Decoherence and estimates on dissipative dynamics

Abstract

In this and the next chapters we discuss the decoherence in the position representation in more detail. In particular, we investigate whether the naive behaviour (14.2) prevails in typical quantum systems. The decoherence in coordinates is related to the decoherent histories of Griffiths [171] and Gell-Mann and Hartle [150] (see also [72]). These authors are interested in space-time classical probability of events. The history h, is defined as a product of projectors P _h =P _{t 1}(B ₁)...P _tn (B _n ), where P(B) denotes a projector on a region B ⊂ℝ^d and P _t (B) means the Heisenberg operator at time t., Then a necessary condition for a consistent definition of a probability P _h (p) of a history h, in a state P,

$${P_h}(\rho ) = Tr({\mathcal{P}_{h\rho }}\mathcal{P}_h^ + ),$$

, can be formulated in terms of the decoherence functional

$$D(h,h') = Tr({\mathcal{P}_{h\rho }}\mathcal{P}_{h'}^ + ).$$

.