TY - JOUR

T1 - Strict Quantization of Polynomial Poisson Structures

AU - Barmeier, Severin

AU - Schmitt, Philipp

N1 - Funding Information: It is a pleasure to thank Jonas Schnitzer for helpful discussions and for making us acquainted in the first place, and Martin Bordemann, Matthias Schötz, Stefan Waldmann and the anonymous referees for various valuable comments. The first named author also gratefully acknowledges the support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the DFG Research Training Group GK1821 “Cohomological Methods in Geometry” at the University of Freiburg.

PY - 2023/3

Y1 - 2023/3

N2 - We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on Rd, generalizing known results for constant and linear Poisson structures to polynomial Poisson structures of arbitrary degree. We give several examples of nonlinear Poisson structures and construct explicit formal star products whose deformation parameter can be evaluated to any real value of ħ, giving strict quantizations on the space of analytic functions on Rd with infinite radius of convergence. We also address further questions such as continuity of the classical limit ħ→ 0 , compatibility with ∗-involutions, and the existence of positive linear functionals. The latter can be used to realize the strict quantizations as ∗-algebras of operators on a pre-Hilbert space which we demonstrate in a concrete example.

AB - We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on Rd, generalizing known results for constant and linear Poisson structures to polynomial Poisson structures of arbitrary degree. We give several examples of nonlinear Poisson structures and construct explicit formal star products whose deformation parameter can be evaluated to any real value of ħ, giving strict quantizations on the space of analytic functions on Rd with infinite radius of convergence. We also address further questions such as continuity of the classical limit ħ→ 0 , compatibility with ∗-involutions, and the existence of positive linear functionals. The latter can be used to realize the strict quantizations as ∗-algebras of operators on a pre-Hilbert space which we demonstrate in a concrete example.

UR - http://www.scopus.com/inward/record.url?scp=85142154061&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2201.03249

DO - 10.48550/arXiv.2201.03249

M3 - Article

AN - SCOPUS:85142154061

VL - 398

SP - 1085

EP - 1127

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -