Some aspects of positive kernel method of quantization
aa r X i v : . [ m a t h - ph ] J a n SOME ASPECTS OF POSITIVE KERNEL METHODOF QUANTIZATION
Anatol Odzijewicz , Maciej Horowski Faculty of MathematicsUniversity of BiałystokK. Ciołkowskiego 1M, 15-245 Białystok, Poland
Abstract
We discuss various aspects of positive kernel method of quantization of the one-parameter groups τ t ∈ Aut ( P, ϑ ) of automorphisms of a G -principal bundle P ( G, π, M ) with a fixed connection form ϑ on its total space P . We show that the generator ˆ F of the unitary flow U t = e it ˆ F being thequantization of τ t is realized by a generalized Kirillov-Kostant-Souriau operator whose domainconsists 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, C ) and M is a non-compactRiemann surface leads to quantization of the arbitrary holomorphic flow τ holt ∈ Aut ( P, ϑ ) . Forthe above case, we present the integral decompositions of the positive kernels on P × P invariantwith respect to the flows τ holt in terms of spectral measure of ˆ F . These decompositions generalizethe ones given by Bochner theorem for a positive kernels on C × C invariant with respect to theone-parameter groups of translations of complex plane. Contents C ∞ ( M, R ) -module corresponding to the group Aut ( P, ϑ )
33 Kirillov-Kostant-Souriau prequantization morphism 54 Positive definite kernels and quantization 105 Extension and reduction 166 Quantization of holomorphic flows on non-compact Riemann surfaces 19 E-mail: [email protected] E-mail: [email protected]
Introduction
From the very beginnings of quantum mechanics the problem of quantization is one of the mostfascinating and crucial ones for understanding of the correspondence between classical and quantumphysics. Excluding the field theory and restricting to the case of mechanics only by quantizationof a Hamiltonian flow R ∋ t σ Ft ∈ SpDiff ( M, ω ) defined on a symplectic manifold ( M, ω ) oneusually understands the construction of corresponding unitary flow R ∋ t e it b F on a Hilbert space H . Additionally one claims that the map Q : P ∞ ( M, R ) ∋ F i b F ∈ L ( D ) which prescribes to aclassical generator the quantum one b F (a self-adjoint operator in H ) is a morphism of some Poissonsubalgebra P ∞ ( M, R ) of the Poisson algebra C ∞ ( M, R ) of smooth functions on M on the Lie algebra L ( D ) generated by the anti-self-adjoint operator i b F having a common domain D dense in H .Among known methods of quantization the Kirillov-Kostant-Souriau geometric quantization [Ki],[Ko], [S] is one of most elegant from geometric point of view and gives precise construction of thequantum generator b F for the given classical one F ∈ P ∞ ( M, R ) . For this construction to workone needs to obtain a σ Ft -invariant complex Lagrangian distribution P ⊂ T C M and the appropriatemeasure (density) on the quotient manifold M/ P ∩ ¯ P . It leads to serious difficulties if one want toquantize concrete mechanical system. In order to omit these difficulties and for deeper understandingof the relationship between the classical ( M, ω ) and quantum ( CP ( H ) , ω F S ) phase spaces in [O1]and [O-H] a method of quantization based on the notion of positive kernel (coherent state map) wasproposed, which in our opinion completes Kirillov-Kostant-Souriau quantization in a natural way. Forexample one can find the application of the coherent state method of quantization to concrete physicalsystems in [H-O], [O-S].For general theory of positive (reproducing) kernels and its role in differential geometry (includingthe Banach differential manifolds and vector bundles over them) and representation theory we addressto [B-G1, B-G2] and to monography [N]. See also the classical paper [A] of N. Aronszajn.Basing partly on [O-H] in Section 2 and Section 3 we shortly discuss how to extend Kirillov-Kostant-Souriau prequantization procedure from the U (1) -principal bundle to the case of arbitrary G -principal bundle P ( G, π, M ) with a fixed connection form ϑ on the total space P . In Section 2 wedefine the Lie-Poisson C ∞ ( M, R ) -module P ∞ G ( P, ϑ ) of generators ( X, F ) ∈ P ∞ G ( P, ϑ ) of generalizedHamiltonian flows τ ( X,F ) t ∈ Aut ( P, ϑ ) , i.e. such ones which are solutions of generalized Hamiltonequations (2.10). In Section 3 we generalize Kirillov-Kostant-Souriau prequantization morphisms tothe morphism Q : P ∞ G ( P, ϑ ) → D Γ ∞ ( M, V ) of P ∞ G ( P, v ) in the C ∞ ( M, R ) -module of differentialoperators of the order less or equal one acting on the smooth sections Γ ∞ ( M, V ) of appropriatesmooth vector bundle V → M over M .In Section 4 we consider the G -equivariant coherent state map K : P → B ( V, H ) and the positivedefined G -eqivariant 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 sectionthe equivalence of the coherent state K and positive kernel K notions is shown and the method ofquantization based on them is investigated. Among others we show that Kirillov-Kostant-Souriaudifferential operator Q ( X,F ) could be treated as a self-adjoint operator ˆ F in the Hilbert space H K which domain is defined by the G -equivariant positive kernel K : P × P → B ( V ) (see (4.30)-(4.31)).The conditions on this kernel which allow the quantization of τ ( X,F ) t ∈ Aut ( P, ϑ ) are presented in(4.24)-(4.25).In Section 5 assuming that G ⊂ GL ( V, C ) is a Lie subgroup of GL ( V, C ) and that there exists acoherent state map K : P → H on the total space of P ( G, π, M ) , we define in a canonical way twoanother principal bundles e P ( GL ( V, C ) , e π, M ) and U ( U ( V ) , π u , M ) over M . Thus we also obtain theconnection forms e ϑ ∈ Γ ∞ ( e P , T ∗ e P ⊗ B ( V )) and ϑ a ∈ Γ ∞ ( U, T ∗ U ⊗ T e U ( V )) as well as the respectivecoherent states maps e K : e P → B ( V, H ) and a : U → B ( V, H ) . Next we show, see Proposition 5.2, thatthe flows τ ( X, ˜ F ) t ∈ Aut ( e P , e ϑ ) and τ ( X,F a ) t ∈ Aut ( U, ϑ a ) have the same quantum counterpart e it ˆ F asthe flow τ ( X,F ) t ∈ Aut ( P, ϑ ) .In Section 6 we quantize the holomorphic one-parameter groups of automorphisms of a holomorphicprincipal bundles P ( GL ( V, C ) , π, M ) over non-compact Riemann surface M . For this relatively simplebut not trivial case the investigated theory is presented in a complete way. In particular we obtain theBochner type integral decompositions of the τ ( X,F ) t -invariant positive kernels on P × P and show theirrelationship with the spectral decomposition of the corresponding quantum generator Q ( X,F ) = ˆ F .2ome physical applications including quantum optics are also mentioned. 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 symplectic form plays the curvature form Ω of afixed connection form ϑ ∈ Γ ∞ ( P, T ∗ P ⊗ T e G ) . We will define the Lie-Poisson C ∞ ( M, R ) -module ( P ∞ G ( P, ϑ ) , {· , ·} ϑ ) with the Lie bracket {· , ·} ϑ given in (2.11), which satisfies the Leibniz propertyin sense of (2.13). The corresponding generalization (2.10) of Hamilton equation is presented. Themodel proposed here in the case when G = U (1) and ϑ has non-singular curvature form reduces tostandard Hamiltonian mechanics on a symplectic manifold.Let Aut ( P, ϑ ) denote the group of diffeomorphisms of P which preserve principal bundle structureof P and the connection form ϑ . We recall that a T e G - valued differential one-form ϑ on P is theconnection form iff ϑ ( pg ) ◦ T κ g ( p ) = Ad g − ◦ ϑ ( p ) , (2.1) ϑ ( p ) ◦ T κ p ( e ) = id T e G , (2.2)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 . Theconnection form ϑ defines the spliting T p P = T vp P ⊕ T hp P, of the tangent space T p P at p ∈ P on the horizontal T hp P := ker ϑ ( p ) and vertical T vp P , i.e. tangentto the fibre π − ( π ( p )) of π : P → M , components (see e.g. [K-N]).Let τ : ( R , +) → ( Aut ( P, ϑ ) , ◦ ) be an one-parameter subgroup of Aut ( P, ϑ ) i.e. the map τ : R × P / / P is a smooth map which satisfies τ t ( pg ) = τ t ( p ) g and τ ∗ t ϑ = ϑ. 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 con-dition (2.3) together with the condition (2.4) we will denote 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 ∞ G ( T h P ) and Γ ∞ G ( T v P ) is defined by C ∞ ( M, R ) × Γ ∞ G ( T P ) ∋ ( f, ξ ) ( f ◦ π ) ξ ∈ Γ ∞ G ( T P ) , 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 horizontalvector 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)In the subsequent we will need the C ∞ ( M, R ) -module C ∞ G ( P, T e G ) which by definition consists ofsmooth functions F : P → T e G satisfying the G -equivariance property F ( pg ) = Ad g − F ( p ) . This module is isomorphic to the module of vertical vector fields Γ ∞ G ( T v P ) on P , where the C ∞ ( M, R ) -module isomorphism ν ∗ : Γ ∞ G ( T v P ) ∼ −→ C ∞ G ( P, T e G ) is defined as follows ν ∗ ( ξ v )( p ) := ϑ ( p )( ξ v ( p )) = ϑ ( p )( ξ ( p )) . (2.6)3ts inverse is given by ( ν ∗ ) − ( F )( p ) = T κ p ( e ) F ( p ) . The correctness of above definitions follows from (2.1) and (2.2), and from
T κ g ( p ) ◦ T κ p ( e ) = T κ pg ( e ) .Taking decomposition ξ = ξ h + ξ v of ξ ∈ Γ ∞ G ( T P ) and isomorphisms (2.5) and (2.6) we define ( X ( π ( p )) , F ( p )) = ( π ∗ × ν ∗ )( ξ )( p ) := ( T π ( p ) ξ h ( p ) , ϑ ( p )( ξ ( p ))) the C ∞ ( M, R ) -module isomorphism π ∗ × ν ∗ : Γ ∞ G ( T P ) ∼ −→ Γ ∞ ( T M ) × C ∞ G ( P, T e G ) . The inverse of π ∗ × ν ∗ is given by ( π ∗ × ν ∗ ) − ( X, F )( p ) = H ∗ ( X )( p ) + T κ p ( e ) F ( p ) , where by H ∗ : Γ ∞ ( T M ) → Γ ∞ G ( T h P ) we denote the horizontal lift, i.e. the module isomorphisminverse to π ∗ .For ξ ∈ Γ ∞ G ( T P ) one has L ξ ϑ = L ξ h ϑ + L ξ v ϑ = d ( ξ h x ϑ ) + ξ h x dϑ + d ( ξ v x ϑ ) + ξ v x dϑ == ξ h x Ω + d ( ξ v x ϑ ) + [ ξ v x ϑ, ϑ ] = H ∗ ( X ) x Ω + dF + [ F, ϑ ] = H ∗ ( X ) x Ω + D F, (2.7)where Ω := D ϑ = dϑ + 12 [ ϑ, ϑ ] is the curvature form of ϑ and D F = dF + [ ϑ, F ] 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 ∗ : Γ ∞ G ( T P ) ∼ −→ Γ ∞ ( T M ) × C ∞ G ( P, T e G ) , we cancarry the Lie bracket [ · , · ] from Γ ∞ G ( T P ) on Γ ∞ ( T M ) × C ∞ G ( P, T e G ) obtaining in this way the Liebracket { ( X , F ) , ( X , F ) } ϑ == ([ X , X ] , − H ∗ ( X )) x D F + H ∗ ( X )) x D F − H ∗ ( X ) , H ∗ ( X )) − [ F , F ]) , (2.8)of ( X i , F i ) = ( π ∗ × ν ∗ )( ξ i ) , i = 1 , , where [ F , F ]( p ) := [ F ( p ) , F ( p )] .It is reasonable to mention here that the Lie algebra (Γ ∞ G ( T P ) , [ · , · ]) is isomorphic with the Liealgebra (Γ ∞ ( T P/G ) , [ · , · ]) of the Atiyah Lie algebroid, being central ingredient of the Atiyah exactsequence of algebroids → P × Ad − T e G l ֒ → T P/G a → T M → , see e.g. [M].Though the language of Lie algebroid theory will not be used later, we note that the projectionon the first component pr : Γ ∞ ( T M ) × C ∞ G ( P, T e G ) → Γ ∞ ( T M ) corresponds to the anchor map a : T P/G → T M of the Atiyah algebroid. Hence, from the defining property of the anchor map wehave pr { ( X , F ) , ( X , F ) } ϑ = [ pr ( X , F ) , pr ( X , F )] and { ( X , F ) , f ( X , F ) } ϑ = f { ( X , F ) , ( X , F ) } ϑ + X ( f )( X , F ) . (2.9)These properties of pr and {· , ·} ϑ can be also obtained directly from their definitions.The above structure of Lie C ∞ ( M ) -module on Γ ∞ ( T M ) × C ∞ G ( P, T e G ) restricts to P ∞ G ( P, ϑ ) :=( π ∗ × ν ∗ )(Γ ∞ G,ϑ ( T P )) making it a Lie C ∞ ( M, R ) -submodule of Γ ∞ ( T M ) × C ∞ G ( P, T e G ) . It followsfrom (2.7) that ( X, F ) ∈ P ∞ G ( P, ϑ ) if and only if H ∗ ( X ) x Ω + D F = 0 . (2.10)Let us note here that the condition (2.10) is invariant with respect to the Lie C ∞ ( M, R ) -moduleoperation. Thus, for ( X , F ) , ( X , F ) ∈ P ∞ ( P, ϑ ) the Lie bracket (2.8) simplifies to the form { ( X , F ) , ( X , F ) } ϑ = ([ X , X ] , H ∗ ( X ) , H ∗ ( X )) − [ F , F ]) == ([ X , X ] , H ∗ ( X ) x D F − [ F , F ]) = ([ X , X ] , H ∗ ( X )( F ) − [ F , F ]) . (2.11)4n the case when the curvature form Ω is a non-singular 2-form, i.e. when ξ h x 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 defineduniquely 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 ∈ ker b if and only if D F = 0 , which does not meanin general that F = const. Substituting X = b ( F ) and X = b ( F ) into (2.11) we obtain the Liebracket { F , F } ϑ := { ( b ( F ) , F ) , ( b ( F ) , F ) } ϑ (2.12)of F , F ∈ C ∞ G ( P, T e G ) , which satisfies the Leibniz property { F , f F } ϑ = f { F , F } ϑ + b ( F )( f ) F (2.13)in sense of C ∞ ( M, R ) -module.Assuming G = U (1) we find that C ∞ G ( P, T e G ) is canonically isomorphic with C ∞ ( M, R ) and thecurvature form Ω is identified with the closed dω = 0 ω on M , which is symplectic form innon-singular case. Hence, formula (2.12) reduces to the symplectic Poisson bracket and (2.13) to itsLeibniz property.Taking the above facts into account, further we will call ( P ∞ G ( P, ϑ ) , {· , ·} ϑ ) the Lie-Poisson C ∞ ( M, R ) -module.In the framework of the assumed terminology it is natural to consider:(i) the equation (2.10) as a generalization on arbitrary Lie group G case of Hamilton equation whichone obtains if G = U (1) ;(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 Hamiltonianflow τ ( X,F ) t ∈ Aut ( P, ϑ ) based on the notion of G -equiuvariant positive kernel on P × P .Though our considerations in subsequent 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 -dimensionalHilbert space V . By h· , ·i : V × V → C we denote the scalar product for V and by G × V ∋ ( g, v ) gv ∈ V the canonical action of G in V . The group of unitary maps of V as usually will be denoted by U ( V ) ∼ = U ( N ) . In this section we generalize Kirillov-Kostant-Souriau prequantization procedure [Ki], [Ko], [S] forthe case of arbitrary Lie group G ⊂ GL ( V, C ) , i.e. we obtain the C ∞ ( M, R ) -module morphism Q : P ∞ G ( P, ϑ ) → D Γ ∞ ( M, V ) of P ∞ G ( P, ϑ ) into the module of differential operators of the order lessor equal one acting on the smooth sections of some complex vector bundle 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 − v ) ∈ P × V of the Lie group G on P × V . Onehas 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 − f ( p ) f or g ∈ G } of G -equivariantsmooth functions on P defined by the one-to-one dependence ψ ( π ( p )) = [( p, f ( p ))] := { ( pg, g − f ( p )) : g ∈ G } (3.1)between ψ ∈ Γ ∞ ( M, V ) and f ∈ C ∞ G ( P, V ) . Using (3.1) we define ( Q ( X,F ) ψ )( π ( p )) := [( p, ξ ( f )( p ))] , (3.2)for any ( X, F ) = ( π ∗ × ν ∗ )( ξ ) ∈ P ∞ G ( P, ϑ ) the differential operator Q ( X,F ) : Γ ∞ ( M, V ) → Γ ∞ ( M, V ) of the order less or equal one. Let us note here that if ξ ∈ Γ ∞ G,ϑ ( T P ) and f ∈ C ∞ G ( P, V ) then ξ ( f ) ∈ C ∞ G ( P, V ) . 5ince π ∗ × ν ∗ : Γ ∞ G,ϑ ( T P ) ∼ −→ P ∞ G ( P, ϑ ) is an isomorphism of the Lie C ∞ ( M, R ) -modules, from(3.1) and (3.2) one obtains [ Q ( X ,F ) , Q ( X ,F ) ] = Q { ( X ,F ) , ( X ,F ) } ϑ , (3.3) Q f ( X,F ) = f Q ( X,F ) , (3.4)where on the left hand side of the equality (3.3) we have the commutator of the differential operatorsand {· , ·} ϑ is the Lie bracket defined in (2.11). Since in the case if dim C V = 1 and G = U (1) the Lie C ∞ ( M, R ) -modules monomorphism Q : P ∞ ( P, ϑ ) → D Γ ∞ ( M, V ) (3.5)is Kirillov-Kostant-Souriau prequantization morphism we will further extend this terminology on gen-eral case. By D Γ ∞ ( M, V ) in (3.5) we denoted the Lie C ∞ ( M, R ) -module of differential operators ofthe order less or equal one acting on Γ ∞ ( M, V ) .In order to establish a more explicite expression for Q ( X,F ) , where ( X, F ) ∈ P ∞ ( P, ϑ ) , we will usethe 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 ht tangent to ξ and ξ h satisfy τ t ( p ) = τ ht ( p ) g ( t, p ) , (3.6)where g : R × P → G is the co-cycle related to the vertical (tangent to ξ v ) flow τ vt by τ vt ( p ) = pg ( t, p ) . Proposition 3.1
The co-cycle g : R × P → G corresponding to ( X, F ) ∈ P ∞ G ( P, ϑ ) by (3.6) has thefollowing form g ( t, p ) = e tF ( p ) . (3.7) Proof
Since for ξ = ( π ∗ × ν ∗ ) − (( X, F )) ∈ Γ ∞ G,ϑ ( T P ) one has ξ ( F ) = ξ ( h ϑ, ξ i ) = h L ξ ϑ, ξ i + h ϑ, [ ξ, ξ ] i = 0 we find that F ( τ t ( p )) = F ( p ) . (3.8)Combining (3.8) with (3.6) we obtain F ( τ ht ( p )) = g ( t, p ) F ( p ) g ( t, p ) − . Applying τ s to both sides of (3.6) we obtain τ s + t ( p ) = τ s ( τ ht ( p )) g ( t, p ) = τ hs ( τ ht ( p )) g ( s, τ ht ( p )) g ( t, p ) = τ hs + t ( p ) g ( s, τ ht ( p )) g ( t, p ) . From the above and from (3.6) we find that g ( s + t, p ) = g ( s, τ ht ( p )) g ( t, p ) . Differentiating the equality above with respect to the parameter s at s = 0 we obtain differentialequation ddt g ( t, p ) = F ( τ ht ( p )) g ( t, p ) = g ( t, p ) F ( p ) (3.9)on the co-cycle g : R × P → G ⊂ GL ( N, C ) . The equality (3.7) is obtained as solution of (3.9) withthe initial condition g (0 , p ) = 11 V . (cid:3) For f ∈ C ∞ G ( P, V ) one has ξ h ( f ) , ξ v ( f ) ∈ C ∞ G ( P, V ) and ξ h ( f )( p ) = h df, ξ h i ( p ) = h D f, ξ i ( p ) , (3.10) ξ v ( f )( p ) = h df, ξ v i ( p ) = ddt f ( pe tF ( p ) ) | t =0 = ddt e − tF ( p ) f ( p ) | t =0 = − F ( p ) f ( p ) , (3.11)where D : C ∞ G ( P, V ) → Γ ∞ G ( P, T ∗ P ⊗ V ) is the covariant derivative of f defined by D f := df ◦ pr hor = df + ϑf. pr hor : T P → T h P we denoted the projection of T P on its horizontal component T h P .Defining the covariant derivative ∇ : Γ ∞ ( M, V ) → Γ ∞ ( M, T ∗ M ⊗ V ) as usually by ∇ ψ ( π ( p )) := [( p, D f ( p ))] = [( p, df ( p ) + ϑ ( p ) f ( p ))] (3.12)and using equations (3.10) and (3.11) one obtains from (3.2) the following expression Q ( X,F ) = ∇ X − F, (3.13)for Kirillov-Kostant-Souriau operator Q ( X,F ) : Γ ∞ ( M, V ) → Γ ∞ ( M, V ) . Let us note here that 0-orderdifferential operator F acts on ψ ∈ Γ ∞ ( M, V ) as follows ( F ψ )( π ( p )) := [( p, F ( p ) f ( p ))] . We note also that F ( pg ) f ( pg ) = g − F ( p ) f ( p ) and ( D ξ f )( pg ) = g − ( D ξ f )( p ) .The Kirillov-Kostant-Souriau operator is the generator Q ( X,F ) ψ ( m ) := lim t → t [(Σ t ψ )( m ) − ψ ( m )] of an one-parameter group Σ t : Γ ∞ ( M, V ) → Γ ∞ ( M, V ) acting on the sections ψ ∈ Γ ∞ ( M, V ) by (Σ t ψ )( m ) := τ V t ψ ( σ − t ( m )) , where the flows τ V t : V → V and σ t : M → M are defined by τ V t [( p, v )] := [( τ t ( p ) , v )] and by σ t ( π ( p )) := π ( τ t ( p )) , respectively. The vector field X ∈ Γ ∞ ( T M ) in (3.13) is tangent to the flow { σ t } t ∈ R .If the curvature form Ω is non-singular the C ∞ ( M, R ) -module morphism ♭ : C ∞ G ( P, T e G ) → Γ ∞ ( T M ) leads to Kirillov-Kostant-Souriau prequantization morphism Q : C ∞ G ( P, T e G ) ∋ F Q F = ∇ X F − F ∈ D Γ ∞ ( M, V ) for the Lie-Poisson C ∞ ( M, R ) -module ( C ∞ G ( P, T e G ) , { ., . } ϑ ) , where X F = ♭ ( F ) and Lie-Poisson bracket { F , F } ϑ of F , F ∈ C ∞ G ( P, T e G ) is defined in (2.12).Furthermore we will need the vector bundle ¯ V := ( P × V ) /G → M associated to P ( G, π, M ) bythe action P × V × G ∋ ( p, v, g ) ( pg, g † v ) ∈ P × V as well as the C ∞ ( M, C ) -module C ∞ ¯ G ( P, V ) := { f ∈ C ∞ ( P, V ) : f ( pg ) = g † f ( p ) f or g ∈ G } (3.14)of the V -valued smooth function on P . Similarly as in (3.1) the equality ¯ ψ ( π ( p )) := [( p, f ( p ))] definesthe isomorphism Γ ∞ ( M, ¯ V ) ∼ = C ∞ ¯ G ( P, V ) of C ∞ ( M, C ) -modules. Using this isomorphism we define ( ¯ Q ( X,F ) ¯ ψ )( π ( p )) := [( p, ξ ( f )( p ))] another Kirillov-Kostant-Souriau differential operator acting now on Γ ∞ ( M, ¯ V ) . For vector field ξ ∈ Γ ∞ G ( T P ) tangent to τ t from (3.6) we find ξ ( f )( p ) = ddt f ( τ t ( p )) | t =0 = ddt f ( τ ht ( p )) | t =0 + ddt g ( t, p ) † | t =0 f ( p ) = ¯ D f ( p ) + F ( p ) † f ( p ) , where ¯ D f := df + ϑ † f . Next, using the covariant derivative ¯ ∇ : Γ ∞ ( M, ¯ V ) → Γ ∞ ( M, T ∗ M ⊗ ¯ V ) defined by ¯ ∇ ¯ ψ ( π ( p )) :=[( p, ¯ D f ( p ))] we obtain ¯ Q ( X,F ) = ¯ ∇ X + F † , (3.15)7he generator of the flow ( ¯Σ t ¯ ψ )( m ) := τ ¯ V t ¯ ψ ( σ − t ( m )) , (3.16)where F † ¯ ψ and τ ¯ V t : ¯ V → ¯ V are defined as follows ( F † ¯ ψ )( π ( p )) := [( p, F ( p ) † f ( p ))] and τ ¯ V t [( p, v )] := [( τ t ( p ) , v )] for [( p, v )] ∈ ¯ V .The covariance properties (3.3) and (3.4) for ¯ Q are proved in an analogous way as for Q .Now let us mention that using the scalar product h· , ·i in V one defines for ¯ ψ ∈ Γ ∞ ( M, ¯ V ) and ψ ∈ Γ ∞ ( M, V ) the smooth function hh ¯ ψ , ψ ii ( π ( p )) := h f ( p ) , f ( p ) i (3.17)on M . One sees from hh ¯ ψ , ψ ii ( σ t ( π ( p ))) = h f ( τ t ( p )) , f ( τ t ( p )) i that the following property X ( hh ¯ ψ , ψ ii ) = hh ¯ Q ( X,F ) ¯ ψ , ψ ii + hh ¯ ψ , Q ( X,F ) ψ ii for the pairing (3.17) is valid.Fixing a local trivialization s α : O α → P (where S α ∈ I O α = M is an open covering of M ) of theprincipal bundle P ( G, π, M ) one defines the local cocycles g α ( t, · ) : O α → G and h α ( t, · ) : O α → G by τ t ( s α ( m )) = s α ( σ t ( m )) g α ( t, m ) , (3.18) τ ht ( s α ( m )) = s α ( σ t ( m )) h α ( t, m ) (3.19)for sufficiently smal t . From (3.6) and (3.7) one has τ t ( s α ( m )) = τ ht ( s α ( m )) e tF ( s α ( m )) . (3.20)Hence, from (3.18)-(3.20) one obtains g α ( t, m ) = h α ( t, m ) exp( tF ( s α ( m ))) . (3.21)From (3.21) and the cocycle properties g α ( t + s, m ) = g α ( s, σ t ( m )) g α ( t, m ) ,h α ( t + s, m ) = h α ( s, σ t ( m )) h α ( t, m ) we have h α ( t, m ) e sF ( s α ( m )) = e sF ( s α ( σ t ( m ))) h α ( t, m ) , which is equivalent to h α ( t, m ) F ( s α ( m )) = F ( s α ( σ t ( m ))) h α ( t, m ) . In order to obtain ddt h α ( t, m ) | t =0 ∈ T e G we note that from (3.19) it follows that ξ h ( s α ( m )) = T s α ( m ) X ( m ) + T κ s α ( m ) ( e ) ddt h