Berezin Toeplitz Quantization By Wiener Regularized Path Integrals
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Berezin-Toeplitz Quantization by Wiener-regularized Path Integrals
ABSTRACT: This dissertation investigates 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. A first concern is the construction of self-adjoint Berezin-Toeplitz operators associated with semibounded quadratic forms. These forms are obtained from the inner product of the Hilbert space by multiplying the underlying measure with sufficiently regular real-valued functions. The semigroups generated by the associated self-adjoint Berezin-Toeplitz operators may under certain conditions be represented in the form of Wiener-regularized path integrals, according to a concept by Daubechies and Klauder. More explicitly, the integration is taken over Brownian-motion paths in phase space in the ultra-diffusive limit. Finally, the probabilistic representation and an invariance property of Brownian motion are combined to yield a relation between resolvents of different Berezin-Toeplitz operators. 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.