02.02. Geometric Subsystem Quantization of the Double Sine-Gordon Kink

We extend the geometric subsystem quantisation programme, initiated for the sine–Gordon kink, to the double sine–Gordon (DSG) equation --- a non‑integrable scalar field theory with a degenerate double‑well potential. The travelling‑wave (kink) sector is parametrised by the centre position a and the velocity v. We immerse this moduli space into the classical phase space (the PTSO~{TSOhierarchy}) of the DSG field; after restriction to a sufficiently small neighbourhood its image is a two‑dimensional symplectic submanifold (a local embedding). The pullback of the canonical field‑theoretic symplectic form is computed explicitly and yields the canonical Darboux form dP da with the relativistic momentum P = Mv/√1-v², where M is the static kink mass. Deformation quantisation via the Moyal product then gives the canonical commutation relation [a,P]=i. The derivation uses only the existence of a finite‑energy topological kink and the Lorentz invariance of the theory; it does not require integrability. The physical result reproduces the standard collective‑coordinate Hamiltonian treatment of the DSG kink~{Willis1987,Sodano1986}, but the method --- direct pullback of the field‑theoretic symplectic form and deformation quantisation --- provides a rigorous geometric foundation for the quantum mechanics of non‑integrable solitons within the PTSO framework.