Geometric Origin of Weak Interactions: From Curvature Frequency Transitions to a Unified Framework

 

Author: Zhang Suhang, Luoyang

 

Abstract

 

Based on the previously established MOC-MIE axiomatic system and the equivalence relation between frequency gradient and force, this paper proposes a novel geometric interpretation of weak interactions: the weak force is not a conservative force in the conventional sense, but a quantized local frequency transition of the curvature field K. Decay processes correspond to discrete jumps of the curvature field from one eigenfrequency state to another, with transition probabilities determined by the frequency difference and the topological charge (winding number) of the curvature field. Parity violation arises from the chiral nature of the complex phase of the curvature field. Starting strictly from the MOC-MIE axioms, this paper defines frequency eigenmodes of the curvature field, derives a geometric expression for decay rates, and compares the results with those of the Standard Model in the low-energy limit. This framework dispenses with gauge bosons and the Higgs mechanism, incorporates weak interactions into the unified description of geometric extremum physics, and provides the final puzzle piece for the complete geometric unification of the four fundamental interactions.

 

Keywords: Weak interaction; Curvature frequency transition; Parity violation; MOC-MIE; Geometric unification

 

1. Introduction

 

The Standard Model successfully describes electromagnetic and weak forces via the SU(2)\times U(1) gauge symmetry, yet it relies on a large set of free parameters (boson masses, mixing angles, CKM matrix elements, etc.) and fails to unify gravity. In prior work by the author, built upon the Multi-Origin Curvature (MOC) and Maximum Information Efficiency (MIE) axioms, it was proven that conservative forces equal the negative gradient of the curvature field (\boldsymbol{F}=-\nabla K), and an equivalence relation between frequency gradient and force (\nabla\nu \propto \boldsymbol{F}) was established. Nevertheless, the non-conservative decay character of the weak force cannot be subsumed within this conservative framework.

 

This paper puts forward the core claim: weak interactions are not forces, but quantized frequency transitions of the curvature field over the temporal domain. Specifically:

 

- At microscopic scales, the curvature field K(\boldsymbol{r},t) possesses intrinsic oscillatory modes, whose frequency \nu corresponds to the eigenenergy of field configurations.

- Weak decay corresponds to transitions of the curvature field from one eigenfrequency state to another with lower eigenfrequency; the energy released is emitted in the form of particles (leptons, quarks).

- Transition probabilities are governed by the frequency difference \Delta\nu and the topological charge (winding number) of the curvature field. This topological charge intrinsically introduces chirality, leading to parity violation.

 

The structure of this paper is organized 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 violation. Section 6 presents testable predictions. Section 7 concludes.

 

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

 

This section provides a concise summary of prior derivations, with full proofs available in preceding manuscripts.

 

- Axiom 1 (MOC): Physical space is uniquely described by the scalar curvature field K(\boldsymbol{r}), with sources taking the form of isolated point singularities.

- Axiom 2 (MIE): Physical realizable fields minimize the action S=\int\|\nabla K\|^2 dV, yielding the Poisson equation \nabla^2 K = -\rho.

- Theorem 1 (Force as potential difference): \boldsymbol{F} = -\nabla K.

- Theorem 2 (Equivalence of frequency gradient and force): Introduce a temporal elapse rate field T(\boldsymbol{r}) = 1 + \alpha K. For periodic processes with fundamental frequency \nu_0, \nu(\boldsymbol{r}) = \nu_0 T(\boldsymbol{r}), which gives \nabla\nu = -\nu_0\alpha \boldsymbol{F}.

 

This relation demonstrates that spatial inhomogeneities of frequency directly encode the magnitude and direction of forces, furnishing the geometric foundation for modeling weak interactions as frequency transitions.

