Continuous-Limit Matching from FCE to the Weak Force Gauge Field Equation

Author: Zhang Suhang

(Luoyang, Henan)

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Abstract

Discrete Order Geometry (DOG) takes the Fundamental Coupling Equation (FCE) as the native expression of its equation layer. Taking the weak interaction SU(2)_L gauge field as an example, this paper gives a continuous-limit matching path from the FCE to the non-Abelian Yang–Mills equation. The path is divided into three steps: limiting operations (continuousization of order distance, order quantity, and coupling coefficients), matching conditions (Lie algebra lifting, gauge connection matching, action matching), and standard derivations (construction of field strength and variation). This paper clearly distinguishes the nature of each step and does not claim that the FCE intrinsically derives the weak force equation. At the end, it explains that the strong force SU(3)_C and electromagnetism U(1) are treated by the same path, replacing SU(2)_L with SU(3)_C and U(1), respectively.

Keywords: Discrete Order Geometry; DOG; Fundamental Coupling Equation; FCE; weak interaction; Yang–Mills equation; continuous limit; matching

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1 Introduction

In previous work, DOG has completed the establishment of the geometric layer, algebraic layer (Zhang matrix), equation layer (FCE), and group-theoretic layer (Zhang group–Lie group emergence). The FCE is a product-type constraint equation:

\lambda_p\,\omega_p
\prod_{q\in\operatorname{Neigh}(p)}
C_{pq}\,\omega_q
=1

This equation is defined on a discrete order lattice, with the Zhang matrix \mathbf{C}=(C_{pq}) encoding the order coupling among lattice points.

The task of this paper is: taking the weak interaction SU(2)_L as an example, to give a continuous-limit matching path from the FCE to the non-Abelian Yang–Mills equation.

This paper does not claim that the FCE intrinsically derives the weak force equation. Each step in the path is clearly marked as: limiting operation, matching condition, or standard derivation.

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2 Review of the FCE

Definition 2.1 (FCE, steady-state form)

Let \mathcal{L} be a DOG discrete order lattice, and let \mathbf{C} be the Zhang matrix. For each lattice point p\in\mathcal{L}:

\lambda_p\,\omega_p
\prod_{q\in\operatorname{Neigh}(p)}
C_{pq}\,\omega_q
=1

Logarithmic form:

\ln\lambda_p+\ln\omega_p+
\sum_{q\in\operatorname{Neigh}(p)}
\ln(C_{pq}\omega_q)=0

where \omega_p is the intrinsic order quantity of the lattice point, whose value is determined by the Zhang group representation.

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3 Limiting Operations

3.1 Continuousization of Lattice Points

p\in\mathcal{L}\;\longrightarrow\;x\in\mathcal{M}

3.2 Continuousization of Order Neighborhood

\operatorname{Neigh}(p)\;\longrightarrow\;
\{y:\ |x-y|<\epsilon\}

3.3 Continuousization of Order Quantity

\omega_p\;\longrightarrow\;\omega(x)

3.4 Continuousization of Coupling Coefficients

C_{pq}\;\longrightarrow\;C(x,y)

The continuous limit of the FCE is:

\ln\lambda(x)+\ln\omega(x)+
\int \ln\!\big(C(x,y)\omega(y)\big)\,dy=0

The above four steps are all standard limiting operations within the DOG framework.

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4 Matching Conditions

4.1 Matching Condition M1: Lie Algebra Lifting

Lift \omega(x) from a scalar to an SU(2)_L Lie-algebra-valued field:

\omega(x)\;\longrightarrow\;\omega(x)=\omega^a(x)T^a

where T^a=\frac{1}{2}\sigma^a (\sigma^a are the Pauli matrices), satisfying:

[T^a,T^b]=i\epsilon^{abc}T^c

Nature: matching condition. The SU(2)_L representation is determined by the Zhang group frequency partition, not intrinsically by the FCE.

4.2 Matching Condition M2: Gauge Connection Matching

Let the phase part of the coupling coefficient C(x,y) along the path x\to y correspond to the SU(2)_L gauge connection:

\ln C(x,y)\;\longrightarrow\;
-ig\int_x^y A_\mu^a(z)T^a\,dz^\mu

that is:

C(x,y)=\exp\!\left(-ig\int_x^y A_\mu^a T^a\,dz^\mu\right)

Nature: matching condition. The reason is that in the logarithmic form of the FCE, \ln C is the quantity connecting two lattice points, and in the continuous limit it corresponds to the phase of parallel transport.

