Construction of self-adjoint Berezin–Toeplitz operators on Kähler manifolds and a probabilistic representation of the associated semigroups
✍ Scribed by Bernhard G. Bodmann
- Publisher
- Elsevier Science
- Year
- 2003
- Tongue
- English
- Weight
- 269 KB
- Volume
- 47
- Category
- Article
- ISSN
- 0393-0440
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✦ Synopsis
We investigate a class of operators resulting from a quantization scheme attributed to Berezin. These so-called Berezin-Toeplitz operators are defined on a Hilbert space of square-integrable holomorphic sections in a line bundle over the classical phase space. As a first goal we develop self-adjointness criteria for Berezin-Toeplitz operators defined via quadratic forms. Then, following a concept of Daubechies and Klauder, the semigroups generated by these operators may under certain conditions be represented in the form of Wiener-regularized path integrals. More explicitly, the integration is taken over Brownian motion paths in phase space in the ultra-diffusive limit. All results are the consequence of a relation between Berezin-Toeplitz operators and Schrödinger operators defined via certain quadratic forms. The probabilistic representation is derived in conjunction with a version of the Feynman-Kac formula.