α ( t, m ) | t =0 . (3.22)Next, applying ϑ ( s α ( m )) to both sides of (3.22) and using (2.2) we obtain ddt h α ( t, m ) | t =0 = − ϑ ( s α ( m ))( T s α ( m ) X ( m )) = − ϑ α ( m )( X ( m )) , where ϑ α ( m ) := ϑ ( s α ( m )) ◦ T s α ( m ) = ( s ∗ α ϑ )( m ) .Introducing the notations φ α ( m ) := ddt 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 φ α ( m ) = −h ϑ α , X i ( m ) + F α ( m ) , (3.23)8hich is useful for finding of the local expression for the infinitesimal generator of the flow Σ t :Γ ∞ ( M, V ) → Γ ∞ ( M, V ) defined in (3.16). Namely, we have (Σ t ψ )( m ) = τ V t ψ ( σ − t ( m )) = τ V t [ s α ( σ − t ( m )) , ( f ◦ s α )( σ − t ( m ))] == [ τ t s α ( σ − t ( m )) , ( f ◦ s α )( σ − t ( m ))] = [ s α ( m ) g α ( t, σ − t ( m )) , ( f ◦ s α )( σ − t ( m ))] == [ s α ( m ) , g α ( t, σ − t ( m ))( f ◦ s α )( σ − t ( m ))] . (3.24)Thus for f α := f ◦ s α : O α → V one has (Σ t f α )( m ) = g α ( − t, m ) − f α ( σ − t ( m )) . (3.25)Differentiating both sides of (3.25) at t = 0 and using (3.23) we obtain the local representation ( Q ( X,F ) f α )( m ) = − X ( f α )( m ) + φ α ( m ) f α ( m ) == − ( X + h ϑ α , X i )( f α )( m ) + F α ( m ) f α ( m ) == − ( ∇ αX f α )( m ) + F α ( m ) f α ( m ) , (3.26)of Kirillov-Kostant-Souriau prequantization operator Q ( X,F ) : Γ ∞ ( M, V ) → Γ ∞ ( M, V ) , where ∇ αX := X + h ϑ α , X i is the local form of the covariant derivative ∇ defined in (3.12). Similarly we have ( ¯ Q ( X,F ) f α )( m ) = − ( ∇ αX f α )( m ) + f α ( m ) F α ( m ) † (3.27)for ¯ Q ( X,F ) : Γ ∞ ( M, ¯ V ) → Γ ∞ ( M, ¯ V ) .In the local gauge the Hamilton equation (2.10) assumes the form X x Ω α + D F α = 0 , (3.28)where Ω α := ( s α ) ∗ Ω = dϑ α + 12 [ ϑ α , ϑ α ] , (3.29) D F α := dF α + [ ϑ α , F α ] . (3.30)Ending this subsection let us mention well known equivariance formulae with respect to the gaugetransformation s β ( m ) = s α ( m ) g αβ ( m ) , (3.31)where g αβ : O α ∩ O β → G is the respective transition co-cycle, i.e. g αβ ( m ) g βγ ( m ) = g αγ ( m ) . Namely,one has ϑ β ( m ) = g − αβ ( m ) ϑ α ( m ) g αβ ( m ) + g − αβ ( m )( dg αβ )( m ) , (3.32) F β ( m ) = g − αβ ( m ) F α ( m ) g αβ ( m ) , (3.33) Ω β ( m ) = g − αβ ( m )Ω α ( m ) g αβ ( m ) , (3.34) f β ( m ) = g − αβ ( m ) f α ( m ) , (3.35) φ β ( m ) = g − αβ ( m ) φ α ( m ) g αβ ( m ) − g − αβ ( m )( Xg αβ )( m ) , (3.36)where m ∈ O α ∩ O β . 9 Positive definite kernels and quantization
In previous section we obtained the formula (3.13) and (3.15) on Kirillov-Kostant-Souriau prequan-tization operators Q ( X,F ) and ¯ Q ( X,F ) as well as their local versions (3.26) and (3.27). Here we willpresent a procedure which allows us to treat them as a self-adjoint operators in a Hilbert space.This quantization procedure is based on the notion of G -equivariant positive kernel, see [O-H]. Inorder to make the paper self-sufficient we present the above procedure in details. Some new resultscomplementary to the ones in [O-H] will be also presented in this section.Let us recall here that we have assumed that dim C V = N < + ∞ and G ⊂ GL ( V ) ∼ = GL ( N, C ) .Further, by B ( V ) ∼ = Mat N × N ( C ) we denote the C ∗ -algebra of linear maps of V and by B ( V, H ) theright Hilbert B ( V ) -module of linear maps Γ : V → H from the Hilbert space V into separable Hilbertspace H . Let us note here that from dim C V < + ∞ follows boundness of Γ : V → H . The B ( V ) -valuedscalar product h· ; ·i : B ( V, H ) × B ( V, H ) → B ( V ) on B ( V, H ) is defined by h Γ; ∆ i := Γ ∗ ∆ for Γ , ∆ ∈ B ( V, H ) , where the operator Γ ∗ : H → V is the one adjoint to Γ : V → H . Definition 4.1
A smooth map K : P → B ( V, H ) will be called G -equivariant coherent state map if ithas the following properties:(i) the G -equivariance property, i.e. K ( pg ) = K ( p ) g (4.1) for any p ∈ P and for any g ∈ G ;(ii) non-singularity, i.e. ker K ( p ) = { } , or equivalently K ( p ) ∗ K ( p ) ∈ GL ( V, C ) (4.2) for any p ∈ P ;(iii) the set K ( P ) V is linearly dense in H , i.e. \ p ∈ P ker K ( p ) ∗ = { } , or equivalently { K ( p ) v : p ∈ P and v ∈ V } ⊥ = { } . (4.3)By K ( p ) ∗ in (4.2) and (4.3) we denoted the map K ( p ) ∗ : H → V adjoint to K ( p ) : V → H , i.e. suchthat h ψ | K ( p ) v i = h K ( p ) ∗ ψ, v i for ψ ∈ H and v ∈ V , where h·|·i is the scalar product in Hilbert space H . For the existence of K : P → B ( V, H ) with the above properties see [N-R]. Definition 4.2
A smooth map K : P × P → B ( V ) will be called G -equivariant positive definite kernelif it has the following properties(i) the G -equivariance property, i.e. K ( p, qg ) = K ( p, q ) g (4.4) for any p, q ∈ P and g ∈ G ;(ii) non-singularity, i.e. K ( p, p ) ∈ GL ( V, C ) ⊂ B ( V ) (4.5) for any p ∈ P ;(iii) positivity, i.e. J X i,j =1 h v i , K ( p i , p j ) v j i ≧ (4.6) for arbitrary finite sequences p , . . . , p J ∈ P and v , . . . , v J ∈ V . K ( p, q ) † = K ( p, q ) and K ( ph, qg ) = h † K ( p, q ) g, where g, h ∈ G and p, q ∈ P .Both defined above structures are mutually depended. Extending rather well known scheme fromthe theory of reproducing (positive) kernels [N], [Ar] to this more complicated in geometric sense casewe shortly describe this dependence. The theory of reproducing kernels for like-Hermitian smoothvector bundles over smooth Banach manifolds and related to them linear connections one can find in[B-G1], [B-G2].Starting from the coherent state map K : P → B ( V, H ) we define the G -equivariant positive definitekernel by K ( p, q ) := K ( p ) ∗ K ( q ) . (4.7)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 D K := ( ¯ f = X i ∈ J K ( · , q i ) v i : q i ∈ P, v i ∈ V ) (4.8)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 todefine the scalar product h ¯ f | ¯ f i K of ¯ f , ¯ f ∈ D K we extend the summations in ¯ f = P i ∈ J K ( · , q i ) v i and ¯ f = P j ∈ J K ( · , q j ) v j to the set of indexes J = J ∪ J and define { p , . . . , p J } := { q , . . . , q J } ∪{ q , . . . , q J } . After that assuming v j = 0 for j ∈ J \ J and v j = 0 for j ∈ J \ J we define the scalarproduct h ¯ f | ¯ f i K := X i,j ∈ J h v i , K ( p i , p j ) v j i , (4.9)of ¯ f k = P j ∈ J K ( · , p j ) v kj , where k = 1 , .In particular case one has h ¯ f | K ( · , p ) v i K = X i ∈ J h v i , K ( p i , p ) v i = h ¯ f ( p ) , v i . (4.10)From (4.10) and Schwartz inequality for the scalar product (4.9) one obtains |h ¯ f ( p ) , v i| = |h ¯ f | K ( · , p ) v i K | ≤ h ¯ f | ¯ f i / K h K ( · , p ) v | K ( · , p ) v i / K ≤≤ k f k K h v, K ( p, p ) v i / ≤ k f k K k K ( p, p ) k / k v k . (4.11)From the inequality (4.11) one sees that k f k K = h f | f i / K = 0 implies f = 0 , so, k · k K is a norm on D K .We conclude from (4.11) that the evaluation functional E p ( ¯ f ) := ¯ f ( p ) satisfies k E p ( ¯ f ) k ≤ k K ( p, p ) k / k ¯ f k K , 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 pre-Hilbert space D K ⊂ C ∞ ¯ G ( P, V ) . It follows from (4.11)that for any equivalence class [ { ¯ f n } ] ∈ H K of Cauchy sequences { ¯ f n } ⊂ D K one defines ¯ f ( p ) := lim n →∞ ¯ f n ( p ) a function ¯ f : P → V depending on [ { ¯ f n } ] only. Hence we see that the Hilbert space H K is realizedin a natural way as a vector subspace of the vector space of V -valued functions on P which satisfythe G -equivalence condition from (3.14). 11ow, rewriting (4.10) as follows h ¯ f | K ( · , p ) v i K = h ¯ f | E ∗ p v i K , where ¯ f ∈ H K , v ∈ V and E ∗ p : V → H K is the conjugation of E p : H K → V , we define the coherentstate map K K : P → B ( V, H K ) by K K ( p ) := E ∗ p = K ( · , p ) . One easily sees that the properties (4.1)-(4.3) for K K : P → B ( V, H K ) follow from the ones for K givenin (4.4)-(4.6). The smoothness of K K : P → B ( V, H K ) follows from the smoothness of the positivekernel K (see Proposition 2.1 in [O-H]).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 , wherethe isomorphism I K : H ∼ −→ H K is defined by I K : H ∋ | ψ i → I K ( | ψ i ) := K ( · ) ∗ | ψ i ∈ H K . (4.12)In order to see that I K is an isomorphism indeed, we note that for | ψ i ∈ D K , where D K ⊂ H is definedby D K := | ψ i = X j ∈ J K ( p j ) v j : p j ∈ P, v j ∈ V , (4.13)one has I K | ψ i = P j ∈ J K ( · , p j ) v j and h I K | ψ i| I K | ψ ii K = h ψ | ψ i H , i.e. I K : D K → D K is a linearisometry of dense linear subspaces D K ⊂ H and D K ⊂ H K , so it extends to the isomorphism of theHilbert spaces.The isometry I K : H → H K defines the Banach spaces isomorphism L I K : B ( V, H ) → B ( V, H K ) by L I K A := I K ◦ A , where A ∈ B ( V, H ) .Therefore, if the coherent state map K and the positive definite kernel K are related by (4.7), then K K = I K ◦ K . If K K = I K ◦ K = I K ◦ K , then K = L I − K ◦ I K K , i.e. the positive definite kernel K defines thecoherent state map K up to an isomorphism of the Hilbert space H .Now let us consider the tautological bundle π N : E N → Grass ( N, H ) over the Grassmanian of N -dimensional subspaces of the Hilbert space H . By definition the total space of this bundle is E N := { ( γ, q ) ∈ H × Grass ( N, H ) : γ ∈ q } and π N := pr | E N , where pr is the projection of H ×
Grass ( N, H ) on its second component.The coherent state map K : P → B ( V, H ) defines the following morphisms ¯ V M V M E N Grass ( N, H ) ❄❄ ❄✲✲✲✲ [ K ] N [ K ] id [ K − ] π ¯ V π V π N (4.14)of vector bundles, where the morphism [ K − ] : ¯ V → V is defined by ¯ V ∋ [( p, v )] −→ [ K − ]([( p, v )]) := [( p, K ( p, p ) − v )] ∈ V . (4.15)The horizontal arrows on the right hand side of (4.14) are defined by M ∋ [ p ] = π − ( π ( p )) −→ [ K ]([ p ]) := K ( p ) V ∈ Grass ( N, H ) and by V ∋ [( p, v )] −→ [ K ] N ([( p, v )]) := ( K ( p ) v, [ K ]([ p ])) ∈ E N . (4.16)Correctness of the above definitions follows from the defining properties (4.1)-(4.3) of the coherent statemap K : P → B ( V, H ) . The bundle morphism defined in (4.15) is an identity covering isomorphism ofvector bundles which inverse is given by [ K ] : V → ¯ V .12estricting the scalar product h·|·i of H to the fibers π − N ( q ) of the tautological vector bundle π N : E N → Grass ( N, H ) and using the vector bundles morphism (4.16) one defines H [ p ] ([( p, v )] , [( p, w )]) := h K ( p ) v | K ( p ) w i = h K ( p, p ) v, w i (4.17)the Hermitian structure on the vector bundle π V : V → M . Proposition 4.1
