Discrete Order Geometry (DOG) Framework for the Geometric Origin of Weak Interactions: From Discrete Chiral Lattice Points to the V–A Current

Author: Zhang Suhang
(Luoyang, Independent Researcher)

Abstract: Based on the four‑dimensional discrete spacetime and spinor order states of Discrete Order Geometry (DOG), this paper constructs a discrete geometric model for weak interactions. By defining short‑range nearest‑neighbor coupling, a chiral projection matrix, and a discrete weak‑interaction evolution equation, we derive from first principles the characteristic properties of weak forces: short range, chiral coupling preference, parity violation, and decay behavior. In the continuum limit, the discrete evolution equation naturally recovers the standard V–A vector‑axial vector current form. We give an explicit construction of the weak‑interaction evolution matrix, reveal the discrete geometric origin of the chiral projection operators, and predict Planck‑scale discrete corrections. This work incorporates weak interactions into the unified DOG framework, completing a full derivation from discrete geometry to the effective theory of weak interactions.

Keywords: Discrete Order Geometry; weak interaction; chirality; parity violation; V–A current; discrete evolution

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1. Introduction

The short range, chiral coupling (only left‑handed particles participate), and parity violation of weak interactions are central features of the Standard Model of particle physics. However, these properties are usually taken as postulates without a geometric origin. Discrete Order Geometry (DOG) views physical spacetime as a four‑dimensional discrete lattice, with all dynamics governed by the discrete evolution equation \boldsymbol{\Psi}_{n+1}=M\boldsymbol{\Psi}_n. The aim of this paper is to show that all special properties of weak interactions emerge naturally from the discrete geometric setting of DOG, without extra assumptions.

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2. Review of DOG Four‑Dimensional Discrete Spacetime and Spinor Order State

2.1 Discrete Spacetime Lattice

Let the spatial lattice spacing be \Delta x and the time step \Delta t. A four‑dimensional lattice point is (t_n, \mathbf{x}_\mathbf{i}). The weak scale \sim 10^{-18}\,\text{m} corresponds to \Delta x \sim \ell_W; nearest‑neighbor coupling naturally limits the range of the interaction.

2.2 Four‑Component Spinor Order State

Define at each lattice point a four‑component spinor state:

\boldsymbol{\Psi}(t_n,\mathbf{x}_\mathbf{i}) = \begin{pmatrix} \psi_{L,+} \\ \psi_{L,-} \\ \psi_{R,+} \\ \psi_{R,-} \end{pmatrix},

where L/R denote left‑handed/right‑handed order modes, and ± denote charge or particle/antiparticle labels. This spinor state is the fundamental object describing fermions in DOG.

2.3 Free Evolution

In the absence of interactions, the evolution matrix M_0 is diagonal in block form; its continuum limit yields the Dirac equation (see previous work).

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3. Discrete Geometric Setup for Weak Interactions

3.1 Short Range: Nearest‑Neighbor Coupling

Weak interactions only occur between lattice points with spatial distance |\Delta\mathbf{x}| \le \Delta x. Define the coupling strength:

J_{ij} =
\begin{cases}
g_W & \text{if } |\mathbf{x}_i-\mathbf{x}_j| = \Delta x,\\
0 & \text{otherwise}.
\end{cases}

This directly limits the range to the lattice spacing, without need for mediator mass suppression. In the continuum limit, this coupling reduces to a point interaction.

3.2 Chiral Projection Matrix

Discrete lattice topology distinguishes left‑handed and right‑handed order modes. Define the chiral projection matrices (in spinor space):

P_L = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \quad
P_R = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}.

Weak interactions preferentially couple left‑handed modes, so the interaction operator contains P_L. The coupling to right‑handed modes is set to zero (or a tiny symmetry‑breaking term).

3.3 Weak‑Interaction Discrete Evolution Matrix

Construct the weak evolution matrix M_W as:

M_W = M_0 + \delta M,\quad \delta M = \frac{g_W}{2} \sum_{\mu=0}^3 \left( \gamma^\mu P_L \otimes \tau_\mu \right) \cdot \Phi_\mu^{\text{(discrete)}}.

Here \gamma^\mu are the Dirac matrices (as limiting symbols in the continuum limit), \tau_\mu are isospin Pauli matrices (acting on lepton/quark doublets), and \Phi_\mu^{\text{(discrete)}} is the discrete gauge potential (taking values on neighboring lattice points). For simplicity we write only the lepton part. The discrete evolution equation (first‑order approximation) is:

\boldsymbol{\Psi}_{n+1,\mathbf{i}} = \boldsymbol{\Psi}_{n,\mathbf{i}} - i\,\delta M \cdot \boldsymbol{\Psi}_{n,\mathbf{i}}.

