Hyperbolic field theory as a Lorentz covariant description for the dissipation

Abstract

Based on the extension of the space of solutions for the D’Alambertian operator on the hyperbolic complex plane, we construct a formalism for the dissipative dynamics of two charged scalar fields, with explicit internal and background Lorentz symmetries, as opposed to the well-established scheme based on damping oscillators physics, which breaks down explicitly such a symmetry. The hyperbolic extension implies that the usual U(1) symmetry for charged scalar fields is extended with the incorporation of a non-compact SO(1,1) symmetry that will allow to describe the damping effects. As response properties of the interaction, the dissipation coefficients in time and in space are determined in the frequency-momentum space; the regimes with and without gapped energy–momentum states emerge; additionally the partition function and an entangled state generated by the dissipation are constructed. The pair of charged fields can be identified with field theories on asymptotic boundaries of gravitational backgrounds, and the possible holographic scenarios are discussed.