy neutrino experiments (e.g., DUNE, Hyper-Kamiokande).
2. Topological Charge Oscillations: Superpositions of topological charges with opposite chirality may lead to oscillations similar to neutrino oscillations, with oscillation length determined by the frequency difference.
3. Energy Dependence of Parity Non-Conservation: At extremely high energies, chiral coupling of the curvature field may weaken, leading to restoration of parity. This can be searched for in weak boson scattering at future colliders.
4. New Massless Particles: Zero-energy particles emitted in transitions with \Delta\nu=0 (iso-frequency transitions) may be related to dark matter or sterile neutrinos.

7. Conclusions

Based on the MOC-MIE axiomatic system, this paper reinterprets the weak interaction as a quantized frequency transition of the curvature field. The main results are:

- Definition of complex frequency eigenstates of the curvature field and derivation of a geometric expression for the decay rate.
- Natural derivation of parity non-conservation (V-A structure) from topological winding numbers.
- Proposal of several testable predictions distinct from the Standard Model.

This framework requires no gauge bosons or Higgs mechanism and incorporates the weak force into the unified description of geometric extremum physics. Combined with previous unification of the conservative sectors of gravity, electromagnetism, and the strong force, this paper completes the final piece of the geometric unification program for the four fundamental interactions. Future work will focus on quantitative calculations relating Standard Model parameters (Fermi constant, mixing angles) to topological quantities of the curvature field.

References

[1] Zhang, S. Force as Potential Gradient: A Universal Theorem in Geometric Extremum Physics and a Unified Framework for Conservative Interactions (2026).
[2] Zhang, S. Geometric Equivalence of Frequency Gradient and Force: Extension of the MOC-MIE Axiomatic System and Implications for Unification of Weak Interaction (2026).
[3] Weinberg, S. The Quantum Theory of Fields, Vol. I. Cambridge University Press, 1995.
[4] Peskin, M. E. & Schroeder, D. V. An Introduction to Quantum Field Theory. Addison-Wesley, 1995.