Rapidity amplitudes and their Fourier transforms

Abstract

Use of the rapidity $y$ as a parameter in high-energy multiparticle scattering appears to be natural. In this paper we point out that since $y$ is a longitudinal boost angle, it has a natural canonical conjugate variable, namely the boost operator $K_3$ in the longitudinal direction, and that the eigenvalue $k_3$ of $K_3$, unlike $y$, is an additive quantum number. Hence it might be useful to expand the scattering amplitude in either $y$ or $k_3$. A field theory in $y$ or $k_3$ space is developed in which simplicities in $y$ can be translated into coordinate space, through the relationship between $k_3$ and $z$. The invariant inclusive cross section, which is known to be a simple function of $y$, is related to the field number operator. This gives us a way of distinguishing in principle between different coordinate-space properties of particle production.