  For many classically chaotic systems it is believed that the quantum wave
functions become uniformly distributed, that is the matrix elements of smooth
observables tend to the phase space average of the observable. In this paper we
study the fluctuations of the matrix elements for the desymmetrized quantum cat
map. We present a conjecture for the distribution of the normalized matrix
elements, namely that their distribution is that of a certain weighted sum of
traces of independent matrices in SU(2). This is in contrast to generic chaotic
systems where the distribution is expected to be Gaussian. We compute the
second and fourth moment of the normalized matrix elements and obtain agreement
with our conjecture.
