Predicting Anomalous g-2 Magnetic Moment of Muon and Other Leptons: A Superior Alternative to the Standard QED Using Sedenionic Gauge Symmetry

2.1. Hypercomplex Gauge and Lagrangian

The foundation of our approach lies in extending the algebraic structure of spacetime beyond complex numbers and standard Clifford algebras. We begin with quaternions, discovered by Hamilton, which extend complex numbers into a 4-dimensional noncommutative system with units {1, i, j, k} satisfying i² = j² = k² = ijk = -1. Octonions, introduced by Graves and Cayley, further extend quaternions to an 8-dimensional system with units {$e_0$,${ e}_1,$…, $e_7$}, characterized by noncommutativity and non-associativity. Sedenions expand this structure to 16 dimensions and include basis elements {$e_0$,${ e}_1,$…, $e_{15}$}, preserving power associativity but losing division algebra properties.

Unlike the Dirac gamma matrices, which form a representation of the Clifford algebra Cl(1,3) used to describe spin-½ particles in Minkowski spacetime, our model uses octonion and sedenion algebra as the gauge structure itself. This allows us to go beyond the Clifford framework by encoding internal degrees of freedom, such as mass and charge, as geometric curvature within a higher-dimensional non-associative space.

We define the octonionic gauge field ${\mathcal{A}}_{\mu }$ as:

$${\mathcal{A}}_{\mu }={\sum }_{i=1}^7{A}_{\mu }^{\left(i\right)}{e}_i,$$

where $A_{\mu }^{\left(i\right)}$ are real-valued gauge components and e₄–e₇ span the internal spinor curvature for first-generation leptons (electron-like). The sedenionic gauge field $ℬ$_μ generalizes this:

$${\mathcal{B}}_{\mu }={\sum }_{i=1}^{15}{B}_{\mu }^{\left(i\right)}{\mathrm{e}}_{\mathrm{i}},$$

allowing generation-dependent curvature sectors: $e_0$ to $e_7$: electron; $e_0$ to $e_{11}$: muon; $e_0$ to $e_{15}$: tau. The quaternion quartet $e_0$ to $e_7$ is responsible for the Coulomb gauge, which is the cornerstone for QED calculations.

The total covariant derivative acting on a spinor Ψ is given by:

$D_{\mu }$Ψ = ${\partial }_{\mu }$Ψ + ${\mathcal{A}}_{\mu }$ Ψ + ${\mathcal{B}}_{\mu }$Ψ, with ${\mathcal{A}}_{\mu }$ and ${\mathcal{B}}_{\mu }$ acting as hypercomplex-valued gauge potentials. The Lagrangian density for a charged lepton in this framework is: $ℒ$ = Ψ̄ (i${\gamma }^{\mu }{D}_{\mu }$- m) Ψ - ¼ Tr($F_{\mathsf{\mu }\mathsf{\nu }}{F}^{\mathsf{\mu }\mathsf{\nu }}$) - ¼ Tr($G_{\mathsf{\mu }\mathsf{\nu }}{G}^{\mathsf{\mu }\mathsf{\nu }}$), where $F_{\mathsf{\mu }\mathsf{\nu }}$ = ${\partial }_{\mu }{\mathcal{A}}_{\nu }$ - ${{\partial }_{\nu }\mathcal{A}}_{\mu }$ + [${\mathcal{A}}_{\mu }$ ,${\mathcal{A}}_{\nu }$ ] and similarly for $G_{\mathsf{\mu }\mathsf{\nu }}$using ${\mathcal{B}}_{\mu }$. The commutators are generalized to accommodate non-associative multiplication using Jordan-like symmetrization.

This Lagrangian encodes both external (Coulomb) and internal curvature, allowing us to derive finite and nonperturbative contributions to the lepton magnetic moment without relying on loop corrections or renormalization.

2.2. Magnetic Moment Derivation

In this framework, the anomalous magnetic moment of a charged lepton emerges as a finite, recursive geometric correction due to curvature in the internal spinor space described by octonion and sedenion gauge fields. Our approach deviates fundamentally from conventional QED, which derives the anomaly perturbatively via loop expansions and vacuum fluctuations. Instead, we model the internal structure of leptons as recursive interactions between spinor modes, each weighted by geometric damping and modulated by triality symmetry.

The total magnetic anomaly a_ℓ for a lepton ℓ ∈ {e, μ, τ} is given by:

$a_{\mathcal{l}}$ = ${a_{\mathcal{l}}}^{\mathrm{Q}\mathrm{E}\mathrm{D}}$ + ${\Delta a_{\mathcal{l}}}^{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{l}}$, ${a_{\mathcal{l}}}^{\mathrm{Q}\mathrm{E}\mathrm{D}}$ includes the known perturbative contributions, and ${\Delta a_{\mathcal{l}}}^{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{l}}$, arises from curvature-induced recursive terms. We define the internal contribution as:

$a_{\mathcal{l}}$ = Σₙ₌₁∞ [cₙ(ε, k) xⁿ] ⋅ (ξ² + cₙ xⁿ ⋅ ξ⁴), where ξ = 1 / (π $\times$137.035999206) is the geometric normalization constant tied to internal compactification, ε is the triality modulation amplitude, and k governs the exponential damping of curvature propagation.

The recursive coefficients cₙ are generation-specific and determined by dominant contributions from different internal contributions on top of the contribution from the Coulomb gauge: e₄–e₇: electron; e₄––e₁₁: muon;e₄–e₁₅: tau. Each recursive layer contributes to the internal curvature field that subtly shifts the effective magnetic moment. Unlike QED, our method produces finite results at each stage without requiring renormalization. The convergence of the series is guaranteed by exponential damping, and the generation hierarchy emerges naturally from the magnitude and structure of internal spinor curvature.

By fitting only a few parameters—triality weight ε, damping k, and anchor curvature λ—we reproduce the known lepton magnetic moments within experimental uncertainties. This result is achieved without invoking virtual particle loops, vacuum polarization, or wavefunction collapse, illustrating the power of internal geometric structure in describing quantum attributes of elementary particles.

2.3. Magnetic Moment Derivation

In this framework, the anomalous magnetic moment of a charged lepton emerges as a finite, recursive geometric correction due to curvature in the internal spinor space described by octonion and sedenion gauge fields. Our approach deviates fundamentally from conventional QED, which derives the anomaly perturbatively via loop expansions and vacuum fluctuations. Instead, we model the internal structure of leptons as recursive interactions between spinor modes, each weighted by geometric damping and modulated by triality symmetry.

The total magnetic anomaly a_ℓ for a lepton ℓ ∈ {e, μ, τ} is given by:

$a_{\mathcal{l}}$ = ${a_{\mathcal{l}}}^{\mathrm{Q}\mathrm{E}\mathrm{D}}$ + ${\Delta a_{\mathcal{l}}}^{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{l}}$, ${ a_{\mathcal{l}}}^{\mathrm{Q}\mathrm{E}\mathrm{D}}$ includes the known perturbative contributions, and ${\Delta a_{\mathcal{l}}}^{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{l}}$, arises from curvature-induced recursive terms. We define the internal contribution as:

$a_{\mathcal{l}}$ = Σₙ₌₁∞ [cₙ(ε, k) xⁿ] ⋅ (ξ² + cₙ xⁿ ⋅ ξ⁴), where ξ = 1 / (π $\times$137.035999206) is the geometric normalization constant tied to internal compactification, ε is the triality modulation amplitude, and k governs the exponential damping of curvature propagation

The recursive coefficients cₙ are generation-specific and determined by dominant contributions from different internal contributions on top of the contribution from the Coulomb gauge: $e_0$–$e_3$; electron; e₄–$e_7$; muon; $e_8$–$e_{11};$tau: $e_{23}$–$e_{15}.$Each recursive layer contributes to the internal curvature field that shifts the effective magnetic moment. Unlike QED, our method produces finite results at each stage without requiring renormalization. The convergence of the series is guaranteed by exponential damping, and the generation hierarchy emerges naturally from the magnitude and structure of internal spinor curvature.

By fitting only a few parameters—triality weight ε, damping k, and anchor curvature λ—we reproduce the known lepton magnetic moments within experimental uncertainties. This result is achieved without invoking virtual particle loops, vacuum polarization, or wavefunction collapse, illustrating the power of internal geometric structure in describing quantum attributes of elementary particles.

2.4. Derivation of Lepton Magnetic Moments

Using the recursive curvature model developed in our framework, we now derive the anomalous magnetic moments for each charged lepton: electron, muon, and tau. Each anomaly receives contributions from generation-dependent internal spinor curvature, with geometric modulation and damping that reflect the compactification of higher-dimensional internal space. The g-2 value for each generation of the charged lepton is given in Table 1.

One of the most significant outcomes of this model is its ability to resolve the longstanding muon g−2 anomaly. We can obtain these values so precisely, in agreement with the experiments for the electron and muon to the exceedingly high precision. Its success lies in carefully adjusting the internal spinor curvature contributions associated with sedenionic triality. In our model calculation, triality weights due to S₃ symmetry: λ₁ = 1.00000, λ₂ = 1.00002, λ₃ = 1.00003, and a global correction factor k = 1.00001 were used. The almost identical λ parameters imply the nearly perfect S₃ spherical symmetry for the electron’s internal structure, like a the s-orbital in the atomic electronic structure. Because the muon and tau have a much greater mass, one may regard them as having asymmetric internal structure like a p- and d-orbital in an atomic electronic structure.

For anisotropic muon and tau, we list In Table 2 the geometric expressions, numerical evaluations, and interpretations of the curvature coefficients and triality weights used in our model. The values of c₂ and c₃ are derived from hype-spherical ratios involving π, while λ₂ and λ₃ are linked to symmetry breaking in internal spinor space, incorporating the fine-structure constant. These parameters are central to the recursive expansion governing lepton magnetic moment predictions.