The intention of these pages is to offer a “distillation of ideas” on the geometric quantization. The studies that have led to the birth of this line of interest are principally due to these authors:     and . On the contrary, the works that inspired this paper are principally two. The first is a recent article of Carosso  , where the author describes in a very detailed way the procedure of the geometric quantization with relevant attention and with professional criticism. The second is the celebrated book of Woodhouse  , considered as the basic text for “beginners”.
The first pages of this article are a review of basic facts on Kähler manifolds and classical mechanics (especially from the mathematical point of view). After introducing this fundamental formalism with a lot of examples, the author proceeds the examination of the “quantization procedure”. The section three is dedicated to the process of prequantization with the description of all prequantum conditions adopted by different authors. In the section four, the process of geometric quantization is described introducing the notion of polarization (real and complex) and examining the particular case of compact Kähler manifolds. In the section five, the square-root bundle is introduced in order to pass to a corrected geometric quantization. BKS pairing, used by different authors, is examined in three cases: in the presence of two real polarizations, one real and one complex and, at the end, two complex polarizations. The last section is dedicated to the Bohr-Sommerfeld subvarieties with recent developments (see  ).
In a subsection of the section five, the Schrödinger equation is derived for the case of the cotangent bundle in a similar way to  with the difference we used a result of Albeverio and Mazzucchi (  ) instead the heat kernel.
The implicit question regarding the future of the geometric quantization is legitimate and necessary. We will see, during these pages, different criticisms on this rigorous procedure together different problems in the construction of a quantization space that is sufficiently satisfactory. Furthermore, at the present state of the art, an application for the relativity theory would be desirable but not possible. What emerge in this direction is that the machinery of geometric quantization seems to be not the rigorous solution to the problem of the unification of the relativity and the quantum theory (at least without future modifications). The problem of the geometric quantization of the general relativity has been recently discussed in  and . An interesting variation is given by the deformation quantization, not treated here, but much more suitable. Another possible way in this direction is the method used by Feynman considering a “summation of histories” or probability amplitudes. For this reason, an interesting development can be based on the relation between the geometric quantization and the Feynman integral. A seed of this idea can be found in  where the author shows how the Feynman integral can be view as a particular case of BKS pairing. This possibility sure is not the end of the history, as other approaches show this is not the only “via regia” (the literature regarding other quantization methods is omitted here due to the large amount to be taken into consideration). A last work that we cite is  where the author studied the connection between the geometric quantization (GQ) process and the quantum logic (QL). In this optics, the geometric quantization can be a considered as a “machinery” that produces Hilbert spaces with interesting properties.
2. Kähler Manifolds, Classical Mechanics and Symmetries
2.1. Kähler Manifolds
A Kähler manifold is represented by a quadruple where M is a complex manifold of complex dimension n, is the symplectic structure locally given by:
where g is the Riemannian metric on M, J is the complex structure associated to M and K is the Kähler potential associated to :
We recall that in a Kähler manifold then locally admits a potential K. Furthermore the metric g is J-invariant and
for every . Furthermore on M it is possible to consider also an hermitian metric, denoted by h and defined by:
The complex structure divides the tangent space in a point , into a direct sum where:
The first and the second .
Sometimes these operators are simply denoted by and . Passing through the 2n real coordinates denoted by (for ), we have:
and respectively, the dual basis is and .
Example 2.2. An example of Kähler manifold is , where:
and the Kähler potential is given by . Here the symplectic form is the standard form .
Example 2.3. Another example of Kähler manifold is , where:
is the Fubini-Study form and the Kähler potential is (here log denotes the natural logarithm).
Example 2.4. The unit disk has the structure of Kähler manifold where:
is the hyperbolic form and .
Example 2.5. A last example is the complex torus where is a lattice on . has the structure of Kähler manifold where:
and is a complex map linear in the first factor and respectively antilinear in the second factor such that and if and only if . In this case .
2.2. Kähler Manifolds and Classical Mechanics
In this section we recall the relation between Kähler manifolds and classical mechanics. Let be a Kähler manifold with and its symplectic form, then it is possible to interpret M as a model of a classical system. For example the cotangent space of a configuration space C of dimension n. So in this case is given in local symplectic coordinates and classical observables are smooth functions f on M. To each observable there is an Hamiltonian vector field defined by the equation:
we underline that this notation is equivalent to other common notations or . In local coordinates:
For any two observables we can define the Poisson bracket as:
where is the Lie derivative along . We recall that the Lie derivative is defined by the flow :
where in the previous formula generally there is a tensorial field instead of f.