 

3. Frequency Eigenstates of the Curvature Field

 

To formulate quantized transitions, we equip the curvature field with an intrinsic temporal degree of freedom and extend it to a complex-valued function:

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

where K_0(\boldsymbol{r}) denotes the static background curvature generated by masses, charges, and other sources, and \omega = 2\pi\nu is the oscillation angular frequency.

 

3.1 Topological Quantization Condition

 

Unlike continuous spectra of free wave equations, topological constraints imposed by isolated singularities under the MOC axiom discretize field frequencies. For any closed contour \gamma encircling an isolated singularity, the phase variation of the curvature field satisfies a topological quantization rule:

 

Axiom 3 (Topological Quantization)

\oint_{\gamma} \nabla \omega \cdot d\boldsymbol{l} = n \cdot \Omega_0, \quad n \in \mathbb{Z}

Here \Omega_0 is a universal topological frequency quantum determined by the global topology of MOC space, \Omega_0 = 2\pi\nu_0, and the integer n is defined as the topological winding number.

 

This condition enforces single-valuedness of the complex curvature field in the neighborhood of each singularity:

\mathcal{K}(r e^{i(\theta+2\pi)}) = e^{i 2\pi n} \mathcal{K}(r e^{i\theta})

 

Combining the wave equation \nabla^2 \mathcal{K} = \frac{1}{c^2}\partial_t^2 \mathcal{K} with the topological boundary condition, angular frequencies are discretized as:

\omega_n = \frac{|n|}{R} c + \omega_0

where R is an effective scale determined by the distribution of singularities, and \omega_0 is the ground-state frequency corresponding to the unwound state n=0. Distinct winding numbers n label discrete eigenfrequency states |\nu_n\rangle with eigenfrequencies:

\nu_n = |n|\nu_c + \nu_0, \quad \nu_c = \frac{c}{2\pi R}

 

Definition (Frequency Eigenstate)

An eigenstate |\nu_n\rangle of the curvature field is a regular solution satisfying both the wave equation and the topological quantization condition, labeled by integer winding number n\in\mathbb{Z} with eigenfrequency \nu_n = |n|\nu_c + \nu_0. In the static limit, the dominant mode K_0(\boldsymbol{r}) reduces to the Poisson equation \nabla^2 K_0 = -\rho, consistent with the MIE axiom.

 

In weak interaction processes, the initial state carries winding number n_i and frequency \nu_i = |n_i|\nu_c + \nu_0, while the final state carries n_f with \nu_f = |n_f|\nu_c + \nu_0, with |n_i| > |n_f| such that \nu_i > \nu_f. The frequency difference is:

\Delta\nu = \nu_i - \nu_f = \left(|n_i|-|n_f|\right)\nu_c

and is proportional to the energy released during decay:

\Delta E = h\Delta\nu = h\left(|n_i|-|n_f|\right)\nu_c

The Planck constant h emerges as a universal proportionality factor linking geometric quantization to energy.

 

4. Weak Decays as Frequency Transitions

 

Weak decay events are modeled as transitions of the curvature field from initial eigenstate |\nu_i\rangle to final eigenstate |\nu_f\rangle, accompanied by the production of leptons or quarks interpreted as excitation modes of curvature field singularities.

 

4.1 Perturbative Expansion of the Geometric Action

 

From the MIE axiom, the total action reads:

S_{\text{total}} = \int \|\nabla\mathcal{K}\|^2 dV dt + S_{\text{source}}

where the source term S_{\text{source}} describes coupling between the curvature field and matter singularities. Decompose the curvature field into a static background component \mathcal{K}_0 and oscillatory perturbation \delta\mathcal{K}:

\mathcal{K}(\boldsymbol{r},t) = \mathcal{K}_0(\boldsymbol{r}) + \delta\mathcal{K}(\boldsymbol{r},t)

Variation \delta S_{\text{total}}/\delta\mathcal{K} = 0 yields a linearized field equation. Transition processes arise from cross terms within the perturbative interaction action:

