topological conditions in geometric and maslov quantization

Topological Conditions in Geometricand Maslov Quantization

Juan Jose Villamarın Castro

A thesis submitted for the degree of

Bachelor in Mathematics

Advisor:

Alexander Cardona Guio Ph.D

Universidad de los Andes

Facultad de Ciencias, Departamento de Matematicas

Bogota, Colombia

2020

Acknowledgements

En primer lugar, quiero agradecer a mi familia por todo el apoyo incondicional a lo

largo de todo este tiempo, sin el cual hubiera sido muy difıcil llegar aca. En especial

a mi mama, mis abuelos y mi tıa por toda la motivacion y el ejemplo a seguir; y a

mi prima por tan buena amistad. Tambien quiero agradecer a todos los maestros

que tuve a lo largo de la carrera, por todo lo que me ensenaron en estos 4 anos.

Especialmente agradezco a mi asesor Alexander Cardona quien fue muy atento,

siempre estuvo ahı para la solucion de dudas y siguio detenidamente mi progreso;

gracias por las numerosas charlas de discusion matematica, en las cuales aprendı

mucho y sin ellas la realizacion de este trabajo serıa imposible. Ademas, tambien

agradezco al profesor Sergio Adarve por sus clases tan provechosas y entretenidas,

y por las charlas no solo de matematicas. Mi interes y gusto por las matematicas y

la geometrıa es gracias a ellos. Para finalizar, quiero agradecer a todos mis amigos,

principalmente a Pedro, Santiago y a Natalia, esta vez por la companıa, las risas y

los inolvidables momentos que pasamos; estos sirvieron para distraernos un poco de

lo academico y hacer mucho mas ameno el tiempo en la universidad.

Contents

1 Introduction 1

2 Symplectic Algebra and Symplectic Geometry 3

2.1 Algebraic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Symplectic Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 3

2.1.2 Symplectic Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The Lagrange-Graßmann Manifold and the Maslov Index . . . . . . . 11

2.2.1 Lagrange-Graßmann Manifold . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Maslov Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Symplectic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Geometric Quantization 21

3.1 Vector Bundles, Connections, Curvature and Chern Classes . . . . . . 21

3.2 Line Bundles and Prequantization . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.1 Kahler Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.2 Real Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.1 Quantization with Real Polarizations . . . . . . . . . . . . . . . 40

3.4.2 Quantization on Kahler Manifolds . . . . . . . . . . . . . . . . . 43

3.5 Half-form Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.5.1 Half-form Correction with Real Polarizations . . . . . . . . . . . 50

3.5.2 Half-form Correction with Kahler Polarizations . . . . . . . . . 57

3.5.3 Corrected Quantization of the One-dimensional Harmonic Os-

cillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

iii

iv CONTENTS

4 Integrality Conditions Including the Maslov Index and Geometric

Quantization 61

4.1 Quantization on Kahler Manifolds . . . . . . . . . . . . . . . . . . . . . 62

4.1.1 Geometric Quantization Including the Maslov Index . . . . . . 63

4.1.2 Quantization of Observables . . . . . . . . . . . . . . . . . . . . . 67

4.2 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Bibliography 75

Chapter 1

Introduction

Among the most important theories in physics is quantum mechanics which, in

contrast to classical mechanics, uses topology in addition to differential geometry.

Specifically, in this work, we will study the well-known geometric quantization pro-

cedure due mainly to B. Kostant and J-M. Souriau, and then compare it to an

alternative geometric quantization method using the Maslov index due to J. Czyz.

Given a symplectic manifold pM,ωq, that generally models a classical physical

system, we wish to quantize the Poisson algebra of observables on it. The idea is

to construct a Hilbert Space H associated to the symplectic manifold and associate

to each f P C8pMq a self-adjoint operator acting on H. This construction is done

in such a way that the Dirac quantization conditions hold. The full process consists

of three steps [Woo91]. The first is called prequantization, in which a topological

condition (on the cohomology class of the symplectic form) gives rise to a geometric

space (a line bundle over M) from which the Hilbert space will be built. The second

step consists of a reduction of the variables involved in quantization, introducing

the so-called polarizations to make sure the wave functions are integrable. Finally,

there is the half-form correction, which is done merely for physical reasons, since

the spectrum of the operators obtained in explicit physical examples results to be

shifted with respect to the known physical spectrum. The main example of this

is the harmonic oscillator, for which the spectrum obtained until the second step

is hn, for a positive integer n, while after the half-form correction we obtain the

usual hpn` 12q spectrum known from physics. Actually, as a matter of fact, both

prequantization and the half-form correction are associated with topological obstruc-

1

tions on the symplectic manifold. In [Czy79] it is shown that both restrictions can

be formulated in a unified way using a topological invariant called the Maslov index.

A subspace L of a symplectic vector space pV,ωq is called lagrangian if it is a

maximal isotropic subspace with respect to ω. The set of all lagrangian subspaces

of V , LpV,ωq, can be given a structure of a smooth manifold. Indeed, a useful rep-

resentation of LpV,ωq is as the homogeneous space UpV qOpLq. This description

allows us to define the Maslov index of a closed curve γ in LpV,ωq, it is the de-

gree of the map det2˝γ ∶ S1 Ñ S1. There is a generalization of the index to closed

curves in a lagrangian submanifold of R2n. This generalization is used in the alter-

native quantization method to define a new topological condition on the manifold.

The advantage of the quantization involving the Maslov index is that the whole

quantization process is done in only one step and, even more importantly, the same

topological condition gives us the corrected spectrum. Thus, both prequantization

and the correction using half-forms are included in a single topological quantization

condition involving the Maslov index.

In this work, we study these two approaches to geometric quantization, with a

particular focus on their equivalence when used to obtain the right physical spec-

trum of the harmonic oscillator. The structure of the document is as follows. In the

next chapter we give an introduction to the symplectic algebra and symplectic ge-

ometry needed to carry out geometric quantization, we also define the Maslov index

and its generalization to lagrangian submanifolds. In the third chapter, we give a

very detailed study of the Kostant-Souriau geometric quantization. We start with a

review of line bundles, hermitian structures, and connections, and then we explain

all the three steps involved in geometric quantization. Although the theory is de-

veloped for the general case, we focus mainly on the cases of Kahler polarizations

(Kahler manifolds) and real polarizations. Each step is illustrated with examples

of important quantum systems such as the harmonic oscillator and a particle with

spin. Finally, in the fourth chapter, we introduce an alternative Kostant-Souriau

quantization and the one by means of the Maslov index, both for the particular

case of compact Kahler manifolds. Then we compute the example of the harmonic

oscillator and show that both quantizations turn out to be equivalent.

2

Chapter 2

Symplectic Algebra and

Symplectic Geometry

In this first chapter we develop the basic theory of symplectic geometry needed to

carry out the geometric quantization program. We also introduce the Maslov index,

which will be important when we study the Maslov quantization. Thus, we begin

with the simple case of symplectic vector spaces and then we define and study the

symplectic group. Later we study the Lagrange-Graßmann manifold, which leads us

to the definition of the Maslov index. Finally, we reproduce what have been done

with symplectic spaces in the general case of symplectic manifolds. For the first part

of the chapter we follow the book [RS13], while for the second we follow [Woo91]

and [Hal13].

2.1 Algebraic Preliminaries

2.1.1 Symplectic Vector Spaces

Definition 2.1. A symplectic vector space is a finite dimensional vector space V

endowed with a non-degenerated, skew-symmetric bilinear form ω, which we will call

the symplectic form.

Example 2.1. Some examples of symplectic vector spaces are the following:

1. V “ R2, and ω “ det ∶ V ˆ V ÐÑ R. Where det “ e1 ^ e2, and e1, e2 is the

dual basis associated to the canonical basis in R2.

3

2.1. Algebraic Preliminaries

2. Let W be any finite dimensional vector space, then V “W‘W ˚ is a symplectic

vector space with ωpv ` α,u` βq “ αpuq ´ βpvq.

The first example can be generalized to R2n taking the symplectic form ω0 “řni“1 e

i ^ ei`n. Note that in every example given above, the symplectic vector

space has even dimension. It is easy to see that every symplectic space V has even

dimension, since if we let Ω be the matrix representation of the symplectic form ω,

and n “ dimV we have:

detpΩq “ detp´Ωtq “ p´1qndetpΩt

q “ p´1qndetpΩq

which implies that n must be even.

The example given above of pR2n, ω0q is very important, in fact, it can be proven

that every symplectic vector space is isomorphic one of them. In order to show this

we need some definitions.

Definition 2.2. Let pV,ωq be a vector space with a bilinear form ω, and W a

subspace of V . We define the orthogonal complement of W with respect to ω as the

subspace given by

Wω“ tv P V ∶ ωpv,wq “ 0 @w PW u.

When ω is a symplectic form on V , we call Wω the symplectic complement of W.

Definition 2.3. Let pV,ωq be a symplectic space. A basis v1, v2, . . . , v2n of V such

that ω “řni“1 v

i ^ vi`n is called a symplectic basis.

Proposition 2.1. Let V be a finite dimensional vector space endowed with an anti-

symmetric bilinear form ω, then there exists a basis in which the matrix representa-

tion of ω is:

rωs “

¨

˚

˝

On In Onˆr

´In On Onˆr

Onˆr Onˆr Or

˛

‹

‚. (2.1)

Proof. We prove this statement by induction on p “ dimV “ 2n` r.

• p “ 1: The condition that ω is anti-symmetric implies that ω “ 0 and the

result is trivial.

4


• pñ p` 1: Assume p ą 1, if ω “ 0 there is nothing to prove and if ω ‰ 0

there exist vectors u, v P V such that ωpu, vq “ 1. Now let W “ Spantu, vu

and consider Wω whose dimension is p´2, by induction hypothesis, there is a

basis v2, . . . vn, vn`2, . . . vp of Wω in which the restriction of ω takes the desired

form. Hence, taking v1 “ u and vn`1 “ v we obtain the basis of V in which

the matrix representation of ω is given by (1.1).

Corollary 2.1. Every symplectic vector space admits a symplectic basis.

Definition 2.4. Given symplectic vector spaces pV1, ω1q and pV2, ω2q, a linear map

T ∶ V1 ÐÑ V2 is called symplectic if T ˚ω2 “ ω1. If T is symplectic and an iso-

morphism it is called a symplectomorphism, and if pV1, ω1q “ pV2, ω2q we call T a

canonical transformation.

Corollary 2.2. If pV,ωq is a symplectic space then every symplectic basis of V

defines a symplectomorphism from pV,ωq to pR2n, ω0q.

Now we are interested in some subspaces that have important characteristics and

will play an important role in what follows.

Definition 2.5. Let pV,ωq be a symplectic space, and W a subspace of V . we say

that W is:

• Isotropic if W ĎWω.

• Coisotropic if Wω ĎW .

• Symplectic if W XWω “H.

• Lagrangian if W “Wω.

It is easy to see that if W is isotropic, then its complement is coisotropic, and

conversely. Also, a lagrangian subspace is the same as an isotropic subspace with

maximal dimension, i.e. with dimension 12dimV . The lagrangian subspaces of a

symplectic vector space are the key to define the Maslov index.

Finally, we see that every symplectic form induces a canonical volume form, also

known as the Liouville form of pV,ωq

5


ε ∶“1

n!p´1q

npn´1q2 ωn, where ωn “ ω ^⋯^ ω.

This defines a volume form since ωn PŹ2n V ˚ and is nonzero because ω is

non-degenerate. Note that every symplectic map preserves also the Liouville form.

2.1.2 Symplectic Group

Now we introduce the notion of the symplectic group, which is important to give a

representation of the Lagrange-Graßmann manifold as a homogeneous space.

Proposition 2.2. Let pV1, ω1q, pV2, ω2q be symplectic vector spaces, and T ∶ V1 Ñ V2

a symplectic map, then

1. T is injective. In particular, if dimV1 “ dimV2, then T is invertible.

2. imT is a symplectic subspace of V2.

3. If pV1, ω1q “ pV2, ω2q, detT “ 1.

Proof. 1. Let v P ker T and let w P V1 be arbitrary; then

ω1pv,wq “ T ˚ω2pv,wq “ ω2pTv,Twq “ ω2p0, Twq “ 0.

Since w was arbitrary and ω1 is non-degenerate we have that v “ 0, and there-

fore T is one-to-one. And if in addition dimV1 “ dimV2, then T is invertible.

2. Let y P imT X pimT qω, then there exists an x P V1 such that Tx “ y, and now

let v P V1 be arbitrary, so

ω1px, vq “ ω2pTx,Tvq “ ω2py, Tvq “ 0.

Which imples that x “ 0 and y “ Tx “ 0, and thus, imT is symplectic.

3. Recall that T preserves the canonical volume form, i.e. T ˚ωn “ ωn, and that

det T is the unique number such that T ˚ωn “ pdet T qωn, then detT “ 1.

6


Now that we know all these properties of symplectic maps, we can consider the

set SppV,ωq of canonical maps of a symplectic vector space pV,ωq. The next propo-

sition shows that actually this set is a Lie Group.

Proposition 2.3. Let pV,ωq be a symplectic vector space, then the set SppV,ωq is a

Lie group and its Lie algebra sppV,ωq consists of all the maps A ∶ V Ñ V such that

ωpAv,wq “ ´ωpv,Awq.

Proof. We show that SppV,ωq is a closed subgroup of GLpV q. First recall that

the bilinear forms on V are in one to one correspondence with the set of matrices

M2nˆ2npRq. Then let Ω be a matrix representation of ω. Then we can identify the

elements of SppV,ωq with the matrices T such that T tΩT “ Ω. Now consider the

map f ∶ M2nˆ2npRq Ñ M2nˆ2npRq, given by A ↦ AtΩA, this f is continuous and

SppV,ωq “ f´1pΩq, so it is a closed subset of GLpV q.

Now we prove that SppV,ωq is a subgroup of GLpV q. Let T,S P SppV,ωq, observe

that

pTSq˚ω “ S˚T ˚ω “ S˚ω “ ω,

so TS P SppV,ωq. Now let T P SppV ωq, then pT´1q˚ω “ pT´1q˚T ˚ω “ pTT´1q˚ω “

Id˚V ω “ ω; thus, T´1 P SppV,ωq. Finally we compute the Lie algebra sppV,ωq.

Recall from basic Lie theory that sppV,ωq “ tA P M2nˆ2npRq ∶ etA P SppV,ωq @t u,then

etAt

ΩetA ðñ AtΩ`ΩA “ O.

Therefore, a map A P sppV,ωq if and only if ωpAv,uq ` ωpv,Auq “ 0.

Now, if a vector space is even dimensional, we can endow it with a complex vector

space structure. In particular, every symplectic vector space can have a complex

structure, but we are only interested in those who behave well with the symplectic

form.

Definition 2.6. Let V be a real even dimensional vector space. A complex structure

on V is a linear map J ∶ V Ñ V such that J2 “ ´IdV .

When a vector space admits a complex structure on it, it acquires a structure of

complex vector space, where the scalar multiplication is given by

7


pa` ibqv “ av ` bJv.

We denote this complex vector space by V C.

Lemma 2.1. In the context of the previous definition, we have that a real linear

map A of V is complex linear if and only if A commutes with the complex structure

J .

Proof.

Apivq “ iAv ðñ AJv “ JAv.

An example of a complex structure on R2 is the matrix

J “

˜

0 ´1

1 0

¸

.

Notice that in fact this is the negative of the matrix representation of det (in the

canonical basis) given in example 2.1.1. In addition, if we compute

rdetsJ “

˜

1 0

0 1

¸

,

it is the euclidean metric in R2; hence, we have detpu, Jvq “ u ¨ v. This example

gives rise to the following definition.

Definition 2.7. Let pV,ωq be a symplectic vector space. We say that a complex

structure J on V is compatible with ω if the bilinear form

gJ ∶“ x¨, ¨yJ ∶ V ˆ V Ñ R, xu, vyJ “ ωpu, Jvq

is an inner product on V . We will denote by OpV, gJq the isometry group of gJ and

by JpV,ωq the set of complex structures compatible with ω.

