Formal Splitting

The basic idea in microscopic theories of dissipation is a decomposition of a total system into a system $S$ and a reservoir part $R$ or $B$, `bath'. The (Hamiltonian) dynamics of the total system is reversible, but the dynamics of the system $S$ is effectively not reversible for times $t\ll T$.

In this lecture, we formulate these ideas for quantum systems. The Hilbert space of the total system is defined by the tensor product

$${\cal H}_{\rm total} = {\cal H}_S \otimes {\cal} {\cal H}_B.$$ (2)

The Hamiltonian of the total system is defined as

$$H_{\rm total}\equiv H\equiv H_S+H_{SB}+H_B$$ (3)

Here and in the following, we will mostly discuss time-independent Hamiltonians. Time-dependent Hamiltonians $H=H(t)$ can be treated as well but require additional techniques (e.g., Floquet theory for period time-dependence; adiabatic theorems for slow time-dependence).