Kink coupled to a truncated meson sector: Geometric symplectic structure and a finite‑dimensional approximation to radiative modes

We extend the geometric subsystem quantisation programme to a kink interacting with a finite number of meson (radiation) modes. Using the φ⁴ model as a concrete example, we embed a field configuration consisting of a moving kink plus a linear combination of N discretised continuum meson modes into the field phase space. The exact pullback of the canonical symplectic form Ω = ∫ δφδπ yields a closed two-form on the $(2+2N)$-dimensional parameter space. All symplectic coefficients are expressed explicitly in terms of overlap integrals of the kink profile and the meson mode functions; within the finite-dimensional truncation the calculation is exact. The diagonal blocks contain the free kink translational term Mγ³\, da dv (which equals -dP da with P=Mγ v) and N blocks 1/γ\, dqⱼ dpⱼ. The off-diagonal coupling terms are given by integrals such as ∫ ηⱼ f₀'', ∫ z f₀' ηⱼ, and ∫ z f₀'' ηⱼ, which are non-zero in general. In the non-relativistic small-amplitude regime all off-diagonal terms are small. Using the Faddeev-Jackiw method and neglecting off-diagonal couplings (a valid approximation when the overlap integrals are small compared to the diagonal entries and the meson amplitudes are sufficiently small), we derive the quantum commutators: [ a, v]=i/(Mγ( v)³) and [ qⱼ, pₖ]=i\,γ( v)\,δⱼₖ at leading order. The construction provides a rigorous classical geometric starting point for the quantisation of meson=soliton interactions within a controlled finite-dimensional truncation. The fundamental limitation of replacing the continuum by a finite set of modes is emphasised throughout.