4.3 Matching Condition M3: Covariant Derivative

Define the covariant derivative:

D_\mu=\partial_\mu+igA_\mu^a T^a

Acting on \omega:

D_\mu\omega=\partial_\mu\omega+ig[A_\mu,\omega]

Nature: matching condition. It follows naturally from the connection structure of M2.

4.4 Matching Condition M4: Action

Take the SU(2)_L Yang–Mills action:

S_{\rm YM}
=-\frac{1}{4}\int \operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})\,d^4x

Nature: matching condition. It is the lowest-order matching of the continuous limit of the FCE.

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5 Standard Derivations

5.1 Field Strength

Take the commutator of covariant derivatives:

[D_\mu,D_\nu]\omega
=igF_{\mu\nu}\omega

The calculation gives:

F_{\mu\nu}
=\partial_\mu A_\nu-\partial_\nu A_\mu+ig[A_\mu,A_\nu]

where A_\mu=A_\mu^a T^a, F_{\mu\nu}=F_{\mu\nu}^a T^a.

Nature: standard derivation.

5.2 Equation of Motion

Vary the action:

\frac{\delta S_{\rm YM}}{\delta A_\mu^a}=0

The standard calculation gives:

D_\mu F^{\mu\nu}=0

If a matter current J^\nu is included:

D_\mu F^{\mu\nu}=J^\nu

Nature: standard derivation.

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6 Summary of the Path

Step Content Nature
FCE discrete constraint equation DOG native
Continuous limit p\to x, \operatorname{Neigh}(p)\to neighborhood Limiting operation
M1 \omega scalar \to SU(2)_L Lie algebra value Matching condition
M2 \ln C\to -ig\int A_\mu^a T^a dz^\mu Matching condition
M3 D_\mu=\partial_\mu+igA_\mu^a T^a Matching condition
M4 Take the Yang–Mills action Matching condition
Standard derivation F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+ig[A_\mu,A_\nu] Standard derivation
Standard derivation D_\mu F^{\mu\nu}=J^\nu Standard derivation

Conclusion: Under the continuous limit, the FCE, after Lie algebra lifting, gauge connection matching, covariant derivative matching, and action matching, can be matched to the SU(2)_L Yang–Mills equation. This is not an "intrinsic derivation" but a "limiting matching."

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7 Strong Force and Electromagnetism

The strong force SU(3)_C and electromagnetism U(1) are treated by the same path:

Strong force: Replace SU(2)_L with SU(3)_C, take the generators T^a as the Gell-Mann matrices, and take the structure constants f^{abc} as the SU(3) structure constants. The path is completely identical, matching to the SU(3)_C Yang–Mills equation. Color confinement and asymptotic freedom are left for future work.

Electromagnetism: Replace SU(2)_L with U(1), take the generator in the one-dimensional representation, and set the structure constants f^{abc}=0. The non-Abelian term ig[A_\mu,A_\nu] automatically vanishes, matching to Maxwell’s equations. Electromagnetism is the Abelian special case and requires no additional mechanism.

Therefore, by the same limiting operations and matching path, replacing the representation SU(2)_L with SU(3)_C and U(1), respectively, one can match to the effective equations of the strong force and electromagnetism, respectively.

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8 Conclusion

Taking the weak interaction SU(2)_L as an example, this paper completes the continuous-limit matching from the FCE to the non-Abelian Yang–Mills equation:

1. Limiting operations: continuousization of lattice points, order neighborhood, order quantity, and coupling coefficients, which are standard steps within the DOG framework;
2. Matching conditions: Lie algebra lifting, gauge connection matching, covariant derivative, and action, which are correspondences between the FCE and continuous field equations;
3. Standard derivations: construction of field strength and variation, whose results hold automatically.

Final result:

D_\mu F^{\mu\nu}=J^\nu,\qquad
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+ig[A_\mu,A_\nu]

Positioning:

· Under the continuous limit, the FCE matches the SU(2)_L Yang–Mills equation;
· Matching conditions are clearly marked, not concealed and not skipped;
· This result is a limiting matching between the FCE and continuous field equations, not an "intrinsic derivation";
· The strong force SU(3)_C and electromagnetism U(1) are treated by the same path, replacing the representation.

Questions to be studied:

1. Uniqueness of the matching conditions;
2. Matching path for Higgs breaking and electroweak mixing;
3. Non-perturbative treatment of color confinement and asymptotic freedom;
4. Continuous limit of the FCE under non-scalar representations.

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References

Omitted.

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Note: This paper is a study of limiting matching from the FCE to the weak force gauge field equation within the DOG framework. All matching conditions are clearly marked; it does not claim that the FCE intrinsically derives the weak force equation. No specific physical applications are involved.