Some aspects of positive kernel method of quantization

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.


Introduction
From the very beginnings of quantum mechanics the problem of quantization is one of the most fascinating and crucial ones for understanding the correspondence between classical and quantum physics. Excluding the field theory and restricting to the case of mechanics only, by quantization of a Hamiltonian flow R ∋ t → σ F t ∈ SpDiff(M, ω) defined on a symplectic manifold (M, ω) one usually understands the construction of a corresponding unitary flow R ∋ t → e it F on a Hilbert space H. Additionally one claims that the map Q : P ∞ (M, R) ∋ F → i F ∈ L(D) which assigns to a classical 1 E-mail: aodzijew@uwb.edu.pl 2 E-mail: horowski@math.uwb.edu.pl generator F the quantum one F (a self-adjoint operator in H) is a morphism of some Lie algebras, where P ∞ (M, R) is a Lie subalgebra of the Poisson algebra C ∞ (M, R) and L(D) is a Lie algebra of anti-self-adjoint operators having a common domain D dense in H.
Among known methods of quantization the Kirillov-Kostant-Souriau geometric quantization [18], [19], [37] is one of the most elegant from a geometric point of view and gives a precise construction of the quantum generator F for the given classical one F ∈ P ∞ (M, R). For this construction one needs to obtain a σ F t -invariant complex Lagrangian distribution P ⊂ T C M and the appropriate measure (density) on the quotient manifold M/P ∩P. However, this leads to serious difficulties if one wants to quantize concrete mechanical systems. In order to omit these difficulties and for deeper understanding of the relationship between the classical (M, ω) and quantum (CP(H), ω F S ) phase spaces in [26] and [29] a method of quantization based on the notion of positive kernel (coherent state map) was proposed, which in our opinion completes the Kirillov-Kostant-Souriau quantization in a natural way. For example one can find the application of the coherent state method of quantization to concrete physical systems in [16], [30].
For a general theory of positive (reproducing) kernels and its role in differential geometry (including Banach differential manifolds and vector bundles over them) and representation theory we address to [5,6] and to the monograph [24]. See also the classical paper [1] of N. Aronszajn. Basing partly on [29], in Section 2 and Section 3 we briefly discuss how to extend the Kirillov-Kostant-Souriau prequantization procedure defined for U (1)-principal bundle to the case of an arbitrary G-principal bundle P (G, π, M ) with a fixed connection form ϑ on the total space P . In Section 2 we define the Poisson C ∞ (M, R)-module P ∞ G (P, ϑ) of generators (X, F ) ∈ P ∞ G (P, ϑ) of generalized Hamiltonian flows τ (X,F ) t ∈ Aut(P, ϑ), i.e. those which are solutions of generalized Hamilton equations (2.10). In Section 3 we generalize the Kirillov-Kostant-Souriau prequantization morphisms to the morphism Q : P ∞ G (P, ϑ) → D 1 Γ ∞ (M, V) of P ∞ G (P, v) in the C ∞ (M, R)-module of differential operators of order less or equal one acting on the smooth sections Γ ∞ (M, V) of an associated smooth vector bundle V → M over M .
In Section 4 we consider the G-equivariant coherent state map K : P → B(V, H) and the positive definite G-equivariant kernel K : P × P → B(V ), where V and H are complex Hilbert spaces and B(V, H) is the right Hilbert B(V )-module of bounded linear maps of V in H. In the same section the equivalence of the coherent state K and positive kernel K notions is shown and the method of quantization based on them is investigated. Among others we show that the Kirillov-Kostant-Souriau differential operator Q (X,F ) can be treated as a self-adjoint operatorF in the Hilbert space H K whose domain is defined by the G-equivariant positive kernel K : P × P → B(V ) (see (4.30) and (4.31)). The conditions on this kernel needed to quantize τ (X,F ) t ∈ Aut(P, ϑ) are presented in (4.24) and (4.25). In Section 5 assuming that G ⊂ GL(V, C) is a Lie subgroup of GL(V, C) and that there exists a coherent state map K : P → B(V, H) on the total space of P (G, π, M ), we define in a canonical way two other principal bundles P (GL(V, C), π, M ) and U (U (V ), π u , M ) over M . The connection forms ϑ ∈ Γ ∞ ( P , T * P ⊗ B(V )) and ϑ a ∈ Γ ∞ (U, T * U ⊗ T e U (V )) as well as the respective coherent states maps K : P → B(V, H) and a : U → B(V, H) are defined on these principal bundles by using of K : P → B(V, H). Next we show, see Proposition 5.2, that the flows τ (X,F ) t ∈ Aut( P , ϑ) and τ (X,F a ) t ∈ Aut(U, ϑ a ) have the same quantum counterpart e itF as the flow τ (X,F ) t ∈ Aut(P, ϑ). In Section 6 we quantize the holomorphic one-parameter groups of automorphisms of a holomorphic principal bundles P (GL(V, C), π, M ) over a non-compact Riemann surface M . For this relatively simple but non-trivial case the investigated theory is presented in a complete way. In particular we obtain a Bochner-type integral decompositions of the τ (X,F ) t -invariant positive kernels on P × P and show their relationship with the spectral decomposition of the corresponding quantum generators Q (X,F ) =F .
Some applications of the coherent state method in physics including quantum optics are shortly discussed in Section 7.
2 The Poisson C ∞ (M, R)-module corresponding to the group Aut(P, ϑ) The main task of this section is the investigation of some variant of Hamiltonian mechanics on a G-principal bundle P (G, π, M ), where the role of the symplectic form is played by the curvature form Ω of a fixed connection form ϑ ∈ Γ ∞ (P, T * P ⊗ T e G). We will define the Poisson C ∞ (M, R)-module (P ∞ G (P, ϑ), {·, ·} ϑ ) with the Lie bracket {·, ·} ϑ given in (2.11), which satisfies the Leibniz property in the sense of (2.13). The corresponding generalization (2.10) of the Hamiltonian equation is presented. The model proposed here in the case when G = U (1) and ϑ has non-singular curvature form reduces to standard Hamiltonian mechanics on a symplectic manifold.
