A reduced inner product for kink states

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