canonical group quantization, foundations and applications

Canonical Group Quantization:

Foundations and Applications

Juan Camilo Orduz B.

Physics Department

Universidad de los Andes

A thesis submitted for the degree of

Bachelor in Physics

November - 2010

mailto:[email protected]

http://fisica.uniandes.edu.co

http://www.uniandes.edu.co

Advisor: Andres Reyes Ph.D.

1. Reviewer: Alonso Botero, Ph.D.

2. Reviewer: Alexander Cardona, Ph.D.

Day of the defense: December 3 - 2010

Signature from Advisor:

ii

mailto:[email protected]

Abstract

We study one quantization scheme known as canonical group quantization

method developed by C.J. Isham. The idea behind this method is to consider

the action of a Lie group on the phase space to relate classical observables

with elements in the Lie algebra via a moment map. We use Mackey’s

theory of representations of semidirect product to find the representation

space, which will be the space of square integrable sections on some vector

bundle. The idea of this document is to explore this method trough explicit

examples where we are interested in the explicit form of the commutation

relations and their representations.

“‘Para estar...”

Acknowledgements

Quiero agradecer principalmente a mi familia por su apoyo incondicional en

este proceso: A mi Mama y a mi Papa como ejemplo de trabajo, respons-

abilidad y dedicacion, definitivamente no hubiera podido hacer este trabajo

sin ellos. Quiero expresar un agradecimiento especial a mi hermano por

simples momentos, que para mi fueron (y son) muy valiosos. En general, a

toda mi gran familia solo quiero decirles: gracias.

Es comun hablar de lo difıcil que fue este proceso, pero esta vez quiero

enfatizar en las cosas que fueron faciles: Fue muy facil encontrar amigos

especiales, de los cuales aprendı muchas mas cosas de las que aprendı en un

salon de clase; en particular que no hay nada que la risa no pueda curar.

Gracias por tan buenos momentos.

Fue muy facil encontrar profesores con los cuales pude aprender mucho mas

que solo fısica: Quiero dar un agradecimiento especial a mi asesor Andres

Reyes quien creyo en mi trabajo desde el principio y me apoyo durante toda

la carrera; este trabajo es fruto de su ejemplo y dedicacion. Quiero agrade-

cer tambien a Juan Carlos Sanabria, Alonso Botero y Francois Leyvraz

quienes tambien dejaron una huella importante en mı.

Fue muy facil encontrar apoyo en toda la gente que estuvo alrededor mıo.

Finalmente quiero agradecer al Grupo de Fısica Teorica de Altas Energıas

(THEP) de la Universidad Johannes Gutenberg, en especial a Florian Scheck

y a Nikolaos Papadopoulos por su invitacion y calida acogida durante el mes

de Julio al Instituto en Mainz, Alemania.

Contents

1 Introduction 1

2 Lie Group Actions on Symplectic Manifolds 5

2.1 Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Towards Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Cohomology of Lie Algebras 15

3.1 Preliminaries and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Interpretation of H1(g, V ) . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Interpretation of H2(g, V ) . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Removing the Obstruction Cocycle . . . . . . . . . . . . . . . . . . . . . 29

4 Canonical Group Quantization: One-Dimensional Examples 33

4.1 Canonical Group Quantization Method . . . . . . . . . . . . . . . . . . 33

4.2 Q = R and the Heisenberg Group . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Semidirect Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4 Q = R+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.5 Q = S1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.6 Uncertainty Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 The General Case: Through Examples 63

5.1 Q = T 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Brief Discussion: The General Case . . . . . . . . . . . . . . . . . . . . . 66

5.3 Q = G/H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

iii

CONTENTS

5.4 Induced Representations of Semidirect Products . . . . . . . . . . . . . 70

5.5 Q = S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6 The Riemann Hypothesis: Spectral Aproach 81

6.1 The Riemann Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2 Landau Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . 86

A Principal Bundles 89

References 93

iv

1

Introduction

It is well known that classical mechanics can be modeled in the cotangent space T ∗Q

of the configuration space Q, taking advantage of the natural symplectic structure

ω = qα ∧ pα on T ∗Q. In this mathematical framework the classical observables are

functions on the cotangent bundle and Hamilton’s equation take the compact form

iXω = dH, where H ∈ C∞(T ∗Q,R) is the Hamiltonian of the system. Moreover, the

symplectic form of the cotangent bundle endows C∞(T ∗Q,R) with a Poisson algebra

structure, in the particular case where Q = Rn, the Poisson bracket is given by

{qα, qβ} = 0 = {pα, pβ} and {qα, pβ} = δβα

On the other hand, in quantum mechanics we work in a Hilbert space H and the

obserbables become self-adjoint operators on H which generally form a Lie algebra. In

the usual cases we have the following commutation relations

[qα, qβ] = 0 = [pα, pβ] and [qα, pβ] = δβα1

However, when the configuration space in not Euclidean, these commutation relations

are no longer valid. Therefore some questions arise:

• If we start from a classical system modeled on a symplectic manifold, what is its

correspondent quantum model? Does it exist? Is it unique?

• For example notice that, as Hilbert spaces, L2(R2, d2~x) ∼= L2(S1, dθ), so where

does the topology of the configuration space appears in the corresponding quan-

tum model?

1

1. INTRODUCTION

• How do the commutation relations depend on the topology of the configuration

space?

These are some questions that motivate the present work. The objective of this

document is to study a quantization programme called Canonical Group Quantization

Method, developed by C.J. Isham [Ish84]. This programme tries to answer this questions

trough the geometric and algebraic background of the physical system. There are two

basic steps in this quantization method:

1. Find a “well-behaved” action of a Lie group (the canonical group) on a symplectic

manifold to relate certain set of observables to elements in the Lie algebra.

2. Find unitary, weakly continuous and unitary representations of the canonical

group in some Hilbert space.

The idea of this document is to understand this method trough examples, so we are

strongly going to use geometric and algebraic tools to do explicit computations. Addi-

tionally we would like to give some perspectives on possible applications of this method

to study spectrums of certain operators in non-trivial configuration spaces.

This document is organized as follows. In the second chapter we will give a short

description of the geometric preliminaries: action of Lie groups on symplectic mani-

folds. The idea of this chapter is not to give an extensive treatment of this topics,

but rather establish notation and conventions that will be used in the rest of the work,

therefore we will assume basic knowledge on differentiable manifolds, Lie groups, differ-

ential forms and de Rham cohomology. In the third chapter we will study the algebraic

framework of a obstruction defined by the action that arise when we want to lift certain

map (moment map). We will define the Lie algebra cohomology and give interpreta-

tion to the particular groups H1(g, V ) and H2(g, V ). The fourth chapter is devoted

to describe the canonical group quantization method and to treat three explicit ex-

amples that will help to understand the spirit of this method. In the fifth chapter

we will describe how to proceed for an arbitrary configuration space, however we will

be particularly interested in the case where this space can be seen as a homogeneous

space. In this part of the document we also describe (very briefly) the formalism used

to study the representations of the canonical group. It should be mentioned that the

2

language of vector and principal fiber bundles will be used in this chapter, that is why

we have included a short appendix treating these topics. Again, the purpose of this

appendix is to establish notation, conventions and important results; not to give an

extensive treatment of this powerful theory. The purpose of the final chapter is to give

some results on a spectral approach to the Riemann hypothesis: these works study a

particular Hamiltonian which, as we will see, will appear in one of the worked examples

in this work.

I hope that this document gives a comprehensive introduction to these topics and

motivate other students to learn about these mathematical tools to attack different

physical phenomena.

3

1. INTRODUCTION

4

2

Lie Group Actions on Symplectic

Manifolds

The main goal of this chapter is to discuss the action of a Lie group over a smooth

manifold, particularly in the case when M carries a symplectic structure. This will be

a basic mathematical tool in the quantization scheme that we will study. The theory

of symplectic manifolds will not be discussed very deeply since we will assume basic

knowledge on this subject (for instance refer to [Ber87]), however we will recall some

definitions, notation and results that will be used in the rest of this work. The main

references for this chapter are [AM87] and [War83].

2.1 Group Actions

Definition 2.1.1. Let M be a smooth manifold. A left action of a Lie group G onM is a smooth map ` : G×M →M such that:

1. `(e, x) = `ex = x ∀x ∈M . (where e is the identity of G)

2. `ghx = `g`hx ∀g, h ∈ G ∀x ∈M .

Definition 2.1.2. Let ` be a left action of G on M. Let x ∈M , we define the orbit ofx by Ox = {`gx | g ∈ G}. The action is said to be transitive if there is just one orbit.

Definition 2.1.3. A left action of G on M is said to be effective if `gx = x ∀x ∈Mimplies that g = e. Although we can consider a weaker condition: a left action is saidto be almost effective if `gx = x ∀x ∈M implies that g ∈ D, where D is a discrete

5

2. LIE GROUP ACTIONS ON SYMPLECTIC MANIFOLDS

subgroup of G. Finally, a left action is said to be free if for each x ∈ M the mapg 7→ `gx is one-to-one.

Remark 2.1.1. If we have an effective left action by G on M we can consider a coveringgroup G of G acting on M. This action will be almost effective since its kernel is justthe kernel of the covering homomorphism G→ G.

2.2 The Adjoint Representation

Theorem 2.2.1. Let ` : G×M →M be a left action of G on M. Assume that x0 ∈Mis fixed by the action, that is, `gx0 = x0 for each g ∈ G. Then the map

ψ : G→ Aut(Tx0M)

defined by

ψ(g) = d`g|Tx0M

is a representation of G

Proof. It follows essentially from the chain rule.

In particular, we can consider the case of a Lie group G acting on itself by the inner

automorphism a(g, h) = agh = ghg−1 = rg−1 lgh. Since the identity is a fixed point of

this action, last theorem asserts that the map

g 7→ dag|TGe∼=g

is a representation of G into Aut(g) known as the adjoint representation denoted

by

Ad : G→ Aut(g).

If we denote by exp : g −→ G the usual exponential map, for each g ∈ G we have

the following commutative diagram.

