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Fock Representations and Deformation Quantization of Kähler Manifolds

Autor(en)
Akifumi Sako, Hiroshi Umetsu
Abstrakt

The goal of this paper is to construct the Fock representation of noncommutative Kähler manifolds. Noncommutative Kähler manifolds studied here are constructed by deformation quantization with separation of variables, which was given by Karabegov. The algebra of the noncommutative Kähler manifolds contains the Heisenberg-like algebras. Local complex coordinates and partial derivatives of a Kähler potential satisfy the commutation relations between creation and annihilation operators. A Fock space is constituted by states obtained by applying creation operators on a vacuum which is annihilated by all annihilation operators. The algebras on noncommutative Kähler manifolds are represented as those of linear operators acting on the Fock space. In representations studied here, creation operators and annihilation operators are not Hermitian conjugates of one other, in general. Therefore, the basis vectors of the Fock space are not the Hermitian conjugates of those of the dual vector space. In this sense, we call the representation the twisted Fock representation. In this presentation, we construct the twisted Fock representations for arbitrary noncommutative Kähler manifolds given by deformation quantization with separation of variables, and give a dictionary to translate between the twisted Fock representations and functions on noncommutative Kähler manifolds concretely.

Organisation(en)
Mathematische Physik
Externe Organisation(en)
Tokyo University of Science, Universität Wien, National Institute of Technology, Ishikawa College
Journal
Advances in Applied Clifford Algebras
Band
27
Seiten
2769-2794
Anzahl der Seiten
26
ISSN
0188-7009
DOI
https://doi.org/10.1007/s00006-016-0753-z
Publikationsdatum
09-2017
Peer-reviewed
Ja
ÖFOS 2012
103036 Theoretische Physik, 103019 Mathematische Physik
Schlagwörter
ASJC Scopus Sachgebiete
Applied Mathematics
Link zum Portal
https://ucrisportal.univie.ac.at/de/publications/79ffd94d-36a9-49b8-ad4a-5a4aadc210ff