Abstract
We discuss various aspects of the positive kernel method of quantization of the one-parameter groups \(\tau _t \in \text{ Aut }(P,\vartheta )\) of automorphisms of a G-principal bundle \(P(G,\pi ,M)\) with a fixed connection form \(\vartheta \) on its total space P. We show that the generator \({\hat{F}}\) of the unitary flow \(U_t = e^{it {\hat{F}}}\) being the quantization of \(\tau _t \) is realized by a generalized Kirillov–Kostant–Souriau operator whose domain consists of sections of some vector bundle over M, which are defined by a suitable positive kernel. This method of quantization applied to the case when \(G=\hbox {GL}(N,{\mathbb {C}})\) and M is a non-compact Riemann surface leads to quantization of the arbitrary holomorphic flow \(\tau _t^{\mathrm{hol}} \in \text{ Aut }(P,\vartheta )\). For the above case, we present the integral decompositions of the positive kernels on \(P\times P\) invariant with respect to the flows \(\tau _t^{\mathrm{hol}}\) in terms of the spectral measure of \({\hat{F}}\). These decompositions generalize the ones given by Bochner’s Theorem for the positive kernels on \({\mathbb {C}} \times {\mathbb {C}}\) invariant with respect to the one-parameter groups of translations of complex plane.
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1 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 \({\mathbb {R}}\ni t\mapsto \sigma _t^F\in \text{ SpDiff }(M,\omega )\) defined on a symplectic manifold \((M,\omega )\) one usually understands the construction of a corresponding unitary flow \({\mathbb {R}}\ni t\mapsto e^{it{\widehat{F}}}\) on a Hilbert space \({{\mathcal {H}}}\). Additionally one claims that the map \(Q: {{\mathcal {P}}}^\infty (M,{\mathbb {R}})\ni F\mapsto i{\widehat{F}}\in {{\mathcal {L}}}({{\mathcal {D}}})\) which assigns to a classical generator F the quantum one \({\widehat{F}}\) (a self-adjoint operator in \({{\mathcal {H}}}\)) is a morphism of some Lie algebras, where \({{\mathcal {P}}}^\infty (M,{\mathbb {R}})\) is a Lie subalgebra of the Poisson algebra \(C^\infty (M,{\mathbb {R}})\) and \({{\mathcal {L}}}({{\mathcal {D}}})\) is a Lie algebra of anti-self-adjoint operators having a common domain \({{\mathcal {D}}}\) dense in \({{\mathcal {H}}}\).
Among known methods of quantization the Kirillov–Kostant–Souriau geometric quantization [17, 18, 36] is one of the most elegant from a geometric point of view and gives a precise construction of the quantum generator \({\widehat{F}}\) for the given classical one \(F\in {{\mathcal {P}}}^\infty (M,{\mathbb {R}})\). For this construction one needs to obtain a \(\sigma _t^F\)-invariant complex Lagrangian distribution \({{\mathcal {P}}}\subset T^{{\mathbb {C}}}M\) and the appropriate measure (density) on the quotient manifold \(M/{{\mathcal {P}}}\cap \bar{{\mathcal {P}}}\). 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 \((M,\omega )\) and quantum \(({\mathbb {C}}{\mathbb {P}}({{\mathcal {H}}}),\omega _{FS})\) phase spaces in [25, 28] 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 [15, 29].
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 [23]. See also the classical paper [1] of N. Aronszajn.
Basing partly on [28], in Sects. 2 and 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 \(P(G,\pi ,M)\) with a fixed connection form \(\vartheta \) on the total space P. In Sect. 2 we define the Poisson \(C^\infty (M, {\mathbb {R}})\)-module \({{\mathcal {P}}}^\infty _G (P,\vartheta )\) of generators \((X, F) \in {{\mathcal {P}}}^\infty _G(P, \vartheta ) \) of generalized Hamiltonian flows \(\tau _t^{(X,F)} \in \text{ Aut }(P, \vartheta )\), i.e. those which are solutions of generalized Hamilton Eq. (10). In Sect. 3 we generalize the Kirillov–Kostant–Souriau prequantization morphisms to the morphism \(Q:{{\mathcal {P}}}^\infty _G(P,\vartheta )\rightarrow {{\mathcal {D}}}^1\Gamma ^\infty (M,{\mathbb {V}})\) of \(\mathcal {P}^\infty _G(P,\vartheta )\) in the Lie \(C^\infty (M, {\mathbb {R}})\)-module of differential operators of order less or equal one acting on the smooth sections \(\Gamma ^\infty (M, {\mathbb {V}})\) of an associated smooth vector bundle \({\mathbb {V}} \rightarrow M\) over M.
In Sect. 4 we consider the G-equivariant coherent state map \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) and the positive definite G-equivariant kernel \(K:P\times P\rightarrow {{\mathcal {B}}}(V)\), where V and \({{\mathcal {H}}}\) are complex Hilbert spaces and \(\mathcal {B}(V,{{\mathcal {H}}})\) is the right Hilbert \(\mathcal {B}(V)\)-module of bounded linear maps of V in \({{\mathcal {H}}}\). In the same section the equivalence of the coherent state map \(\kappa \) 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 operator \({\hat{F}}\) in the Hilbert space \({{\mathcal {H}}}_K\) whose domain is defined by the G-equivariant positive kernel \(K:P\times P\rightarrow {{\mathcal {B}}}(V)\) (see (79) and (80)). The conditions on this kernel needed to quantize \(\tau ^{(X,F)}_t\in \text{ Aut }(P,\vartheta )\) are presented in (73) and (74).
In Sect. 5 assuming that \(G\subset GL(V,{\mathbb {C}})\) is a Lie subgroup of \(GL(V, {\mathbb {C}})\) and that there exists a coherent state map \({\mathfrak {K}}: P \rightarrow B(V, \mathcal {H})\) on the total space of \(P(G,\pi ,M)\), we define in a canonical way two other principal bundles \({\widetilde{P}}(GL(V,{\mathbb {C}}),{\widetilde{\pi }},M)\) and \(U(U(V),\pi ^u,M)\) over M. The connection forms \({\widetilde{\vartheta }}\in \Gamma ^\infty ({\widetilde{P}}, T^*{\widetilde{P}}\otimes {{\mathcal {B}}}(V))\) and \(\vartheta ^a \in \Gamma ^\infty (U,T^*U\otimes T_e U(V))\) as well as the respective coherent states maps \(\widetilde{{\mathfrak {K}}}:{\widetilde{P}}\rightarrow B(V, \mathcal {H})\) and \({\mathfrak {a}}: U\rightarrow B(V, \mathcal {H})\) are defined on these principal bundles by using of \({\mathfrak {K}}: P \rightarrow B(V, \mathcal {H})\). Next we show, see Proposition 5, that the flows \(\tau ^{(X, {\tilde{F}})}_t \in \text{ Aut }({\widetilde{P}}, {\widetilde{\vartheta }})\) and \(\tau _t^{(X, F^a)}\in \text{ Aut } (U, \vartheta ^a)\) have the same quantum counterpart \(e^{it{\hat{F}}}\) as the flow \(\tau _t^{(X, F)} \in \text{ Aut }(P, \vartheta )\).
In Sect. 6 we quantize the holomorphic one-parameter groups of automorphisms of a holomorphic principal bundles \(P(GL(V,{\mathbb {C}}),\pi ,M)\) 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 \(\tau _t^{(X,F)}\)-invariant positive kernels on \(P\times P\) and show their relationship with the spectral decomposition of the corresponding quantum generators \(Q_{(X,F)} = {\hat{F}}\).
Some applications of the coherent state method in physics including quantum optics are shortly discussed in Sect. 7.
2 The Poisson \(C^\infty (M, {\mathbb {R}})\)-Module Corresponding to the Group \(\text{ Aut }(P,\vartheta )\)
The main task of this section is the investigation of some variant of Hamiltonian mechanics on a G-principal bundle \(P(G,\pi ,M)\), where the role of the symplectic form is played by the curvature form \(\varOmega \) of a fixed connection form \(\vartheta \in \Gamma ^\infty (P, T^*P\otimes T_eG)\). We will define the Poisson \(C^\infty (M, {\mathbb {R}})\)-module \(({{\mathcal {P}}}^\infty _G (P, \vartheta ), \{\cdot , \cdot \}_{\vartheta })\) with the Lie bracket \(\{\cdot , \cdot \}_{\vartheta }\) given in (11), which satisfies the Leibniz property in the sense of (13). The corresponding generalization (10) of the Hamiltonian equation is presented. The model proposed here in the case when \(G= U(1)\) and \(\vartheta \) has non-singular curvature form reduces to standard Hamiltonian mechanics on a symplectic manifold.
Let \(\text{ Aut }(P,\vartheta )\) denote the group of diffeomorphisms of P which preserve the principal bundle structure of P and the connection form \(\vartheta \). We recall that a \(T_eG\)-valued differential one-form \(\vartheta \) on P is a connection form if and only if
for any \(p\in P\) and \(g\in G\), where \(\kappa : P\times G\rightarrow P\) is the right-action \(\kappa (p,g)=:pg\) of the Lie group G on P and \(T_eG\) is the tangent space to G at the unit element \(e\in G\), i.e. the Lie algebra of G. By \(\kappa _g: P\rightarrow P\) and \(\kappa _p: G\rightarrow P\) we denoted the maps defined by \(\kappa _g(p):=pg\) and \(\kappa _p(g):=pg\). The connection form \(\vartheta \) defines the spliting
of the tangent space \(T_pP\) at \(p\in P\) on the horizontal \(T_p^hP:=\text{ ker }\;\vartheta (p)\) and vertical \(T_p^vP\), i.e. tangent to the fibre \(\pi ^{-1}(\pi (p))\) of \(\pi :P\rightarrow M\), components (see e.g. [20]).
Let \(\tau :({\mathbb {R}},+)\rightarrow (\text{ Aut }(P,\vartheta ),\circ )\) be a one-parameter subgroup of \(\text{ Aut }(P,\vartheta )\) i.e. the map \(\tau : {\mathbb {R}} \times P\rightarrow P\) is a smooth map which satisfies
Then for the vector field \(\xi \in \Gamma ^\infty (TP)\) tangent to the flow \(\{\tau _t\}_{t\in {\mathbb {R}}}\) one has
where \({{\mathcal {L}}}_\xi \) is the Lie derivative with respect to \(\xi \).
The \(C^\infty (M,{\mathbb {R}})\)-modules of vector fields \(\xi \in \Gamma ^\infty (TP)\) which satisfy the condition (3) and the condition (3) together with the condition (4) will be denoted by \(\Gamma _G^\infty (TP)\) and \(\Gamma _{G,\vartheta }^\infty (TP)\), respectively. The \(C^\infty (M,{\mathbb {R}})\)-module structure on \(\Gamma _G^\infty (TP)\), and hence on its submodules \(\Gamma _G^\infty (T^hP)\) and \(\Gamma _G^\infty (T^vP)\), is defined by
where by \(T^hP\) and \(T^vP\) we denote the horizontal and vertical vector subbundles of TP, respectively.
The tangent map \(T\pi :TP\rightarrow TM\) defines an isomorphism of \(C^\infty (M,{\mathbb {R}})\)-modules of horizontal vector fields \(\Gamma _G^\infty (T^hP)\) on P and vector fields \(\Gamma ^\infty (TM)\) on M by
In the sequel we will need the \(C^\infty (M,{\mathbb {R}})\)-module \(C^\infty _G(P,T_eG)\) which by definition consists of smooth functions \(F:P\rightarrow T_eG\) satisfying the G-equivariance property
This module is isomorphic to the module of vertical vector fields \(\Gamma _G^\infty (T^vP)\) on P, where the \(C^\infty (M,{\mathbb {R}})\)-module isomorphism \(\nu ^*:\Gamma _G^\infty (T^vP){\mathop {\longrightarrow }\limits ^{\sim }} C^\infty _G(P,T_eG)\) is defined as follows
Its inverse is given by
The correctness of the definitions above follows from (1) and (2), and from \(T\kappa _g(p)\circ T\kappa _p(e)=T\kappa _{pg}(e)\).
Taking the decomposition \(\xi =\xi ^h+\xi ^v\) of \(\xi \in \Gamma _G^\infty (TP)\) on the horizontal and vertical components and isomorphisms (5) and (6) we define
a \(C^\infty (M,{\mathbb {R}})\)-module isomorphism \(\pi ^*\times \nu ^*: \Gamma _G^\infty (TP){\mathop {\longrightarrow }\limits ^{\sim }}\Gamma ^\infty (TM)\times C^\infty _G(P,T_eG)\). The inverse of \(\pi ^*\times \nu ^*\) is given by
where by \(H^*:\Gamma ^\infty (TM)\rightarrow \Gamma _G^\infty (T^hP)\) we denote the horizontal lift, i.e. the module isomorphism inverse to \(\pi ^*\).
For \(\xi \in \Gamma _G^\infty (TP)\) one has
where
is the curvature form of \(\vartheta \) and
is the covariant derivative of F.
On \(\Gamma _G^\infty (TP)\) one has the structure of Lie algebra given by the Lie bracket \([\cdot , \cdot ]\) of vector fields. Using the \(C^\infty (M, {\mathbb {R}})\)-modules isomorphism \(\pi ^*\times \nu ^*: \Gamma _G^\infty (TP){\mathop {\longrightarrow }\limits ^{\sim }}\Gamma ^\infty (TM)\times C^\infty _G(P,T_eG),\) we can carry the Lie bracket \([\cdot , \cdot ]\) from \(\Gamma _G^\infty (TP)\) to \(\Gamma ^\infty (TM)\times C^\infty _G(P,T_eG)\) obtaining in this way the Lie bracket
of \((X_i,F_i)=(\pi ^*\times \nu ^*)(\xi _i)\), \(i=1,2\), where \([F_1,F_2](p):=[F_1(p),F_2(p)]\).
It is reasonable to mention here that the Lie algebra \((\Gamma _G^\infty (TP), [\cdot , \cdot ])\) is isomorphic with the Lie algebra \((\Gamma ^\infty (TP/G), [\cdot , \cdot ])\) of the Atiyah Lie algebroid, being a central ingredient of the Atiyah exact sequence of algebroids
see e.g. [22].
Though the language of Lie algebroid theory will not be used later, we note that the projection on the first component \(\text{ pr}_1: \Gamma ^\infty (TM)\times C^\infty _G(P, T_eG) \rightarrow \Gamma ^\infty (TM)\) corresponds to the anchor map \(a: TP/G \rightarrow TM \) of the Atiyah algebroid. Hence, from the defining property of the anchor map we have
and
These properties of \(\text{ pr}_1\) and \(\{\cdot , \cdot \}_{\vartheta }\) can also be obtained directly from their definitions.
The above structure of Lie \(C^\infty (M)\)-module on \(\Gamma ^\infty (TM) \times C^\infty _G (P, T_eG)\) restricts to \({{\mathcal {P}}}^\infty _G(P,\vartheta ):=(\pi ^*\times \nu ^*)(\Gamma _{G,\vartheta }^\infty (TP))\) making it a Lie \(C^\infty (M, {\mathbb {R}})\)-submodule of \(\Gamma ^\infty (TM) \times C^\infty _G (P, T_eG)\). It follows from (7) that \((X,F)\in {{\mathcal {P}}}^\infty _G(P,\vartheta )\) if and only if
Let us note here that the condition (10) is invariant with respect to the Lie \(C^\infty (M, {\mathbb {R}})\)-module operation. Thus, for \((X_1, F_1), (X_2, F_2) \in {{\mathcal {P}}}^\infty (P,\vartheta )\) the Lie bracket (8) simplifies to the form
In the case when the curvature form \(\varOmega \) is a non-singular 2-form, i.e. when \(\xi ^h\llcorner \varOmega =0\) implies \(\xi ^h=0\), one has from (10) that for \((X,F)\in {{\mathcal {P}}}^\infty _G(P,\vartheta )\) the vector field \(X \in \Gamma ^\infty (TM)\) is defined uniquely by the function \(F\in C^\infty _G(P,T_eG)\). So, in this case we have the \(C^\infty (M, {\mathbb {R}})\)-modules morphism \({\mathfrak {b}}: C^\infty _G(P,T_eG) \rightarrow {{\mathcal {P}}}^\infty _G(P,\vartheta )\). Note here that \(F\in \text{ ker }{\mathfrak {b}}\) if and only if \(\mathbf{D} F =0\), which does not mean in general that \(F= \text{ const }\). Substituting \(X_1 = {\mathfrak {b}}(F_1)\) and \(X_2 = {\mathfrak {b}}(F_2)\) into (11) we obtain the Lie bracket
of \(F_1, F_2 \in C^\infty _G(P,T_eG)\), which satisfies the Leibniz property
in sense of \(C^\infty (M, {\mathbb {R}})\)-module.
