NDLI: Berezin–Toeplitz Quantization and Invariant Symbolic Calculi

Content Provider	Springer Nature Link
Author	Engliš, Miroslav
Copyright Year	2003
Abstract	It is well known that one can often construct an invariant star-product by expanding the product of two Toeplitz operators asymptotically into a series of another Toeplitz operators multiplied by increasing powers of the Planck constant h. This is the Berezin–Toeplitzquantization. We show that on bounded symmetric domains (Hermitian symmetric spaces of noncompact type), one can in fact obtain in a similar way any invariant star-productwhich is G-equivalent to the Berezin–Toeplitz star-product, by using, instead of Toeplitz operators, other suitable assignments f↦Q $_{ f }$ from compactly supported C $^{∞}$ functions f to bounded linear operators Q $_{ f }$ on the corresponding Hilbert spaces. (This procedureis referred to as prime quantization by some authors.) Along the way, we establish two technical results which are of interest in their own right, namely a controlled-growth parameter generalization of the classical theorem of Borel on the existence of a function with prescribed derivatives of all orders at a point, and the fact that any invariant bi-differential operator (Hochschild two-cochain) on a bounded symmetric domain automatically maps the Schwartz space into itself.
Starting Page	59
Ending Page	74
Page Count	16
File Format	PDF
ISSN	03779017
Journal	Letters in Mathematical Physics
Volume Number	65
Issue Number	1
e-ISSN	15730530
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2003-01-01
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Group Theory and Generalizations Geometry Mathematical and Computational Physics Statistical Physics
Content Type	Text
Resource Type	Article
Subject	Statistical and Nonlinear Physics Mathematical Physics

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