Gauge Interactions and the Standard Model from the Axioms of Self-Variation Theory

1. Introduction

Despite its extraordinary empirical success, the Standard Model (SM) rests on postulated gauge symmetries, coupling constants, and fields whose physical origin remains unexplained [1,2,3]. Gauge invariance, renormalization, confinement, and spontaneous symmetry breaking are introduced as structural principles rather than derived consequences [2].

In this work, we adopt an alternative starting point: intrinsic self-variation of particle properties, consistently constrained by energy–momentum conservation and causal propagation. In this framework, interactions are not fundamental forces but manifestations of spacetime momentum flows that compensate intrinsic self-variation.

We demonstrate that, starting from a set of four axioms, the full gauge structure of the Standard Model emerges naturally.

2. Axiomatic Foundations

• Axiom I — Self-Variation 

Let $q$ denote a generalized intrinsic property of a material particle, whose specific physical realization may correspond to electric charge, rest mass, or generators of an internal symmetry.

The intrinsic self-variation of $q$ is defined through its spacetime transport and is accompanied by a compensating spacetime effective momentum flow $P_{\mu }$, according to

$${\partial }_{\mu }q={\displaystyle \frac{b}{\hslash }}{P}_{\mu }q,$$

(1)

where $b$ is a dimensionless constant and $\hslash$ is the reduced Planck constant.

Equation (1) defines a local and Lorentz-covariant transport law for the intrinsic quantity $q$, with $P_{\mu }$ acting as an effective spacetime connection-like structure generated by self-variation.

• Axiom II — Conservation of Total Momentum 

The material particle and the spacetime effective momentum flow form a unified dynamical system (generalized particle). The total four-momentum

$$C_{\mu }=J_{\mu }+P_{\mu }$$

(2)

where $J_{\mu }$ denotes the particle four-momentum, is conserved along the particle worldline,

$${\displaystyle \frac{d{C}_{\mu }}{d\tau }}=0,$$

(3)

Equation (3) expresses the exchange of four-momentum between the particle and spacetime along the worldline. Interaction is thus described as momentum transfer within the unified system, rather than as a fundamental force.

• Axiom III — Definition of Rest Mass 

The invariant rest mass of the generalized system is defined by

$$M^2{c}^2=C_{\mu }{C}^{\mu },$$

(4)

where the Minkowski metric is used to raise the index.

This rest mass is invariant under the local self-variation of $P_{\mu }$ and $J_{\mu }$, even though these components may vary individually along the particle worldline.

• Axiom IV — Causal Propagation via Action Principle 

There exists an action functional

$$S\left[q\right]={\displaystyle \int L\left(q,{\partial }_{\mu }q,P_{\mu },J_{\mu }\right) }{d}^{4}x,$$

(5)

whose variation with respect to the generalized quantities yields the coupled equations of motion for both the particle and the spacetime momentum flow. This guarantees causal propagation of self-variation through spacetime, while maintaining local Lorentz invariance.

3. Emergence of Abelian Gauge Structure

Defining the spacetime connection

$$A_{\mu }={\displaystyle \frac{b}{\hslash }}{P}_{\mu }.$$

(6)

Equation (1) becomes a covariant transport law for the intrinsic quantity q:

$${\partial }_{\mu }q=A_{\mu }q.$$

(7)

Consistency of mixed derivatives requires the antisymmetric field strength tensor

$$F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu },$$

(8)

which reproduces Maxwell’s equations and ensures charge conservation [1]. Gauge transformations naturally arise as local redefinitions of the phase of $q$,

$$q\to e^{ia\left(x\right)}q, A_{\mu }\to A_{\mu }+{\partial }_{\mu }a\left(x\right),$$

leaving (7) and (8) invariant. Quantum Electrodynamics (QED) emerges as the effective quantum description of this Abelian gauge structure.

4. Renormalization as Self-Variation Dynamics

In conventional QED, renormalization compensates ultraviolet divergences arising from point-like fields.

In the self-variation framework, the intrinsic quantities $q$ are dynamically distributed in spacetime via the effective momentum flow $P_{\mu }$ (or equivalently $A_{\mu }={\displaystyle \frac{b}{\hslash }}{P}_{\mu }$), which prevents singular self-energies. The running of couplings and vacuum polarization arise naturally from the spacetime evolution of $q$, without invoking virtual particle creation. Thus, renormalization is interpreted as a phenomenological effect of self-variation dynamics, rather than a separate postulate [4].