Let Aut(P, ϑ) denote the group of diffeomorphisms of P which preserve the principal bundle structure of P and the connection form ϑ. We recall that a T e G-valued differential one-form ϑ on P is a connection form if and only if ϑ(pg) • T κ g (p) = Ad g −1 • ϑ(p), for any p ∈ P and g ∈ G, where κ : P × G → P is the right-action κ(p, g) =: pg of the Lie group G on P and T e G is the tangent space to G at the unit element e ∈ G, i.e. the Lie algebra of G. By κ g : P → P and κ p : G → P we denoted the maps defined by κ g (p) := pg and κ p (g) := pg. The connection form ϑ defines the spliting of the tangent space T p P at p ∈ P on the horizontal T h p P := ker ϑ(p) and vertical T v p P , i.e. tangent to the fibre π −1 (π(p)) of π : P → M , components (see e.g. [21]).
Then for the vector field ξ ∈ Γ ∞ (T P ) tangent to the flow {τ t } t∈R one has T κ g (p)ξ(p) = ξ(pg), (2.3) L ξ ϑ = 0, (2.4) where L ξ is the Lie derivative with respect to ξ. The C ∞ (M, R)-modules of vector fields ξ ∈ Γ ∞ (T P ) which satisfy the condition (2.3) and the condition (2.3) together with the condition (2.4) will be denoted by Γ ∞ G (T P ) and Γ ∞ G,ϑ (T P ), respectively. The C ∞ (M, R)-module structure on Γ ∞ G (T P ), and hence on its submodules Γ ∞ , where by T h P and T v P we denote the horizontal and vertical vector subbundles of T P , respectively.
The tangent map T π : T P → T M defines an isomorphism of C ∞ (M, R)-modules of horizontal vector fields Γ ∞ G (T h P ) on P and vector fields Γ ∞ (T M ) on M by (π * ξ h )(π(p)) := T π(p)ξ h (p).
(2.5) Its inverse is given by The correctness of the definitions above follows from (2.1) and (2.2), and from T κ g (p) • T κ p (e) = T κ pg (e).
Taking the decomposition ξ = ξ h + ξ v of ξ ∈ Γ ∞ G (T P ) on the horizontal and vertical components and isomorphisms (2.5) and (2.6) we define where by H * : we denote the horizontal lift, i.e. the module isomorphism inverse to π * .
For ξ ∈ Γ ∞ G (T P ) one has where is the curvature form of ϑ and is the covariant derivative of F . On Γ ∞ G (T P ) one has the structure of Lie algebra given by the Lie bracket [·, ·] of vector fields. Using the C ∞ (M, R)-modules isomorphism π * × ν * : It is reasonable to mention here that the Lie algebra (Γ ∞ G (T P ), [·, ·]) is isomorphic with the Lie algebra (Γ ∞ (T P/G), [·, ·]) of the Atiyah Lie algebroid, being a central ingredient of the Atiyah exact sequence of algebroids [23].
Though the language of Lie algebroid theory will not be used later, we note that the projection on the first component pr 1 : corresponds to the anchor map a : T P/G → T M of the Atiyah algebroid. Hence, from the defining property of the anchor map we have These properties of pr 1 and {·, ·} ϑ can also be obtained directly from their definitions. The above structure of Lie It follows from (2.7) that (X, F ) ∈ P ∞ G (P, ϑ) if and only if H * (X) Ω + DF = 0. (2.10) Let us note here that the condition (2.10) is invariant with respect to the Lie C ∞ (M, R)-module operation. Thus, for (X 1 , F 1 ), (X 2 , F 2 ) ∈ P ∞ (P, ϑ) the Lie bracket (2.8) simplifies to the form In the case when the curvature form Ω is a non-singular 2-form, i.e. when ξ h Ω = 0 implies ξ h = 0, one has from (2.10) that for (X, F ) ∈ P ∞ G (P, ϑ) the vector field X ∈ Γ ∞ (T M ) is defined uniquely by the function F ∈ C ∞ G (P, T e G). So, in this case we have the C ∞ (M, R)-modules morphism b : C ∞ G (P, T e G) → P ∞ G (P, ϑ). Note here that F ∈ kerb if and only if DF = 0, which does not mean in general that F = const. Substituting X 1 = b(F 1 ) and X 2 = b(F 2 ) into (2.11) we obtain the Lie bracket , which satisfies the Leibniz property is canonically isomorphic with C ∞ (M, R) and the curvature form Ω is identified with the closed dω = 0 2-form ω on M , which is a symplectic form in the non-singular case. Hence, formula (2.12) reduces to the symplectic Poisson bracket and (2.13) to its Leibniz property.
