Structural Conservation and the Unified Reconstruction of the Four Fundamental Forces
Author: Zhang Suhang

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Abstract
Unifying gravity, electromagnetism, the weak nuclear force, and the strong nuclear force has been a core goal of theoretical physics since Einstein. In his later years, Einstein attempted to unify gravity and electromagnetism but stopped at the metric level; the Standard Model successfully integrates electromagnetism, the weak force, and the strong force, yet cannot incorporate gravity. This paper proposes that the inherent predicament of unifying the four forces stems from traditional theories always attempting to achieve unification at the metric level. The four fundamental interactions obey the law of structural conservation. Gravity obeys the conservation of the diffeomorphism group of the spacetime manifold, while gauge interactions obey the conservation of the symmetry groups of their respective fiber bundles. The differences among the four forces arise solely from the different group structures corresponding to the base space and the fiber bundles, and all interactions at the underlying level jointly obey the principle of structural conservation.
This paper establishes a paradigm of jointly obeying the law of structural conservation and dividing computational labor: unification lies in the principle of structural conservation, and it is not necessary to force the sharing of a single set of field equations for numerical computation. Gravity continues to use general relativity, and gauge forces continue to use the Standard Model, each maintaining validity within its applicable range. The problem that quantum gravity is non-renormalizable is essentially a contradiction brought about by forcibly merging the two systems at the metric level; the framework of structural conservation shows that unification should be built on the principle of structural conservation, while at the computational level the respective dynamical equations of the two systems are retained.
Keywords: structural conservation; unification of the four forces; gauge field theory; general relativity; quantum gravity; invariants; fiber bundles

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1 Introduction: A Century of Predicament in Unification
Unifying the four fundamental interactions is a long-sought goal of theoretical physics.
In his later years, Einstein attempted a geometrized unification of gravity and electromagnetism, incorporating the electromagnetic field into extra dimensions of the spacetime metric, but this scheme ultimately failed at the metric level. The Yang-Mills non-Abelian gauge field theory, later combined with quantum chromodynamics to construct the Standard Model, completed the unification of the electromagnetic, weak, and strong interactions, but gravity has always been impossible to incorporate into the framework. Quantum gravity proposals such as string theory, loop quantum gravity, and asymptotic safety still attempt to reconcile general relativity and quantum field theory at the metric level, and to date no generally accepted complete theory has been obtained.
The two types of systems differ fundamentally: general relativity is a background-independent theory of spacetime geometry, while quantum field theory belongs to a background-dependent operator-spectrum description; the two are difficult to reconcile at the metric level, and the non-renormalizability of quantum gravity is precisely the technical manifestation of this contradiction. This paper argues that the root of this predicament is placing the goal of unification at the metric level.
What this paper calls "structural conservation" refers to the fact that each physical system, under its own transformations, keeps the structure of its geometric carrier unchanged. Each of the four forces has its own structure, and each structure remains unchanged; all obey the rule of structural conservation, rather than sharing one and the same structure.

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2 The Contradictions of the Four Forces at the Metric Level
2.1 Metric Characteristics of the Four Forces
Interaction | Metric characteristic | Symmetry group | Propagator | Range
Gravity | Spacetime metric | Diffeomorphism group | Graviton (theoretical hypothesis) | Infinity
Electromagnetism | Phase potential | U(1) | Photon | Infinity
Weak nuclear force | Weak isospin potential | SU(2) | W/Z bosons | Short range
Strong nuclear force | Color potential | SU(3) | Gluons | Short range
The coupling strengths, ranges, and propagating particles are all different, and under the metric representation the four forces appear as mutually independent dynamical systems.
2.2 Limitations of Unification Schemes at the Metric Level
Einstein's geometric unification scheme could not accommodate quantum effects; the Standard Model can describe only gauge interactions; string theory is mathematically self-consistent but lacks observable predictions; loop quantum gravity achieves quantization of spacetime geometry but is difficult to make compatible with gauge fields. All such schemes attempt to construct a single dynamical equation at the metric level, thereby bringing about irreconcilable conflicts between systems.

