Construction of Self-adjoint Berezin-toeplitz Operators on Kähler Manifolds and a Probabilistic Representation of the Associated Semigroups

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. In this paper we consider self-adjoint Berezin-Toeplitz operators associated with semibounded quadratic forms. 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.