G

=

ag // G

g

exp

OO

Adg // g

exp

OO (2.2.1)

6


We can also consider its differential ad = D(Ad), which makes the following diagram

commute:

G

=

Ad // Aut(g)

g

exp

OO

ad // End(g)

exp

OO(2.2.2)

Now we will show an explicit expression for ad(X) = adX , which will be an impor-

tant tool to compute the commutators.

Proposition 2.2.1. Let G be a Lie group with Lie algebra g, and let X,Y ∈ g. ThenadXY = [X,Y ].

Proof. We compute

adXY =(d

dt

∣∣∣∣t=0

Adexp tX

)Y

=d

dt

∣∣∣∣t=0

Adexp tX(Y )

=d

dt

∣∣∣∣t=0

D(aexp tX)(Y ).

It suffices to prove it at the identity so, if ϕXt denotes the 1-parameter group ofdiffeomorphisms associated with X, then

adXY (e) =d

dt

∣∣∣∣t=0

D(rexp(−t))D(lexp tX)Y (e)

=d

dt

∣∣∣∣t=0

D(rexp(−t))Y (exp tX)

=d

dt

∣∣∣∣t=0

D(ϕX−t)Y (ϕXt (e))

= LXY (e)

= [X,Y ](e),

where LX denotes the Lie derivative in the direction of the vector field X.

7


Remark 2.2.1 ([Sch10]). Let {Tj}nj=1 be a basis for g, then there exist constants Ckij ,called structure constants, such that

[Ti, Tj ] = iCkijTk, (2.2.3)

where there is a sum over repeated indices. The following relation can be easily shown,as a result of the Jacobi identity on the Lie bracket,

Cmij Clmk + CmjkC

lmi + CmkiC

lmj = 0.

We can define a representation U of g on Cn by

Ulm(Tk) = −iCmkl . (2.2.4)

The only non-trivial thing to verify is that this representations preserves de Lie bracket,

Ulm([Ti, Tj ]) = −iCkijUlm(Tk)

= CkijCmkl

= −CkjlCmki − CkliCmkj= (−i)Ckil(−i)Cmjk − (−i)Ckjl(−i)Cmik= Ulk(Ti)Ukm(Tj)− Ulk(Tj)Ukm(Ti)

= [U(Ti), U(Tj)]lm.

This representation is actually the derivative of the adjoint representation, that isU = ad. However it may not be clear how U(A) ∈ End(g) for each A ∈ g. To see thisjust express A as a linear combination in the basis {Tj}nj=1, then put the associatedcoefficients as a row vector and then operate by matrix multiplication on the right. Forexample, let us evaluate U(Ti)Tj : In this basis Tj is represented by the column vectoreTj , seen as an element of the canonical basis of Rn. Therefore, in the chosen basis wehave

(U(Ti)Tj)k = δmjUmk(Ti) = Ujk(T i) = −iCkij = −([Ti, Tj ])k.

This result should be compared with Proposition (2.2.1).

Example 2.2.1. Now we are going to consider the explicit example g = su(2). Wechoose the basis Ti = σi/2, where the σs are the Pauli matrices

σ1 =

(0 11 0

)σ2 =

(0 −ii 0

)σ3 =

(1 00 −1

)

8


In this case we have that [Ti, Tj ] = iεijkTk, therefore the representation discussedabove can be written as

U(T1) =

0 0 00 0 −i0 i 0

U(T2) =

0 0 i

0 0 0−i 0 0

U(T3) =

0 −i 0i 0 00 0 0

Definition 2.2.1. Let ` : G ×M → M be a left action of G on M. Let A ∈ g, thent 7→ `(exp tA,x) defines a curve on M. The corresponding vector field defined by

β(A)(x) = βA(x) =d

dt

∣∣∣∣t=0

`(exp tA, x) (2.2.5)

is called the infinitesimal generator of the action corresponding to A.

Proposition 2.2.2. For every g ∈ G and A,B ∈ g we have

1. β(AdgA) = d`g−1β(A).

2. β[A,B] = −[βA, βB].

Proof. 1. We compute

β(AdgA)(x) =d

dt

∣∣∣∣t=0

`(exp tAdgA, x)

=d

dt

∣∣∣∣t=0

`(g(exp tA)g−1, x)

=d

dt

∣∣∣∣t=0

`g ◦ `(exp tA, `g−1x)

= d`g,`g−1 (x)d

dt

∣∣∣∣t=0

`(exp tA, `g−1x)

= d`g,`g−1 (x)β(A)(`g−1x)

= (d`g−1β(A))(x)

2. In particular, take g = exp tB in 1.,

β(Adexp tBA) = d`exp(−tB)β(A). (2.2.6)

If we take the derivative with respect to t at t = 0, the left-hand side last equationis β[B,A]. Notice that `exp(−tB) is the flow of β(−B) so

9


d

dt

∣∣∣∣t=0

dèxp(−tB)β(A)(x) =d

dt

∣∣∣∣t=0

dèxp tB,èxp(−tB)xβ(A)(èxp(−tB)x)

= [−βB, βA](x).

We can state this last result in other terms

Proposition 2.2.3. The map γ : g → Γ(TM) defined by γA = −βA is a Lie algebrahomomorphism.

A natural question is under which circumstances this map one-to-one. This will

fail when there are elements A ∈ g such that on a neighborhood around t = 0 we have

`(exp tA, x) = x ∀x ∈M . A strong condition to ensure that γ is one-to-one is that the

action is effective. However we will only require this to happen locally around t = 0,

so actually it is sufficient to consider an almost-effective action.

2.3 Symplectic Manifolds

Definition 2.3.1. Let M be a smooth manifold and ω a 2-form over M such thatit is closed and non-degenerate. Then, the pair, (M,ω) will be called a symplecticmanifold.

Definition 2.3.2. Let (M,ω) and (M, ω) be two symplectic manifolds. A smooth mapf : M → M is called symplectic if f∗ω = ω. In particular a left action ` : G×M →M

of a Lie group G on M is said to be symplectic if the map `g : M → M is symplecticfor each g ∈ G.

Example 2.3.1 (Phase Space). Let Q be a smooth manifold (the configuration space),we shall see that the cotangent bundle T ∗Q (the phase space) has a natural symplecticstructure: Let α ∈ T ∗Q and v ∈ Tα(T ∗Q), then we define the Liouville 1-form θ

on T ∗Q by 〈θα, v〉 = 〈α, (π∗v)〉. Where π∗ denotes the pushforward of the projectionπ : T ∗Q→ Q.Then we can take the exterior derivative two obtain the symplectic form ω = −dθ. Inlocal coordinates q1, ..., qn, p1, ..., pn it can be written as ω = dqi ∧ dpi.

Now, if φ ∈ Diff(Q) clearly the pullback φ∗ ∈ Diff(T ∗Q), but there is an additional

10

2.3 Symplectic Manifolds

nice property: φ∗ preserves the Liouville 1-form. To prove it let us denote ψ = φ∗ andcompute for a vector field v on T ∗Q:〈(ψ∗θ)α, vα〉 = 〈θψ(α), ψ∗vα〉 = 〈ψ(α), π∗ψ∗v〉 = 〈φ∗(α), φ−1

∗ π∗vα〉 = 〈α, π∗vα〉 = 〈θα, vα〉where we have used the fact that π(φ∗α) = φ−1(π(α)).

Symplectic geometry arises naturally in classical mechanics since, if H is the Hamil-

tonian of a classical system (which is a function in the phase space),the equations of

motion of a free system are given by iXω = dH, where ω is the symplectic form of the

cotangent bundle of the configuration space mentioned above.

Definition 2.3.3. Let (M,ω) be a symplectic manifold, and f ∈ C∞(M,R). ThenXf ∈ Γ(TM) is said to be (globally) Hamiltonian with respect to f if

iXfω = df. (2.3.1)

The set of Hamiltonian vector fields over M is denoted by HamVF(M).

Remark 2.3.1. Notice that the Hamiltonian vector field Xf always exists since ω isnon-degenerate.

Example 2.3.2. Consider the symplectic manifold (T ∗R, dq ∧ dp) (T ∗R) ∼= R2 and letf be a real smooth function over T ∗R. Notice that

(dq ∧ dp)(∂pf∂q − ∂qf∂p) = ∂pdp+ ∂qfdq = df,

hence Xf = ∂pf∂q − ∂qf∂p is the corresponding Hamiltonian vector field.

The following is a remarkable useful theorem

Theorem 2.3.1 ([Ber87]). Let Xf be a Hamiltonian vector field, and ϕXft be its flow.Then ϕ

Xft is symplectic. In particular LXfω = 0.

Moreover, if X is a vector field over M such that φXt is symplectic then LXω = 0,

which implies that iXω is closed since diXω = (LX − iXd)ω = 0. In this case, does the

vector field X is Hamiltonian? In general the answer is no. However, in the particular

case where H1dR(M), the first cohomology group of M , vanishes then X is always

Hamiltonian. Nevertheless, X is always locally Hamiltonian (as a result of Poincare

Lemma) and will show a nice property of these vector fields:

Proposition 2.3.1. Let X and Y two be locally Hamiltonian vector fields, then [X,Y ]is globally Hamiltonian.

11


Proof. Recall form Cartan’s formulas

i[X,Y ]ω = LXiY ω − iY LXω

= (diX + iXd)iY ω − iY (diX + iXd)ω

= d(iXiY ω).

Now we will recall a familiar concept from classical mechanics.

Definition 2.3.4. The map {, } : C∞(M,R) × C∞(M,R) → C∞(M,R) defined by{f, g} = ω(Xf , Xg) is called the Poisson Bracket.

Among other reasons, the importance of the Poisson bracket is given by the following

two propositions.

Proposition 2.3.2 ([Ber87]). The pair (C∞(M), {, }) has a Lie algebra structure.

Proposition 2.3.3. The pair (HamVF(M), [, ]) has a Lie algebra structure. Moreover,let j : C∞(M) → HamVF(M) be the map defined by j(f) = −Xf , then j is a Liealgebra homomorphism.

Proof. Notice that LXf g = {g, f}, so if we compute d{f, g} = LXg(iXfω) = i[Xg ,Xf ]ω.

