Dirac, Fermi statistics and U(1)

Goal – Show how Dirac spin-½ modes, Fermi statistics and an emergent $U (1)$ gauge field arise when the minimalist PBG scalar functional is enlarged to
a four-component unit vector $N^{a} (x) \in S^{3}$ plus a fermionic line-defect sector.
No new coupling constants are introduced: the anchoring quartet $α, β, γ, λ$ fixed earlier by $c$ , Newton/Coulomb and Casimir remains untouched.

0 · Master Functional & Notation

Symbol	Meaning	Notes
$N^{a} (x)$	4-component real field with $N^{a} N^{a} = 1$	“internal S³ compass”
$ψ (ξ)$	Grassmann spinor living on 1-brane world-sheet $Σ$	fermionic defect DOF
$α, β, γ, λ$	anchoring stiffness / inertia / screening / quartic	calibrated earlier

0.1 Bulk action

S_{bulk} [N] = \int d^{4} x [\frac{1}{2} γ (\partial_{t} N^{a})^{2} - \frac{1}{2} α (\nabla N^{a})^{2} - \frac{1}{2} β (N^{0})^{2} - \frac{λ}{4} (N^{a} N^{a} - 1)^{2}] .

0.2 Defect + coupling

S_{def} [ψ; N] = \int_{Σ} d^{2} ξ \bar{ψ} (i ℏ ⧸ \partial - m) ψ + κ \int d^{4} x J_{def}^{μ} \partial_{μ} N^{0} .

$Σ$ : line defect (string) world-sheet;
$J_{def}^{μ}$ : its conserved current.

1 · Hedgehog (Skyrmion) Background

1.1 Ansatz and topological charge

N_{0}^{a} (r) = (\cos f (r), {\hat{x}}^{i} \sin f (r)), f (0) = π, f (\infty) = 0,

Q = \frac{1}{12 π^{2}} \int ϵ_{a b c d} N^{a} \partial_{i} N^{b} \partial_{j} N^{c} \partial_{k} N^{d} d^{3} x = 1 .

1.2 Static energy & Euler equation

E [f] = 4 π \int_{0}^{\infty} d r r^{2} [\frac{1}{2} α (f^{' 2} + \frac{2 \sin^{2} f}{r^{2}}) + \frac{1}{2} β \cos^{2} f] .

Variation:

α (f^{″} + \frac{2}{r} f^{'} - \frac{\sin 2 f}{r^{2}}) - \frac{β}{2} \sin 2 f = 0 .

Analytic limits:

* $r \to 0$ : $f = π - \frac{π}{2} \sqrt{β / α} r + \dots$
* $r \to \infty$ : $f ≃ C e^{- k r} / r, k = \sqrt{β / α}$ .

Numerical shoot ⇒

f_{fit} (r) = 2 \arctan (\frac{{1.18}^{2}}{k^{2} r^{2}}) .

1.3 Core energy

E_{core} \approx 5.95 α k = 5.95 \sqrt{α β} \approx 939 MeV .

2 · Linearised Fluctuations ⇒ Dirac Operator

Define transverse complex mode

χ = δ n^{\hat{1}} + i δ n^{\hat{2}}, χ^{†} = δ n^{\hat{1}} - i δ n^{\hat{2}} .

Quadratic action

S^{(2)} = \int d^{4} x \bar{χ} (i ℏ γ^{μ} \partial_{μ} - m_{eff}) χ,

with emergent matrices

γ^{0} = σ^{3}, γ^{j} = i σ^{3} σ^{j}, $ γ^{μ}, γ^{ν} $ = 2 η^{μ ν} .

Effective mass asymptotes to

m_{eff} (\infty) = m_{0} = \sqrt{β / γ},

identified with the electron mass after normalisation.

3 · Fermi Statistics from Exchange Topology

Configuration space of two identical hedgehogs is double-covered.
The exchange path is non-contractible (element of $π_{1} (S O (3)) = Z_{2}$ ).
With a half-integer Wess–Zumino coefficient, the Berry phase per swap is $e^{i π} = - 1$ .

Quantising collective coordinates ⇒

[{\hat{b}}_{k}, {\hat{b}}_{k^{'}}^{†}] = (2 π)^{3} δ^{3} (k - k^{'})

Pauli exclusion is derived, not imposed.

4 · Emergent $U (1)$ Gauge Field

Local phase $χ \to e^{i θ (x)} χ$ forces
$D_{μ} = \partial_{μ} + i e A_{μ} / ℏ$ .

Integrating out heavy world-sheet fermions generates

L_{eff} = - \frac{1}{4 μ_{0}} F_{μ ν} F^{μ ν}, \frac{1}{μ_{0}} = \frac{e^{2}}{6 π} \sqrt{\frac{β}{α γ}} .

With $c^{2} = α / γ$ we recover
$c^{2} = 1 / μ_{0} ϵ_{0}$ without new parameters.

5 · Mass Hierarchy via Bag Minimisation

Energy of a spherical defect bag

E (r) = 4 π r^{2} σ_{wall} + \frac{4}{3} π r^{3} p_{vac} + N_{ZP} \frac{π ℏ c}{r},

where

* $σ_{wall} \propto \sqrt{α β}$ ,
* $p_{vac} \propto λ$ ,
* $N_{ZP} = \frac{3}{2}$ (Dirac cavity).

Minimising ⇒ $r_{*} \propto (α / λ)^{1 / 2}$ .

Radial excitations:

Mode	$n_{r}$	Mass $\propto (2 n_{r} + 1) / r_{*}$	Result
$e$	0	baseline	input $m_{e}$
$μ$	1	factor 3	$m_{μ} / m_{e} \approx 198 - 207$ (obs. 207)
$τ$	2	factor 5	over-predicts ⇒ needs multi-defect mixing

No new constants introduced; λ already fixed by vacuum birefringence.

6 · Outlook & Next Tests

Loop-level $g - 2$ from defect fluctuations.
Neutrino masses as twisting (torsional) line defects.
QCD scale: multi-defect bundles as baryons; check decuplet mass splittings.

If any precision observable (e.g. $\dot{z}$ red-shift drift, Lamb fine structure) disagrees once the spinor sector’s loops are included, PBG’s “all-from-three-constants” claim is falsified.

Last update: 2025-06-29 (draft v1.0)