  The nearest-neighbor quantum-antiferromagnetic (AF) Heisenberg model for spin
1/2 on a two-dimensional square lattice is studied in the auxiliary-fermion
representation. Expressing spin operators by canonical fermionic particles
requires a constraint on the fermion charge Q=1 on each lattice site, which is
imposed approximately through the thermal average. The resulting interacting
fermion system is first treated in mean-field theory (MFT), which yields an AF
ordered ground state and spin waves in quantitative agreement with conventional
spin-wave theory. At finite temperature a self-consistent approximation beyond
mean field is required in order to fulfill the Mermin-Wagner theorem. We first
discuss a fully self-consistent approximation, where fermions are renormalized
due to fluctuations of their spin density, in close analogy to FLEX. While
static properties like the correlation length come out correctly, the dynamical
response lacks the magnon-like peaks which would reflect the appearance of
short-range order at low T. This drawback, which is caused by overdamping, is
overcome in a `minimal self-consistent approximation' (MSCA), which we derive
from the equations of motion. The MSCA features dynamical scaling at small
energy and temperature and is qualitatively correct both in the regime of
order-parameter relaxation at long wavelengths and in the short-range-order
regime. We also discuss the impact of vertex corrections and the problem of
pseudo-gap formation in the single-particle density of states due to long-range
fluctuations. Finally we show that the (short-range) magnetic order in MFT and
MSCA helps to fulfill the constraint on the local fermion occupancy.