Taking the above facts into account, further we will call (P ∞ In the framework of the assumed terminology it is natural to consider: (i) the equation (2.10) as a generalization of Hamilton's equation to the arbitrary Lie group G case; (ii) the one-parameter group τ (X,F ) t ∈ Aut(P, ϑ) as a generalized Hamiltonian flow generated by (X, F ) ∈ P ∞ G (P, ϑ) (in non-singular case by F ∈ C ∞ G (P, T e G)). In the next two sections we propose and investigate a method of quantization of the Hamiltonian flow τ (X,F ) t ∈ Aut(P, ϑ) based on the notion of G-equivariant positive kernel on P × P . Though our considerations below are valid for an arbitrary Lie group G we will assume that G ⊂ GL(V, C) ∼ = GL(N, C) is a Lie subgroup of the linear group GL(V, C) of a complex N -dimensional Hilbert space V . By ·, · : V × V → C we denote the scalar product for V and by the canonical action of G in V . The group of unitary maps of V as usually will be denoted by

Kirillov-Kostant-Souriau prequantization morphism
In this section we generalize the Kirillov-Kostant-Souriau prequantization procedure [18], [19], [37] for the case of an arbitrary Lie group G ⊂ GL(V, C), i.e. we obtain the Lie C ∞ (M, R)-module morphism Q : P ∞ G (P, ϑ) → D 1 Γ ∞ (M, V) of the Poisson module P ∞ G (P, ϑ) into the Lie module of differential operators of the order less or equal one acting on the smooth sections of some complex vector bundle V → M over M .
To this end we define the smooth complex bundle V := (P × V )/G → M over M associated to P (G, π, M ) by the action P ×V ×G ∋ (p, v, g) → (pg, g −1 v) ∈ P ×V of the Lie group G on P ×V . One has the natural C ∞ (M, R)-module isomorphism between the module Γ ∞ (M, V) of smooth sections of V → M and the module C ∞ G (P, V ) := {f ∈ C ∞ (P, V ) : f (pg) = g −1 f (p) f or g ∈ G} of G-equivariant smooth functions on P defined by the one-to-one dependence ψ(π(p)) = [(p, f (p))] := {(pg, g −1 f (p)) : g ∈ G} (3.1) between ψ ∈ Γ ∞ (M, V) and f ∈ C ∞ G (P, V ). Using (3.1) we define for any (X, F ) = (π * × ν * )(ξ) ∈ P ∞ G (P, ϑ) the differential operator Q (X,F ) : Γ ∞ (M, V) → Γ ∞ (M, V) of order less or equal one. Let us note here that if ξ ∈ Γ ∞ G,ϑ (T P ) and is an isomorphism of the Lie C ∞ (M, R)-modules, from (3.1) and (3.2) one obtains 3) where on the left hand side of the equality (3.3) we have the commutator of the differential operators and {·, ·} ϑ is the Lie bracket defined in (2.11). Since in the case dim C V = 1 and G = U (1) the Lie C ∞ (M, R)-modules monomorphism is the Kirillov-Kostant-Souriau prequantization morphism we will further extend this terminology to the general case. By D 1 Γ ∞ (M, V) in (3.5) we denote the Lie C ∞ (M, R)-module of differential operators of order less or equal one acting on Γ ∞ (M, V).
In order to establish a more explicit expression for Q (X,F ) , where (X, F ) ∈ P ∞ (P, ϑ), we will use the decomposition ξ = ξ h + ξ v of ξ ∈ Γ ∞ G,ϑ (T P ) on the horizontal and vertical components. The flows τ t and τ h t tangent to ξ and ξ h satisfy where g : R × P → G is the cocycle related to the vertical (tangent to ξ v ) flow τ v t by τ v t (p) = pg(t, p).
Differentiating the equality above with respect to the parameter s at s = 0 we obtain differential equation d dt g(t, p) = F (τ h t (p))g(t, p) = g(t, p)F (p) (3.9) on the cocycle g : R × P → G ⊂ GL(N, C). The equality (3.7) is obtained as a solution of (3.9) with the initial condition g(0, p) = 1 1 V .
In order to obtain d dt h α (t, m)| t=0 ∈ T e G we note that from (3.19) it follows that Next, applying ϑ(s α (m)) to both sides of (3.22) and using (2.2) we obtain where ϑ α (m) := ϑ(s α (m)) • T s α (m) = (s * α ϑ)(m). Introducing the notations φ α (m) := d dt g α (t, m)| t=0 and F α (m) := F • s α (m), after differentiating (3.21) at t = 0 we obtain the equality which is useful for finding the local expression for the infinitesimal generator of the flow Σ t : (3.16). Namely, we have Differentiating both sides of (3.25) at t = 0 and using (3.23) we obtain the local representation In the local gauge the Hamilton equation (2.10) assumes the form where Ending this subsection let us mention the well known equivariance formulae with respect to the gauge transformation

Positive definite kernels and quantization
In the previous section we obtained the formulas (3.13) and (3.15) on Kirillov-Kostant-Souriau prequantization operators Q (X,F ) andQ (X,F ) as well as their local versions (3.26) and (3.27). Here we will present a procedure which allows us to treat them as self-adjoint operators in a Hilbert space. This quantization procedure is based on the notion of G-equivariant positive kernel, see [29]. In order to make the paper self-sufficient we present the above procedure in detail. Some new results complementary to the ones in [29] will also be presented in this section. The theory of reproducing kernels for like-Hermitian smooth vector bundles over smooth Banach manifolds and related linear connections one can find in [5], [6].
Let us recall here that we have assumed that we denote the C * -algebra of linear maps of V and by B(V, H) the right Hilbert B(V )-module of linear maps Γ : V → H from the Hilbert space V into the separable Hilbert space H. Let us note here that from dim C V < +∞ follows boundness of Γ : Definition 4.1 A smooth map K : P → B(V, H) will be called a G-equivariant coherent state map if it has the following properties: for any p ∈ P and for any g ∈ G; (ii) non-singularity, i.e.
for any p ∈ P ; (iii) the set K(P )V is linearly dense in H, i.e.
By K(p) * in (4.2) and (4.3) we denoted the map K(p) * : where ·|· is the scalar product in Hilbert space H. For the existence of K : P → B(V, H) with the above properties see [25].
From the positivity condition (4.6) one obtains the following properties of the kernel and where g, h ∈ G and p, q ∈ P . The above structures are mutually dependent. Extending the rather well known scheme from the theory of reproducing (positive) kernels [24], [3] to this more geometrically complicated setting we shortly describe this dependence.