The coherent state map K : P → B ( V, H ) defines by ϑ ( p ) K := ( K ( p ) ∗ K ( p )) − K ( p ) ∗ d K ( p ) (4.18) the metric (with respect to 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 ϑ K ( pg ) = ( K ( pg ) ∗ K ( pg )) − K ( pg ) ∗ d K ( pg ) == g − ( K ( p ) ∗ K ( p )) − ( g ∗ ) − g ∗ K ( p ) ∗ d K ( p ) g = g − ϑ K ( p ) g, for g ∈ G , so ϑ K satisfies (2.1).In order to show the condition (2.2) we take X := ddt g ( t ) | t =0 , where ] − ε, ε [ ∋ t g ( t ) ∈ G is asmooth 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 ϑ K ( p )( T κ p ( e ) X ) = ( K ( p ) ∗ K ( p )) − K ( p ) ∗ d K ( p )( T κ p ( e ) X ) == ( K ( p ) ∗ K ( p )) − K ( p ) ∗ d ( K ◦ κ p )( e ) X = ( K ( p ) ∗ K ( p )) − K ( p ) ∗ ddt K ( pg ( t )) | t =0 == ( K ( p ) ∗ K ( p )) − K ( p ) ∗ K ( p ) ddt g ( t ) | t =0 = X. (4.19)From (4.19) we see also that T p P = ker ϑ K ( p ) ⊕ T vp P and h ϑ K ( p ) , ξ ( p ) i ∈ T e G for ξ ( p ) ∈ T p P .It follows from dK ( p, p ) = K ( p, p ) ϑ K ( p ) + ϑ K ( p ) ∗ K ( p, p ) , (4.20)where K ( p, p ) = K ( p ) ∗ K ( p ) , that ϑ K is a metric connection with respect the Hermitian structure(4.17). In order to see this let us take f , f ∈ C ∞ G ( P, V ) . Then from (4.20) one obtains d h f ( p ) , K ( p, p ) f ( p ) i == h df ( p ) + ϑ ( p ) f ( p ) , K ( p, p ) f ( p ) i + h f ( p ) , K ( p, p )( df ( p ) + ϑ ( p ) f ( p )) i . (4.21)Since arbitrary sections ψ , ψ ∈ Γ ∞ ( M, V ) of the vector bundle π V : V → M one can write as ψ i ( π ( p )) = [( p, f i ( p ))] , i = 1 , , from (4.21) we obtain dH ( ψ , ψ ) = H ( ∇ ψ , ψ ) + H ( ψ , ∇ ψ ) , where ∇ : Γ ∞ ( M, V ) → Γ ∞ ( M, T ∗ M ⊗ V ) is the covariant derivative defined in (3.12). (cid:3) After these preliminary considerations we propose a method of quantization of the generalizedHamiltonian flow τ ( X,F ) t : P → P tangent to the vector field ξ = ( π ∗ × ν ∗ ) − ( X, F ) ∈ Γ ∞ G,ϑ ( T P ) ,where ( X, F ) ∈ P ∞ G ( P, ϑ ) . Definition 4.3
A strongly continuos unitary representation U ( X,F ) : ( R , +) → ( Aut H , ◦ ) of theadditive group ( R , +) is the positive kernel (coherent state) quantization of a generalized Hamiltonianflow τ ( X,F ) : ( R , +) → Aut ( P, ϑ ) if there exists a coherent state map K : P → B ( V, H ) such that:(i) the connection form ϑ is equal ϑ = ϑ K to ϑ K defined in (4.18);(ii) the equivariance property K ( τ ( X,F ) t ( p )) = U ( X,F ) t K ( p ) (4.22) for K : P → B ( V, H ) is fulfilled with respect to both considered flows. τ ( X,F ) t defines U ( X,F ) t in anunique way.Let us also note that if K : P → B ( V, H ) is an one-to-one smooth map and the flow τ t ∈ Aut ( P ) satisfies K ( τ t ( p )) = U t K ( p ) for certain unitary flow U t , then τ t ∈ Aut ( P, ϑ K ) and, thus there exists ( X, F ) ∈ Γ ∞ ( T M ) × C ∞ G,ϑ K ( P, T e G ) such that τ t = τ ( X,F ) t add U t = U ( X,F ) t . Additionally ( X, F ) satisfies the generalized Hamilton equation (2.10) for the curvature form Ω K defined by ϑ K . If Ω K isnon-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 τ Ft and U Ft instead τ ( X,F ) t and U ( X,F ) t .It follows from Stone theorem, see e.g. [R-S], that there exists a self-adjoint operator b F with domain D b F dense in H such that U ( X,F ) t = e it b F . From (4.22) we see that D K defined in (4.13) is a U ( X,F ) t -invariant dense linear subspace of H . From (4.22) it also follows that functions R ∋ t U t | ψ i ∈ H ,where | ψ i ∈ D K , are differentiable. So, see Theorem VII.11 in [R-S], D K is an essential domain of theinfinitesimal generator b F of U ( X,F ) t .Differentiating the both sides of the equation (4.22) with respect to t ∈ R we obtain i b F K ( p ) = ξ ( K )( p ) = (cid:0) ( π ∗ × ν ∗ ) − ( X, F ) K (cid:1) ( p ) = ( H ∗ ( X ) K )( p ) + K ( p ) F ( p ) . (4.23)Note here that for any p ∈ P one has Ran K ( p ) ⊂ D K ⊂ D b F . We also note that the last equality in(4.23) follows from (3.6)-(3.7) and (4.22).The symmetricity of b F on the domain D K is equivalent to the equation H ∗ ( X ) K )( p ) + K ( p ) F ( p )] ∗ K ( q ) + K ∗ ( p )[( H ∗ ( X ) K )( q ) + K ( q ) F ( q )] , 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 h·i : b F
7→ h b F i defined by h b F i := ( K ( p ) ∗ K ( p )) − K ∗ ( p ) b F K ( p ) = − iF ( p ) is inverse to the quantization Q : F b F of the classical generator F of the Hamiltonian flow τ Ft .Using the isomorphism I K : H → H K we can represent the quantum flow U ( X,F ) t and its generator b F in terms of the Hilbert space H K . Namely, we have I K ◦ U ( X,F ) t ◦ I − K = ¯Σ t and I K ◦ b F ◦ I − K = ¯ Q ( X,F ) , where ¯Σ t and ¯ Q ( X,F ) are defined in (3.16) and (3.15), respectively.Having in mind the Hilbert spaces isomorphism I K : H → H K , defined in (4.12), we see that D K defined in (4.8) is a common essential domain of the self-adjoint operators ¯ Q ( X,F ) being the generatorsof the flows U ( X,F ) t .According to the Definition 4.3 the problem of quantization of the classical flow R ∋ t τ ( X,F ) t ∈ Aut ( P, ϑ ) which solves the generalized Hamilton equation (2.10) reduces to the finding of coherentstate map K : P → B ( V, H ) satisfying the conditions ϑ = ϑ K and (4.22). Taking into account that K : P × P → B ( V ) defines K : P → B ( V, H ) up to an unitary map U : H → H it is reasonable toformulate these conditions in terms of the positive definite kernel: ϑ ( p ) = ϑ K ( p ) = K ( p, p ) − d q K ( p, q ) | q = p , (4.24) K ( p, q ) = K ( τ ( X,F ) t ( p ) , τ ( X,F ) t ( q )) . (4.25)Now, let us describe the quantization conditions (4.24) and (4.25) in the terms of local represen-tations K α := K ◦ s α : O α → B ( V, H ) and K ¯ αβ := K ∗ α ◦ K β : O α × O β → B ( V ) of K : P → B ( V, H ) and K : P × P → B ( V ) . 14 roposition 4.2 (i) The conditions (4.24) and (4.25) written in terms of a local representation s α : O α → P assume the following form ϑ α ( m ) = K ¯ αα ( m, m ) − d n K ¯ αα ( m, n ) | n = m , (4.26) K ¯ αβ ( m, n ) = g α ( t, m ) † K ¯ αβ ( σ ( X,F ) t ( m ) , σ ( X,F ) t ( n )) g β ( t, n ) (4.27) where ϑ α := s ∗ α ϑ and the local co-cycle g α ( t, m ) and the flow σ ( X,F ) t : M → M are related by(3.18), if we put τ t = τ ( X,F ) t and σ t = σ ( X,F ) t .(ii) The infinitesimal version of the condition (4.27) is the following X ( K ¯ αβ )( m, n ) + φ α ( m ) K ¯ αβ ( m, n ) + K ¯ αβ ( m, n ) φ β ( n ) = 0 , (4.28) where X ( K ¯ αβ )( m, n ) := ddt K ¯ αβ ( σ ( X,F ) t ( m ) , σ ( X,F ) t ( n )) | t =0 , i.e. X ( m, n ) = ( X ( m ) , X ( n )) , and φ α ( m ) , which is defined in (3.23), satisfies the equation L X ϑ α + dφ α + [ ϑ α , φ α ] = 0 (4.29) equivalent to the eq. (3.28).Proof i) The equality (4.27) follows from (3.18).ii) One obtains (4.28) by differentieting of (4.27) with respect to the parameter t ∈ R at t = 0 .The condition (4.29) we obtain from (3.28) using (3.23), (3.29) and (3.30). (cid:3) Let us also mention the transformations formulas K β ( m ) = K α ( m ) g αβ ( m ) , and K ¯ βγ ( m, n ) = g αβ ( m ) † K ¯ αδ ( m, n ) g δγ ( n ) , where g αβ : O α ∩ O β → G is the co-cycle defined in (3.31) and m ∈ O α ∩ O β , n ∈ O γ ∩ O δ , betweentwo local representations.The domain D b F of the generator b F contains D K ⊂ D b F the dense subset D K defined in (4.13) whichconsists of | ψ i = P j ∈ J K β j ( n j ) v j ∈ D b F , where n j ∈ O β j , v j ∈ V and J is finite. So, the local versionof the formula (4.23) is the following i b F K β ( n ) v = X ( K β )( n ) v + K β ( n ) φ β ( n ) v. (4.30)The domains D ( Q ( X,F ) ) and D ( ¯ Q ( X,F ) ) of Kirillov-Kostant-Souriau operators Q ( X,F ) : D ( Q ( X,F ) ) → Γ ∞ ( M, V ) and ¯ Q ( X,F ) : D ( ¯ Q ( X,F ) ) → Γ ∞ ( M, ¯ V ) defined in (3.13) and (3.15), respectively, in the localrepresentation consists the vectors ψ = P j ∈ J K ¯ α j β ( m j , · ) v j ∈ D ( Q ( X,F ) ) and ¯ ψ = P j ∈ J K ¯ βα j ( · , n j ) v j ∈D ( ¯ Q ( X,F ) ) .The local version of (3.13) and (3.15) are Q ( X,F ) K ¯ αβ ( m, · ) v = X ( K ¯ αβ )( m, · ) v + K ¯ αβ ( m, · ) φ β ( · ) v (4.31)and ¯ Q ( X,F ) K ¯ αβ ( · , n ) v = X ( K ¯ αβ )( · , n ) v + φ α ( · ) † K ¯ αβ ( · , n ) v. (4.32)Summing up the above facts, we see that the problem of quantization of classical flow τ ( X,F ) t described by the generalized Hamilton equation (2.10) (in local representation by the equations (3.28)or by (4.29)) is reduced to the obtaining of a solution K ¯ αβ : O α × O β → B ( V ) of the equations (4.26)and (4.28), where X , F α and ϑ α are fixed and satisfy (4.29). In general this is rather hard task.However, it is possible to do this for some particular cases. For this reason see Section 6.15 Extension and reduction
It turns out that if G ⊂ GL ( V, C ) , then having a principal bundle P ( G, π, M ) one can define ina canonical way two other principal bundles e P ( GL ( V, C ) , e π, M ) and U ( U ( V ) , π u , M ) over M withthe structural groups GL ( V, C ) and U ( N ) , respectively. Moreover the coherent state method ofquantization of the generalized Hamiltonian flows on P ( G, π, M ) extended uniquely to each of theseprincipal 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 e P ( GL ( V, C ) , e π, M ) over M in the following way:a) the total space e P is the quotient e P := ( P × GL ( V, C )) /G defined by the action e Φ g : P × GL ( V, C ) ∋ ( p, e g ) ( pg, g − e g ) ∈ P × GL ( V, C ) (5.1)of G on the product P × GL ( V, C ) ;b) the bundle projection e π : e P → M is defined by π ([( p, e g )]) := π ( p ) , where [( p, e g )] := { ( pg, g − e g ) : g ∈ G } ∈ e P ;c) the right action e κ : e P × GL ( V, C ) → e P of GL ( V, C ) on e P one defines by e κ e h ([( p, e g )]) = [( p, e g )] e h := [( p, e g e h )] . One has natural principal bundles morphism E : P → e P defined by E ( p ) := [( p, e )] , which covers the identity map id : M → M of M . Note here that E ( pg ) = [( pg, e )] = [( p, g )] = [( p, e ]) g = E ( p ) g for g ∈ G ⊂ GL ( V, C ) . Hence, according to the Proposition 6.1 in Chapter II of [K-N], there existsuniquely defined connection e H ∗ : T e π ( e p ) M → T e p e P which in the case considered here is related to theconnection H ∗ : T π ( p ) M → T p P on P by e H ∗ e p := T ( e κ e g ◦ E )( p ) ◦ H ∗ p , (5.2)where e p = [( p, e g )] ∈ e P .The connection form e ϑ ∈ Γ ∞ ( e P , T ∗ e P ⊗ B ( V )) corresponding to (5.2) is the following e ϑ ([( p, e g )]) := e g − ϑ ( p ) e g + e g − d e g. Having the principal bundles morphism E : P → e P we can extend a G -equivariant coherent statemap K : P → B ( V, H ) as well as an automorphism τ ∈ Aut ( P ) of P ( G, π, M ) to the ones defined on e P . These extensions are defined as follows e K ([( p, e g )]) := K ( p ) e g, e τ ([( p, e g )]) := [( τ ( p ) , e g )] . (5.3)The correctness of above definitions including their independence on the choice of representative ( pg, g − e g ) ∈ [( p, e g )] , where g ∈ G , one can check easily.If we extend the map F : P → T e G , defined in (2.6), to the map e F : e P → B ( V ) by e F ([( p, e g )]) := e g − F ( p ) e g, (5.4)then the following relations ( e D e F )( e p ) = e g − D F ( p ) e g, (5.5) e H ∗ e p ( X ) x e Ω e p = e g − ( H ∗ p ( X ) x Ω p ) e g (5.6)are fulfilled, where e D and e Ω are the covariant derivative and the curvature form corresponding to e ϑ .The above allows us to formulate the following proposition.16 roposition 5.1 A generalized