Covariant Derivative & Spin Connection

Covariant Derivative & Spin Connection in PBG

29 Jun 2025 · draft v0.9

Context – Following Appendix AC, we proved the global Clifford relation
${γ^{μ} (x), γ^{ν} (x)} = 2 η^{μ ν} 1 .$
Goal here – Construct the spin connection $ω_{μ}^{\hat{a} \hat{b}}$ that carries the hedgehog’s internal S³ geometry and show that the resulting Dirac operator
$\slashed D χ = γ^{μ} (\partial_{μ} + Γ_{μ}) χ$
reproduces (i) flat-space free propagation outside the core, and (ii) Zeeman-type coupling when an emergent $U (1)$ gauge field $A_{μ}$ is present.

1 · Recap: internal frame & γ’s

We keep the orthonormal frame

e_{\hat{a}}^{b} (x), \hat{a} = 0, 1, 2, 3, e_{\hat{a}}^{b} e_{\hat{b}}^{c} δ_{b c} = η_{\hat{a} \hat{b}} .

Dirac matrices in curved internal space:

γ^{μ} (x) = e_{\hat{a}}^{μ} (x) Γ^{\hat{a}}, {Γ^{\hat{a}}, Γ^{\hat{b}}} = 2 η^{\hat{a} \hat{b}} 1 .

2 · Spin connection from the S³ frame

Define the connection 1-form

ω_{μ}^{\hat{a} \hat{b}} \equiv e_{c}^{\hat{a}} \nabla_{μ} e^{\hat{b} c} = e_{c}^{\hat{a}} (\partial_{μ} e^{\hat{b} c} + Γ_{d μ}^{c} e^{\hat{b} d}) .

External metric is Minkowski ⇒ Christoffels vanish:
$Γ_{d μ}^{c} = 0$ .
Hence in PBG’s hedgehog background

ω_{μ}^{\hat{a} \hat{b}} = e_{c}^{\hat{a}} \partial_{μ} e^{\hat{b} c} .

3 · Fermi connection and covariant derivative

Dirac spinors couple via

Γ_{μ} = \frac{1}{4} ω_{μ}^{\hat{a} \hat{b}} Γ_{\hat{a}} Γ_{\hat{b}}, D_{μ} = \partial_{μ} + Γ_{μ} .

Property

[D_{μ}, γ^{ν}] = 0 \Rightarrow (D_{μ} γ^{ν}) χ = γ^{ν} D_{μ} χ,

ensuring covariance of the Dirac equation.

4 · Free Dirac equation outside the core

In the vacuum region ( $r ≫ k^{- 1}$ ) the frame approaches a constant
and $ω_{μ}^{\hat{a} \hat{b}} \to 0$ .
The Dirac operator reduces to

(i ℏ γ^{μ} \partial_{μ} - m_{0}) χ = 0, m_{0} = \sqrt{β / γ},

matching the free-particle electron equation.

5 · Minimal $U (1)$ coupling (emergent photon)

Local phase $χ \to e^{i θ (x)} χ$ requires the covariant derivative

D_{μ} ⟶ D_{μ}^{(A)} = \partial_{μ} + Γ_{μ} + \frac{i e}{ℏ} A_{μ} .

A one-loop world-sheet determinant (Appendix D) produced the Maxwell term, so the full Dirac–Maxwell system

(i ℏ γ^{μ} D_{μ}^{(A)} - m_{0}) χ = 0, \partial_{μ} F^{μ ν} = 0

is now embedded in PBG without new constants:
$e$ fixed by the long-range Coulomb kernel, $μ_{0}^{- 1}$ by
$α, β, γ$ .

6 · Zeeman & Stern–Gerlach test

Linearise $A_{μ} = (0, A)$ with a uniform $ \mathbf B=\nabla\times\mathbf A$.
Standard Foldy-Wouthuysen expansion yields magnetic moment

μ = \frac{e ℏ}{2 m_{0}},

matching the Dirac $g = 2$ value once $m_{0}$ was identified with $m_{e}$ .
Hence spin-splitting and SG deflection emerge with the correct Landé factor.

7 · Key take-aways

Spin connection $ω_{μ}^{\hat{a} \hat{b}}$ follows directly from the internal S³ frame; no external curvature needed.
The covariant derivative $D_{μ}$ preserves the γ-algebra globally, proving full Clifford consistency.
Coupling to the emergent $U (1)$ potential gives the Dirac–Maxwell system with no extra constants beyond the anchoring quartet.

8 · Remaining formal checks

Loop renormalisation: quantify whether internal torsion yields Schwinger’s $α / 2 π$ correction.
Anomaly cancellation: verify axial anomaly vanishes when left- and right-handed hedgehog modes are counted (ongoing).

(End Appendix AD)