A reduced inner product for kink states

Abstract

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.

(Anti-)Stokes scattering on kinks

“…In ref. [27] we describe how such inner products may be calculated systematically, by dividing the numerator and denominator by the translation group. There are corrections with respect to the naive manipulations above, as a result of the nondiagonal action of the translation operator in the kink frame.…”

“…However, the same matrix elements appear in the numerator and denominator, and so fortunately they cancel. In a companion paper [27] we treat such ratios more carefully, dividing by the translation symmetry so that the numerator and denominator are both finite. We find that indeed there are corrections to the results obtained via a naive cancellation.…”

“…If one wishes to compute loop corrections, however, the corrections found in ref. [27] must be included as they enter at the same order.…”

Form factors for meson-kink scattering

“…The results here justify some assumptions made in Refs. [23,25,24] regarding quantum corrections to the initial and final states in inelastic meson-kink scattering. In the short term, this was the last missing ingredient needed for the study of several problems.…”

“…[23] and by beginning with a full Hamiltonian eigenstate in Ref. [25], reflecting the two strategies in nonrelativistic quantum mechanics described above.…”