The classical limit for quantum mechanical correlation functions

Abstract

For quantum systems of finitely many particles as well as for boson quantum field theories, the classical limit of the expectation values of products of Weyl operators, translated in time by the quantum mechanical Hamiltonian and taken in coherent states centered in x-and p-space around h~x l2 (coordinates of a point in classical phase space) are shown to become the exponentials of coordinate functions of the classical orbit in phase space. In the same sense, h~112 [(quantum operator) (ί) -(classical function) (ί)] converges to the solution of the linear quantum mechanical system, which is obtained by linearizing the non-linear Heisenberg equations of motion around the classical orbit.

Norm Approximation for Many-Body Quantum Dynamics and Bogoliubov Theory

“…There are grand canonical analogues of (2) related to the fluctuations around coherent states in Fock space [29,21,22,26,27,24,30]. In particular, our Theorem 1 is comparable to the Fock-space result of Kuz [30].…”

“…In the context of the ground state problem, this has been done successfully for one and two-component Bose gases [35,36,49], for the Lee-Huang-Yang formula of homogeneous, dilute gases [19,28,50] and for the excitation spectrum in the mean-field regime [47,23,32,14,44]. In the context of the dynamical problem, Bogoliubov theory has been used widely to study the quantum dynamics of coherent states in Fock space [29,21,22,46,26,27,24,30,9,25]. Very recently, Lewin, Schlein and one of us [31] were able to justify Bogoliubov theory as a norm approximation for the N-particle quantum dynamics in the mean-field regime.…”

Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential

“…This strategy has already been used to derive the nonlinear Hartree equations for the effective dynamics of so-called mean-field systems (see [27,13,4,9]) to derive the cubic nonlinear Schrödinger equation with different (and simpler) scalings of the interaction potential (see [8,11]) and to derive the nonlinear Schrödinger equation in a one-dimensional setting (see [1,2]). We remark that the first derivation of the Hartree equation was obtained using a different method in [17,14]. With this method the speed of convergence was recently estimated in [25].…”

Beyond mean field: On the role of pair excitations in the evolution of condensates

“…See also [2]. It continued with the mathematically rigorous work of Hepp [20], and Ginibre and Velo [15].…”