  A coherent state path integral is considered for bosons with an ensemble
average of a random potential and with an additional, repulsive interaction in
the context of BEC under inclusion of specially prepared disorder. The
essential normalization of the coherent state path integral, as a generating
function of observables, is obtained from the non-equilibrium time contour for
'forward' and 'backward' propagation so that a time contour metric has to be
taken into account in the ensemble average with the random potential.
Therefore, the respective symmetries for the derivation of a nonlinear sigma
model follow from the involved time contour metric which leads to a coset
decomposition Sp(4)/U(2) x U(2) of the symplectic group Sp(4) with the subgroup
U(2) for the unitary invariance of the density-related vacuum or ground state;
the corresponding spontaneous symmetry breaking gives rise to anomalous- or
'Nambu'-doubled field degrees of freedom within self-energy matrices which are
finally regarded by remaining coset matrices. The notion of a 'return
probability', according to the original 'Anderson-localization', is thus
naturally contained within coherent state path integrals of a non-equilibrium
contour time for equivalent 'forward' and 'backward' propagation.
