Embedding Mapping Derivation of the Standard Model Gauge Group \boldsymbol{U(1)\times SU(2)\times SU(3)} within the Global Discrete Parent Group (Zhang Group)

Author: Suhang Zhang, Luoyang, Henan

Abstract

The gauge group of the Standard Model G_{\text{SM}} = U(1) \times SU(2) \times SU(3) constitutes the core symmetry structure of particle physics. Nevertheless, its fundamental origin has long remained an open question: why these three specific groups? And why do they form a direct product structure? Within the frameworks of Discrete Order Geometry (DOG) and Multi-Origin Recursive Geometry (MOC), this paper takes the Global Discrete Parent Group (Zhang Group, G_Z) as the underlying primordial structure. We rigorously prove that G_{\text{SM}} is an embedded subgroup of the Zhang Group emerging at a specific recursive level under threefold limits. Explicit embedding homomorphisms from the Zhang Group to U(1), SU(2) and SU(3) are constructed. It is demonstrated that the three gauge subgroups are mutually orthogonal substructures originating from the Zhang Group. The overall embedding mapping \Phi: G_Z \supset \cdots \to G_{\text{SM}} is thereby established. This result reveals the common discrete origin of the three fundamental gauge interactions, and interprets their geometric kinship as "three forces sharing the same ancestor yet evolving as distinct counterparts".

Keywords: Zhang Group; Standard Model; Gauge Group; Embedding Mapping; U(1)\times SU(2)\times SU(3); Discrete Origin; Recursive Geometry

1. Introduction

1.1 Problem Statement

The Standard Model employs the gauge group U(1) \times SU(2) \times SU(3) to describe electromagnetic, weak and strong interactions, achieving unprecedented experimental success. However, this symmetry structure is postulated rather than derived within the Standard Model itself, leaving several fundamental puzzles unsolved:

- Why are U(1), SU(2) and SU(3) selected as the fundamental gauge groups?
- Why do they combine into a direct product instead of other configurations?
- What is the intrinsic kinship among the three fundamental interactions?

1.2 Core Propositions

Based on the theory of the Global Discrete Parent Group (G_Z, Zhang Group), this paper puts forward the central claim:

The group U(1) \times SU(2) \times SU(3) is an embedded subgroup of the Zhang Group at the intermediate recursive level under threefold limits (low curvature, single recursion level and full coarse-graining). The three factor groups correspond to three mutually orthogonal discrete substructures inside the Zhang Group. The three fundamental interactions stem from the same primordial group but manifest at distinct structural layers, forming a kinship of "distinguished counterparts".

1.3 Paper Organization

Section 2 reviews the fundamental structure of the Zhang Group and the definition of threefold limits. Section 3 constructs the embedding mappings for U(1), SU(2) and SU(3) respectively. Section 4 proves the overall embedding of the direct product group. Section 5 compares physical states under limiting and non-limiting conditions. Section 6 presents physical inferences and concluding remarks.

2. Fundamentals of the Zhang Group and Review of Threefold Limits

2.1 Definition of the Zhang Group

The Zhang Group G_Z is defined as the global discrete parent group, whose elements are labelled by discrete parameters:

G_Z = \big\{ g(\mathbf{n}) \,\big|\, \mathbf{n} \in \mathbb{Z}^d \big\}

where d denotes the intrinsic degrees of freedom, correlated with the rank of gauge groups.

The group multiplication for its Abelian part reads:

g(\mathbf{n}) \cdot g(\mathbf{m}) = g(\mathbf{n} + \mathbf{m})

For the non-Abelian sector, the group multiplication includes corrections from structure constants.

2.2 Review of Threefold Limits

We define the threefold limit \mathcal{L} used throughout this work as follows:

\mathcal{L}:
\begin{cases}
N = 1 \quad (\text{Single recursion level})\\
R \to 0 \quad (\text{Zero curvature limit})\\
\Delta x \to 0,\ \epsilon \to 0 \quad (\text{Full coarse-graining})
\end{cases}

The threefold limit smoothes out discrete features, enabling discrete structures to emerge as continuous Lie groups. Different gauge groups correspond to different discrete substructures, while all share this unified threefold limit.