Assuming \(G=U(1)\) we find that \(C^\infty _G(P,T_eG)\) is canonically isomorphic with \(C^\infty (M,{\mathbb {R}})\) and the curvature form \(\varOmega \) is identified with the closed \(d \omega =0\) 2-form \(\omega \) on M, which is a symplectic form in the non-singular case. Hence, formula (12) reduces to the symplectic Poisson bracket and (13) to its Leibniz property.
Taking the above facts into account, further we will call \(({{\mathcal {P}}}^\infty _G(P,\vartheta ), \{\cdot , \cdot \}_{\vartheta })\) the Poisson \(C^\infty (M, {\mathbb {R}})\)-module.
In the framework of the assumed terminology it is natural to consider:
-
(i)
the Eq. (10) as a generalization of Hamilton’s equation to the arbitrary Lie group G case;
-
(ii)
the one-parameter group \(\tau _t^{(X, F)} \in \text{ Aut }(P,\vartheta )\) as a generalized Hamiltonian flow generated by \((X, F) \in {{\mathcal {P}}}^\infty _G(P,\vartheta )\) (in non-singular case by \(F\in C^\infty _G(P,T_eG)\)).
In the next two sections we propose and investigate a method of quantization of the Hamiltonian flow \(\tau _t^{(X, F)} \in \text{ Aut }(P,\vartheta )\) based on the notion of G-equivariant positive kernel on \(P\times P\).
Though our considerations below are valid for an arbitrary Lie group G we will assume that \(G\subset GL(V,{\mathbb {C}})\cong GL(N,{\mathbb {C}})\) is a Lie subgroup of the linear group \(GL(V,{\mathbb {C}})\) of a complex N-dimensional Hilbert space V. By \(\langle \cdot ,\cdot \rangle :V\times V\rightarrow {\mathbb {C}}\) 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 \(U(V)\cong U(N)\).
3 Kirillov–Kostant–Souriau Prequantization Morphism
In this section we generalize the Kirillov–Kostant–Souriau prequantization procedure [17, 18, 36] for the case of an arbitrary Lie group \(G\subset GL(V, {\mathbb {C}})\), i.e. we obtain the Lie \(C^\infty (M,{\mathbb {R}})\)-module morphism \(Q: {{\mathcal {P}}}^\infty _G(P,\vartheta ) \rightarrow {{\mathcal {D}}}^1\Gamma ^\infty (M,{\mathbb {V}})\) of the Poisson module \({{\mathcal {P}}}^\infty _G(P,\vartheta )\) into the Lie module of differential operators of the order less or equal one acting on the smooth sections of some complex vector bundle \({\mathbb {V}}\rightarrow M\) over M.
To this end we define the smooth complex bundle \({\mathbb {V}}:= (P\times V)/G\rightarrow M\) over M associated to \(P(G,\pi ,M)\) by the action \(P\times V\times G\ni (p,v,g)\mapsto (pg,g^{-1}v)\in P\times V\) of the Lie group G on \(P\times V\). One has the natural \(C^\infty (M,{\mathbb {R}})\)-module isomorphism between the module \(\Gamma ^\infty (M,{\mathbb {V}})\) of smooth sections of \({\mathbb {V}}\rightarrow M\) and the module \(C^\infty _G(P,V):=\{f\in C^\infty (P,V): f(pg)=g^{-1}f(p)\;for\;g\in G\}\) of G-equivariant smooth functions on P defined by the one-to-one dependence
between \(\psi \in \Gamma ^\infty (M,{\mathbb {V}})\) and \(f\in C^\infty _G(P,V)\). Using (14) we define
for any \((X,F)=(\pi ^*\times \nu ^*)(\xi )\in {{\mathcal {P}}}^\infty _G(P,\vartheta )\) the differential operator \(Q_{(X,F)}:\Gamma ^\infty (M,{\mathbb {V}})\rightarrow \Gamma ^\infty (M,{\mathbb {V}})\) of order less or equal one. Let us note here that if \(\xi \in \Gamma _{G,\vartheta }^\infty (TP)\) and \(f\in C^\infty _G(P,V)\) then \(\xi (f)\in C^\infty _G(P,V)\).
Since \(\pi ^*\times \nu ^*: \Gamma _{G,\vartheta }^\infty (TP){\mathop {\longrightarrow }\limits ^{\sim }}{{\mathcal {P}}}^\infty _G(P,\vartheta )\) is an isomorphism of the Lie \(C^\infty (M,{\mathbb {R}})\)-modules, from (14) and (15) one obtains
where on the left hand side of the equality (16) we have the commutator of the differential operators and \(\{\cdot ,\cdot \}_\vartheta \) is the Lie bracket defined in (11). Since in the case \(\dim _{{\mathbb {C}}}V=1\) and \(G=U(1)\) the Lie \(C^\infty (M,{\mathbb {R}})\)-modules monomorphism
is the Kirillov–Kostant–Souriau prequantization morphism we will further extend this terminology to the general case. By \({{\mathcal {D}}}^1\Gamma ^\infty (M,{\mathbb {V}})\) in (18) we denote the Lie \(C^\infty (M,{\mathbb {R}})\)-module of differential operators of order less or equal one acting on \(\Gamma ^\infty (M,{\mathbb {V}})\).
In order to establish a more explicit expression for \(Q_{(X,F)}\), where \((X,F)\in {{\mathcal {P}}}^\infty (P,\vartheta )\), we will use the decomposition \(\xi =\xi ^h+\xi ^v\) of \(\xi \in \Gamma _{G,\vartheta }^\infty (TP)\) on the horizontal and vertical components. The flows \(\tau _t\) and \(\tau ^h_t\) tangent to \(\xi \) and \(\xi ^h\) satisfy
where \(g:{\mathbb {R}}\times P\rightarrow G\) is the cocycle related to the vertical (tangent to \(\xi ^v\)) flow \(\tau ^v_t\) by \(\tau ^v_t(p)=pg(t,p)\).
Proposition 1
The cocycle \(g:{\mathbb {R}}\times P\rightarrow G\) corresponding to \((X,F)\in {{\mathcal {P}}}^\infty _G(P,\vartheta )\) by (19) has the following form
Proof
Since for \(\xi =(\pi ^*\times \nu ^*)^{-1}((X, F))\in \Gamma _{G,\vartheta }^\infty (TP)\) one has \(\xi (F)=\xi (\langle \vartheta ,\xi \rangle )=\langle L_\xi \vartheta ,\xi \rangle +\langle \vartheta ,[\xi ,\xi ]\rangle =0\) we find that
Combining (21) with (19) we obtain
Applying \(\tau _s\) to both sides of (19) we obtain
From the above and from (19) we find that
Differentiating the equality above with respect to the parameter s at \(s=0\) we obtain differential equation
on the cocycle \(g:{\mathbb {R}}\times P\rightarrow G\subset GL(N,{\mathbb {C}})\). The equality (20) is obtained as a solution of (22) with the initial condition \(g(0,p)=1\!\!1_V\). \(\quad \square \)
For \(f\in C^\infty _G(P,V)\) one has \(\xi ^h(f),\xi ^v(f)\in C^\infty _G(P,V)\) and
where \(\mathbf{D} :C^\infty _G(P,V)\rightarrow \Gamma _G^\infty (P,T^*P\otimes V)\) is the covariant derivative of f defined by
By \(pr^{hor}:TP\rightarrow T^hP\) we denoted the projection of TP on its horizontal component \(T^hP\).
Defining the covariant derivative \(\nabla :\Gamma ^\infty (M,{\mathbb {V}})\rightarrow \Gamma ^\infty (M,T^*M\otimes {\mathbb {V}})\) as usual by
where \(\psi \) is defined in (14), and using Eqs. (23) and (24) one obtains from (15) the following expression
for the Kirillov–Kostant–Souriau operator \(Q_{(X,F)}:\Gamma ^\infty (M,{\mathbb {V}})\rightarrow \Gamma ^\infty (M,{\mathbb {V}})\). Let us note here that the 0-order differential operator F acts on \(\psi \in \Gamma ^\infty (M,{\mathbb {V}})\) as follows
We note also that \(F(pg) f(pg)=g^{-1}F(p) f(p)\) and \((\mathbf{D} _\xi f)(pg)=g^{-1}(\mathbf{D} _\xi f)(p)\).
The Kirillov–Kostant–Souriau operator is the generator
of the one-parameter group \(\Sigma _t:\Gamma ^\infty (M,{\mathbb {V}})\rightarrow \Gamma ^\infty (M,{\mathbb {V}})\) acting on the sections \(\psi \in \Gamma ^\infty (M,{\mathbb {V}})\) by
where the flows \(\tau ^{\mathbb {V}}_t:{\mathbb {V}}\rightarrow {\mathbb {V}}\) and \(\sigma _t:M\rightarrow M\) are defined by
and by
respectively. The vector field \(X\in \Gamma ^\infty (TM)\) in (26) is tangent to the flow \(\{\sigma _t\}_{t\in {\mathbb {R}}}\).
If the curvature form \(\varOmega \) is non-singular the \(C^\infty (M,{\mathbb {R}})\)-module morphism \({\mathfrak {b}}:C^\infty _G(P, T_eG)\rightarrow \Gamma ^\infty (TM)\) leads to the Kirillov–Kostant–Souriau prequantization morphism
for the Poisson \(C^\infty (M,{\mathbb {R}})\)-module \((C^\infty _G(P,T_eG),\{.,.\}_\vartheta )\), where \(X_F=\flat (F)\) and Poisson bracket \(\{F_1,F_2\}_\vartheta \) of \(F_1,F_2\in C^\infty _G(P,T_eG)\) is defined in (12).
Furthermore we will need the vector bundle \(\bar{{\mathbb {V}}}:= (P\times V)/G\rightarrow M\) associated to \(P(G,\pi ,M)\) by the action
as well as the \(C^\infty (M,{\mathbb {C}})\)-module
of the V-valued smooth function on P. Similarly as in (14) the equality \({\bar{\psi }}(\pi (p)):=[(p,{\overline{f}}(p))]\) defines the isomorphism \(\Gamma ^\infty (M,\bar{{\mathbb {V}}})\cong C^\infty _{{\bar{G}}}(P,V)\) of \(C^\infty (M,{\mathbb {C}})\)-modules. Using this isomorphism we define
another Kirillov–Kostant–Souriau differential operator acting now on \(\Gamma ^\infty (M,\bar{{\mathbb {V}}})\). For vector field \(\xi \in \Gamma _{G}^\infty (TP)\) tangent to \(\tau _t\) from (19) we find
where
Next, using the covariant derivative \({\bar{\nabla }}:\Gamma ^\infty (M,\bar{{\mathbb {V}}})\rightarrow \Gamma ^\infty (M,T^*M\otimes \bar{{\mathbb {V}}})\) defined by \({\bar{\nabla }}{\bar{\psi }}(\pi (p)):=[(p,\bar{\mathbf{D }}{\overline{f}}(p))]\) we obtain
the generator of the flow
where \(F^\dagger {\bar{\psi }}\) and \(\tau ^{\bar{{\mathbb {V}}}}_t:\bar{{\mathbb {V}}}\rightarrow \bar{{\mathbb {V}}}\) are defined as follows
and
for \([(p,v)]\in \bar{{\mathbb {V}}}\).
The covariance properties (16) and (17) for \({\bar{Q}}\) are proved in an analogous way as for Q.
Now let us mention that using the scalar product \(\langle \cdot ,\cdot \rangle \) in V one defines for \({\bar{\psi }}_1\in \Gamma ^\infty (M,\bar{{\mathbb {V}}}) \) and \(\psi _2\in \Gamma ^\infty (M,{\mathbb {V}})\) the smooth function
on M. One sees from \(\langle \!\langle {\bar{\psi }}_1,\psi _2 \rangle \!\rangle (\sigma _t(\pi (p)))=\langle {\overline{f}}_1(\tau _t(p)),f_2(\tau _t(p))\rangle \) that the following property
for the pairing (30) is valid.
Fixing a local trivialization \(s_\alpha :{{\mathcal {O}}}_\alpha \rightarrow P\) (where \(\bigcup _{\alpha \in I} {{\mathcal {O}}}_\alpha =M\) is an open covering of M) of the principal bundle \(P(G,\pi ,M)\) one defines the local cocycles \(g_\alpha (t,\cdot ):{{\mathcal {O}}}_\alpha \rightarrow G\) and \(h_\alpha (t,\cdot ):{{\mathcal {O}}}_\alpha \rightarrow G\) by
for sufficiently small t. From (19) and (20) one has
Hence, from (31)–(33) one obtains
From (34) and the cocycle properties
we have
which is equivalent to
In order to obtain \(\frac{d}{dt}h_\alpha (t,m)|_{t=0}\in T_eG\) we note that from (32) it follows that
Next, applying \(\vartheta (s_\alpha (m))\) to both sides of (35) and using (2) we obtain
where \(\vartheta _\alpha (m):=\vartheta (s_\alpha (m))\circ Ts_\alpha (m)=(s_\alpha ^*\vartheta )(m)\).
Introducing the notations \(\phi _\alpha (m):=\frac{d}{dt}g_\alpha (t,m)|_{t=0}\) and \(F_\alpha (m):=F\circ s_\alpha (m)\), after differentiating (34) at \(t=0\) we obtain the equality
which is useful for finding the local expression for the infinitesimal generator of the flow \(\Sigma _t:\Gamma ^\infty (M,{\mathbb {V}})\rightarrow \Gamma ^\infty (M,{\mathbb {V}})\) defined in (29). Namely, we have
Thus for \(f_\alpha :=f\circ s_\alpha :{{\mathcal {O}}}_\alpha \rightarrow V\) one has
Differentiating both sides of (38) at \(t=0\) and using (36) we obtain the local representation
of the Kirillov–Kostant–Souriau prequantization operator \(Q_{(X,F)}:\Gamma ^\infty (M,{\mathbb {V}})\rightarrow \Gamma ^\infty (M,{\mathbb {V}})\), where \(\nabla _X^\alpha :=X+\langle \vartheta _\alpha ,X\rangle \) is the local form of the covariant derivative \(\nabla \) defined in (25). Similarly we have
for \({\bar{Q}}_{(X,F)}:\Gamma ^\infty (M,\bar{{\mathbb {V}}})\rightarrow \Gamma ^\infty (M,\bar{{\mathbb {V}}})\).
In the local gauge the Hamilton Eq. (10) assumes the form
where
Ending this subsection let us mention the well known equivariance formulae with respect to the gauge transformation
where \(g_{\alpha \beta }:{{\mathcal {O}}}_\alpha \cap {{\mathcal {O}}}_\beta \rightarrow G\) is the respective transition cocycle, i.e. \(g_{\alpha \beta }(m)g_{\beta \gamma }(m)=g_{\alpha \gamma }(m)\). Namely, one has
where \(m\in {{\mathcal {O}}}_\alpha \cap {{\mathcal {O}}}_\beta \).
4 Positive Definite Kernels and Quantization
In the previous section we obtained the formulas (26) and (28) on Kirillov–Kostant–Souriau prequantization operators \(Q_{(X,F)}\) and \({\bar{Q}}_{(X,F)}\) as well as their local versions (39) and (40). 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 [28]. In order to make the paper self-sufficient we present the above procedure in detail. Some new results complementary to the ones in [28] will also be presented in this section.
The theory of reproducing kernels for 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 \(\dim _{\mathbb {C}} V = N <+\infty \) and \(G \subset GL(V)\cong GL(N, {\mathbb {C}})\). Further, by \(\mathcal {B}(V) \cong \text{ Mat}_{N \times N} ({\mathbb {C}})\) we denote the \(C^*\)-algebra of linear maps of V and by \(\mathcal {B}(V, \mathcal {H})\) the right Hilbert \(\mathcal {B}(V)\)-module of linear maps \(\Gamma : V \rightarrow \mathcal {H}\) from the Hilbert space V into the separable Hilbert space \(\mathcal {H}\). Let us note here that from \(\dim _{\mathbb {C}} V<+\infty \) follows boundness of \(\Gamma :V\rightarrow \mathcal {H}\). The \(\mathcal {B}(V)\)-valued scalar product \(\langle \cdot ; \cdot \rangle : \mathcal {B}(V, \mathcal {H})\times \mathcal {B}(V, \mathcal {H}) \rightarrow \mathcal {B}(V)\) on \(\mathcal {B}(V, \mathcal {H})\) is defined by
for \(\Gamma , \Delta \in \mathcal {B}(V, \mathcal {H})\), where the operator \(\Gamma ^*: \mathcal {H}\rightarrow V\) is the one adjoint to \(\Gamma :V\rightarrow \mathcal {H}\).