Note that if J P JpV,ωq, then it is in both SppV,ωq and OpV, gJq, since

ωpJu, Jvq “ xJu, vyJ “ xv, JuyJ “ ωpv, J2uq “ ´ωpv, uq “ ωpu, vq,

xJu, JvyJ “ ωpJu, J2vq “ ´ωpJu, vq “ ωpv, Juq “ xv, uyJ “ xu, vyJ .

8


Theorem 2.1. Let pV,ωq be a symplectic vector space and J a complex structure

compatible with ω. Then the following holds.

1. J also defines a hermitian inner product on V C, given by

hJ ∶“ xx¨, ¨yyJ ∶ V ˆ V Ñ C, xxu, vyyJ ∶“ xu, vyJ ` iωpu, vq.

and for the corresponding isometry group UpV,hJq we have the following rela-

tions:

UpV,hJq “ SppV,ωq XGLCpV q “ OpV, gJq XGLCpV q “ OpV, gJq X SppV,ωq.

(2.2)

2. For every lagrangian subspace L of pV,ωq we have that JL is a complementary

lagrangian, i.e. V “ L‘JL. Moreover, JL corresponds precisely with LK, the

orthogonal complement with respect to gJ .

3. Every endomorphism A of a lagrangian L can be extended to a complex linear

endomorphism of V C, by Apu ` Jvq ∶“ Au ` JAv, where u, v P L. Further-

more, every complex linear endomorphism of V C is completely determined by

its values on L.

4. Every gJ -orthonormal basis ei of a lagrangian subspace L, is an hJ -orthonormal

basis of V C, and the basis e1, . . . , en, Je1, . . . , Jen is a symplectic orthonormal

basis of V .

Proof. 1. The bilinear form hJ clearly defines a hermitian inner product on V C.

We show only the equalities in (2.2).

(a) UpV,hJq “ SppV,ωq XOpV, gJq:

xxTu,TvyyJ “ xxu, vyyJ ðñ xTu,TvyJ ` iωpTu,Tvq “ xu, vyJ ` iωpu, vq.

Note that this also shows one of the inclusions in (b) and (c).

(b) UpV,hJq “ SppV,ωq XGLCpV q: (Ě)Let T P SppV,ωq XGLCpV q, so it is

complex linear and then by Lemma 2.1 we have

xxTu,TvyyJ “ xTu,TvyJ ` iωpTu,Tvq “ ωpTu, JTvq ` iωpu, vq,

9


“ ωpTu,TJvqìωpu, vq “ ωpu, Jvqìωpu, vq “ xu, vyJìωpu, vq “ xxu, vyyJ .

(c) UpV,hJq “ OpV, gJq X GLCpV q: (Ě) If T is orthogonal and complex

invertible, then

xxTu,TuyyJ “ xTu,TuyJ ` iωpTu,Tuq “ xTu,TuyJ “ xu,uyJ ,

“ xu,uyJ ` iωpu,uq “ xxu,uyyJ .

Hence, T is unitary.

2. Let L be a lagrangian subspace, then JL is lagrangian since J is symplectic.

Now, we prove that JL “ LK. Let u, v P L, then xu, JvyJ “ ωpu, J2vq “

´ωpu, vq “ 0, this proves one of the inclusions. For the other one let u P LK,

then J´1u P L, since ωpv, J´1uq “ ωpJv, uq “ ´xu, vyJ “ 0. Therefore, J´1u P

Lω “ L, and V “ L‘ JL.

3. Let A be an endomorphism of L and define A ∶ V Ñ V by Apv ` Juq ∶“

Av ` JAu. This is clearly well defined and R-linear. It commutes with J ,

since AJu “ JAu “ JAu for any u P L; thus, it is C-linear. Now a complex

linear endomorphism A is completely determined by its values on L, since A

commutes with J , and V “ L‘ JL.

4. If e1, . . . en is a gJ -orthonormal basis of L, then xxei, ejyyJ “ xei, ejyJ , there-

fore it is a hJ -orthonormal basis of V C. Finally, e1, . . . , en, Je1, . . . , Jen is a

symplectic orthonormal basis of V since

ωpei, Jejq “ xei, ejyJ “ δij , ωpei, ejq “ ωpJei, Jejq “ xJei, ejyJ “ 0.

10

2.2. The Lagrange-Graßmann Manifold and the Maslov Index

2.2 The Lagrange-Graßmann Manifold and the Maslov

Index

2.2.1 Lagrange-Graßmann Manifold

Now we are going to study the geometry of the space LpV,ωq of all lagrangian

subspaces of a symplectic space pV,ωq, giving it a description as a homogeneous

space. To this end, let J P JpV,ωq, L P LpV,ωq be fixed, and OpLq be the isometry

group of L with respect to the restriction gJ |L. Due to Theorem 2.1.3 we can identify

OpLq with a Lie subgroup of UpV,hJq

OpLq “ tT P UpV,hJq ∶ TL “ Lu.

Theorem 2.2. In the context given above, every L P LpV,ωq gives rise to a diffeo-

morphism

ψ ∶ UpV,hJqOpLq Ñ LpV,ωq, rT s↦ TL. (2.3)

for the proof of this theorem we need first the following lemma, for a proof of it

you can see [Lee12].

Lemma 2.2. Let X be a set, and G a Lie group that acts transitively on X, such

that there is a point p PX whose isotropy group Gp is closed in G. Then, there exists

a unique smooth structure on X with respect to which the given action is smooth,

and the map ψ ∶ GGpÑX, rgs↦ g ¨ p defines a diffeomorphism.

Proof of the Theorem 2.2. Using Lemma 2.2, it suffices to show that UpV,hJq acts

transitively on LpV,ωq and that for any lagrangian L, OpLq is a closed subgroup. The

latter holds by definition, since OpLq “ F´1pgJ |Lq, where F ∶MnˆnpRq ÑMnˆnpRq

is the map T ↦ T tgJ |LT (here we identify linear maps and bilinear forms with their

matrix representation).

For the transitivity, let L,K be lagrangians, let e1, . . . , en and f1, . . . , fn be

gJ´orthonormal basis of L and K respectively, complete them as in Theorem 2.1.4.

Then, the map that takes one basis to the other is both symplectic and orthogonal;

hence, it is unitary, and in particular, it takes L to K.

11


Now that we know that LpV,ωq is a manifold, we will call it from now on the

Lagrange- Graßmann Manifold.

2.2.2 Maslov Index

Now it is time to introduce the notion of the Maslov index of a curve in the

Lagrange-Graßmann manifold corresponding to a symplectic vector space pV,ωq.

Let pV,ωq be a symplectic vector space, and let J be a compatible complex

structure. Now, consider the map det ∶ UpV,hJq Ñ S1 which is clearly continuous.

If we choose a lagrangian L of pV,ωq, then the elements of OpLq are mapped to ˘1,

since we can identify OpLq with Opnq. Recall from basic topology, that if q ∶X Ñ Y

is a quotient map, then a continuous map f ∶ X Ñ Z passes to the quotient if

it is constant on the fibers (of q). Since det does not satisfy this property, we

consider det2 instead, so that it passes to the quotient, and thus, we obtain a map

det2L ∶ LpV,ωq Ñ S1.

Definition 2.8. In the context of the previous paragraph, the Maslov index µpγq of

a closed curve γ ∶ S1 Ñ LpV,ωq is defined as the degree of the map det2L ˝γ ∶ S1 Ñ S1.

Lemma 2.3. The Maslov index is a homotopy invariant, i.e. it is constant on the

homotopy classes of closed curves.

Proof. Since det2L is continuous and the degree is homotopy invariant, it follows that

the Maslov index so does.

Informally, we can think of the degree of a map f ∶ S1 Ñ S1 as the number of

times the map f wraps the circle around itself. The fact that the degree is an integer

number, implies that the Maslov index defines a homomorphism µ ∶ π1pLpV,ωqq Ñ Z,

which we denote by the same symbol. The following proposition tells us that in fact

this is an isomorphism.

Proposition 2.4. The homomorphism µ ∶ π1pLpV,ωqq Ñ Z induced by the Maslov

index, is an isomorphism.

Before we prove this proposition, we need to state the following lemma, its proof

can be found in [Pic08].

12


Lemma 2.4. Let f ∶ GÑ G1 be a Lie group homomorphism, and let H Ď G, H 1 Ď G1

be closed subgroups such that fpHq ĎH 1. If the map

∶ GH Ñ G1H 1, pgHq “ fpgqH 1,

induced from f to the quotient, is surjective then it is a differentiable fibration with

typical fiber f´1pH 1qH.

Proof of the Proposition 2.4. Using the previous lemma, we have that det2L ∶ LpV,ωq Ñ

S1 is a differentiable fibration with typical fiber kerpdet2LqOpLq. We will use the

homotopy long exact sequence

⋯Ñ π1pkerpdet2LqOpLqq Ñ π1pLpV,ωqq Ñ π1pS1

q Ñ π0pkerpdet2LqOpLqq Ñ ⋯

(2.4)

to prove the statement. First, we prove that kerpdet2LqOpLq is diffeomorphic to

SUpV,hJqSOpLq, for this use the Lemma 2.2 with the transitive action of SUpV,hJq

on kerpdet2LqOpLq by left translation. It is clear that the isotropy group of the

identity element IL ¨OpLq is SUpV,hJq XOpLq “ SOpLq. Recall that SUpV,hJq is

simply connected, because it can be identified with SUpnq, and using the homotopy

long exact sequence

π1pSUpV,hJqq Ñ π1pSUpV,hJqSOpLqq Ñ π0pSOpLqq Ñ π0pSUpV,hJqq

we conclude that SUpV,hJqSOpLq is also simply connected. Finally, from p2.4q we

get the exact sequence:

0Ñ π1pLpV,ωqq Ñ ZÑ 0.

Hence µ is an isomorphism.

Now we will give another approach to the Maslov index. Let φ be the standard

angle coordinate on S1, and consider the 1-form

µ ∶“1

2πpdet2

Lq˚dφ

on LpV,ωq.

13


Proposition 2.5. Let γ ∶ S1 Ñ LpV,ωq be a closed curve, then

µpγq “

ż

γµ.

Proof.ż

γµ “

ż

S1

γ˚µ “

ż

S1

γ˚1

2πpdet2

Lq˚dφ “

ż

S1

1

2πpdet2

L ˝ γq˚dφ

“1

2πdegpdet2

L ˝ γq

ż

S1

dφ “ degpdet2L ˝ γq.

Because of the relation of this 1-form µ with the Maslov index, from now on we

will also denote it by µ, and we will call it the Universal Maslov class.

Example 2.2. Consider pR2, detq, in there every 1-dimensional subspace is a la-

grangian subspace, thus LpR2, detq is diffeomorphic to S1. To see this we can also

apply Theorem 2.2, that yields

LpR2, detq “ Up1qOp1q “ S1t1,´1u » S1.

For m P Z, let γm ∶ r0, πs Ñ LpR2, detq be the closed path defined by γmpφq “

cospmφqe1` sinpmφqe2. In terms of the differomorphism given above γmpφq “ eimθ.

We now proceed to compute its Maslov index, then

µpγmq “

ż

γm

µ “

ż

r0,πs

1

2πγ˚mpdet

2q˚dφ “

ż

r0,πs

1

2πpdet2peimφqq˚dφ

“

ż

r0,πs

1

2πpe2imφ

q˚dφ “

ż

r0,πs

1

2πdp2mφq

“m

π

ż

r0,πsdφ “ m.

Notice that we have just done another proof for the fact that µ is an isomorphism

for the case of V “ R2.

14

2.3. Symplectic Geometry

2.3 Symplectic Geometry

Now we end this chapter with a review of symplectic geometry, which is the proper

language for the study of hamiltonian systems. We will establish a relation between

smooth functions on a symplectic manifold and vector fields, called hamiltonian

vector fields, and then we will study their properties. The general version of the

Darboux theorem is stated, which tells us that locally every symplectic manifold

looks like R2n with its canonical symplectic structure, in other words, there are

no local invariants in symplectic geometry. Finally we give a generalization of the

Maslov index for closed curves in lagrangian submanifols of pR2n, ω0q

Definition 2.9. A closed non-degenerate 2-form ω on a manifold M is called a

symplectic form, and the pair pM,ωq is called a symplectic manifold.

Definition 2.10. Let pM,ωq be a symplectic manifold and U an open subset of M .

A 1-form θ P Ω1pUq is said to be a symplectic potential if dθ “ ω, when U “M , θ

is called a global symplectic potential.

Note that @p P M we have that ωp is a symplectic structure on TpM , then the

symplectic form turns each tangent space into a symplectic vector space. This allows

us to classify submanifolds of M as we did previously for subspaces of a symplectic

space.

Definition 2.11. Let pM,ωq be a symplectic manifold, and N an immersed or

embedded submanifold of M , we say that N is:

• Isotropic if TpN is an isotropic subspace of TpM for all p P N ,

• Coisotropic if TpN is a coisotropic subspace of TpM for all p P N ,

• Lagrangian if TpN is a lagrangian subspace of TpM for all p P N .

• Symplectic if TpN is a symplectic subspace of TpM for all p P N .

Definition 2.12. Let pM1, ω1q and pM2, ω2q be two symplectic manifolds. A smooth

map F ∶M1 ÑM2 is called a symplectic if F ˚ω2 “ ω1, and a (local) symplectomor-

phism if in addition F is a (local) diffeomorphism.

Example 2.3. 1. Every symplectic vector space pV,ωq can be considered as a

symplectic manifold with constant symplectic form ω.

15


2. We will generalize the case for V ‘ V ˚ to any manifold Q as follows. Let

M “ T ˚Q. Now we construct a symplectic form ω in M . Define the 1-form θ

on M by θpp,qqpξq ∶“ ppdπξq, where π ∶M Ñ Q is the canonical projection and

ξ P Tpp,qqM . In coordinates θ “ř

j pjdqj or using Einstein notation θ “ pjdq

j.

Taking ω0 “ dθ we obtain a symplectic form on T ˚Q since it is exact and

therefore closed. Identifying open subsets of M with R2n we have that ω0 is

just the canonical symplectic structure on R2n. This example is very important

because we can think of Q as the configuration space of some physical system,

and T ˚Q is then the phase space of the system.

3. Every orientable surface M is a symplectic manifold together with its volume

form, this provide us with a lot of examples of symplectic manifolds.

4. By (3) we know that S2 is a symplectic manifold. Now we ask which other

spheres Sn are also symplectic manifolds. The answer is that, as a matter of

fact, S2 is the only symplectic sphere. This follows simply because if pSn, ωqwould be a symplectic manifold, then the cohomology class rωs in H2

dRpSnq

would be non-zero, but recall that the only non-zero cohomology groups for Sn

are 0 and n. This lefts us only with the option n “ 2.

Recall that when a manifold is endowed with a riemannian metric we can asso-

ciate a vector field gradf to the differential df by raising an index. In symplectic

geometry we can do a similar process but with the symplectic structure, this leads

us to the following definition.

Definition 2.13. Let pM,ωq be a symplectic manifold, and f P C8pMq. Define the

hamiltonian vector field Xf by

Xf ω “ ´df.

Lemma 2.5. In the context of the definition above, the flow of any hamiltonian

vector field preserves the symplectic form, so it is a one-parameter group of local

symplectomorphisms.

Proof.

LXfω “Xf dω ` dpXf ωq “ 0´ d2f “ 0.

16


Another interesting property of the hamiltonian vector fields is the following:

Proposition 2.6. Let γ be an integral curve of a hamiltonian vector field Xf , then

f is constant along γ.

Proof. Let γ be such a curve, then

d

dtfpγptqq “ dfp 9γq “ ´Xf ωp 9γq “ ´ωpXf , 9γq “ ´ωpXf ,Xf q “ 0.

