Emergence of a Wiener process as a result of the quantum mechanical interaction with a macroscopic medium

Abstract

We analyze a modified version of the Coleman-Hepp model, that is able to take into account energy-exchange processes between the incoming particle and the linear array made up of $N$ spin-1/2 systems. We bring to light the presence of a Wiener dissipative process in the weak-coupling, macroscopic ($N \rightarrow \infty$) limit. In such a limit and in a restricted portion of the total Hilbert space, the particle undergoes a sort of Brownian motion, while the free Hamiltonian of the spin array serves as a Wiener process. No assumptions are made on the spectrum of the Hamiltonian of the spin system, and no partial trace is computed over its states. The mechanism of appearance of the stochastic process is discussed and contrasted to other noteworthy examples in the literature. The links with van Hove's ``$λ^2 T$ limits are emphasized.