CONVERGENCE PROPERTIES OF THE CLUSTER-EXPANSION FOR EQUAL-TIME GREEN-FUNCTIONS IN SCALAR THEORIES

We investigate the convergence properties of the cluster expansion of equal-time Green functions in scalar theories with quartic self-coupli ng in (0 + 1), (1 + 1), and (2 + 1) space-time dimensions. The computa tions are carried out within the equal-time correlation dynamics appro ach, which consists in a closed set of coupled equations of motion for connected Green functions as obtained by a truncation of the BBGKY hi erarchy. We find that the cluster expansion shows good convergence as long as the system is in a localized state (single phase configuration ) and that it breaks down in a non-localized state (two phase configur ation), as one would naively expect. Furthermore, in the case of dynam ical calculations with a time dependent Hamiltonian for the evaluation of the effective potential we find two timescales determining the adi abaticity of the propagation; these are the time required for adiabati city in the single phase region and the time required for tunneling in to the non-localized lowest energy state in the two phase region. Our calculations show a good convergence for the effective potentials in ( 1 + 1) and (2 + 1) space-time dimensions since tunneling is suppressed in higher space-time dimensions.