As an easy corollary of this proposition we have a very important result in

physics, the Theorem of Conservation of Energy. Observe that integral curves of Xf

are exactly the solutions of the Hamilton’s equations for the hamiltonian f . Thus,

energy is conserved along the trajectories of the system.

Since for each function f P C8pMq there is a vector field Xf associated to it,

now we wish to define a product on functions similar to the Lie bracket in vector

fields, turning the vector space of smooth functions into a Lie algebra.

Definition 2.14. Let pM,ωq be a symplectic manifold, and f, g P C8pMq define

their Poisson bracket as tf, gu ∶“Xfg “ ωpXf ,Xgq.

Proposition 2.7. The Poisson bracket has the following properties:

• Bilinearity: it is bilinear over R in both f, g,

• Antisymmetry: tf, gu “ ´tg, fu,

• Jacobi Identity:

tf, tg, huu “ ttf, gu, hu ` tg, tf, huu,or equivalently

ttf, gu, hu ` ttg, hu, fu ` tth, fugu “ 0.

The previous proposition tells us that in fact C8pMq is a Lie algebra with the

Poisson bracket. In fact more is true, due to the Jacobi identity one can show that

the map f ↦Xf is a Lie algebra homomorphism:

rXf ,Xgsh “ pXfXg ´XgXf qh “Xftg, hu ´Xgtf, hu “ tf, tg, huu ´ tg, tf, huu

17


“ ttf, gu, hu “Xtf,guh,

since f, g, h P C8pMq were arbitrary functions we obtain that Xtf,gu “ rXf ,Xgs

which is the desired result. The algebra C8pMq together with the product t, u is

called the Poisson algebra of M . Now we state, without a proof, a generalization

of the Corollary 2.2 known as the Darboux theorem. This theorem says that lo-

cally every symplectic manifold is isomorphic to R2n with its canonical symplectic

structure. Therefore, there are no obstructions for flatness on a symplectic manifold

and given two symplectic manifolds of the same dimension they can differ at most

globally.

Theorem 2.3 (Darboux). Let pM,ωq be a symplectic manifold and let p PM . Then

there exists a neighborhood U of p and a coordinate system ppi, qjq, i, j “ 1, . . . n

such that ω “ dpj ^ dqj. In other words, there exists a symplectomorphism between

pU,ω|U q and pR2n, ω0q.

We end this chapter with a generalization of the Maslov index in the context of

lagrangian submanifolds. If M is a lagrangian submanifold of pR2n, ω0q the assigna-

tion p↦ TpL defines a map c ∶M Ñ LpR2n, ω0q.

Definition 2.15. Let M be a lagrangian submanifold of pR2n, ω0q and let L be a

fixed lagrangian subspace. The Maslov index µpγq of a closed path γ ∶ S1 ÑM is

defined as the degree of the map det2L ˝c ˝ γ ∶ S1 Ñ S1. In other words, is the Maslov

index of c ˝ γ.

Let us study the Maslov index of curves when M is a submanifold of R4, which

is a very instructive case. To this end, we use an equivalent definition of the Maslov

index; and thus, we follow [Sou73]. In R4 identify each lagrangian subspace L with

the matrix P of the projection over it along its orthogonal complement. Consider

the closed path P ∶ r0, πs Ñ LpR4, ω0q given by

P pφq “

¨

˚

˚

˚

˚

˝

cos2 φ 0 cosφ sinφ 0

0 0 0 0

cosφ sinφ 0 sin2 φ 0

0 0 0 1

˛

‹

‹

‹

‹

‚

. (2.5)

We claim that this loop is the generator of π1pLpR4, ω0qq. Now, a closed path

γ ∶ S1 Ñ M induces a closed path in LpR4, ω0q by composing with c. Hence c ˝ γ

18


is homotopic to some multiple of P , that is, rc ˝ γs “ mrP s. This integer m is the

Maslov index of γ. Now we proceed to show that m P 2Z if and only if M is an

oriented submanifold of R4. M can be considered as a riemannian manifold with the

metric induced by the euclidean metric in R4. Now, choose an orthonormal basis S0

in TpM , where p is the base point of γ, and parallel transport it along γ. This gives

rise to another orthonormal basis S1 in TpM , hence the matrix S1S´1o is orthogonal.

Therefore, detpS1S´10 q “ 1 for each γ if and only if M is oriented. For example, you

can take the plane torus embedded in R4. In that case the induced metric is again

euclidean, so that parallel transport corresponds with its usual notion, and S1 “ S0.

On the other hand, we can repeat this process with P in LpR4, ω0q. Now the parallel

transport is given by the solution of the differential equation

dS

dφ“dP

dφS. (2.6)

It is very important to remark that when P is not given by p2.5q but is the path

induced by c ˝ γ this equation gives the relation between the information of γ seen

as a curve in M and that of γ seen in LpR4, ω0q. The solution of (2.6) gives two

orthonormal basis S0, S1, and once again S1S´10 is orthogonal. Then, detpS1S

´10 q “

˘1. For the P given in (2.5) the computation shows that detpS1S´10 q “ ´1. The

number detpS1S´10 q is a homotopy invariant, thus we have that

p´1qm “ 1.

Implying m must be an even number. The preceding process can be generalized to

R2n by modifying the matrix P in (2.5). Thus, we have just shown the following

result.

Proposition 2.8. The Maslov index of a curve in M is even if and only if M is

oriented.

19


20

Chapter 3

Geometric Quantization

In this chapter, we study in detail the Kostant-Souriau quantization program.

This procedure consists of three steps: In the first one, it is defined what does

it mean for a symplectic manifold pM,ωq to be quantizable. Then one constructs

a Hilbert space H and self-adjoint (or at least symmetric) operators acting on it

related to smooth functions on M . The vectors on H corresponds to the quantum

states of the system and the operators to the quantum observables. The preceding

process is known as prequantization. In the second step, the notion of a polarization

is introduced to obtain the irreducibility of the Hilbert space H under the action of

the momentum and position operators; and also integrability of the wave functions

for the Kahler case. And the last step is the half-form correction, it is done to

correct the spectrum of the operators. For example, in the harmonic oscillator it

reproduces the well know spectrum of pn` 12qh. We follow mainly the ideas that

appear in [Woo91]. There are also another references with detailed explanation of

geometric quantization such as [EEea98] and [Hal13].

3.1 Vector Bundles, Connections, Curvature and Chern

Classes

Before starting with the quantization procedure we need a brief review or in-

troduction to the concepts of vector bundles (specially of line bundles), connections

and Chern classes, since henceforth they are the context we are going to deal with.

For further reading see [EEea98] and Chap.1 of [Nic13].

21

3.1. Vector Bundles, Connections, Curvature and Chern Classes

Definition 3.1 (Vector Bundle). Let M be a smooth manifold. A vector bundle

over M of rank r is a manifold E with a surjective submersion π ∶ E ÑM satisfying

the following conditions:

1. All the fibers have a r-dimensional K-vector space structure, where K “ C,R.

2. @p PM there exists a neighborhood U of p and a diffeormorphism ϕ ∶ UˆKr Ñ

π´1pUq that restricted to fibers is a linear isomorphism.

Definition 3.2. For us, a line bundle over M is a complex vector bundle of rank 1.

Equivalently, given an open cover U “ pUαqαPA of M together with maps gαβ ∶

Uαβ Ñ GLrpKq satisfying the cocycle conditions gααpuq “ Id and gαγ “ gαβgβγ in Uαβγ ,

we can define a vector bundle of rank r over M as follows. Let

E “

˜

ž

αPA

Uα ˆKr

¸

M

„

where puα, vαq „ puβ, vβq if and only if uα “ uβ P Uαβ and vα “ gαβpuαqvβ, the maps

gαβ are called transition functions. Note that for the case of a line bundle the gαβ’s

are just functions with values in K˚.

Definition 3.3. Let π ∶ E ÑM be a vector bundle, a connection on E is a map

∇ ∶ XpMq ˆ ΓpEq Ñ ΓpEq, pX,sq↦ ∇Xs,

that is K-linear in s, C8pMq-linear in X and satisfies the following product rule:

∇Xfs “ pXfqs` f∇Xs.

A pair pE,∇q is called a vector bundle with connection.

The map ∇X ∶ ΓpEq Ñ ΓpEq is known as the covariant derivative along X. Note

that a connection ∇ on E induces a 1´form ∇s with values in ΓpEq for every section

s of E. Given a local trivialization pU,ϕq of a line bundle pL,∇q, let s0 “ ϕp¨,1q be

the unit section, then

∇s0 “ ´iθs0

for a 1´form θ P Ω1CpUq.

22


Definition 3.4. In the context above, the 1´form θ is called a (local) potential

1´form of the connection. And the 2-form ω “ dθ is called the curvature of the

connection.

Remark 3.1. Although the potential 1-form might be a local differential form and

the curvature depends on it, the curvature is actually a global 2-form.

Definition 3.5. In general, for an arbitrary vector bundle pE,∇q, we define the

Riemann curvature as follows

RpX,Y qs “ r∇X ,∇Y ss´∇rX,Y ss. (3.1)

Remark 3.2. For line bundles ω and R are related by R “ ´iω.

Since we are interested in constructing Hilbert spaces we need to introduce more

structure in the line bundles. More concretely we endow each fiber of L with a

hermitian metric and additionally we need the connection to be compatible with the

metric. This leads us to the following definitions.

Definition 3.6. A hermitian line bundle is a line bundle π ∶ LÑM together with a

hermitian inner product hp on each fiber Lp that varies smoothly, in the sense that

the map LÑ C, vp ↦ hppvp, vpq is a smooth function or, what is equivalent, for every

section s of L, hps, sq is a smooth function on M .

Definition 3.7. Let pL,hq be a hermitian vector bundle over M , we say that a

connection ∇ on L is compatible with the hermitian metric if

Xhps1, s2q “ hp∇Xs1, s2q ` hps1,∇Xs2q,@X P XpMq and s1, s2 P ΓpLq.

The triple pL,h,∇q is called a hermitian vector bundle with connection if pL,hq is a

hermitian line bundle, pL,∇q is a bundle with connection and ∇ is compatible with

h.

Now, given two (complex or real) line bundles L,L1 over M whose transition

functions are gαβ and g1αβ respectively we can construct another line bundles with

the following operations:

• Tensor product: L b L1 defined as the line bundle with transition functions

gαβg1αβ,

23


• Dual: L˚ whose transition functions are pgαβq´1,

• Conjugate: L with transition functions gαβ.

Taking tensor products of L with itself we can construct any integer power of L, even

more, we can construct non integral powers of L. For example, we can construct?L

taking the square root of the transition functions. Clearly when it is not possible to

choose the square roots so that the cocycle conditions still hold?L does not exist.

Alternatively, the square root of L is a line bundle?L such that

?L b

?L “ L.

Furthermore, if L,L1 are line bundles with connection, there is an induced connection

on the new line bundles.

Proposition 3.1. Let pL,h,∇q be a hermitian vector bundle with connection, then

the curvature of the connection is a real 2-form.

Proof. Let X P XpMq and s a trivializing unit section, then:

Xhps, sq “ hp∇Xs, sq ` hps,∇Xsq “ hp´iθpXqs, sq ` hps,´iθpXqsq

“ iθpXqhps, sq ´ iθpXqhps, sq “ i`

θ´ θ˘

pXqhps, sq.

Ôñ θ´ θ “ idhps, sq

hps, sq“ idplnphps, sqq

Which implies that dθ´ dθ “ 0 so that ω is real.

Remark 3.3. • Note that if we have a trivialization where the trivializing sec-

tions satisfy hps0, s0q “ 1 then θ´θ “ 0; therefore, the connection 1-forms are

also real.

• Let pUα, sαq be such a trivialization, then:

hpsα, sαq “ hpgαβsβ, gαβsβq “ gαβgαβhpsβ, sβq,

hence gαβgαβ “ 1 so that every hermitian line bundle admits transition func-

tions with values in S1.

Definition 3.8 (Holonomy). Let M be a smooth manifold endowed with a hermitian

line bundle with connection pL,h,∇q. Given a closed curve γ ∶ ra, bs Ñ M , we can

construct a parallel section s of L along γ, solving the differential equation ∇ 9γs “ 0.

24


In general, spaq ‰ spbq, but they differ by a number z P S1. We call this number the

holonomy of γ, and denote it by Holpγq.

Now one would ask, when is it possible for a hermitian line bundle to have a

connection compatible with the metric? We will find out that this is related with a

condition on the cohomology class of the curvature.

Theorem 3.1. Let M be a smooth manifold and ω a closed 2-form on M , then there

exists a hermitian line bundle with connection pL,h,∇q such that the curvature is ω

if and only if“

ω2π

‰

PH2dRpMq is in the image of H2pM,Zq.

Proof. pÐÑq Suppose there exist such a line bundle and let pUα, sαq be a trivializa-

tion such that hpsα, sαq “ 1. Thus, we know that ω and the potential 1-forms θα

associated with it are real, so there exist functions uαβ such that:

θα ´ θβ “ 2πduαβ

Now since pθα´θβq`pθβ´θγq`pθγ´θαq “ 0 we have that duαβ`duβγ`duγα “ 0.

Hence, uαβ ` uβγ ` uγα “ fαβγ P R. Recall that on Uαβ we have that sα “ gαβsβ

then

∇sα “ ∇gαβsβ “ dgαβsβ ` gαβ∇sβ

Ôñ´igαβθα “ dgαβ ´ igαβθβ

so that we have just obtained another expression for θα´θβ “ idgαβ

gαβand we deduce

that 2πuαβ “ i lnpgαβq, in other words gαβ “ e´2πiuαβ . The cocycle conditions of

gαβ imply that fαβγ is in fact in Z, and thus, the cohomology class associated with

ω2π is integer.

pÐÐq Suppose the cohomology class“

ω2π

‰

is integer, that is, there is a contractible

open cover U “ pUαqαPA of M and smooth functions uαβ such that 2πduαβ “ θα´θβ

and uαβ ` uβγ ` uγα P Z, where the θα are the 1-form potentials. Define gαβ ∶“

e´2πiuαβ , then it follows that the gαβ satisfy the cocycle condition. Therefore, the

gαβ’s define a line bundle L Ñ M . Define a connection ∇ on L given by ∇sα “

´iθαsα where sα is the unit section in Uα and extend by the product rule to any

section. The curvature of this connection is ω by construction. Now we only need

to construct a hermitian metric on L such that the connection is compatible with

25


it. To this purpose, define hp as the canonical complex inner product on C for each

p P Uα, that is, hppsp, s1pq “ zz1 where sp “ pp, zq and s1 “ pp, z1q. Now let s, s1 P ΓpLq

and let f, f 1 be their local representatives, i.e. s|Uα “ fsα and s1|Uα “ f 1sα. Then,

for any p P Uα:

Xhps, s1qppq “Xppff1q “ pXpfqf

1ppq ` fpXpf

1qppq

“ hppXfqsα, f1sαqppq ` hpfsα,Xf

1sαqppq.

On the other hand:

hp∇Xs, s1qppq “ hp∇Xfsα, f

1sαqppq “ hppXfqsα ´ ifθαpXqsα, f1sαqppq

“ hppXfqsα, f1sαqppq ` iθαpXqhpfsα, f

1sαqppq , and

hps,∇Xs1qppq “ hpfsα,∇Xf

1sαqppq “ hpfsα, pXf1qsα ´ if

1θαpXqsαqppq

“ hpfsα, pXf1qsαqppq ´ iθαpXqhpfsα, f

1sαqppq

Adding this last equations we obtain that

Xhps, s1q “ hp∇Xs, s1q ` hps,∇Xs

1q.

so that ∇ is compatible with the metric and pL,h,∇q is a hermitian line bundle with

connection.

It can also be shown that if M is simply connected, then the line bundle con-

structed above is unique up to isomorphism. On the other hand, if M is not simply

connected then the classes of equivalent line bundles are in correspondence with

H1pM,S1q. Furthermore, by duality, the hypothesis of r ω2π s being in the image of

H2pM,Zq is equivalent to the condition that the integral of ω over any 2-cocycle in

M is an integer multiple of 2π.

Now we introduce characteristic classes which will be important in the study of

the geometric quantization including the Maslov index in chapter 4.

Definition 3.9. We say that a polynomial f ∶ MnpCq Ñ C is invariant if for any

A P GLnpCq and X P MnpCq we have fpA´1XAq “ fpXq. The space of invariant

polynomials is denoted by IPn

26


Definition 3.10. Define the polynomials ck for k “ 1, . . . n by

detpI ` tXq “ 1` c1pXqt`⋯` cnpXqtn.

Note that the invariance of det imply that the polynomials defined above are

actually invariant polynomials.

Proposition 3.2. The space IPn is generated by the polynomials c1, . . . cn and also

by the polynomials s1, . . . , sn, where skpXq ∶“ TrpXkq.

With this in mind we can now define the characteristic classes.

Theorem 3.2 (Chern - Weil). Let π ∶ E Ñ M be a complex vector bundle with

connection ∇ and curvature ω, and let f P IPn. Then:

• fpωq defines a global differential form,

• fpωq is a closed form,

• the class rfpωqs P HevendR pMq does not depend on the connection ∇ but on the

class of the vector bundle E.

The class fpEq ∶“ rfpωqs is called the characteristic class of E related to f . When

the polynomial f is one of the ck’s, the class corresponding to it has a special name:

Definition 3.11. Given a vector bundle π ∶ E Ñ M with connection ∇ the class

ckpEq is called the k-th Chern class and the one corresponding to fpωq “ detp1`ωq

is the total Chern class.

Due to the Proposition 3.2 we have that for any polynomial f P IPn the charac-

teristic class fpEq can by written as a polynomial in the Chern classes of E.

There is also an axiomatic definition of the Chern classes, which happens to be

equivalent to this one.

Definition 3.12. Let E be a complex vector bundle over a smooth manifold M .

Then, the Chern classes cipEq and the total Chern class cpEq ∶“

rn2sÿ

k“0

ckpEq are

cohomology classes satisfying:

1. cipEq PH2ipM,Zq and c0pEq “ 1.

27


2. If F ∶ N ÑM is smooth, then cipF˚Eq “ F ˚pcipEqq.

3. If E “ E1‘E2, then cipEq “iÿ

k“0

ckpe1q ¨ci´kpE2q, or equivalently, for the total

Chern class we have cpEq “ cpE1q ¨ cpE2q.

4. Let Ln be the tautological bundle over CPn, then cpLnq “ 1 ` α, where α is

the generator of H2pCPn,Zq corresponding to positive orientation.

Lemma 3.1. The Chern classes satisfy the following properties:

1. if L1, L2 are line bundles over M , then c1pL1 bL2q “ c1pL1q ` c1pL2q.

2. cipE˚q “ ´p1qicipLq

3. If E1,E2 are vector bundles over M such that E1 ‘M ˆCl » E2 ‘M ˆCk,

then cpE1q “ cpE2q.

28

3.2. Line Bundles and Prequantization

3.2 Line Bundles and Prequantization

This is the first step done in geometric quantization. Given a symplectic man-

ifold pM,ωq that satisfies some topological condition it is constructed a Hilbert

space H and operators corresponding to smooth functions on M . There are certain

conditions, known as the Dirac’s quantization conditions, that H and the induced

operators should satisfy:

Q1) The map f ↦ Qpfq is linear,

Q2) If f is constant, then Qpfq is the operator multiplication by f ,

Q3) If tf1, f2u “ f3, then rQpf1q,Qpf2qs “ íhQpf3q.

pQ1q and pQ2q tells us that the map f ↦ Qpfq is a Lie algebra homomorphism

(up to a constant), and thus, H is a representation of C8pMq. In other words, the

Poisson bracket is the classical analogue to the quantum commutator. Furthermore,

pQ2q ensures that for example, the uncertainty principle holds. One could ask for

more conditions on the operators and the Hilbert space such as self-adjointness or

some kind of irreducibility respectively. Even though it seems impossible to achieve

such a quantization, geometric quantization comes as close as possible.

A good guess for starting is to set H “ L2pMq with inner product

xψ,φy “

ż

Mψφε,

where ε is the Liouville form of M , and Qpfq “ íhXf . Because of Proposition 2.7,

this assignment satisfies pQ1q and pQ3q, but this does not satisfy pQ2q since constants

are mapped to zero. We correct this by adding a new term to the operator i.e.

Qpfq “ íhXf ` f.

Now constants are mapped to their multiplication operator, nevertheless pQ3q

does not hold anymore. The following assignment is the one that does the trick.

Qpfqψ ∶“ íhrXf ´ ih´1θpXf qsψ ` fψ. (3.2)

29


It is linear, and all the constant are still acting by multiplication. Now we prove

that it satisfies pQ3q. For this, note that the expression in brackets corresponds to

that of a connection, in this case, in the bundle of smooth functions in M . In view

of this, we can rewrite expression (3.2) as follows

Qpfqψ “ íh∇Xfψ ` fψ, (3.3)

and due to the definition of curvature and the fact that r∇X , f s “Xf , we have

rQpfq,Qpgqs “ ríh∇Xf ` f,íh∇Xg ` gs “ ´h2r∇Xf ,∇Xg s ´ ihr∇Xf , gs ` ihr∇Xg , f s

“ ´h2píh´1ωpXf ,Xgq `∇rXf ,Xgsq ´ ihXfg ` ihXgf

“ ihtf, gu ´ h2∇Xtf,gu ´ ihtf, gu ` ihtg, fu

“ íhpíh∇Xtf,gu ` tf, guq “ íhQptf, guq.

The definition above suggests that instead of working with L2pMq it is better to

work with square integrable sections of a hermitian line bundle with connection over

M , whose curvature is ωh. Since we want the potentials of the connection to be

the symplectic potentials we need that ωh satisfies the condition of Theorem 3.1,

leading us to the following definition.

Definition 3.13. A symplectic manifold pM,ωq is prequantizable if r ω2πh s is in the

image of H2pM,Zq.

Given a quantizable manifold pM,ωq there exist a hermitian line bundle with

connection pL,h,∇q which will be called the Prequantum line bundle. We define

the prequantum Hilbert space, denoted by Hpre, as the completion of the square

integrable smooth sections of L with respect to the inner product

xψ1, ψ2y ∶“

ż

Mhpψ1, ψ2qε. (3.4)

Definition 3.14. Given a smooth function f on M, we define the prequantum op-

erator on the subspace of compactly supported smooth sections by

Qprepfqψ ∶“ íh∇Xfψ ` fψ.

Proposition 3.3. For any f P C8pMq the operator Qprepfq satisfies the Dirac

quantization conditions.

30


Proof. Copy the proof we have done before.

Proposition 3.4. The operator Qprepfq, for f P C8pMq is symmetric.

Proof. We prove here a more general statement. For smooth functions f, g, where g

has compact support, we have that

ż

Mtf, guε “ 0,

For instance,

ż

Mtf, guε “

ż

MXfgε “

ż

MLXf gε` gLXf ε

“

ż

MLXf pgεq “

d

dt

ż

Mgε “ 0.

Thus, taking g “ hpψ,ψ1q we compute

0 “

ż

Mtf, hpψ,ψ1quε “

ż

MXfhpψ,ψ

1qε “

ż

Mhp∇Xfψ,ψ

1qε`

ż

Mhpψ,∇Xfψ

1qε.

Therefore, ∇Xf is antisymmetric which implies that Qprepfq is symmetric.

Example 3.1 (Canonical Operators). Consider pM,ω0q as in Example 2.3.2 that

is, M “ T ˚Q and ω0 “ dpj ^ dqj. Take as potential 1-forms θ “ pjdq

j. Then the

prequantized operators corresponding to position and momentum are given by

Qpreppjq “ ´ihB

Bqj, and (3.5)

Qprepqjq “ qj ` ih

B

Bpj. (3.6)

Proof. In the first place, pM,ω0q is prequantizable because the integral of ω over any

2-cycle in M is 0 (by Stokes theorem). Then there is a natural choice of L and ∇,

choose L “M ˆC and ∇ “ d´ ih´1θ. We now compute the associated hamiltonian

vector fields. Let Xqj “XaBpa ` YaBqa , then

ω0pXqj , Bqbq “ ´dqjpBqbq ñXb

“ ´δjb ,

ω0pXqj , Bpbq “ ´dqjpBpbq ñ ´Y b

“ 0.

31


Hence, Xqj “ ´Bpj . A similar process yields that Xpj “ Bqj . Now given a smooth

section ψ “ fs0 of L, where s0 is a trivializing section, we obtain

Qprepqjqψ “ íh∇X

qjψ` qjψ “ íh∇Bpj fs0 ` q

jfs0

“ ihB

Bpjfs0 ´ ihf∇Bpj s0 ` q

jfs0

“ ihB

Bpjfs0 ` fθ

ˆ

B

Bpj

˙

s0 ` qjfs0

“ ihB

Bpjfs0 ` q

jfs0

and for the prequantum momentum operator, we have

Qpreppjqψ “ íh∇Xpjψ ` pjψ “ íh∇Bqj fs0 ` pjfs0

“ íhB

Bqjfs0 ´ ihf∇B

qjs0 ` pjfs0

“ íhB

Bqjfs0 ´ fθ

ˆ

B

Bqj

˙

s0 ` pjs0

“ íhB

Bqjfs0 ´ pjfso ` pjfs0 “ íh

B

Bqjfs0.

Finally, the result holds identifying ψ with its local representative.

The preceding example is one of the reasons why the prequantization process

is not a good quantization procedure. Note that though the momentum operators

correspond to the usual ones, there is an extra term in the position operators. In

the next section, we fix this problem by introducing polarizations. Also, we will

construct some subspaces of Hpre that remain invariant under the action of the

position and momentum operators, showing that Hpre is far from being irreducible.

The next example shows another reason why we need more steps in the quantization

process.

Example 3.2 (Harmonic Oscillator). Consider pR2, ω0q as before, with the hamil-

tonian Hpp, qq “`

p2 ` q2˘

2, and potential θ “ 12ppdq ´ qdpq. Then for all n P Z,

nh is an eigenvalue for QprepHq.

Proof. In this case, the hamiltonian vector field is XH “ pBq ´ qBp. In fact, the

32


prequantum operator is just QprepHq “ íhXH , since

QprepHqψ “ íh∇XHψ `Hψ

“ íhpXH ´ ih´1θpXHqqψ `Hψ

“ íhXHψ ´1

2ppdq ´ qdpqppBq ´ qBpqψ `Hψ

“ íhXHψ ´1

2pp2` q2

qψ `Hψ

“ íhXHψ.

Changing to polar coordinates, we have that XH “ Bθ. Now, define

ψnpr, θq “ fnprqeinθ

where fn is a smooth function satisfying

ż

R|fn|

2rdr ă8.

Therefore

QprepHqψn “ íhB

Bθfnprqe

inθ“ íhfnprqine

inθ

“ nhfnprqeinθ“ nhψn.

So that for each n P Z ψn is an eigenvector of QprepHq with eigenvalue nh.

As said before, the result above suggests that prequantization is not enough and

we need to improve our quantization procedure. Recall from canonical quantization,

that the spectrum of the harmonic oscillator is pn` 12qh, for non-negative n. But

the spectrum of QprepHq is not even bounded below and does not consist of half-

integer multiples of h. We fix the first problem in the next section, with a suitable

choice of a polarization. However, for the second, we have to wait until section (3.5)

where the space of half-forms is introduced.

33

3.3. Polarizations

3.3 Polarizations

We now introduce the so-called polarizations. The motivations for this concept are

the problems appearing in the examples of the last section. What causes them is

that in canonical quantization the wave functions depend on n variables, while the

elements in Hpre depend on 2n variables. Choosing a polarization and a subspace of

Hpre corresponding to it, fixes the problem. But, before that, we develop the theory

of polarizations.

Definition 3.15. A distribution D on a smooth manifold M , is a subbundle of

the tangent bundle TM . A complex distribution is a subbundle of the complexified

tangent bundle TCM

Remark 3.4. A (complex) distribution D is called integrable if every point in M is

contained in a submanifold N such that Dp “ TpN (or Dp “ TCp N) for all p P N . By

the Frobenious theorem, a necessary and sufficient condition for D to be integrable

is that for any X,Y P ΓpDq we have rX,Y s P ΓpDq. This last property is called

involutivity.

Definition 3.16. Given a symplectic manifold pM,ωq, a polarization on M is a

complex distribution P such that

• Pp is a lagrangian subspace of TCp M , i.e. dimCPp “ n and for all Xp, Yp P Pp

ωpXp, Ypq “ 0 (where ω has been extented to TCp M),

• P is involutive (integrable).

• The dimension of Pp X P p X TpM is constant.

Observe that if P is a polarization, then P is also a polarization. In addition, the

distribution P XP is invariant under conjugation; thus, D ∶“ P XP X TM is a real

distribution whose complexification DC is P X P . Since Pp is lagrangian for all p,

Dp is an isotropic subspace of TpM . We call D the isotropic distribution (associated

to P ). Now, consider the complex distribution P ` P , as before, it is invariant

under conjugation. Then, P ` P is the complexification of the real distribution

E “ pP ` P q X TM . Note that for each p we have Dωpp “ Ep, therefore Ep is

coisotropic and we call E the coisotropic distribution.

Definition 3.17. Let P be a complex polarization on a symplectic manifold pM,ωq.

We say that P is strongly integrable if the coisotropic distribution E is integrable.

34

3.3. Polarizations

Now, we can define a pseudo hermitian form h for ΓpP q by

hpX,Y q “ ´iωpX,Y q. (3.7)

Note that ker h “ P XP . One of the inclusion holds because P XP is isotropic,

for the other, if X P ΓpP q is in ker h we have that ωpX,Y q “ 0 for all Y P ΓpP q,

or which is the same ωpX,Y q “ 0 for all Y P ΓpP q. Hence, X P ΓpP q because P is

also lagrangian. So, h induces a non-degenerate hermitian form h in the quotient

P pP X P q. This h leads us to the following definition.

Definition 3.18. We say that a polarization P on pM,ωq is of type pr, sq if the

hermitian form h defined above has signature pr, sq.

The complex dimension of P pP XP q is r` s, so that the dimension of P XP is

n´pr`sq, which is also the (real) dimension of D. So, we call the number n´pr`sq

the real directions of P .

Definition 3.19. Let pM,ωq be a symplectic manifold together with a hermitian line

bundle with connection pL,h,∇q. A polarization P is adapted to the connection if

for each p P M there is a neighborhood and a local potential 1-form θ such that

θpP q “ 0. We also say that θ is adapted to P .

We now state without a proof an important result relating strongly integrable

polarizations with the adapted ones. For a prove of it you can see [Woo91].

Proposition 3.5. If P is a strongly integrable polarization on a symplectic manifold

pM,ωq endowed with a hermitian line bundle with connection pL,h,∇q, then it is

adapted to the connection.

We are interested specially in two kind of polarizations, Kahler and real, which

will be important for the quantization procedure.

3.3.1 Kahler Polarizations

Definition 3.20. Let pM,ωq be a symplectic manifold and P a polarization on M .

P is called a Kahler polarization if P X P “ 0 and P has type pn,0q.

A Kahler polarizations is also called completely complex, since it has no real

directions. The fact that P X P “ 0 implies that for this type of polarization we

have that TCp M “ Pp ‘ P p.

35

3.3. Polarizations

Now we establish an important relation between Kahler polarizations and Kahler

manifolds. It will be shown that they are equivalent, in the sense that a symplectic

manifold admitting a Kahler polarization is a Kahler manifold and, for a Kahler

manifold the spaces of holomorphic and antiholomorphic sections are Kahler polar-

izations.

