Bulk-boundary asymptotic equivalence of two strict deformation quantizations

Abstract

The existence of a strict deformation quantization of $$X_k=S(M_k({\mathbb {C}}))$$ X k = S ( M k ( C ) ) , the state space of the $$k\times k$$ k × k matrices $$M_k({\mathbb {C}})$$ M k ( C ) which is canonically a compact Poisson manifold (with stratified boundary), has recently been proved by both authors and Landsman (Rev Math Phys 32:2050031, 2020. 10.1142/S0129055X20500312). In fact, since increasing tensor powers of the $$k\times k$$ k × k matrices $$M_k({\mathbb {C}})$$ M k ( C ) are known to give rise to a continuous bundle of $$C^*$$ C ∗ -algebras over $$I=\{0\}\cup 1/\mathbb {N}\subset [0,1]$$ I = { 0 } ∪ 1 / N ⊂ [ 0 , 1 ] with fibers $$A_{1/N}=M_k({\mathbb {C}})^{\otimes N}$$ A 1 / N = M k ( C ) ⊗ N and $$A_0=C(X_k)$$ A 0 = C ( X k ) , we were able to define a strict deformation quantization of $$X_k$$ X k à la Rieffel, specified by quantization maps $$Q_{1/N}:/ \tilde{A}_0\rightarrow A_{1/N}$$ Q 1 / N : / A ~ 0 → A 1 / N , with $$\tilde{A}_0$$ A ~ 0 a dense Poisson subalgebra of $$A_0$$ A 0 . A similar result is known for the symplectic manifold $$S^2\subset \mathbb {R}^3$$ S 2 ⊂ R 3 , for which in this case the fibers $$A'_{1/N}=M_{N+1}(\mathbb {C})\cong B(\text {Sym}^N(\mathbb {C}^2))$$ A 1 / N ′ = M N + 1 ( C ) ≅ B ( Sym N ( C 2 ) ) and $$A_0'=C(S^2)$$ A 0 ′ = C ( S 2 ) form a continuous bundle of $$C^*$$ C ∗ -algebras over the same base space I, and where quantization is specified by (a priori different) quantization maps $$Q_{1/N}': \tilde{A}_0' \rightarrow A_{1/N}'$$ Q 1 / N ′ : A ~ 0 ′ → A 1 / N ′ . In this paper, we focus on the particular case $$X_2\cong B^3$$ X 2 ≅ B 3 (i.e., the unit three-ball in $$\mathbb {R}^3$$ R 3 ) and show that for any function $$f\in \tilde{A}_0$$ f ∈ A ~ 0 one has $$\lim _{N\rightarrow \infty }||(Q_{1/N}(f))|_{\text {Sym}^N(\mathbb {C}^2)}-Q_{1/N}'(f|_{_{S^2}})||_N=0$$ lim N → ∞ | | ( Q 1 / N ( f ) ) | Sym N ( C 2 ) - Q 1 / N ′ ( f | S 2 ) | | N = 0 , where $$\text {Sym}^N(\mathbb {C}^2)$$ Sym N ( C 2 ) denotes the symmetric subspace of $$(\mathbb {C}^2)^{N \otimes }$$ ( C 2 ) N ⊗ . Finally, we give an application regarding the (quantum) Curie–Weiss model.

The classical limit and spontaneous symmetry breaking in algebraic quantum theory

“…N now implies that the algebraic vector state ω (0) 1/N is strictly invariant under Z 2 : no SSB occurs for any finite N . In a similar fashion as above (see for example [35] and references therein) one can prove that the algebraic vector state ω…”

The classical limit of mean-field quantum spin systems

“…In what follows we consider mean field quantum spin systems whose Hamiltonians H 1/N are restricted to this subspace, since quantum spin systems arising in that way are tipically of the form Q 1/N (h N ) for some (N-dependent) real polynomial function h N on M = S 2 given by (1.26) and Q 1/N given by (1.20) -(1.21) (see e.g. [19,Theorem 3.1] for an example). Such spin systems are widely studied in (condensed matter) physics, but also in mathematical physics they form an important field of research.…”

“…Let us first stress that the limit N → ∞ can be taken in two entirely different ways, which depends on the class of observables one considers, namely either quasi-local observables or macroscopic observables. The former are the ones traditionally studied for quantum spin systems, but the latter relate these systems to strict deformation quantization, since macroscopic observables are precisely defined by (quasi-) symmetric sequences which form the continuous cross sections of a continuous bundle of C * -algebras [16,19,15]. In [19,Theorem 2.3] it has been shown that the quantization maps (1.20) -(1.21) define quasi-symmetric sequences, and hence macroscopic observables.…”

The classical limit of Schrödinger operators in the framework of Berezin quantization and spontaneous symmetry breaking as emergent phenomenon

“…However, different approaches were used and no limit of a sequence of eigenvectors was taken. More recent studies [14,17,30,25] have shown a well-established relation between the classical limit of a sequence of eigenvectors associated to mean-field quantum spin system Hamiltonians and spontaneous symmetry breaking. In these relatively simple systems, where the Hamiltonian is always a bounded operator, the symmetry group is typically finite.…”