Quantum Corrections to the Kink Dispersion Relation from an Internal Shape Mode

We compute the leading quantum correction to the energy‑momentum relation of a kink that possesses an internal shape mode. The analysis is model‑independent: it assumes only the existence of a linearised internal mode (which may be approximate, as in the double sine–Gordon wobbling kink) and the coupling term derived from the geometric subsystem quantisation programme. After a perturbative Darboux diagonalisation of the symplectic form, the classical Hamiltonian in the non‑relativistic, small‑amplitude regime takes the universal form \ (H = M + P^22M + 12 (p^2+^2q^2) + M\, P\, q\), where \ (q, p\) are the shape‑mode canonical variables, \ (\) is the mode frequency, and \ (\) is a model‑dependent coupling constant. Quantisation via the Moyal product yields a Schr\"odinger operator with the same form. Second‑order perturbation theory in \ (\) gives a momentum‑dependent energy shift \ (E (P) = -^22M^{2^2}\, P^2 + O (P^4) \). This term reduces the curvature of the energy‑momentum relation, corresponding to an increase in the kink’s effective kinetic mass. The result provides the first concrete quantum observable computed entirely within the geometric subsystem programme, applicable to any kink with a genuine or approximate internal mode.