An observable is conserved when . In particular there is a smooth function , called Hamiltonian, such that determines the trajectories of a classical system via the Hamilton’s equations:
for a point .
The system is called Hamiltonian system. If is an integral curve of then the energy function is constant for all t and .
Note that .
Further information on Hamiltonian mechanics can be found in .
From the fact that is closed we have that locally admits a potential . From the potential we can define the lagrangian associate to an observable f defined as:
The lagrangian mechanics is associated to lagrangian submanifolds of M. When the function f is the Hamiltonian then is the Legendre transform formula and by the Cartan’s formula we have that:
We recall that generates a diffeomorphism that preserve . From the fact that we have that so is closed and locally exact. Thus , where S is a function on . Let be an integral curve of with parametrization t, then we have that:
because the pushforward of along itself is the identity and, the form is exact. By integration:
where is a local phase function (because is complete). We obtain thus a function called the generating function:
where and and . The function generates a submanifold where and where .
Such manifolds are called lagrangian submanifolds. The fiber coordinates on are given by:
for . In order to pass from to we use the Equations (2.14).
Details on lagrangian and hamiltonian mechanics are in  and .
2.3. Kähler Manifolds and Classical Mechanics with Symmetries
LetG be a finite dimensional compact Lie group and an action on the Kähler manifold . Let be the Lie algebra associated to G and the dual space. If we have an n-form on M it is associated a map defined as:
where is (observe that if the action of G is symplectic then ). Furthermore if is closed, then also ( ) and . If in addition then there is a unique element in such that:
There is a unique map such that:
We can be more explicit with . In fact, under our assumptions, there is a homeomorphism between the Lie algebra and the hamiltonian vector fields on M (see  ). In particular there is a unique homeomorphism such that where:
The map depends linearly by so we can consider given by:
where is the pairing between and . In other words:
for every where is called the moment map. The previous equation can be written in the following form:
where for every and .
The moment map has different properties. The first is that is the transpose map of the valuation . We have that:
where is the subspace of vectors . We have that G acts on M transitively so for every . We have also that:
is the stabilizator of . The stabilizator is discrete if and only if is surjective. The last property is a conseguence of the fact that G acts on by the adjoint representation, val is a G-morphism, thus is a G-morphism:
for every and .
Example 2.25. Let us consider with . Let us consider as G the circle that acts with rotations. The generator of the action is
the field . Thus is the moment map.
In general on where , with the action of
and heights , the action has generator the vector field and moment map given by:
3.1. Prequatization Conditions
Recalling that a closed surface of M is a surface that is compact and without boundary, we state the prequantum condition PC1 as the following.
PC1 The integral of over any closed 2-surface is an integral multiple of .
This prequantum condition PC1 is necessary for the existence of the hermitian line bundle L over M. In the case of M simply connected PC1 is also sufficient. Assume M simply connected and be a base point. Let us consider the following set:
and the equivalence relation ~ defined as:
where is any oriented 2-surface with boundary made up of (from m0 to m) and (from m to m0). The surface exists because M is simply connected. We define the line bundle L as:
We can define operations of addition and scalar multiplication between the fibres:
with and . Trivializations of L are determined locally by symplectic potentials. In fact let us assume that is a symplectic
potential on a simply connected open set of a collection . Let
us consider a point and a curve from m0 to m1. We define locally a section s of L in by
where is any curve from m1 to m in and is the curve from m0 to m obtained from and . We observe that a different choice of gives the same value of and that a different choice of m1 or gives the same section multiplied by a constant of modulus one. The effect of replacing by
, with , is .
Now we can assume that the line bundle exists. Let us consider the parallel transport of a section s respect to around a loop of . Assume that is the boundary of a 2-surface contained in the domain of a symplectic potential. Solving the parallel transport equation , the result is equivalent to a linear transformation given by the multiplication
of that, by the Stokes’ theorem, is equivalent to . Now let us consider a second surface with boundary such that is a closed
2-surface in M (in other terms gluing together and in we obtain the closed 2-surface ). In a similar way the parallel transport gives a linear
transformation by the multiplication of , where the minus is because the
boundary of is . By the uniqueness of the solution of the differential equation associated to the parallel transport we have that:
the last equation is equivalent to PC1 because .
The prequantum condition is related to the existence of the hermitian line bundle L over M also through the Weil theorem.
