  The AMAS group at the Paul Scherrer Institute developed an object oriented
library for high performance simulation of high intensity ion beam transport
with space charge. Such particle-in-cell (PIC) simulations require a method to
generate multivariate particle distributions as starting conditions.
  In a preceeding publications it has been shown that the generators of
symplectic transformations in two dimensions are a subset of the real Dirac
matrices (RDMs) and that few symplectic transformations are required to
transform a quadratic Hamiltonian into diagonal form.
  Here we argue that the use of RDMs is well suited for the generation of
multivariate normal distributions with arbitrary covariances. A direct and
simple argument supporting this claim is that this is the "natural" way how
such distributions are formed. The transport of charged particle beams may
serve as an example: An uncorrelated gaussian distribution of particles
starting at some initial position of the accelerator is subject to linear
deformations when passing through various beamline elements. These deformations
can be described by symplectic transformations.
  Hence, if it is possible to derive the symplectic transformations that bring
up these covariances, it is also possible to produce arbitrary multivariate
normal distributions without Cholesky decomposition. The method allows the use
of arbitrary uncoupled distributions. The functional form of the coupled
multivariate distributions however depends in the general case on the type of
the used random number generator. Only gaussian generators always yield
gaussian multivariate distributions.
