Exponentian problem in the construction of an effective low-momentum Hamiltonian for bosons

The problem of exponentiation of connected-graph contributions C, when one carries out only a partial trace of the density matrix of an assembly of bosons in order to construct an effective, low-momentum Hamiltonian, is examined. It is found that besides accounting for the exponentiation of connected graphs, disconnected graphs contribute certain terms D to connected-graph contributions. The D-terms diminish as the number of iterations increases in the Singh's renormalization-group theory for the present system. Therefore, these terms play no role in determining critical behaviour of the system. (author)