Fractional integration and uniform densities in quantum mechanics Kingsley R. W. Jones Physics Department, University of Queensland, St Lucia 4072, Brisbane, Australia. May 11, 1992 Abstract In both the classical theory of statistical inference, and in classical ergodic theory, the uniform density over states plays an important role to describe circumstances of no prior knowledge. Here we develop the analogous object for quantum mechanics and describe some of its uses. In one class of physical problems it is necessary to calculate the average of an arbitrary function on quantum states with respect to the uniform density. This problem reveals an interesting role for fractional integration in the reduction of a multidimensional integral on Hilbert space to a single fractional integral. 1