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.

arXiv comment46 pages, no figures

References in corpus (23)

• Cosmic Structure as the Quantum Interference of a Coherent Dark Wave
• Quantum Radiation of Oscillons
• Anomaly and quantum corrections to solitons in two-dimensional theories with minimal supersymmetry
• Solvable self-dual impurity models
• Manifestly Finite Derivation of the Quantum Kink Mass
• Two-Loop Scalar Kinks
• Quantum Corrections to Soliton Energies
• Quantum Instabilities of Solitons
• Spectral Walls at One Loop
• A Smörgåsbord of Skyrmions
• Spontaneous Emission from Excited Quantum Kinks
• Normal Ordering Normal Modes
• The Forced Soliton Equation and Semiclassical Soliton Form Factors
• Wobbling kinks in a two-component scalar field theory: Interaction between shape modes
• Excited Kinks as Quantum States
• Normal modes for two-dimensional gravitating kinks
• Quantum Corrections to Solitons in the ${\Phi }^8$ Model
• Singular Pöschl-Teller II potentials and gravitating kinks
• Accelerating Solitons
• The semiclassical energy density of kinks and solitons
• Vacuum polarization energy of the Shifman-Voloshin soliton
• Leading Quantum Correction to the ${\Phi }^4$ Kink Form Factor
• Analytical computation of quantum corrections to non-topological soliton (bright soliton) within the saddle-point approximation

Cited by in corpus (1)

• (Anti-)Stokes Scattering on Kinks