Hamiltonian flow τ ( X,F ) t ∈ Aut ( P, ϑ ) extends by (5.3) to the flow e τ ( X,F ) t ∈ Aut ( e P , e ϑ ) and one has the equality e τ ( X,F ) t = τ ( X, e F ) t of the flows, where e F : e P → B ( V ) isdefined in (5.4), i.e. the extension e τ ( X,F ) t is a generalized Hamiltonian flow τ ( X, e F ) t on e P .Proof It follows from (5.5) and (5.6) that ( X, e F ) satisfies equation e H ∗ ( X ) x e Ω + e D e F = 0 (5.7)if and only if ( X, F ) satisfies equation (2.10). (cid:3) In order to define the principal bundle U ( U ( V ) , π u , M ) let us note that having done a G -equivariantcoherent state map K : P → B ( V, H ) we can define by Φ K ( p, e g ) := ( p, K ( p, p ) e g ) , where K ( p, p ) = K ( p ) ∗ K ( p ) , the map Φ K : P × GL ( V, C ) → P × GL ( V, C ) . This map intertwines Φ ug ◦ Φ K = Φ K ◦ e Φ g (5.8)the action (5.1) with the action Φ ug ( p, e g ) := ( pg, c ( p, g ) − e g ) , (5.9)where the U ( V ) -valued co-cycle c : P × G → U ( V ) is defined by c ( p, g ) := K ( p, p ) gK ( pg, pg ) − . The co-cycle property c ( p, g ) c ( pg, h ) = c ( p, gh ) and the unitary property c ( p, g ) † c ( p, g ) = 11 V for c : P × G → U ( V ) follow from (4.4).From (5.8) we see that Φ K defines an automorphism [Φ K ] : e P → e P of the principal bundle e P ( GL ( V, C ) , e π, M ) which covers the identity map of the base M . The inverse [Φ K ] − = [Φ K − ] ofthis automorphism transforms e ϑ and e F in the following way ϑ u ([( p, e g )]) := ([Φ K − ] ∗ e ϑ )([( p, e g )]) == e g − h K ( p, p ) ϑ ( p ) K ( p, p ) − + K ( p, p ) ( dK − )( p, p ) i e g + e g − d e g, (5.10)and F u ([( p, e g )]) := ( e F ◦ [Φ K − ])([( p, e g )]) := e g − K ( p, p ) F ( p ) K ( p, p ) − e g, respectively.It follows from c ( p, g ) ∈ U ( V ) that the submanifold P × U ( V ) ⊂ P × GL ( V, C ) is invariant withrespect 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 which bundle projection map π u : U → M and the right action κ u : U × U ( V ) → U of U ( V ) are defined as follows π u ([( p, c )]) := π ( p ) ,κ u ([( p, e c )] , c ) := [( p, e cc )] . Let us note here that [( p, e c )] ∈ ( P × U ( V )) /G is defined by [( p, e c )] := { ( pg, c ( p, g ) − e c ) : g ∈ G } .Taking the Lie algebra U ( V ) := { Y ∈ B ( V ) : h Y v, w i + h v, Y w i = 0 for v, w ∈ V } of U ( V ) and thereal vector space H ( V ) of the Hermitian h Hv, w i = h v, Hw i endomorphisms H ∈ B ( V ) one obtains theAd ( U ( V )) -invariant splitting B ( V ) = U ( V ) ⊕ H ( V ) of B ( V ) . So, according to Propsition 6.4 of ChapterII in [K-N], the anti-Hermitian part of ϑ u after restriction ϑ a := ( ϑ u − ( ϑ u ) † ) | U to U ⊂ e P defines aconnection form on the U ( V ) -principal bundle U ( U ( V ) , π u , M ) . Restricting F a := ( F u − ( F u ) † ) | U to U ⊂ e P one obtains U ( V ) -valued U ( V ) -equivariant function on the total space of U ( U ( V ) , π u , M ) .Therefore, 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 e P described by (5.7) to the one defined on U by ϑ a and F a .Now, we discuss the problem of quantization of the generalized Hamiltonian flows τ ( X,F ) t , τ ( X, e F ) t and τ ( X,F a ) t on P , e P and U , respectively, which are reciprocally related.The main facts being under our interest we collect in the subsequent proposition.17 roposition 5.2 If a flow τ ( X,F ) t ∈ Aut ( P, ϑ ) is quantized in sense of Definition 4.3, then:(i) The flow τ ( X, e F ) t is also quantized in sense of Definition 4.3, i.e. one has e ϑ = ϑ e K and e U ( X,F ) t e K ([( p, e g )]) = e K ( τ ( X, e F ) [( p, e g )]) . Additionally one has the equality e U ( X,F ) t = U ( X,F ) t of the quantum flows.(ii) The total space U of U ( U ( V ) , π u , M ) is invariant with respect to τ ( X, e F ) t ∈ Aut ( e P , e ϑ ) and τ ( X, e F ) t | U = τ ( X,F a ) t ∈ Aut ( U, ϑ a ) . The connection form ϑ a satisfies ϑ a = ϑ a a := a ∗ d a (5.11) the quantization condition (i) of Definition 4.3, for the coherent state map a : U → U ( V, H ) ⊂B ( V, H ) defined by a ([( p, e c )]) := K ( p ) K ( p, p ) − e c, (5.12) where U ( V, H ) is the set of partial isometries of Hilbert space V into the Hilbert space H .(iii) The generalized Hamiltonian flow τ ( X,F a ) t ∈ Aut ( U, ϑ a ) is quantized in sense of Definition 4.3and the quantum flow U ( X,F a ) t corresponding to it is equal U ( X,F a ) t = U ( X,F ) t to the quantumflow U ( X,F ) t .Proof (i) From ϑ = ϑ K we have e ϑ ([( p, e g )]) = e g − ( K ( p ) ∗ K ( p )) − K ( p ) ∗ d K ( p ) e g + e g − d e g == ( e K ([( p, e g )]) ∗ e K ([( p, e g )])) − e K ([( p, e g )]) ∗ d e K ([( p, e g )]) = ϑ e K ([( p, e g )]) . Next, from (4.22) and (5.3) taken for τ = τ ( X,F ) t we obtain U t e K ([( p, e g )]) = U t K ( p ) e g = K ( τ ( X,F ) t ( p )) e g = e K ( τ ( X, e F ) t ([( p, e g )])) . (ii) The quantization property for τ ( X,F ) t implies that K ( τ ( X,F ) t ( p ) , τ ( X,F ) t ( p )) = K ( p, p ) . (5.13)This condition is equivalent to the following one Φ K ◦ ( τ ( X,F ) t × id ) = ( τ ( X,F ) t × id ) ◦ Φ K . 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 ug ( P × U ( V )) ⊂ P × U ( V ) we find that U ⊂ e P isinvariant with respect to the flow τ ( X, e F ) t . From (5.13) one obtains that a : U → U ( V, H ) defined in(5.12) satisfies U t a ([( p, e c )]) = a ( τ ( X, e F ) ([( p, e c )])) . (5.14)Restricting ϑ u = [Φ K − ] ∗ e ϑ = [Φ K − ] ∗ ϑ e K to U ⊂ e P and using (5.10) one obtains (5.11).(iii) The quantization of τ ( X,F a ) t = τ ( X, e F ) t | U as well as the equality U t = U ut follows from (5.11)and (5.14). (cid:3) According to (4.13) the essential domains of generators be F : D e K → H and c F a : D a → H of flows U t = e it be F and U t = e it c F a are the following D e K := (X i ∈F e K ( e p i ) v i : e p i ∈ e P , v i ∈ V ) and D a := (X i ∈F a ( e p i ) v i : e p i ∈ U, v i ∈ V ) , F is a finite subset of Z .Since e K ( e p i ) v i = K ( p i ) e g i v i , a ( e p i ) v i = K ( p i ) K ( p i , p i ) − e c i v i and v i ∈ V are chosen in an arbitraryway we obtain that D e K = D a = D K . So, we have also the equalities be F = c F a = b F for the generators.However, the formula (4.23) taken for be F and c F a is different from the one for b F . Namely, we have i be F e K ( e p ) = ( H ∗ ( X ) e K )( e p ) + e K ( e p ) e F ( e p ) , (5.15)where e p = [( p, e g )] ∈ e P , and i c F a a ( e p ) = ( H a ∗ ( X ) a )( e p ) + a ( e p ) F a ( e p ) , (5.16)where e p = [( p, e c )] ∈ U .From (5.15) and (5.16) we easily see that the mean value functions h be F i and h c F a i for these generatorson the coherent states are equal h be F i ([( p, e g )]) = e g − h b F i ( p ) g = e F ([( p, e g )]) , h c F a i ([( p, e c )]) = ( b F ◦ [Φ K − ])([( p, e g )]) to the generators e F and F a of the Hamiltonian flows τ ( X, e F ) t and τ ( X,F a ) t , respectively.Taking the positive kernels e K ( e p, e q ) = e K ( e p ) ∗ e K ( e q ) = e g † K ( p, q ) e h, where e p = [( p, e g )] , e q = [( q, e h )] ∈ e P and A ( e p, e q ) = a ( e p ) ∗ a ( e q ) = e c † K ( p, p ) − K ( p, q ) K ( q, q ) − e b, where e p = [( p, e c )] , e q = [( q, e b )] ∈ U , we obtain the domains D e K ⊂ C ∞ GL ( V, C ) ( e P , V ) ∼ = Γ ∞ ( M, ¯ V ) and D A ⊂ C ∞ U ( V ) ( U, V ) ∼ = Γ ∞ ( M, ¯ V ) , see (4.8), of the corresponding Kirillov-Kostant-Souriau operators ¯ Q ( X, e F ) : D e K → Γ ∞ ( M, ¯ V ) and ¯ Q ( X,F a ) : D A → Γ ∞ ( M, ¯ V ) .Summing up the above considerations we conclude that there are three equivalent ways of quan-tizing of the flow σ ( X,F ) t by positive kernel method based on the principal bundles P ( G, π, M ) , e P ( GL ( V, C ) , e π, M ) and U ( U ( V ) , π u , M ) over M , respectively. The choice of one of these ways dependson the physical as well as mathematical aspects of the model under investigation.For example, the quantization based on U ( U ( V ) , π u , M ) is directly related to interpretation of thepositive kernel A : U × U → B ( V ) as the matrix valued transition amplitude kernel. More precisely,let us take such v, w ∈ V that k v k = k w k = 1 , then the vectors a ( e p ) v, a ( e q ) w ∈ H have norm equal1 also, i.e. they describe pure states of the system and the transition amplitude between of themis h a ( e p ) v | a ( e q ) w i = h v | A ( e p, e q ) w i . So, one can interpreters A ( e p, e q ) as the transition amplitude matrixbetween the states e p and e q . For more exhaustive discussion of these physical aspects we address to[O1].Ending, let us mention that if the base M of the principal bundle P ( G, π, M ) is a complex analyticmanifold then it is resonable to use approach based on the principal bundle e P ( GL ( V, C ) , e π, M ) . Asan example of such situation see Section 6. 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 relativesimplicity what allows to solve the system of differential equations (4.26), (4.28) on the kernel K ¯ αβ : O α × O β → B ( V ) under assumption that ( X, F α ) ∈ P ∞ G ( P, ϑ ) . The above, as it was shown in Section19, allows us to quantize the flow τ ( X,F ) t ∈ Aut ( P, ϑ ) , using the kernel K ¯ αβ obtained in such a way. Thesecond reason is that this type of kernels (equivalently coherent state maps) occur in various problemsof quantum optics, e.g. see [H-O-T],[T-O-H-J-Ch]. We omit here the subcase when M is a compactRiemann surface, since then the Hilbert space H postulated in Definition 4.1. has finite dimension,what makes the theory less interesting from mathematical point of view, but not necessarily fromphysical one, e.g. see [O-W1], [O-W2], [O-W3], [H-Ch-O-T].Using the invariants of the flows τ ( X,F ) t ∈ Aut ( P, ϑ ) and the appropriate gauge transformation, wewill reduce the equations (4.26) and (4.28) to the linear ordinary differential equation (6.32), whichis 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 thepositive kernels K ¯ ββ (¯ v, z ) invariant with respect to the flows τ ( X,F ) t on the positive kernels K ¯ ββ (¯ v, z ; λ ) presented in Proposition 6.8. The relationship between the B ( V ) -valued measures d (Γ ∗ E Γ )( λ ) usedfor these decompositions and spectral measure of the generator ˆ F of the quantum flow U ( X,F ) t = e it ˆ F is described also.One proves, eg. see Section III par. 30 in [F], that for non-compact Riemann surface M onehas H ( M, GL ( N, O )) = 0 , i.e. any GL ( N, O ) -valued holomorphic transition 1-cocycle ( g αβ ) ∈ Z ( {O} α ∈ I , GL ( N, O )) is solvable g αβ = δ α δ − β , where the holomorphic maps δ α : O α → GL ( N, C ) , α ∈ I , represent a holomorphic 0-cocycle. Therefore, all holomorphic vector bundles V → M and aswell as the holomorphic principal bundles P ( GL ( V, C ) , π, M ) over M are trivial.