Assumption (factorising initial condition):

$${\chi}(t=0) = R_0 \otimes \rho(t=0)$$ (18)

$$R_0 \equiv {\rm Tr}_S [\chi(t=0)],\quad \rho(t=0)\equiv {\rm Tr}_B [{\chi}(t=0)].$$

This factorisation assumption is key to most of the results that follow. Its validity has been discussed and criticised in the past (see Weiss book for further references). Some of the issues are:

• Does the factorisation assumption only affect transient or also the long-time behaviour of the density matrix?
• Are there exactly solvable models where these issues can be clarified?

A theoretical formulation of time-evolution for arbitrary initial condition is sometimes possible: `preparation function' (exact solution of dissipative quantum oscillator; Grabert, Ingold et al); generalisation of many-body Keldysh GF (three-by-three matrix instead of two-by-two matrix, M. Wagner).