Starting from the coherent state map K : P → B(V, H) we define the G-equivariant positive definite kernel by The smoothness of the kernel (4.7) and the properties (4.4)-(4.6) follow from the smoothness of K : P → B(V, H) and its properties mentioned in (4.1)-(4.3). The opposite dependence needs longer considerations. Firstly let us define the vector space of V -valued functions on P , where J is a finite set of indices, i.e. the vector subspace D K ⊂ C ∞ G (P, V ) which consists of linear combinations of the functions K(·, q)v indexed by q ∈ P and v ∈ V . In order to define the scalar product In particular one has (4.10) From (4.10) and the Schwarz inequality for the scalar product (4.9) one obtains From the inequality (4.11) one sees that f K = f |f We conclude from (4.11) that the evaluation functional E p (f ) :=f (p) satisfies i.e. it is a bounded functional on D K for every p ∈ P . So, we can extend it to the Hilbert space H K ⊃ D K being the abstract extension of the pre-Hilbert space D K ⊂ C ∞ G (P, V ). It follows from (4.11) that for any equivalence class Hence we see that the Hilbert space H K is realized in a natural way as a vector subspace of the vector space of V -valued functions on P which satisfy the G-equivalence condition from (3.14). Now, rewriting (4.10) as follows . One easily sees that the properties (4.1)-(4.3) for K K : P → B(V, H K ) follow from the ones for K given in (4.4)-(4.6). The smoothness of K K : P → B(V, H K ) follows from the smoothness of the positive kernel K (see Proposition 2.1 in [29]).
As we see from (4.7) a coherent state map K : P → B(V, H) defines the positive kernel K : P × P → B(V ). The Hilbert space H K for this kernel is isomorphic to the Hilbert space H, where the isomorphism I K : H ∼ −→ H K is defined by (4.12) In order to see that I K is indeed an isomorphism we note that for |ψ ∈ D K , where D K ⊂ H is defined by Therefore, if the coherent state map K and the positive definite kernel K are related by (4.7), then The horizontal arrows on the right hand side of (4.14) are defined by Restricting the scalar product ·|· of H to the fibers π −1 N (q) of the tautological vector bundle π N : E N → Grass(N, H) and using the vector bundles morphism (4.16) one defines the Hermitian structure on the vector bundle π V : V → M .
the metric (with respect to the Hermitian structure (4.17)) connection form ϑ K ∈ Γ ∞ (T * P ⊗ T e G) on the G-principal bundle P (G, π, M ).

Proof
From the equivariance property (4.1) one has In order to show the condition (2.2) we take X : ∈ G is a smooth curve in G such that g(0) = e, and substitute T κ p (e)X into both sides of the definition (4.18). Thus we obtain where K(p, p) = K(p) * K(p), that ϑ K is a metric connection with respect to the Hermitian structure (4.17). In order to see this let us take Since arbitrary sections ψ 1 , ψ 2 ∈ Γ ∞ (M, V) of the vector bundle π V : V → M can be written as After these preliminary considerations we propose a method of quantization of the generalized Hamiltonian flow τ (i) the connection form ϑ is equal to ϑ K defined in (4.18); (ii) the equivariance property for K : P → B(V, H) is fulfilled with respect to both considered flows.
From the condition (iii), see (4.3) of the Definition 4.1 it follows that τ in an unique way.
Let us also note that if K : P → B(V, H) is a one-to-one smooth map and the flow τ t ∈ Aut(P ) satisfies K(τ t (p)) = U t K(p) for a certain unitary flow U t , then τ t ∈ Aut(P, ϑ K ) and thus there exists Additionally (X, F ) satisfies the generalized Hamilton equation (2.10) for the curvature form Ω K defined by ϑ K . If Ω K is non-singular then the vector field X ∈ Γ ∞ (T M ) is uniquely defined by F ∈ C ∞ G (P, T e G). Since for the non-singular curvature form Ω the equation (2.10) allows one to define X ∈ Γ ∞ (T M ) by F ∈ C ∞ G (P, T e G), so, in this case we will use the notation It follows from Stone's Theorem, see e.g. [36], that there exists a self-adjoint operator F with Differentiating both sides of the equation (4.22) with respect to t ∈ R we obtain (4.23) Note here that for any p ∈ P one has Ran K(p) ⊂ D K ⊂ D F . We also note that the last equality in (4.23) follows from (3.6)-(3.7) and (4.22). The symmetricity of F on the domain D K is equivalent to the equation where H * (X) is the horizontal lift of X ∈ Γ ∞ (T M ) with respect to ϑ. From (4.23) and (4.18) we find that the mean values map · : F → F defined by is inverse to the quantization Q : F → F of the classical generator F of the Hamiltonian flow τ F t . Using the isomorphism I K : H → H K we can represent the quantum flow U (X,F ) t and its generator F in terms of the Hilbert space H K . Namely, we have where ϑ α := s * α ϑ, the local cocycle g α (t, m) and the flow σ e. X (m, n) = (X(m), X(n)), and φ α (m), defined in (3.23), satisfy the equation equivalent to the eq. (3.28).
Let us also mention the transformation formulas and Kβ γ (m, n) = g αβ (m) † Kᾱ δ (m, n)g δγ (n), The domain D F of the generator F contains D K ⊂ D F the dense subset D K defined in (4.13) which consists of |ψ = j∈J K βj (n j )v j ∈ D F , where n j ∈ O βj , v j ∈ V and J is finite. So, the local version of the formula (4.23) is the following The local version of (3.13) and (3.15) are , where X, F α and ϑ α are fixed and satisfy (4.29). In general this is a rather hard task. However, it is possible to do this for some particular cases. For this reason see Section 6.