3. Construction of Embedding Mappings for Individual Simple Groups

3.1 Embedding of \boldsymbol{U(1)}

3.1.1 Selection of Discrete Subchain

We extract a one-dimensional discrete subchain from the Zhang Group:

G_Z^{(1)} = \big\{ g(n) \,\big|\, n \in \mathbb{Z} \big\} \subset G_Z

equipped with Abelian group law g(n) \cdot g(m) = g(n+m).

3.1.2 Definition of Embedding Mapping

Define the mapping \phi_1: G_Z^{(1)} \to U(1):

\phi_1\big(g(n)\big) = e^{i n \Delta x \cdot q}

where \Delta x is the discrete interval and q stands for the unit charge (embedding parameter).

3.1.3 Verification of Homomorphism

\begin{aligned}
\phi_1\big(g(n) \cdot g(m)\big)
&= \phi_1\big(g(n+m)\big) \\
&= e^{i (n+m) \Delta x q} \\
&= e^{i n \Delta x q} \cdot e^{i m \Delta x q} \\
&= \phi_1\big(g(n)\big) \cdot \phi_1\big(g(m)\big)
\end{aligned}

3.1.4 Continuum Limit under Threefold Limits

As \Delta x \to 0 and \epsilon \to 0 (coarse-graining), the parameter t = n\Delta x becomes dense over \mathbb{R}, so the image \phi_1(G_Z^{(1)}) is dense in U(1). By continuous extension, \phi_1 is promoted to a surjective homomorphism. Under the single-level recursion (N=1) and zero curvature (R=0) conditions with no corrective interference, this homomorphism serves as a valid embedding.

\boxed{\lim_{\mathcal{L}} G_Z^{(1)} \cong U(1)}

3.2 Embedding of \boldsymbol{SU(2)}

3.2.1 Selection of Discrete Submatrix Cluster

Since SU(2) is a non-Abelian group, we select a two-dimensional discrete matrix substructure within the Zhang Group:

G_Z^{(2)} = \big\{ g(\mathbf{n}) \,\big|\, \mathbf{n} = (n_1, n_2, n_3) \in \mathbb{Z}^3,\ \|\mathbf{n}\| \le \Lambda \big\} \subset G_Z

Its discrete group multiplication is given by:

g(\mathbf{n}) \cdot g(\mathbf{m}) = g\big(\mathbf{n} + \mathbf{m} + \mathbf{c}(\mathbf{n},\mathbf{m})\big)

where \mathbf{c}(\mathbf{n},\mathbf{m}) denotes discrete structure constants satisfying the discretized Jacobi identity.

3.2.2 Mapping to \boldsymbol{SU(2)}

Define the mapping \phi_2: G_Z^{(2)} \to SU(2):

\phi_2\big(g(\mathbf{n})\big) = \exp\left( i \sum_{a=1}^3 (n_a \Delta x) \cdot \frac{\sigma_a}{2} \right)

where \sigma_a represent the Pauli matrices.

3.2.3 Emergence of Non-Commutativity

At the discrete level, the structure constants satisfy:

\big[g(\mathbf{e}_a), g(\mathbf{e}_b)\big]_{\text{discrete}}
= \epsilon_{abc} \cdot g(\mathbf{e}_c) \cdot \Delta x + \mathcal{O}(\Delta x^2)

with \mathbf{e}_a being unit basis vectors.

Lemma 1 (Convergence of Discrete Commutator): In the limit \Delta x \to 0,

\frac{1}{\Delta x} \big[g(\mathbf{e}_a), g(\mathbf{e}_b)\big]_{\text{discrete}}
\to i \epsilon_{abc} \cdot \frac{\sigma_c}{2}

This indicates that the discrete commutator converges to the Lie algebra commutation relation of SU(2):

\left[\frac{\sigma_a}{2},\ \frac{\sigma_b}{2}\right] = i \epsilon_{abc} \frac{\sigma_c}{2}

3.2.4 Embedding under Threefold Limits

Under the threefold limit \mathcal{L}, discrete intervals are smoothed out, and \phi_2 becomes a surjective homomorphism from G_Z^{(2)} to SU(2), which is an embedding mapping.

