AbstractWe discuss various aspects of the positive kernel method of quantization of the one-parameter groups $$\tau _t \in \text{ Aut }(P,\vartheta )$$
τ
t
∈
Aut
(
P
,
ϑ
)
of automorphisms of a G-principal bundle $$P(G,\pi ,M)$$
P
(
G
,
π
,
M
)
with a fixed connection form $$\vartheta $$
ϑ
on its total space P. We show that the generator $${\hat{F}}$$
F
^
of the unitary flow $$U_t = e^{it {\hat{F}}}$$
U
t
=
e
i
t
F
^
being the quantization of $$\tau _t $$
τ
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 $$G=\hbox {GL}(N,{\mathbb {C}})$$
G
=
GL
(
N
,
C
)
and M is a non-compact Riemann surface leads to quantization of the arbitrary holomorphic flow $$\tau _t^{\mathrm{hol}} \in \text{ Aut }(P,\vartheta )$$
τ
t
hol
∈
Aut
(
P
,
ϑ
)
. For the above case, we present the integral decompositions of the positive kernels on $$P\times P$$
P
×
P
invariant with respect to the flows $$\tau _t^{\mathrm{hol}}$$
τ
t
hol
in terms of the spectral measure of $${\hat{F}}$$
F
^
. These decompositions generalize the ones given by Bochner’s Theorem for the positive kernels on $${\mathbb {C}} \times {\mathbb {C}}$$
C
×
C
invariant with respect to the one-parameter groups of translations of complex plane.
Some Aspects of Positive Kernel Method of Quantization