5. Non-Abelian Generalization and QCD

The intrinsic quantity $q$ is generalized to a vector or matrix in an internal space:

$$q\in {ℂ}^N,$$

(9)

$$A_{\mu }={\displaystyle \frac{b}{\hslash }}{P}_{\mu }^{\mathrm{int}}.$$

(10)

The corresponding field strength tensor becomes non-Abelian:

$$F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+\left[A_{\mu },A_{\nu }\right],$$

(11)

reproducing the Yang-Mills structure of QCD [5]. Gauge transformations act as

$$q\to U\left(x\right)q, A\to U{A}_{\mu }{U}^{-1}+\left({\partial }_{\mu }U\right)U^{-1}, U\left(x\right)\in SU\left(N\right),$$

leaving (9)–(11) invariant.

Self-variation requires complete internal momentum balance. Isolated color configurations would generate uncompensated flows and are forbidden, leading naturally to confinement. Stable eigenmodes of the self-variation equations correspond to hadronic states, whose masses are dominated by the spacetime momentum flows.

Confinement

Self-variation requires complete internal momentum balance in the generalized system. Isolated color configurations would generate uncompensated spacetime momentum flows $P_{\mu }^{\mathrm{int}}$ (or equivalently $A_{\mu }$) and are therefore forbidden. Confinement thus emerges naturally as a kinematic consequence of the self-variation dynamics: only color-neutral, globally balanced configurations are allowed [6].

6. Hadron Masses and Spectra

Hadronic states correspond to stable self-variation configurations of the generalized system that satisfy global internal momentum balance.

Their masses are determined by the total spacetime momentum energy of the configuration:

$$m_{hadron}^2=C_{\mu }{C}^{\mu }=\left(J_{\mu }+P_{\mu }^{\mathrm{int}}\right)\left(J^{\mu }+P^{\mu \mathrm{int}}\right),$$

(12)

where $J_{\mu }$ is the particle momentum and $P_{\mu }^{\mathrm{int}}$ is the compensating internal momentum flow. This explains why hadron masses are dominated by interaction energy (spacetime momentum flows) rather than the quark rest masses [6].

7. Chiral Structure and the Electroweak Sector

Time-oriented self-variation of the intrinsic quantity $q$ breaks left–right symmetry at the fundamental level.

Decomposing q into left- and right-handed components,

$$q_L={\displaystyle \frac{1}{2}}\left(1-{\gamma }^5\right)q\mathrm{and} q_R={\displaystyle \frac{1}{2}}\left(1+{\gamma }^5\right)q,$$

the self-variation dynamics naturally lead to a chiral gauge structure for the electroweak interactions:

$$q_L\to U{\left(2\right)}_{L}doublets, q_R\to U{\left(1\right)}_{Y}singlets,$$

without imposing parity violation by hand [7]. The effective gauge fields $A_{\mu }$ couple differently to left- and right-handed components, reproducing the chiral structure observed in the Standard Model [3].

8. Higgs Phenomenon as Vacuum Self-Variation

Mass generation arises from a stable vacuum configuration of self-variation of the intrinsic quantity $q$. In this state, the compensating spacetime momentum flows $P_{\mu }^{\mathrm{int}}$ (or equivalently the connection

$A_{\mu }$) acquire persistent non-zero values, corresponding to gauge boson masses:

$$m_A^2\sim <P_{\mu }^{\mathrm{int}}{P}^{\mu \mathrm{int}}{>}_{vacuum}.$$

This provides a geometric interpretation of the Higgs mechanism, where masses of the $W$ and $Z$ bosons arise naturally from the vacuum self-variation of the generalized system, without introducing an independent scalar field [8].

9. Fermion Generations and Neutrino Mixing

Multiple stable eigenmodes of the self-variation equation for the intrinsic quantity $q$ naturally correspond to the observed three fermion generations. Each eigenmode represents a distinct stable configuration of $q$ and its associated spacetime momentum flow $P_{\mu }^{\mathrm{int}}$.

Neutrino mixing arises from the non-orthogonality of these self-variation eigenstates:

$$<q_i\mid q_j>\ne 0$$

which leads to flavor oscillations without introducing additional ad hoc parameters [9].