Weak-Coupling Limit. II On the Quantum Fokker-Planck Equation

TL;DR

This paper derives a general Quantum Fokker-Planck Equation as the generator of a quantum dynamical semigroup in the weak-coupling limit, providing a new quantum analogue of the Fermi Golden Rule.

Contribution

It specializes a contraction semigroup approach to $W^*$-algebras, establishing a universal quantum dynamical semigroup generator called the Quantum Fokker-Planck Equation.

Findings

Derivation of the Quantum Fokker-Planck Equation as the generator of the limit dynamics.

Introduction of a quantum generalization of the Fermi Golden Rule.

Demonstration of the approach's independence from spectral properties or dimensions.

Abstract

In a recent work we have found a contraction semigroup able to correctly approximate a projected and perturbed one-parameter group of isometries in a generic Banach space, in the limit of weak-coupling. Here we study its generator by specializing to $W^*$-algebras: after defining a Physical Subsystem in terms of a completely positive projecting conditional expectation, we find that it generates a Quantum Dynamical Semigroup. As a consequence of uniqueness and strong generality (well defined dynamics, irrespective of the Physical Subsystem spectral properties or dimensions), its generator deserves to be referred as "the" Quantum Fokker-Planck Equation. We then provide important examples of the limit dynamics, one of which constitutes a new Quantum generalization of the celebrated Fermi Golden Rule.