Now, suppose that Xf = Xg, then f and g must differ by a constant. Hence we can

construct the following short exact sequence of Lie algebras:

0 // R ı // C∞(M,R)j // Ham VF(M) // 0. (2.3.2)

2.4 Towards Quantization

We would like to restrict ourselves to an action of a Lie group G on a symplectic

manifold (M,ω) such that γA is a globally Hamiltonian vector field for each A ∈ g.

The reason is that we would like to relate classical observables with the elements of g

via the homomorphism j. Two common cases would be when H1dR(M) = 0 or when

g = [g, g] as seen by Proposition 2.3.1.

In this context it is natural to ask when does the dotted lift P : g −→ C∞(MR) in the

diagram (2.4.1) exist such that it is a linear Lie algebra homomorphism and makes the

12

2.4 Towards Quantization

diagram commute (this map is known as a momentum map for the action), that is

γA = j(PA) = −XPA :

0 // R i // C∞(M,R)j // Ham VF(M) // 0.

g

γ

OO

P

hhP P P P P P P

(2.4.1)

It is not difficult to find a linear map that does this job. However the Lie homo-

morphism requirement would need a little work. For instance, if we compute

j(P [A,B]) = γ[A,B] = [γA, γB] = [j(PA), j(PB)] = j({PA, PB}), (2.4.2)

we notice that the problem may be solved up to a constant. Define z(A,B) ∈ R by

z(A,B) = {PA, PB} − P [A,B]. (2.4.3)

Can we make this constant vanish for all A,B ∈ g? This question and the meaning

of this constant will be treated more carefully in the next chapter where we will develop

some algebraic language to understand the problem.

Suppose that two linear maps P, P ′ : g → C∞(M,R) are given such that the diagram

(2.4.1) commutes. Then there exists h ∈ g∗ such that

P′

= P + h. (2.4.4)

Is it possible to choose h such that {PA, PB} = P [A,B]? Let us compute

{P ′A, P ′B} − P ′[A,B] = {PA + h(A), PB + h(B)} − P [A,B] − h([A,B])

= {PA, PB} − P [A,B] − h([A,B])

= z(A,B)− h([A,B]).

We see that the problem will be solved if we could find h ∈ g∗ such that z(A,B) =

h([A,B]) for all A,B ∈ g. The existence of such h will depend on purely algebraic

properties of g that should become clear at the end of the following chapter.

From the time being, we will end this section with some properties of this bilinear

map z : g× g −→ R:

13


Proposition 2.4.1. Let A,B,C ∈ g, then

1. z(A,B) = −z(A,B).

2. z(A, [B,C]) + z(B, [C,A]) + z(C, [A,B]) = 0 (Jacobi Identity).

Proof. The proof of 1. is trivial. For 2. we compute

z(A, [B,C]) + z(B, [C,A]) + z(C, [A,B])

= {PA, P [B,C]} − P [A,[B,C]] + {PB, P [C,A]} − P [B,[C,A]] + {PC , P [A,B]} − P [C,[A,B]]

= {PA, P [B,C]}+ {PB, P [C,A]}+ {PC , P [A,B]} − P [A,[B,C]]+[B,[C,A]]+[C,[A,B]]

= {PA, P [B,C]}+ {PB, P [C,A]}+ {PC , P [A,B]}

= {PA, ({PB, PC} − z(B,C))}+ {PB, ({PC , PA} − z(C,A))}+ {PC , ({PA, PB} − z(A,B))}

= {PA, {PB, PC}}+ {PB, {PC , PA}}+ {PC , {PA, PB}}

= 0.

14

3

Cohomology of Lie Algebras

This chapter is devoted to study the algebraic interpretation of the obstruction z for

the moment map discussed before. We will define cohomology groups for Lie algebras

and study the interpretation of the first two groups (namely H1(g, V ) and H2(g, V )),

in particular we will understand how H2(g, V ) will help us classify central extensions.

The main references for this part of the work are the lecture notes of B. Dietrich [Die05]

and the classic article of Chevalley and Eilenberg [CE48].

3.1 Preliminaries and Definitions

Definition 3.1.1. Let g be a Lie algebra and V a finite dimensional vector space overa field K (which for simplicity will be taken of characteristic zero). We say that V is ag-module if there exists a map

g× V // V

(g, v) � // g ·m(3.1.1)

such that for r, s ∈ K, and x, y ∈ g and v, w ∈ V we have

1. (rx+ sy) · v = rx · v + sy · v.

2. x · (rv + sw) = rx · v + sy · w.

3. [x, y] · v = x · (y · v)− y · (x · v).

15

3. COHOMOLOGY OF LIE ALGEBRAS

Now we will recall some properties of the tensor product: We will use the following

notation

Vr,s = V ⊗ V ⊗ · · · ⊗ V︸︷︷︸r

⊗V ∗ ⊗ V ∗ ⊗ · · · ⊗ V s︸︷︷︸s

.

Mr,s(V ) = {T : V × V × · · · × V︸︷︷︸r

×V ∗ × V ∗ × · · · × V s︸︷︷︸s

−→ K | T multilinear}.

Mr,s(V,W ) = {T : V × V × · · · × V︸︷︷︸r

×V ∗ × V ∗ × · · · × V s︸︷︷︸s

−→W | T multilinear},

where W is any vector space.

From the universal property that defines the tensor product if T ∈ Mr,s(V ), then

there is a unique linear lift T that makes the following diagram commute for any other

vector space W :

V × V × · · · × V × V ∗ × V ∗ × · · · × V ∗

π

��

T // W

Vr,s

T

44hhhhhhhhhhhhhhhhh

(3.1.2)

Then we can construct an isomorphism Mr,s(V,W ) ∼= Hom(Vr,s,W ).

If we consider the particular case where r = n and s = 0

V n

π

��

T // W

⊗n V

T

55llllllll

(3.1.3)

and consider Alt(V n,W ) = {T ∈ Mn,0(V,W ) | T alternating}, then we will have

an induced isomorphism Alt(V n,W ) ∼= Hom(∧nV,W ).

Definition 3.1.2. Let g be a Lie algebra and V a g-module, then for each p ∈ Z wedefine the space of p-chains by

Cp(g, V ) =

HomK(∧pg, V ) p > 0,

V p = 0,

0 p < 0.

16

3.1 Preliminaries and Definitions

Clearly Cp(g, V ) has a vector space structure but moreover it actually has a g-modulestructure.Notice that Cp(g, V ) = HomK(Λpg, V ) ∼= Alt(gp, V ).

Definition 3.1.3. For each p ∈ Z we define the coboundary operator dp : Cp(g, V ) −→Cp+1(g, V ) by

(dpω)(x0 ∧ · · · ∧ xp) =p∑i=0

(−1)ixi · (ω(x0 ∧ · · · ∧ xi ∧ · · · ∧ xp))

+∑

0≤i<j≤p(−1)i+jω([xi, xj ] ∧ x0 ∧ · · · ∧ xi ∧ · · · ∧ xp).

Notice that dp is a linear transformation and in particular we have (d0v)x = x · v.

Definition 3.1.4. For each x ∈ g we define the contraction operator ix : Cp(g, V ) −→Cp+1(g, V ) by

(ixω)(x1 ∧ · · · ∧ xp−1) = ω(x ∧ x1 ∧ · · · ∧ xp−1).

Definition 3.1.5. For each x ∈ g we define a linear transformation ρ(x) : Cp(g, V ) −→Cp(g, V ), which will be the analogue of the Lie derivative on differential forms, by

(ρ(x)ω)(x1 ∧ · · · ∧ xp) = x · ω(x1 ∧ · · · ∧ xp)−p∑j=1

ω(x1 ∧ · · · ∧ [x, xj ] ∧ · · · ∧ xp).

Notice that the formulas that we have obtained are exactly the same as the formulas

for differential forms, for this reason is not surprising that we have the following results.

The proofs in the case of differential forms can be found in [Mor01].

Proposition 3.1.1. For all x, y ∈ g and if we denote dp just by d, we have the followingrelations:

1. d2 = 0.

2. ρ([x, y]) = [ρ(x), ρ(y)] for all x, y ∈ g.

3. ρ(x) = d ◦ ix + ix ◦ d.

4. [ρ(x), d] = 0.

5. i[x,y] = [ix, ρ(y)].

As in the case of differential forms we can denote Zp(g, V ) = Ker(dp) the set of

p-cocycles and Bp(g, V ) = Im(dp−1) the set of p-coboundaries. Since d2 = 0 we

can consider the quotient space Hp(g, V ) = Zp(g, V )/Bp(g, V ) which will be called the

cohomology group at level p of g with coefficients in V .

17


3.2 Interpretation of H1(g, V )

Definition 3.2.1. Let V1, V and W be finite dimensional vector spaces. Consider thefollowing short exact sequence

0 // Wα // V1

β // V // 0 (3.2.1)

(that is, α is one-to-one, β is onto and Im(α) = Ker(β)). We say that the sequencesplits if there exists a linear transformation τ : V −→ V1 such that β ◦ τ = idV .

Proposition 3.2.1. The short exact sequence (3.2.1) splits if and only if then V1∼=

W ⊕ V .

Proof. Clearly if V1∼= W⊕V the sequence splits. For the other direction define a linear

transformation φ : W ⊕V −→ V1 by φ(w, v) = α(w) + τ(v). We want to show that thismap is an isomorphism. Suppose that φ(w, v) = 0, then 0 = β(0) = β(α(w) + τ(v)) =0 + v, hence v = 0 and since α is one-to-one it follows that w = 0.To show that this map is onto let v1 ∈ V1, then v1 = β(v) for some v ∈ V . Sincev1 − τ(v) ∈ Ker(β) there exists w ∈ W such that α(w) = v1 − τ(v), hence v1 =φ(w, v).

Proposition 3.2.2. Every short exact sequence of vector spaces splits.

Proof. If we consider the short exact sequence (3.2.1) we want to show that V1∼= W⊕V .