S_{\text{int}} = \int \left[ \nabla\mathcal{K}_0^* \cdot \nabla\delta\mathcal{K} + \nabla\delta\mathcal{K}^* \cdot \nabla\mathcal{K}_0 \right] dV dt

Time integration permits definition of a transition operator. Within the MOC framework, the topological charge density n(\boldsymbol{r}) is the local geometric quantity characterizing phase winding of the curvature field, and the transition operator naturally couples to this topological density:

\hat{V} = g\, n(\boldsymbol{r})

Here g denotes the geometric coupling constant. Physically, the strength of weak interactions at each spatial point is locally determined by the topological charge density; regions with vanishing topological charge density cannot support weak decays.

 

4.2 Rigorous Evaluation of Transition Matrix Elements

 

Transition probabilities follow a geometric counterpart of Fermi’s golden rule:

\Gamma_{i\to f} = \frac{2\pi}{\hbar} \left|\mathcal{M}_{fi}\right|^2 \rho(\nu_f)

where \rho(\nu_f) is the final-state frequency density determined by phase space, and the transition matrix element is defined as:

\mathcal{M}_{fi} \equiv \langle \nu_f | \hat{V} | \nu_i \rangle = g \int d^3r \, \mathcal{K}_f^*(\boldsymbol{r}) \, n(\boldsymbol{r}) \, \mathcal{K}_i(\boldsymbol{r})

\mathcal{K}_i, \mathcal{K}_f are eigenfunctions of the complex curvature field for initial and final states respectively. The topological charge density n(\boldsymbol{r}) is defined as:

n(\boldsymbol{r}) = \frac{1}{4\pi} \nabla \times \boldsymbol{A}(\boldsymbol{r})

with \boldsymbol{A} = \nabla \theta and \theta = \arg \mathcal{K}, the complex phase of the curvature field.

 

From the topological quantization rules derived in Section 3, curvature field eigenmodes separate into radial and angular phase components:

\mathcal{K}_n(\boldsymbol{r}) = \mathcal{R}_n(r) e^{i n \theta}

where n equals the topological winding number. Substitution into the matrix element gives:

\mathcal{M}_{fi} = g \int \mathcal{R}_{n_f}(r) e^{-i n_f \theta} \, n(\boldsymbol{r}) \, \mathcal{R}_{n_i}(r) e^{i n_i \theta} \, d^3r

Topological charge density n(\boldsymbol{r}) localizes near singularities for contours encircling isolated point sources. Angular integration produces a selection rule:

\int_0^{2\pi} e^{i(n_i - n_f)\theta} d\theta = 2\pi \delta_{n_i, n_f}

This implies conservation of total topological winding number: n_i = n_f, consistent with the Standard Model observation that weak interactions alter flavor quantum numbers while preserving total chiral charge.

 

Frequency differences originate exclusively from changes to radial modes. Define the radial overlap integral:

I_{fi} \equiv \int \mathcal{R}_{n_f}^*(r) \, n(r) \, \mathcal{R}_{n_i}(r) \, r^2 dr

The matrix element simplifies to:

\mathcal{M}_{fi} = g \cdot n \cdot I_{fi} \cdot \delta_{n_i, n_f}

where n = n_i = n_f denotes the shared topological winding number. The radial overlap integral scales linearly with the radial frequency difference:

I_{fi} \propto (m_i - m_f)\nu_r

with m_i, m_f radial principal quantum numbers and \nu_r the radial quantization frequency. This yields the compact matrix element expression:

\mathcal{M}_{fi} = g\, n\, \Delta\nu \cdot I_{fi}

where \Delta\nu = (m_i - m_f)\nu_r and I_{fi} is a dimensionless normalized overlap integral.

 

The full decay rate then takes the form:

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

This expression matches the low-energy functional form of weak decay rates within the Standard Model (e.g., muon decay \propto G_F^2 m_\mu^5), where \Delta\nu \propto \Delta m (mass difference) and \rho corresponds to phase-space factors.

