03. Semiclassical quantisation of the sine–Gordon kink: Quartic kinetic vertex, Bose symmetrisation, and a finite two‑loop energy shift

The rigorous local Marsden–Weinstein symplectic reduction of the sine–Gordon phase space is combined with an explicit flat Darboux parametrisation of the reduced meson phase space. The exact quantum Hamiltonian in canonical meson variables is derived, revealing new interaction vertices beyond the standard linearised expansion. The functional measure is shown to be the flat Liouville measure. The two‑meson matrix element of the quartic kinetic vertex is symmetrised according to Bose statistics, yielding a fully symmetric, smooth amplitude free of Dirac deltas and proportional to (p²-k²) ² up to momentum‑dependent denominators. The four‑meson interaction Hamiltonian is obtained with the correct combinatorial factors, leading to an explicit, absolutely convergent three‑dimensional integral for the two‑loop energy shift coming from this vertex. No counterterm subtraction is needed for this contribution. The one‑loop kink mass correction is recovered correctly, and the two‑loop result is presented in a manifestly finite form ready for numerical evaluation.