Symplectic Geometry and Canonical Quantization of the φ⁴ Kink Coupled to Its Shape Mode

We construct the exact finite-dimensional symplectic structure of a kink coupled to its discrete shape modes in $(1+1)$-dimensional scalar field theory, on the collective-coordinate ansatz that retains the kink position, its scale (Derrick) parameter, and the discrete shape-mode amplitudes. This work is part of the geometric subsystem quantisation programme. We develop the theory in two stages. First, treating the scale (Derrick) parameter b as an independent collective coordinate, we compute the exact kinetic metric gᵢⱼ on the space (a,b,qA), derive its explicit components for the reflectionless P\"oschl--Teller family, and prove that the pullback of the field-theoretic symplectic form is closed, with explicit Darboux coordinates. For the reflectionless P\"oschl--Teller family with a single shape mode (the φ⁴ kink), the metric is shown to be non-degenerate for all q and $b>0$; the general multimode case is left open. Second, under the velocity-dependent adiabatic specialization b=γ(v) (not a holonomic constraint) that fixes the kink profile to be Lorentz-contracted, the configuration space reduces to (a,qA), the Lagrangian becomes L=-A(q)/γ+ q²/(2γ) with A=12 Meff+W, and the canonical momentum acquires a pq² correction. We compute the exact pullback of the field-theoretic symplectic form to the constraint surface b=γ(v), show that it is closed and non-degenerate outside a small ultrarelativistic degeneracy locus, and construct its Darboux transformation; it is not equal to the canonical form dpₐ da+dpq dq of the reduced Lagrangian: the two agree at $q=0$ in the strict non-relativistic limit v→ 0, q→ 0, while for general q they differ even in that limit by the Derrick discrepancy (12 Meff(q)-W(q))\,dv da. We extend the reduction to include acceleration corrections a≠ 0; the resulting Lagrangian is higher-derivative in a and admits either an Ostrogradsky or a Dirac treatment. Quantization is performed by Weyl symmetrization in the Darboux variables of the reduced Lagrangian; an alternative Laplace--Beltrami prescription is discussed, agreeing at leading order in and differing at O(²) by geometric terms. We show that the leading-order Hamiltonian has no ordering ambiguity, while the next-to-leading order acquires both - and ²-corrections. Essential self-adjointness of the quantum Hamiltonian is left as an open problem, as are the numerical spectrum, the multimode extension, the continuum contribution, the quantum acceleration corrections, and the Dirac analysis of the velocity-dependent constraint b=γ(v).