A projection operator approach to a dissipative two-level system

The equations of motion of a system with a few degrees of freedom that is coupled to a dissipative bath are obtained with the introduction of a new projection operator. When expanded to second order in an interaction term, the projected equation of motion can be used to discuss the two-state system coupled to a boson bath (the spin-boson Hamiltonian). The expansion parameter is the splitting of the states of the two-state system. We obtain an equation of motion for the elements of the density matrix of the two-state system, which can be solved to yield the non-interacting blip approximation discussed by Legget et al. [A.J. Legget, S. Chakravaty, A.T. Dorsey, M.P.A. Fisher and W. Zwerger, Rev. Mod. Phys. 59 (1987) 1]. The case of an ohmic bath is also treated in a direct fashion.