Definition 1
A smooth map \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) will be called a G-equivariant coherent state map if it has the following properties:
-
(i)
the G-equivariance property, i.e.
$$\begin{aligned} {\mathfrak {K}}(pg)={\mathfrak {K}}(p)g \end{aligned}$$(50)for any \(p\in P\) and for any \(g\in G\);
-
(ii)
non-singularity, i.e.
$$\begin{aligned} \ker {\mathfrak {K}}(p)=\{0\},\quad \hbox {or equivalently}\quad {\mathfrak {K}}(p)^*{\mathfrak {K}}(p)\in GL(V,{\mathbb {C}}) \end{aligned}$$(51)for any \(p\in P\);
-
(iii)
the set \({\mathfrak {K}}(P)V\) is linearly dense in \({\mathcal {H}}\), i.e.
$$\begin{aligned} \bigcap _{p\in P}\ker {\mathfrak {K}}(p)^*=\{0\},\quad \hbox {or equivalently}\quad \{{\mathfrak {K}}(p)v: p\in P\,\,\hbox {and}\,\, v\in V \}^\perp =\{0\}.\nonumber \\ \end{aligned}$$(52)
By \({\mathfrak {K}}(p)^*\) in (51) and (52) we denoted the map \({\mathfrak {K}}(p)^*: {{\mathcal {H}}}\rightarrow V\) adjoint to \({\mathfrak {K}}(p):V\rightarrow {{\mathcal {H}}}\), i.e. such that \(\langle \psi |{\mathfrak {K}}(p)v\rangle =\langle {\mathfrak {K}}(p)^*\psi ,v\rangle \) for \(\psi \in {{\mathcal {H}}}\) and \(v\in V\), where \(\langle \cdot |\cdot \rangle \) is the scalar product in Hilbert space \({{\mathcal {H}}}\). For the existence of \({\mathfrak {K}}:P \rightarrow B(V, \mathcal {H})\) with the above properties see [24].
Definition 2
A smooth map \(K:P\times P\rightarrow {{\mathcal {B}}}(V)\) will be called a G-equivariant positive definite kernel if it has the following properties
-
(i)
the G-equivariance property, i.e.
$$\begin{aligned} K(p,qg)= K(p,q)g \end{aligned}$$(53)for any \(p,q\in P\) and \(g\in G\);
-
(ii)
non-singularity, i.e.
$$\begin{aligned} K(p,p)\in GL(V,{\mathbb {C}})\subset \mathcal {B}(V) \end{aligned}$$(54)for any \(p\in P\);
-
(iii)
positivity, i.e.
$$\begin{aligned} \sum _{i,j=1}^J \langle v_i, K(p_i,p_j)v_j\rangle \geqq 0 \end{aligned}$$(55)for arbitrary finite sequences \(p_1,\ldots ,p_J\in P\) and \(v_1,\ldots ,v_J\in V\).
From the positivity condition (55) one obtains the following properties of the kernel
and
where \(g,h\in G\) and \(p,q\in P\).
The above structures are mutually dependent. Extending the rather well known scheme from the theory of reproducing (positive) kernels [3, 23] to this more geometrically complicated setting we shortly describe this dependence.
Starting from the coherent state map \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) we define the G-equivariant positive definite kernel by
The smoothness of the kernel (56) and the properties (53)–(55) follow from the smoothness of \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) and its properties mentioned in (50)–(52).
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 \({{\mathcal {D}}}_K\subset C^\infty _{{\bar{G}}}(P,V)\) which consists of linear combinations of the functions \(K(\cdot ,q)v\) indexed by \(q\in P\) and \(v\in V\). In order to define the scalar product \( \langle {\bar{f}}_1|{\bar{f}}_2\rangle _{K}\) of \({\bar{f}}_1,{\bar{f}}_2\in {{\mathcal {D}}}_K\) we extend the summations in \({\bar{f}}_1=\sum _{i\in J_1}K(\cdot ,q^1_i)v^1_i\) and \({\bar{f}}_2=\sum _{j\in J_2}K(\cdot ,q^2_j)v^2_j\) to the set of indexes \(J=J_1\cup J_2\) and define \(\{p_1,\ldots ,p_J \}:=\{q^1,\ldots ,q^1_{J_1} \}\cup \{q^2_1,\ldots ,q^2_{J_2} \}\). After that assuming \(v^1_j=0\) for \(j\in J\setminus J_1\) and \(v^2_j=0\) for \(j\in J\setminus J_2\) we define the scalar product
of \({\bar{f}}_k=\sum _{j\in J}K(\cdot ,p_j)v^k_j\), where \(k=1,2\).
In particular one has
From (59) and the Schwarz inequality for the scalar product (58) one obtains
From the inequality (60) one sees that \(\Vert f\Vert _K=\langle f| f\rangle _K^{1/2}=0\) implies \(f=0\), so, \(\Vert \cdot \Vert _K\) is a norm on \({{\mathcal {D}}}_K\).
We conclude from (60) that the evaluation functional \(E_p({\bar{f}}):={\bar{f}}(p)\) satisfies
i.e. it is a bounded functional on \({{\mathcal {D}}}_K\) for every \(p\in P\). So, we can extend it to the Hilbert space \({{\mathcal {H}}}_{K}\supset {{\mathcal {D}}}_K\) being the abstract extension of the pre-Hilbert space \({{\mathcal {D}}}_K\subset C^\infty _{{\bar{G}}}(P,V)\). It follows from (60) that for any equivalence class \([\{{\bar{f}}_n\}]\in {{\mathcal {H}}}_{K}\) of Cauchy sequences \(\{{\bar{f}}_n\}\subset {{\mathcal {D}}}_K\) one defines
a function \({\bar{f}}:P\rightarrow V\) depending on \([\{{\bar{f}}_n\}]\) only. Hence we see that the Hilbert space \({{\mathcal {H}}}_{K}\) 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 (27).
Now, rewriting (59) as follows
where \({\bar{f}}\in {{\mathcal {H}}}_{K}\), \(v\in V\) and \(E_p^*:V\rightarrow {{\mathcal {H}}}_{K} \) is the conjugation of \(E_p:{{\mathcal {H}}}_{K}\rightarrow V \), we define the coherent state map \({\mathfrak {K}}_K:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}}_{K})\) by
One easily sees that the properties (50)–(52) for \({\mathfrak {K}}_K:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}}_{K})\) follow from the ones for K given in (53)–(55). The smoothness of \({\mathfrak {K}}_K:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}}_{K})\) follows from the smoothness of the positive kernel K (see Proposition 2.1 in [28]).
As we see from (56) a coherent state map \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) defines the positive kernel \(K:P\times P\rightarrow {{\mathcal {B}}}(V)\). The Hilbert space \({{\mathcal {H}}}_{K}\) for this kernel is isomorphic to the Hilbert space \({{\mathcal {H}}}\), where the isomorphism \(I_K:{{\mathcal {H}}}{\mathop {\longrightarrow }\limits ^{\sim }}{{\mathcal {H}}}_{K}\) is defined by
In order to see that \(I_K\) is indeed an isomorphism we note that for \(|\psi \rangle \in {{\mathcal {D}}}_{{\mathfrak {K}}}\), where \({{\mathcal {D}}}_{{\mathfrak {K}}}\subset {{\mathcal {H}}}\) is defined by
one has \(I_K|\psi \rangle =\sum _{j\in J} K(\cdot ,p_j)v_j \) and \(\langle I_K|\psi \rangle |I_K|\psi \rangle \rangle _K=\langle \psi |\psi \rangle _{{{\mathcal {H}}}}\), i.e. \(I_K:{{\mathcal {D}}}_{{\mathfrak {K}}}\rightarrow {{\mathcal {D}}}_K\) is a linear isometry of dense linear subspaces \({{\mathcal {D}}}_{{\mathfrak {K}}}\subset {{\mathcal {H}}}\) and \({{\mathcal {D}}}_K\subset {{\mathcal {H}}}_K\), so it extends to the isomorphism of the Hilbert spaces.
The isometry \(I_K:{{\mathcal {H}}}\rightarrow {{\mathcal {H}}}_{K}\) defines the isomorphism of Banach spaces \(L_{I_K}:\mathcal {B}(V,{{\mathcal {H}}})\rightarrow \mathcal {B}(V,{{\mathcal {H}}}_{K})\) by \(L_{I_K}A:={I_K}\circ A\), where \(A\in \mathcal {B}(V,{{\mathcal {H}}})\).
Therefore, if the coherent state map \({\mathfrak {K}}\) and the positive definite kernel K are related by (56), then
If \({\mathfrak {K}}_K=I_{K_1}\circ {\mathfrak {K}}_1=I_{K_2}\circ {\mathfrak {K}}_2\), then \({\mathfrak {K}}_2=L_{{I_{K_2}^{-1}}\circ I_{K_1}}{\mathfrak {K}}_1\), i.e. the positive definite kernel K defines the coherent state map \({\mathfrak {K}}\) up to an isomorphism of the Hilbert space \(\mathcal {H}\).
Now let us consider the tautological bundle \(\pi _{N}:{\mathbb {E}}_N\rightarrow \text{ Grass }(N,{{\mathcal {H}}})\) over the Grassmanian of N-dimensional subspaces of the Hilbert space \({{\mathcal {H}}}\). By definition the total space of this bundle is
and \(\pi _{N}:= pr_2|_{{\mathbb {E}}_N}\), where \(pr_2\) is the projection of \({{\mathcal {H}}}\times \text{ Grass }(N,{{\mathcal {H}}})\) on its second component.
The coherent state map \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) defines the following morphisms
of vector bundles, where the morphism \([K^{-1}]:\bar{{\mathbb {V}}}\rightarrow {\mathbb {V}}\) is defined by
The horizontal arrows on the right hand side of (63) are defined by
and by
Correctness of the above definitions follows from the defining properties (50)–(52) of the coherent state map \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\). The bundle morphism defined in (64) is an identity covering isomorphism of vector bundles with inverse given by \([K]:{\mathbb {V}}\rightarrow \bar{{\mathbb {V}}}\).
Restricting the scalar product \(\langle \cdot |\cdot \rangle \) of \({{\mathcal {H}}}\) to the fibers \(\pi _N^{-1}(q)\) of the tautological vector bundle \(\pi _{N}:{\mathbb {E}}_N\rightarrow \text{ Grass }(N,{{\mathcal {H}}})\) and using the vector bundles morphism (65) one defines
the Hermitian structure on the vector bundle \(\pi _{{\mathbb {V}}}:{\mathbb {V}}\rightarrow M\).
Proposition 2
The coherent state map \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) defines by
the metric (with respect to the Hermitian structure (66)) connection form \(\vartheta _{{\mathfrak {K}}}\in \Gamma ^\infty (T^*P\otimes T_eG)\) on the G-principal bundle \(P(G,\pi ,M)\).
Proof
From the equivariance property (50) one has
for \(g\in G\), so \(\vartheta _{{\mathfrak {K}}}\) satisfies (1).
In order to show the condition (2) we take \(X:=\frac{d}{dt}g(t)|_{t=0}\), where \(]-\varepsilon ,\varepsilon [\ni t\mapsto g(t)\in G\) is a smooth curve in G such that \(g(0)=e\), and substitute \(T\kappa _p(e)X\) into both sides of the definition (67). Thus we obtain
From (68) we see also that \(T_pP=\ker \vartheta _{{\mathfrak {K}}}(p)\oplus T^v_p P\) and \(\langle \vartheta _{{\mathfrak {K}}}(p),\xi (p)\rangle \in T_eG\) for \(\xi (p)\in T_pP\).
It follows from
where \(K(p,p)={\mathfrak {K}}(p)^*{\mathfrak {K}}(p)\), that \(\vartheta _{{\mathfrak {K}}}\) is a metric connection with respect to the Hermitian structure (66). In order to see this let us take \(f_1,f_2\in C_G^\infty (P,V)\). Then from (69) one obtains
Since arbitrary sections \(\psi _1,\psi _2\in \Gamma ^\infty (M,{\mathbb {V}})\) of the vector bundle \(\pi _{{\mathbb {V}}}:{\mathbb {V}}\rightarrow M\) can be written as \(\psi _i(\pi (p))=[(p,f_i(p))]\), \(i=1,2\), from (70) we obtain
where \(\nabla :\Gamma ^\infty (M,{\mathbb {V}})\rightarrow \Gamma ^\infty (M,T^*M\otimes {\mathbb {V}})\) is the covariant derivative defined in (25). \(\quad \square \)
After these preliminary considerations we propose a method of quantization of the generalized Hamiltonian flow \(\tau _t^{(X,F)}:P\rightarrow P\) tangent to the vector field \(\xi =(\pi ^*\times \nu ^*)^{-1}(X, F)\in \Gamma _{G,\vartheta }^\infty (TP)\), where \((X,F)\in {{\mathcal {P}}}^\infty _G(P,\vartheta )\).
Definition 3
A strongly continuos unitary representation \(U^{(X,F)}:({\mathbb {R}},+)\rightarrow (\text{ Aut }\;{{\mathcal {H}}},\circ )\) of the additive group \(({\mathbb {R}},+)\) is the positive kernel (coherent state) quantization of a generalized Hamiltonian flow \(\tau ^{(X,F)}:({\mathbb {R}},+)\rightarrow \text{ Aut }(P,\vartheta )\) if there exists a coherent state map \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) such that:
-
(i)
the connection form \(\vartheta \) is equal to \(\vartheta _{{\mathfrak {K}}}\) defined in (67);
-
(ii)
the equivariance property
$$\begin{aligned} {\mathfrak {K}}(\tau _t^{(X,F)}(p))=U^{(X,F)}_t{\mathfrak {K}}(p) \end{aligned}$$(71)for \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) is fulfilled with respect to both considered flows.
From the condition (iii), see (52) of the Definition 1 it follows that \(\tau _t^{(X,F)}\) defines \(U^{(X,F)}_t\) in an unique way.
Let us also note that if \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) is a one-to-one smooth map and the flow \(\tau _t\in \text{ Aut }(P)\) satisfies \({\mathfrak {K}}(\tau _t(p))=U_t{\mathfrak {K}}(p)\) for a certain unitary flow \(U_t\), then \(\tau _t\in \text{ Aut }(P,\vartheta _{{\mathfrak {K}}})\) and thus there exists \((X,F)\in \Gamma ^\infty (TM)\times C^\infty _{G,\vartheta _{{\mathfrak {K}}}}(P,T_eG)\) such that \(\tau _t=\tau _t^{(X,F)}\) and \(U_t=U^{(X,F)}_t\). Additionally (X, F) satisfies the generalized Hamilton Eq. (10) for the curvature form \(\varOmega _{{\mathfrak {K}}}\) defined by \(\vartheta _{{\mathfrak {K}}}\). If \(\varOmega _{{\mathfrak {K}}}\) is non-singular then the vector field \(X\in \Gamma ^\infty (TM)\) is uniquely defined by \(F\in C^\infty _G(P,T_eG)\).
Since for the non-singular curvature form \(\varOmega \) the Eq. (10) allows one to define \(X\in \Gamma ^\infty (TM)\) by \(F\in C^\infty _G(P,T_eG)\), so, in this case we will use the notation \(\tau ^F_t\) and \(U^F_t\) instead of \(\tau ^{(X,F)}_t\) and \(U^{(X,F)}_t\).
It follows from Stone’s Theorem, see e.g. [35], that there exists a self-adjoint operator \({\widehat{F}}\) with domain \({{\mathcal {D}}}_{{\widehat{F}}}\) dense in \({{\mathcal {H}}}\) such that \(U^{(X,F)}_t=e^{it{\widehat{F}}}\). From (71) we see that \({{\mathcal {D}}}_{{\mathfrak {K}}}\) defined in (62) is a \(U^{(X,F)}_t\)-invariant dense linear subspace of \({{\mathcal {H}}}\). From (71) it also follows that the functions \({\mathbb {R}}\ni t \mapsto U_t|\psi \rangle \in {{\mathcal {H}}}\), where \(|\psi \rangle \in {{\mathcal {D}}}_{{\mathfrak {K}}}\), are differentiable. So, see Theorem VII.11 in [35], \({{\mathcal {D}}}_{{\mathfrak {K}}}\) is an essential domain of the infinitesimal generator \({\widehat{F}}\) of \(U^{(X,F)}_t\).
Differentiating both sides of the Eq. (71) with respect to \(t\in {\mathbb {R}}\) we obtain
Note here that for any \(p\in P\) one has \(Ran\;{\mathfrak {K}}(p)\subset {{\mathcal {D}}}_{{\mathfrak {K}}}\subset {{\mathcal {D}}}_{{\widehat{F}}}\). We also note that the last equality in (72) follows from (19)–(20) and (71).
The symmetricity of \({\widehat{F}}\) on the domain \({{\mathcal {D}}}_{{\mathfrak {K}}}\) is equivalent to the equation
where \(H^*(X)\) is the horizontal lift of \(X\in \Gamma ^\infty (TM)\) with respect to \(\vartheta \).