Theorem 3.1 (Weil, 1958,  ). Let M be a smooth manifold and a real,
closed 2-form whose cohomology class is integral. Then there is a unique hermitian line bundle L over M with unitary connection so that .
The converse is also true. In fact let be a contractible open cover of M. By
assumption there is a collection of 1-forms such that
on . We can find such that whenever . On (whenever ) we have that , so the function is constant. They define a Cěch cohomology class in . Now whenever and, the de Rham isomorphism send to
By assumption are transition functions and must satisfy the cocycle condition:
We have a second version of the prequantization condition PC2.
Note that in PC2 it is not required M to be simply connected, so it is more general than PC1.
Another prequantization of a symplectic manifold consists in a -bundle with the projection and an invariant 1-form such that satisfy the prequantization condition PC3.
The -bundle it is also called the circle bundle and is defined as:
where h is the hermitian metric on L. We have that is also called a contact manifold.
The last way to state the prequantization condition derive from PC3 and consists relating the curvature form of the connection of the line bundle L and the symplectic form . In other terms if is the curvature form of the unique covariant derivative on L compatible with both the complex and hermitian structures, we have the prequantization condition PC4.
Example 3.2. Let be a complex linear map in the first factor and antilinear in the second. Assume also that and that is zero if and only if . Let us define:
then is a Kähler form for the complex torus that is invariant under the action of the lattice . Furthermore it is possible to prove that the torus is quantizable if and only if . This condition it is in fact equivalent to the integrability condition PC4. We can try to see this in one direction. Let us assume , then the image of through H is in . This condition ensure the existence of a complex line bundle where is a semicharacter associated to H. On L we can define an hermitian structure defined as:
where is the holomorphic section of L, also called theta function such that:
and A is a factor of automorphy on (a good reference is  ). The second term in the definition of h is that is a map defined as:
It is possible to see that with this definition, the function h is invariant under the action of and h defines an hermitian structure on L. Now analyzing the curvature form of the line bundle L we find that it is equal to . This last term is equal to that is the condition PC4 with the Planck constant equal to 1. The converse of the proof is proved in .
Example 3.7. Let be the sphere in of radius r. The sphere is a Kähler manifold with symplectic structure:
that is a 2-form on where, , and . In order to be quantizable the integral:
must be equal to for some . We conclude that not all spheres satisfy the prequantization condition PC1 but only the spheres with .
Example 3.9. Let be the sphere realized as section of the light-cone by the hyperplane . It is show in  that M can be quantized in the “sense of Souriau”. The quantization is obtained as -fiber bundle where are the so called “KS-transformations” (Kustaankeimo and Stiefel transformations for the regularization of the Kepler problem) which associates to each vector in the light-cone a one-index spinor. In practice, in  , it is described how from the KS-trasformations we obtain the Hopf fibering of the sphere . In this case the quantization condition is the same of the previous example for :
Example 3.10. Let be the unit disk. It is a Kähler manifold with the hyperbolic form and, the prequantization condition PC1, is trivially satisfied. The line bundle is trivial and is the hermitian structure with . An analogue argument shows that the complex space is prequantizable, in the same way.
Example 3.11. Let be a compact Riemann surface of genus . We can think the Riemann surface as the quotient , where is the unit disk of and G is the subgroup of of fractional linear transformation. An element is represented by a matrix:
such that . The action on is defined by . We have
that with is a Kähler manifold because is invariant by the action of G.
Let us consider be a complex atlas on and be holomorphic functions on . We can define global sections such that on we have that . Now from complex analysis we have that:
that is, defining , equivalent to for every
. Then there exists a canonical bundle K. If we consider the projection the pull-back is holomorphically trivial and its holomorphic global sections are of the form on . We can think to global holomorphic sections on K as 1-forms of type on invariant under the action of G. They are forms such that:
for every . We can proceed as in the case of the torus defining an hermitian structure:
that is invariant for every section of K and .
At the end we see that the curvature form of the line bundle K is equal to
. This last term is equal to that is the condition PC4 with the Planck constant equal to 1.
Example 3.15. A very remarkable example is the complex projective space . In this case we can start to show that satisfy the condition PC2 with the convention that (in this case it is a Riemann surface and the result it is true). To see this we can use these relations:
where in this case is the hyperplane bundle. Now we can directly perform the integration:
Thus we have that , that is PC2. The result is true for general n and to see this it is possible to use the chain of isomorphisms: , in order to have .
The result that the Kähler form is integral is also showed in .
