Fock Representations and Deformation Quantization of Kähler Manifolds

• Published in: Advances in applied Clifford algebras 2017-09, Vol.27 (3), p.2769-2794
• Main Authors: Sako, Akifumi, Umetsu, Hiroshi
• Format: Article
• Language: English
• Subjects: Algebra; Applications of Mathematics; Commutation; Conjugates; Deformation; Linear operators; Manifolds; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Measurement; Physics; Physics and Astronomy; Representations; Separation; Theoretical; Vectors (mathematics)
• ISSN: 0188-7009; 1661-4909
• DOI: 10.1007/s00006-016-0753-z

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.