Vertex Operators and Soliton Time Delays in Affine Toda Field Theory

TL;DR

This paper investigates how soliton collisions in affine Toda field theory cause time delays or advances, revealing that these are related to vertex operator normal ordering and indicating attractive interactions.

Contribution

It establishes a direct link between soliton time delays and the logarithm of a vertex operator-derived factor, providing new insights into soliton interactions in affine Toda theories.

Findings

Time delay proportional to the logarithm of a vertex operator factor

Vertex operator factor X ranges between 0 and 1

Negative time delay indicates attractive soliton forces

Abstract

In a space-time of two dimensions the overall effect of the collision of two solitons is a time delay (or advance) of their final trajectories relative to their initial trajectories. For the solitons of affine Toda field theories, the space-time displacement of the trajectories is proportional to the logarithm of a number $X$ depending only on the species of the colliding solitons and their rapidity difference. $X$ is the factor arising in the normal ordering of the product of the two vertex operators associated with the solitons. $X$ is shown to take real values between $0$ and $1$. This means that, whenever the solitons are distinguishable, so that transmission rather than reflection is the only possible interpretation of the classical scattering process, the time delay is negative and so an indication of attractive forces between the solitons.