XXXVIII Workshop on Geometric Methods in Physics 30.06-6.07.2019 VIII School on Geometry and Physics 24-28.06.2019
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# Stephen B. Sontz

## Coherent States for a Toeplitz Quantization of the Manin Plane

In the theory of Toeplitz quantization of algebras coherent states are defined as eigenvectors of a Toeplitz annihilation operator. These coherent states are studied in the case when the algebra is the generically non-commutative Manin plane. We introduce the resolution of the identity, upper and lower symbols as well as a coherent state quantization, which in turn quantizes the Toeplitz quantization. We thereby have a curious combination of quantization schemes which might be a novelty. We proceed by identifying a generalized Segal–Bargmann space $\mathcal{SB}$ of square-integrable, anti-holomorphic functions as the image of a coherent state transform. Then $\mathcal{SB}$ has a reproducing kernel function which allows us to define a secondary Toeplitz quantization, whose symbols are functions.

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