Berezin-Toeplitz quantization on Lie groups

Details Berezin-Toeplitz quantization on Lie groups Berezin-Toeplitz quantization on Lie groups

2008-06-18

arxiv.org/abs/0806.3063v1

Journal of Functional Analysis, Vol. 255 (2008), 2488-2506

Physics

Mathematical Physics

To appear in Journal of Functional Analysis

Scientific paper

10.1016/j.jfa.2008.06.022

Let K be a connected compact semisimple Lie group and Kc its complexification. The generalized Segal-Bargmann space for Kc, is a space of square-integrable holomorphic functions on Kc, with respect to a K-invariant heat kernel measure. This space is connected to the "Schrodinger" Hilbert space L^2(K) by a unitary map, the generalized Segal-Bargmann transform. This paper considers certain natural operators on L^2(K), namely multiplication operators and differential operators, conjugated by the generalized Segal-Bargmann transform. The main results show that the resulting operators on the generalized Segal-Bargmann space can be represented as Toeplitz operators. The symbols of these Toeplitz operators are expressed in terms of a certain subelliptic heat kernel on Kc. I also examine some of the results from an infinite-dimensional point of view based on the work of L. Gross and P. Malliavin.

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