  In this first of a series of four articles, it is shown how a hamiltonian
quantum dynamics can be formulated based on a generalization of classical
probability theory using the notion of quasi-invariant measures on the
classical phase space for our description of dynamics. This is based on certain
distributions rather than invariant Gibbs measures. The first quantization is
by functorial analytic continuation of real probability amplitudes,
mathematically effecting the introduction of correlation between otherwise
independent subsystems, and whose physical consequence is the incorporation of
Breit-Wigner resonances associated to Gamow vectors into our description of
dynamics. The resulting quantum dynamics admits a natural field theory
interpretation. This basic formalism will be employed in subsequent
installments of the series.