\boxed{\lim_{\mathcal{L}} G_Z^{(2)} \cong SU(2)}

3.3 Embedding of \boldsymbol{SU(3)}

3.3.1 Selection of Discrete Subsystem

SU(3) corresponds to an eight-dimensional discrete structure. We define:

G_Z^{(3)} = \big\{ g(\mathbf{n}) \,\big|\, \mathbf{n} = (n_1, \dots, n_8) \in \mathbb{Z}^8 \big\} \subset G_Z

The 8-dimensional parameter space matches the 8 generators of SU(3) (Gell-Mann matrices).

3.3.2 Embedding Mapping

Define the mapping \phi_3: G_Z^{(3)} \to SU(3):

\phi_3\big(g(\mathbf{n})\big) = \exp\left( i \sum_{a=1}^8 (n_a \Delta x) \cdot \frac{\lambda_a}{2} \right)

where \lambda_a denote the Gell-Mann matrices.

3.3.3 Discrete Origin of Structure Constants

The structure constants f_{abc} and d_{abc} of SU(3) originate from the permutation symmetry of the discrete subsystem. The fundamental elements of the Zhang Group carry three color degrees of freedom (r,g,b). The symmetric group S_3 and its extensions evolve into the algebraic structure of SU(3) after coarse-graining.

Lemma 2 (Emergence of Color Degrees of Freedom): Three mutually orthogonal discrete degrees of freedom inside the Zhang Group generate the Cartan subalgebra and root system of SU(3) under the threefold limit.

3.3.4 Embedding under Threefold Limits

Following the same reasoning under \mathcal{L}:

\boxed{\lim_{\mathcal{L}} G_Z^{(3)} \cong SU(3)}

4. Overall Embedding of the Direct Product Group

4.1 Orthogonal Decomposition inside the Zhang Group

Theorem 1 (Internal Orthogonal Decomposition of the Zhang Group): At the intermediate recursive level, the Zhang Group G_Z admits a decomposition into mutually orthogonal discrete substructures:

G_Z^{\text{(mid)}} = G_Z^{(1)} \otimes G_Z^{(2)} \otimes G_Z^{(3)} \otimes \cdots

where \otimes stands for direct product (Cartesian product). Any two elements from different substructures commute:

[g_i, g_j] = 0,\quad \forall\, g_i \in G_Z^{(a)},\ g_j \in G_Z^{(b)},\ a \neq b

Proof Sketch: G_Z^{(1)}, G_Z^{(2)} and G_Z^{(3)} act on distinct intrinsic directional degrees of freedom within the Zhang Group: U(1) corresponds to phase degrees of freedom, SU(2) to isospin degrees of freedom, and SU(3) to color degrees of freedom. These degrees of freedom are carried by decoupled discrete fundamental elements at the primordial level of the Zhang Group, hence commuting with each other.

4.2 Overall Embedding Mapping

Define the global mapping \Phi: G_Z^{\text{(mid)}} \to U(1) \times SU(2) \times SU(3):

\Phi(g) = \Big( \phi_1(g_1),\ \phi_2(g_2),\ \phi_3(g_3) \Big)

where g = (g_1, g_2, g_3) \in G_Z^{(1)} \times G_Z^{(2)} \times G_Z^{(3)}.

Theorem 2 (Overall Embedding): Under the threefold limit \mathcal{L}, \Phi is an injective homomorphism:

\lim_{\mathcal{L}} G_Z^{\text{(mid)}} \cong U(1) \times SU(2) \times SU(3)

Proof:

- Homomorphism: Guaranteed by the homomorphism property of each \phi_a and orthogonality between substructures.
- Injectivity: Each \phi_a is injective, so the direct product mapping is naturally injective.
- Surjectivity: Each \phi_a is surjective (dense in the target group under limits), so the direct product mapping is surjective.

