We extend the geometric subsystem quantisation programme to a kink interacting with a finite number of meson (radiation) modes. Using the ^4 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 = 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 M^3\, da dv (which equals -dP da with P=M v) and N blocks 1\, dqⱼ dpⱼ. The off-diagonal coupling terms are given by integrals such as ⱼ f₀'', z f₀' ⱼ, and z f₀'' ⱼ, 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/ (M (v) ³) and qⱼ, pₖ=i\, (v) \, ₉₊ 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.
Timmermans et al. (Thu,) studied this question.