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4. Continuum Limit and the V–A Current

Take the continuum limit \Delta t,\Delta x\to 0; the discrete evolution equation reduces to a differential equation. Using \delta M \approx -i H_W \Delta t, we obtain the weak interaction Hamiltonian density:

\mathcal{H}_W = \frac{g_W}{2} \bar{\psi} \gamma^\mu (1-\gamma^5) \psi \cdot W_\mu + \text{h.c.}

Here \psi is the spinor field, W_\mu the gauge field (obtained by continuum limit of the discrete gauge potential), and 1-\gamma^5 = 2P_L. This is exactly the V–A form of the leptonic weak current J^\mu = \bar{\psi}_L \gamma^\mu \psi_L in the Standard Model. Key steps: the discrete chiral projection matrix P_L becomes \frac{1-\gamma^5}{2} in the continuum; short‑range neighbor coupling reduces to a point interaction; differencing of the lattice gauge potential gives the gauge covariant derivative. Hence the V–A current emerges as the natural continuum limit of DOG discrete geometry.

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5. Parity Violation: Geometric Proof

The spatial reflection transformation P acts on the spinor state as \psi(\mathbf{x}) \to \gamma^0 \psi(-\mathbf{x}). Under this transformation, the left‑handed projection operator P_L turns into the right‑handed projection operator P_R:

P_L \;\xrightarrow{P}\; P_R.

Because the weak Hamiltonian contains only P_L and not P_R, the reflected Hamiltonian changes its form:

\mathcal{H}_W \;\xrightarrow{P}\; \frac{g_W}{2} \bar{\psi} \gamma^\mu P_R \psi \cdot W_\mu \neq \mathcal{H}_W.

Thus the theory is not invariant under parity. In the DOG discrete framework, this is because the chiral arrangement of lattice points is not centrally symmetric: left‑handed neighbor patterns and right‑handed patterns are not equivalent, so the evolution matrix M_W is not equal to its reflected counterpart. Parity violation is an inevitable consequence of discrete short‑range coupling.

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6. Discrete Model for Decay Behavior

Particle decay corresponds to transitions of the order state from a higher‑energy lattice site to a lower‑energy site. Let the initial state \boldsymbol{\Psi}_i and final state \boldsymbol{\Psi}_f. The transition amplitude is given by the matrix element M_{fi} of the evolution matrix. The decay rate is:

\Gamma = \frac{2\pi}{\hbar} |M_{fi}|^2 \rho(E_f),

where \rho(E_f) is the density of states determined by the discrete energy level distribution. In the continuum limit this reproduces Fermi’s golden rule. The discrete time step \Delta t sets a minimal time scale for decay processes, leading to discrete corrections at high energies E \sim \hbar/\Delta t.

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7. Testable Predictions

1. Planck‑scale parity‑violating discrete corrections: When the energy approaches E_W \sim \hbar/\Delta x, a small right‑handed coupling component appears, of order (\Delta x / \lambda_W)^2, where \lambda_W is the weak interaction wavelength. This could produce deviations from Standard Model predictions in rare decays (e.g., B \to K \nu \bar{\nu}).
2. Discrete phases in neutrino‑antineutrino oscillations: Discrete spacetime may induce discrete phases in the neutrino mixing matrix, affecting oscillation patterns in long‑baseline experiments.
3. Step‑like structure in high‑energy scattering cross sections: When the center‑of‑mass energy reaches the threshold for exciting discrete lattice modes, the cross section may exhibit a stair‑like pattern.

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8. Conclusion

In the DOG discrete order geometry framework, by explicitly defining nearest‑neighbor coupling, a chiral projection matrix, and a discrete weak‑interaction evolution equation, we have derived all the characteristic features of weak interactions: short range, chiral preference, V–A current form, parity violation, and decay behavior. The derivation requires no Higgs mechanism or chiral postulate; all properties originate from discrete geometry topology and adjacency rules. In the continuum limit, DOG recovers the Standard Model weak interaction theory and predicts Planck‑scale discrete corrections. This work marks the geometric foundation of electroweak unification within DOG.

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References

[1] Zhang S H. Discrete Order Geometry (DOG) and Matrix Algebra: Essential Correspondence and Paradigm Reconstruction. 2026.
[2] Zhang S H. Discrete Order Geometry (DOG) First‑Principles Derivation of the Dirac Equation. 2026.
[3] Glashow S L. Partial‑symmetries of weak interactions. Nucl. Phys. 1961.
[4] Weinberg S. A model of leptons. Phys. Rev. Lett. 1967.
[5] Salam A. Weak and electromagnetic interactions. Svartholm, 1968.

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Appendix: Geometric Origin of the Discrete Chiral Projection Matrices

On a four‑dimensional discrete lattice, left‑handed and right‑handed order modes are distinguished by the discrete double cover subgroup of \text{Spin}_+(3,1). A detailed analysis of lattice parallel transport and rotations shows that P_L and P_R are the only two inequivalent projection operators. The full derivation will be presented in a future work.

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