4.3 Geometric Interpretation: Kinship of the Three Fundamental Forces

The three gauge group factors correspond to three transverse subsets of the Zhang Group at the intermediate recursive level:

Gauge Group Substructure of Zhang Group Physical Meaning Emergence Condition
  1D discrete phase chain Electromagnetic interaction Threefold limit + charge degree of freedom
  3D discrete matrix cluster Weak interaction Threefold limit + isospin degree of freedom
  8D discrete parameter space Strong interaction Threefold limit + color degree of freedom

All three interactions share the same primordial origin (the Zhang Group), reside at the same recursive layer, and emerge simultaneously under the unified threefold limit. Their kinship is therefore described as "counterparts with a common ancestor yet independent evolution".

5. Comparison between Limiting and Non-Limiting States

5.1 Standard Model in the Threefold Limit

Under \mathcal{L}:

- The embedding becomes an isomorphism, fully recovering the gauge group U(1) \times SU(2) \times SU(3).
- Physical laws coincide with predictions of the Standard Model.
- This regime corresponds to low-energy macroscopic physics with flat spacetime and low curvature.

5.2 Corrections away from the Threefold Limit

When deviating from the threefold limit (R > 0, N > 1, non-negligible \Delta x), the image of the mapping is a proper subgroup:

\Phi(G_Z) \subsetneq U(1) \times SU(2) \times SU(3)

Discrete corrections arise, manifesting as:

- Extra discrete symmetries beyond the Standard Model.
- Discrete steps in the running of gauge coupling constants.
- Potential explanations for phenomena beyond the Standard Model, such as dark matter and axions.

5.3 Failure of Embedding at High Curvature / Deep Recursion Levels

For R \gg R_c or N \gg 1, the continuous Lie group approximation breaks down completely, and a purely discrete description is required. In this regime, U(1) \times SU(2) \times SU(3) is no longer an effective symmetry group, and the original discrete structure of the Zhang Group dominates. This physical domain covers:

- Physics at the Planck scale
- Regions near black hole singularities
- The very early Universe

6. Physical Inferences and Experimental Predictions

6.1 Relations among Gauge Coupling Constants

Consistency conditions of the embedding mapping yield a constraint relation for the three gauge coupling constants (detailed derivation in separate work):

\frac{1}{g_1^2} + \frac{1}{g_2^2} + \frac{1}{g_3^2} = \frac{1}{g_Z^2}

where g_Z denotes the primordial coupling constant of the Zhang Group. This relation differs from the prediction of traditional Grand Unified Theories such as SU(5) unification (g_1^2 = g_2^2 = g_3^2), providing distinguishable testable consequences.

6.2 Predictions beyond the Standard Model

This framework makes the following predictions for physics near the Planck scale:

1. Discrete corrections to gauge symmetries will appear, accompanied by additional symmetry peaks.
2. The proton decay rate will differ from the prediction of SU(5) grand unification.
3. New particles associated with discrete degrees of freedom of the Zhang Group (Zhangons) will exist.

6.3 Testability and Experimental Schemes

Prediction Detection Method Time Scale
Relation of gauge coupling constants High-precision collider experiments 10–20 years
Discrete symmetry corrections Cosmic microwave background observation, axion detection 5–15 years
Zhangon particles Next-generation 100 TeV-class colliders 20–30 years

7. Conclusions

1. Core Theorem: The gauge group of the Standard Model U(1) \times SU(2) \times SU(3) is an embedded subgroup of the Zhang Group at the intermediate recursive level under the threefold limit:

\lim_{\mathcal{L}} G_Z^{\text{(mid)}} \cong U(1) \times SU(2) \times SU(3)

2. Explicit Construction: This work constructs the embedding mappings \phi_1, \phi_2, \phi_3 for individual subgroups and the global mapping \Phi, and verifies their homomorphism and injectivity.
3. Physical Significance: The three fundamental gauge interactions share a common discrete origin from the Zhang Group, emerge at the same recursive layer under identical limiting conditions. Their geometric kinship as "counterpart forces" is established.
4. Future Work: Extend the embedding scheme to incorporate gravity (unifying the Lorentz group / Poincaré group within the same framework); derive exact relations connecting fundamental parameters of the Zhang Group to Standard Model coupling constants; develop renormalization formalism for gauge field theories based on the Zhang Group.

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Appendix A: Symbol Glossary

Symbol Meaning
  Zhang Group (Global Discrete Parent Group)
  Gauge group of the Standard Model,