05. Geometric Subsystem Quantization of the Sine-Gordon Breather

We quantize the moduli space of the sine--Gordon breather by the geometric subsystem method. The correct canonical internal coordinate is identified as the phase evaluated at the breather centre, \(\theta = \varphi - \omega\gamma v a\). Using translation invariance and the explicit symplectic form on the static slice, together with a rigorous global extension via the Hamiltonian action of Lorentz boosts, we prove that the pulled‑back 2‑form is globally \[ \omega_{br}= da\wedge dP + d\theta\wedge dI, \] with \(P = \frac{16v\sqrt{1-\omega^{2}}}{\sqrt{1-v^{2}}}\) and \(I = 16\arccos\omega\). Global Darboux coordinates are therefore \((Q,P,\theta,I)\). The Moyal product in these coordinates gives a rigorous formal deformation quantization. All gaps of previous versions are closed, and the derivation is largely self‑contained.