Details
| Original language | English |
|---|---|
| Pages (from-to) | 1085-1127 |
| Number of pages | 43 |
| Journal | Communications in Mathematical Physics |
| Volume | 398 |
| Issue number | 3 |
| Early online date | 17 Nov 2022 |
| Publication status | Published - Mar 2023 |
Abstract
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.
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematics(all)
- Mathematical Physics
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In: Communications in Mathematical Physics, Vol. 398, No. 3, 03.2023, p. 1085-1127.
Research output: Contribution to journal › Article › Research › peer review
}
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 -