Strict Quantization of Polynomial Poisson Structures

We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on R d , generalizing known results for constant and linear Poisson structures to polynomial Poisson structures of arbitrary degree. We give several examples of nonlinear P...

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Bibliographic Details
Published in:Communications in mathematical physics 2023-03, Vol.398 (3), p.1085-1127
Main Authors: Barmeier, Severin, Schmitt, Philipp
Format: Article
Language:English
Subjects:
Analytic functions
Classical and Quantum Gravitation
Combinatorial analysis
Complex Systems
Continuity (mathematics)
Hilbert space
Mathematical analysis
Mathematical and Computational Physics
Mathematical Physics
Operators (mathematics)
Physics
Physics and Astronomy
Polynomials
Quantum Physics
Relativity Theory
Theoretical
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Summary:We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on R d , 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 R d 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.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-022-04541-4