Let {w1, · · · , wm} a basis forW , since α is one-to-one it follows that {α(w1), · · · , α(wm)}is a basis for Im(α) ≤ V1. We can complete this set {α(w1), · · · , α(wm), x1, ..., xk} tomake it a basis for V1, so if x ∈ V1 it can be written uniquely as x =

∑mi=1 aiα(wi) +∑k

j=1 bjxj . Now we will define a linear transformation φ : V1 −→ V ⊕W by φ(x) =(∑m

i=1 aiα(wi),∑k

j=1 biβ(xi)). We want to show that this map is an isomorphism.Suppose that φ(x) = 0, then since {wi}mi=1 is a basis for W it follows that ai = 0for i = 1, · · · ,m. On the other hand there exists w =

∑mi=1 λiwi ∈ W such that∑k

j=1 bjxj = α(w) since β(∑k

j=1 bj) = 0. This implies that all bj and λi should be zeroby linear independence. We have shown that φ is one-to-one.To show that φ is onto notice that if n = dimV then k ≤ n since if c1β(x1) + · · · +ckβ(xk) = 0 implies that c1x1 + · · ·+ ckxk = α(w) for some w ∈W . But since β is ontowe have k = n.

Definition 3.2.2. Let V and W be g-modules. A g-module V1 is said to be an ex-tension of V by W if there exists a short exact sequence like (3.2.1) where the mapsα and β are g-module homomorphisms.

18


Notice that, a priori, not every short exact sequence of g-modules splits since τ

should be itself a homomorphism.

Definition 3.2.3. If V1 and V2 are extensions of V by W we say that these extensionsare equivalent if there exists a g-module homomorphism φ : V1 −→ V2 such that thefollowing diagram commutes:

0 // W

id

��

α // V1

φ

��

β // V

id

��

// 0

0 // Wα′ // V2

β′ // V // 0

(3.2.2)

(this defines an equivalence relation).

Definition 3.2.4. Ext(V,W ) = {equivalence classes of extension of V by W}

We will consider the special case V = K as a trivial g-module, that is, x · λ = 0

for all x ∈ g, λ ∈ K. If W is a g-module we would like to show a bijective relation

H1(g,W )←→ Ext(K,W ).

Remark 3.2.1. Let ω ∈ C1(g,W ) = Hom(g,W ), then for all x, y ∈ g we have

dω(x, y) = x · (ω(y))− y · (ω(x))− ω([x, y])

Then Z1(g,W ) = {ω ∈ C1(g,W )|ω([x, y]) = x · (ω(y))− y · (ω(x)) ∀x, y ∈ g}So if ω ∈ Z1(g,W ) we can consider the associated cohomology class [ω] ∈ H1(g,W ).Suppose that ω′ ∈ [ω] then ω(x)− ω′(x) = x · v for some v ∈W .

Fix [ω] ∈ H1(g,W ), we would like to associate it an extension of K by W . Let us

define a g-module Wω = (K×W, ·) by

g× (K×W ) // K×W

(x, (λ,w)) � // x · (λ,w) = (0, x · w + λω(x))

.

First of all we should make sure that this definition only depends on the cohomology

class: Choose ω′ ∈ [ω], we need to construct a g-module homomorphism φ : K×W −→K×W such that the following diagram commutes:

0 // W

id

��

α // Wω

φ��

β // K

id

��

// 0

0 // Wα // Wω′

β // K // 0,

(3.2.3)

19


where α(w) = (0, w) and β(λ,w) = λ.

Since α ◦ id = φ ◦α we have φ(0, w) = (0, w). In the same way β ◦ id = β ◦φ implies

that λ = β(λ,w) = (β ◦ φ)(λ, φ) so φ must have the form φ(λ,w) = (λ, f(λ,w)) where

f(0, w) = w. Now we will explore the homomorphism condition of φ:

φ(x · (λ,w)) = φ(0, x · w + λω(x))

= (0, x · w + λω(x))

= x · φ(λ,w) (homomorphism condition)

= x · (λ, f(λ,w))

= (0, x · f(λ,w) + λω′(x))

= (0, x · f(λ,w) + λω(x) + λx · v). (by Remark 3.2.1)

So for instance if we choose f(λ,w) = w − λv we are sure that Wω is well defined.

However we still need to show that indeed Wω is a g-module. Conditions 1 and 2

are easy to prove, we will work out condition 3.

x · (y · (λ,w)− y · (x · (λ,w) = x · (0, y · w + λω(y))− y · (0, x · w + λω(x))

= (0, x · y · w + λx · ω(y))− (0, y · x · w + λy · ω(x))

= (0, x · y · w − y · x · w + λ(x · ω(y)− y · (ω(x)))

= (0, [x, y]w − λω([x, y])) (by Remark 3.2.1)

= [x, y] · (λ, ω).

We have constructed a map η : H1(g,W ) −→ Ext(K,W ) which associates to each

cohomology class [ω] the equivalence class of the extension

0 // Wα // Wω

β // K // 0. (3.2.4)

Now we will construct a map in the other direction: Consider an extension

0 // Wα // V1

β // K // 0 . (3.2.5)

Let v ∈ V such that β(v1) = 1. Since K is a trivial g-module then for every x ∈ g we

have 0 = x · 1 = x · β(v1), but we know that β is an homomorphism so there exists

20


w ∈W such that α(w) = x ·v1. Notice that this element is unique since α is one-to-one

and obviously depends on x and v so we will denote it by w = w(x, v1).

Define ω1 : g −→W by ω1(x) = w(x, v). First of all we will show that this map defines

a cohomology class in H1(g,W ).

α(ω1([x, y])− x · ω1(y) + y · ω1(x)) = α(ω1([x, y]))− α(x · ω1(y)) + α(y · ω1(x))

= [x, y] · v − x · (y · v) + y · (x · v)

= 0.

We will now make sure that the cohomology class that ω1 defines does not depend

on v1, that is, if β(v2) = 1 then [ω1] = [ω2]. Since β(v1 − v2) = 0 there exists u ∈ Wsuch that v1 − v2 = α(u), so

α(ω1(x)− ω2(x)) = x · (ω1 − ω2)

= x · α(u)

= α(x · u).

Notice that since α is one-to-one and it we take into account Remark 3.2.1 we are

done.

Finally will show that [ω1] does not depend on the representative of the equivalence

class of (3.2.5). Consider the following commutative diagram

0 // W

id

��

α // V1

φ

��

β // K

id

��

// 0

0 // Wα′ // V2

β′ // K // 0

, (3.2.6)

we can associate to each vertical level of the diagram two cohomology classes [ω1]

and [ω2] with v1 ∈ V1 and v2 ∈ V2. Notice that we can choose v2 = φ(v1) since

β′(v2) = (β′ ◦ φ)(v1) = β(v1) = 1, hence

α′(ω2(x)− ω1(x)) = x · v2 − α′(ω1(x))

= x · φ(v1)− (φ ◦ α)(ω1(x))

= x · φ(v1)− φ(x · v1)

= 0.

21


and again, since α′ is one-to-one we have shown that ω2(x) = ω1(x).

We have constructed a well defined map ψ : Ext(K,W ) −→ H1(g,W ) which asso-

ciates to the equivalence class of (3.2.5) the cohomology class [ω1].

Recall that our objective is to show the correspondence between H1(g,W ) ←→Ext(K,W ), we will do so by checking that η ◦ ψ = id and ψ ◦ η = id.

η ◦ ψ = id

Lets start with an extension like (3.2.5), then we can construct [ω1] as we described

before. We need to construct a homeomorphism φ : Wω1 −→ V1 such that the following

diagram commutes

0 // W

id

��

i // Wω1

φ

��

π1 // K

id

��

// 0

0 // Wα // V1

β // K // 0,

(3.2.7)

where i(w) = (0, w) and π1(λ,w) = λ.

Define φ(λ, ω) = λv1+α(w). We need to check that it is actually an homomorphism:

φ(x · (λ,w)) = φ(0, x · w + λω1(x))

= α(x · w + λω1(x))

= x · α(w) + λx · v1

= x · (α(w) + λv1)

= x · φ(λ,w).

Finally we should verify that the diagram commutes:

• (φ ◦ i)(w) = φ(0, w) = α(w).

• (β ◦ φ)(λ, ω) = β(λv1 + α(w)) = λβ(v1) + 0 = λ = (id ◦ π1)(λ).

ψ ◦ η = id

Let [ω] ∈ H1(g,W ). Then we can associate it an extension like (3.2.4) via η. Using

22


the notation on the construction of ψ we can choose v1 = (1, 0) and therefore we have

α(w) = x · (1, 0) = (0, x · w + 1ω(x)) = (0, x · w + ω(x)). Finally, if we compute:

α(ω(x)− ω(x)) = x · (1, 0)− α(ω(x))

= (0, x · w + ω(x))− (0, ω(x))

= (0, x · w)

= α(x · w).

So we conclude that [ω] = [ω1] again by Remark 3.2.1.


Now we would like to give an interpretation to H2(g, V ) in a similar way that we did

in the last section.

Definition 3.3.1. Let g be a Lie algebra. A linear map δ : g −→ g is said to be aderivation if

δ([x, y]) = [δ(x), y] + [x, δ(y)] ∀x, y ∈ g (3.3.1)

We define Der(g) = {δ : g −→ g | δ is a derivation}.

Proposition 3.3.1. Der(g) has a Lie algebra structure with respect to the commutator.

Proof. We already know that gl(g) has a Lie algebra structure, so it suffices to showthat Der(g) is closed under the commutator:

[δ, δ′]([x, y]) =δ(δ′([x, y]))− δ′(δ([x, y]))

=δ([δ′(x), y] + [x, δ′(y)])− δ′([δ(x), y] + [x, δ′(y)])

=[δ(δ′(x)), y] + [δ′(x), δ(y)] + [δ(x), δ′(y)] + [x, δ(δ′(y))]

− [δ′(δ(x)), y]− [δ(x), δ′(y)]− [δ′(x), δ(y)]− [x, δ′(δ(y))]

=[[δ, δ′](x), y] + [x, [δ, δ′](y)].

Definition 3.3.2. We define a the adjoint representation ad : g −→ Der(g) byad(x)(y) = [x, y].

