Bulk-boundary asymptotic equivalence of two strict deformation quantizations
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Bulk-boundary asymptotic equivalence of two strict deformation quantizations Valter Moretti1
· Christiaan J. F. van de Ven1,2
Received: 9 May 2020 / Revised: 14 September 2020 / Accepted: 21 September 2020 / Published online: 1 October 2020 © The Author(s) 2020
Abstract The existence of a strict deformation quantization of X k = S(Mk (C)), the state space of the k × k matrices Mk (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 k × k matrices Mk (C) are known to give rise to a continuous bundle of C ∗ -algebras over I = {0} ∪ 1/N ⊂ [0, 1] with fibers A1/N = Mk (C)⊗N and A0 = 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 : / A˜ 0 → A1/N , with A˜ 0 a dense Poisson subalgebra of A0 . A similar result is known for the symplectic manifold S 2 ⊂ R3 , for which in this case the fibers A1/N = M N +1 (C) ∼ = B(Sym N (C2 )) and A0 = 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 : A˜ 0 → A1/N . In this paper, we focus on the particular case X 2 ∼ = B 3 (i.e., the unit three-ball in R3 ) and show that for any function f ∈ A˜ 0 one has lim N →∞ ||(Q 1/N ( f ))| N 2 − Sym (C )
Q 1/N ( f | S2 )|| N = 0, where Sym N (C2 ) denotes the symmetric subspace of (C2 ) N ⊗ . Finally, we give an application regarding the (quantum) Curie–Weiss model.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2942 1.1 Strict deformation quantization maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2942 1.2 Spin systems and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2944
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Valter Moretti [email protected] Christiaan J. F. van de Ven [email protected]
1
Department of Mathematics, University of Trento, INFN-TIFPA, via Sommarive 14, 38123 Povo, Trento, Italy
2
Istituto Nazionale di Alta Matematica, Rome, Italy
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V. Moretti, C. J. F. van de Ven
2 Interplay of bulk quantization map Q 1/N and boundary quantization map Q 1/N 2.1 Preparatory results on Q 1/N and harmonic polynomials . . . . . . . . . . . 2.2 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Subsidiary technical results . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Application to the quantum Curie–Weiss model . . . . . . . . . . . . . . . . . . Appendix A: Continuous bundle of C ∗ -algebras . . . . . . . . . . . . . . . . . . . Appendix B: Coherent spin states . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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