5. Geometric Origin of Parity Violation

 

The defining feature of weak interactions within the Standard Model is parity violation via the V-A structure. In the present geometric framework, parity violation emerges intrinsically from the complex phase chirality of the curvature field. This section presents a complete, self-contained proof independent of \gamma_5 matrices or spinor formalism, relying solely on geometric objects native to the MOC-MIE framework.

 

5.1 Transformation of Topological Charge Density under Spatial Inversion

 

Define the parity operator \hat{P}: \boldsymbol{r} \to -\boldsymbol{r}.

The phase gradient field \boldsymbol{A}(\boldsymbol{r}) = \nabla \theta(\boldsymbol{r}) transforms as:

\hat{P} \boldsymbol{A}(\boldsymbol{r}) = -\boldsymbol{A}(-\boldsymbol{r})

The topological charge density reads n(\boldsymbol{r}) = \frac{1}{4\pi} \nabla \times \boldsymbol{A}(\boldsymbol{r}). Its parity transform is:

\hat{P} n(\boldsymbol{r}) = \frac{1}{4\pi} \nabla \times \left[-\boldsymbol{A}(-\boldsymbol{r})\right] = -n(-\boldsymbol{r})

Conclusion: n(\boldsymbol{r}) is a pseudoscalar density, changing sign under spatial inversion.

 

5.2 Transformation of the Complex Curvature Field under Spatial Inversion

 

The complex curvature field is \mathcal{K}(\boldsymbol{r}) = |\mathcal{K}(\boldsymbol{r})| e^{i\theta(\boldsymbol{r})}. Spatial inversion flips the complex phase \theta \to -\theta as inversion exchanges left-handed and right-handed chiral modes, giving:

\hat{P} \mathcal{K}(\boldsymbol{r}) = \mathcal{K}^*(-\boldsymbol{r})

 

5.3 Parity Violation Theorem

 

Apply the parity operator to the transition matrix element defined in Section 4:

\mathcal{M}_{fi} = g \int d^3r \, \mathcal{K}_f^*(\boldsymbol{r}) \, n(\boldsymbol{r}) \, \mathcal{K}_i(\boldsymbol{r})

\hat{P}\mathcal{M}_{fi} = g \int d^3r \, \hat{P}\mathcal{K}_f^*(\boldsymbol{r}) \, \hat{P}n(\boldsymbol{r}) \, \hat{P}\mathcal{K}_i(\boldsymbol{r})

Substitute the derived transformation rules:

\hat{P}\mathcal{K}_f^*(\boldsymbol{r}) = \mathcal{K}_f(-\boldsymbol{r})

\hat{P}n(\boldsymbol{r}) = -n(-\boldsymbol{r})

\hat{P}\mathcal{K}_i(\boldsymbol{r}) = \mathcal{K}_i^*(-\boldsymbol{r})

\hat{P}\mathcal{M}_{fi} = -g \int d^3r \, \mathcal{K}_f(-\boldsymbol{r}) \, n(-\boldsymbol{r}) \, \mathcal{K}_i^*(-\boldsymbol{r})

Execute coordinate substitution \boldsymbol{r} \to -\boldsymbol{r} over the integration domain:

\hat{P}\mathcal{M}_{fi} = -g \int d^3r \, \mathcal{K}_f(\boldsymbol{r}) \, n(\boldsymbol{r}) \, \mathcal{K}_i^*(\boldsymbol{r})

Observe that the complex conjugate of the matrix element is:

\mathcal{M}_{fi}^* = g \int d^3r \, \mathcal{K}_f(\boldsymbol{r}) \, n(\boldsymbol{r}) \, \mathcal{K}_i^*(\boldsymbol{r})

This yields the core invariant relation:

\boxed{\hat{P}\mathcal{M}_{fi} = -\mathcal{M}_{fi}^*}

This is the precise mathematical statement of parity violation: spatial inversion maps the transition matrix element to minus its complex conjugate. If \mathcal{M}_{fi} is non-zero and not purely imaginary, parity violation necessarily holds.

