Some new mathematical features in cascade theory

Abstract

The fluctuation problem in cascade theory is viewed from the standpoint of the invariant imbedding method of Bellman et al. [1]. The correlation functions that are used in the description of the electromagnetic cascades are shown to obey a simple system of two component vector differential equations. The advantage of the present method over the conventional approach of writing down Kolmogorov forward equations for these functions lies in that we encounter 2×2 matrices of a particular type only. In view of the simplicity of the structure it is possible to generalize the equations to correlation functions of arbitrary order. The reduction in dimension from 2n×2n matrices to only 2×2 matrices which may, at first sight, appear perplexing is due to the fact that each of the 2n elements that appear in the single system of differential equations, corresponding to the two different initial conditions, can be obtained by considering 2n disjoint systems of two component vector equations. The imbedding technique is also used to arrive at the independent differential equations satisfied by sequent product densities that are encountered in more comprehensive description of electromagnetic cascades.