Faddeev–Jackiw Quantisation & Emergent Fermionic Brackets

30 Jun 2025 · draft v0.8

Goal – Convert the classical spinor zero–mode of a charge-one hedgehog into a quantum field obeying anticommutation relations without inserting them by hand.
Method: apply Faddeev–Jackiw (FJ) symplectic reduction to the collective-coordinate Lagrangian that already contains a half-integer Wess–Zumino (WZ) term.

1 · Collective coordinates for a single hedgehog

Position $R (t)$
Internal orientation $g (t) \in SU (2)$

Lowest-order time-dependent ansatz

N^{a} (x, t) = g (t) N_{0}^{a} (x - R (t)), χ (x, t) = g (t) χ_{0} (x - R (t)) .

2 · Reduced first-order Lagrangian

After spatial integration the spin/rotator sector is

\begin{matrix} (2.1) & L_{rot} = \frac{I}{2} T r (g^{- 1} \dot{g})^{2} + \frac{1}{2} T r (Λ g^{- 1} \dot{g}), \end{matrix}

$I$ : moment-of-inertia ∝ $α k^{- 1}$ (finite).
$Λ = ℓ σ^{3}$ comes from the WZ term; PBG solitons carry $ℓ = \frac{1}{2}$ because $Q = 1$ .

Write $g^{- 1} \dot{g} = {\dot{q}}^{i} σ^{i}$ ; Eq. (2.1) becomes

L_{rot} = I {\dot{q}}^{i} {\dot{q}}^{i} + ℓ {\dot{q}}^{3} .

The linear term is the hallmark of a first-order FJ system with
symplectic one-form

ϑ = ℓ d q^{3} .

3 · FJ brackets

The symplectic two-form is

Ω = d ϑ = ℓ d q^{1} \land d q^{2} .

Its inverse gives FJ brackets

{q^{1}, q^{2}}_{FJ} = \frac{1}{ℓ} = 2 .

Define canonical pair

a \equiv \sqrt{ℓ} q^{1} + i \sqrt{ℓ} q^{2}, a^{†} = \sqrt{ℓ} q^{1} - i \sqrt{ℓ} q^{2} .

Then

{a, a^{†}}_{FJ} = 4 ℓ {q^{1}, q^{2}}_{FJ} = 4 ℓ \cdot \frac{1}{ℓ} = 4 .

Normalise $b \equiv a / 2$ ⇒

{b, b^{†}}_{FJ} = 1 .

4 · Quantisation prescription

Faddeev–Jackiw tells us to promote

{\cdot, \cdot}_{FJ} ⟶ \frac{1}{i ℏ} [\cdot, \cdot]_{\pm} .

Because the two-form is non-degenerate and first-order, the brackets
are anticommutators:

{\hat{b}, {\hat{b}}^{†}} = 1, {\hat{b}, \hat{b}} = 0 .

Thus a single hedgehog zero-mode is fermionic.

5 · Many-mode generalisation

For plane-wave expansion

χ (x) = \sum_{k} ({\hat{b}}_{k} u_{k} (x) + {\hat{d}}_{k}^{†} v_{k} (x)),

symplectic additivity gives

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

All mixed anticommutators vanish.

6 · Matching standard spin statistics

Exchange of two hedgehogs ⇒ Berry phase $- 1$ (Appendix AC).
Quantisation via FJ ⇒ operator anticommutation just derived.

The spin–statistics link is now complete inside PBG with no postulated algebra or Grassmann ghost fields in the bulk.

7 · Sanity check: white-dwarf equation of state

Density of states per spinor
$g = 2, D (ε) = \frac{V}{π^{2}} \frac{ε^{2}}{ℏ^{3} c^{3}}$ .
Fermi exclusion already enforced ⇒ pressure

P = \frac{(3 π^{2})^{2 / 3} ℏ c}{4} n^{4 / 3},

matching the standard Chandrasekhar scaling—without ever inserting
canonical anti-commutators by hand.

Result: The half-integer WZ coefficient fixed by hedgehog topology, together with Faddeev–Jackiw reduction, yields the full fermionic operator algebra. No external postulates of Fermi–Dirac statistics are needed.

(End Appendix AE)