Dissipation and Topologically Massive Gauge Theories in the Pseudo-Euclidean Plane

In pseudo-Euclidean metrics the Chern–Simons gauge theory in the infrared region is found to be associated with dissipative dynamics. In the infrared limit the Lagrangian of (2+1)-dimensional pseudo-euclidean topologically massive electrodynamics has indeed the same form as the Lagrangian of the damped harmonic oscillator. On the hyperbolic plane a set of two damped harmonic oscillators, each time-reversed from the other, is shown to be equivalent to a single undamped harmonic oscillator. The equations for the damped oscillators are proven to be the same as the ones for the Lorentz force acting on two particles carrying opposite charge in a constant magnetic field and in the electric harmonic potential. This provides an immediate link with Chern–Simons-like dynamics of Bloch electrons in solids propagating along the lattice plane with a hyperbolic energy surface. The symplectic structure of the reduced theory is finally discussed in the Dirac constrained canonical formalism and in the Faddeev–Jackiw symplectic formalism.