Solitons in classical field theories correspond to states in quantum field theories. If the spatial dimension is infinite, then momentum eigenstates are not normalizable. This leads to infrared divergences, which are generally regularized via wave packets or by compactification. However, in some applications both possibilities are undesirable. In the present note, we introduce a finite inner product on translation-invariant kink states that allows us to compute probabilities involving these nonnormalizable states. Essentially, it is the quotient of the usual inner product by the translation group. We present a surprisingly simple formula for the reduced inner product, which requires no knowledge of the zero-mode dependence of the states but includes a correction which accounts for the mixing between zero modes and normal modes as the kink moves. As an application, we show that initial and final state corrections to meson multiplication vanish. However, we find that the pole of the subleading term in the initial state requires an infinitesimal imaginary shift.

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At leading order, there are three inelastic scattering processes beginning with a quantum kink and a fundamental meson. Meson multiplication, in which the final state is a kink and two mesons, was treated recently. In this note we treat the other two, (anti)-Stokes scattering, in which the kink’s shape mode is (de-)excited and the final state contains one meson. In the case of a general scalar kink, we find analytic formulas for the forward and backward scattering amplitudes and probabilities as functions of the momentum of the incident meson. The general results are then specialized to the kink of the ϕ4 double-well model.