Definition 3.21. An almost complex structure in a smooth manifold M is a selec-

tion for each p PM of a complex structure in TpM that varies smoothly with respect

to p. In other words, is a p1,1q´tensor field J on M , such that J2 “ ÍdTM .

As it happens with vector spaces, the existence of a complex structure in a

manifold implies it has even dimension. In the case M is a symplectic manifold we

want a complex structure to behave well with the symplectic structure. Observe

that when Jp is extended to TCp M it has eigenvalues ˘i, therefore

TCp M “ T p1,0qp M ‘ T p0,1qp M.

Where Tp1,0qp M is the eigenspace corresponding to i and T

p0,1qp M the one correspond-

ing to í. This spaces are called the holomorphic and antiholomorphic subspaces

respectively. They define two distributions T p1,0qM and T p0,1qM called holomorphic

and atiholomrphic distributions, and their sections are denoted by Xp1,0q and Xp0,1q.

Definition 3.22. An almost complex structure J on M is a complex structure if

the distributions defined above are integrable.

Remark 3.5. The fact that J is a complex structure on M is equivalent to have a

complex coordinate system pzj , zjq such that

JB

Bzj“ i

B

Bzjand J

B

Bzj“ í

B

Bzj,

or for some real coordinates ppj , qjq

JB

Bpj“

B

Bqj, J

B

Bqj“ ´

B

Bpj.

Definition 3.23. A symplectic manifold pM,ωq endowed with an (almost) complex

structure J is called an (almost) Kahler manifold if the p0,2q tensor field

gpX,Y q ∶“ ωpX,JY q (3.8)

36

3.3. Polarizations

is a riemannian metric on M .

Proposition 3.6. If pM,ω,Jq is a Kahler manifold then T p1,0qM and T p0,1qM are

Kahler polarizations. Conversely, if P is a Kahler polarization then there exist a

complex structure J for which pM,ω,Jq is a Kahler manifold.

Proof. The first part follows immediately, since for X,Y holomorphic vectors

ωpX,Y q “ ωpJX,JY q “ ωpiX, iY q “ ´ωpX,Y q.

Thus, ωpX,Y q “ 0. Involitivity follows by definition of a Kahler manifold. Finally,

note that T p1,0qM “ T p0,1qM , which implies that dimTp1,0qp M XT

p1,0qp M XTpM “ 0.

So that T p1,0qM is a polarization, we call it the holomorphic polarization. In the

case of the antiholomorphic distribution a similar process yields the same result.

Conversely, if P is a Kahler polarization, we have that TCp M “ Pp ‘ P p. Define Jp

on TCp M by JpX “ iX if X P Pp and JpX “ ´iX if X P P p. Note that if X P TpM

we can write it as X “ Z ` Z, for Z P Pp. Then JpX “ iZ ´ iZ. It is easy to see

that J is a complex structure on M . Now we prove it is compatible with ω.

gppX,Y q “ ωpX,JpY q “ ωpW `W, iZ ´ iZq

“ ´iωpW,Zq ` iωpW,Zq “ iωpZ,W q ´ iωpZ,W q

“ ωpZ `Z, iW ´ iW q “ gppY,Xq,

gppX,Xq “ ωpX,JpXq “ ωpW `W, iW ´ iW q

“ ´2iωpW,W q “ 2hpW,W q

ě 0.

Therefore, pM,ω,Jq is a Kahler manifold.

Remark 3.6. In the context given above, more is true. In fact, it is possible to

show that there is a coordinate system pzj , zjq on M such that P corresponds exactly

with the holomorphic polarization.

3.3.2 Real Polarizations

Definition 3.24. A polarization P on a symplectic manifold pM,ωq is a real polar-

ization if P “ P .

37

3.3. Polarizations

As opposed to the latter case, this kind of polarization has only real directions,

hence its name. As a matter of fact, for a real polarization, the isotropic distribution

D is actually lagrangian. Conversely, if D is a lagrangian distribution on M , its

complexification DC is a real polarization. We conclude that real polarizations are

in correspondence with lagrangian distributions. The following proposition shows

another advantage of working with real polarizations.

Proposition 3.7. Let P be a real polarization on a symplectic manifold pM,ωq.

Then, D “ P X TM has a frame consisting only of hamiltonian vector fields.

38

3.4. Quantization

3.4 Quantization

As said in previous sections the idea is to reduce the number of variables on which

the states depend. To this end, we consider the sections that are covariantly constant

in the directions of a polarization P . Though considering these sections fixes the

number of variables, these sections do not need to be square integrable. Moreover,

it is possible that even if ψ is such a section, then Qprepfqψ is no more covariantly

constant for some f . Thus, we also have to consider a subset of the functions that

will be quantized. Then, in the quantization step, the new quantum Hilbert space,

and the functions whose associated operators will act on it are defined. Through all

this section let pM,ωq be a prequantizable symplectic manifold, with prequantum

line bundle pL,h,∇q and let P be a fixed polarization on M .

Definition 3.25. A smooth section ψ of L is called a polarized section (with respect

to P ) if

∇Xψ “ 0 (3.9)

for all X P P . The space of polarized sections is denoted by ΓP pLq.

It is important to mention that the choice of P or P in p3.9q is not standard and

depends on the author.

Definition 3.26. The polarized Hilbert space HP is the closure of the smooth

square integrable polarized section of L.

Remark 3.7. Recall that every closed subspace of a Hilbert space is again a Hilbert

space, then HP is really a Hilbert space.

Definition 3.27. A function f P C8pMq is quantizable (with respect to P ) if

Qprepfq preserves the the subspace of polarized sections. In other words Qpreψ P

ΓP pLq provided ψ P ΓP pLq. If f is quantizable we denote the restriction of Qprepfq

to HP by QP pfq.

Now we want to find a condition that determines the quantizability of a function

in terms of its hamiltonian vector field. This lead us to the following definition.

Definition 3.28. A vector field X (possibly complex) on M is said to preserve the

polarization P if Y P ΓpP q implies rX,Y s P ΓpP q.

39

3.4. Quantization

Note that for example if X P P , the involutivity condition on P implies that X

preserves P . An equivalent definition for real vector fields to preserve a polarization

is that its flow φt preserves the polarization, in the sense that dφtpPpq “ Pφtppq. This

follows from the identity rX,Y s “ LXY . In this case, X preserves P if and only if

it also preserves P .

Proposition 3.8. Let f P C8pMq. If Xf preserves P , then f is quantizable.

Proof. Assume f is such that its hamiltonian vector field Xf preserves P , and let ψ

be a polarized section and X P P , then

∇XQprepfqψ “ ∇X`

íh∇Xfψ ` fψ˘

“ íh∇X∇Xfψ `∇Xfψ

“ íhpíh´1ωpX,Xf qψ `∇Xf∇Xψ `∇rX,Xf sψq ` pXfqψ ` f∇Xψ

“ ωpXf ,Xqψ` pXfqψ

“ 0.

(The third equality follows form the definition of the Riemann curvature).

Note that since f is real valued, we could have put Xf preserves P instead.

3.4.1 Quantization with Real Polarizations

After having studied the general theory of the quantization step, we can now com-

pute some examples. In this case we consider a quantizable symplectic manifold

pM,ωq, with prequantum line bundle pL,h.∇q and a fixed real polarization P . The

main example in this case, is the vertical polarization, for which the position and

momentum operators have the desired form.

Example 3.3 (Canonical Operators Revisited). Consider M “ T ˚Q, ω0 “ dpj^dqj

and the global potential θ “ pjdqj. Let P be the vertical polarization, that is, P “

SpantBpju. Then, the position and momentum operators act on HP by

QP pqjqψ “ qjψ, and (3.10)

QP ppjqψ “ íhB

Bqjψ. (3.11)

Proof. The first thing we are going to prove is that the polarized sections are of the

form

ψpp, qq “ φpqq. (3.12)

40

3.4. Quantization

For this, notice that

∇BBpj “B

Bpj´ ih´1θ

ˆ

B

Bpj

˙

“B

Bpj.

Thus, the sections that are covariantly constant in the Bpj directions are just the

sections constant in these directions. In the second place, we need to prove that the

function pj and qj are quantizable. So

Qpreppjqφ “ ´ihB

Bqjφ,

Qprepqjqφ “ ih

B

Bpjφ` qjφ “ 0` qjφ “ qjφ.

Both Qprepqjqφ and Qpreppjqφ are again polarized since they are only dependent on

the qj ’s. The last computation also shows that the action of the canonical operators

is the one desired.

In the last example, the HP is known as the position representation, or the

Schrodinger representation of pM,ω0q. We can also consider the case for the po-

larization spanned by tBqju. For that polarization HP is called the momentum

representation.

Example 3.4. Take pM,ω0q as before, but now take θ “ ´qjdpj and let P “

SpantBqju. Then, the canonical operators act as follows

QP ppjqψ “ pjψ, and (3.13)

QP pqjqψ “ ih

B

Bpjψ. (3.14)

Proof. This time the polarized sections can be written as

ψpp, qq “ φppq, (3.15)

and a similar process gives the result.

Remark 3.8. 1. Note that in each representation, its corresponding operator acts

only by multiplication. This happens due to the choice of the adapted potential in each

case. In fact, if for the momentum representation we would have chosen θ “ pjdqj

41

3.4. Quantization

again, then the polarized sections would be of the form

ψpp, qq “ eipaqahφppq. (3.16)

In this case we have

QP ppjq “ eipaqahpjφppq, and (3.17)

QP pqjq “ iheipaq

ah B

Bpjφppq. (3.18)

2. Note that there are no non-zero square integrable polarized section, since

xψ,ψy “

ż

Mφ2ε (3.19)

diverges. To see this, notice that the integral over the p directions (when ψ is in

the Schrodinger representation) does not converge. This happens because we are still

integrating over 2n variables while the ψ’s now just depend on n. This is one of the

reasons of the introduction of the half-forms in the next section.

Now we give another example where the existence of non-zero polarized sections

is again a problem. But before doing so, we need first a definition.

Definition 3.29. A lagrangian submanifold L of M is said to be Bohr-Sommerfeld

whenever the holonomy (see Definition 3.8) of every closed curve in L is trivial,

meaning it is equal to 1.

Example 3.5. Let M “ T ˚S1 » S1 ˆ R, ω0 “ dx ^ dφ (here x denotes the p

coordinate and φ the q coordinate), and pL,h,∇q be the trivial line bundle. Take

θ “ xdφ a potential 1-form and let P be the polarization spanned by Bφ. Then the

integral manifolds S1 ˆ txu of P are Bohr-Sommerfeld if and only if xh P Z. This

implies that every polarized section must be zero.

Proof. Let γx ∶ r0,2πs Ñ M be given by γxptq “ peit, xq. We are going to compute

the holonomy of each γx. It is easy to see that the solution of the parallel transport

equation ∇ 9γxs “ 0 is

sptq “ s0 exp

˜

i

h

ż γxptq

γxp0qθ

¸

(3.20)

42

3.4. Quantization

Thus, the holonomy is given by

Holpγxq “ exp

ˆ

i

h

ż

γx

θ

˙

“ exp

ˆ

i

h

ż 2π

0γ˚xxdφ

˙

“ exp

ˆ

ix2π

h

˙

.

Implying the holonomy is trivial only when xh is an integer number. Now we

compute the polarized sections. Let ψ be such a section, then

∇Bφψ “ 0 ðñBψ

Bφ´i

hxψ “ 0.

The solution is given by ψpx,φq “ Cpxq exppixφhq. Hence, ψ must vanish whenever

xh is not an integer, otherwise Cp0q ‰ Cp2πq and ψ is not a smooth section (it

is not even continuous). Since this subset is dense in M , we conclude that every

polarized section is the zero section.

3.4.2 Quantization on Kahler Manifolds

Let us now focus on the study of quantization with Kahler polarizations. The main

example this polarizations is the harmonic oscillator, which is explained below.

Example 3.6. (Harmonic Oscillator Revisited) Consider pR2, ω0q with the hamil-

toian Hpp, qq “ pp2 ` q2q2. Let L “ R2 ˆ C be the trivial line bundle, and take

θ “ ppdq ´ qdpq2 as in Example 3.2. Let J be the complex structure given by

JBp “ Bq and JBq “ ´Bp, and let P be the holomorphic polarization. Then, QP pHq

has nh as eigenvalue for each n P N.

Proof. Introduce the complex coordinates z “ p` iq and z “ p´ iq, therefore

B

Bz“B

Bp´ i

B

Bq, and

B

Bz“B

Bp` i

B

Bq. (3.21)

Now we claim that every polarized section is of the form

ψpz, zq “ φpzq exp´

´zz

4h

¯

. (3.22)

43

3.4. Quantization

Where φ is a holomorphic function. Notice that

∇Bz “B

Bzí

hθ

ˆ

B

Bz

˙

“B

Bzí

h

1

4ppdq ´ qdpq

ˆ

B

Bp` i

B

Bq

˙

“B

Bz´

i

4hpip´ qq “

B

Bz`

z

4h.

So that the ψ given in (3.22) is a polarized section. Now let ψ be a polarized

section. Every section can be written in the form ψpz, zq “ φpz, zqe´zz4h since the

exponential factor in non-zero. Now we compute

∇Bzψ “B

Bz

´

φpz, zqe´zz4h¯

`z

4hφpz, zqe´zz4h

“Bφ

Bze´zz4h ´

z

4hφe´zzh `

z

4hφpz, zqe´zz4h

“Bφ

Bz.

This implies that φ must be a holomoprhic function, because ψ is polarized.

As before, QprepHq “ íhppBq ´ qBpq, thus

QprepHqφpzqe´zz4h

“ íhppBq ´ qBpqφpp` iqqe´H2h

“ íh

ˆ

ipBφ

Bz´pq

hφ´ q

Bφ

Bz`pq

hφ

˙

e´H2h

“ zhBφ

Bz.

Showing that H is quantizable. Finally, taking ψnpzq “ zne´zz4h for n P N we

obtain QP pHqψn “ nhψn, as we wanted.

Remark 3.9. Notice that the spectrum obtained in the previous example does not

correspond to the physical spectrum of the harmonic oscillator. It is shifted be a

factor of 12 h.

Example 3.7 (Segal-Bargmann Representation). The idea is to generalize the one-

dimensional harmonic oscillator of the example above to the n-dimensional harmonic

oscillator and also introduce the ladder operators. Then consider pM,ω0q as in

Example 3.3, but this time choose instead the holomorphic polarization given by the

44

3.4. Quantization

complex structure

J

ˆ

B

Bpj

˙

“B

Bqj, and J

ˆ

B

Bqj

˙

“ ´B

Bpj.

That is, P “ SpantBzju, where zj “ pj ` iqj. Then the complex functions zj , zj and

H “ zj zj2 are quantizable and their operators can be written in the form

QP pzjq “ zj , QP pzjq “ 2hB

Bzj, and QP pHq “ zj h

B

Bzj. (3.23)

Proof. First notice that in complex coordinates ω0 “i2dzj^dzj , and let us take the

potential θ “ i4pzjdzj ´ zjdzjq. For simplicity we will use from now on the notation

B

Bzj“ Bj and

B

Bzj“ Bj . (3.24)

A similar process to that of the preceding example shows that the polarized sections

are of the form

ψpzj , zjq “ φpzjq exp

ˆ

´zjzj

4h

˙

An easy computation shows that the hamiltonian vector fields are

Xzj “ ´2iBj , Xzj “ 2iBj and Xzj zj2 “ i`

zjBj ´ zj Bj˘

.

