Work

Strict Quantization of Polynomial Poisson Structures

Year: 2022

Type: article

Abstract: We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on $${\mathbb {R}}^d$$ 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 $$\hbar $$ ħ , giving strict quantizations on the space of analytic functions on $${\mathbb {R}}^d$$ Rd with infinite radius of convergence. We also address further questions such as continuity of the classical limit $$\hbar \rightarrow 0$$ 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.

Source: Communications in Mathematical Physics

DOI: 10.1007/s00220-022-04541-4

Authors: Severin Barmeier, Philipp Schmitt

Institutions: University of Freiburg, University of Cologne, Leibniz University Hannover

Language: English

Topic: Homotopy and Cohomology in Algebraic Topology

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,790

Funders: Deutsche Forschungsgemeinschaft, Albert-Ludwigs-Universität Freiburg