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 aholomorphic map.The existence of a holomorphic flow σ Xt = π ( τ ( X,F ) t ) ∈ Aut ( M ) on a Riemann surface M radicallyrestricts the class of non-compact Riemann surfaces with this property. Namely, see eg. [F-K], oneproves that any non-compact Riemann surface M which admits a non-discrete group of automorphismsis biholomorphic to the one listed bellow:(i) the Gauss plane C = C \ {∞} ,(ii) the punctured Gauss plane C ∗ := C \ {∞ , } ,(iii) the unit disc D := { z ∈ C : | z | < } ,(iv) the punctured unit disc D ∗ := D \ { } ,(v) an annulus A r := { z ∈ C : r < | z | < } , where < r < .Since the groups Aut ( M ) of automorphisms of M = C , C ∗ , D , D ∗ , A r ⊂ C ∼ = CP (1) can be consideredas the subgroups of Aut ( CP (1)) ∼ = SL (2 , C ) / Z we find that:(i) Aut ( C ) = { z αz + β : α ∈ C ∗ , β ∈ C } ∼ = C ∗ ⋉ C ,(ii) Aut ( C ∗ ) = (cid:8) z αz or z αz : α ∈ C ∗ (cid:9) ∼ = Z ⋉ C ∗ ,(iii) Aut ( D ) = n z αz + ββz + α : | α | − | β | = 1 , α, β ∈ C o ∼ = SU (1 , / Z ,(iv) Aut ( D ∗ ) = (cid:8) z αz : α ∈ S (cid:9) ∼ = S ,(v) Aut ( A r ) = (cid:8) z αz or z r αz : a ∈ S (cid:9) ∼ = Z ⋉ S .For all these cases M is a circularly symmetric open subset in C . So, the inclusion map M ∋ m ι ( m ) =: z ∈ C defines the global chart which is common for all Riemann surfaces considered here.Hence, the vector field X ∈ Γ ∞ ( T M ) tangent to a holomorphic flow σ Xt ∈ Aut ( M ) is given by X = w ( z ) ∂∂z + w ( z ) ∂∂ ¯ z , (6.1)where w ( z ) is a second order polynomial w ( z ) = cz + az + b (6.2)with coefficients satisfying the following conditions20i) c = b = 0 if M = C ∗ , D ∗ , A r ,(ii) c = 0 if M = C ,(iii) c = − b and a = 2 iω , ω ∈ R , if M = D .From above conditions we see that if b = 0 then one has also c = 0 in (6.2).Taking into account the above facts we present below the list of possible holomorphic flows σ Xt onM. Proposition 6.1
The following holomorphic flows are possible:(i) for M = C ∗ , D ∗ , A r one has σ Xt ( z ) = e at z, (6.3) where a ∈ C \ { } for C ∗ and a = iω , ω ∈ R \ { } , for D ∗ and A r ;(ii) for M = C one has σ Xt ( z ) = e at z + ba ( e at − , (6.4) where a ∈ C \ { } , note here that for a = 0 the formula (6.4) reduces to σ Xt ( z ) = z + bt ; (6.5) (iii) for M = D one has σ Xt ( z ) = z ( ̺ cosh ̺t + iω sinh ̺t ) + b sinh ̺tzb sinh ̺t + ̺ cosh ̺t − iω sinh ̺t , (6.6) if ̺ := p − ω + | b | = 0 and σ Xt ( z ) = z (1 + iωt ) + btzbt + 1 − iωt , (6.7) if ̺ = 0 . For the formula (6.6) it is reasonable to distinguish the following two subcases ̺ ∈ R \ { } ( − ω + | b | > and ̺ ∈ i R \ { } ( − ω + | b | < . In order to quantize the flows τ ( X,F ) t , whose projections π ( τ ( X,F ) t ) = σ Xt 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 holomorphic section s α : M → P and the correspondingtrivialization K α : M → B ( V, H ) of the coherent state map which are defined on whole M . Theequations (4.29), (4.26) and (4.28) in this trivialization assume the following forms X ( F α )(¯ z, z ) + [ F α (¯ z, z ) , φ α ( z )] = 0 , ∂φ α ∂ ¯ z (¯ z, z ) = 0 , (6.8) ϑ α ( z, z ) = K ¯ αα (¯ z, z ) − ∂K ¯ αα ∂z (¯ z, z ) dz, (6.9) X ( K ¯ αα )(¯ z, z ) + K ¯ αα (¯ z, z ) φ α ( z ) + φ α ( z ) † K ¯ αα (¯ z, z ) = 0 , (6.10)respectively. Note here that the positive kernel K ¯ αα (¯ z, z ) := K α ( z ) ∗ K α ( z ) is anti-holomorphic in thefirst 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 φ α ( z ) = − ( X x ϑ α )(¯ z, z ) + F α (¯ z, z ) . (6.11)Substituting ϑ α (¯ z, z ) , given by (6.9), into (6.11) we obtain the equation K ¯ αα (¯ z, z ) F α (¯ z, z ) = K ¯ αα (¯ z, z ) φ α ( z ) + w ( z ) ∂K ¯ αα ∂z (¯ z, z ) (6.12)on the kernel K ¯ αα : M × M → B ( V ) complementary to the equation (6.10). Remark 6.1
Summarizing, we see that the flow τ ( X,F ) t ∈ Aut ( P, ϑ ) can be quantizable iff the kernel K ¯ αα satisfies (6.9), (6.10) and (6.12) for given classical data ϑ α , F α , w . ϑ β (¯ z, z ) = g − αβ ( z ) ϑ α (¯ z, z ) g αβ ( z ) + g − αβ ( z )( dg αβ )( z ) ,F β (¯ z, z ) = g − αβ ( z ) F α (¯ z, z ) g αβ ( z ) ,φ β ( z ) = g − αβ ( z ) φ α ( z ) g αβ ( z ) − w ( z ) g − αβ ( z ) ∂g αβ ∂z ( z ) . (6.13)We see from (6.13) that if the equation w ( z ) ∂g αβ ∂z ( z ) = φ α ( z ) g αβ ( z ) (6.14)has a holomorphic solution g αβ : M → GL ( V, C ) for the given φ α and w , then there exists holomorphicsection s β : M → P such that the equations (6.10) and (6.12) reduce to the following ones X ( K ¯ ββ )(¯ z, z ) = 0 , (6.15) K ¯ ββ (¯ z, z ) F β (¯ z, z ) = w ( z ) ∂K ¯ ββ ∂z (¯ z, z ) (6.16)and equation (6.8) reduces to X ( F β )(¯ z, z ) = 0 . (6.17)Existence of a holomorphic solution g αβ ∈ O ( M, GL ( V, C )) of the equation (6.14) on M depends onthe w φ α which is a holomorphic function at least on the domain M := M \ { z , z } , where z , z ∈ M are the roots of the polynomial w (the cases z = z or { z , z } = ∅ are admissible also).For the cases mentioned in Proposition 6.1 one has(i) if M = C ∗ , D ∗ , A r then M = M ;(ii) if M = C then M = C \ {− ba } for a = 0 and M = C for a = 0 ;(iii) if M = D then M = D \ { z } for | b | − ω < and M = D for | b | − ω ≥ .From the above we have:(a) for the cases (ii) and (iii) if the function w φ α extends as a holomorphic function to C and to D ,respectively, then (6.14) has a holomorphic solution;(b) for the remainder cases we note that the pull-back of the equation (6.14) on the universal covering c M of M has always a holomorphic solution and if this solution is invariant with respect tothe natural action of the group Deck ( c M /M ) on c M 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 bebrought to φ β = 0 . Further we will investigate this case only.Let us note that for φ β = 0 the local form (3.28) of Hamilton equation (2.10) is L X ϑ β = X x dϑ β + dF β where the connection form ϑ β (¯ z, z ) = e ϑ β (¯ z, z ) dz is related by F β (¯ z, z ) = ( X x ϑ β )(¯ z, z ) = w ( z ) e ϑ β (¯ z, z ) (6.18)to the function F β : M → B ( V ) . See equalities (3.23) and (4.29) for this reason. So, in order toconsider the holomorphic flow τ ( X,F ) t as a Hamiltonian flow generated by ( X, F ) we define e ϑ β (¯ z, z ) by (6.18). Therefore the kernel K ¯ ββ quantizes the Hamiltonian flow π ( τ ( X,F ) t ) = σ Xt if and only if itsatisfies the equations (6.15) and (6.16), while F β satisfies equation (6.17).Due to the circular symmetry of M ⊂ C we expand K β : M → B ( V, H ) into Laurent series K β ( z ) = X n ∈ J Γ n z n , (6.19)22y definition convergent in the norm topology of the Banach space B ( V, H ) . Let us mention thatbecause of finite dimension of V the norm convergence of (6.19) is equivalent to its strong convergence.The set of indices J , which numerate = Γ n ∈ B ( V, H ) in (6.19), is an infinite subset J ⊂ Z of thering of integer numbers Z . For M = C , D , in particular, one has J = N ∪ { } . The condition (4.3) onthe coherent state map K β : M → B ( V, H ) implies { Γ n v : n ∈ J and v ∈ V } ⊥ = { } . (6.20)Taking into account the norm convergence of (6.19) we can express Γ n = 12 πi I S ρ K β ( z ) z n +1 dz (6.21)the coefficients Γ n by K β : M → B ( V, H ) , where S ρ := { z ∈ C : | z | = ρ } ⊂ M .From (6.19) we obtain that K ¯ ββ (¯ z, z ) = K β ( z ) ∗ K β ( z ) = X m,n ∈ J Γ ∗ m Γ n ¯ z m z n . (6.22)Using this expansion we find that the equation (6.15) is equivalent to the two-variable second orderdifference equation c ( m − ∗ m − Γ n + c ( n − ∗ m Γ n − + ( an + am )Γ ∗ m Γ n ++ b ( n + 1)Γ ∗ m Γ n +1 + b ( m + 1)Γ ∗ m +1 Γ n = 0 (6.23)on the coefficients Γ ∗ m Γ n ∈ B ( V ) , where one assumes that Γ − = 0 if J = N ∪ { } .In the next proposition we present some properties of the coefficients Γ n ∈ B ( V, H ) whenever theysatisfy the equation (6.23). Proposition 6.2 (i) If b = c = 0 and Re a = 0 , then Γ n = 0 for n ∈ J \ { } .(ii) If b = c = 0 and a = iω ∈ i R \ { } , then Γ ∗ m Γ n = Γ ∗ n Γ n δ mn , (6.24) for m, n ∈ J .(iii) If b = 0 then J = N ∪ { } and Γ n are linearly independent in B ( V, H ) .Proof :For b = c = 0 the equations (6.23) reduces to ( an + am )Γ ∗ m Γ n = 0 (6.25)for all m, n ∈ J . Thus, for Re a = 0 one obtains that Γ n = 0 if n = 0 . This proves (i). In the case Re a = 0 the equations (6.25) gives Γ ∗ m Γ n = 0 for m = n , so, (ii) holds.(iii). If b = 0 let us assume that operators Γ n , n ∈ N ∪ { } , are linearly dependent. Then thereexists N ∈ N such that Γ N = N − X n =0 s n Γ n (6.26)Let we put in (6.23) n = N and rewrite this equation as follows [ c ( m − m − + b ( m + 1)Γ m +1 ] ∗ Γ N = − Γ ∗ m [ c ( N − N − + ( aN + ¯ am )Γ N + b ( N + 1)Γ N +1 ] , (6.27)Next substituting Γ N defined by (6.26) into the left hand side of (6.27) and using eq. (6.23) again wefind that b ( N + 1)Γ ∗ m Γ N +1 = − Γ ∗ m [( aN + ¯ am )Γ N + c ( N − N − ] + Γ ∗ m " N − X n =0 s n [ c ( n − n − + ( an + ¯ am )Γ n + b ( n + 1)Γ n +1 ] for arbitrary m ∈ N ∪ { } . Thus, due to (6.20), we have b ( N + 1)Γ N +1 = − ( aN + ¯ am )Γ N − c ( N − N − ++ N − X n =0 s n [ c ( n − n − + ( an + ¯ am )Γ n + b ( n + 1)Γ n +1 ] , what means that Γ N +1 is a linear combination of { Γ , . . . , Γ N − } , too. Repeating the above procedurewe conclude that dim H ≤ N , which leads to the contradiction with the assumption that dim H = ∞ .Ending, let us note that for b = 0 one has Γ m = 0 for all m ∈ N ∪ { } . Since, assuming Γ m = 0 forcertain m we obtain from eq. (6.23) that c ( m − m − + b ( m + 1)Γ m + = 0 . What leads up to thelinear dependence of { Γ n } n ∈ N ∪{ } (cid:3) Corollary 6.1
