Perturbative Darboux Transformation and Deformation Quantization of the Kink-Antikink System - Explicit diagonalisation and the star product for interacting solitons

Starting from the exact canonical symplectic form and Hamiltonian on the four‑dimensional parameter space of a kink‑antikink superposition ansatz, we construct an explicit perturbative Darboux transformation to all orders in the small parameter = e^- s, where s is the kink‑antikink separation and the decay rate of the static kink. Using Moser's trick and the homotopy operator on the phase space, we obtain an explicit primitive for the off‑diagonal part of the symplectic form and solve for the generating vector field. The resulting Darboux coordinates are used to define a rigorous deformation quantisation of the interacting two‑kink system: the Moyal star product in the Darboux coordinates is pulled back to the original parameter space, providing an associative star product that deforms the exact Poisson bracket. The classical Hamiltonian is transformed accordingly, yielding the quantum Hamiltonian operator to leading order in. The construction is model‑independent and applies to any relativistic scalar field theory with a topological kink.