  We study a family of physical observable quantities in quantum gravity. We
denote them W functions, or n-net functions. They represent transition
amplitudes between quantum states of the geometry, are analogous to the n-point
functions in quantum field theory, but depend on spin networks with n connected
components. In particular, they include the three-geometry to three-geometry
transition amplitude. The W functions are scalar under four-dimensional
diffeomorphism, and fully gauge invariant. They capture the physical content of
the quantum gravitational theory. We show that W functions are the natural
n-point functions of the field theoretical formulation of the gravitational
spin foam models. They can be computed from a perturbation expansion, which can
be interpreted as a sum-over-four-geometries. Therefore the W functions bridge
between the canonical (loop) and the covariant (spinfoam) formulations of
quantum gravity. Following Wightman, the physical Hilbert space of the theory
can be reconstructed from the W functions, if a suitable positivity condition
is satisfied. We compute explicitly the W functions in a "free" model in which
the interaction giving the gravitational vertex is shut off, and we show that,
in this simple case, we have positivity, the physical Hilbert space of the
theory can be constructed explicitly and the theory admits a well defined
interpretation in terms of diffeomorphism invariant transition amplitudes
between quantized geometries.