23


Definition 3.3.3. Consider the following short exact sequence of Lie algebras

0 // hα // g

β // q // 0. (3.3.2)

We say that the sequence splits if there exists a Lie algebra homomorphism τ : q −→ g

such that β ◦ τ = id. The notion of equivalent sequences is similar to the case ofg-modules.

Definition 3.3.4. Ext(q, h) = {equivalence classes of extensions of q by h that split}

Remark 3.3.1. Since β[α(h), x] = 0 for all n ∈ n and x ∈ g then α(h) is an ideal of g,that is [α(h), g] ⊆ α(h). However, since α is one-to-one, we can identify α(h) ∼ h andrephrase the last result as: [h, g] ⊆ h.

The objective of this section is to show the correspondence between Ext(q, V ) and

H2(q, V ). We will need to introduce the notion of semidirect sum of Lie algebras to

achieve our goal. This will be also helpful since it is very similar to the notion of

semidirect products of groups, that will be essential in the quantization scheme that

we will discuss in the next chapter.

Definition 3.3.5. Let q and h be two Lie algebras and ϕ : q −→ Der(h) a Lie algebrahomomorphism. Define a bracket bilinear [, ]ϕ on q× h by

[(x1, y1), (x2, y2)]ϕ = ([x1, x2], [y1, y2] + ϕ(x1)(y2)− ϕ(x2)(y1)). (3.3.3)

Proposition 3.3.2. The pair q nϕ h = (q× h, [, ]ϕ), which is a semidirect sum, hasa Lie algebra structure.

Example 3.3.1. Let q = Γ(TM) the set of vector fields over a smooth manifold M

with the usual Lie bracket, h = Γ(T ∗M) the set of 1-forms with a trivial Lie bracketand ϕ(X) = LX the Lie derivative in the direction of the vector field X. Then we have

[(X,ω), (Y, η)]ϕ = ([X,Y ], LXη − LY ω).

However, if we add a new term:

[(X,ω), (Y, η)]c = ([X,Y ], LXη − LY ω −12d(iXη − iY ω))

it defines what is known as the Courant bracket which determines the integrabilitycondition of Dirac structures over manifolds. However, this new bracket is not a Liebracket. (The relevance and meaning of this example is explained in detail in theauthor’s Mathematics Bachelor Thesis [Ord10]).

24


Lemma 3.3.1. Every short exact sequence of Lie algebras that splits is equivalent toan extension of a semidirect q nϕ h of q by h.

Proof. Consider a short exact sequence like (3.3.2) that splits. Define ϕτ : q −→ Der(h)by ϕτ (x)(h) = α−1([τ(x), α(h)]) = [τ(x), h], where the last equality follows from theidentification of Remark 3.3.1.Now we can define a map ψ : q nϕτ h −→ g by ψ(x, h) = τ(x) + α(h). Notice that ψ isan homomorphism

ψ([(x1, h1), (x2, h2)]ϕτ )

= ψ([x1, x2], [h1, h2] + ϕτ (x1)(h2)− ϕτ (x2)(h1))

= τ([x1, x2]) + α([h1, h2] + ϕτ (x1)(h2)− ϕτ (x2)(h1))

= τ([x1, x2]) + α([h1, h2] + α−1([τ(x1), α(h2)])− α−1([τ(x2), α(h2)]))

= τ([x1, x2]) + α([h1, h2]) + [τ(x1), α(h2)]− [τ(x2), α(h2)])

= [τ(x1), τ(x2)] + [α(h1), α(h2)] + [τ(x1), α(h2)] + [α(h2), τ(x2)])

= [τ(x1) + α(h1), τ(x2) + α(h2)]

= [ψ(x1, h1), ψ(x2, h2)].

Consider now the following diagram

0 // h

id

��

i // q nϕτ h

ψ

��

π1 // qτ0

oo

id

��

// 0

0 // hα // g

β // qτ

oo // 0,

with τ0(x) = (x, 0).

Remark 3.3.2. Let V a q-module. Then V can be viewed as an Abelian Lie algebraif we set [a, b] = 0 ∀a, b ∈ V .

Lemma 3.3.2. Let V a q-module and ω ∈ C2(q, V ). Define a bracket on q × V by[(x, a), (y, b)]ω = ([x, y], x · b− y · a+ ω(x, y)).Then we have the following properties:

1. [, ]ω is a Lie bracket if and only if ω ∈ Z2(q, V ).

2. In such a case we obtain a short exact sequence of Lie algebras

0 // Vi // qω

π1 // q // 0, (3.3.4)

where qω = (q× V, [, ]ω).

25


3. The last sequence splits if and only if ω ∈ B2(q, V ).

Proof. Recall that

dω(x, y, z) = x ·ω(y, z)− y ·ω(x, z) + z ·ω(x, y)−ω([x, y], z) +ω([x, z], y)−ω([y, z], x).

So if we compute

[(x, a), [(y, b), (z, c)]] = [(x, a), ([y, z], y · c− z · b+ ω(y, z))]

= ([x, [y, z]], x · (y · c− z · b+ ω(y, z))− [y, z] · a+ ω(x, [y, z])),

[(y, b), [(z, c), (x, a)]] = [(y, b), ([z, x], z · a− x · c+ ω(z, x))]

= ([y, [z, x]], y · (z · a− x · c+ ω(z, x))− [z, x] · b+ ω(y, [z, x])),

[(z, c), [(x, a), (y, b)]] = [(z, c), ([x, y], x · b− y · a+ ω(x, y))]

= ([z, [x, y]], z · (x · b− y · a+ ω(x, y))− [x, y] · c+ ω(z, [x, y])),

we have then

[(x, a), [(y, b), (z, c)]] + [(y, b), [(z, c), (x, a)]] + [(z, c), [(x, a), (y, b)]] = (0, dω(x, y, z)).

Now suppose ω ∈ Z2(g, V ), we want to find a Lie algebra homomorphism τ : q −→ q×Vsuch that π ◦ τ = id. Suppose τ is of the form τ(x) = (f1(x), f2(x)), we require

τ([x, y]) = (f1([x, y]), f2([x, y]))

= [τ(x), τ(y)]ω

= [(f1(x), f2(x)), (f1(y), f2(y))]ω

= ([f1(x), f1(y)], f1(x) · f2(y)− f1(y) · f2(x) + ω(f1(x), f1(y))).

But, since π ◦ τ = id we must have f1 = id. Finally we conclude that

f2([x, y]) = x · f2(y)− y · f2(x) + ω([x, y]).

Hence ω = −df by Remark 3.2.1.

Proposition 3.3.3. Suppose that the sequence (3.3.4) splits, then qω is equivalent toa semidirect sum of q and V .

26


Proof. Let τ : q −→ qω be such that π1 ◦ τ = id. Define φτ : q −→ Der(V ) andψ : q nφτ V −→ qω by

φτ (x)(v) = i−1([τ(x), i(v)]ω) = i−1([(x, f2(x)), (0, v)]ω) = i−1(0, x · v) = x · v

and

ψ(x, a) = τ(x) + i(a) = (x, f2(x)) + (0, a) = (x, f2(x) + a).

We will show that ψ is a homomorphism

[ψ(x, a), ψ(y, b)]ω = [(x, f2(x) + a), (y, f2(y) + b)]ω

= ([x, y], x · (f2(y) + b)− y · (f2(x) + a) + ω(x, y))

= ([x, y], x · (f2(y) + b)− y · (f2(x) + a)− df2(x, y))

= ([x, y], x · b− y · a+ f2([x, y]))

= τ([x, y]) + i(x · b− y · a)

= ψ([x, y], x · b− y · a)

= ψ([(x, a), (x, b)]φτ ).

Remark 3.3.3. Consider the short exact sequence (3.3.2) of Lie algebras and supposeh is Abelian. We can induce a q-module structure on h by q · h = α−1([β−1(q), α(h)]).This definition makes sense by Remark 3.3.1, but we need to check that is well defined:Let x1, x2 ∈ β−1(q), then x1 − x2 = α(h) for some h ∈ h. So if we compute

α(α−1([x1, α(h)])− α−1([x2, α(h)])) = [α−1(x1), h]− [α−1(x2), h] = [h, h] = 0.

Theorem 3.3.1. There is a one-to-one correspondence between Ext(q, V ) and H2(q, V ).(HereV is viewed as a q-module as seen in Remark 3.3.3)

Proof. Consider an extension of q by g like (3.3.2) (with h = V ). Choose a lineartransformation τ : q −→ g such that β ◦ τ = id and define ω ∈ C2(q, V ) by

ωτ (x, y) = ω(x, y) = α−1([τ(x), τ(y)]− τ([x, y])).

Notice that [τ(x), τ(y)]− τ([x, y] ∈ Ker(β) = Im(α), so the definition of ω makes sense.Our next step is to show that ω ∈ Z2(q, V ). In Remark 3.3.3 we can choose β−1(x) as

27


τ(x), so if we compute

dω(x, y, z) =x · ω(y, z)− y · ω(x, z) + z · ω(x, y)− ω([x, y], z) + ω([x, z], y)− ω([y, z], x)

=α−1([τ(x), α(ω(y, z))])− α−1([τ(y), α(ω(x, z))]) + α−1([τ(z), α(ω(x, y))])

− ω([x, y], z) + ω([x, z], y)− ω([y, z], x)

=α−1([τ(x), [τ(y), τ(z)]])− α−1([τ(x), τ([y, z])])− α−1([τ(y), [τ(x), τ(z)]])

+ α−1([τ(y), τ([x, z])]) + α−1([τ(z), [τ(x), τ(y)]])− α−1([τ(z), τ([x, y])])

− α−1([τ([x, y]), τ(z)]) + α−1(τ([[x, y], z])) + α−1([τ([x, z]), τ(y)])

− α−1(τ([[x, z], y]))− α−1([τ([y, z]), τ(x)]) + α−1(τ([[y, z], x]))

=0.