From (72) and (67) we find that the mean values map \(\langle \cdot \rangle :{\widehat{F}}\mapsto \langle {\widehat{F}}\rangle \) defined by
is inverse to the quantization \(Q:F \mapsto {\widehat{F}}\) of the classical generator F of the Hamiltonian flow \(\tau _t^F\).
Using the isomorphism \(I_K:{{\mathcal {H}}}\rightarrow {{\mathcal {H}}}_{K}\) we can represent the quantum flow \(U^{(X,F)}_t\) and its generator \({\widehat{F}}\) in terms of the Hilbert space \({{\mathcal {H}}}_{K}\). Namely, we have
and
where \({\bar{\Sigma }}_t\) and \({\bar{Q}}_{(X,F)}\) are defined in (29) and (28), respectively.
Having in mind the Hilbert spaces isomorphism \(I_K:{{\mathcal {H}}}\rightarrow {{\mathcal {H}}}_{K}\), defined in (61), we see that \({{\mathcal {D}}}_{K}\) defined in (57) is a common essential domain of the self-adjoint operators \({\bar{Q}}_{(X,F)}\) being the generators of the flows \(U^{(X,F)}_t\).
According to Definition 3 the problem of quantization of the classical flow \({\mathbb {R}}\ni t\mapsto \tau ^{(X,F)}_t\in \text{ Aut }(P,\vartheta )\) which solves the generalized Hamilton Eq. (10) reduces to the finding of a coherent state map \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) satisfying the conditions \(\vartheta =\vartheta _{{\mathfrak {K}}}\) and (71). Taking into account that \(K:P\times P\rightarrow {{\mathcal {B}}}(V)\) defines \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) up to a unitary map \(U:{{\mathcal {H}}}\rightarrow {{\mathcal {H}}}\) it is reasonable to formulate these conditions in terms of the positive definite kernel:
Now, let us describe the quantization conditions (73) and (74) in terms of local representations \({\mathfrak {K}}_\alpha := {\mathfrak {K}}\circ s_\alpha :{{\mathcal {O}}}_\alpha \rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) and \(K_{{\bar{\alpha }}\beta }:={\mathfrak {K}}_\alpha ^*\circ {\mathfrak {K}}_\beta :{{\mathcal {O}}}_\alpha \times {{\mathcal {O}}}_\beta \rightarrow {{\mathcal {B}}}(V)\) of \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) and \(K:P\times P\rightarrow {{\mathcal {B}}}(V)\).
Proposition 3
-
(i)
The conditions (73) and (74) written in terms of a local representation \(s_\alpha :{{\mathcal {O}}}_\alpha \rightarrow P\) assume the following form
$$\begin{aligned} \vartheta _\alpha (m)=K_{{\bar{\alpha }}\alpha }(m,m)^{-1}d_nK_{{\bar{\alpha }}\alpha }(m,n)|_{n=m}, \end{aligned}$$(75)$$\begin{aligned} K_{{\bar{\alpha }}\beta }(m,n)= g_\alpha (t,m)^\dagger K_{{\bar{\alpha }}\beta }(\sigma ^{(X,F)}_t (m),\sigma ^{(X,F)}_t (n))g_\beta (t,n) \end{aligned}$$(76)where \(\vartheta _\alpha :=s_\alpha ^*\vartheta \), the local cocycle \(g_\alpha (t,m)\) and the flow \(\sigma ^{(X,F)}_t:M\rightarrow M\) are related by (31), if we put \(\tau _t=\tau ^{(X,F)}_t\) and \(\sigma _t=\sigma ^{(X,F)}_t\).
-
(ii)
The infinitesimal version of the condition (76) is the following
$$\begin{aligned} {{\mathcal {X}}}(K_{{\bar{\alpha }}\beta })(m,n)+ \phi _\alpha (m)K_{{\bar{\alpha }}\beta }(m,n)+K_{{\bar{\alpha }}\beta }(m,n) \phi _\beta (n)=0, \end{aligned}$$(77)where \({{\mathcal {X}}}(K_{{\bar{\alpha }}\beta })(m,n):=\frac{d}{dt}K_{{\bar{\alpha }}\beta }(\sigma ^{(X,F)}_t (m),\sigma ^{(X,F)}_t (n))|_{t=0}\), i.e. \({{\mathcal {X}}}(m,n)=(X(m),X(n))\), and \(\phi _\alpha (m)\), defined in (36), satisfy the equation
$$\begin{aligned} {{\mathcal {L}}}_X\vartheta _\alpha +d\phi _\alpha +[\vartheta _\alpha ,\phi _\alpha ]=0 \end{aligned}$$(78)equivalent to the Eq. (41).
Proof
- (i)
-
(ii)
One obtains (77) by differentiating of (76) with respect to the parameter \(t\in {\mathbb {R}}\) at \(t=0\). The condition (78) we obtain from (41) using (36), (42) and (43). \(\square \)
Let us also mention the transformation formulas
and
where \(g_{\alpha \beta }:{{\mathcal {O}}}_\alpha \cap {{\mathcal {O}}}_\beta \rightarrow G\) is the cocycle defined in (44) and \(m\in {{\mathcal {O}}}_\alpha \cap {{\mathcal {O}}}_\beta \), \(n\in {{\mathcal {O}}}_\gamma \cap {{\mathcal {O}}}_\delta \), between two local representations.
The domain \({{\mathcal {D}}}_{{\widehat{F}}}\) of the generator \({\widehat{F}}\) contains \({{\mathcal {D}}}_{{\mathfrak {K}}}\subset {{\mathcal {D}}}_{{\widehat{F}}}\) the dense subset \({{\mathcal {D}}}_{{\mathfrak {K}}}\) defined in (62) which consists of \(|\psi \rangle =\sum _{j\in J} {\mathfrak {K}}_{\beta _j}(n_j)v_j\in {{\mathcal {D}}}_{{\widehat{F}}}\), where \(n_j\in {{\mathcal {O}}}_{\beta _j}\), \(v_j\in V\) and J is finite. So, the local version of the formula (72) is the following
The domains \({{\mathcal {D}}}(Q_{(X,F)})\) and \({{\mathcal {D}}}({\bar{Q}}_{(X,F)})\) of the Kirillov–Kostant–Souriau operators \(Q_{(X,F)}:{{\mathcal {D}}}(Q_{(X,F)})\rightarrow \Gamma ^\infty (M,{\mathbb {V}})\) and \({\bar{Q}}_{(X,F)}:{{\mathcal {D}}}({\bar{Q}}_{(X,F)})\rightarrow \Gamma ^\infty (M,\bar{{\mathbb {V}}})\) defined in (26) and (28), respectively, in the local representation consists of the vectors \(\psi =\sum _{j\in J} K_{{\bar{\alpha }}_j\beta }(m_j,\cdot )v_j \in {{\mathcal {D}}}(Q_{(X,F)})\) and \({\bar{\psi }}=\sum _{j\in J} K_{{\bar{\beta }} \alpha _j}(\cdot ,n_j)v_j\in {{\mathcal {D}}}({\bar{Q}}_{(X,F)})\).
The local version of (26) and (28) are
and
Summing up the above facts, we see that the problem of quantization of classical flow \(\tau ^{(X,F)}_t\) described by the generalized Hamilton Eq. (10) (in local representation by the Eqs. (41) or by (78)) is reduced to the problem of finding a solution \(K_{{\bar{\alpha }}\beta }:{{\mathcal {O}}}_\alpha \times {{\mathcal {O}}}_\beta \rightarrow {{\mathcal {B}}}(V)\) of the Eqs. (75) and (77), where X, \(F_\alpha \) and \(\vartheta _\alpha \) are fixed and satisfy (78). In general this is a rather hard task. However, it is possible to do this for some particular cases. For this reason see Sect. 6.
5 Extension and Reduction
It turns out that if \(G\subset GL(V,{\mathbb {C}})\), then having a principal bundle \(P(G,\pi ,M)\) and a coherent state map \({\mathfrak {K}}: P \rightarrow B(V, \mathcal {H})\) one can define in a canonical way two other principal bundles \({\widetilde{P}}(GL(V, {\mathbb {C}}), {\widetilde{\pi }}, M)\) and \(U(U(V), \pi ^u, M)\) over M with the structural groups \(GL(V,{\mathbb {C}})\) and U(N), respectively. Moreover the coherent state method of quantization of the generalized Hamiltonian flows on \(P(G,\pi ,M)\) extends uniquely to each of these principal bundles giving the same quantum flows as in the case of \(P(G,\pi ,M)\).
Indeed, since G is a Lie subgroup of \(GL(V,{\mathbb {C}})\) one can define the \(GL(V,{\mathbb {C}})\)-principal bundle \({\widetilde{P}}(GL(V,{\mathbb {C}}),{\widetilde{\pi }},M)\) over M in the following way:
-
(a)
the total space \({\widetilde{P}}\) is the quotient \({\widetilde{P}}:=(P\times GL(V,{\mathbb {C}}))/G\) defined by the action
$$\begin{aligned} {\widetilde{\varPhi }}_g: P\times GL(V,{\mathbb {C}})\ni (p,{\widetilde{g}})\mapsto (pg,g^{-1}{\widetilde{g}})\in P\times GL(V,{\mathbb {C}}) \end{aligned}$$(82)of G on the product \(P\times GL(V,{\mathbb {C}})\);
-
(b)
the bundle projection \({\widetilde{\pi }}:{\widetilde{P}}\rightarrow M\) is defined by
$$\begin{aligned} \pi ([(p,{\widetilde{g}})]):=\pi (p), \end{aligned}$$where \([(p,{\widetilde{g}})]:=\{(pg,g^{-1}{\widetilde{g}}): g\in G \}\in {\widetilde{P}}\);
-
(c)
the right action \({\widetilde{\kappa }}:{\widetilde{P}}\times GL(V,{\mathbb {C}})\rightarrow {\widetilde{P}}\) of \(GL(V,{\mathbb {C}})\) on \({\widetilde{P}}\) one defines by
$$\begin{aligned} {\widetilde{\kappa }}_{{\widetilde{h}}}([(p,{\widetilde{g}})])=[(p,{\widetilde{g}})]{\widetilde{h}}:=[(p,{\widetilde{g}}{\widetilde{h}})]. \end{aligned}$$
One has a natural principal bundles morphism \(E:P\rightarrow {\widetilde{P}}\) defined by
which covers the identity map \(id:M\rightarrow M\) of M. Note here that
for \(g\in G\subset GL(V,{\mathbb {C}})\). Hence, according to the Proposition 6.1 in Chapter II of [20], there exists uniquely defined connection \({\widetilde{H}}^*: T_{{\widetilde{\pi }}({\widetilde{p}})}M\rightarrow T_{{\widetilde{p}}}{\widetilde{P}}\) which in the case considered here is related to the connection \(H^*: T_{\pi (p)}M\rightarrow T_{p}P\) on P by
where \({\widetilde{p}}=[(p,{\widetilde{g}})]\in {\widetilde{P}}\).
The connection form \({\widetilde{\vartheta }}\in \Gamma ^\infty ({\widetilde{P}}, T^*{\widetilde{P}}\otimes {{\mathcal {B}}}(V))\) corresponding to (83) is the following
Having the principal bundles morphism \(E:P\rightarrow {\widetilde{P}}\) we can extend a G-equivariant coherent state map \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) as well as an automorphism \(\tau \in \text{ Aut }(P)\) of \(P(G,\pi ,M)\) to the ones defined on \({\widetilde{P}}\). These extensions are defined as follows
The correctness of the above definitions including their independence on the choice of representative \((pg,g^{-1}{\widetilde{g}})\in [(p,{\widetilde{g}})]\), where \(g\in G\), one can check easily.
If we extend the map \(F:P\rightarrow T_eG\), defined in (6), to the map \({\widetilde{F}}:{\widetilde{P}}\rightarrow {{\mathcal {B}}}(V)\) by
then the following relations
are fulfilled, where \(\widetilde{\mathbf{D }}\) and \({\widetilde{\varOmega }}\) are the covariant derivative and the curvature form corresponding to \({\widetilde{\vartheta }}\).
The above allows us to formulate the following proposition.
Proposition 4
A generalized Hamiltonian flow \(\tau ^{(X,F)}_t\in \text{ Aut }(P,\vartheta )\) extends by (84) to the flow \({\widetilde{\tau }}^{(X,F)}_t\in \text{ Aut }({\widetilde{P}},{\widetilde{\vartheta }})\) and one has the equality \({\widetilde{\tau }}^{(X,F)}_t=\tau ^{(X,{\widetilde{F}})}_t\) of the flows, where \({\widetilde{F}}:{\widetilde{P}}\rightarrow {{\mathcal {B}}}(V)\) is defined in (85), i.e. the extension \({\widetilde{\tau }}^{(X,F)}_t\) is a generalized Hamiltonian flow \(\tau ^{(X,{\widetilde{F}})}_t\) on \({\widetilde{P}}\).
Proof
It follows from (86) and (87) that \((X,{\widetilde{F}})\) satisfies equation
if and only if (X, F) satisfies Eq. (10). \(\quad \square \)
In order to define the principal bundle \(U(U(V),\pi ^u,M)\) let us note that having a G-equivariant coherent state map 
we can define the map 
where \(K(p,p)={\mathfrak {K}}(p)^*{\mathfrak {K}}(p)\). This map intertwines
the action (82) with the action
where the U(V)-valued cocycle \(c:P\times G\rightarrow U(V)\) is defined by
The cocycle property \(c(p,g)c(pg,h)=c(p,gh)\) and the unitary property \(c(p,g)^\dag c(p,g)=1\!\!1_V\) for \(c:P\times G\rightarrow U(V)\) follow from (53).
From (89) we see that \(\varPhi _K\) defines an automorphism \([\varPhi _K ]:{\widetilde{P}}\rightarrow {\widetilde{P}}\) of the principal bundle \({\widetilde{P}}(GL(V,{\mathbb {C}}),{\widetilde{\pi }},M)\) which covers the identity map of the base M. The inverse \([\varPhi _K ]^{-1}=[\varPhi _{K^{-1}}]\) of this automorphism transforms \({\widetilde{\vartheta }}\) and \({\widetilde{F}}\) in the following way
and
respectively.
It follows from \(c(p,g)\in U(V)\) that the submanifold \(P\times U(V)\subset P\times GL(V,{\mathbb {C}})\) is invariant with respect to the action (90). Therefore, one can consider the quotient manifold \(U:=(P\times U(V))/G\) as the total space of a U(V)-principal bundle \(U(U(V),\pi ^u,M)\) over M with bundle projection map \(\pi ^u:U\rightarrow M\) and the right action \(\kappa ^u: U\times U(V)\rightarrow U\) of U(V) defined as follows
Let us note here that \([(p,{\widetilde{c}})]\in (P\times U(V))/G\) is defined by \([(p,{\widetilde{c}})]{:=}\{(pg,c(p,g)^{-1}{\widetilde{c}}): g\in G \}\).
Taking the Lie algebra \({{\mathfrak {U}}}(V):=\{Y\in {{\mathcal {B}}}(V): \langle Yv,w\rangle +\langle v,Yw\rangle =0 \;\text{ for }\;v,w\in V \}\) of U(V) and the real vector space \({{\mathfrak {H}}}(V)\) of the Hermitian \(\langle Hv,w\rangle =\langle v,Hw\rangle \) endomorphisms \(H\in {{\mathcal {B}}}(V)\) one obtains the Ad(U(V))-invariant splitting \({{\mathcal {B}}}(V)={{\mathfrak {U}}}(V)\oplus {{\mathfrak {H}}}(V)\) of \({{\mathcal {B}}}(V)\). So, according to Proposition 6.4 of Chapter II in [20], the anti-Hermitian part of \(\vartheta ^u\) after restriction \(\vartheta ^a:=\frac{1}{2}(\vartheta ^u-(\vartheta ^u)^\dag )|_U\) to \(U\subset {\widetilde{P}}\) defines a connection form on the U(V)-principal bundle \(U(U(V),\pi ^u,M)\). Restricting \(F^a:=\frac{1}{2}(F^u-(F^u)^\dag )|_U\) to \(U\subset {\widetilde{P}}\) one obtains a \({{\mathfrak {U}}}(V)\)-valued U(V)-equivariant function on the total space of \(U(U(V),\pi ^u,M)\).
Therefore, by fixing a coherent state map \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) on the total space of principal bundle \(P(G,\pi ,M)\) we reduce a generalized Hamilton system on \({\widetilde{P}}\) described by (88) to the one defined on U by \(\vartheta ^a\) and \(F^a\). Let us stress here that this reduction depends on the choice of \({\mathfrak {K}}:P\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) in a canonical way.
