Averages on the unitary group and applications to the problem of disordered conductors

In a recent approach to the study of disordered conductors a maximum-entropy criterion is used to perform statistical calculations directly on the transfer matrix for the full physical system. Quantities of great importance in that approach are averages of products of the u and u* matrix elements (u being a unitary matrix) evaluated with the invariant measure of the unitary group. The authors evaluate those averages that contain up to four u and four u* matrix elements, since they are the ones needed in the calculation of the average and covariance of transmission and reflection coefficients. The evaluation involves no direct integration at all, but makes use solely of the invariance of the measure, the fact that the matrix elements ua alpha are commuting c-numbers and the unitarity of u. The physical consequences for the average and covariance of transmission and reflection coefficients are briefly discussed.