Example 3.18. Let be the product of spheres with the same radius r. The manifold M is a Kähler manifold with symplectic structure:
where are the projections on the two spheres with radius and we define the symplectic structure as the pullback respectively of the two Kähler structures on . In order to be quantizable we have that the integral:
for every , must be equal to for some . We conclude that , in this case the product of the two sphere is prequantizable.
We observe also that if the radii of the two spheres are incommensurable then is not prequantizable!
Example 3.20. In this example we describe the prequantization of the Kepler manifold X defined as:
where is the usual Euclidean scalar product. Usually, the manifold X is denoted also with . The Kepler manifold has the following symplectic form:
We can think X as a complex manifold with the identification of X with the complex light cone:
The identification can be realized using the map: such that . We have that is a Kähler manifold where J is the pullback of the complex structure of and
The Kähler form is exact, trivially integral and there is a quantum line bundle L. Moreover for we have that X is simply connected and L is holomorphically trivial over X (details are in  ).
Example 3.23. Let be the cylinder. It can be identified with . It is a Kähler manifold with . There is a symplectic potential and is globally exact and the prequantum line bundle L is trivial (but not unique!).
4. Geometric Quantization
4.1. The Dirac Axioms and Quantum Operators
In his work  , P. Dirac defines the quantum Poisson bracket of any two variablesu andv as:
where is the Plank constant h over 2π. The formula (4.1) is one of the basic postulates of quantum mechanics. We can summarize these postulates as follows. To start we fix a symplectic manifold of dimension n, with the corresponding symplectic structure and an Hilbert space . The quantization is a “way” to pass from the classical system to the quantum system. In this case the classical system (or phase space) is described by the symplectic manifold M and the Poisson algebra of smooth function on M denoted by . The quantum system is described by . We define “quantization” a map Q from the subset of the commutative algebra of observables to the space of operators in . Let be an observable we have that is the corresponding quantum operator. We can summarize the quantum axioms in this scheme:
1) Linearity: , for every scalars and observables;
2) Normality: , where id is the identity operator;
3) Hermiticity: ;
4) (Dirac) quantum condition: ;
5) Irreducibility condition: for a given set of observables , with the property that for every other , such that for all ,
then g is constant. We can associate a set of quantum operators
such that for every other operator Q that commute with all of them is a multiple of the identity.
The last postulate states that in the case we consider a connected Lie group G we say that is a group of symmetries of the physical system if we have the two following irreducible representation: one as symplectomorphisms acting on and another as unitary transformations acting on . For many details about these postulates look .
Example 4.2 (Schrodinger quantization). Let and be the canonical coordinates of position and momentum. In this case that
acts as multiplication and . The Hilbert space and there are the following relations of commutations:
Let us examine the form of these quantum operators. We can start considering the case where with C the configuration space. Now M is a symplectic space with as symplectic form and is prequantizable with an hermitian line bundle . Proceeding in the choice of a potential it is possible to construct for each observable satisfying the quantum axioms. This operator has the following form:
These operators can be “glued” in order to form a global operator on the sections of the corresponding line bundle L.
In general if we have a Kähler manifold M that satisfies the prequantum condition PC4, and if s is a section of the line bundle L, for every , we have an Hamiltonian vector field and the operator that acts on . So if we define the operator:
now it is an hermitian linear operator and, when f is constant, is only the multiplication by f. The formula (4.4) satisfies the Dirac postulate 4.
4.2. Kähler Polarizations
Polarizations of symplectic manifolds are introduced in order to have a dependence of sections of L (that are waves functions) by half the coordinates of the configuration space.
A complex Kähler polarization of M (of complex dimension n) is a smooth complex distribution D (subbundle of TM) that is a map that to each point assigns a linear subspace of , such that:
4) D is involutive, ;
5) is isotropic, that is ;
6) For every in D, .
First note that if D is a polarization, also is a polarization. Second by the Frobenius theorem D is integrale: that is for each point there is an integrable submanifold N (of M) whose tangent space at m is . These N are called leaves of D.
A section s of the line bundle L over M is called polarized if:
for all .
A polarization is real if .
A reference for Kähler polarization is . Furthermore, if a Kähler manifold has one Kähler polarization then it has many, that is changing polarization the Hilbert space change. A good tool in order to consider possible relations between these Hilbert spaces is the pairing we will see next.
Example 4.6 (Kahler polarization). Let be a Kähler manifold, then a Kähler polarization D consists in a submanifold spanned by vectors such that . For example in the Boson-Fock space of dimension 1, that is
with and . The Kähler polarization
is spanned by the antiholomorphic basis. In this case the polarized sections are simply the holomorphic functions. This example it is also called the holomorphic quantization.