Now, we discuss the problem of quantization of the generalized Hamiltonian flows \(\tau ^{(X,F)}_t\), \(\tau ^{(X,{\widetilde{F}})}_t\) and \(\tau ^{(X,F^a)}_t\) on P, \({\widetilde{P}}\) and U, respectively, which are reciprocally related.
We collect the main facts under our consideration in the subsequent proposition.
Proposition 5
If a flow \(\tau ^{(X,F)}_t\in \text{ Aut }(P,\vartheta )\) is quantized in the sense of Definition 3, then:
-
(i)
The flow \(\tau ^{(X,{\widetilde{F}})}_t\) is also quantized in sense of Definition 3, i.e. one has \({\widetilde{\vartheta }}=\vartheta _{\widetilde{{\mathfrak {K}}}}\) and \({\widetilde{U}}^{(X,F)}_t\widetilde{{\mathfrak {K}}}([(p,{\widetilde{g}})])=\widetilde{{\mathfrak {K}}}(\tau ^{(X,{\widetilde{F}})}[(p,{\widetilde{g}})])\). Additionally one has the equality \({\widetilde{U}}^{(X,F)}_t=U^{(X,F)}_t\) of the quantum flows.
-
(ii)
The total space U of \(U(U(V),\pi ^u,M)\) is invariant with respect to \(\tau ^{(X,{\widetilde{F}})}_t\in \text{ Aut }({\widetilde{P}},{\widetilde{\vartheta }})\) and \(\tau ^{(X,{\widetilde{F}})}_t|_U=\tau ^{(X,F^a)}_t\in \text{ Aut }(U,\vartheta ^a)\). The connection form \(\vartheta ^a\) satisfies
$$\begin{aligned} \vartheta ^a=\vartheta ^a_{{\mathfrak {a}}}:={\mathfrak {a}}^*d{\mathfrak {a}} \end{aligned}$$(92)the quantization condition (i) of Definition 3, for the coherent state map \({\mathfrak {a}}:U\rightarrow \mathcal {U}(V,{{\mathcal {H}}})\subset \mathcal {B}(V,{{\mathcal {H}}})\) defined by
$$\begin{aligned} {\mathfrak {a}}([(p,{\widetilde{c}})]):={\mathfrak {K}}(p)K(p,p)^{-\frac{1}{2}}{\widetilde{c}}, \end{aligned}$$(93)where \(\mathcal {U}(V,{{\mathcal {H}}})\) is the set of isometries of Hilbert space V into the Hilbert space \({{\mathcal {H}}}\).
-
(iii)
The generalized Hamiltonian flow \(\tau ^{(X,F^a)}_t\in \text{ Aut }(U,\vartheta ^a)\) is quantized in the sense of Definition 3 and the quantum flow \(U^{(X,F^a)}_t\) corresponding to it is equal \(U^{(X,F^a)}_t=U^{(X,F)}_t\) to the quantum flow \(U^{(X,F)}_t\).
Proof
-
(i)
From \(\vartheta =\vartheta _{{\mathfrak {K}}}\) we have
$$\begin{aligned} {\widetilde{\vartheta }}([(p,{\widetilde{g}})])= & {} {\widetilde{g}}^{-1} ({\mathfrak {K}}(p)^*{\mathfrak {K}}(p))^{-1}{\mathfrak {K}}(p)^*d{\mathfrak {K}}(p){\widetilde{g}}+ {\widetilde{g}}^{-1}d{\widetilde{g}}\\= & {} (\widetilde{{\mathfrak {K}}}([(p,{\widetilde{g}})])^*\widetilde{{\mathfrak {K}}}([(p,{\widetilde{g}})]))^{-1} \widetilde{{\mathfrak {K}}}([(p,{\widetilde{g}})])^*d\widetilde{{\mathfrak {K}}}([(p,{\widetilde{g}})])= \vartheta _{\widetilde{{\mathfrak {K}}}}([(p,{\widetilde{g}})]). \end{aligned}$$Next, from (71) and (84) taken for \(\tau =\tau ^{(X,F)}_t\) we obtain
$$\begin{aligned} U_t\widetilde{{\mathfrak {K}}}([(p,{\widetilde{g}})])=U_t{\mathfrak {K}}(p){\widetilde{g}}= {\mathfrak {K}}(\tau ^{(X,F)}_t(p)){\widetilde{g}}= \widetilde{{\mathfrak {K}}}(\tau ^{(X,{\widetilde{F}})}_t([(p,{\widetilde{g}})])). \end{aligned}$$ -
(ii)
The quantization property for \(\tau ^{(X,F)}_t\) implies that
$$\begin{aligned} K(\tau ^{(X,F)}_t(p),\tau ^{(X,F)}_t(p))=K(p,p). \end{aligned}$$(94)This condition is equivalent to the following one
$$\begin{aligned} \varPhi _K\circ (\tau ^{(X,F)}_t\times id)=(\tau ^{(X,F)}_t\times id)\circ \varPhi _K. \end{aligned}$$Since the flow \(\tau ^{(X,F)}_t\times id:P\times GL(V,{\mathbb {C}})\rightarrow P\times GL(V,{\mathbb {C}})\) also commutes with the action \(\varPhi ^u:P\times GL(V,{\mathbb {C}})\times G\rightarrow P\times GL(V,{\mathbb {C}})\) of G and \(\varPhi ^u_g(P\times U(V))\subset P\times U(V)\) we find that \(U\subset {\widetilde{P}}\) is invariant with respect to the flow \(\tau ^{(X,{\widetilde{F}})}_t\). From (94) one obtains that \({\mathfrak {a}}:U\rightarrow \mathcal {U}(V,{{\mathcal {H}}})\) defined in (93) satisfies
$$\begin{aligned} U_t{\mathfrak {a}}([(p,{\widetilde{c}})])={\mathfrak {a}}(\tau ^{(X,{\widetilde{F}})}([(p,{\widetilde{c}})])). \end{aligned}$$(95)Restricting \(\vartheta ^u=[\varPhi _{K^{-1}}]^*{\widetilde{\vartheta }}= [\varPhi _{K^{-1}}]^*\vartheta _{\widetilde{{\mathfrak {K}}}}\) to \(U\subset {\widetilde{P}}\) and using (91) one obtains (92).
-
(iii)
The quantization of \(\tau ^{(X,F^a)}_t=\tau ^{(X,{\widetilde{F}})}_t|_U\) as well as the equality \(U_t=U^u_t\) follows from (92) and (95).\(\square \)
According to (62) the essential domains of the generators \(\widehat{{\widetilde{F}}}:{{\mathcal {D}}}_{\widetilde{{\mathfrak {K}}}}\rightarrow {{\mathcal {H}}}\) and \(\widehat{F^a}:{{\mathcal {D}}}_{{\mathfrak {a}}}\rightarrow {{\mathcal {H}}}\) of flows \(U_t=e^{it\widehat{{\widetilde{F}}}}\) and \(U_t=e^{it\widehat{F^a}}\) are the following
and
where \(\mathcal {F}\) is a finite subset of \({\mathbb {Z}}\).
Since \(\widetilde{{\mathfrak {K}}}({\widetilde{p}}_i)v_i={\mathfrak {K}}(p_i){\widetilde{g}}_iv_i\), \({\mathfrak {a}}({\widetilde{p}}_i)v_i={\mathfrak {K}}(p_i)K(p_i,p_i)^{-\frac{1}{2}}{\widetilde{c}}_iv_i\) and \(v_i\in V\) are chosen in an arbitrary way we obtain that \({{\mathcal {D}}}_{\widetilde{{\mathfrak {K}}}}={{\mathcal {D}}}_{{\mathfrak {a}}}={{\mathcal {D}}}_{{\mathfrak {K}}}\). So, we have also the equalities \(\widehat{{\widetilde{F}}}=\widehat{F^a}={\widehat{F}}\) for the generators. However, the formula (72) taken for \(\widehat{{\widetilde{F}}}\) and \(\widehat{F^a}\) is different from the one for \({\widehat{F}}\). Namely, we have
where \({\widetilde{p}}=[(p,{\widetilde{g}})]\in {\widetilde{P}}\), and
where \({\widetilde{p}}=[(p,{\widetilde{c}})]\in U\).
From (96) and (97) we easily see that the mean value functions \(\langle \widehat{{\widetilde{F}}}\rangle \) and \(\langle \widehat{F^a}\rangle \) for these generators on the coherent states are equal
to the generators \({\widetilde{F}}\) and \(F^a\) of the Hamiltonian flows \(\tau ^{(X,{\widetilde{F}})}_t\) and \(\tau ^{(X,F^a)}_t\), respectively.
Taking the positive kernels
where \({\widetilde{p}}=[(p,{\widetilde{g}})],{\widetilde{q}}=[(q,{\widetilde{h}})]\in {\widetilde{P}}\) and
where \({\widetilde{p}}=[(p,{\widetilde{c}})],{\widetilde{q}}=[(q,{\widetilde{b}})]\in U\), we obtain the domains \({{\mathcal {D}}}_{{\widetilde{K}}}\subset C^\infty _{\overline{GL(V,{\mathbb {C}})}}({\widetilde{P}},V)\cong \Gamma ^\infty (M,\bar{{\mathbb {V}}})\) and \({{\mathcal {D}}}_{A}\subset C^\infty _{\overline{U(V)}}(U,V)\cong \Gamma ^\infty (M,\bar{{\mathbb {V}}})\), see (57), of the corresponding Kirillov–Kostant–Souriau operators \({\bar{Q}}_{(X,{\widetilde{F}})}:{{\mathcal {D}}}_{{\widetilde{K}}}\rightarrow \Gamma ^\infty (M,\bar{{\mathbb {V}}})\) and \({\bar{Q}}_{(X,F^a)}:{{\mathcal {D}}}_{A}\rightarrow \Gamma ^\infty (M,\bar{{\mathbb {V}}})\).
Summing up the above considerations we conclude that there are three equivalent ways of quantizing of the flow \(\sigma ^{(X,F)}_t\) by the positive kernel method based on the principal bundles \(P(G,\pi ,M)\), \({\widetilde{P}}(GL(V,{\mathbb {C}}),{\widetilde{\pi }},M)\) and \(U(U(V),\pi ^u,M)\) over M, respectively. The choice of one of these ways depends on the physical as well as mathematical aspects of the model under investigation.
For example, the quantization based on \(U(U(V),\pi ^u,M)\) is directly related to interpretation of the positive kernel \(A:U\times U\rightarrow {{\mathcal {B}}}(V)\) as the matrix valued transition amplitude kernel. More precisely, let us take such \(v,w\in V\) that \(\Vert v\Vert =\Vert w\Vert =1\). Then the vectors \({\mathfrak {a}}({\widetilde{p}})v,{\mathfrak {a}}({\widetilde{q}})w\in {{\mathcal {H}}}\) have norm equal to 1 also, i.e. they describe pure states of the system and the transition amplitude between them is \(\langle {\mathfrak {a}}({\widetilde{p}})v| {\mathfrak {a}}({\widetilde{q}})w\rangle =\langle v|A({\widetilde{p}},{\widetilde{q}})w \rangle \). So, one can interpret \(A({\widetilde{p}},{\widetilde{q}})\) as the transition amplitude matrix between the states \({\widetilde{p}}\) and \({\widetilde{q}}\). For more exhaustive discussion of these physical aspects we address to [25, 26].
Ending, let us mention that if the base M of the principal bundle \(P(G,\pi ,M)\) is a complex analytic manifold then it is resonable to use an approach based on the principal bundle \({\widetilde{P}}(GL(V,{\mathbb {C}}),{\widetilde{\pi }},M)\). As an example of such a situation see Sect. 6.
6 Quantization of Holomorphic Flows on Non-compact Riemann Surfaces
In this section we will apply the method of quantization presented in Sect. 4 to the case when \(P(GL(V,{\mathbb {C}}),\pi ,M)\) is a holomorphic \(GL(V, {\mathbb {C}})\)-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 Eqs. (75), (77) on the kernel \(K_{{\bar{\alpha }}\beta }:{{\mathcal {O}}}_\alpha \times {{\mathcal {O}}}_\beta \rightarrow {{\mathcal {B}}}(V)\) under the assumption that \((X,F_\alpha )\in {{\mathcal {P}}}^\infty _G (P, \vartheta )\). The above assumption, as was shown in Sect. 4, allows us to quantize the flow \(\tau ^{(X, F)}_t \in \text{ Aut }(P, \vartheta )\), using the kernel \(K_{{\bar{\alpha }}\beta }\) 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 [16, 38]. We omit here the subcase when M is a compact Riemann surface, since then the Hilbert space \(\mathcal {H}\) 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 [14, 30,31,32].
Using the invariants of the flows \(\tau ^{(X, F)}_t \in \text{ Aut }(P, \vartheta )\) and the appropriate gauge transformation, we will reduce the Eqs. (75) and (77) to the linear ordinary differential Eq. (129), which is solvable for \((X, F) \in {{\mathcal {P}}}^\infty _G(P, \vartheta )\). The solutions of (129) are presented through the formula (126) in Proposition 9 and Proposition 10. We will also obtain the integral decompositions (151) of the positive kernels \( K_{{\bar{\beta }}\beta } ({\bar{v}}, z)\) invariant with respect to the flows \(\tau ^{(X, F)}_t\) on the positive kernels \(K_{{\bar{\beta }}\beta } ({\bar{v}}, z;\lambda )\) presented in Proposition 13. The relationship between the \(\mathcal {B}(V)\)-valued measures \(d(\Gamma _0^*E\Gamma _0)(\lambda ) \) used for these decompositions and the spectral measure of the generator \({\hat{F}}\) of the quantum flow \(U_t^{(X, F)}= e^{it{\hat{F}}}\) is described as well.
One proves, eg. see Section III par. 30 in [9], that for a non-compact Riemann surface M one has \(H^1(M,GL(N,{{\mathcal {O}}}))=0\), i.e. any \(GL(N,{{\mathcal {O}}})\)-valued holomorphic transition 1-cocycle \((g_{\alpha \beta })\in Z^1(\{{{\mathcal {O}}}\}_{\alpha \in I},GL(N,{{\mathcal {O}}}))\) is solvable \(g_{\alpha \beta }=\delta _\alpha \delta _\beta ^{-1}\), where the holomorphic maps \(\delta _\alpha :{{\mathcal {O}}}_\alpha \rightarrow GL(N,{\mathbb {C}})\), \(\alpha \in I\), represent a holomorphic 0-cocycle. Therefore, all holomorphic vector bundles \({\mathbb {V}}\rightarrow M\) as well as the holomorphic principal bundles \(P(GL(V,{\mathbb {C}}),\pi ,M)\) over M are trivial.
Being in the framework of the above category we will quantize the holomorphic flows \(\tau ^{(X,F)}_t\in \text{ Aut }(P,\vartheta )\) only. We will assume also that the coherent state map \({\mathfrak {K}}:M\times GL(V,{\mathbb {C}})\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) is a holomorphic map.
The existence of a holomorphic flow \(\sigma _t^X=\pi (\tau ^{(X,F)}_t)\in \text{ Aut }(M)\) on a Riemann surface M radically restricts the class of non-compact Riemann surfaces with this property. Namely, see eg. [10], one proves that any non-compact Riemann surface M which admits a non-discrete group of automorphisms is biholomorphic to the one listed below:
-
(i)
the Gauss plane \({\mathbb {C}}=\overline{{\mathbb {C}}}\setminus \{\infty \}\),
-
(ii)
the punctured Gauss plane \({\mathbb {C}}^*:= \overline{{\mathbb {C}}}\setminus \{\infty ,0\}\),
-
(iii)
the unit disc \({\mathbb {D}}:= \{z\in {\mathbb {C}}:|z|<1\}\),
-
(iv)
the punctured unit disc \({\mathbb {D}}^*:= {\mathbb {D}}\setminus \{0\}\),
-
(v)
an annulus \({\mathbb {A}}_r:=\{z\in {\mathbb {C}}:r<|z|<1\}\), where \(0<r<1\).
