Concentration of cylindrical Wigner measures

Abstract

In this note we aim to characterize the cylindrical Wigner measures associated to regular quantum states in the Weyl C*-algebra of canonical commutation relations. In particular, we provide conditions at the quantum level sufficient to prove the concentration of all the corresponding cylindrical Wigner measures as Radon measures on suitable topological vector spaces. The analysis is motivated by variational and dynamical problems in the semiclassical study of bosonic quantum field theories.

Magnetic Schrödinger operators as the quasi-classical limit of Pauli–Fierz-type models

“…the semiclassical parameter. The analysis of such convergence is carried out in detail in [Fal16]; in the Proposition below we provide a partial result that covers what is needed in this paper. then the set of cluster points Cluster P εÑ0 pΨ ε , ε P p0, 1qq w.r.t.…”

“…§ 1.1 for the definitions of the objects appearing below) of a more general result, proved in [Fal17, Theorem 3.3]. 5 P is the weak topology on quantum states (Ψε defines uniquely the state xΨε |¨ΨεyΓ s ) of the C*-algebra of canonical commutation relations, with respect to compactly supported and smooth cylindrical observables, see [Fal16] for additional details. 6 T is the (preimage of the) topology of pointwise convergence of the generating functional xΨε|W p¨qΨεy Γs associated to Ψε, where W p¨q is the Weyl operator [Fal16].…”

“…In this Appendix we briefly outline some of the tools of bosonic semiclassical analysis in infinite dimensions used throughout the text, see, e.g., [AN08, AN09, AN11, AN15, AF14, ABN17, AFP16, AF17, LP14, Fal16, Fal17] for an overview of the theory and its applications. We essentially adopt the notation of [AN08], and to a lesser extent of [Fal16,Fal17]. Throughout the rest of this section, let us denote by Z a generic separable complex Hilbert space.…”

“…. , x N q are M -integrable as cylindrical functions [Fal17]. Finally, if the measure µ M is a linear map from L p loc pΛˆH; W q into L p loc pΛ; W q for p " 1, 2 (for any W finite dimensional real vector space), we get…”

“…Now, removing Assumption A3 altogether we still have convergence, up to an eventual subsequence extraction, of Ψ ε to a cylindrical measure M on L 2 pK, νq, i.e., to a Radon probability measure µ M on the Hausdorff completion H " L 2 pK, νq weak of L 2 endowed with the weak topology. In addition, all the cylindrical measures are reached by suitable sequences of quantum states [Fal16]. Furthermore, since the symbols involved in the limit are cylindrical (finite rank), as discussed in the proof of Lemma 2.2, it is possible to prove pointwise convergence for generic cylindrical measures, provided a p¨q px 1 , .…”