Example 4.7 (Real polarization). Let where C is a configuration space. An example of real polarization is the vertical polarization associated to the cotangent bundle. The spann of D is given by the momentum basis . In this case, the wave functions depend only by the position and the condition of preservation of D on an observable f is that:
The vertical polarization is typically associated to the Schödinger representation of quantum mechanics. The holomorphic quantization brings to the Fock-Segal-Bergmann representation.
4.3. Kähler Quantization, Holomorphic Sections and Szegö Kernels
In this subsection let us consider to be a compact Kähler manifold of complex dimension n. Assume we have the prequantization condition:
Observation 4.9. We must do some considerations. The first is that the condition (4.8) is slightly different from PC4, this is due by a different definition of but the geometrical essence is the same. Second, the condition (4.8) ensures that the Dirac postulate is satisfied and, the fact that is integral, ensures the existence of the hermitian line bundle with (4.8) satisfied.
We call the triple the prequantization bundle over . We observe that with the hermitian product h and the volume form , n times defined on M, we can consider the space of smooth holomorphic sections s of M such that:
is finite. The L2-completition of this space is an Hilbert space denoted by or also .
In the compact Kähler case from (4.8) we have that L is a positive line bundle and by the Kodaira embedding theorem there exists a positive tensor power with and global holomorphic sections that give the following embedding:
where . The set is a basis for the space of holomorphic sections of . The .
In the contest of geometric quantization the parameter k can be understood as
a quantum parameter in the sense that . If we imagine that then
and we refind the semiclassical limit. Let us consider the circle bundle X of (defined previously) with to X that for simplicity we denote in the same way. The circle bundle is the boundary of that is a strictly pseudoconvex domain in L. We denote with the induced norm by h so we have that where is defined as where we write and is a smooth function over a and is a local coframe over U. We have a circle action on X denoted by
with infinitesimal generator . As in  we consider the
holomorphic and respectively antiholomorphic subspaces and the correspondent differentials for f smooth on D.
has a Cauchy Riemann structure and, the vectors on D that are elements of (resp. ), are of the form (resp. ). We can choose a basis for these vector spaces and consider the Cauchy Riemann operator defined as . If we define and the Volume form we have that is a contact manifold.
Definition 4.11. We define the Hardy space that admit the following decomposition:
where the subspaces
are called k-Hardy spaces.
Observation 4.14. On the Hardy spaces we have an hermitian product because there is an unitary isometry between sections (between and with ). The notation is used to denote the equivariant smooth section defined on . In analogue way on determines a .
Definition 4.15 (Equivariant Szegö projector). We define the equivariant Szegö projector where we have that:
where is an orthonormal basis of .
Expanding , we have this other definition.
Definition 4.16 (Equivariant Szegö kernel). The equivariant Szegö kernel is:
By a theorem of  it is possible to represent the Szegö kernel as a complex Fourier integral operator (FIO representation).
Theorem 4.17. Let be the Szegö kernel of X, the boundary of a strictly pseudoconvex domain in L. Then there exists a symbol that admit the following expansion:
where such that ( define D), vanish to infinite order along the diagonal and .
We have that is a Fourier integral operator with complex phase and the canonical relation is generated by the phase on . In fact the canonical relation is the lagrangian submanifold of that has as generating function the phase function . The condition that must be true for the parametrization of the lagrangian submanifold is that:
that is when and, on the diagonal we have . Let and let be the symplectic cone generated by the contact form , the real points of consist in the diagonal . We say that has a Toeplitz structure on the symplectic cone .
Observation 4.21. The canonical relation covers an important rule in the theory of quantization as Guillemin and Sternberg remark in  “the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds”.
Now we are ready to recall an important result due to  that uses the method of stationary phase and the microlocal analysis of Szegö and Bergman kernels.
Theorem 4.22 (Zelditch, Tian, Yau). Let M a compact complex projective manifold of dimension n, and let a positive hermitian holomorphic line bundle. Let a Kähler metric on M and a Kähler form. For each , h induces a hermitian metric on . Let be any orthonormal basis of with . Then there exists a complete asymptotic expansion:
for some smooth with .