Extension and reduction
It turns out that if G ⊂ GL(V, C), then having a principal bundle P (G, π, M ) and a coherent state map K : P → B(V, H) one can define in a canonical way two other principal bundles P (GL(V, C), π, M ) and U (U (V ), π u , M ) over M with the structural groups GL(V, C) and U (N ), respectively. Moreover the coherent state method of quantization of the generalized Hamiltonian flows on P (G, π, M ) extends uniquely to each of these principal bundles giving the same quantum flows as in the case of P (G, π, M ). Indeed, since G is a Lie subgroup of GL(V, C) one can define the GL(V, C)-principal bundle P (GL(V, C), π, M ) over M in the following way: a) the total space P is the quotient P := (P × GL(V, C))/G defined by the action for g ∈ G ⊂ GL(V, C). Hence, according to the Proposition 6.1 in Chapter II of [21], there exists uniquely defined connection H * : T π( p) M → T p P which in the case considered here is related to the connection H * : T π(p) M → T p P on P by where p = [(p, g)] ∈ P . The connection form ϑ ∈ Γ ∞ ( P , T * P ⊗ B(V )) corresponding to (5.2) is the following ϑ([(p, g)]) := g −1 ϑ(p) g + g −1 d g.
Having the principal bundles morphism E : P → P we can extend a G-equivariant coherent state map K : P → B(V, H) as well as an automorphism τ ∈ Aut(P ) of P (G, π, M ) to the ones defined on P . These extensions are defined as follows are fulfilled, where D and Ω are the covariant derivative and the curvature form corresponding to ϑ.
The above allows us to formulate the following proposition. In order to define the principal bundle U (U (V ), π u , M ) let us note that having a G-equivariant coherent state map K : P → B(V, H) we can define the map Φ K : P × GL(V, C) → P × GL(V, C) where K(p, p) = K(p) * K(p). This map intertwines It follows from c(p, g) ∈ U (V ) that the submanifold P × U (V ) ⊂ P × GL(V, C) is invariant with respect to the action (5.9). Therefore, one can consider the quotient manifold U := (P × U (V ))/G as the total space of a U (V )-principal bundle U (U (V ), π u , M ) over M with bundle projection map π u : U → M and the right action κ u : U × U (V ) → U of U (V ) defined as follows Therefore, by fixing a coherent state map K : P → B(V, H) on the total space of principal bundle P (G, π, M ) we reduce a generalized Hamilton system on P described by (5.7) to the one defined on U by ϑ a and F a . Let us stress here that this reduction depends on the choice of K : P → B(V, H) in a canonical way. Now, we discuss the problem of quantization of the generalized Hamiltonian flows τ (X,F ) t , τ (X, F ) t and τ (X,F a ) t on P , P and U , respectively, which are reciprocally related. We collect the main facts under our consideration in the subsequent proposition. ).
(ii) The quantization property for τ (X,F ) t implies that This condition is equivalent to the following one Since the flow τ (X,F ) t × id : P × GL(V, C) → P × GL(V, C) also commutes with the action Φ u : P × GL(V, C) × G → P × GL(V, C) of G and Φ u g (P × U (V )) ⊂ P × U (V ) we find that U ⊂ P is invariant with respect to the flow τ (5.14) Restricting ϑ u = [Φ K −1 ] * ϑ = [Φ K −1 ] * ϑ K to U ⊂ P and using (5.10) one obtains (5.11).
(iii) The quantization of τ (X,F a ) t = τ (X, F ) t | U as well as the equality U t = U u t follows from (5.11) and (5.14).
According to (4.13) the essential domains of the generators F : D K → H and F a : D a → H of flows U t = e it F and U t = e it F a are the following where F is a finite subset of Z. Since and v i ∈ V are chosen in an arbitrary way we obtain that D K = D a = D K . So, we have also the equalities F = F a = F for the generators.
However, the formula (4.23) taken for F and F a is different from the one for F . Namely, we have where p = [(p, g)] ∈ P , and i F a a( p) = (H a * (X)a)( p) + a( p)F a ( p), From (5.15) and (5.16) we easily see that the mean value functions F and F a for these generators on the coherent states are equal to the generators F and F a of the Hamiltonian flows τ For example, the quantization based on U (U (V ), π u , M ) is directly related to interpretation of the positive kernel A : U × U → B(V ) as the matrix valued transition amplitude kernel. More precisely, let us take such v, w ∈ V that v = w = 1. Then the vectors a( p)v, a( q)w ∈ H have norm equal to 1 also, i.e. they describe pure states of the system and the transition amplitude between them is a( p)v|a( q)w = v|A( p, q)w . So, one can interpret A( p, q) as the transition amplitude matrix between the states p and q. For more exhaustive discussion of these physical aspects we address to [26,27].
Ending, let us mention that if the base M of the principal bundle P (G, π, M ) is a complex analytic manifold then it is resonable to use an approach based on the principal bundle P (GL(V, C), π, M ). As an example of such a situation see Section 6.

Quantization of holomorphic flows on non-compact Riemann surfaces
In this section we will apply the method of quantization presented in Section 4 to the case when P (GL(V, C), π, M ) is a holomorphic GL(V, C)-principal bundle over a non-compact Riemann surface M . There are two reasons which motivated us to consider this case. The first one is its relative simplicity what allows to solve the system of differential equations (4.26), (4.28) on the kernel Kᾱ β : ϑ). The above assumption, as was shown in Section 4, allows us to quantize the flow τ (X,F ) t ∈ Aut(P, ϑ), using the kernel Kᾱ β obtained in such a way. The second reason is that this type of kernel (equivalently coherent state map) occurs in various problems of quantum optics, e.g. see [17], [39]. We omit here the subcase when M is a compact Riemann surface, since then the Hilbert space H postulated in Definition 4.1. has finite dimension, which makes the theory less interesting from a mathematical point of view, but not necessarily from a physical one, e.g. see [31], [32], [33], [15].
