  Discretizations of the Feynman-Kac path integral representation of the
quantum mechanical density matrix are investigated. Each infinite-dimensional
path integral is approximated by a Riemann integral over a finite-dimensional
function space, by restricting the integration to a subspace of all admissible
paths. Using this process, a wide class of methods can be derived, with each
method corresponding to a different choice for the approximating subspace. The
traditional ``short-time'' approximation and ``Fourier discretization'' can be
recovered from this approach, using linear and spectral basis functions
respectively. As an illustration, a novel method is formulated using cubic
elements and is shown to have improved convergence properties when applied to a
simple model problem.