The proof is on  but we recall briefly the scheme. First we observe that are Fourier coefficients of , so we have the following expression:
where we used the FIO representation of and denote the action on X. Changing variables we have an oscillating integral:
with phase . To simplify the phase we consider an
holomorphic coframe and . We write any as
and for the coordinates we have , with is an almost analytic function on
such that . On X we have and . So for we have:
and on the diagonal we have and . The critical points for are and , the Hessian at this point is . Applying the stationary phase method we find that:
with differential operator of order 2j defined by:
The hypothesis of the theorem 7.7.5 of  is satisfied because the phase has nonnegative imaginary part and critical points are real and independent by x. So we have that has the following form:
The term was expressed in  using that is a projection and
on the diagonal one has that with the restriction of the Levi form to the maximal complex subspace of TX. So:
The main coefficient is a nonzero constant times . Comparing with the leading term of the Hirzebruch Riemann Roch polynomial we have that . This concludes the proof.
The method of the asymptotic expansion of the Szegö kernel in presence of symmetries can be generalized in different ways. The interested reader can consult the following literature     and .
5. The Corrected Geometric Quantization: The Case of Complex Cotangent Bundle
5.1. The Square-Root Bundle, the Half Form Hilbert Space and the Harmonic Oscillator
From now we restrict our attention for the case where with Q a configuration space of dimension n. Now is a Kähler manifold with Kähler form , complex structure J and Kähler potential K. We will consider associated to M a Kähler polarization D.
In this subsection we recall some facts about determinant bundles, “square-root bundles” and metaplectic correction.
Definition 5.1. Let D be a Kähler polarization of a Kähler manifold . The determinant bundle is the complex line bundle for which the sections are n-forms satisfying:
A section is said polarized if
Definition 5.4. By square-root of we will denote the complex line bundle over M such that is isomorphic to .
On we have a partial connection defined for and given by . descends from to . So we have a connection of L denoted by and a partial connection on denoted by . In conclusion we have a partial connection on .
Proposition 5.5. If is an -form on M, then for each point m in M the 2n-form:
is a non negative multiple of the volume form . There is an unique hermitian structure on such that for each section s of :
Let be the holomorphic coordinates on M, then:
the second member is equivalent to (5.6) with and .
Both the hermitian structure on L and on gives in a natural way an hermitian structure on .
Definition 5.10. We define the half form Hilbert space as the space of square integrable polarized sections of .
Before to see how observables are quantizable in this new Hilbert space we recall some properties of sections of . First we can decompose a section s of as . For we have that:
where is a vector field that preserve and .
We have that polarized sections of are such that:
Now if f is an observable on M such that preserves then the quantum operator is given by:
where is the prequantum operator and is a section of . Note that:
If we look for , that is exactly the previous result. Then:
Example 5.16. In the case of , with hamiltonian , the prequantum operator was:
with eigenfunctions (sections) and eigenvalues . These values of the energy are not the desired values. Let us consider a polarization D spanned by . Then
and the corrected quantization gives:
This operators gives , the correct eigenvalues.
Details are in  and .
5.2. The BKS Pairing
Let us consider two Hilbert spaces and given by two different polarizations and on the same line bundle . The two Hilbert spaces are both part of the prequantum Hilbert space . Let be the orthogonal projection. Let be the inner product on , we define the pairing by:
The map is not unitary but if the flow of an observable preserves both polarizations then:
Note that when and are determinated by two complex structures on the symplectic space then a multiple of the projector is in fact unitary.
5.3. The BKS Pairing on the Cotangent Bundle, the Time Evolution and the Schrödinger Equation
In this subsection we enter in details examining the case of cotangent bundles , for some configuration space C. In this case M is a model for classical mechanics, a canonical transformation preserves the symplectic structure . We can consider an Hamiltonian vector field that generates a canonical flow . This flow induces a time evolution of observable via (pull-back for every ). Now this time evolution is lifted, through , at the level of sections of an hermitian line bundle . In particular the lifting flow is:
where (details are in  ) and this lift is unitary. Given an observable that preserve a polarization D then we describe the unitary time evolution by:
Considering the half-form construction, that is the line bundle , we use a pull-back in order to define an evolution in t:
The derivative at reproduce the quantum operator corrected. Now let us assume on M two transverse real polarizations and . Then . We can write M as the product of two configuration spaces , one for position q and the second as the momentum space . Then we have that:
with the generating function. We can define a pairing between and :
where are n-form of the determinant bundle. Then the BKS pairing between sections of and is:
The Liouville measure is and a computation shows that .
Let be the canonical generator, then the determinant will be the identity, corresponds to the base space of whose leaves are surfaces of constant and, because the polarization is transverse to the surfaces of constant q are leaves of . At the end the pairing is given by:
where and are complex wave functions. The projector corresponds effectively to the Fourier transform:
with value in .
