Bulk-boundary asymptotic equivalence of two strict deformation quantizations

Abstract

The existence of a strict deformation quantization of $X_k=S(M_k(\mathbb{C}))$, the state space of the $k\times k$ matrices $M_k(\mathbb{C})$ which is canonically a compact Poisson manifold (with stratified boundary) has recently been proven by both authors and K. Landsman \cite{LMV}. In fact, since increasing tensor powers of the $k\times k$ matrices $M_k(\mathbb{C})$ are known to give rise to a continuous bundle of $C^{\ast }$-algebras over $I=\{0\}\cup 1/\mathbb{N}\subset [0,1]$ with fibers $A_{1/N}=M_k(\mathbb{C}{)}^{\otimes N}$ and $A_0=C(X_k)$, we were able to define a strict deformation quantization of $X_k$ à la Rieffel, specified by quantization maps $Q_{1/N}:{\tilde{A}}_0\to A_{1/N}$, with ${\tilde{A}}_0$ a dense Poisson subalgebra of $A_0$. A similar result is known for the symplectic manifold $S^2\subset {\mathbb{R}}^3$, for which in this case the fibers $A_{1/N}^{\prime }=M_{N+1}(\mathbb{C})\cong B({\text{Sym}}^N({\mathbb{C}}^2))$ and $A_0^{\prime }=C(S^2)$ form a continuous bundle of $C^{\ast }$-algebras over the same base space $I$, and where quantization is specified by (a priori different) quantization maps $Q_{1/N}^{\prime }:{\tilde{A}}_0^{\prime }\to A_{1/N}^{\prime }$. In this paper we focus on the particular case $X_2\cong B^3$ (i.e the unit three-ball in ${\mathbb{R}}^3$) and show that for any function $f\in {\tilde{A}}_0$ one has ${\lim }_{N\to \infty }||(Q_{1/N}(f)){|}_{{\text{Sym}}^N({\mathbb{C}}^2)}-Q_{1/N}^{\prime }(f{|}_{{}_{S^2}})|{|}_N=0$, were ${\text{Sym}}^N({\mathbb{C}}^2)$ denotes the symmetric subspace of $({\mathbb{C}}^2{)}^{N\otimes }$. Finally, we give an application regarding the (quantum) Curie-Weiss model.

arXiv comment27 pages no figures, minor changes, accepted for publication in Letters in Mathematical Physics

References in corpus (5)

• Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to \infty$
• Understanding quantum measurement from the solution of dynamical models
• The Phase Diagram of the Quantum Curie-Weiss Model
• Strict deformation quantization of the state space of $M_k(\mathbb{C})$ with applications to the Curie-Weiss model
• Ground states for mean field models with a transverse component

Cited by in corpus (6)

• The classical limit and spontaneous symmetry breaking in algebraic quantum theory
• The classical limit of mean-field quantum spin systems
• Injective tensor products in strict deformation quantization
• Classical and quantum KMS states on spin lattice systems
• Strict deformation quantization and local spin interactions
• The classical limit of Schrödinger operators in the framework of Berezin quantization and spontaneous symmetry breaking as emergent phenomenon