  The classical and quantum formulations for open systems related to
dissipative dynamics are constructed on a complex hyperbolic ring, following
universal symmetry principles, and considering the double thermal fields
approach for modeling the system of interest, and the environment. The
hyperbolic rotations are revealed as an underlying internal symmetry for the
dissipative dynamics, and a chemical potential is identified as conjugate
variable to the charge operator, and thus a grand partition function is
constructed. As opposed to the standard scheme, there are not patologies
associated with the existence of many unitarity inequivalent representations on
the hyperbolic ring, since the whole of the dissipative quantum dynamics is
realized by choosing only one representation of the field commutation
relations. Entanglement entropy operators for the subsystem of interest and the
environment, are constructed as a tool for study the entanglement generated
from the dissipation. The holographic perspectives of our results are
discussed.