Since the groups \(\text{ Aut }(M)\) of automorphisms of \(M={\mathbb {C}}, {\mathbb {C}}^*,{\mathbb {D}}, {\mathbb {D}}^*, {\mathbb {A}}_r\subset \overline{{\mathbb {C}}}\cong {\mathbb {C}}{\mathbb {P}}(1)\) can be considered as the subgroups of \(\text{ Aut }({\mathbb {C}}{\mathbb {P}}(1))\cong SL(2,{\mathbb {C}})/{\mathbb {Z}}_2\) we find that:
-
(i)
\(\text{ Aut }({\mathbb {C}})=\{z\mapsto \alpha z+\beta : \alpha \in {\mathbb {C}}^*, \beta \in {\mathbb {C}}\}\cong {\mathbb {C}}^*\ltimes {\mathbb {C}}\),
-
(ii)
\(\text{ Aut }({\mathbb {C}}^*)=\left\{ z\mapsto \alpha z\;\text{ or }\;z\mapsto \frac{\alpha }{z}: \alpha \in {\mathbb {C}}^*\right\} \cong {\mathbb {Z}}_2\ltimes {\mathbb {C}}^*\),
-
(iii)
\(\text{ Aut }({\mathbb {D}})=\left\{ z\mapsto \frac{\alpha z+\beta }{{\overline{\beta }}z+{\overline{\alpha }}}: |\alpha |^2-|\beta |^2=1, \alpha ,\beta \in {\mathbb {C}}\right\} \cong SU(1,1)/{\mathbb {Z}}_2\),
-
(iv)
\(\text{ Aut }({\mathbb {D}}^*)=\left\{ z\mapsto \alpha z: \alpha \in S^1 \right\} \cong S^1\),
-
(v)
\(\text{ Aut }({\mathbb {A}}_r)=\left\{ z\mapsto \alpha z\; \text{ or }\;z\mapsto r\frac{\alpha }{z}: \alpha \in S^1\right\} \cong {\mathbb {Z}}_2\ltimes S^1\).
For all these cases M is a circularly symmetric open subset in \({\mathbb {C}}\). So, the inclusion map \(M\ni m\mapsto \iota (m)=:z\in {\mathbb {C}}\) defines the global chart which is common for all Riemann surfaces considered here. Hence, the vector field \(X\in \Gamma ^\infty (TM)\) tangent to a holomorphic flow \(\sigma _t^X\in \text{ Aut }(M)\) is given by
where w(z) is a second order polynomial
with coefficients satisfying the following conditions
-
(i)
\(c=b=0\) if \(M={\mathbb {C}}^*,{\mathbb {D}}^*, {\mathbb {A}}_r\),
-
(ii)
\(c=0\) if \(M={\mathbb {C}}\),
-
(iii)
\(c=-{\overline{b}}\) and \(a=2i\omega \), \(\omega \in {\mathbb {R}}\), if \(M={\mathbb {D}}\).
From the above conditions we see that if \(b=0\) then one also has \(c=0\) in (99).
Taking into account the above facts we present below the list of possible holomorphic flows \(\sigma _t^X\) on M.
Proposition 6
The following holomorphic flows are possible:
-
(i)
for \(M={\mathbb {C}}^*,{\mathbb {D}}^*, {\mathbb {A}}_r\) one has
$$\begin{aligned} \sigma _t^X(z)=e^{a t}z, \end{aligned}$$(100)where \(a\in {\mathbb {C}}\setminus \{0\}\) for \({\mathbb {C}}^*\) and \(a=i\omega \), \(\omega \in {\mathbb {R}}\setminus \{0\}\), for \({\mathbb {D}}^*\) and \({\mathbb {A}}_r\);
-
(ii)
for \(M={\mathbb {C}}\) one has
$$\begin{aligned} \sigma _t^X(z)=e^{a t}z+\frac{b}{a}(e^{a t}-1), \end{aligned}$$(101)where \(a\in {\mathbb {C}}\setminus \{0\}\). Note here that for \(a=0\) the formula (101) reduces to
$$\begin{aligned} \sigma _t^X(z)=z+b t; \end{aligned}$$(102) -
(iii)
for \(M={\mathbb {D}}\) one has
$$\begin{aligned} \sigma _t^X(z)=\frac{z(\varrho \cosh \varrho t + i\omega \sinh \varrho t) +b\sinh \varrho t}{z{\overline{b}}\sinh \varrho t + \varrho \cosh \varrho t - i\omega \sinh \varrho t}, \end{aligned}$$(103)if \(\varrho :=\sqrt{-\omega ^2+|b|^2}\ne 0\) and
$$\begin{aligned} \sigma _t^X(z)=\frac{z(1 + i\omega t) +b t}{z{\overline{b}}t + 1 - i\omega t}, \end{aligned}$$(104)if \(\varrho =0\). For the formula (103) it is reasonable to distinguish the following two subcases \(\varrho \in {\mathbb {R}}\setminus \{0\}\) \((-\omega ^2+|b|^2>0)\) and \(\varrho \in i{\mathbb {R}}\setminus \{0\}\) \((-\omega ^2+|b|^2<0)\).
Remark 1
One can consider the flows (100–104) as a one-parameter subgroups of the Möbius group and in accordance with the standard classification of its elements we obtain:
-
(i)
The flows (100) are elliptic for \(a= i\omega \in i {\mathbb {R}}\) and loxodromic in all other cases.
-
(ii)
The flows (101) are parabolic for \(b\ne 0\) and for \(b=0\) we obtain (i).
-
(iii)
The flows (102) are parabolic.
-
(iv)
The flows (103) are hyperbolic for \(\rho ^2>0\) and elliptic for \(\rho ^2<0\).
-
(v)
The flows (104) are parabolic.
In order to quantize the flows \(\tau ^{(X,F)}_t\) whose projections \(\pi (\tau ^{(X,F)}_t)=\sigma _t^X\) on M are listed in (100)–(104), we recall that for every non-compact Riemann surface M the holomorphic principal bundle \(P(GL(V,{\mathbb {C}}),\pi ,M)\) is trivial. So, there exist a holomorphic section \(s_\alpha :M\rightarrow P\) and the corresponding trivialization \({\mathfrak {K}}_\alpha :M\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) of the coherent state map which are defined on the whole of M. The Eqs. (78), (75) and (77) in this trivialization assume the following forms
respectively. Note here that the positive kernel \(K_{{\bar{\alpha }}\alpha }({\bar{z}},z):={\mathfrak {K}}_\alpha (z)^*{\mathfrak {K}}_\alpha (z)\) is anti-holomorphic in the first variable and holomorphic in the second one.
The relation (36) between the classical data \(\vartheta _\alpha \), \(F_\alpha \), X in this case is the following
Substituting \(\vartheta _\alpha ({\bar{z}},z)\) given by (106), into (108) we obtain the equation
on the kernel \(K_{{\bar{\alpha }}\alpha }:M\times M\rightarrow {{\mathcal {B}}}(V)\) complementary to the Eq. (107).
Remark 2
Summarizing, we see that the flow \(\tau ^{(X,F)}_t\in \text{ Aut }(P,\vartheta )\) can be quantizable iff the kernel \(K_{{\bar{\alpha }}\alpha }\) satisfies (106), (107) and (109) for given classical data \(\vartheta _\alpha \), \(F_\alpha \), w.
For simplification of the Eqs. (107) and (109) we make use of the gauge transformations listed in (44)–(49), which in this case are given by
We see from (110) that if the equation
has a holomorphic solution \(g_{\alpha \beta }:M\rightarrow GL(V,{\mathbb {C}})\) for the given \(\phi _\alpha \) and w, then there exists a holomorphic section \(s_{\beta }:M\rightarrow P\) such that the Eqs. (107) and (109) reduce to the following ones
and Eq. (105) reduces to
The existence of a holomorphic solution \(g_{\alpha \beta }\in {{\mathcal {O}}}(M, GL(V,{\mathbb {C}}))\) of the Eq. (111) on M depends on \(\frac{1}{w}\phi _\alpha \) which is a holomorphic function at least on the domain \(M_0:=M\setminus \{z_1,z_2\}\), where \(z_1,z_2\in M\) are the roots of the polynomial w (the cases \(z_1=z_2\) or \(\{z_1,z_2\}=\emptyset \) are admissible also).
For the cases mentioned in Proposition 6 one has
-
(i)
if \(M={\mathbb {C}}^*,{\mathbb {D}}^*, {\mathbb {A}}_r\) then \(M_0=M\);
-
(ii)
if \(M={\mathbb {C}}\) then \(M_0={\mathbb {C}}\setminus \{-\frac{b}{a} \}\) for \(a\ne 0\) and \(M_0= {\mathbb {C}}\) for \(a=0\);
-
(iii)
if \(M={\mathbb {D}}\) then \(M_0={\mathbb {D}}\setminus \{z_1 \}\) for \(|b|^2-\omega ^2<0\) and \(M_0={\mathbb {D}}\) for \(|b|^2-\omega ^2\ge 0\).
From the above we have:
-
(a)
for the cases (ii) and (iii) if the function \(\frac{1}{w}\phi _\alpha \) extends as a holomorphic function to \({\mathbb {C}}\) and to \({\mathbb {D}}\), respectively, then (111) has a holomorphic solution;
-
(b)
for the remaining cases we note that the pull-back of the Eq. (111) on the universal covering \(\widehat{M_0}\) of \(M_0\) always has a holomorphic solution and if this solution is invariant with respect to the natural action of the group \(Deck(\widehat{M_0}/M_0)\) on \(\widehat{M_0}\) then it defines a holomorphic solution of (111) on M.
Hence we see that by the gauge transformation (110) the large class of functions \(\phi _\alpha \) could be brought to \(\phi _\beta =0\). Further we will investigate this case only.
Let us note that for \( \phi _\beta =0\) the local form (41) of Hamilton Eq. (10) is
where the connection form \(\vartheta _\beta ({\bar{z}},z)={\widetilde{\vartheta }}_\beta ({\bar{z}},z)dz\) is related by
to the function \(F_\beta :M\rightarrow {{\mathcal {B}}}(V)\). See equalities (36) and (78) for this reason. So, in order to consider the holomorphic flow \(\tau ^{(X,F)}_t\) as a Hamiltonian flow generated by (X, F) we define \({\widetilde{\vartheta }}_\beta ({\bar{z}},z)\) by (115). Therefore the kernel \(K_{{\bar{\beta }}\beta }\) quantizes the Hamiltonian flow \(\pi (\tau ^{(X,F)}_t)=\sigma ^X_t\) if and only if it satisfies the Eqs. (112) and (113), while \(F_\beta \) satisfies Eq. (114).
Due to the circular symmetry of \(M\subset {\mathbb {C}}\) we expand \({\mathfrak {K}}_\beta :M\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) as a Laurent series
by definition convergent in the norm topology of the Banach space \({{\mathcal {B}}}(V,{{\mathcal {H}}})\). Let us mention that because V is finite-dimensional the norm convergence of (116) is equivalent to its strong convergence. The set of indices J, which numerate \(0\ne \Gamma _n\in {{\mathcal {B}}}(V,{{\mathcal {H}}})\) in (116), is an infinite subset \(J\subset {\mathbb {Z}}\) of the ring of integer numbers \({\mathbb {Z}}\). For \(M={\mathbb {C}}, {\mathbb {D}}\), in particular, one has \(J={\mathbb {N}}\cup \{0\}\). The condition (52) on the coherent state map \({\mathfrak {K}}_\beta :M\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\) implies
Taking into account the norm convergence of (116) we can express
the coefficients \(\Gamma _n\) by \({\mathfrak {K}}_\beta :M\rightarrow \mathcal {B}(V,{{\mathcal {H}}})\), where \(S^1_\rho :=\{z\in {\mathbb {C}}:\;|z|=\rho \}\subset M\).
From (116) we obtain that
Using this expansion we find that the Eq. (112) is equivalent to the two-variable second order difference equation
on the coefficients \(\Gamma _m^*\Gamma _{n}\in {{\mathcal {B}}}(V)\), where one assumes that \(\Gamma _{-1}=0\) if \(J={\mathbb {N}}\cup \{0\}\).
In the next proposition we present some properties of the coefficients \(\Gamma _n\in {{\mathcal {B}}}(V,{{\mathcal {H}}})\) whenever they satisfy the Eq. (120).
Proposition 7
-
(i)
If \(b=c=0\) and \(Re\;a\ne 0\), then \(\Gamma _n=0\) for \(n\in J\setminus \{0\}\).
-
(ii)
If \(b=c=0\) and \(a=i\omega \in i{\mathbb {R}}\setminus \{0\}\), then
$$\begin{aligned} \Gamma _m^*\Gamma _{n}= \Gamma _n^*\Gamma _{n} \delta _{mn}, \end{aligned}$$(121)for \(m,n\in J\).
-
(iii)
If \(b\ne 0\) then \( J={\mathbb {N}}\cup \{0\}\) and \(\Gamma _n\) are linearly independent in \({{\mathcal {B}}}(V,{{\mathcal {H}}})\).
Proof
For \(b=c=0\) the Eq. (120) reduces to
for all \(m,n\in J\). Thus, for \(Re\;a \ne 0\) one obtains that \(\Gamma _n=0\) if \(n\ne 0\). This proves (i). In the case \(Re\;a =0\) the Eq. (122) gives \(\Gamma _m^*\Gamma _{n}= 0\) for \(m\ne n\), so (ii) holds.
(iii) If \(b \ne 0\) let us assume that operators \(\Gamma _n\), \(n\in {\mathbb {N}}\cup \{0\}\), are linearly dependent. Then there exists \(N\in {\mathbb {N}}\) such that
Let us put in (120) \(n=N\) and rewrite this equation as follows
Next substituting \(\Gamma _N\) defined by (123) into the left hand side of (124) and using Eq. (120) again we find that
for arbitrary \(m\in {\mathbb {N}}\cup \{0\}\). Thus, due to (117), we have
which means that \(\Gamma _{N+1}\) is a linear combination of \(\{\Gamma _{0},\ldots ,\Gamma _{N-1} \}\) too. Repeating the above procedure we conclude that \(\dim {{\mathcal {H}}}\le N\), which leads to the contradiction with the assumption that \(\dim {{\mathcal {H}}}=\infty \). Ending, let us note that for \(b\ne 0\) one has \(\Gamma _m\ne 0\) for all \(m\in {\mathbb {N}}\cup \{0\}\). Since, assuming \(\Gamma _m=0\) for certain m we obtain from Eq. (120) that \(c(m-1)\Gamma _{m-1}+b(m+1)\Gamma _{m_+1}=0\). It leads to the linear dependence of \(\{\Gamma _n\}_{n\in {\mathbb {N}}\cup \{0\}}\). \(\square \)
Corollary 1
The vector space
is dense in \({{\mathcal {H}}}\), where \(\mathcal {F}\) is a finite subset of \({\mathbb {Z}}\).
Let us observe that for \(M={\mathbb {C}}\) the flows (100) and (101) are conjugated by the translation \(T_{\frac{b}{a}}(z):=z+\frac{b}{a}\). From this and from the point (i) of Proposition 7 we have:
Corollary 2
If \(Re\;a\ne 0\), then \({\mathfrak {K}}_\beta (z)=\gamma _0=const\). Therefore, the flows \(\sigma _t^X=\pi (\tau ^{(X,F)}_t)\) 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 \(a=i\omega \), where \(\omega \in {\mathbb {R}}\).
In order to find a solution \(K_{{\bar{\beta }}\beta }(\bar{z},z)\) of the differential Eqs. (112) and (113) we note that using (112) and (114) one can write \(K_{{\bar{\beta }}\beta }\) and \(F_{\beta }\) as the power series of a real variable \(I\in \Delta \subset {\mathbb {R}} \):
where \(C_n=C_n^\dagger , Q_n\in {{\mathcal {B}}}(V)\). By definition we assume the norm convergence of these power series expansions of the functions \(\varPhi _\beta :\Delta \rightarrow {{\mathcal {B}}}(V)\) and \(\varPsi _\beta :\Delta \rightarrow {{\mathcal {B}}}(V)\) defined on the range \(I(M)=:\Delta \) of an invariant \(I:M\rightarrow {\mathbb {R}}\):
of the flow \(\sigma _t^X\) tangent to the vector field X defined in (98).
Substituting \(K_{{\bar{\beta }}\beta }\) and \(F_\beta \) given by (126) and (127) into the Eq. (113) we reduce this equation to the ordinary linear equation on the function \(\varPhi _\beta \)
where \(\varPsi _\beta \) is defined by \(F_\beta \) through the Eq. (127), and the \(i{\mathbb {R}}\)-valued function \(\nu :\Delta \rightarrow i{\mathbb {R}}\) is defined
by the invariant \(I({\overline{z}},z)\). The correctness of this definition follows from the Eq. (128) and from \([w\frac{\partial }{\partial z},X]=[w\frac{\partial }{\partial z},w\frac{\partial }{\partial z}+{\bar{w}}\frac{\partial }{\partial {\bar{z}}}]=0\).
Since \(\varPhi _\beta =\varPhi _\beta ^\dagger \) and \(\nu =-{\bar{\nu }}\) we have from (129) that \(\varPhi _\beta \varPsi _\beta +\varPsi _\beta ^\dagger \varPhi _\beta =0\) which in the case \(M={\mathbb {C}},{\mathbb {D}}\) is equivalent to \(\sum _{l=0}^k(C_lQ_{k-l}+Q_{k-l}^\dagger C_l)=0\).
Summarizing the above facts we conclude:
Corollary 3
The flow \(\tau ^{(X,F)}_t\) is quantized by the positive kernel \(K_{{\bar{\beta }}\beta }=\varPhi _\beta \circ I\) if and only if the function \(\varPhi _\beta :\Delta \rightarrow {{\mathcal {B}}}(V)\) is a solution of the Eq. (129), where \(\varPsi _\beta \) and \(\nu \) are related to \(F_\beta \) and X through (127) and (128), respectively.