Hence, their associated operators are

Qprepzjq “ íh∇´2iBj` zj “ zj ,

Qprepzjq “ íh`

2iBj ´ ih´1θp2iBjq

˘

` zj “ 2hBj `zj

2

QprepHq “ íh´

Xzj zj2 ` ih´1θpi

`

zjBj ´ zj Bj˘

¯

`zj zj

2“ íhXH

This shows that the functions zj , zj are quantizable, and for H we have

íhXHψpzj , zjq “ h`

zjBj ´ zjBj˘

φpzjqe´zkzk4h

“ h

ˆ

zjBφ

Bzj´zjzj

4hφ`

zjzj

4hφ

˙

e´zkzk4h

“ hzjBφ

Bzje´zkzk4h.

45

3.4. Quantization

So that H is also quantizable. We now find eigenvectors for the operator QP pHq.

As before, Nh with N P N is an eigenvalue for it. For each n-tuple pN1, . . . ,Nnq P Nn

define

φN1⋯Nnpzjq “ zN11 zN2

2 ⋯zNnn , and ψN1⋯Nnpzj , zjq “ φN1⋯Nne´zjzj4h. (3.25)

Then QP pHqψN1⋯Nn “ NhψN1⋯Nn where N “ N1 ` ⋯ ` Nn. This means that

the space of homogeneous polynomials of degree N are eigenvectors of QP pHq with

corresponding eigenvalue Nh. Moreover, if ψ has degree N then QP pzjqψ has eigen-

value pN ` 1qh and QP pzjqψ has eigenvalue pN ´ 1qh. Then they are known as

raising and lowering operators.

Finally we quantize the angular momentum. One way of understanding the

following example is with the orbit method, due to Aleksandr Kirılov. Nevertheless,

we follow a simpler approach.

Example 3.8 (Spin). Consider the sphere of radious ρ, S2ρ with the symplectic

structure

ωρ “ 2iρ3

pρ2 ` zzq2dz ^ dz. (3.26)

Here z is given by the stereographic projection. Note that ωρ is just the volume

form associated with riemannian metric induced by R3 divided by ρ. Then, S2ρ is

quantizable only when ρ is a half-integer multiple of h. Choose P the holomor-

phic polarization given by the complex coordinates pz, zq. Let x1, x2, x3 represent

the cartesian coordinates in S2ρ , then they are quantizable and satisfy the following

commutation relation

rQP pxaq,QP px

bqs “ ´ihεabcQP px

cq. (3.27)

Moreover, HP is finite dimensional and is an irreducible representation of sup2q.

Proof. The first part follows by direct calculation,

ż

S2ρ

ωρ “1

ρV olpS2

ρq “ 4πρ.

46

3.4. Quantization

Let pL,h,∇q be the line bundle constructed in Theorem 3.1. and choose the potential

θρ “iρ

ρ2 ` zzpzdz ´ zdzq.

It is easy to see that the polarized sections are of the form ψpz, zq “ φpzqe´K2h,

where φ is a holomorphic function and K “ 2ρ lnpρ2 ` zzq. In this coordinates we

have that

x1“pz ` zqρ2

ρ2 ` zz, x2

“ípz ´ zqρ2

ρ2 ` zzand x3

“ ρzz ´ ρ2

zz ` ρ2.

The hamiltonian vector field for for x1 is

Xx1 “i

2ρ

`

pρ2´ z2

qBz ´ pρ2´ z2

qBz˘

,

and with the chosen potential, we have that the prequantum operator is

Qprepx1q “ íhpXx1 ´ ih´1θρXx1q ` x1

“ íhXx1 íρ

ρ2 ` zzpzdz ´ zdzq

i

2ρrpρ2

´ z2qBz ´ pρ

2´ z2

qBzs ` x1

“ íhXx1 ´1

2pρ2 ` zzqpzpρ2

´ z2q ` zpρ2

´ z2qq `

ρ2pz ` zq

ρ2 ` zz

“ íhXx1 ´1

2pρ2 ` zzqpρ2pz ` zq ´ zzpz ` zqq `

ρ2pz ` zq

ρ2 ` zz

“ íhXx1 `z ` z

2.

Similar calculations for x2 and x3 yield

Xx2 “ ´1

2ρ

`

pρ2` z2

qBz ` pρ2` z2

qBz˘

,

Xx3 “ i pzBz ´ zBzq ,

and

Qprepx2q “ íhXx2 ´

ipz ´ zq

2,

Qprepx3q “ íhXx3 ´ ρ.

We now prove that x3 is a quantizable function, for x1 and x1 the computations are

47

3.4. Quantization

long and we won’t do them. Then,

Qprepx3q

´

φe´K2h¯

“ hpzBz ´ zBzqφe´K2h

´ ρφe´K2h

“ h

ˆ

zBφ

Bz´

zzρφ

hpρ2 ` zzq`

zzρφ

hpρ2 ` zzq

˙

e´K2h ´ ρφe´K2h

“

ˆ

zhBφ

Bz´ ρφ

˙

e´K2h,

Which shows the desired result. We also have that

QP px1qφe´K2h “

ˆ

h

2ρpρ2´ z2

qBφ

Bz` zφ

˙

e´K2h,

QP px2qφe´K2h “

ˆ

ih

2ρpρ2` z2

qBφ

Bz´ izφ

˙

e´K2h

The commutation relations of (3.27) follows directly by the commutation relations

of the Poisson bracket. Now, notice that we can solve the eigenvalue equation for

QP px3q. For n P N define ψnpz, zq “ zne´K2h, then QP px

3qψn “ pnh´ ρqψn. Since

the ψ1ns must be square integrable, there is a greatest n for which this holds. In

fact, if ρ “ N2 h, we have

xψn, ψny “

ż

S2ρ

pzzqne´Kh2iρ3

pρ2 ` zzq2dz ^ dz

“

ż

S2ρ

pzzqn2iρ3

pρ2 ` zzqN`2dz ^ dz

“

ż

R2

px2 ` y2qn4ρ3

pρ2 ` x2 ` y2qN`2dxdy

Which converges if and only if n ă N ` 1, showing that HP is finite dimensional

and is generated by ψ0, . . . , ψN . Finally, consider the operators L`, L´, L3 on HP

defined by

L`φe´K2h

“`

QP px1q ` iQP px

2q˘

φe´K2h “

ˆ

´z2h

ρBφ` 2zφ

˙

e´K2h,

L´φe´K2h

“`

QP px1q ´ iQP px

2q˘

φe´K2h “ pρhBφq e´K2h,

L3φe´K2h

“ ´QP px3qφe´K2h.

Notice that the eigenvectors ψn now have eigenvalues ρ´nh, and L` and L´ act as

48

3.4. Quantization

follows:

L`ψn “ p´nhρ` 2qψn`1

L´ψn “ ρhnψn´1.

As before, L` and L´ are known as the raising and lowering operators. The com-

mutation relations in (3.27) imply

rL3, L˘s “ r´QP px3q,QP px

1q ˘ iQP px

2qs

“ ´`

rQP px3q,QP px

1qs ˘ irQP px

3q,QP px

2qs˘

“ ´`

´ihQP px2q ˘ i2hQP px

1q˘

“ ˘hL˘,

rL`, L´s “ ´2irQP px1q,QP px

2qs

“ 2hQP px3q

which are the commutation relations (up to the constant h) of slp2q “ sup2qC. The

computations above show that in fact, HP is isomorphic to the irreducible represen-

tation of sup2q of dimension N ` 1, as we wanted to prove.

49

3.5. Half-form Correction

3.5 Half-form Correction

The half-form correction is the last step to accomplish the Kostant-Souriau geometric

quantization of a symplectic manifold. The main motivation for this step is to

correct the spectrum obtained in the case of known physical systems, such as the

harmonic oscillator. Specifically, in the case of the harmonic oscillator (see Remark

3.9) after the correction the spectrum is shifted by a factor of 12 h, giving us the known

spectrum from physics. To this purpose, we consider instead of L the line bundle

LbδP , the tensor product between the prequantum line bundle and the square root

of the canonical bundle (defined later). Thus, it is a topological correction of the

spectrum. This last step fixes another problem, as a matter of fact, the polarized

sections of L b δP are also square integrable for the real case, thus, we obtain a

non-trivial quantum Hilbert space.

3.5.1 Half-form Correction with Real Polarizations

Definition 3.30. Let pM,ωq be a symplectic manifold and a polarization P on M ,

we denote the space of leaves by E (endowed with the quotient topology), that is,

every point of E is identified with a maximal integral submanifold of the polarization

P .

From now on we assume that E is a smooth manifold of dimension n, that the

quotient map Π ∶ M Ñ E is a smooth and that kerdΠp “ PRp “ Pp X TpM . This

assumption is done because we want to integrate over E . The idea is to relate an

n-forms of E with special n-forms of M allowing us to integrate.

Definition 3.31. Let pM,ωq be a quantizable manifold with a real polarization P .

We define the canonical bundle of P as the real line bundle KP whose sections are

the real n-forms α such that X α “ 0, for every X P P “ P . We say that a section

α of KP is polarized whenever X dα “ 0 for all X P P .

Remark 3.10. Note that KP is a line bundle since the its sections are in correspon-

dence with the alternating forms of top degree in TpMPRp which is n dimensional.

Before continuing with the theory we give an illuminating example for the case

when M is a cotangent bundle.

50


Example 3.9. Let M “ T ˚Rn with the canonical symplectic form ω0. Take the

vertical polarization P “ SpantBpju. Then the sections of KP are of the form

α “ fpp, qqdq1^⋯^ dqn, (3.28)

and α is polarized if f is independent of the pj’s.

Now we give an important relation between n-forms of E and M , as we mentioned

before.

Proposition 3.9. A section α of KP is polarized if and only if there is an n-form

α of E such that α “ Π˚α.

Proof (Sketch). pðq If α is an n-form on E and X P P , then X P kerdΠ, which

implies that X dα “ dpΠ˚αq “ Π˚dα “ 0.

pñq M is locally symplectomorphic to T ˚Rn, and by the preceding example a po-

larized section α has the form

α “ fpqqdq1^⋯^ dqn,

defining a form α in Rn whose pullback is equal to α. Notice that Rn plays the

role of the leave space, that is, locally the leaf space is diffeomorphic to Rn, so that

α is the desired form, it is easy to check that this process in fact defines a global

differential form α such that Π˚α “ α.

Proposition 3.10. For each vector field X on M preserving P , there exists a Π-

related vector field Y on E such that for a polarized section α “ Π˚α of KP we have

that

LXα “ Π˚ pLY αq . (3.29)

Now the idea is to have the notion of a connection and Lie derivative on KP that

preserves it. For a vector field X P P we define the partial connection ∇ on KP ,

by

∇Xα “X dα. (3.30)

Notice that since α is a section of KP , then ∇Xα “ LXα.

Proposition 3.11. Let X be a vector field preserving P , i.e. rX,Y s P P for every

Y P P . Then if α is a section of KP then LXα is again a section of the canonical

bundle. Furthermore, if α is polarized, so does LXα.

51


Proof. Let α be section of canonical bundle, X a vector field preserving P , and

X1, . . . ,Xn vector fields on M , where X1 P P . Then

LXαpX1, . . . ,Xnq “ XαpX1, . . . ,Xnq ´ αprX,X1s, . . . ,Xnq

´

nÿ

j“2

αpX1, . . . , rX,Xjs, . . . ,Xnq

“ 0.

The firs term vanishes because X1 P P , for the same reason the terms in the sum

with j ě 2 also vanish; for j “ 1 rX,X1s P P , then αprX,X1s, . . . ,Xnq “ 0. Now

suppose α is a polarized section, by the Cartan’s magic formula we have

d pLXαq “ d pdpX αq `X dαq “ d2pX αq “ 0.

Hence, Y pdLXαq “ 0 for any Y P P .

Now we are going to define the square root of the canonical bundle. This new

bundle, (when it exists) is tensorized with the prequantum line bundle L, producing

the new bundle where the whole quantization is done. We want to induce the Lie

derivative and the partial connection from KP on the square root bundle, this will

allow us to define the Quantum Hilbert Space and the operators on it.

Definition 3.32. Let P be a real polarization on a quantizable manifold pM,ωq.

The square root of the canonical bundle, is a line bundle δP such that there

exists an isomorphism between δP b δP and KP . The section of δP are the so-called

half-forms.

If in addition we assume that the leaf space E is orientable, we obtain that the

canonical bundle KP is trivial, with trivializing section Π˚η where η is an oriented

volume form on E . This implies that the transition functions on KP are real and

positive, allowing us to construct?KP as in Section 3.1, taking the square root of

the transition functions. Thus, we can define δP as?KP . From now on we will also

take this assumption.

Proposition 3.12. Let X be a vector field on M preserving P , then the Lie deriva-

52


tive LX on KP induces a Lie derivative on δP , such that

LXpµb νq “ LXpµq b ν ` µbLXpνq, (3.31)

LXpfµq “ pXfqµ` fLXµ (3.32)

for all µ, ν sections of δP . If X P P , then the partial connection on KP is also

induced on δP , and satisfies

∇Xpµb νq “ ∇Xpµq b ν ` µb∇Xpνq, (3.33)

∇Xpfµq “ pXfqµ` f∇Xµ. (3.34)

Proof. Here we prove the result for the Lie derivative, the calculations for the partial

connection are very similar. Let µ0 be a trivializing section of δP , if such a Lie

derivative exists then (3.31) must be satisfied and

LXpµ0 b µ0q “ LXµ0 b µ0 ` µ0 bLXµ0 “ 2LXµ0 b µ0.

Recall that for a 1 dimensional vector space V , and a non-zero v0, the map V Ñ

V b V , v ↦ v b v0 is an isomorphism. Therefore, define LXµ0 as the inverse (via

the preceding map) of LXpµ0 b µ0q2. Now, for each µ section of δP , we have that

µ “ fµ0, for some function f . Thus, define LXµ applying (3.32). Finally we verify

our definition satisfies (3.31), let µ “ fµ0 and ν “ gµ0, so

LXpµb νq “ LXpfµ0 b gµ0q “ LXpfgµ0 b µ0q

“ LXpfgqµ0 b µ0 ` fgLXpµ0 b µ0q

“ pXfqµ0 b ν ` µb pXgqµ0 ` fLXµ0 b ν ` µb gLXµ0

“ LXµb ν ` µbLXν.

Remark 3.11. Both, the Lie derivative and the partial connection can be naturally

extended to sections of the complexified canonical bundle δCP ∶“ δP bC. We also say

that a section α of δCP is polarized, whenever ∇Xα “ 0 for all X P P .

Now that we have developed the theory of half-forms, we are going to define the

Quantum Hilbert Space corresponding to our quantization.

53


Definition 3.33. Let pM,ωq be a quantizable manifold with pre-quantum line bundle

pL,∇, hq, P a real polarization on M and δP a square root of the canonical bundle

of P . Let LP be the line bundle Lb δCP , with a pairing x, y (with values in KP ) given

by

xψ b µ,φb νy ∶“ hpψ,φqµb ν. (3.35)

We also define a partial connection on LP given by

∇Xpψ b µq “ p∇Xψq b µ` ψ b∇Xµ, (3.36)

for all X P P . The triple pLP “ LbδCP , x, y,∇q is called the quantum line bundle.

Definition 3.34. A section s “ ψbµ of the quantum line bundle is polarized if for

all X P P we have

∇Xpψ b µq “ 0. (3.37)

Definition 3.35. The Half-form Space is the vector space of polarized section of

LP , together with the norm

s2 ∶“

ż

EĆxs, sy, (3.38)

where s “ ψbµ. We define the Quantum Hilbert Space HQ as the completion of

the polarized square integrable sections (in the sense of (3.38)) of LP , with hermitian

metric

xxs1, s2yy “

ż

EČxs1, s2y. (3.39)

Remark 3.12. In (3.38) and (3.39), by Čxs1, s2y we mean the differential form α on

E corresponding to xs1, s2y as in Proposition 3.10.

