Mathematical Research Letters

Volume 20 (2013)

Number 1

The classical limit of the Heisenberg and time-dependent Hartree–Fock equations: the Wick symbol of the solution

Pages: 119 – 139



Laurent Amour (Université de Reims Moulin de la Housse, Reims, France)

Mohamed Khodja (Université de Reims Moulin de la Housse, Reims, France)

Jean Nourrigat (Université de Reims Moulin de la Housse, Reims, France)


This paper is concerned with the Wick symbol of time evolving quantum observables. The time dynamics is following either the Heisenberg equation relative to the Schrödinger Hamiltonian, or the time-dependent Hartree–Fock equation. Under very weak assumptions, we prove that the Wick symbol approximatively follows the classical mechanics laws when the semiclassical parameter $h$ tends to zero. For the Heisenberg equation, this is a form of what is commonly called the Ehrenfest theorem. These statements have to be understood in a weaker sense than usual and in return, we do not assume that the Weyl symbol of the initial observable belongs to a class allowing the use of the Egorov theorem.

Published 20 September 2013