Geometric Subsystem Quantization of the Double Sine-Gordon Wobbling Kink: A two‑subkink approximation and exact symplectic pullback

We quantize the internal shape mode (wobbling mode) of the double sine–Gordon kink within the geometric subsystem programme. Using the two‑subkink approximation, which becomes accurate for small coupling, the static kink profile depends on a separation parameter R, and an approximate shape mode function is obtained as (x) = f₀/ R evaluated at the equilibrium separation R₀. This function is smooth, exponentially localized, and orthogonal to the translational zero mode. We embed a linearised wobbling ansatz into the field phase space and rigorously pull back the canonical symplectic form, obtaining a canonical pair (Q, P) for the wobbling amplitude and phase. The classical Hamiltonian, evaluated to quadratic order in the amplitude within the two‑subkink approximation, reduces to that of a harmonic oscillator with an effective frequency b. Deformation quantisation via the Moyal product then yields a quantum harmonic oscillator describing the internal excitation spectrum. The derivation is exact at the level of the symplectic pullback, while the harmonic oscillator reduction relies on the two‑subkink approximation; the resulting quantum mechanics is an effective description of the wobbling mode.