Example 3.10. Let M “ T ˚Rn with the usual symplectic structure ω0, take P

the vertical polarization and the symplectic potential θ “ pjdqj. In this case E is

diffeomorphic to Rn, so let α “ dq1^⋯^ dqn be the volume form in Rn. Thus, KP

is trivial with trivializing section α “ Π˚α, therefore we can take δP “?KP with

trivializing section?α. Then every polarized section s of LP has the form ψ b

?α

where ψ is a polarized section of L.

Proof. Note that every section LP has the form ψ b?α, now suppose ψ b

?α is

polarized, that is

p∇Bpjψq b?α` ψ b∇Bpj

?α “ 0. (3.40)

54


First we calculate the second term, for this recall Proposition 3.13, then

∇Bpj

?αb

?α “ ∇Bpjα “ Bpj dα “ Bpj Π˚pdαq “ 0.

Hence, (3.40) holds if and only if ∇Bpjψ “ 0, i.e. ψ is a polarized section of L.

Furthermore, in this case, the the norm in the Half-form space is given by

xxψ b?α,ψb

?αyy “

ż

Rn|ψpqq|2dq1

^⋯^ dqn.

Showing that there are non-zero square integrable polarized sections, that is, HQ is

not trivial. This solves the problem illustrated in Remark 3.8.2, recall that HP “ t0u

before the half-form correction..

After defining our quantum Hilbert space, we need to extend the definition of

the operator QP pfq for a quantizable function f to a new operator Qpfq on HQ.

For this, we use the extension of the Lie derivative from KP to its square root, as

the following definition says.

Definition 3.36. Let pM,ωq be a quantizable manifold with quantum line bundle LP

and quantum Hilbert space HQ. For a quantizable function f we define the operator

Qpfq on the smooth sections of HQ by

Qpfqpψ b µq ∶“ QP pfqψ b µ´ ihψ bLXfµ. (3.41)

Proposition 3.13. The assignation f ↦ Qpfq satisfies the Dirac quantum condi-

tions.

Proof. For this notice that every section of LP can be written as ψ b?α, where

α “ Π˚α for a volume form α in E . Observe that ψb?α is polarized if and only if

ψ is a polarized section of L, since

∇Xpψ b?αq “ p∇Xψq b

?α` ψ b∇X

?α “ p∇Xψq b

?α.

Now we need to prove that Qpfq preserves polarized sections. Let ψ b?α be a

55


polarized section, then

∇XQpfqpψ b?αq “ ∇XpQP pfqψ b

?α´ ihψ bLXf

?αq

“ ∇XQpfqPψb?α`QpfqPψ b∇X

?α

íh∇Xψ b?α´ ihψ b∇XLXf

?α.

The first three terms vanish because ψ and?α are polarized, and for the last term

we have

∇XLXf?α “ LXLXf

?α “ LXfLX

?α` rLX ,LXf s

?α

“ LXfLX?α`LrX,Xf s

?α “ 0.

Finally, it is clear that it satisfies Q1 and Q2 from Section 3.2 , it only remains to

show Q3:

rQpfq,Qpgqsψ b?α “ QpfqpQpgqpψb

?αqq ´QpgqpQpfqpψ b

?αqq

“ Qpfq`

QP pgqψ b?α´ ihψ bLXg

?α˘

´Qpgq`

QP pfqψ b?α´ ihψ bLXf

?α˘

“ QP pfqQP pgqψ b?α´ ihQP pgqψ bLXf

?α

íh`

QP pfqψbLXg?α´ ihψ bLXfLXg

?α˘

´QP pgqQP pfqψ b?α` ihQP pfqψ bLXg

?α

ìh`

QP pgqψ bLXf?α´ ihψ bLXgLXf

?α˘

“ rQP pfq,QP pgqsψ b?α` píhq2ψ b rLXf ,LXg s

?α

“ íhQtf,guψ b?α` píhq2ψbLXtf,gu

?α

“ íhQtf, gupψ b?αq.

Moreover, it can be shown that for a real valued function f P C8pMq preserving

the polarization P , the operator Qpfq is symmetric [Hal13].

Example 3.11 (Canonical Operators). As in Example 3.3 let M “ T ˚Q with the

canonical symplectic form ω0 “ dpj ^ dqj and symplectic potential θ “ pjdqj. Let

P be the vertical polarization, and choose δP the trivial square root with trivializing

56


section?µ for a volume form µ on Q. Then HQ is nonzero and the position and

momentum operators act as follows

Qpqjqpψ b?µq “ qjψ b

?µ, (3.42)

Qppjqpψ b?µq “ ´ih

B

Bqjψ b

?µ. (3.43)

Proof. Recall that a section ψb?µ is polarized whenever ψ is polarized, that is, ψ

only depends the qj ’s. Now notice that polarized square integrable sections exists,

since (identifying a function ψ constant along the fibers with a function on Q) we

have

xxψ b?µ,ψ b

?µyy “

ż

Q|ψ|2µ

Taking ψ P L2pQq we obtain a polarized square integrable section of LP . Finally, we

compute

Qpqjqpψ b?µq “ QP pq

jqψb

?µ´ ihψ bLX

qj

?µ

“ qjψ b?µ´ ihb∇´Bpj

?µ “ qjψ b

?µ,

Qppjqpψ b?µq “ QP ppjqψ b

?µ´ ihψ bLXpj

?µ

“ ´ihB

Bqjψ b µ,

where the last equality follows from the fact that LXpjµ “Xpj dµ` dpX µq “ 0,

which implies LXpj?µ “ 0.

Remark 3.13. We can prove even more. All the quantizable functions have the

form f “ f1 ` f2 where f1 is constant on the fibers and f2 is linear on the fibers.

Furthermore, when M “ R2 » T ˚R we have that Qpfq is the Weyl quantization of

f .

3.5.2 Half-form Correction with Kahler Polarizations

It is important to mention that in this case this last step is merely done for physical

reasons, since polarized square integrable sections of L exists. Indeed, carrying out

this step reproduces the exact spectrum of the harmonic oscillator and makes this

case more similar to the real one. In fact, there are no big differences between the

definitions involving real polarizations and those involving Kahler polarizations.

57


Definition 3.37. Let pM,ω,Jq be a Kahler manifold and let P be the holomorphic

polarization. The canonical bundle KP is the complex line bundle of n-forms α

such that X α “ 0 for all X P P . A section of KP is called polarized if X dα “ 0

for all X P P .

Definition 3.38. Define the pairing x, y ∶ ΓpKP q ˆ ΓpKP q Ñ C8C pMq by

inxα,βyωn “ α^ β. (3.44)

As before, a square root of the canonical bundle is a line bundle δP such

that tensored with itself is isomorphic to KP . The only difference is that here δP is

a complex line bundle and in the real case it was real. We can also define on the

canonical bundle the partial connection and the Lie derivative, for vector fields in

P and preserving P respectively. Of course they induce a partial connection and a

Lie derivative on the square root δP . Also the pairing x, y of Definition 3.38 induces

a pairing on δP by

xµ, νy “a

xµb µ, ν b νy. (3.45)

Definition 3.39. Let pM,ω,Jq be a quantizable Kahler manifold with pre-quantum

line bundle pL,∇, hq, let P be the holomorphic polarization on M and δP a square

root of the canonical bundle of P . Let LP be the line bundle Lb δP . We also define

a partial connection on LP given by

∇Xpψ b µq “ p∇Xψq b µ` ψ b∇Xµ, (3.46)

for all X P P . The pairing from equation (3.45) induces a hermitian form on LP

given by

hP pψ1 b µ1, ψ2 b µ2q ∶“ hpψ1, ψ2qxµ1, µ2y. (3.47)

The triple pLP “ Lb δP ,∇, hP q is called the quantum line bundle.

As before, a polarized section s of LP is one that satisfies ∇Xs “ 0 for all X P P .

Then, the Quantum Hilbert Space HQ is the completion of the polarized square

integrable sections of LP given by the inner product

xxψ1 b µ1, ψ2 b µ2yy “

ż

MhP pψ1 b µ1, ψ2 b µ2qε. (3.48)

58


The operators on HQ are given again by the equation (3.41), that is

Qpfqpψ b µq “ QP pfqψ b µ´ ihψ bLXfµ, (3.49)

for any f P C8pMq quantizable function.

3.5.3 Corrected Quantization of the One-dimensional Harmonic

Oscillator

Consider the Kahler manifold pR2, ω0, Jq with hamiltonian Hpz, zq “ zz2. Let

pL,h,∇q be given by L “ R2 ˆC, ∇ “ d´ ih´1θ and h is the usual hermitian form

in C » R2. Let P be the holomorphic polarization, i.e. P “ SpantBu, and choose δP

the trivial square root of KP . We will see that the spectrum of QpHq is pn` 12qh

for n P N.

Recall form Example 3.6 that the polarized section of L have the form

ψpzq “ φpzqe´zz4h,

foe a polarized function φ. Note that KP is spanned by dz; thus, a polarized section

of LP can be written as

s “ φpzqe´zz4h b?dz.

Now we compute the action of QpHq on a polarized section:

QpHqpψ b?dzq “ QP pHqψ b

?dz ´ ihψ bLXH

?dz

“ zhBzφe´zz4h

b?dz ´ ihψ bLXH

?dz.

For the second term, we need to compute the Lie derivative, then

2LXH?dz b

?dz “ LXHdz “ dppBq ´ qBpqz

“ dpip´ qq “ idz “ i?dz b

?dz.

59


Therefore, LXH?dz “ i

?dz2, so that

QpHqpψ b?dzq “ zhBzφe

´zz4hb?dz `

h

2ψb

?dz

“ hpzBzφ` φ2qe´zz4h

b?dz.

Finally, letting sn “ zne´zz4h b?dz we have that QpHqsn “ pn` 12qhsn, as we

wanted to prove.

60

Chapter 4

Integrality Conditions Including

the Maslov Index and

Geometric Quantization

In this chapter, we give another approach to geometric quantization after the correc-

tion using half-forms. In Kostant-Souriau quantization we divide the process into

three steps, now, the idea is to establish a quantization procedure involving in a

single topological condition the full procedure described in Chapter 3. To this end,

we introduce an modification of the Kostant-Souriau quantization in which there is

no need of the half-form correction and a quantization step after the prequantiza-

tion process; because polarizations are taken into account from the beginning and

the geometric correction due to half-forms is replaced by a topological condition in

the manifold. All these conditions are encoded in a cohomology class q, and if it is

integral then there is hermitian line bundle with connection Q with class q, which

we will call later a quantization bundle. After this, we introduce another way of con-

structing a line bundle involving the Maslov index; thus, the topological obstruction

in the manifold is now replaced by a condition on the orbits and its Maslov index.

We end the chapter with a detailed example of the harmonic oscillator using the

modified KS-quantization and the one including the Maslov index. The content of

this chapter is based on the ideas appearing in [Czy79].

61

4.1. Quantization on Kahler Manifolds

4.1 Quantization on Kahler Manifolds

This is a modification of the quantization presented in Section 3.4 for the case of

Kahler manifolds. Recall that for Kahler manifolds there exist two canonical Kahler

polarizations, the holomorphic and antiholomorphic polarizations. Conversely for a

sympletic manifold with a Kahler polarization there exists a complex structure J

turning the manifold a Kahler manifold.

Definition 4.1. Let pM,ω,Jq be a Kahler manifold, a complex line bundle Q over

M is called a quantization bundle if the following conditions hold:

q1) The class of the bundle Q satisfies q ∶“`

rω2πhs ` 12c1pMq

˘

PH2pM,Zq, where

c1pMq means c1pTCPn´1q, the first Chern class of the tangent bundle TCPn´1

(See Section 2.1).

q2) signpxrω2πhs, rcsyq “ signpxq, rcsyq for all rcs PH2pM,Zq.

When this Q exists we say M is quantizable.

Remark 4.1. • In he preceding definition c1pMq turns out to be also equal to

c1pKP q the first Chern class of the canonical bundle corresponding to the holo-

morphic polarization.

• As a consequence of the Kodaira embedding theorem we have that every com-

pact Kahler manifold admitting a quantization bundle is algebraic, meaning

that it is an analytic submanifold of CP r for some r ě dimCM .

Recall from Chapter 2 that in KS-quantization, we have the prequantization

step that is an integrality condition on the cohomology class of ω2πh; and then we

include the polarizations in the quantization step. On the contrary, in this mod-

ification we omit prequantization and pass directly to quantization. Furthermore,

the integrality condition now includes c1pMq2, thus, the quantization bundle Q

includes already the bundle of polarized half-forms with respect to the holomor-

phic polarization. Now we establish a relation between KS-quantization of Kahler

manifolds and this alternative quantization.

Proposition 4.1. Assume pM,ω,Jq is a Kahler manifold that admits a KS-quantization

and let LP “ L b δP be the quantum line bundle where P is the holomorphic po-

larization. If lP “ c1pLP q P H satisfies signpxrωs, rcsyq “ signpxlp, rcsyq for each

rcs PH2pM,Rq, then pM,ω,Jq admits a quantization bundle Q with q “ lP .

62


Proof. Define the cohomology class

q ∶” ω

2πh

ı

`1

2c1pMq.

By properties of the Chern classes we have that c1pL1bL2q “ c1pL1q` c1pL2q, thus

lP “ c1pLP q “ c1pLb δP q “ c1pLq ` c1pδP q “” ω

2πh

ı

`1

2c1pKP q.

Since the square root of the canonical bundle δP exists if and only if c1pKP q is divis-

ible by 2, the computations above show that q PH2pM,Zq. Choose a representative

form η2π of q, then η is a closed 2-form. Applying Theorem 3.1 we conclude that

there exists a hermitian line bundle with connection pQ,h,∇q whose curvature is

η. Finally, q1q and q2q are satisfied because q “ lp, therefore Q is a quantization

bundle over M .

Thus, if pM,ω,Jq is quantizable in the Kostant-Souriau sense then it admits a

quantization bundle Q.

4.1.1 Geometric Quantization Including the Maslov Index

Now we are going to introduce the Maslov quantization, this is, geometric quanti-

zation by means of the Maslov index. We are going to quantize a Kahler manifold

that can be seen as the quotient of an embedded submanifold of R2n of codimension

1.

Proposition 4.2. Let h be a hamiltonian on the sympletic manifold pR2n, ω0q. Let

ME “ h´1pEq be a compact connected and simply connected submanifold. Assume

all the integral curves of Xh are closed, and let M ∶“ME „ be the space of orbits

(integral curves), with projection map pr ∶ME ÑM . Then:

1. M is simply connected.

2. There exists a 2-form ω on M such that pM,ωq is symplectic manifold for

which pr˚ω “ i˚ω0.

3. Let θ “ pjdqj, then the integral

ż

Oθ, (4.1)

where O is any orbit in ME, is independent of the orbit.

63


4. For each closed simply connected 2 dimensional surface a in M there is a

surface b in ME such that prpbq “ a and

ż

aω “

ż

bω0 “

ż

Bbθ. (4.2)

Proof. 1. Consider the homotopy long exact sequence

⋯Ñ π1pMEq Ñ π1pMq Ñ π0pS1q Ñ π0pMEq Ñ ⋯.

Using that ME is simply connected and that S1 and ME are connected we ob-

tain that π1pMEq » π0pS1q » π0pMEq » 0. Therefore, M is simply connected.

2. This is a consequence of symplectic reduction with respect to the one-parameter

group generated by Xh.

3. Notice that H1pMEq “ t0u because ME is simply connected, so that any 2

orbits o1 and o2 are in the same homology class. Now let γ ∶ r0,1s ÑM be a

path connecting 2 orbits o1 and o2, then S ∶“ pr´1pγq is a surface such that

BS “ o1Yo2. Observe that ω0 “ 0 in S, since for all p P S, TpS “ SpantXh,p, vu

for some tangent vector v and ωpXh,p, vq “ ´dhpvq “ 0. Hence, by Stokes

theorem we obtainż

o1

θ “

ż

o2

θ.