In the next proposition we will present the invariants I and \(\nu \circ I\) in correspondence with flows \(\sigma _t^X\) listed in Proposition 6.
Proposition 8
-
(i)
For \(M={\mathbb {C}}^*,{\mathbb {D}}^*, {\mathbb {A}}_r\) one has \(w(z)=i\omega z\). Thus
$$\begin{aligned} I({\overline{z}},z)={\bar{z}}z,\;\;\nu (I)=i\omega I, \end{aligned}$$(130)and \(I({\mathbb {C}}^*)=\Delta =]0,\infty [\), \(I({\mathbb {D}}^*)= \Delta =]0,1[\) or \(I({\mathbb {A}}_r)= \Delta =]r^2,1[\), respectively.
-
(ii)
For \(M={\mathbb {C}}\) one has \(w(z)=i\omega z+b\). Thus
$$\begin{aligned} I({\overline{z}},z)=\omega {\bar{z}}z+ i{\overline{b}} z-ib{\overline{z}},\;\;\nu (I)=i(\omega I+|b|^2), \end{aligned}$$(131)and \(I({\mathbb {C}})= \Delta ={\mathbb {R}}\) if \(\omega =0\), \(I({\mathbb {C}})=\Delta =[-\frac{|b|^2}{\omega },\infty [\) if \(\omega >0\) and \(I({\mathbb {C}})= \Delta =]-\infty , -\frac{|b|^2}{\omega }]\) if \(\omega <0\).
-
(iii)
For \(M={\mathbb {D}}\) one has \(w(z)=-{\bar{b}} z^2+2i\omega z+b\). Thus
$$\begin{aligned} I({\overline{z}},z)=\frac{2\omega z{\overline{z}}+ i{\overline{b}}z-ib{\overline{z}}}{1-z{\overline{z}}}, \;\;\nu (I)=i(I^2+2\omega I+|b|^2), \end{aligned}$$(132)and \(I({\mathbb {D}})=\Delta ={\mathbb {R}}\) for \(b\ne 0\), \(I({\mathbb {D}})=\Delta =]-\infty ,0[\) for \(b=0\), \(\omega <0\) and \(\Delta =]0,\infty [\) for \(b=0\), \(\omega >0\).
Proof
By straightforward verification. \(\quad \square \)
Now, based on the formulas given in Proposition 8 we will find the dependence of \(\Gamma _m^*\Gamma _{n}\in {{\mathcal {B}}}(V)\) on the \(C_{n}\in {{\mathcal {B}}}(V)\). This task is equivalent to solving the difference Eq. (120) with the \(\{C_n\}_{n\in J}\) as initial data, see (138). We will investigate the subcases mentioned in (130), (131) and (132) separately.
Proposition 9
For the case \(b=0\), see (130), which concerns with an arbitrary \(M={\mathbb {C}}, {\mathbb {D}}, {\mathbb {C}}^*,{\mathbb {D}}^*, {\mathbb {A}}_r\) we have
for \(m,n\in J\subset {\mathbb {Z}}\).
Proof
See formula (121) of Proposition 7. \(\quad \square \)
Now, let us shortly discuss the subcases presented in Proposition 9. From (133) it follows that the Hilbert space \({{\mathcal {H}}}\) can be decomposed
into the orthogonal \(\Gamma _nV\bot \Gamma _mV\), for \(n\ne m\), eigenspaces of \({\widehat{F}}\). The eigenvalue of \({\widehat{F}}\) corresponding to \(\Gamma _nV\) is \(n\omega \in \omega J\), so the subset \(\omega J\subset {\mathbb {R}}\) is the spectrum of \({\widehat{F}}\). We note here that \(J\subset {\mathbb {Z}}\) could be chosen as an arbitrary infinite subset of \({\mathbb {Z}}\). Thus the spectral decomposition of the operator \({\widehat{F}}\) is given by
where \({\hat{P}}_n\) are the orthogonal projectors on the eigenspaces \(\Gamma _nV\subset \mathcal {H}\). The kernels \(K_{{\bar{\beta }}\beta }(\bar{v},z)\) for all these subcases are given by the same formula
where \(0<C_n\in {{\mathcal {B}}}(V)\). For \(\dim _{\mathbb {C}}V=1\) this type of kernels was investigated in [27], where their relationship with the theory of q-special functions was also shown.
For the remaining subcases, i.e. if \(b\ne 0\) we will obtain the dependence of \(\Gamma _m^*\Gamma _{n}\) on the \(C_{n}\) comparing the coefficients in front of monomials \({\bar{v}}^m z^n\) occurring in the equality
valid for arbitrary \(v,z\in {\mathbb {C}},{\mathbb {D}}\), which follows from (119) and (126).
Proposition 10
-
(i)
For the case \(M={\mathbb {C}}\) and \(b\ne 0\) we have
$$\begin{aligned} \Gamma _m^*\Gamma _{n}=\sum _{l=n}^{n+m}\beta ^l_{mn}C_{l}, \end{aligned}$$(134)where
$$\begin{aligned} {\beta ^{l_{mn}}}=(i{\bar{b}})^{n-m}{m \atopwithdelims ()l-n}{l\atopwithdelims ()m}{\omega ^{m+n-l}}|b|^{2l-2n}, \end{aligned}$$(135)if \(m\le n\).
-
(ii)
For the case \(M={\mathbb {D}}\) and \(b=-{\bar{c}}\ne 0\) we have
$$\begin{aligned} {\Gamma _{m^{*}}}{\Gamma _{n}}=\sum _{l=n-m}^{n+m}\beta ^l_{mn}C_{l}, \end{aligned}$$(136)where
$$\begin{aligned} \beta ^l_{mn}= & {} i^{n-m} \sum _{j=0}^m{l-1+j \atopwithdelims ()l-1}{l \atopwithdelims ()2l+2j-n-m} \nonumber \\&\quad \times {2l+2j-n-m \atopwithdelims ()j+l-n} (2\omega )^{n+m-l-2j}b^{j+l-n} {{\bar{b}}}^{j+l-m} \end{aligned}$$(137)if \(m\le n\). The respective formulas for \(m>n\) one obtains by conjugation of the ones presented in (135) and (137) and transposition of the indices.
-
(iii)
For both cases described above one has
$$\begin{aligned} \Gamma _0^*\Gamma _n=(i\bar{b})^n C_{n}. \end{aligned}$$(138)
Proof
By straightforward verification. \(\quad \square \)
Next proposition describes the action of generator \({\widehat{F}}\) of the quantum flow \(U^{(X, F)}_t = e^{it{\hat{F}}}\) on the coefficients \(\Gamma _n\in {{\mathcal {B}}}(V,{{\mathcal {H}}})\) of the Laurent expansion (116).
Proposition 11
The vector space \({{\mathcal {D}}}_\Gamma \) defined in (125) is contained \({{\mathcal {D}}}_\Gamma \subset {{\mathcal {D}}}_{{\widehat{F}}}\) in the domain of \({\widehat{F}}\) and one has
for \(n\in J\), where we assume \(\Gamma _{-1}=0\) if \(J={\mathbb {N}}\cup \{0\}\).
Proof
After applying \(e^{it{\widehat{F}}}\) to both sides of equality (118) we obtain
Since one has \(\Vert \frac{1}{z^{n+1}}e^{it{\widehat{F}}}{\mathfrak {K}}_\beta (z) \Vert \le \frac{1}{\rho ^{n+1}}\sup _{z\in S^1_\rho }\Vert {\mathfrak {K}}_\beta (z) \Vert <\infty \), so due to Lebesgue’s dominated convergence theorem the derivative \(\frac{d}{dt}|_{t=0}\) at \(t=0\) of the right-hand side of (140) commutes with the integral over \(S^1_\rho \). Thus, we have
Hence, using also Stone’s Theorem, we find that the rank of \(\Gamma _n\) belongs to \({{\mathcal {D}}}_{{\widehat{F}}}\) and thus (139) is valid. To obtain the successive equalities in (141) we have used the norm convergence of the series (116). \(\quad \square \)
The expression (79) for the generator \({\widehat{F}}:{{\mathcal {D}}}_{{\widehat{F}}}\rightarrow {{\mathcal {H}}}\) and the expressions (80) and (81) for the Kirillov–Kostant–Souriau operators \(Q_{(X,F)}:{{\mathcal {D}}}(Q_{(X,F)})\rightarrow \Gamma ^{hol}(M,{\mathbb {V}})\) and \({\bar{Q}}_{(X,F)}:{{\mathcal {D}}}({\bar{Q}}_{(X,F)})\rightarrow \Gamma ^{antihol}(M,\bar{{\mathbb {V}}})\) in the case under consideration, i.e. when \(\phi _\beta =0\), assume the following forms
and
The essential domains of the above operators are given by
and by
respectively, where \({{\mathcal {F}}}\) is a finite subset of \({\mathbb {Z}}\).
Proposition 12
If \(b\ne 0\), then \(\Gamma _0={\mathfrak {K}}_\beta (0)\in {{\mathcal {B}}}(V,{{\mathcal {H}}})\) is a generating element for \(i{\widehat{F}}\) in the Hilbert \({{\mathcal {B}}}(V)\)-module \({{\mathcal {B}}}(V,{{\mathcal {H}}})\), i.e. the elements \((i{\widehat{F}})^n\Gamma _0\), where \(n\in {\mathbb {N}}\cup \{0\}\), are linearly independent and they span a linearly dense subspace of \({{\mathcal {B}}}(V,{{\mathcal {H}}})\). Moreover, one has
where the polynomials
are defined by the recurrence
with the initial conditions \(K_{-1}(i\lambda )\equiv 0\) and \(K_{0}(i\lambda )\equiv 1\).
Proof
The linear dependence between \(\Gamma _0, i{\widehat{F}}\Gamma _0,\ldots ,(i{\widehat{F}})^n\Gamma _0\in {{\mathcal {B}}}(V,{{\mathcal {H}}})\) and \(\Gamma _0, \Gamma _1,\ldots ,\Gamma _n\in {{\mathcal {B}}}(V,{{\mathcal {H}}})\), where \(n\in {\mathbb {N}}\cup \{0\}\), given by the equations
where \(k=0,1,\ldots ,n\), is invertible. Hence, and from the linear independence of \(\Gamma _n\in {{\mathcal {B}}}(V,{{\mathcal {H}}})\) we conclude that the vectors \((i{\widehat{F}})^n\Gamma _0\in {{\mathcal {B}}}(V,{{\mathcal {H}}})\), \(n\in {\mathbb {N}}\cup \{0\}\), are linearly independent.
From Proposition 7 it follows that the operators \(\Gamma _n\) span a dense subset of \({{\mathcal {B}}}(V,{{\mathcal {H}}})\), so, \((i{\widehat{F}})^n\Gamma _0\) span too. \(\quad \square \)
Now, let us describe the relationship between the coherent state representation (142) and the spectral representation of the generator \({\widehat{F}}\) of the quantum flow \(U^{(X,F)}_t=e^{it{\widehat{F}}}\). Therefore, let \(E:{\mathbb {R}}\rightarrow {{\mathcal {L}}}({{\mathcal {H}}})\) denote the resolution of identity or equivalently the spectral measure \(E:{{\mathcal {B}}}({\mathbb {R}})\rightarrow {{\mathcal {L}}}({{\mathcal {H}}})\) of the self-adjoint operator \({\widehat{F}}\), i.e. \(\psi \in {{\mathcal {D}}}_{{\widehat{F}}}\) if and only if
and one has
for \(\psi \in {{\mathcal {D}}}_{{\widehat{F}}}\), see Chapter VI §66 in [2] for details. Above by \({{\mathcal {L}}}({{\mathcal {H}}})\) and by \({{\mathcal {B}}}({\mathbb {R}})\) we denoted the lattices of orthogonal projections of \({{\mathcal {H}}}\) and Borel subsets of \({\mathbb {R}}\), respectively.
From Proposition 11 follows that \(\Gamma _nV\subset {{\mathcal {D}}}_{{\widehat{F}}}\), so, using (143) and (145) we find that
Next, substituting \(\Gamma _n\) given by (146) into (138) we obtain
where \(\mu _l\in {{\mathcal {B}}}(V)\) defined by
are the moments of the positive \({{\mathcal {B}}}(V)\)-valued measure
Summing up we conclude from (126) and (147) that through the Hamburger moment problem defined by (148) one obtains the relationship between the positive kernel \(K_{{\bar{\beta }}\beta }(\bar{v},z)\) and the resolution of identity \(E:{\mathbb {R}}\rightarrow {{\mathcal {L}}}({{\mathcal {H}}})\) of \({\widehat{F}}\).
Let us define the Hilbert \({{\mathcal {B}}}(V)\)-module \(L^2({\mathbb {R}},d(\Gamma _0^*E\Gamma _0))\) of \({{\mathcal {B}}}(V)\)-valued Borel square integrable functions \(\gamma :{\mathbb {R}}\rightarrow {{\mathcal {B}}}(V)\), i.e. such ones that \(\langle \gamma ;\gamma \rangle _{L^2}\le M1\!\!1_V\), where \(0< M\in {\mathbb {R}}\), in sense of \({{\mathcal {B}}}(V)\)-valued scalar product
of the square integrable \(\mathcal {B}(V)\)-valued functions \(\gamma ,\delta \in L^2({\mathbb {R}},d(\Gamma _0^*E\Gamma _0))\). As it follows from (125) and (143) \(\Gamma _0\in {{\mathcal {B}}}(V,{{\mathcal {H}}})\) is a generating element in \({{\mathcal {B}}}(V,{{\mathcal {H}}})\) for \({\widehat{F}}\), so we have the isomorphism
of the defined above Hilbert \({{\mathcal {B}}}(V)\)-modules.
Using the isomorphism \(\mathcal {I}\) and (143) we find that
and, thus
where the function \({\mathfrak {K}}_\beta (z;\cdot )\in L^2({\mathbb {R}},d(\Gamma _0^*E\Gamma _0))\) is defined by the power series
convergent in the norm \(\Vert \cdot \Vert _{L^2}:= \Vert \langle \cdot , \cdot \rangle _{L^2}\Vert \), where by \(\Vert \cdot \Vert \) we denoted the norm on \({{\mathcal {B}}}(V)\). Later on we will see in Proposition 13 that it is also point-wise convergent.
We note here that the equivariance condition
written in terms of \({\mathfrak {K}}_\beta (z;\lambda )\) assumes the following form
Taking into account (149) and that \(\langle {\mathfrak {K}}_\beta (v);{\mathfrak {K}}_\beta (z) \rangle =\langle {\mathfrak {K}}_\beta (v;\cdot ));{\mathfrak {K}}_\beta (z;\cdot ))\rangle _{L^2}\) we obtain the integral decomposition
where
of the kernel \(K_{{\bar{\beta }}\beta }(\bar{v},z)={\mathfrak {K}}_\beta (v)^*{\mathfrak {K}}_\beta (z)\) invariant with respect to the flow \(\sigma ^X_t\) quantized by \(e^{it{\widehat{F}}}\).
Combining (126) and (147) we obtain the expression
on the kernel \(K_{{\bar{\beta }}\beta }(\bar{v},z;\lambda )\) which is different to (152).
The equivalence of (152) and (153) follows from the equality
valid for the polynomials \(K_n(i\lambda )\), where \(\beta ^l_{mn}\) are given: by (135) and \(L=n\) for \(M={\mathbb {C}}\); by (137) and \(L=n-m\) for \(M={\mathbb {D}}\).
Comparing the right-hand sides of (150) and (153) we obtain
where \(I({\bar{z}}, z)\) is the \(\sigma ^X_t\)-invariant presented in (131) and (132) of the Proposition 8. In the next proposition we will present expressions on \(K_n(i\lambda )\), \({\mathfrak {K}}_\beta (z)\) and \(K_{{\bar{\beta }}\beta }(\bar{v},z;\lambda )\) for the cases when \(b\ne 0\).
