Effective field theory of time-translational symmetry breaking in
  nonequilibrium open system
  We develop the effective field theoretical (EFT) approach to
time-translational symmetry breaking of nonequilibrium open systems based on
the Schwinger-Keldysh formalism. In the Schwinger-Keldysh formalism, all the
symmetries of the microscopic Lagrangian are doubled essentially because the
dynamical fields are doubled to describe the time-evolution along the
closed-time-path. The effective Lagrangian for open systems are then obtained
by coarse-graining the microscopic Schwinger-Keldysh Lagrangian. As a
consequence of coarse-graining procedure, there appear the noise and
dissipation effects, which explicitly break the doubled time-translational
symmetries into a diagonal one. We therefore need to incorporate this symmetry
structure to construct the EFT for Nambu-Goldstone bosons in symmetry broken
phases of open systems. Based on this observation together with the consistency
of the Schwinger-Keldysh action, we construct and study the general EFT for
time-translational symmetry breaking in particular, having in mind applications
to synchronization, time crystal, and cosmic inflation.
