The diagonalisation of the Lund fragmentation model I

We will in this note show that it is possible to diagonalise the Lund fragmentation model. We show that the basic original result, the Lund area law, can be factorised into a product of transition operators, each describing the production of a single particle and the two adjacent break up points (vertex positions) of the string field. The transition operator has a discrete spectrum of (orthonormal) eigenfunctions, describing the vertex positions (which in a dual way correspond to the momentum transfers between the particles produced) and discrete eigenvalues, which only depend upon the particle produced. The eigenfunctions turn out to be the well-known two-dimensional harmonic oscillator functions and the eigenvalues are the analytic continuations of these functions to timelike values (corresponding to the particle mass). In this way all observables in the model can be expressed in terms of analytical formulas. In this note only the 1+1-dimensional version of the model is treated, but we end with remarks on the extensions to gluonic radiation, transverse momentum generation etc., to be performed in future papers. Received: 7 April 2000 / Published online: 18 May 2000