Abstract
The existence of a strict deformation quantization of X_k=S(M_k({mathbb {C}})), the state space of the ktimes k matrices M_k({mathbb {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. https://doi.org/10.1142/S0129055X20500312). In fact, since increasing tensor powers of the ktimes k matrices M_k({mathbb {C}}) are known to give rise to a continuous bundle of C^*-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}_0rightarrow A_{1/N}, with tilde{A}_0 a dense Poisson subalgebra of A_0. A similar result is known for the symplectic manifold S^2subset mathbb {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)) and A_0'=C(S^2) form a continuous bundle of 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}'. In this paper, we focus on the particular case X_2cong B^3 (i.e., the unit three-ball in mathbb {R}^3) and show that for any function fin tilde{A}_0 one has lim _{Nrightarrow infty }||(Q_{1/N}(f))|_{text {Sym}^N(mathbb {C}^2)}-Q_{1/N}'(f|_{_{S^2}})||_N=0, where 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.
Highlights
An important field of research within mathematical physics concerns the relation between classical theories viewed as limits of quantum theories, for example, classical mechanics of a particle on the phase space R2n versus quantum mechanics on the Hilbert space L2(Rn), or classical thermodynamics of a spin system versus statistical mechanics of a quantum spin system on a finite lattice [14]
In order the one associated to relate Cc∞(R2n) to a quantum theory described on some Hilbert space, one needs to deform Cc∞(R2n) into non-commutative C∗-algebras exploiting a family of quantization maps
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 bysymmetric sequences which form the continuous cross sections of a continuous bundle of C∗-algebras
Summary
An important field of research within mathematical physics concerns the relation between classical theories viewed as limits of quantum theories, for example, classical mechanics of a particle on the phase space R2n versus quantum mechanics on the Hilbert space L2(Rn), or classical thermodynamics of a spin system versus statistical mechanics of a quantum spin system on a finite lattice [14]. In these examples, the relation between both (different) theories can be described by a continuous bundle of algebras of observables equipped with certain quantization maps. A modern way establishing a link between both theories is based on the concept of strict deformation quantization, i.e., the mathematical formalism that describes the transition from a classical theory to a quantum theory [13,22,23] in terms of deformations of (commutative) Poisson algebras (representing the classical theory) into non-commutative C∗ algebras characterizing the quantum theory
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