  We present a restricted path integral approach to the 2D and 3D repulsive
Hubbard model. In this approach the partition function is approximated by
restricting the summation over all states to a (small) subclass which is chosen
such as to well represent the important states. This procedure generalizes mean
field theory and can be systematically improved by including more states or
fluctuations. We analyze in detail the simplest of these approximations which
corresponds to summing over states with local antiferromagnetic (AF) order. If
in the states considered the AF order changes sufficiently little in space and
time, the path integral becomes a finite dimensional integral for which the
saddle point evaluation is exact. This leads to generalized mean field
equations allowing for the possibility of more than one relevant saddle points.
In a big parameter regime (both in temperature and filling), we find that this
integral has {\em two} relevant saddle points, one corresponding to finite AF
order and the other without. These degenerate saddle points describe a phase of
AF ordered fermions coexisting with free, metallic fermions. We argue that this
mixed phase is a simple mean field description of a variety of possible
inhomogeneous states, appropriate on length scales where these states appear
homogeneous. We sketch systematic refinements of this approximation which can
give more detailed descriptions of the system.
