  We study the Caldeira-Leggett model where a quantum Brownian particle
interacts with an environment or a bath consisting of a collection of harmonic
oscillators in the path integral formalism. Compared to the contours that the
paths take in the conventional Schwinger-Keldysh formalism, the paths in our
study are deformed in the complex time plane as suggested by the recent study
[C. Aron, G. Biroli and L. F. Cugliandolo, SciPost Phys.\ {\bf 4}, 008 (2018)].
This is done to investigate the connection between the symmetry properties in
the Schwinger-Keldysh action and the equilibrium or non-equilibrium nature of
the dynamics in an open quantum system. We derive the influence functional
explicitly in this setting, which captures the effect of the coupling to the
bath. We show that in equilibrium the action and the influence functional are
invariant under a set of transformations of path integral variables. The
fluctuation-dissipation relation is obtained as a consequence of this symmetry.
When the system is driven by an external time-dependent protocol, the symmetry
is broken. From the terms that break the symmetry, we derive a quantum
Jarzynski-like equality for quantum mechanical work given as a function of
fluctuating quantum trajectory. In the classical limit, the transformations
becomes those used in the functional integral formalism of the classical
stochastic thermodynamics to derive the classical fluctuation theorem.