5.4. Reconstruction of Schrödinger Equation on the Cotangent Bundle
Let be a -polarized section, an hamiltonian vector field and the lift of the canonical flow in . Then:
this is for every where and for we assume equal to . We have that:
We must to evaluate the following expression:
In our situation, we consider the free particle on , with , the uniform motion on flat space (curvature term is 0) with equation: and . We observe that: . Let be the coordinates with and . Then:
After these refinements we must to evaluate the integrals:
In order to evaluate the integral we start as in  to expand the function around the point q.
The contribution given by the second term in the expansion to the initial integral is 0 due by the parity of the integrand. Substituting inside the integral:
Now we can evaluate the two integral respecting the momentum variable p. The first is:
where is a phase that depends by the metaplectic correction. This integral
as been estimated using the stationary phase method on multidimensional Fresnel integrals (the reference is  ). The second integral to estimate is the following:
where the only terms different from zero inside the integral are when , so we can consider this integral as one dimensional integral with amplitude (evaluating using the result of  ) and a second integral as before where is obtained by p excluding the j coordinate. We find that:
where . After calculations and semplifications the integral (5.32) assume the following form:
that we can write as:
with . Now after a differentiation respect to t at and taking the complex conjugate, we find for every section of :
the Schrödinger equation.
5.5. The Klein-Gordon Equation and the Cotangent Bundle
Let Y be a space-time manifold. Assuming that Y is an orientable manifold, we have that is a symplectic manifold with symplectic form:
where F is the electromagnetic field (a closed 2-form on Y), is the symplectic potential, e is the charge of a particle in an external electromagnetic field and .
The metric g is the Lorentz metric with signature . Let us consider the function defined on . The square root of N represents the mass of a particle in the classical state x. Let us consider the complete hamiltonian vector field and the flux generated by .
In this case the prequantum condition reads as:
for every compact oriented 2-surface on Y with empty boundary.
We can consider a vertical polarization on and the associated Hilbert space that is the .
Under the assumption that and are transverse, it is possible to evaluate . The method is similar as before and is showed in  in the chapter 10. Let us denote with the corrected quantum line bundle, then the BKS kernel is given by:
for local sections .
What the author find at the end of calculations (in  ), is the following expression for the quantum operator:
where is the d’Alembert operator and R is the scalar curvature of the metric g. The operator N is also called the “mass-squared operator” and, the equation:
is the Klein-Gordon equation and the wave function.
In  has been discovered that the critical metrics of the expectation value of have to satisfy Einstein’s equation for suitable energy—momentum tensor.
5.6. Considerations on the BKS Pairing
If the first problem of the prequantization procedure was to have the dependence on half variables by the wave functions (sections of the line bundle), solved by introducing the concept of polarizations, the second problem was the treatment of second order, or higher operators, as the free particle and harmonic oscillator. This second problem has been solved using the BKS construction with the half-forms bundle and the lift of the flow that describes the dynamic of the system. As focused also by  this seems the “exact” choice of the quantum operators, with a suitable metaplectic correction and considering the Hamiltonian dynamics (so the presence of symmetries). The final form of our observables , in the particular case of cotangent bundles for section in the corrected bundle, seems to be:
where P is the usual Fourier transform. As observed always in  generalization on operators of order bigger then 2 is a problem. Generally the Dirac axiom is not true, the time evolution can be not unitary, operators are not always self-adjoint and operators can be not linear. These problems have been showed not only in  but also in the deep study of .