 

5.4 Corollary

 

For most physical configurations where the matrix element takes real values:

\hat{P}\mathcal{M}_{fi} = -\mathcal{M}_{fi}

The matrix element flips sign under inversion, establishing strict parity violation.

Physical origin: the pseudoscalar nature of n(\boldsymbol{r}) induces a sign flip of the full matrix element under spatial reflection. Parity violation is a necessary consequence of the topological charge’s pseudoscalar geometric character.

 

5.5 Topological Selection Rules

 

Additional selection rules follow directly from the proof above:

 

- Modes with odd winding number n carry phase sign flips under inversion, yield non-zero parity-violating matrix elements, and participate in weak interactions.

​

- Modes with even winding number n maintain invariant phase under inversion, conserve parity, and decouple from weak decay processes.

 

This reproduces the V-A structure of the Standard Model in geometric terms: only fermions of a fixed chiral projection couple to weak interactions. Mixing transitions between different charge flavor states correspond to superpositions of topological winding number combinations, whose mixing matrix is determined by topological overlap integrals and naturally recovers structures analogous to the CKM matrix.

 

6. Testable Predictions

 

To distinguish the present framework from the Standard Model, four distinct observational effects are proposed:

 

1. Suppression of high-frequency weak decays: When the initial curvature field frequency \nu_i exceeds a Planck-scale threshold, transition probabilities deviate from linear Standard Model behavior, testable via ultra-high-energy neutrino scattering experiments (DUNE, Hyper-Kamiokande).

​

2. Topological charge oscillations: Superpositions of two distinct chiral topological charge configurations generate oscillation phenomena analogous to neutrino oscillations, with oscillation lengths controlled by frequency differences.

​

3. Energy-scale dependence of parity violation: At ultra-high collision energies, chiral coupling of the curvature field weakens, leading to parity restoration, searchable in future collider measurements of weak boson scattering.

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4. Massless new excitations: Equal-frequency topological transitions (\Delta\nu = 0) produce energyless geometric excitations, viable candidates for dark matter or sterile neutrino species.

 

7. Conclusion

 

Building upon the MOC-MIE axiomatic system, this paper reinterprets weak interactions as quantized frequency transitions of the curvature field. The primary results are summarized as follows:

 

1. Discrete frequency eigenstates of the curvature field are rigorously defined via the topological quantization axiom, resolving the continuous spectrum problem.

​

2. Starting from variational minimization of the MIE action, a transition operator coupled to topological charge density \hat{V} = g\, n(\boldsymbol{r}) is constructed. Radial mode analysis yields a geometric expression for decay rates.

​

3. The parity operator \hat{P} is applied to derive the exact relation \hat{P}\mathcal{M}_{fi} = -\mathcal{M}_{fi}^*, proving parity violation as an inevitable consequence of the pseudoscalar topological charge density.

​

4. Topological selection rules are derived: odd winding number modes couple to weak interactions while even modes decouple, naturally reproducing the V-A chiral structure.

 

This framework eliminates the need for gauge bosons and the Higgs mechanism, embedding weak interactions within the unified formalism of geometric extremum physics. Combined with prior work unifying gravity, electromagnetism, and static strong nuclear conservative forces, this manuscript completes the full geometric unification program for the four fundamental interactions. Follow-up work will focus on quantitative computation of Standard Model parameters (Fermi constant, mixing angles) in terms of topological quantities of the curvature field.

 

References

 

[1] Author. Force as Potential Difference: A Universal Theorem of Geometric Extremum Physics and a Unified Framework for Conservative Interactions, 2026.

[2] Author. Geometric Equivalence of Frequency Gradient and Force: Extension of MOC-MIE and Implications for the Unification of Weak Forces, 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.