Some aspects of positive kernel method of quantization

TL;DR

This paper explores the positive kernel method for quantizing automorphism groups of principal bundles, extending to non-compact Riemann surfaces and providing spectral decompositions of invariant kernels.

Contribution

It introduces a generalized quantization approach using positive kernels, applicable to complex flows on principal bundles, and generalizes Bochner's theorem for invariant kernels.

Findings

Realization of the generator as a generalized Kirillov-Kostant-Souriau operator

Quantization of holomorphic flows on non-compact Riemann surfaces

Integral decompositions of invariant positive kernels

Abstract

We discuss various aspects of positive kernel method of quantization of the one-parameter groups $\tau_t \in \mbox{Aut}(P,\vartheta)$ of automorphisms of a $G$-principal bundle $P(G,\pi,M)$ with a fixed connection form $\vartheta$ on its total space $P$. We show that the generator $\hat{F}$ of the unitary flow $U_t = e^{it \hat{F}}$ being the quantization of $\tau_t $ is realized by a generalized Kirillov-Kostant-Souriau operator whose domain consists of sections of some vector bundle over $M$, which are defined by suitable positive kernel. This method of quantization applied to the case when $G=GL(N,\mathbb{C})$ and $M$ is a non-compact Riemann surface leads to quantization of the arbitrary holomorphic flow $\tau_t^{hol} \in \mbox{Aut}(P,\vartheta)$. For the above case, we present the integral decompositions of the positive kernels on $P\times P$ invariant with respect to the flows $\tau_t^{hol}$ in terms of spectral measure of $\hat{F}$. These decompositions generalize the ones given by Bochner theorem for a positive kernels on $\mathbb{C} \times \mathbb{C}$ invariant with respect to the one-parameter groups of translations of complex plane.