  Cluster Dynamical Mean Field Theories are analyzed in terms of their
semiclassical limit and their causality properties, and a translation invariant
formulation of the cellular dynamical mean field theory, PCDMFT, is presented.
The semiclassical limit of the cluster methods is analyzed by applying them to
the Falikov-Kimball model in the limit of infinite Hubbard interaction U where
they map to different classical cluster schemes for the Ising model.
Furthermore the Cutkosky-t'Hooft-Veltman cutting equations are generalized and
derived for non translation invariant systems using the Schwinger-Keldysh
formalism. This provides a general setting to discuss causality properties of
cluster methods. To illustrate the method, we prove that PCDMFT is causal while
the nested cluster schemes (NCS) in general and the pair scheme in particular
are not. Constraints on further extension of these schemes are discussed.
