The Unified Curvature Equation (MOC) Rigorously Derives the Yang–Mills Equation within the Framework of Geometric Extremum and Fiber Bundle

Author: Zhang Suhang, Luoyang

The unified curvature equation (MOC) can rigorously derive the Yang–Mills equation under the framework of geometric extremum and fiber bundle theory. Its derivation is more concise and more unified than the conventional approach. The core logic, derivation procedure, and key distinctions are elaborated as follows.

1. Core Logic: Curvature Is the Gauge Field

- Yang–Mills (YM): Defined on a principal G-bundle. Given a connection A, the corresponding curvature is F = dA + A\wedge A. The action is constructed as S=\int|F|^2, and variation yields the Yang–Mills equation d_A^*F=0.
- Unified Curvature (MOC): Spacetime curvature and gauge curvature are unified into the total curvature of a multi-origin manifold. The extremum principle directly constrains the total curvature, which naturally decomposes into two components: gravitational spacetime curvature and fiber curvature of the gauge field.

2. Three-Step Derivation of the Yang–Mills Equation from MOC

2.1 Geometric Setup

Consider the product of four-dimensional spacetime M and the fiber of a gauge group G.
The total connection is written as:

\mathbb{A} = \text{spacetime connection } \Gamma \;+\; \text{gauge connection } A

The total curvature decomposes as:

\mathbb{R} = \text{spacetime curvature } R \;+\; \text{gauge curvature } F

with cross terms vanishing.

2.2 The Core MOC Equation

The MOC extremum principle reads:

\delta\int\|\mathbb{R}\|^2\sqrt{g}\,d^4x=0

Variation leads to the total curvature extremum condition:

D^*\mathbb{R}=0

where D denotes the total covariant derivative.

2.3 Extraction of the Yang–Mills Equation

- Projection onto the fiber direction:

D^*F=0 \quad\Rightarrow\quad \text{standard Yang–Mills equation } d_A^*F=0

- Projection onto the spacetime direction:

D^*R=0 \quad\Rightarrow\quad \text{Einstein field equation (including cosmological term)}

3. Key Differences from the Traditional Derivation

- Conventional theory: Gravitation and gauge fields are treated separately; the Yang–Mills equation is obtained solely from variation on the fiber bundle.
- MOC framework: A single unifying equation governs both sectors. The Yang–Mills equation emerges naturally as a branch of the total curvature extremum, without the need to postulate an independent gauge action.

4. Conclusion

The derivation is valid and rigorous:

- The total curvature extremum condition of MOC, when projected onto the fiber bundle structure, is strictly equivalent to the Yang–Mills equation.
- This constitutes the core advantage of the geometric extremum unified field theory: one fundamental geometric principle derives the field equations of all fundamental interactions.