Using the invariants of the flows τ (X,F ) t ∈ Aut(P, ϑ) and the appropriate gauge transformation, we will reduce the equations (4.26) and (4.28) to the linear ordinary differential equation (6.32), which is solvable for (X, F ) ∈ P ∞ G (P, ϑ). The solutions of (6.32) are presented through the formula (6.29) in Proposition 6.4 and Proposition 6.5. We will also obtain the integral decompositions (6.54) of the positive kernels Kβ β (v, z) invariant with respect to the flows τ Being in the framework of the above category we will quantize the holomorphic flows τ (X,F ) t ∈ Aut(P, ϑ) only. We will assume also that the coherent state map K : M × GL(V, C) → B(V, H) is a holomorphic map.
The existence of a holomorphic flow σ X t = π(τ (X,F ) t ) ∈ Aut(M ) on a Riemann surface M radically restricts the class of non-compact Riemann surfaces with this property. Namely, see eg. [11], one proves that any non-compact Riemann surface M which admits a non-discrete group of automorphisms is biholomorphic to the one listed below: (v) an annulus A r := {z ∈ C : r < |z| < 1}, where 0 < r < 1.
Since the groups Aut(M ) of automorphisms of M = C, C * , D, D * , A r ⊂ C ∼ = CP(1) can be considered as the subgroups of Aut(CP(1)) ∼ = SL(2, C)/Z 2 we find that:

For all these cases M is a circularly symmetric open subset in C. So, the inclusion map M ∋ m →
In order to quantize the flows τ (X,F ) t whose projections π(τ (X,F ) t ) = σ X t on M are listed in (6.3)-(6.7), we recall that for every non-compact Riemann surface M the holomorphic principal bundle P (GL(V, C), π, M ) is trivial. So, there exist a holomorphic section s α : M → P and the corresponding trivialization K α : M → B(V, H) of the coherent state map which are defined on the whole of M . The equations (4.29), (4.26) and (4.28) in this trivialization assume the following forms respectively. Note here that the positive kernel Kᾱ α (z, z) := K α (z) * K α (z) is anti-holomorphic in the first variable and holomorphic in the second one. The relation (3.23) between the classical data ϑ α , F α , X in this case is the following Substituting ϑ α (z, z) given by (6.9), into (6.11) we obtain the equation on the kernel Kᾱ α : M × M → B(V ) complementary to the equation (6.10).
For simplification of the equations (6.10) and (6.12) we make use of the gauge transformations listed in (3.31)-(3.36), which in this case are given by We see from (6.13) that if the equation has a holomorphic solution g αβ : M → GL(V, C) for the given φ α and w, then there exists a holomorphic section s β : M → P such that the equations (6.10) and (6.12) reduce to the following ones and equation (6.8) reduces to X(F β )(z, z) = 0. From the above we have: (a) for the cases (ii) and (iii) if the function 1 w φ α extends as a holomorphic function to C and to D, respectively, then (6.14) has a holomorphic solution; (b) for the remaining cases we note that the pull-back of the equation (6.14) on the universal covering M 0 of M 0 always has a holomorphic solution and if this solution is invariant with respect to the natural action of the group Deck( M 0 /M 0 ) on M 0 then it defines a holomorphic solution of (6.14) on M .
Hence we see that by the gauge transformation (6.13) the large class of functions φ α could be brought to φ β = 0. Further we will investigate this case only.
Next substituting Γ N defined by (6.26) into the left hand side of (6.27) and using eq. (6.23) again we find that ) corresponding to this subcase are not quantizable by the coherent state map method.
Hence taking the above statement into account we will assume subsequently that a = iω, where ω ∈ R.
In order to find a solution Kβ β (z, z) of the differential equations (6.15) and (6.16) we note that using (6.15) and (6.17) one can write Kβ β and F β as the power series of a real variable I ∈ ∆ ⊂ R: Kβ β (z, z) = Φ β (I(z, z)) = n∈J C n I(z, z) n , (6.29) where Ψ β is defined by F β through the equation (6.30), and the iR-valued function ν : ∆ → iR is defined by the invariant I(z, z). The correctness of this definition follows from the equation (6.31) and from Summarizing the above facts we conclude: In the next proposition we will present the invariants I and ν • I in correspondence with flows σ X t listed in Proposition 6.1.  Now, based on the formulas given in Proposition 6.3 we will find the dependence of Γ * m Γ n ∈ B(V ) on the C n ∈ B(V ). This task is equivalent to solving the difference equation (6.23) with the {C n } n∈J as initial data, see (6.41). We will investigate the subcases mentioned in (6.33), (6.34) and (6.35) separately.
Proposition 6.4 For the case b = 0, see (6.33), which concerns with an arbitrary M = C, D, C * , D * , A r we have Γ * m Γ n = C m δ mn (6.36) for m, n ∈ J ⊂ Z.
Now, let us shortly discuss the subcases presented in Proposition 6.4. From (6.36) it follows that the Hilbert space H can be decomposed into the orthogonal Γ n V ⊥Γ m V , for n = m, eigenspaces of F The eigenvalue of F corresponding to Γ n V is nω ∈ ωJ, so the subset ωJ ⊂ R is the spectrum of F . We note here that J ⊂ Z could be chosen as an arbitrary infinite subset of Z. Thus the spectral decomposition of the operator F is given by F = n∈J nωP n whereP n are the orthogonal projectors on the eigenspaces Γ n V ⊂ H. The kernels Kβ β (v, z) for all these subcases are given by the same formula where 0 < C n ∈ B(V ). For dim C V = 1 this type of kernels was investigated in [28], where their relationship with the theory of q-special functions was also shown. For the remaining subcases, i.e. if b = 0 we will obtain the dependence of Γ * m Γ n on the C n comparing the coefficients in front of monomialsv m z n occurring in the equality ∞ m,n=0 valid for arbitrary v, z ∈ C, D, which follows from (6.22) and (6.29).