Now suppose that we choose a different τ ′ : q −→ g such that β ◦ τ ′ = id and letσ = τ − τ ′, then β ◦σ = 0, which implies that σ(q) ⊆ α(V ). Since α−1 ◦σ ∈ Hom(q, V )we have

d(α−1 ◦ σ)(x, y) = x · (α−1 ◦ σ)(y)− y · (α−1 ◦ σ)(x)− (α−1 ◦ σ)([x, y]).

If we compute

(ωτ − ωτ ′)(x, y) =α−1([τ(x), τ(y)]− τ([x, y]))− α−1([τ ′(x), τ ′(y)]− τ ′([x, y]))

=α−1([τ(x), τ(y)]− [τ ′(x), τ(y)] + [τ ′(x), τ(y)]

− τ([x, y])− [τ ′(x), τ ′(y)] + τ ′([x, y]))

=α−1([σ(x), τ(y)] + [τ ′(x), σ(y)]− σ([x, y]))

=α−1([σ(x), τ(y)] + [(τ − σ)(x), σ(y)]− σ([x, y]))

=α−1([σ(x), τ(y)] + [τ(x), σ(y)]− σ([x, y]))

=x · (α−1 ◦ σ)(y)− y · (α−1 ◦ σ)(x)− (α−1 ◦ σ)([x, y])

=d(α−1 ◦ σ)(x, y)

We see that ωτ and ωτ ′ belong to the same cohomology class. Recall that by Proposi-tion 3.3.3 and lemmas 3.3.1 and 3.3.2 we have associated to each equivalence class ofsequences that split a cohomology class in H2(q, V ).

Now suppose that ω, ω ∈ Z2(g, V ) are given. By Lemma (3.3.2) we have a shortexact sequence of Lie algebras like (3.3.4) associated to ω and ω. We want to show thatif the extensions are equivalent, then [ω] = [ω]. Suppose that the following commutative

28

3.4 Removing the Obstruction Cocycle

diagram is given:

0 // V

id

��

i // qω

φ

��

π1 // q

id

��

// 0

0 // Vα // qω

β // q // 0.

There exists a linear map φ : q −→ V such that φ(x, a) = (x, a+ φ(x)). Now we explorethe homomorphism condition of φ: by one hand

φ[(x, a), (y, b)] = φ([x, y], x · b− y · a+ ω(x, y))

= ([x, y], x · b− y · a+ ω(x, y) + φ(x, y)).

and on the other

[φ(x, a), φ(y, b)] = [(x, a+ φ(x)), (y, b+ φ(y))]

= ([x, y], x · (b+ φ(y))− y · (a+ φ(x)) + ω(x, y)).

So we conclude that

(ω − ω)(x, y) = x · φ(y)− y · φ(x) + φ([x, y]) = dφ(x, y).

Therefore [ω] = [ω].

Remark 3.3.4. As a result of the last theorem we conclude that the elements ofH2(q,K) classify the central extensions of q by K, which are extension where V = Kis considered as a trivial q-module:

0 // K i // qωπ1 // q // 0.

The Lie bracket in this case is given by (see Lemma 3.3.2)

[(x, r), (y, s)] = ([x, y], ω(x, y)). (3.3.5)

Notice that the image of i lies in the center of qω.


Recall from last chapter that the obstruction for the lift map P : g −→ C∞(M,R) to

be a Lie algebra homomorphism is measured by a bilinear map z : g × g −→ R which

29


is antisymmetric and satisfies the Jacobi identity.

If we consider R as a trivial g-module, then is easily seen that z ∈ C2(g,R) and there-

fore it defines a cohomology class in H2(g,R). We showed that this 2-cocycle can be

made to vanish if we could find an element h ∈ g∗ such that z(A,B) = h([A,B]) for all

A,B ∈ g, however this equivalent as requiring that the cohomology class defined by z

is the zero class.

If the cohomology class defined by the action of the group G is not the zero class

(as it will be the case for Q = R as the configuration space) we can make a little trick,

as Isham says: we can enlarge the group in such a way that the Poisson algebra bracket

the new group does close. More precisely, we can consider a central extension of g by

R.

0 // R α //

id

��

g⊕ R β //

P

��

g //

γ

��

0

0 // R i // C∞(M,R)j // Ham VF(M) // 0,

(3.4.1)

where α(r) = (0, r) and β(A, r) = A.

Notice that the Lie bracket of g⊕ R is given by equation (3.3.5):

[(A, r), (B, s)] = ([A,B], z(A,B)). (3.4.2)

If we denote the original map defined in the last chapter by P : g −→ C∞(M,R) ,

then the new map P is defined by P (A,r) = PA + r. Clearly, this definition makes the

above diagram commutative, but we need to make sure that this time P is indeed a

Lie algebra homomorphism:

{P (A,r), P (B,s)} = {PA + r, PB + s}

= {PA, PB}

= P [A,B] + z(A,B)

= P ([A,B],z(A,B))

= P [(A,r),(B,s)].

30


Remark 3.4.1. In the where we require to do a central extension to find the desiredmoment map, the new canonical group will be the unique simply connected Lie groupwhose Lie algebra is g⊕ R with the Lie bracket defined above.

31


32

4

Canonical Group Quantization:

One-Dimensional Examples

The goal of this chapter is to describe the Canonical Group Quantization Method and

illustrate it by means of three examples in which the configuration space is a one-

dimensional manifold. The notion of semidirect product is introduced in this chapter

too, since it is essential for the understanding of the general case.

4.1 Canonical Group Quantization Method

We are going to describe the quantization method presented by C.J Isham in a beautiful

text [Ish84] that we highly recommend to refer for further details and remarks. We are

going to describe the main points on this quantization scheme, although it is expected

to be understood through the worked examples that we will explore afterwards.

We will develop the method for a given symplectic manifold (M,ω), but it is helpful

to think of M as a cotangent bundle of a configuration space. The scheme is divided,

as Isham points out, in two main parts:

1. The first one has a purely geometric point of view: we are going to look for a

connected Lie group G, which we will call the canonical group that will act

(appropriately) on M such that its Lie algebra g is:

(a) a sub-algebra of the Poisson algebra on smooth real functions on M (under

an appropriate identification) and

33

4. CANONICAL GROUP QUANTIZATION: ONE-DIMENSIONALEXAMPLES

(b) big enough (this is to be defined properly) to generate a sufficiently class of

classical observables.

2. The second part is focused on representation theory: we will find irreducible,

weakly continuous, unitary representations of G to use the self-adjoint generators

to provide representations of g on some Hilbert space.

We have been unprecise with some notions on the discussion above. We will now

explain the scheme, step by step, in a similar way Isham does it in his text. Again,

we highly recommend to go to the primary source [Ish84] to have a long and detailed

description of the quantization programme, since it is not out purpose to rewrite com-

plete text but rather make some comments and be more explicit in some computations.

Step 1: We are given a symplectic manifold (M,ω) and we need to find a con-

nected finite-dimensional Lie group G that acts via symplectic transformations on M .

Nevertheless, we will impose some restrictions on this action:

1. We want a transitive action: If this were not the case we could decompose M

into G-orbits and construct a quantum theory on each one and it will not be clear

how to treat them as a whole global theory.

The transitivity of the action will also be justified when we discuss the size of the

family of the chosen observables.

2. We want an effective action: We can require this without lost of generality since,

if the action is not effective, we can work with the quotient G/K, where

K = {g ∈ G | `gx = x ∀x ∈M} (which is a normal subgroup).

3. For each A ∈ G we require that γA ∈ Ham VF(M). (Recall the definitions

presented in Chapter 2).

4. We want that the map γ to be one-to-one, thus we can relax condition 2 and ask

for an almost-effective action.

Step 2 We want to construct a moment map P : g 7−→ C∞(M,R) such that it is a

34

4.1 Canonical Group Quantization Method

Lie algebra homomorphism and makes the following diagram commute:

0 // R i // C∞(M,R)j // Ham VF(M) // 0.

g

γ

OO

P

hhP P P P P P P

(4.1.1)

We have seen that this is possible if the cohomology class in H2(g,R) defined by the

action is the zero class. If it is not the case we have to perform a central extension of

g, as we presented in Chapter 3.

Step 3 This is not actually a step on the programme but rather a discussion on the

notion of a big enough family of observables. We will discuss two possible approaches:

1. Strong Generating Principle (SG): Let {A1, ..., Ak} be a basis for g. Given

any function f ∈ C∞(M,R) there exists a function Ff ∈ C∞(Rk,R), where

k = dim(g), such that f(s) = Ff (PA1(s), ..., PAk(s)). This requirement is equiv-

alent to the existence of a smooth embedding F : M 7−→ Rk, defined by

F(s) = (PA1(s), ..., PAk(s)), such that f factors trough F via Ff .

Recall that by the Whitney embedding theorem if dim(M) = 2n, then such em-

bedding always exists if k ≥ 4n, however the family of chosen observables may

not close, in general, under the Poisson bracket.

As Isham points out, we have no a priori physical knowledge of what the gener-

ating principle should be, he proposes a local point of view.

2. Local Generating Principle (LG): Around any point s0 ∈ M there exists an

open neighborhood Us0 such that, given any f ∈ C∞(M,R) whose support is con-

tained in Us0, there exists a function Ff ∈ C∞(RkR) such that, for all s ∈ Us0we have f(s) = Ff (PA

1(s), ..., PAk(s)). This is equivalent to that the map F is a

local embedding, that is, a local diffeomorphism onto its image.

We will now justify why a transitive action implies (LG). When G acts transitively

on M , the vector fields γA will span TsM . Suppose that this were not the case,

35


then we would not be able to construct a 1-parameter subgroup of G to move

the points in the missed direction that is not spanned by the γA’s, which will

contradict the transitivity of the action.

Notice that, since iγAω = dPA, and ω is non-degenerate, the 1-forms {dPA|A ∈ g}

will span the cotangent space T ∗sM . This means that the map F : M −→ Rk has

maximal rank k, so it follows that it is an immersion. Finally, by the implicit

function theorem we know that every immersion in a local embedding.

