  We study certain aspects of the effective, occasionally called collective,
description of complex quantum systems within the framework of the path
integral formalism, in which the environment is integrated out. Generalising
the standard Feynman-Vernon Caldeira-Leggett model to include a non-linear
coupling between ``particle'' and environment, and considering a particular
spectral density of the coupling, a coordinate-dependent mass (or
velocity-dependent potential) is obtained. The related effective quantum
theory, which depends on the proper discretisation of the path integral, is
derived and discussed. As a result, we find that in general a simple effective
low-energy Hamiltonian, in which only the coordinate-dependent mass enters,
cannot be formulated. The quantum theory of weakly coupled superconductors and
the quantum dynamics of vortices in Josephson junction arrays are physical
examples where these considerations, in principle, are of relevance.