The respective formulas for m > n one obtains by conjugation of the ones presented in (6.38) and (6.40) and transposition of the indices.
(iii) For both cases described above one has By straightforward verification.
Next proposition describes the action of generator F of the quantum flow U (X,F ) t = e itF on the coefficients Γ n ∈ B(V, H) of the Laurent expansion (6.19).

Proof
After applying e it F to both sides of equality (6.21) we obtain (6.43) Since one has 1 z n+1 e it F K β (z) ≤ 1 ρ n+1 sup z∈S 1 ρ K β (z) < ∞, so due to Lebesgue's dominated convergence theorem the derivative d dt | t=0 at t = 0 of the right-hand side of (6.43) commutes with the integral over S 1 ρ . Thus, we have Hence, using also Stone's Theorem, we find that the rank of Γ n belongs to D F and thus (6.42) is valid. To obtain the successive equalities in (6.44) we have used the norm convergence of the series (6.19). and The essential domains of the above operators are given by and by respectively, where F is a finite subset of Z.

Proof
The linear dependence between Γ 0 , i F Γ 0 , . . . , (i F ) n Γ 0 ∈ B(V, H) and Γ 0 , Γ 1 , . . . , Γ n ∈ B(V, H), where n ∈ N ∪ {0}, given by the equations where k = 0, 1, . . . , n, is invertible. Hence, and from the linear independence of Γ n ∈ B(V, H) we conclude that the vectors (i F ) n Γ 0 ∈ B(V, H), n ∈ N ∪ {0}, are linearly independent. From Proposition 6.2 it follows that the operators Γ n span a dense subset of B(V, H), so, (i F ) n Γ 0 span too. Now, let us describe the relationship between the coherent state representation (6.45) and the spectral representation of the generator F of the quantum flow U for ψ ∈ D F , see Chapter VI §66 in [2] for details. Above by L(H) and by B(R) we denoted the lattices of orthogonal projections of H and Borel subsets of R, respectively. From Proposition 6.6 follows that Γ n V ⊂ D F , so, using (6.46) and (6.48) we find that Next, substituting Γ n given by (6.49) into (6.41) we obtain i l−n (b) −n a n l µ l , (6.50) where µ l ∈ B(V ) defined by are the moments of the positive B(V )-valued measure Summing up we conclude from (6.29) and (6.50) that through the Hamburger moment problem defined by (6.51) one obtains the relationship between the positive kernel Kβ β (v, z) and the resolution of identity E : of the square integrable B(V )-valued functions γ, δ ∈ L 2 (R, d(Γ * 0 EΓ 0 )). As it follows from (6.28) and (6.46) Γ 0 ∈ B(V, H) is a generating element in B(V, H) for F , so we have the isomorphism of the defined above Hilbert B(V )-modules.
Using the isomorphism I and (6.46) we find that and, thus where the function K β (z; ·) ∈ L 2 (R, d(Γ * 0 EΓ 0 )) is defined by the power series K n (iλ)z n 1 1 V , (6.53) convergent in the norm · L 2 := ·, · L 2 , where by · we denoted the norm on B(V ). Later on we will see in Proposition 6.8 that it is also point-wise convergent. We note here that the equivariance condition K β (σ X t (z)) = e it F K β (z) written in terms of K β (z; λ) assumes the following form Taking into account (6.52) and that K β (v); K β (z) = K β (v; ·)); K β (z; ·)) L 2 we obtain the integral decomposition invariant with respect to the flow σ X t quantized by e it F . Combining (6.29) and (6.50) we obtain the expression on the kernel Kβ β (v, z; λ) which is different to (6.55). The equivalence of (6.55) and (6.56) follows from the equality valid for the polynomials K n (iλ), where β l mn are given: by (6.38) and L = n for M = C; by (6.40) and L = n − m for M = D.
For b = 0 and |b| 2 − ω 2 = 0 we have that the polynomials K n are expressed by (non-orthogonal) Laguerre polynomials L where I(v, z) is given in (6.35).
To prove (6.65), (6.66), (6.68) and (6.69) it is enough to observe that (6.53) and (6.56) is nothing else than the generating function for the family of polynomials {K n } ∞ n=0 , which for Meixner-Pollaczek and Laguerre polynomials may be found in [22] as well.
The integral decomposition (6.54) taken for the case dim C V = 1, b = 0 and a = 0 leads to Bochner's Theorem, see e.g. [36], which is one of most important instruments in the operator theory [2] as well as the probability theory. So, as a by-product of our method of quantization applied to the case M = C, D we obtain a family of Bochner type integral decompositions presented in Proposition 6.8 for the positive definite kernels invariant with respect to suitable holomorphic flows σ X t on the Riemann surfaces C and D as well as on the ones which are biholomorphic to them. We stress here that these decompositions are valid for the arbitrary dimension of the Hilbert space V . Now let us discuss in details the case when dim C V = 1. In this case one has the natural isomorphism B(V, H) ∼ = H. Therefore, after applying the Gram-Schmidt orthonormalization procedure to the elements F n Γ 0 ∈ H, where n ∈ N ∪ {0}, which according to Proposition 6.7 are linearly independent and span the vector subspace D Γ ⊂ D F ⊂ H dense in H, we obtain the orthonormal basis |n := P n ( F )Γ 0 ∈ D Γ (6.71) in H. The polynomials P n (λ) of degree n appearing in (6.71) are orthogonal with respect to the positive measure d(Γ * 0 EΓ 0 )(λ). They satisfy the three term recurrence λP n (λ) = b n−1 P n−1 (λ) + a n P n (λ) + b n P n+1 (λ) defined by infinite Jacobi matrix J. One can express the coefficients a n and b n of this matrix as well as the polynomials P n (λ) in terms of the moments µ n , see (6.51), of the measure d(Γ * 0 EΓ 0 )(λ). For the respective formulas see Chapter I of [1].
