Strict quantization of coadjoint orbits

Schmitt, Philipp

doi:10.4171/jncg/429

Cited by 3 publications

(1 citation statement)

References 33 publications

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“…In this case, convergence on polynomials is immediate, since for all f, g ∈ P(R d ) their formal star product f ⋆ g is a polynomial in the formal parameter which can be evaluated to any value of . The continuity of star products, and the properties of the algebra obtained by completion, was studied successfully for constant and linear Poisson structures [25,10], also in infinite-dimensional, field-theoretic [23] and "global" settings, such as on coadjoint orbits of Lie groups [17,21] or on cotangent bundles of Lie groups [13] (see [26] for a review). Yet, although constant and linear Poisson structures are important classes of Poisson structures, one cannot expect them to cover all physically relevant phase spaces.…”

Section: Introductionmentioning

confidence: 99%

Strict quantization of polynomial Poisson structures

Barmeier,

Schmitt

2022

Preprint

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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 , 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 → 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.

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Section: Introductionmentioning

confidence: 99%

Strict quantization of polynomial Poisson structures

Barmeier,

Schmitt

2022

Preprint

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Classical and Quantised Resolvent Algebras for the Cylinder

van Nuland,

Stienstra

2024

Ann. Henri Poincaré

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Buchholz and Grundling (Commun Math Phys 272:699–750, 2007) introduced a $$\hbox {C}^*$$ C ∗ -algebra called the resolvent algebra as a canonical quantisation of a symplectic vector space and demonstrated that this algebra has several desirable features. We define an analogue of their resolvent algebra on the cotangent bundle $$T^*\mathbb {T}^n$$ T ∗ T n of an n-torus by first generalising the classical analogue of the resolvent algebra defined by the first author of this paper in earlier work (van Nuland in J Funct Anal 277:2815–2838, 2019) and subsequently applying Weyl quantisation. We prove that this quantisation is almost strict in the sense of Rieffel and show that our resolvent algebra shares many features with the original resolvent algebra. We demonstrate that both our classical and quantised algebras are closed under the time evolutions corresponding to large classes of potentials. Finally, we discuss their relevance to lattice gauge theory.

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Strict Quantization of Polynomial Poisson Structures

Barmeier

Schmitt

2022

Commun. Math. Phys.

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We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on $${\mathbb {R}}^d$$ 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 $$\hbar $$ ħ , giving strict quantizations on the space of analytic functions on $${\mathbb {R}}^d$$ R d 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.

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Strict quantization of coadjoint orbits

Cited by 3 publications

References 33 publications

Strict quantization of polynomial Poisson structures

Strict quantization of polynomial Poisson structures

Classical and Quantised Resolvent Algebras for the Cylinder

Strict Quantization of Polynomial Poisson Structures

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