The Berezin-Simon quantization for Kähler manifolds and their path integral representations

The Berezin–Simon (BS) quantization is a rigorous version of the ``operator formalism’’ of quantization procedure. The goal of the paper is to present a rigorous real-time (not imaginary-time) path-integral formalism corresponding to the BS operator formalism of quantization; Here we consider the classical systems whose phase space $M$ is a (possibly non-compact) Kähler manifold which satisfies some conditions, with a Hamiltonian $H:M\to \mathbb{R}$. For technical reasons, we consider only the cases where $H$ is smooth and bounded. We use Güneysu’s extended version of the Feynman–Kac theorem to formulate the path-integral formula.