The self-adjoint operator F expressed in the basis (6.71) assumes the three-diagonal form F |n = b n−1 |n − 1 + a n |n + b n |n + 1 , Proposition 6.9 The domain D Γ * of the operator Γ * adjoint to Γ contains D Γ , which is also the range of Γ. Hence D Γ * is dense in H.

Proof
For ϕ ∈ H and ψ = n∈F c n |n ∈ D Γ one has Thus we see that if n∈J | ϕ|Γ n | 2 < ∞ then ϕ ∈ D(Γ * ), so, we need to prove that n∈J | Γ m |Γ n | 2 < ∞ (6.73) for any m ∈ J. Let us consider three subcases mentioned in Proposition 6.4 and Proposition 6.5 separately.
To prove (6.73) for the subcase (i) of Proposition 6.5 where M = C let us observe that the quantities β l m,n , l = n, . . . , n + m, given by (6.38) form, up to the factor (ib) n , a finite family of polynomials of the variable n of degree no greater than m with coefficients depends on b, ω, m. Thus from (6.37) we obtain that for fixed m ∈ N ∪ {0} one has n Γ m |Γ n ≤ n n+m l=n |β l m,n ||C l | − −−− → n→∞ 0 where the last limit follows from the fact that the right hand side of (6.29) is convergent for arbitrary I ∈ R (see (6.34)), i.e. n |C n | → 0, and n |β l m,n | → |b|. Finally, (6.73) holds due to the root test for the convergence of a series.
The proof of (6.73) for the subcase (ii) of Proposition 6.5 where M = D is similar to the previous case. Namely, from (6.40) follows that the quantities β l m,n , l = n − m, . . . , n + m form, up to the factor (ib) n , a finite family of polynomials of the variable n of degree no greater than 3m with coefficients depending on b, ω, m. Thus n Γ m |Γ n ≤ |b| n n+m l=n−m |β l m,n ||C l | − −−− → n→∞ 0 because the right hand side of (6.29) is convergent for arbitrary grates |I| ∈ R (see (6.35)), i.e. n |C n | → 0.
We see from the above proposition that the assumption of the Theorem VIII.1 in [36] are fulfilled and thus we have: This definition is correct since n √ n → 1 for n → ∞. Because z ∂ψ(z) ∂z = φ(z) − p 0 , then for φ 0 = p 0 the holomorphic function g αβ (z) = e 1 α ψ(z) is a solution of (6.75) holomorphic on M .
Corollary 6.4 If Re φ 0 = 0, then the flows mentioned in Proposition 6.11 are not quantizable.
In the next section we will shortly discuss a possible physical applications of the obtained results.

Remarks about physical applications
In the theory of quantum mechanical systems there are two naturally distinguished ways of representing quantum Hamiltonians. The first one is by Schrödinger differential operator having domain in the Hilbert space L 2 (R N , d N x) of square-integrable functions. The second one, called Fock representation, is given by using the creation and annihilation operators which are the weighted shift operators acting in an abstract Hilbert space. The Schrödinger approach is used if one defines a quantum system starting from its classical counterpart (Schrödinger quantization). The Fock approach is usually applied to systems which do not have the classical equivalents. This for example happens in quantum optics [12,13,40] and nuclear physics [20], where the annihilation operators describe the quantum amplitudes of distinguished modes of a quantum physical system.
In order to integrate a quantum system, i.e. to obtain its evolution in time, one needs to find the spectral resolution of the Hamiltonian. This is the main mathematical task of quantum mechanics leading to the spectral representation of a quantum Hamiltonian.
The coherent states representation of the physical system investigated in this paper was initiated by E. Schrödinger in 1926 in the paper [38] and next was investigated by V. Fock [9] and V. Bargmann [4], and is known in quantum mechanics as the Bargmann-Fock representation. Later it was revitalized in quantum optics by R.J. Glauber [13]. Let us also mention also the contribution of A. Perelomov, see [35], to this subject, i.e. the construction of coherent state maps through the irreducible representations of Lie groups.
In the papers [26,27] a method of quantization of an arbitrary Hamiltonian system based on the notion of coherent state map was proposed and its generalization to the case of an arbitrary G-principal bundle we investigated here. Therefore, the illustration of the above method by its application to concrete physical systems is desirable. The coherent state method of quantization of the harmonic oscillator [38] is the most known and one can find it also in the textbooks of quantum mechanics. The two cases related to atomic physics crucial from the physical point of view, i.e. Kepler and MIC-Kepler systems, were quantized by the coherent state map method in [16] and [30], respectively. One can find a large class of systems quantizable by the coherent state method in optics [15,17,39], where one usually considers a finite number of modes of an electromagnetic field self-interacting through a nonlinear medium [12,34,40]. In the papers [31,32,33] the classical and quantum reduction procedures were applied to the system of nonlinearly coupled harmonic oscillators (modes) which leads to quantization of the Hamiltonian systems on circularly symmetric surfaces called Kummer shapes [14,31]. This is a case to which one can apply the results obtained in Section 6. The detailed discussion of all mentioned cases would require considerable extension of the paper, so we plan to make it the subject of a subsequent publication.
Finally, let us mention our belief that the kernel decomposition (6.54) presented in Proposition 6.8, which generalizes the one considered in Bochner's Theorem to arbitrary noncompact Riemann surfaces, will find applications in probability theory problems.