  The connection between the Hamilton and the standard Lagrange formalism is
established for a generic Quantum Field Theory with vanishing vacuum
expectation values of the fundamental fields. The Effective Actions in both
formalisms are the same if and only if the fundamental fields and the momentum
fields are related by the stationarity condition. These momentum fields in
general differ from the canonical fields as defined via the Effective Action.
By means of functional methods a systematic procedure is presented to identify
the full correlation functions, which depend on the momentum fields, as
functionals of those usually appearing in the standard Lagrange formalism.
Whereas Lagrange correlation functions can be decomposed into tree diagrams the
decomposition of Hamilton correlation functions involves loop corrections
similar to those arising in n-particle effective actions. To demonstrate the
method we derive for theories with linearized interactions the propagators of
composite auxiliary fields and the ones of the fundamental degrees of freedom.
The formalism is then utilized in the case of Coulomb gauge Yang-Mills theory
for which the relations between the two-point correlation functions of the
transversal and longitudinal components of the conjugate momentum to the ones
of the gauge field are given.
