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Subalgebras with converging star products in deformation quantization: An algebraic construction for \(\mathbb{C} P^n\). (English) Zbl 0923.58024
In an earlier article [Lett. Math. Phys. 36, No. 4, 357-371 (1996; Zbl 0849.58035)], the authors gave a closed formula for a star product of the Wick type complex projective space. Here they use this formula to construct a subalgebra \({\mathcal U}\) of the formal algebra consisting of converging power series of the formal parameter. When substituting this parameter by a real number \(\alpha\), the quotient of \({\mathcal U}\) by the kernel of the substitution homomorphism turns out to be infinite dimensional, except for the case \(\alpha= {1\over n}\), \(n\) some positive integer, where it turns out to be finite dimensional and can be identified with an associative algebra of quantum observables of the isotropic harmonic oscillator with \(n + 1\) degrees of freedom. This procedure is tested on other phase spaces like \(\mathbb{C}^{n+l}\), the \(2n\)-torus or the Poincaré disk.

MSC:
53D50 Geometric quantization
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
81Q99 General mathematical topics and methods in quantum theory
46L60 Applications of selfadjoint operator algebras to physics
46N50 Applications of functional analysis in quantum physics
81S10 Geometry and quantization, symplectic methods
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