Some Aspects of Positive Kernel Method of Quantization

Abstract

We discuss various aspects of the positive kernel method of quantization of the one-parameter groups tau _t in text{ 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 a suitable positive kernel. This method of quantization applied to the case when G=hbox {GL}(N,{mathbb {C}}) and M is a non-compact Riemann surface leads to quantization of the arbitrary holomorphic flow tau _t^{mathrm{hol}} in text{ Aut }(P,vartheta ). For the above case, we present the integral decompositions of the positive kernels on Ptimes P invariant with respect to the flows tau _t^{mathrm{hol}} in terms of the spectral measure of {hat{F}}. These decompositions generalize the ones given by Bochner’s Theorem for the positive kernels on {mathbb {C}} times {mathbb {C}} invariant with respect to the one-parameter groups of translations of complex plane.