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

Zied Ammari (1) , Francis Nier (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: Mean field; Semiclassical; Wigner measure; BBGKY hierarchy; Second quantization; Fock space

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]

Zied Ammari, Francis Nier. Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states. Journal de Mathématiques Pures et Appliquées, 2011, 95 (6), pp.585-626. ⟨10.1016/j.matpur.2010.12.004⟩. ⟨hal-00462536v2⟩