  Schwinger Keldysh field theory is a widely used paradigm to study
non-equilibrium dynamics of quantum many-body systems starting from a thermal
state. We extend this formalism to describe non-equilibrium dynamics of quantum
systems starting from arbitrary initial many-body density matrices. We show how
this can be done for both Bosons and Fermions, and for both closed and open
quantum systems, using additional sources coupled to bilinears of the fields at
the initial time, calculating Green's functions in a theory with these sources,
and then taking appropriate set of derivatives of these Green's functions
w.r.t. initial sources to obtain physical observables. The set of derivatives
depend on the initial density matrix. The physical correlators in a dynamics
with arbitrary initial conditions do not satisfy Wick's theorem, even for
non-interacting systems. However our formalism constructs intermediate
``n-particle Green's functions'' which obey Wick's theorem and provide a
prescription to obtain physical correlation functions from them. This allows us
to obtain analytic answers for all physical many body correlation functions of
a non-interacting system even when it is initialized to an arbitrary density
matrix. We use these exact expressions to obtain an estimate of the violation
of Wick's theorem, and relate it to presence of connected multi-particle
initial correlations in the system. We illustrate this new formalism by
calculating density and current profiles in many body Fermionic and Bosonic
open quantum systems initialized to non-trivial density matrices. We have also
shown how this formalism can be extended to interacting many body systems.