Step 4 This is where representation theory becomes essential: we want to find

irreducible, weakly continuous, unitary representations U : G −→ H, where H is a

Hilbert space. Recall tat a weakly continuous representation means that the maps

g 7−→ 〈ψ,U(g)φ〉 and g 7−→ 〈U(g)ψ, φ〉 are continuous for all ψ, φ ∈ H and g ∈ G. This

conditions guarantee the existence of self-adjoint operators U(exp(A)) = exp(−iKA)

for all A ∈ g. Notice that

[−iKA,−iKB] =d

dt

∣∣∣∣t=0

Adexp(−itKA)(−iKB)

=d

dt

∣∣∣∣t=0

AdU(exp(tA))(−iKB)

=d

dt

∣∣∣∣t=0

AdU(exp(tA))d

dτ

∣∣∣∣τ=0

U(exp(τB))

= −iK [A,B].

So [KA, KB] = iK [A,B], therefore we can define a quantizing map:

PA ˆ7−→1iKA, (4.1.2)

which is consistent between the Poisson bracket and the operator commutator:

{PA, PB} = P [A,B]

=1iK [A,B]

= −[KA, KB]

=[

1iKA,

1iKB

]= [PA, PB].

36

4.2 Q = R and the Heisenberg Group


Our first example for the quantization method that we discussed is the topologically

trivial configuration space Q = R. The cotangent bundle is trivial, that is T ∗R = R×R.

We can use global coordinates (which, in general, is not always possible):

q(s) = pr1(s) p(s) = pr2(s) s ∈ T ∗R, (4.2.1)

where pr1 is the projection onto the first component and pr2 is the projection onto the

second component. As we saw, the symplectic form on T ∗R is given by ω = dq ∧ dp.

A natural choice for the canonical group is R× R acting on T ∗R by

`(u,v)(p, q) = (q + u, p− v). (4.2.2)

Notice that this action is clearly transitive and effective. To show it is symplectic

we compute

`∗(u,v)ω = `∗(u,v)(dq ∧ dp)

= d(`∗(u,v)q) ∧ d(`∗(u,v)p)

= d(q ◦ `(u,v)) ∧ d(p ◦ `(u,v))

= d(q + v) ∧ d(p− v)

= dq ∧ dp

= ω.

Now let us denote the Lie algebra of R × R by L(R × R) = R × R. So the one

parameter subgroup generated by (a, b) ∈ L(R×R) is just t 7−→ (ta, tb). Hence the its

associated infinitesimal generator is obtained by

γ(a,b)(p, q) = − d

dt

∣∣∣∣t=0

`(ta,tb)(p, q)

= − d

dt

∣∣∣∣t=0

(q + ta, p− tb)

= −a ∂∂q

+ b∂

∂p.

37


We need to verify that this vector field is Hamiltonian, but this is not a problem

since H1dR(T ∗R) = 0. But actually observe that if we define f : T ∗R −→ R by f(s) =

ap(s) + bq(s) then

iγ(a,b)ω = iγ(a,b)(dq ∧ dp)

= (dq ∧ dp)(−a ∂

∂q+ b

∂

∂p

)= −adp− bdq

= −df.

So we have shown that the map γ : L(R× R) −→ Ham VF(T ∗R) is well defined.

Now we need to compute the cohomology class defined by the action. A natural

map P : L(R× R) −→ C∞(T ∗R,R) to choose is P (a,b)(s) = ap(s) + bq(s). First of all

we compute the Poisson bracket

{P (a1,b1), P (a2,b2)} = {a1p+ b1q, a2p+ b2q}

= (dq ∧ dp)(a1

∂

∂q− b1

∂

∂p, a2

∂

∂q− b2

∂

∂p

)= b1a2 − b2a1.

On the other hand, since L(R× R) is Abelian ,we conclude that

z((a1, b1), (a2, b2)) = b1a2 − b2a1. (4.2.3)

The cohomology class defined by this 2-cocycle is not the zero class since for all

h ∈ L(R× R)∗ we have h([(a1, b1), (a2, b2)]) = 0.

As discussed in the last section, to continue with the quantization scheme we need

to employ a central extension of L(R×R). In this case the new Lie bracket on the Lie

algebra L(R× R)⊕ R is given by

[(a1, b1, c1), (a2, b2, c2)] = (0, 0, b1a2 − b2a1), (4.2.4)

and the new momentum map is then

P (a,b,r)(s) = ap(s) + bq(s) + r. (4.2.5)

38


It is natural to ask which is the (simply connected) Lie group whose Lie algebra is

given by (4.2.4). As we will see, this group is known as the Heisenberg group, which

we will describe following Guillemin’s approach. [GS84].

Definition 4.2.1. Let (V, ω) a symplectic vector space and S1 ⊆ C. We define a groupstructure on V × S1 by

(v1, z1)(v2, z2) =(v1 + v1, z1z2 exp

(i

2ω(v1, v2)

)). (4.2.6)

This group is called the Heisenberg group and will be denoted by Heisω(V ).

We can identify the its Lie algebra L(Heisω(V )) ∼= V × iR since the most general

1-parameter subgroup is given by t 7−→ (tv, exp(itθ)).

(tv, exp(itθ))(sv, exp(isθ)) = (tv + sv, exp(itθ) exp(isθ) exp((i/2)ω(tv, sv))

= ((t+ s)v, exp(i(t+ s)θ)).

Now we want to compute the explicit form of the Lie bracket on L(Heisω(V )) using

the formulas of the adjoint representation presented in Chapter 2. First of all we need

to explore the conjugation on the group

(v1, z1)(v2, z2)(v1, z1)−1 = (v1, z1)(v2, z2)(−v1, z−11 )

= (v1, z1)(v2 − v1, z2z−11 exp((i/2)ω(v2,−v1)))

= (v2, z2 exp((i/2)ω(v2,−v1)) exp((i/2)ω(v1, v2 − v1)))

= (v2, z2 exp(iω(v2, v2))).

Now we find an explicit formula for the adjoint representation

Ad(v1,z1)(u2, iθ2) =d

dt

∣∣∣∣t=0

(v1, z1)(tu2, exp(itθ2))(v1, z1)−1

=d

dt

∣∣∣∣t=0

(tu2, exp(itθ2) exp(iω(v1, tu2)))

=d

dt

∣∣∣∣t=0

(tu2, exp(it(θ2 + ω(v1, u2))))

= (u2, i(θ2 + ω(v1, u2))).

39


Finally we compute its derivative

[(u1, iθ1), (u2, iθ2)] = ad(u1,iθ1)(u2, iθ2)

=d

dt

∣∣∣∣t=0

Ad(tu1,exp(itθ1))(u2, iθ2)

=d

dt

∣∣∣∣t=0

(u2, i(θ2 + ω(tu1, u2)))

= (0, iω(u1, u2)).

Keep on mind that we are looking for a simply connected Lie group, one realizes

that V × S1 will not work. However, we can go to the universal cover: Recall that Ris the universal cover of S1 and the covering map is given by exp(i ) : R −→ S1. So

the Heisenberg group structure on R × R × R (taking V = T ∗R = R × R) induced by

equation (4.2.6) is

(u1, v1, t1), (u2, v2, t2) =(u1 + u2, v1 + v1, t1 + t2 +

12

(v1u2 − v2u1)), (4.2.7)

which is the definition that Isham gives arguing that with the Baker-Cambell-Hausdorff

formula and exp(A) exp(B) = exp(A+B+ (1/2)[A,B] + higher order commutators) it

can be shown that its Lie algebra structure is given by equation (4.2.4).

The second part on the quantization programme is to find weakly continuous, ir-

reducible and unitary representations g 7−→ U ′(g). We will now define the operators

U(a) = U ′(exp(a, 0, 0)) V (b) = U ′(exp(0, b, 0)) (4.2.8)

Notice that for R, the central subgroup, the irreducible representations are of the form

U ′(exp(0, 0, r)) = exp(−iµr) for some µ ∈ R (4.2.9)

Now we will prove the following formulas which are known as Weyl commutation

relations:

Proposition 4.2.1. The operators defined above satisfy the following relations

1. U(a1)U(a2) = U(a1 + a2).

2. V (b1)V (b2) = U(b1 + b2).

40


3. U(a)V (b) = V (b)U(a) exp(iµab).

Proof. The first two relations are easy to prove, we will work the last one: Notice thatfrom equation (4.2.7) we have

(u1, 0, 0)(0, v2, 0) = (u1, v1,−(1/2)v2u1)

= (0, v2, 0)(u1, 0, 0)(0, 0, u1v2).

Choose in particular (u1, 0, 0) = exp(a, 0, 0) = (a, 0, 0) and (0, v2, 0) = exp(0, b, 0) =(0, 0, b).

Now we will rewrite those operators in terms of exponentials of self-adjoint operators

and find their the commutation relation

U(a) = exp(−iap) V (b) = exp(−ibq) (4.2.10)

Proposition 4.2.2. The commutation relation of the self-adjoint operators defined byequation (4.2.10) is

[q, p] = iµ. (4.2.11)

Proof. We will use, as usual, the formulas for the adjoint representation and Proposition4.2.1. First of all we compute

AdV (b)−1(−ip) =d

dt

∣∣∣∣t=0

V (b)−1U(t)V (b)

=d

dt

∣∣∣∣t=0

U(t) exp(iµtb)

= −ip+ iµb.

Hence,

[iq,−ip] =d

dt

∣∣∣∣t=0

AdV (t)−1(−ip)

=d

dt

∣∣∣∣t=0

(−ip+ iµt)

= iµ.

Notice that this commutation relation is consistent with the quantizing map

P (a,b,r) = ap+ bq + r ˆ7−→1i(ap+ bq + rµ). (4.2.12)

41


Remark 4.2.1. If we compute from equation (4.2.4)

[(1, 0, 0), (0, 1, 0)] = (0, 0,−1),

we see that the commutation relation [q, p] = iµ could be “read off” from the bracketof the Lie algebra.

We have not talked at all about the role of Planck’s constant ~. This constant will

arise when we fix the dimensions. Notice that we can define new operators with, as

Isham says, dimensional eigenvalues:

qphys = q0q pphys = p0p (4.2.13)

Thus [qphys, pphys] = iq0p0µ, and therefore we can identify q0p0µ = ~. So Plank’s con-

stant parameterizes a one-parameter family of unitary representations of the central

group R.

