Vertex Operators and Soliton Time Delays in Affine Toda Field Theory

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.