Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states

TL;DR

This paper extends the propagation of Wigner measures to infinite-dimensional bosonic systems, overcoming challenges posed by the non-equivalence of Weyl and Wick quantizations, and provides a rigorous mean field analysis.

Contribution

It develops a novel approach leveraging both Weyl and Wick calculuses to prove the propagation of Wigner measures in infinite-dimensional bosonic states.

Findings

Propagation of Wigner measures is established for general bosonic states.

The approach overcomes the non-equivalence of Weyl and Wick quantizations in infinite dimensions.

Standard semiclassical results are extended to the infinite-dimensional setting.

Abstract

Contrary to the finite dimensional case, Weyl and Wick quantizations are no more asymptotically equivalent in the infinite dimensional bosonic second quantization. Moreover neither the Weyl calculus defined for cylindrical symbols nor the Wick calculus defined for polynomials are preserved by the action of a nonlinear flow. Nevertheless taking advantage carefully of the information brought by these two calculuses in the mean field asymptotics, the propagation of Wigner measures for general states can be proved, extending to the infinite dimensional case a standard result of semiclassical analysis.