  We derive a generalized Luttinger-Ward expression for the Free energy of a
many body system involving a constrained Hilbert space. In the large $N$ limit,
we are able to explicity write the entropy as a functional of the Green's
functions. Using this method we obtain a Luttinger sum rule for the Kondo
lattice. One of the fascinating aspects of the sum rule, is that it contains
two components, one describing the heavy electron Fermi surface, the other, a
sea of oppositely charged, spinless fermions. In the heavy electron state, this
sea of spinless fermions is completely filled and the electron Fermi surface
expands by one electron per unit cell to compensate the positively charged
background, forming a ``large'' Fermi surface. Arbitrarily weak magnetism
causes the spinless Fermi sea to annihilate with part of the Fermi sea of the
conduction electrons, leading to a small Fermi surface. Our results thus enable
us to show that the Fermi surface volume contracts from a large, to a small
volume at a quantum critical point. However, the sum rules also permit the
possible formation of a new phase, sandwiched between the antiferromagnet and
the heavy electron phase, where the charged spinless fermions develop a true
Fermi surface.
