A Reduced Inner Product for Kink States

TL;DR

This paper introduces a finite inner product for kink states in quantum field theories, enabling probability calculations for non-normalizable states without regularization, and applies it to meson multiplication analysis.

Contribution

It presents a simple, zero-mode independent formula for a reduced inner product of kink states, addressing infrared divergence issues in infinite spatial dimensions.

Findings

Initial and final state corrections to meson multiplication vanish.

The pole of the subleading term requires an infinitesimal imaginary shift.

The method simplifies calculations involving non-normalizable 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.