5.7. The BKS Pairing for One Complex and One Real Polarization: the Segal-Bargmann Transform
Let us consider for M a Kähler manifold and two polarizations: one a real polarization and a Kähler polarization. Let us assume , as in the example of the harmonic oscillator. We have that is spanned by and corresponds to q = constant. Let us denote the two Hilbert spaces by and and by and the respective coordinates. The following symplectic potential:
is associated to the Kähler form: . The line bundle L over M is trivial and the space corresponds to the Fock space of holomorphic function with scalar product:
A section s of the line bundle has squared norm and s have the following form:
where is an holomorphic function. A section for the real polarization is the complex function:
Let us consider the half form bundle in order to ensure the metaplectic correction. We can define and the projections given by the BKS pairing. The pairing between the two hilbert spaces is given by the following formula:
where C is a constant we specify at the end. Now we can give and explicit form for P and P’ using the reproducing property of holomorphic function in the second hilbert space:
Now inserting this expression inside the BKS pairing (5.47), we can integrate respect p on the expression in order to have:
Then the expression for the two projections is:
We observe that functions in are mapped in eigenfunctions of the harmonic oscillator. This is the Segal-Bargmann transform and the ground state 1 is mapped to:
5.8. The BKS Pairing between Two Complex Polarizations: The Bogoliubov Transformation
The last case is the case of two complex polarizations and determined respectively by two complex structures . In this case a multiple of the projector operator is unitary and the existence is guarantee by the Stone Von Neumann theorem (see  ). In this brief summary we consider the case in one dimension with and without metaplectic correction. To the polarization we have the Hilbert space that is the Fock space with holomorphic sections:
where is holomorphic respect and . The inner product is:
Now let be a second complex structure that determines and a second hilbert space (always a Fock space) with sections:
where is holomorphic respect (the inner product is the same of the previous case). The BKS pairing is given by:
The pairing determines a projection given by:
where . The projection of (the ground state in ) is:
where and . A good reference is  where also the infinite dimensional case is explained. The same topic is present in .
6. The Bohr-Sommerfeld (Sub)manifolds
Let us consider a lagrangian submanifold of a cotangent space . It is
possible to interpret the quantity as a covariantly constant section s of the
line bundle . Now there is a natural definition of square root of a volume form on replacing by . The section is a section of where is the line bundle of the complex volume forms on . We can consider the triple and to work with the machinery of geometric quantization. Following  , we can study the asymptotic expansion of the squared norm of an “isotropic state” in particular conditions. An isotropic state is a family of sections of an Hilbert space associated to the lagrangian submanifold. Another problem is the study of finding a simultaneous WKB eigenstate of a set of classical observables in involution.
Assuming that must be of the form with that are constant for every . We can consider the volume form of . Assuming for simplicity that is simply connected, the wave functions are well defined if is a single valued global section of . The existences of this section depend by the validity of the Bohr-Sommerfeld condition PC5. Assume that and that is the symplectic potential, then the corrected quantization condition is that:
a section of , admits a single values square root with values in . Here is any loops in ( ). There are other formulations of PC5, for example in  it is given by:
where , as in the first quantization condition PC1, with the difference that there is the quantity d associated to the holonomy from the flat connection of the bundle of half-forms. Authors who worked in the setting of geometric quantization of Bohr-Sommerfeld (sub)manifolds are   and  (where the last two concentrated the attention on the equivariant case).
The real contribution of this paper consists of a new way to derive the Schödinger equation, as in  , with the difference we used a result of Albeverio and Mazzucchi (  ) on asymptotic expansion of Fresnel integrals instead the heat kernel. All possible expressions of quantization conditions have been examined in detail. These notes provide a complete, detailed summary on the state of the art on the geometric quantization showing possible connection between mathematics and physics.
 Kostant, B. (1970) Quantization and Unitary Representations. In: Taam, C.T., Ed., Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics, Vol. 170, Springer, Berlin, 87-208.
 Woodhouse, N.M.J. (1977) Geometric Quantization and the WKB Approximation. In: Bleuler, K., Reetz, A. and Petry, H.R., Eds., Differential Geometrical Methods in Mathematical Physics II, Lecture Notes in Mathematics, Vol. 676, Springer, Berlin, 295-309.
 McClain, T. (2020) Obstacles to the Quantization of General Relativity Using Symplectic Structures. In: Slagter, R.J. and Keresztes, Z., Eds., Spacetime: 1909-2019, Minkowski Institute Press, Montréal.
 Simms, D.J. (1979) Geometric Aspects of the Feynman Integral. In: García, P.L., Pérez-Rendón, A. and Souriau, J.M., Eds., Differential Geometrical Methods in Mathematical Physics, Lecture Notes in Mathematics, Vol. 836, Springer, Berlin, 167-170.
 Rawnsley, J., Cahen, M. and Gutt, S. (1990) Quantization of Kähler Manifolds I: Geometric Interpretation of Berezin’s Quantization. Journal of Geometry and Physics, 7, 45-62.
 Galasso, A. and Paoletti, R. (2018) Equivariant Asymptotics of Szegö Kernels under Hamiltonian U(2)-Actions, Annali di Matematica Pura ed Applicata, 198, 639-683.
 Debernardi, M. and Paoletti, R. (2006) Equivariant Asymptotics for Bohr-Sommerfeld Lagrangian Submanifolds. Communications in Mathematical Physics, 267, 227-263.