Applications of Thirring Model to Inhomogenous Rolling Tachyon and Dissipative Quantum Mechanics

We study the rolling tachyon and the dissipative quantum mechanics using the Thirring model with a boundary mass. We construct a boundary state for the dissipative quantum system in one dimension, which describes the system at the off-critical points as well as at the critical point. Then we extend the Thirring model with a boundary mass in order to depict the time evolution of an unstable D-branes with one direction wrapped on a circle of radius $R$, which is termed the inhomogeneous rolling tachyon. The analysis based on the Thirring model shows that the time dependent evolution of the inhomogeneous tachyon is possible only when $\frac{2}{\sqrt{3}}< R < 2$.