4. Let a be a closed simply connected 2-dimensional surface in M . It is easy to see

that pr´1paq has a structure of a S1-bundle with the action of the hamiltonian

flow. Then, since a´m0 is a disk and any bundle over it is trivial, there is a

section s over it such that spa´m0q can be identified with a surface b in ME ,

for which prpBbq “ m0. Hence, Bb is an orbit or the empty set. Finally, we

obtain

ż

aω “

ż

a´m0

ω “

ż

prpbqω

“

ż

bpr˚ω “

ż

bω0 “

ż

Bbθ.

Definition 4.2. Let ME and M be as in the previous proposition and L be a la-

64


grangian submanifold of R2n contained in ME. We say L satisfies the Maslov

quantization condition if for any c closed curve in L the following equality holds

ż

c

θ

2πh“ k´

1

4µpcq, k P Z, (4.3)

where µpcq is the Maslov index of c (see Definition 2.15).

Since any 2 orbits in L are in the same homology class we find that µpcq is is

independent of the orbit. The idea is to construct a cohomology class q (in M) based

on the Maslov index, and then use the Maslov quantization condition to construct

a line bundle whose class is precisely q. Taking this into account, we would want to

have a notion of Maslov index for an arbitrary orbit o on ME . Thus, using the fact

that µpcq is constant for orbits in L we define the Maslov index of any orbit o in

ME as µpoq ∶“ µpcLq where cL is any orbit that is contained in L. Define the map q

given by

qpaq “

ż

a

ω

2πh´

1

4µpBbq,

for a a closed and simply connected surface in M i.e. a sphere.

Figure 4.1: Connected sum of 2 torus

Now we have to extend q to a map q ∶ Z2pM,Rq Ñ R (where Z2pM,Rq is the

space of 2-cycles on M). Since each closed surface in M is homeomorphic to a

connected sum of torus we just have to prove that the torus can be decomposed

in closed simply connected surfaces; and then extend q by linearity to any closed

65


surface a in M . To this end, let T be a torus and remove 2 circles form it to obtain 2

cylinders (c1 and c2). We can glue disks d1 and d2 to close each cylinder and form 2

surfaces that are actually homeomorphic to a sphere (those disks always exist since

M is simply connected). Thus, the torus is decomposed by 2 spheres a1 and a2 (see

Figure 3.2). We denote this by T zC “ a1 e a2, where C is the disjoint union of the

removed circles. Observe that the integral of ω over T is equal to the sum of the

integral aver a1 and a2 since

ż

Tω “

ż

T zCω “

ż

c1

ω `

ż

c2

ω

“

ż

d1

ω ´

ż

d1

ω `

ż

d2

ω ´

ż

d2

ω `

ż

c1

ω `

ż

c2

ω

“

ż

c1

ω `

ż

d1

ω `

ż

d2

ω `

ż

c2

ω ´

ż

d1

ω ´

ż

d2

ω

“

ż

a1

ω `

ż

a2

ω

and hence, we define

qpT q ∶“ qpa1q ` qpa2q.

In general, if a » T1#⋯#Tk define

qpaq ∶“ÿ

i

qpTiq. (4.4)

T c1 c2

d1

d2

a1 a2 a1 a2

Figure 4.2: Gluing process of the disks on the torus

Therefore, (3.4) defines the desired map q ∶ Z2pM,Rq Ñ R. Note that if ras “ 0

in H2pM,Rq, then qpaq “ 0 because of the Stokes theorem and Bb is hence the empty

66


set. Thus, q descends to the quotient H2pM,Rq and is a well defined cohomology

class q in H2pM,Rq.

Recall that there is a bundle Q with class q when it is integral, that is, q is in

the image of H2pM,Zq. The following theorem says that the Maslov quantization

condition ensures the existence of such a bundle.

Theorem 4.1. Let n ą 1 and suppose that for some lagrangian orientable subman-

ifold L the Maslov quantization condition holds. Then, q PH2pM,Zq.

Proof. By Proposition 2.8 we have that the Maslov index is even. Now let a be

a closed simply connected surface in M . Thus, by the previous proposition there

exists a surface b in ME satisfying (3.2). Then :

qpaq “

ż

a

ω

2πh´

1

4µpBbq “

ż

b

ω0

2πh´

1

4µpBbq

“

ż

Bb

θ

2πh´

1

4µpBbq “ k´

1

4µpBbq ´

1

4µpBbq

“ k´1

2µpBbq P Z.

Since any closed surface a in M can be decomposed by closed simply connected

surfaces we conclude that q PH2pM,Zq.

The construction of the following section can also be done for the bundle Q

whose class is q if the quotient manifold M is also a Kahler manifold and pq2q holds

for q too, i.e. signpxrω2πhs, rasyq “ signpxq, rasyq.

4.1.2 Quantization of Observables

As in Chapter 3 after defining the quantization bundle, we define the quantum

Hilbert space an the operators on it corresponding to smooth function on M . To

this end, we select η2π to be harmonic representative of q and construct a hermitian

line bundle with connection pQ,h,∇q applying Theorem 3.1. From now on, the 2-

form η will be called the quantization form. After this, we define the Quantum

Hilbert Space H to be the completion of the space of square integrable polarized

sections with respect to the inner product

xψ1, ψ2y “

ż

Mhpψ1, ψ2qε. (4.5)

67


It is important to remark that here h is a hermitian metric on Q and ∇ is the

connection compatible with h such that the curvature is ηh (NOT ωh). Now we

define the quantizable functions.

Definition 4.3. Let f be smooth function, and let Xηf be the vector field given by

Xηf η “ ´df. (4.6)

A quantizable function is a function f such that Xηf preserves the antiholomorphic

polarization. For a quantizable function f define the operator Qpfq on smooth sec-

tions in H by

Qpfqψ ∶“ íh∇Xηfψ ` fψ. (4.7)

Proposition 4.3. Let s be a polarized section and f a quantizable function. Then,

s is again a polarized section.

Proof. Let X be an antiholomorphic vector field, thus

∇XpQpfqsq “ ∇Xpíh∇Xηfs` fsq “ íh∇X∇Xη

fs`∇Xfs

“ íhr∇X ,∇Xηfss´ ih∇Xη

f∇Xs`Xfs` f∇Xs

“ íhr∇X ,∇Xηfss`Xfs

“ ípíqηpX,Xηf qs´ ih∇rX,Xη

fss`Xfs

“ ´ηpX,Xηf qs`Xfs “ 0.

Remark 4.2. Notice that the quantization form η is not necessarily non-degenerate,

therefore Xηf could be the zero vector field.

As a final comment, before our main example, is important to mention that this

quantization procedure can be generalized to symplectic manifolds diffeomorphic to

a product M1ˆM2 with a polarization P of general type (see Definition 3.18). More

specifically, P induces a Kahler polarization on M1 and a real polarization on M2.

This quantization procedure consists of applying the quantization by means of the

Maslov index on M1 and the KS-quantization on M2. For further details we refer

the reader to [Czy79].

68

4.2. Harmonic Oscillator

4.2 Harmonic Oscillator

In this section we quantize the complex projective space using the techniques dis-

cussed previously. The resulting system will be the quantum harmonic oscillator.

We will see that the alternative KS-quantization of CPn is equivalent to that one

using the Maslov Index.

Consider the symplectic manifold pR2n, ω0q with the hamiltonian

hpp, qq “1

2p|p|2 ` |q|2q.

Let ME “ h´1pEq be the E level set i.e. the 2n ´ 1-sphere of radius 2E. The

space of orbits must be identified with the projective space CPn´1. On it consider

the sympletic form 2πEωFS , where ωFS is the Fubini Study Kahler form. In Uj “

trz1, . . . , zns ∶ zj ‰ 0u use the standard coordinates, that is

rz1, . . . , zns↦ pw1, . . . , wj , . . . ,wnq, wk “zk

zj.

In Un ωFS has the following coordinate expression

ωFS “i

2πBB ln

`

1` |w|2˘

“i

2π

p1` |w|2qřn´1j“1 dw

j ^ dwj ´wjwidwi ^ dwj

p1` |w|2q2.

Notice that for this form the following identities hold:

i˚ω0π “ pr˚ωFS and

ż

CPn´1

pωFSqn´1

“ 1, (4.8)

where i ∶ S2n´1 Ñ R2n is the inclusion and pr∶ S2n´1 Ñ CPn´1 is the quotient map.

Thus, rωFSs is a positive generator of H2pCPn´1,Zq and ωE satisfies i˚ω0 “ pr˚ωE

where i ∶ME Ñ R2n is the inclusion and pr∶ME Ñ CPn´1 is the quotient map.

Now we proceed to compute the c1pCPn´1q, that is, c1pTCPn´1q. To this purpose,

we are going to use the fact that TCPn´1 “ HompLn´1,En´1q, where Ln´1 is the

tautological bundle over CPn´1 and En´1 is its orthogonal complement in the trivial

69


bundle CPn´1 ˆCn. Observe that

TCPn´1‘ pLn´1 bL

˚n´1q » HompLn´1,En´1q ‘EndpLn´1q

HompLn´1,En´1 ‘Ln´1q » HompLn´1,CPn´1ˆCnq »

nà

i“1

L˚n´1.

Therefore, TCPn´1 is stably equivalent toÀn

i“1L˚n´1. By Lemma 3.1 we obtain

that

cpTCPn´1q “ cp

nà

i“1

L˚n´1q “ cpL˚n´1qn“ p1´ c1pLn´1qq

n,

hence, c1pTCPn´1q “ ńc1pLn´1q. Furthermore, by Definition 3.12.4 c1pLn´1q is

the generator of H2pCPn´1,Zq correponding to positive orientation, then we con-

clude that c1pTCPn´1q “ ńrωFSs.

Notice that

q “E

hrωFSs ´

n

2rωFSs.

Thus, in order to satisfy pq2q, that is, signpxrωE2πhs, rcsyq “ signpxq, rcsyq, we must

have thatE

hń

2P Zą0,

because ωE is a positive multiple of ωFS . Then, pME , ωEq admits a quantization

bundle Q with class q if and only if

E “ EN “ h´

N `n

2

¯

, N P Zą0. (4.9)

Now, assume E is given by equation (3.9), and let pQN ,∇, hq be the quantization

bundle. For the quantization form we choose η “ NωFS , and for the potential

1-forms let

θj “iN

4π

`

B ´ B˘

lnp1` |w|2q. (4.10)

On each Uj let sj be the trivializing section given by

sj “`

1` |w|2˘Ń4πh

. (4.11)

70


Note that it is a polarized section, since

∇Bwasj “B

Bwa`

1` |w|2˘Ń4πh

`N

4πh

`

1` |w|2˘Ń4πh wa

1` |w|2

“ Ń

4πh

`

1` |w|2˘Ń4πh´1

wa `N

4πh

`

1` |w|2˘Ń4πh wa

1` |w|2“ 0.

So that every polarized section ψ of QN can be written (in Ujq as ψ “ φsj for a

holomorphic function φ.

Now we are going to construct the quantum bundle Q using the Maslov Index.

Let L be the Lagrangian submanifold of R2n given by

L “ tpp1, . . . , pn, q1, . . . , qnq ∶ ppkq

2` pqkq2 “ a2

k, ak ą 0,ÿ

a2k “ 2Eu.

Observe that the last condition implies L is contained in ME . We can see this is an

embedded orientable submanifold applying the Regular Level Set Theorem to the

map

F ∶ R2nÑ Rn

given by F pp, qq “`

pp1q2 ` pq1q2, . . . , ppnq

2 ` pqnq2˘

. To see L is lagrangian, we use

the fact that Tpp,qqL “ kerpdFpp,qqq, and

rdFpp,qqs “ 2

¨

˚

˚

˚

˚

˚

˝

p1 0 ⋯ 0 q1 0 ⋯ 0

0 p2 ⋯ 0 0 q2 ⋯ 0...

.... . .

......

.... . .

...

0 0 ⋯ pn 0 0 ⋯ qn

˛

‹

‹

‹

‹

‹

‚

,

so that Tpp,qqL is spanned by the vectors ´qjBBpj ` pjBBqj , which is clearly la-

grangian in Tpp,qqR2n. The orbits in L can be parametrized as follows

γptq “ pa1 cos t, a2 cos t, . . . , an cos t, a1 sin t, a2 sin t, . . . , an sin tq, t P r0,2πs.

Now, in this case, we want to find E such that the Maslov quantization condition

holds. We first compute the integral

ż

γ

θ

2πh“

1

2πh

ż 2π

0γ˚θ “

1

2πh

ż 2π

0

ÿ

a2k cos2 tdt “

E

h.

71


Second, we compute the Maslov index of γ, we will see that µpγq “ ´2n. To this

end we need to find the unitary matrix taking L0 “ SpantBBpiu Ď R2n identifying,

for each t P r0,2πs, TγptqR2n with R2n. Let

Pγptq ∶ R2nÑ L0

be the projection map corresponding to the decomposition L0 ‘ TγptqL. An easy

calculation gives us that, point-wise,

Pγptq

ˆ

B

Bpi

˙

“B

Bpi, Pγptq

ˆ

B

Bqi

˙

“ tan tB

Bpi.

Now let Aγptq be the extension to Cn of the endomorphism PγptqJ of L0, as in

Theorem 2.1.3. The matrix representation is given by rAγptqs “ tan tIn and the

unitary matrix we are looking for is given by the polar decomposition of Aγptq.

After the polar decomposition we obtain the unitary matrix

Uγptq “i` tan t?

1` tan2 tIn.

Therefore,

pdetpUγqq2“

ˆ

i` tan t?

1` tan2 t

˙2n

“

ˆ

tan2 t´ 1` 2i tan t

1` tan2 t

˙n

“ psin2 t´ cos2 t` 2i sin t cos tqn “ exppinπ ´ 2nitq.

With this information we can finally compute the Maslov index

µpγq “

ż

γppdet˝cq2q˚

dφ

2π“

1

2π

ż 2π

0dpnπ ´ 2ntqq “ ´2n.

The Maslov quantization condition states that that

ż

γ

θ

2πh“ k´

1

4µpγq, k P Z,

which implies that Eh “ N `n2 and we obtain once again the same condition on

the energy, that is,

EN “ h´

N `n

2

¯

, N P Zą0

72


adding the condition pq2q to the class q. Then, if E is given by the last equation,

there is a quantization line bundle QN whose class is q. Moreover, notice that

bundles QN and QN have the same class, so that they are isomorphic. Hence, the

modified KS-quantization and the one by means of the Maslov index are equivalent

for the harmonic oscillator.

Remark 4.3. It can be shown that the quantization form of QN is pN ` nqωFS,

therefore, we obtain the relation QN » QN bLnn´1.

73


74

Bibliography

[Czy79] J. Czyz. On geometric quantization and its connections with the Maslov

theory. Rep. Math. Phys., 15(1):pp.57–97, 1979.

[EEea98] A. Echeverrıa-Enrıquez et al. Mathematical foundations of geometric

quantization. Extracta Math., 13(2):pp.135–238, 1998.

[Hal13] B. Hall. Quantum theory for mathematicians. Springer, first edition, 2013.

[Lee12] J. Lee. Introduction to smooth manifolds. Springer-Verlag New York,

second edition, 2012.

[Nic13] L. Nicolaescu. Notes on the Atiyah-Singer index theorem. University of

Notre Dame, nov 2013.

[Pic08] P. Piccione. A students guide to symplectic spaces, grassmannians and

Maslov index. Universidad de Sao Paulo, oct 2008.

[RS13] G. Rudolph and M. Schmidt. Differential geometry and mathematical

physics, Part I. Springer, first edition, 2013.

[Sou73] J-M. Souriau. Indice de Maslov des varietes lagrangiennes orientables. C.

R. Acad. Sci. Paris Ser. A-B, 276:pp. A1025–1026, 1973.

[Woo91] N.M.J. Woodhouse. Geometric quantization. Oxford mathematical mono-

graphs. Oxford University Press, 1991.

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topological conditions in geometric and maslov quantization

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