The vector space D Γ := ( X n ∈F Γ n v n : v n ∈ V ) (6.28) is dense in H , where F is a finite subset of Z . Let us observe that for M = C the flows (6.3) and (6.4) are conjugated by the translation T ba ( z ) := z + ba . From this and from the point (i) of Proposition 6.2 we have: Corollary 6.2 If Re a = 0 , then K β ( z ) = γ = const . Therefore, the flows σ Xt = π ( τ ( X,F ) t ) corre-sponding to this subcase are not quantizable by the coherent state map method. Hence taking the above statement into account we will assume in subsequent that a = iω , where ω ∈ R .In order to find a salution K ¯ ββ (¯ z, z ) of the differential equations (6.15) and (6.16) we note thatsince of (6.15) and (6.17) one can write K ¯ ββ and F β as the power series of a real variable I ∈ ∆ ⊂ R : K ¯ ββ (¯ z, z ) = Φ β ( I ( z, z )) = X n ∈ J C n I ( z, z ) n , (6.29) F β (¯ z, z ) = Ψ β ( I ( z, z )) = X n ∈ J Q n I ( z, z ) n , (6.30)where C n = C † n , Q n ∈ B ( V ) . By definition we assume the norm convergence of these power seriesexpansions of the functions Φ β : ∆ → B ( V ) and Ψ β : ∆ → B ( V ) defined on the range I ( M ) =: ∆ ofan invariant I : M → R : X ( I )(¯ z, z ) = 0 , (6.31)of the flow σ Xt tangent to the vector field X defined in (6.1).Substituting K ¯ ββ and F β given by (6.29) and (6.30) into the equation (6.16) we reduce this equationto the ordinary linear equation on the function Φ β ν ( I ) ddI Φ β ( I ) = Φ β ( I )Ψ β ( I ) , (6.32)where the Ψ β is defined by F β through the equation (6.30), and the i R -valued function ν : ∆ → i R isdefined ν ( I ( z, z )) := w ( z ) ∂I∂z (¯ z, z ) by the invariant I ( z, z ) . The correctness of this definition follows from the equation (6.31) and from [ w ∂∂z , X ] = [ w ∂∂z , w ∂∂z + ¯ w ∂∂ ¯ z ] = 0 .Since Φ β = Φ † β and ν = − ¯ ν we have from (6.32) that Φ β Ψ β +Ψ † β Φ β = 0 what in the case M = C , D is equivalent to P kl =0 ( C l Q k − l + Q † k − l C l ) = 0 .Summarizing the above facts we conclude: 24 orollary 6.3 The flow τ ( X,F ) t is quantized by the positive kernel K ¯ ββ = Φ β ◦ I if and only if thefunction Φ β : ∆ → B ( V ) is a solution of the equation (6.32), where Ψ β and ν are related to F β and X through (6.30) and (6.31), respectively. In the next proposition we will present the invariants I and ν ◦ I in correspondence with flows σ Xt listed in Proposition 6.1. Proposition 6.3 (i) For M = C ∗ , D ∗ , A r one has w ( z ) = iωz . Thus I ( z, z ) = ¯ zz, ν ( I ) = iωI, (6.33) and I ( C ∗ ) = ∆ =]0 , ∞ [ , I ( D ∗ ) = ∆ =]0 , or I ( A r ) = ∆ =] r , , respectively.(ii) For M = C one has w ( z ) = iωz + b . Thus I ( z, z ) = ω ¯ zz + ibz − ibz, ν ( I ) = i ( ωI + | b | ) , (6.34) and I ( C ) = ∆ = R if ω = 0 , I ( C ) = ∆ = [ − | b | ω , ∞ [ if ω > and I ( C ) = ∆ =] − ∞ , − | b | ω ] if ω < .(iii) For M = D one has w ( z ) = − ¯ bz + 2 iωz + b . Thus I ( z, z ) = 2 ωzz + ibz − ibz − zz , ν ( I ) = i ( I + 2 ωI + | b | ) , (6.35) and I ( D ) = ∆ = R for b = 0 , I ( D ) = ∆ =] − ∞ , for b = 0 , ω < and ∆ =]0 , ∞ [ for b = 0 , ω > .Proof By straightforward verification. (cid:3)
Now, basing 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 arbitrary M = C , D , C ∗ , D ∗ , A r wehave Γ ∗ m Γ n = C m δ mn (6.36) for m, n ∈ J ⊂ Z .Proof See formula (6.24) of Proposition 6.2. (cid:3)
Now, let us shortly discuss the subcases presented in Proposition 6.4. Since of (6.36) the Hilbertspace H is decomposed H = M n ∈ J Γ n V on the orthogonal Γ n V ⊥ Γ m V , for n = m , eigensubspaces of b F The eigenvalue of b F correspondingto Γ n V is nω ∈ ωJ , so, the subset ωJ ⊂ R is the spectrum of b F . We note here that J ⊂ Z couldbe chosen as an arbitrary infinite subset of Z . Thus the spectral decompositions of the operator b F isgiven by b F = X n ∈ J nω ˆ P n where ˆ P n are the orthogonal projectors on the eigensubspaces Γ n V ⊂ H . The kernels K ¯ ββ (¯ v, z ) forall these subcases are given by the same formula K ¯ ββ (¯ v, z ) = X n ∈ J C n (¯ vz ) n , < C n ∈ B ( V ) . For dim C V = 1 this type kernels was investigated in [O3], where theirrelationship 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 monomials ¯ v m z n occurring in the equality ∞ X m,n =0 Γ ∗ m Γ n ¯ v m z n = ∞ X n =0 C n I ( v, z ) n , valid for arbitrary v, z ∈ C , D , which follows from (6.22) and (6.29). Proposition 6.5 (i) For the case M = C and b = 0 we have Γ ∗ m Γ n = n + m X l = n β lmn C l , (6.37) where β lmn = ( i ¯ b ) n − m (cid:18) ml − n (cid:19)(cid:18) lm (cid:19) ω m + n − l | b | l − n , (6.38) if m ≤ n .(ii) For the case M = D and b = − ¯ c = 0 we have Γ ∗ m Γ n = n + m X l = n − m β lmn C l , (6.39) where β lmn = i n − m m X j =0 (cid:18) l − jl − (cid:19)(cid:18) l l + 2 j − n − m (cid:19) ×× (cid:18) l + 2 j − n − mj + l − n (cid:19) (2 ω ) n + m − l − j b j + l − n ¯ b j + l − m (6.40) if m ≤ n .The respective formulas for m > n one obtains by conjugation the ones presented in (6.38) and(6.40) and transposition of the indices.(iii) For both cases described above one has Γ ∗ Γ n = ( i ¯ b ) n C n . (6.41) Proof
By straightforward verification. (cid:3)
Next proposition describes the action of generator b F of the quantum flow U ( X,F ) t = e it ˆ F on thecoefficients Γ n ∈ B ( V, H ) of the Laurent expansion (6.19). Proposition 6.6
The vector space D Γ defined in (6.28) is contained D Γ ⊂ D b F in the domain of b F and one has i b F Γ n = c ( n − n − + an Γ n + b ( n + 1)Γ n +1 , (6.42) for n ∈ J , where we assume Γ − = 0 if J = N ∪ { } .Proof After applying e it b F to both sides of equality (6.21) we obtain e it b F Γ n = e it b F πi I S ρ z n +1 K β ( z ) dz = 12 πi I S ρ z n +1 e it b F K β ( z ) dz. (6.43)26ince one has k z n +1 e it b F K β ( z ) k ≤ ρ n +1 sup z ∈ S ρ k K β ( z ) k < ∞ , so, due to Lebesgue’s dominatedconvergence theorem the derivative ddt | t =0 at t = 0 of the right-hand side of (6.43) commutes with theintegral over S ρ . Thus, we have i b F Γ n = ddt πi I S ρ z n +1 e it b F K β ( z ) dz ! | t =0 == 12 πi I S ρ z n +1 ddt (cid:0) K β ( σ Xt ( z )) (cid:1) | t =0 dz = 12 πi I S ρ z n +1 X ( K β )( z ) dz == 12 πi I S ρ z n +1 X l ∈ J ( clz l +1 + alz l + blz l − )Γ l dz == 12 πi I S ρ z n +1 X l ∈ J [ c ( l − l − + al Γ l + b ( l + 1)Γ l +1 ] z l dz == c ( n − n − + an Γ n + b ( n + 1)Γ n +1 . (6.44)Hence, using also Stone theorem, we find that the rank of Γ n belongs to D b F and thus (6.42) isvalid. To obtain the successive equalities in (6.44) we have used the norm convergence of the series(6.19). (cid:3) The expression (4.30) for the generator b F : D b F → H and the expressions (4.31) and (4.32) forthe Kirillov-Kostant-Souriau operators Q ( X,F ) : D ( Q ( X,F ) ) → Γ hol ( M, V ) and ¯ Q ( X,F ) : D ( ¯ Q ( X,F ) ) → Γ antihol ( M, ¯ V ) in the case under consideration, i.e. when φ β = 0 , assume the following forms i b F K β ( z ) v = w ( z ) (cid:18) ∂∂z K β (cid:19) ( z ) v, (6.45)and Q ( X,F ) K ¯ ββ (¯ z, · ) v = w ( z ) (cid:18) ∂∂z K ¯ ββ (cid:19) (¯ z, · ) v, ¯ Q ( X,F ) K ¯ ββ ( · , z ) v = w ( z ) (cid:18) ∂∂ ¯ z K ¯ ββ (cid:19) ( · , z ) v The essential domains of the above operators are given by D K = ψ = X j ∈F K β ( z j ) v j : z j ∈ M, v j ∈ V, , and by D ( Q ( X,F ) ) = ψ = X j ∈F K ¯ ββ (¯ z j , · ) v j : z j ∈ M, v j ∈ V, , D ( ¯ Q ( X,F ) ) = ¯ ψ = X j ∈F K ¯ ββ ( · , z j ) v j : z j ∈ M, v j ∈ V, , respectively, where F is a finite subset of Z . Proposition 6.7 If b = 0 , then Γ = K β (0) ∈ B ( V, H ) is a generating element for i b F in the Hilbert B ( V ) -module B ( V, H ) , i.e. the elements ( i b F ) n Γ , where n ∈ N ∪ { } , are linearly independent andthey span a linearly dense subspace of B ( V, H ) . Moreover, one has Γ n = K n ( i b F )Γ , (6.46) where the polynomials K n ( iλ ) = n X l =0 a nl ( iλ ) l re defined by the recurrence K n +1 ( iλ ) = 1( n + 1) b [ iλK n ( iλ ) − naK n ( iλ ) − ( n − cK n − ( iλ )] (6.47) with the initial conditions K − ( iλ ) ≡ and K ( iλ ) ≡ .Proof The linear dependence between Γ , i b F Γ , . . . , ( i b F ) n Γ ∈ B ( V, H ) and Γ , Γ , . . . , Γ n ∈ B ( V, H ) ,where n ∈ N ∪ { } , given by the equations Γ k = k X l =0 a kl ( i b F ) k Γ , where k = 0 , , . . . , n , is invertible. Hence, and from the linear independence of Γ n ∈ B ( V, H ) weconclude that the vectors ( i b F ) n Γ ∈ B ( V, H ) , n ∈ N ∪ { } , are linearly independent.From Proposition 6.2 it follows that the operators Γ n span dense subset of B ( V, H ) , so, ( i b F ) n Γ span too. (cid:3) Now, let us describe the relationship between the coherent state representation (6.45) and thespectral representation of the generator b F of the quantum flow U ( X,F ) t = e it b F . Therefore, let E : R →L ( H ) denotes the resolution of identity or equivalently the spectral measure E : B ( R ) → L ( H ) of theself-adjoint operator b F , i.e. ψ ∈ D b F if and only if Z R λ d h Eψ | ψ i ( λ ) < ∞ and one has b F ψ = Z R λd ( Eψ )( λ ) (6.48)for ψ ∈ D b F , see Chapter VI §66 in [A-G] for details. Above, by L ( H ) and by B ( R ) we denoted thelattices of orthogonal projections of H and Borel subsets of R , respectively.From Proposition 6.6 follows that Γ n V ⊂ D b F , so, using (6.46) and (6.48) we find that Γ n = K n ( i b F )Γ = Z R K n ( iλ ) d ( E Γ )( λ ) . (6.49)Next, substituting Γ n given by (6.49) into (6.41) we obtain C n = ( i ¯ b ) − n Γ ∗ Γ n = Z R ( i ¯ b ) − n K n ( iλ ) d (Γ ∗ E Γ )( λ ) = n X l =0 i l − n (¯ b ) − n a nl µ l , (6.50)where µ l ∈ B ( V ) defined by µ n := Γ ∗ b F n Γ = Z R λ n d (Γ ∗ E Γ )( λ ) (6.51)are the moments of the positive B ( V ) -valued measure d (Γ ∗ E Γ )( λ ) := d [( E Γ ) ∗ ( E Γ )]( λ ) . Summing up we conclude from (6.29) and (6.50) that through the Hamburger moment problemdefined by (6.51) one obtains the relationship between the positive kernel K ¯ ββ (¯ v, z ) and the resolutionof identity E : R → L ( H ) of b F .Let us define the Hilbert B ( V ) -module L ( R , d (Γ ∗ E Γ )) of B ( V ) -valued Borel square integrablefunctions γ : R → B ( V ) , i.e. such ones that h γ ; γ i L ≤ M V , where < M ∈ R , in sense of B ( V ) -valued scalar product h γ ; δ i L := Z R γ ( λ ) ∗ d (Γ ∗ E Γ )( λ ) δ ( λ )
28f the square integrable B ( V ) -valued functions γ, δ ∈ L ( R , d (Γ ∗ E Γ )) . As it follows from (6.28) and(6.46) Γ ∈ B ( V, H ) is a generating element in B ( V, H ) for b F , so we have the isomorphism I : L ( R , d (Γ ∗ E Γ )) ∋ γ ∼ −→ Z R d ( E Γ )( λ ) γ ( λ ) =: Γ ∈ B ( V, H ) of the defined above Hilbert B ( V ) -modules.Using the isomorphism I and (6.46) we find that Γ n = I ( K n ( i · )11 V ) and, thus K β ( z ) = ∞ X n =0 Γ n z n = I ( K β ( z ; · )) , (6.52)where the function K β ( z ; · ) ∈ L ( R , d (Γ ∗ E Γ )) is defined by the power series K β ( z ; λ ) := ∞ X n =0 K n ( iλ ) z n ! V , (6.53)convergent in the norm k · k L := kh· , ·i L k , where by k · k we denoted the norm on B ( V ) . Later on wewill see in Proposition 6.8 that it is also point-wise convergent.We note here that the equivariance condition K β ( σ Xt ( z )) = e it b F K β ( z ) written in terms of K β ( z ; λ ) assumes the following form K β ( σ Xt ( z ); λ ) = e itλ K β ( z ; λ ) . Taking into account (6.52) and that h K β ( v ); K β ( z ) i = h K β ( v ; · )); K β ( z ; · )) i L we obtain the integraldecomposition K ¯ ββ (¯ v, z ) = Z R K ¯ ββ (¯ v, z ; λ ) d (Γ ∗ E Γ )( λ ) , (6.54)where K ¯ ββ (¯ v, z ; λ ) = K β ( v ; λ ) † K β ( z ; λ ) = ∞ X m,n =0 K m ( iλ ) K n ( iλ )¯ v m z n ! V , (6.55)of the kernel K ¯ ββ (¯ v, z ) = K β ( v ) ∗ K β ( z ) invariant with respect to the flow σ Xt quantized by e it b F .Combining the (6.29) and (6.50) we obtain the expression K ¯ ββ (¯ v, z ; λ ) := ∞ X n =0 K n ( iλ ) 1( i ¯ b ) n I (¯ v, z ) n ! V (6.56)on the kernel K ¯ ββ (¯ v, z ; λ ) other than (6.55).The equivalence of (6.55) and (6.56) follows from the equality K m ( iλ ) K n ( iλ ) = m + n X l = L i ¯ b ) l β lmn K l ( iλ ) , valid for the polynomials K n ( iλ ) , where β lmn are given: by (6.38) and L = n for M = C ; by (6.40)and L = n − m for M = D .Comparing the right-hand sides of (6.53) and (6.56) we obtain K ¯ ββ (¯ v, z ; λ ) = K β (cid:18) i ¯ b I (¯ v, z ); λ (cid:19) , (6.57)where I (¯ z, z ) is the σ Xt -invariant presented in (6.34) and (6.35) of the Proposition 6.3. In the nextproposition we will present expressions on K n ( iλ ) , K β ( z ) and K ¯ ββ (¯ v, z ; λ ) for the cases when b = 0 .29 roposition 6.8 (i) If M = C and b = 0 then we have K n ( iλ ) = ( − iω ) n − n ! b n (cid:18) − ω λ (cid:19) n , (6.58) where ( x ) n = x ( x + 1) . . . ( x + n − is the Pochhammer symbol, K β ( z ; λ ) = iω F (cid:18) − ω λ ; − iωb z (cid:19) V , (6.59) K ¯ ββ (¯ v, z ; λ ) = iω F (cid:18) − ω λ ; − ω | b | ¯ vz − iωb z + iω ¯ b v ) (cid:19) V . (6.60) If a = iω = 0 the above formulas takes the form K n ( iλ ) = 1 n ! (cid:18) iλb (cid:19) n , (6.61) K β ( z ; λ ) = e i λb z V , (6.62) K ¯ ββ (¯ v, z ; λ ) = e iλ ( zb − ¯ v ¯ b )11 V . (6.63) (ii) If M = D and b = 0 then for | b | − ω = 0 we find that polynomials K n are given by (non-orthogonal) Maixner-Pollaczek polynomials P (0) n ( x ; ϕ ) as follows K n ( iλ ) = (cid:18) A ib sin ϕ (cid:19) n P (0) n ( − iλ/A ; ϕ ) == (cid:18) Ae iϕ ib sin ϕ (cid:19) n (2 µ ) n n ! F (cid:18) − n, µ − iλA ; 2 µ ; 1 − e − iϕ (cid:19) | µ =0 (6.64) where A := − ω | ω | p | b | − ω and cos ϕ := | ω || b | (if | b | − ω > , then ϕ ∈ [0 , π/ , if | b | − ω < ,then A and ϕ are imaginary, A, ϕ ∈ i R , and ϕ/i ∈ [1 , ∞ [ ) and K β ( z ; λ ) = (cid:18) ib sin ϕ − Ae − iϕ z ib sin ϕ − Ae iϕ z (cid:19) iλA V , (6.65) K ¯ ββ (¯ v, z ; λ ) = (cid:18) | b | sin ϕ + Ae − iϕ I (¯ v, z )2 | b | sin ϕ + Ae iϕ I (¯ v, z ) (cid:19) iλA V . (6.66) where I (¯ v, z ) is given in (6.35).For b = 0 and | b | − ω = 0 we have that polynomials K n are expressed by (non-orthogonal)Laguerre polynomials L ( − n ( x ) by K n ( iλ ) = (cid:16) ωib (cid:17) n L ( − n ( λ/ω ) == (cid:16) ωib (cid:17) n ( α + 1) n n ! F ( − n ; α + 1; λ/ω ) | α = − , (6.67) and K β ( z ; λ ) = exp (cid:18) λzωz − ib (cid:19) V , (6.68) K ¯ ββ (¯ v, z ; λ ) = exp (cid:18) λI (¯ v, z ) ωI (¯ v, z ) + | b | (cid:19) V , (6.69) where I (¯ v, z ) is given in (6.35). roof For M = C one can put in (6.47) c = 0 (see (6.2)). Thus it is easy to check that (6.61) and(6.58) are solutions of this recurrence equation for a = 0 and a = iω = 0 , respectively. The relations(6.62), (6.63), (6.59) and (6.60) follows immediately from (6.53), (6.56) and the definition of thehypergeometric functions.For M = D one has c = − ¯ b and a = 2 iω . Let us introduce the polynomials Q n defined by Q n ( λ ) := n ! b n i n A n K n ( − iAλ ) , (6.70)where A ∈ C \ { } . Then (6.47) takes the form λQ n ( λ ) = Q n +1 ( λ ) − n ωA Q n ( λ ) + n ( n − | b | A bQ n − ( λ ) . This is a three-term recurrence formula on monic polynomials Q n with the initial conditions Q − ( λ ) ≡ and Q ( λ ) ≡ . By the Favard’s theorem, see e.g. [Ch] Theorem 4.4, the solutions of (6.71) are not-orthogonal polynomials described in [K-S]. Namely, for | b | − ω = 0 and A = − ω | ω | p | b | − ω the poly-nomials Q n are the (non-orthogonal) monic Meixner-Pollaczek polynomials Q n ( λ ) = n !(2 sin ϕ ) n P (0) ( λ ; ϕ ) and for | b | − ω = 0 and A = − ω the polynomials Q n are the (non-orthogonal) monic Laguerre poly-nomials Q n ( λ ) = n !( − n L ( − ( λ ) . This proves (6.64) and (6.67).To prove (6.65), (6.66), (6.68) and (6.69) it is enough to observe that (6.53) and (6.56) is nothingthat the generating function for the family of polynomials { K n } ∞ n =0 , which for Meixner-Pollaczek andLaguerre polynomials may be find in [K-S], too. (cid:3) The integral decomposition (6.54) taken for the case dim C V = 1 , b = 0 and a = 0 leads to Bochnertheorem, see e.g. [R-S], which is the one of most important instrument in the operator theory [A-G]as well as the probability theory. So, as a by-product of our method of quantization applied to thecase M = C , D we obtain a family of Bochner type integral decompositions presented in Proposition6.8 for the positive definite kernels invariant with respect to suitable holomorphic flows σ Xt on theRiemann surfaces C and D as well as on the ones which are biholomorphic to them. We stress herethat 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 isomor-phism B ( V, H ) ∼ = H . Therefore, after applying Gram-Schmidt orthonormalization procedure to theelements b F n Γ ∈ H , where n ∈ N ∪ { } , which according to Proposition 6.7 are linearly independentand span the vector subspace D Γ ⊂ D b F ⊂ H dense in H , we obtain the orthonormal basis | n i := P n ( b F )Γ ∈ D Γ (6.71)in H .The polynomials P n ( λ ) of degree n appearing in (6.71) are orthogonal with respect the positivemeasure d (Γ ∗ E Γ )( λ ) . They satisfy the three therm recurrence λP n ( λ ) = b n − P n − ( 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 wellas the polynomials P n ( λ ) in terms of the moments µ n , see (6.51), of the measure d (Γ ∗ E Γ )( λ ) . Forthe respective formulas see Chapter I of [A].The self-adjoint operator b F expressed in the basis (6.71) assumes the three-diagonal form b F | n i = b n − | n − i + a n | n i + b n | n + 1 i , (6.72)as well as in the base Γ n , n ∈ N ∪ { } , see (6.42).We summarize the facts mentioned above defining Γ | n i := Γ n ,P ( b F n Γ ) := P n ( b F )Γ = | n i ,K ( b F n Γ ) := K n ( i b F )Γ = Γ n D Γ P ւ ց K D Γ Γ −→ D Γ which by definition intertwine the bases { Γ n } ∞ n =0 , { b F n Γ } ∞ n =0 , and {| n i} ∞ n =0 of the Hilbert space H . Proposition 6.9
The domain D Γ ∗ of the operator Γ ∗ adjoint to Γ contains D Γ , which is also therange of Γ . Hence D Γ ∗ is dense in H .Proof For ϕ ∈ H and ψ = P n ∈F c n | n i ∈ D Γ one has |h ϕ | Γ ψ i| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ∈F c n h ϕ | Γ n i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ h ψ | ψ i X n ∈F |h ϕ | Γ n i| ≤ h ψ | ψ i X n ∈ J |h ϕ | Γ n i| Thus we see that if P n ∈ J |h ϕ | Γ n i| < ∞ then ϕ ∈ D (Γ ∗ ) , so, we need to prove that X n ∈ J |h Γ m | Γ n i| < ∞ (6.73)for any m ∈ J . Let us consider three subcases mentioned in Proposition 6.4 and Proposition 6.5separately.For the subcase of Proposition 6.4 it follows from (6.36) that P n ∈ J |h Γ m | Γ n i| = |h Γ m | Γ m i| < ∞ .To prove (6.73) for the subcase (i) of Proposition 6.5 where M = C let us observe that the quantities β lm,n , l = n, . . . , n + m , given by (6.38) form, up to the factor ( i ¯ b ) n , a finite family of polynomials ofthe variable n of degree no greater than m with coefficients depends on b, ω, m . Thus from (6.37) weobtain that for fixed m ∈ N ∪ { } one has n p h Γ m | Γ n i ≤ n vuut n + m X l = n | β lm,n || C l | −−−−→ n →∞ where the last limit follows from the facts that the right hand side of (6.29) is convergent for arbitrarygrates | I | ∈ R (see (6.34)), i.e. n p | C n | → , and n q | β lm,n | → | b | . Finally, (6.73) holds due to the roottest for the convergence of a series.The prove of (6.73) for the subcase (ii) of Proposition 6.5 where M = D is similar to the previouscase. Namely, from (6.40) follows that the quantities β lm,n , l = n − m, . . . , n + m form, up to the factor ( i ¯ b ) n , a finite family of polynomials of the variable n of degree no greater than m with coefficientsdepends on b, ω, m . Thus n p h Γ m | Γ n i ≤ | b | n vuut n + m X l = n − m | β lm,n || C l | −−−−→ n →∞ because the right hand side of (6.29) is convergent for arbitrary grates | I | ∈ R (see (6.35)), i.e. n p | C n | → . (cid:3) We see from the above proposition that the assumption of the Theorem VIII.1 in [R-S] are fulfilledand thus we have:
Proposition 6.10 (i) The adjoint operator Γ ∗ is closed.(ii) The operator Γ is closable and one has ¯Γ = Γ ∗∗ , (¯Γ) ∗ = Γ ∗ .(iii) The operator ¯Γ ∗ ¯Γ = Γ ∗ Γ ∗∗ defined on the dense domain D ¯Γ ∗ ¯Γ = { ψ ∈ D ¯Γ : ¯Γ ψ ∈ D ¯Γ ∗ } isself-adjoint (see Exercise 45 in Chapter VIII of [R-S]). K β : D → H whichquantize a holomorphic flow σ Xt : D → D on the disc. Namely, let us define K : D → H by K ( z ) := ∞ X n =0 z n | n i . (6.74)From (6.74) and from the clousability of Γ : D Γ → D Γ we find that K β ( z ) = ¯Γ K ( z ) . The above allows us to represent K ¯ ββ (¯ v, z ) = ∞ X m,n =0 h m | ¯Γ ∗ ¯Γ | n i ¯ v m z n the positive kernel K ¯ ββ in terms of the matrix elements h m | ¯Γ ∗ ¯Γ | n i of the positive self-adjoint operator Γ ∗ Γ ∗∗ = ¯Γ ∗ ¯Γ . Proposition 6.11 If dim V = 1 then for the flow σ t ( z ) = e at z , z ∈ M = C , D , C ∗ , D ∗ , A r there existsa holomorphic section s β : M → P ( GL (1 , C ) , π, M ) for which φ β ( z ) =: φ = const .Proof :We need to show that when dim V = 1 the equation (6.13) has solution g αβ : M → C \ { } for φ β ( z ) = φ . Let us rewrite this equation in following form ∂g αβ ∂z ( z ) = − φ α ( z ) − φ az g αβ ( z ) . (6.75)Because each M is circularly symmetric domain in C , then holomorphic function φ α : M → C isglobally defined by its Laurent expansion φ α ( z ) = X n ∈ J p n z n . Let us define an holomorphic function on Mψ ( z ) := X n ∈ J \{ } n p n z n . This definition is correct since n √ n → for n → ∞ .Because z ∂ψ ( z ) ∂z = φ ( z ) − p , then for φ = p the holomorphic function g αβ ( z ) = e α ψ ( z ) is aholomorphic on M solution of (6.75). (cid:3) Corollary 6.4 If Re φ = 0 , then the flows mentioned in Proposition 6.11 are not quantizable.Proof :If φ β ( z ) = φ β (0) = φ = 0 , then for b = 0 the equations (6.10) and (6.19) give [ δ ( m + n ) + 2 µ + iω ( m − n )] h Γ m | Γ n i = 0 , where δ := Re a , ω := Im a and µ := Re φ . hence, we find that Γ n = 0 iff n = − µδ ∈ Z . So, K β ( z ) = Γ n z n and from defining property of K β : M → H follows that dim C H = 1 . The above iscontradicted with the postulates of Definiction 4.1. (cid:3) At the end we shortly discuss a possible physical applications of the results obtained in this section.For this reason we use the formula (6.72) which leads to representation of b F by an infinite Jacobimatrix, see [A]. This allows us to express b F in terms of the annihilation and creation operators, see[H-O-T] and [O-W1], and, thus interpret it as a Hamilton (Schrödinger) operator of some multimodequantum system. In particular such type quantum Hamiltonians are responsible for the interaction of33hotons with non-linear medium. Hence, they describe many quantum optical non-linear phenomena,see [F-P], [P-L], [W-M]. These Hamiltonians also appear in nuclear physics where they describemotion of fermion pairs if one treats these pairs as bosons, see the review paper [K-M].Let us mention that the Gaussian coherent states were introduced into quantum optics at itsbeginning by R. J. Glauber [G]. In quantum mechanics they where firstly considered by E. Schrödinger[Sch]. Therefore, one can say that the coherent state method of quantization has a long tradition inquantum physics. References [A] N.I. Ahiezer: "The Classical Moment Problem", Hafner Publ. Co., N.Y., 1965[A-G] N.I. Ahiezer, I.M. Glazman: "Theory of Linear Operators in Hilbert Space" Nauka, Moscow,1966 (in Russian)[Ar] N. Aronszajn, "Theory of reproducing kernels", Trans. Am. Math. Soc. (1950) 337-401.[B-G1] D. Beltiţˇa, J. 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