An internal Lorentz symmetry induces the background Lorentz symmetry in the dissipative dynamics

We show that a dissipative field theory with background Lorentz symmetry underlies the field theory with global U(1)×SO(1,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U(1)\times SO(1,1)$$\end{document} symmetry constructed on a hyperbolic ring; the theory represents a dissipative model for a bipartite system compound of Klein-Gordon fields with different masses; the infrared limit corresponds to the usual dissipative field theory with a constant dissipative parameter, and with broken background Lorentz symmetry; in the ultraviolet limit the fields behave as free fields with unobservable dissipative effects. In this hyperbolic ring-based formulation, the observables correspond to Hermitian quantities, encoding two real quantities, which are appropriate for describing bipartite system; thus, the Lagrangian is constructed as a Hermitian U(1)×SO(1,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U(1)\times SO(1,1)$$\end{document} invariant quantity, and the two real potentials are identified with the subsystem-plus-reservoir system. The potentials can be identified with elliptic and hyperbolic paraboloids by adjusting a real parameter that is interpolating between pure U(1) and pure SO(1, 1) symmetries. At the end we address the problem of constructing a propagator on the hyperbolic ring.