Proposition 13
-
(i)
If \(M={\mathbb {C}}\) and \(b\ne 0\) then we have
$$\begin{aligned} K_n(i\lambda )=\frac{(-i\omega )^{n-1}}{n!b^n}\left( -\frac{1}{\omega }\lambda \right) _n, \end{aligned}$$(155)where \((x)_n=x(x+1)\ldots (x+n-1)\) is the Pochhammer symbol,
$$\begin{aligned}&{\mathfrak {K}}_\beta (z;\lambda )= \frac{i}{\omega } \, _1F_0\left( -\frac{1}{\omega }\lambda ; -\frac{i\omega }{b}z\right) 1\!\!1_V, \end{aligned}$$(156)$$\begin{aligned}&\quad K_{{\bar{\beta }}\beta }(\bar{v},z;\lambda )= \frac{i}{\omega } \, _1F_0\left( -\frac{1}{\omega }\lambda ; -\frac{\omega ^2}{|b|^2}{\bar{v}}z-\frac{i\omega }{b} z+\frac{i\omega }{{\bar{b}}}{\overline{v}})\right) 1\!\!1_V. \end{aligned}$$(157)If \(a=i\omega =0\) the formulas above take the form
$$\begin{aligned}&K_n(i\lambda )=\frac{1}{n!}\left( \frac{i\lambda }{b}\right) ^n, \end{aligned}$$(158)$$\begin{aligned}&{\mathfrak {K}}_\beta (z;\lambda )=e^{i\frac{\lambda }{b}z }1\!\!1_V, \end{aligned}$$(159)$$\begin{aligned}&K_{{\bar{\beta }}\beta }(\bar{v},z;\lambda )=e^{i\lambda \left( \frac{z}{b} -\frac{{\bar{v}}}{{\bar{b}}}\right) }1\!\!1_V. \end{aligned}$$(160) -
(ii)
If \(M={\mathbb {D}}\) and \(b\ne 0\) then for \(|b|^2-\omega ^2\ne 0\) we find that polynomials \(K_n\) are given by (non-orthogonal) Meixner–Pollaczek polynomials \(P^{(0)}_n\left( x;\varphi \right) \) as follows
$$\begin{aligned} K_n(i\lambda )= & {} \left( \frac{A }{2ib\sin \varphi } \right) ^n P^{(0)}_n\left( -i\lambda /A;\varphi \right) \nonumber \\= & {} \left( \frac{A e^{i\varphi }}{2ib\sin \varphi } \right) ^n\frac{(2\mu )_n}{n!}\;_2F_1\left( -n,\mu -\frac{i\lambda }{A};2\mu ;1-e^{-2i\varphi } \right) |_{\mu =0} \nonumber \\ \end{aligned}$$(161)where \(A:=-\frac{2\omega }{|\omega |}\sqrt{|b|^2-\omega ^2}\) and \(\cos \varphi :=\frac{|\omega |}{|b|}\) (if \(|b|^2-\omega ^2>0\), then \(\varphi \in [0,\pi /2]\), if \(|b|^2-\omega ^2<0\), then A and \(\varphi \) are imaginary, \(A,\varphi \in i{\mathbb {R}}\), and \(\varphi /i\in [1,\infty [\)) and
$$\begin{aligned} {\mathfrak {K}}_\beta (z;\lambda )= & {} \left( \frac{2ib\sin \varphi -Ae^{-i\varphi }z}{2ib\sin \varphi -Ae^{i\varphi }z} \right) ^{\frac{i\lambda }{A}}1\!\!1_V, \end{aligned}$$(162)$$\begin{aligned} K_{{\bar{\beta }}\beta }(\bar{v},z;\lambda )= & {} \left( \frac{2|b|^2\sin \varphi +Ae^{-i\varphi }I({\bar{v}}, z)}{2|b|^2\sin \varphi +Ae^{i\varphi }I({\bar{v}}, z)} \right) ^{\frac{i\lambda }{A}}1\!\!1_V, \end{aligned}$$(163)where \(I({\bar{v}}, z)\) is given in (132). For \(b\ne 0\) and \(|b|^2-\omega ^2=0\) we have that the polynomials \(K_n\) are expressed by (non-orthogonal) Laguerre polynomials \(L^{(-1)}_n(x)\) by
$$\begin{aligned} K_n(i\lambda )= & {} \left( \frac{\omega }{ib}\right) ^n L^{(-1)}_n(\lambda /\omega )\nonumber \\= & {} \left( \frac{\omega }{ib}\right) ^n\frac{(\alpha +1)_n}{n!}\;_1F_1\left( -n;\alpha +1;\lambda /\omega \right) |_{\alpha =-1}, \end{aligned}$$(164)and
$$\begin{aligned} {\mathfrak {K}}_\beta (z;\lambda )= & {} \exp {\left( \frac{\lambda z}{\omega z -ib}\right) }1\!\!1_V, \end{aligned}$$(165)$$\begin{aligned} K_{{\bar{\beta }}\beta }(\bar{v},z;\lambda )= & {} \exp {\left( \frac{\lambda I({\bar{v}}, z)}{\omega I({\bar{v}}, z)+|b|^2}\right) }1\!\!1_V, \end{aligned}$$(166)where \(I({\bar{v}}, z)\) is given in (132).
Proof
For \(M={\mathbb {C}}\) one can put in (144) \(c=0\) (see (99)). Thus it is easy to check that (158) and (155) are solutions of this recurrence equation for \(a=0\) and \(a=i\omega \ne 0\), respectively. The relations (159), (160), (156) and (157) follow immediately from (150), (153) and the definition of the hypergeometric functions.
For \(M={\mathbb {D}}\) one has \(c=-{\bar{b}}\) and \(a=2i\omega \). Let us introduce the polynomials \(Q_n\) defined by
where \(A\in {\mathbb {C}}\setminus \{0\}\). Then (144) takes the form
This is a three-term recurrence formula on monic polynomials \(Q_n\) with the initial conditions \(Q_{-1}(\lambda )\equiv 0\) and \(Q_{0}(\lambda )\equiv 1\). By Favard’s Theorem, see e.g. [7] Theorem 4.4, the solutions of (167) are non-orthogonal polynomials described in [21]. Namely, for \(|b|^2-\omega ^2\ne 0\) and \(A=-\frac{2\omega }{|\omega |}\sqrt{|b|^2-\omega ^2}\) the polynomials \(Q_n\) are the (non-orthogonal) monic Meixner–Pollaczek polynomials \(Q_n(\lambda )= \frac{n!}{(2\sin \varphi )^n}P^{(0)}(\lambda ;\varphi )\) and for \(|b|^2-\omega ^2=0\) and \(A=-\omega \) the polynomials \(Q_n\) are the (non-orthogonal) monic Laguerre polynomials \(Q_n(\lambda )= n!(-1)^nL^{(-1)}(\lambda )\). This proves (161) and (164).
To prove (162), (163), (165) and (166) it is enough to observe that (150) and (153) is nothing else than the generating function for the family of polynomials \(\{K_n\}_{n=0}^\infty \), which for Meixner–Pollaczek and Laguerre polynomials may be found in [21] as well. \(\quad \square \)
The integral decomposition (151) taken for the case \(\dim _{\mathbb {C}}V=1, b\ne 0\) and \(a=0\) leads to Bochner’s Theorem, see e.g. [35], 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 \(M={\mathbb {C}},{\mathbb {D}}\) we obtain a family of Bochner type integral decompositions presented in Proposition 13 for the positive definite kernels invariant with respect to suitable holomorphic flows \(\sigma ^X_t\) on the Riemann surfaces \({\mathbb {C}}\) and \({\mathbb {D}}\) 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 \(\dim _{\mathbb {C}}V=1\). In this case one has the natural isomorphism \({{\mathcal {B}}}(V,{{\mathcal {H}}})\cong {{\mathcal {H}}}\). Therefore, after applying the Gram–Schmidt orthonormalization procedure to the elements \({\widehat{F}}^n\Gamma _0\in {{\mathcal {H}}}\), where \(n\in {\mathbb {N}}\cup \{0\}\), which according to Proposition 12 are linearly independent and span the vector subspace \({{\mathcal {D}}}_\Gamma \subset {{\mathcal {D}}}_{{\widehat{F}}}\subset {{\mathcal {H}}}\) dense in \({{\mathcal {H}}}\), we obtain the orthonormal basis
in \({{\mathcal {H}}}\).
The polynomials \(P_n(\lambda )\) of degree n appearing in (168) are orthogonal with respect to the positive measure \(d(\Gamma _0^*E\Gamma _0)(\lambda )\). They satisfy the three term recurrence
defined by infinite Jacobi matrix J. One can express the coefficients \(a_n\) and \(b_n\) of this matrix as well as the polynomials \(P_n(\lambda )\) in terms of the moments \(\mu _n\), see (148), of the measure \(d(\Gamma _0^*E\Gamma _0)(\lambda )\). For the respective formulas see Chapter I of [1].
The self-adjoint operator \({\widehat{F}}\) expressed in the basis (168) assumes the three-diagonal form
as well as in the basis \(\Gamma _n\), \(n\in {\mathbb {N}}\cup \{0\}\), see (139).
We summarize the facts mentioned above defining
the operators
which by definition intertwine the bases \(\{\Gamma _n \}_{n=0}^\infty \), \(\{{\widehat{F}}^n\Gamma _0 \}_{n=0}^\infty \), and \(\{|n\rangle \}_{n=0}^\infty \) of the Hilbert space \({{\mathcal {H}}}\).
Proposition 14
The domain \({{\mathcal {D}}}_{\Gamma ^*}\) of the operator \(\Gamma ^*\) adjoint to \(\Gamma \) contains \({{\mathcal {D}}}_\Gamma \), which is also the range of \(\Gamma \). Hence \({{\mathcal {D}}}_{\Gamma ^*}\) is dense in \({{\mathcal {H}}}\).
Proof
For \(\varphi \in {{\mathcal {H}}}\) and \(\psi =\sum _{n\in {{\mathcal {F}}}}c_n |n\rangle \in {{\mathcal {D}}}_\Gamma \) one has
Thus we see that if \(\sum _{n\in J}|\langle \varphi |\Gamma _{n}\rangle |^2<\infty \) then \(\varphi \in {{\mathcal {D}}}(\Gamma ^*)\), so, we need to prove that
for any \(m\in J\). Let us consider three subcases mentioned in Proposition 9 and Proposition 10 separately.
For the subcase of Proposition 9 it follows from (133) that \(\sum _{n\in J}|\langle \Gamma _m |\Gamma _{n}\rangle |^2=|\langle \Gamma _m |\Gamma _{m}\rangle |^2<\infty \).
To prove (170) for the subcase (i) of Proposition 10 where \(M={\mathbb {C}}\) let us observe that the quantities \(\beta ^l_{m,n}\), \(l=n,\ldots ,n+m\), given by (135) form, up to the factor \((i{{\bar{b}}})^{n}\), a finite family of polynomials of the variable n of degree no greater than m with coefficients depends on \(b,\omega , m\). Thus from (134) we obtain that for fixed \(m\in {\mathbb {N}}\cup \{0\}\) one has
where the last limit follows from the fact that the right hand side of (126) is convergent for arbitrary \(I\in {\mathbb {R}}\) (see (131)), i.e. \(\root n \of {|C_n|}\rightarrow 0\), and \(\root n \of {|\beta ^l_{m,n}| }\rightarrow |b|\). Finally, (170) holds due to the root test for the convergence of a series.
The proof of (170) for the subcase (ii) of Proposition 10 where \(M={\mathbb {D}}\) is similar to the previous case. Namely, from (137) follows that the quantities \(\beta ^l_{m,n}\), \(l=n-m,\ldots ,n+m\) form, up to the factor \((i{{\bar{b}}})^{n}\), a finite family of polynomials of the variable n of degree no greater than 3m with coefficients depending on \(b,\omega , m\). Thus
because the right hand side of (126) is convergent for arbitrary grates \(|I|\in {\mathbb {R}}\) (see (132)), i.e. \(\root n \of {|C_n|}\rightarrow 0\). \(\quad \square \)
We see from the above proposition that the assumption of the Theorem VIII.1 in [35] are fulfilled and thus we have:
Proposition 15
-
(i)
The adjoint operator \(\Gamma ^*\) is closed.
-
(ii)
The operator \(\Gamma \) is closable and one has \({\bar{\Gamma }}=\Gamma ^{**}\), \(({\bar{\Gamma }})^*=\Gamma ^*\).
-
(iii)
The operator \({{\bar{\Gamma }}}^*{\bar{\Gamma }}=\Gamma ^{*}\Gamma ^{**}\) defined on the dense domain \({{\mathcal {D}}}_{{{\bar{\Gamma }}}^*{\bar{\Gamma }}}=\{\psi \in {{\mathcal {D}}}_{{\bar{\Gamma }}}: {\bar{\Gamma }}\psi \in {{\mathcal {D}}}_{{\bar{\Gamma }}^*} \}\) is self-adjoint (see Exercise 45 in Chapter VIII of [35]).
Let us mention an interesting possibility to describe the coherent state map \({\mathfrak {K}}_\beta :{\mathbb {D}}\rightarrow {{\mathcal {H}}}\) which quantizes a holomorphic flow \(\sigma ^X_t:{\mathbb {D}}\rightarrow {\mathbb {D}}\) on the disc. Namely, let us define \({\mathfrak {K}}_0:{\mathbb {D}}\rightarrow {{\mathcal {H}}}\) by
From (171) and from the closability of \(\Gamma :{{\mathcal {D}}}_\Gamma \rightarrow {{\mathcal {D}}}_\Gamma \) we find that
The above allows us to represent
the positive kernel \(K_{{\bar{\beta }}\beta }\) in terms of the matrix elements \(\langle m|{{\bar{\Gamma }}}^*{\bar{\Gamma }} |n \rangle \) of the positive self-adjoint operator \(\Gamma ^{*}\Gamma ^{**}={{\bar{\Gamma }}}^*{\bar{\Gamma }}\).
Proposition 16
If \(\dim V=1\) then for the flow \(\sigma _t(z)=e^{a t}z\), \(z\in M={\mathbb {C}}, {\mathbb {D}}, {\mathbb {C}}^*,{\mathbb {D}}^*, {\mathbb {A}}_r\) there exists a holomorphic section \(s_\beta :M\rightarrow P(GL(1,{\mathbb {C}}),\pi ,M)\) for which \(\phi _\beta (z)=:\phi _0= const\).
Proof
We need to show that when \(\dim V=1\) the Eq. (110) has solution \(g_{\alpha \beta }:M\rightarrow {\mathbb {C}}\setminus \{0\}\) for \(\phi _\beta (z)= \phi _0\). Let us rewrite this equation in the following form
Because each M is a circularly symmetric domain in \({\mathbb {C}}\), the holomorphic function \(\phi _\alpha :M\rightarrow {\mathbb {C}}\) is globally defined by its Laurent expansion
Let us define a holomorphic function on M
This definition is correct since \(\root n \of {n}\rightarrow 1\) for \(n\rightarrow \infty \).
Because \(z\frac{\partial \psi (z)}{\partial z}=\phi (z)-p_0\), then for \(\phi _{0}=p_0\) the holomorphic function \(g_{\alpha \beta }(z)=e^{\frac{1}{\alpha }\psi (z)}\) is a solution of (172) holomorphic on M. \(\quad \square \)
Corollary 4
If \(Re\, \phi _0\ne 0\), then the flows mentioned in Proposition 16 are not quantizable.
Proof
If \(\phi _\beta (z) = \phi _\beta (0) = \phi _0 \ne 0\), then for \(b= 0\) the Eqs. (107) and (116) give
where \(\delta := Re\, a\), \(\omega := Im\, a \) and \(\mu := Re\, \phi _0\). Hence, we find that \(\Gamma _n \ne 0\) iff \(n = - \frac{\mu }{\delta }\in {\mathbb {Z}}\). So, \({\mathfrak {K}}_\beta (z) = \Gamma _n z^n\) and from defining property of \({\mathfrak {K}}_\beta :M \rightarrow \mathcal {H}\) it follows that \(\dim _{\mathbb {C}} \mathcal {H} = 1\). The above contradicts the postulates of Definition 1. \(\quad \square \)
In the next section we will shortly discuss a possible physical applications of the obtained results.
7 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 \(L^2({\mathbb {R}}^N, d^Nx)\) 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 [11, 12, 39] and nuclear physics [19], 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 [37] and next was investigated by Fock [8] and Bargmann [4], and is known in quantum mechanics as the Bargmann–Fock representation. Later it was revitalized in quantum optics by Glauber [12]. Let us also mention also the contribution of Perelomov, see [34], to this subject, i.e. the construction of coherent state maps through the irreducible representations of Lie groups.
In the papers [25, 26] 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 [37] 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 [15, 29], respectively. One can find a large class of systems quantizable by the coherent state method in optics [14, 16, 38], where one usually considers a finite number of modes of an electromagnetic field self-interacting through a nonlinear medium [11, 33, 39]. In the papers [30,31,32] 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 [13, 30]. This is a case to which one can apply the results obtained in Sect. 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 (151) presented in Proposition 13, which generalizes the one considered in Bochner’s Theorem to arbitrary noncompact Riemann surfaces, will find applications in probability theory problems.
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Odzijewicz, A., Horowski, M. Some Aspects of Positive Kernel Method of Quantization. Commun. Math. Phys. 389, 933–975 (2022). https://doi.org/10.1007/s00220-021-04158-z
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DOI: https://doi.org/10.1007/s00220-021-04158-z