  A quantum field theoretic formulation of the dynamics of the Contact Process
on a regular graph of degree z is introduced. A perturbative calculation in
powers of 1/z of the effective potential for the density of particles phi(t)
and an instantonic field psi(t) emerging from the quantum formalism is
performed. Corrections to the mean-field distribution of densities of particles
in the out-of-equilibrium stationary state are derived in powers of 1/z.
Results for typical (e.g. average density) and rare fluctuation (e.g. lifetime
of the metastable state) properties are in very good agreement with numerical
simulations carried out on D-dimensional hypercubic (z=2D) and Cayley lattices.