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3 The Four Forces Obey the Law of Structural Conservation
3.1 Gravity: Conservation of the Group Structure of the Spacetime Manifold
The core underlying general relativity is not metric information such as the metric tensor $g_{\mu\nu}$, but the structural conservation of the diffeomorphism group of the spacetime manifold. Under arbitrary coordinate transformations, the spacetime system maintains the invariance of its own group-structure rules, while the metric tensor belongs to variable metric information and does not affect the underlying rule of structural conservation. Gravity is the physical effect corresponding to the structural conservation of the spacetime diffeomorphism group.
3.2 Gauge Forces: Conservation of the Group Structure of Fiber Bundles
Electromagnetism, the weak force, and the strong force correspond respectively to different fiber-bundle symmetry groups: electromagnetism corresponds to the U(1) group, the weak interaction to the SU(2) group, and the strong interaction to the SU(3) group. The gauge potential $A_\mu$ belongs to metric information and may undergo local transformations; but the structural rules of each symmetry group are strictly conserved. The gauge field is essentially the compensating field that maintains the conservation of the corresponding group structure under local internal symmetry transformations.
The gauge field strength and the Riemann curvature possess a consistent mathematical structure:
F = dA + A \wedge A
R = d\Gamma + \Gamma \wedge \Gamma
The two are unified into a general structural equation:
\mathcal{F} = d\mathcal{A} + \mathcal{A} \wedge \mathcal{A}
where $\mathcal{A}$ is the unified generalized connection and $\mathcal{F}$ is the generalized curvature, uniformly describing the structural form of all interactions. At low energy scales, symmetry breaking occurs, and the unified structural group decomposes into the subgroups of $\text{Diff}(M) \times SU(3) \times SU(2) \times U(1)$, corresponding respectively to the four fundamental interactions.
It should be noted that Yang-Mills theory itself already adopted this paradigm: the curvature formula $F = dA + A \wedge A$ remains unchanged, and only the symmetry group is replaced, describing respectively U(1) electromagnetism, SU(2) weak, and SU(3) strong interactions. The unified structural equation $\mathcal{F} = d\mathcal{A} + \mathcal{A} \wedge \mathcal{A}$ proposed in this paper continues this idea. It must still be emphasized: formal unification is not the same as structural identity—the gravitational connection takes values in the spacetime tangent bundle, while the gauge connection takes values in internal fiber bundles; the two geometric carriers are each independent, and they merely share the same curvature expression form and the rule of structural conservation.
3.3 The Four Forces Obey the Law of Structural Conservation
Interaction | Conservation mechanism | Geometric carrier
Gravity | Conservation of diffeomorphism group structure | Spacetime manifold
Electromagnetism | Conservation of U(1) group structure | U(1) fiber bundle
Weak nuclear force | Conservation of SU(2) group structure | SU(2) fiber bundle
Strong nuclear force | Conservation of SU(3) group structure | SU(3) fiber bundle
The differences among the four forces come from the different group structures corresponding to the base space and the fiber bundles, and all interactions strictly obey the law of structural conservation.

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4 Structural Conservation, Division of Computational Labor
4.1 Structural Conservation: Each Structure Remains Unchanged
Structural conservation is a universal physical principle. The four fundamental interactions need not have the same geometric structure or the same metric system; they need only each maintain the structural conservation of their own corresponding symmetry group, and that is the underlying physical unification.
4.2 Concrete Calculations Are Not the Same
At the level of numerical computation, it is not necessary to share a single set of dynamical equations: gravity adopts the Einstein field equations, and gauge interactions continue to use the Yang-Mills equations together with the Higgs mechanism, each remaining valid in its applicable domain.
Both types of systems satisfy the dynamical constraints corresponding to structural conservation:
· The Yang-Mills system satisfies covariant conservation $D_\mu J^\mu = 0$;
· The general relativity system satisfies energy-momentum tensor conservation $\nabla_\mu T^{\mu\nu} = 0$.
Yang-Mills | General relativity
Structural equation $F = dA + A \wedge A$ | $R = d\Gamma + \Gamma \wedge \Gamma$
Dynamical equation Yang-Mills field equations | Einstein gravitational field equations
Matter flow Color current, weak current | Energy-momentum tensor
Conservation condition $D_\mu J^\mu = 0$ | $\nabla_\mu T^{\mu\nu} = 0$
4.3 Core Argument: Unified Rules Do Not Require Unified Equations
The mistake of traditional unified theories is to mistake the principle of structural conservation for structural identity, and to implement it at the computational level, pursuing a single universal field equation. This paper argues: structural conservation is unification at the level of principle; division of computational labor is a strategy at the operational level, and the two proceed in parallel. Unification requires only obeying a common law of structural conservation, not using one single formula for everything.

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5 A Structural-Conservation Explanation of the Quantum Gravity Predicament
The non-renormalizability of quantum gravity has its root in forcibly merging, at the metric level, a background-independent geometric theory and a background-dependent quantum field theory. Forcibly combining two completely different metric systems into a single equation system is an artificial theoretical contradiction, not a contradiction inherent in nature.

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6 Relation to Existing Unification Programs
Theoretical program | Level of unification | Computational approach | Position of this paper
Einstein's geometric unification | Metric level | Seeking a unified field equation | Failed at the metric level, did not touch the principle of structural conservation
Standard Model | Metric level (gauge fields) | Unifying weak, electromagnetic, and strong interactions | Succeeded at the metric level, consistent with the law of structural conservation
String theory | Metric level (higher-dimensional spacetime) | Unified field equations | Relies on metric reconstruction, not unification at the level of principle
Loop quantum gravity | Metric level (spin networks) | Quantization of spacetime | Belongs to metric modification, not underlying unification at the level of principle
This paper does not deny the technical achievements of existing theories, but makes clear: the foundation of the unification of the four forces is the principle of structural conservation, and the computational level need not insist on merging equations.

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7 Conclusion
Relying on the principle of structural conservation, this paper reconstructs the unification logic of the four fundamental interactions. The core conclusions are as follows:

1. The differences among the four forces at the metric level are superficial. Gravity obeys the conservation of spacetime group structure, and gauge interactions obey the conservation of the group structures of their respective fiber bundles;
2. The distinctions among the four forces come only from the base space and the fiber symmetry groups, and all follow the principle of structural conservation;
3. The predicament of quantizing gravity stems from forcibly unifying, at the metric level, two systems with different structures. It is like putting small shoes on an elephant—the problem is not the elephant, but the approach. The correct path to unification is to stand on structural conservation and let each domain handle its own calculations;
4. This framework is isomorphic to Yang-Mills theory: different forces correspond to the structural conservation of different groups and physically manifest as their respective conserved quantities, and therefore adopt their respective computational formulas.

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