  We propose a new field theoretic method for calculating Renyi entropy of a
sub-system of many interacting Bosons without using replica methods. This
method is applicable to dynamics of both open and closed quantum systems
starting from arbitrary initial conditions. Our method identifies the Wigner
characteristic of a reduced density matrix with the partition function of the
whole system with a set of linear sources turned on only in the subsystem and
uses this to calculate the subsystem's Renyi entropy. We use this method to
study evolution of Renyi entropy in a non-interacting open quantum system
starting from an initial Fock state. We find a relation between the initial
state and final density matrix which determines whether the entropy shows
non-monotonic behaviour in time. For non-Markovian dynamics, we show that the
entropy approaches its steady state value as a power law with exponents
governed by non-analyticities of the bath. We illustrate that this
field-theoretic method can be used to study large bosonic open quantum systems.
