09.01. Exact Symplectic Form and Hamiltonian on the Kink-Antikink Superposition Ansatz

We compute the exact pullback of the canonical field‑theoretic symplectic form and the exact field energy to the four‑dimensional parameter space of a kink–antikink superposition ansatz in a general relativistic scalar field theory with degenerate vacua. The diagonal blocks of the symplectic matrix reproduce the free‑particle forms $dP_i\wedge da_i$ with the relativistic momenta $P_i = M\gamma_i v_i$. The six off‑diagonal interaction components are expressed in closed form through overlap integrals of the static kink profile. The Hamiltonian splits naturally into the sum of the free relativistic energies of two kinks plus an exact interaction potential $\mathcal{V}$ that vanishes exponentially with the separation. In the asymptotic regime of large separation the off‑diagonal symplectic couplings are exponentially suppressed, enabling a formal Darboux diagonalisation. We perform the transformation to the canonical momentum variables and discuss the resulting complete classical system as the rigorous starting point for deformation quantisation of the interacting two‑kink dynamics. This work opens the path to a rigorous deformation quantisation of the interacting kink–antikink system.