  We describe a low-temperature approach to the two-dimensional half-filled
Hubbard model which allows us to study both antiferromagnetism and
single-particle properties. This approach ignores amplitude fluctuations of the
antiferromagnetic (AF) order parameter and is valid below a crossover
temperature $T_X$ which marks the onset of AF short-range order. Directional
fluctuations (spin waves) are described by a non-linear sigma model
(NL$\sigma$M) that we derive from the Hubbard model. At zero temperature and
weak coupling, our results are typical of a Slater antiferromagnet. The AF gap
is exponentially small; there are well-defined Bogoliubov quasi-particles
(QP's) (carrying most of the spectral weight) coexisting with a high-energy
incoherent excitation background. As $U$ increases, the Slater antiferromagnet
progressively becomes a Mott-Heisenberg antiferromagnet. The Bogoliubov bands
evolve into Mott-Hubbard bands separated by a large AF gap. A significant
fraction of spectral weight is transferred from the Bogoliubov QP's to
incoherent excitations. At finite temperature, there is a metal-insulator
transition between a pseudogap phase at weak coupling and a Mott-Hubbard
insulator at strong coupling. Finally, we point out that our results
straightforwardly translate to the half-filled attractive Hubbard model, where
the ${\bf q}=(\pi,\pi)$ charge and ${\bf q}=0$ pairing fluctuations combine to
form an order parameter with SO(3) symmetry.
