31.02. Exact local non‑linear symplectic reduction for the sine–Gordon kink: rigorous separation of the translational zero mode without truncation

We construct a local Marsden–Weinstein reduction of the sine–Gordon phase space (built on the Fr\'echet space of smooth functions with all derivatives square‑integrable) by the translation group. The result is an exact symplectic manifold \(\mathcal{M}\) — the meson phase space — together with the exact reduced Hamiltonian, both defined in a neighbourhood of the kink. No linearisation or truncation is used; the translational zero mode is separated exactly. At the linearised level the reduced symplectic form coincides with the canonical P"oschl–Teller meson form. The Hamiltonian expands into the classical kink mass, a quadratic meson term, and higher‑order interactions, with no term linear in the meson variables. The construction avoids global quotient issues by working solely with a local slice; it provides a rigorous geometric foundation for the quantisation of the kink sector.