The following is a very important remark:

Remark 4.2.2. There is an extremely powerful result which is known as the Stone-von Newmann theorem that asserts that, up to an unitary equivalence, there isa unique weakly continuous, irreducible representation of the Weyl relations given byProposition 4.2.1. The representation space is H = L2(R, dq) and the explicit expres-sions are:

(U(a)ψ)(q) = ψ(q − µa) (V (b)ψ)(q) = exp(−ibq)ψ(q) (4.2.14)

Now we will work out how to obtain the action of the self-adjoint operators:

(qψ)(q) =d

dt

∣∣∣∣t=0

iV (t)ψ(q)

=d

dt

∣∣∣∣t=0

i exp(−itq)ψ(q)

= qψ(q),

in the same way

(pψ)(q) =d

dt

∣∣∣∣t=0

iU(t)ψ(q)

=d

dt

∣∣∣∣t=0

iψ(q − µt)

= −iµdψdq

(q).

42

4.3 Semidirect Products

Finally we will quote a comment made by Isham [Ish84] that exhibits the importance

of this result: It can be said that the Stone-Von Newmann theorem lies at the heart of

the success of quantum theory. If there were many irreducible representations, each one

would generate different physical predictions and the rules of quantum mechanics would

need to be supplemented with a specification of how the correct representation was to be

selected.


This section is based on my lecture notes of the course Particles and Fields II given by

professor Dr. Andres Reyes, at Universidad de los Andes, in the first semester of 2010.

The motivation for discussing semidirect products was the study of the representation

of the Poincare group. I want to thank again professor Dr. Andres Reyes for a com-

prehensive introduction to these topics.

Definition 4.3.1. Let N and K be two groups and ϕ : K −→ Aut(N) a group homo-morphism. Then ϕ can induce a group structure on N ×K given by the multiplicationrule

(n, k)(n, k) = (nϕk(n), kk). (4.3.1)

The group under this operation is called the semidirect product of N and K and willbe denoted by N oϕ K.

This definition of semidirect product seems a little strange, however in the next

example we will see that it arises naturally from a physical point of view.

Example 4.3.1 (Poincare Group). Consider the proper orthochronous Lorentz groupL↑+. Let a ∈ R4 be a four vector and Λ ∈ L↑+ be a Lorentz transformation. We wouldlike to give a group structure to the product R4 × L↑+ such that (a,Λ) · x = a + Λx isan action on the Minkowski space. If we compute

((a1,Λ1)(a2,Λ2)) · x = (a1,Λ1) · (a2 + Λ2x)

= a1 + Λ1(a2 + Λ2x)

= (a1 + Λ1a2) + Λ1Λ2x,

we can read that the group structure should be

(a1,Λ1)(a2,Λ2) = (a1 + Λ1a2,Λ1Λ2), (4.3.2)

43


which is nothing more that the semidirect product R4 oϕ L↑+, where ϕ(Λ) = Λ. This

group is known as the Poincare group.

The last example illustrates a particular interesting case of semidirect products

where N = V is a vector space and K = G is a Lie group, and therefore the map

ϕ : G −→ Aut(V ) can be seen as a representation.

In this case the product on V oϕ G is given by

(v1, g1)(v2, g2) = (v1 + ϕ(g1)v2, g1g2). (4.3.3)

We would like to study the Lie bracket of L(V oϕ G) ∼= V × g induced by the

semidirect product. Let {Jα}nα=1 a basis for g and {ei}mi=1 be a basis for V (which is

its same Lie algebra). We want to compute the bracket [(ei, Jα), (ej , Jβ)].

We will prove one technical lemma, which for simplicity, will be the special case

where the Lie group is a subgroup of Mn(R), the group of n× n matrices over R.

Lemma 4.3.1. Let H ⊆ Mn(R) be a Lie group and h its Lie algebra, then for allx, y ∈ h we have

[x, y] = limε→0

1ε2

(exp(εx) exp(εy)− exp(εy) exp(εx)), (4.3.4)

where exp : h −→ H is the usual exponential map for matrices.

Proof. We just compute

exp(εx) exp(εy)− exp(εy) exp(εx)

=(1 + εx+12!ε2x2 + · · · )(1 + εy +

12!ε2y2 + · · · )

− (1 + εy +12!ε2y2 + · · · )(1 + εx+

12!ε2x2 + · · · )

=ε2(xy − yx) + O(ε3).

Remark 4.3.1. We will assume that there exists a representation the Lie algebra (seediagram (2.2.2)) ϕ : g −→ End(V ) such that

ϕ(exp(x)) = exp ϕ(x) ∀x ∈ g. (4.3.5)

44


We will use the above results and the fact that for ε � 1 we can approximate

exp(εx) ≈ 1 + εx to compute the following commutator:

[(e1, 0), (0, Jβ)] = limε→0

1ε2

(exp(ε(ei, 0))(exp(ε(0, Jβ)))− exp(ε(ei, 0))(exp(ε(0, Jβ))))

= limε→0

1ε2

((εei, e)(0, exp(εJβ))− (0, exp(εJβ))(εei, 0))

= limε→0

1ε2

((εei + ϕ(0), exp(εJβ))− (0 + ϕ(exp(εJβ))(εei), exp(εJβ)))

= limε→0

1ε2

((εei, exp(εJβ))− (ϕ(exp(εJβ))(εei), exp(εJβ)))

≈ limε→0

1ε2

((εei, 1 + εJβ)− (ϕ(1 + εJβ)(εei), 1 + εJβ))

≈ limε→0

1ε2

(−ε2ϕ(Jβ)(ei), 0)

≈ −(ϕ(Jβ)(ei), 0).

Now we are ready to compute a general commutator

[(ei, Jα), (ej , Jβ)] = [(ei, 0) + (0, Jα), (ej , 0) + (0, Jβ)]

= [(ei, 0), (ej , 0)] + [(ei, 0), (0, Jβ)] + [(0, Jα), (ej , 0)] + [(0, Jα), (0, Jβ)]

= −(ϕ(Jβ)(ei), 0) + (ϕ(Jα)(ej), 0) + (0, [Jα,Jβ ])

= (ϕ(Jα)(ej)− ϕ(Jβ)(ei), [Jα, Jβ]).

The following equation will be continuously used during the next sections:

[(ei, Jα), (ej , Jβ)] = (ϕ(Jα)(ej)− ϕ(Jβ)(ei), [Jα, Jβ]) (4.3.6)

Example 4.3.2 (Poincare Group II). If we denote by J1, J2 and J3 the generators ofrotations in three dimensions, and by K1,K2 and K3 the generator of boosts, we havethe following commutation relations for the Lie algebra of L↑+:

[Ji, Jj ] = εijkJk [Ji,Kj ] = εijkKk [Ki,Kj ] = −εijkJk

Using equation (4.3.6) we find that the commutation relations for the Lie algebra ofthe Poincare group are

[Ji, Jj ] = εijkJk [Ji,Kj ] = εijkKk [Ki,Kj ] = −εijkJk [eµ, eν ] = 0

[Ji, ej ] = εijkek [Ji, e0] = 0 [Ki, ej ] = δije0 [Ki, e0] = ei

45


4.4 Q = R+

We are going to discuss another example to illustrate the quantization method. The

configuration space is going to be Q = R+ = {r ∈ R | r > 0}. We could attempt to

impose the commutation relations [x, p] = i~ and define operators (which are actually

the unique operators that satisfy the commutation relation in the representation space

H = L2(R+, dx)) by

(xψ)(x) = xψ(x) (pψ)(x) = −i~dψdx

(x) (4.4.1)

but there would be important inconsistencies:

1. There would be a whole family of square integrable functions, ψk(x) = exp−kxwith k > 0, such that are eigenvectors of p† with purely imaginary eigenvalues.

2. Suppose p is self-adjoint, then U(a) = exp(−iap) would be a unitary operator

that would act over functions as a translation operator (U(a)ψ)(x) = ψ(x− ~a).

This will be impossible since we could take a sufficiently large value of a to take

the function out the configuration space.

This is why we should be careful in the choice of the commutation relations. Notice

that R is also topologically trivial (it is contractible) but it is not a vector space. This

global property affects its quantum theory.

We will now proceed with the quantization program step by step: The cotangent

bundle of R+ is trivial, T ∗R+ = R+ × R, and therefore we can also consider a global

system of coordinates

x(s) = pr1(s) p(s) = pr2(s) s ∈ T ∗R+, (4.4.2)

where pr1 is the projection onto the first component and pr2 is the projection onto

the second component, and therefore the symplectic form on T ∗R+ ca be written as

ω = dx ∧ dp.The canonical group will be choose to the semidirect product R oϕ R+, where ϕ :

R+ −→ Aut(R) is defined by ϕ(λ)(x) = λ−1x, and the multiplication law (given by

equation (4.3.3))

(v1, λ1)(v2, λ2) = (v1 + λ−11 v2, λ1λ2). (4.4.3)

46

4.4 Q = R+

The action of R nϕ R+ on T ∗R+ is defined by

`(v,λ)(x, p) = (λx, λ−1p− v). (4.4.4)

First of all we need to check that this actually defines an action:

`(v1,λ1)`(v2,λ2)(x, p) = `(v1,λ1)(λ2x, λ−12 p− v2)

= (λ1λ2x, λ−11 (λ−1

2 p− v2)− v1)

= (λ1λ2x, (λ1λ2)−1p− (v1 + λ−11 v2))

= `(v1,λ1)(v2,λ1)(x, p).

Notice the action is clearly transitive and effective since

(v1, λ1)(λ1(v2 − v1), λ−11 λ2) = (v2, λ2).

Finally we need to show that the action is symplectic

`∗(v,λ)(dx ∧ dp) = d(`∗(v,λ)x) ∧ d(`∗(v,λ)p)

= d(λx) ∧ d(λ−1p− v)

= (λdx) ∧ (λ−1dp)

= dx ∧ dp.

Now we will find the commutation relations of the the Lie algebra L(RoϕR+) using

the equation (4.3.6). Therefore we need to compute explicitly ϕ : L(R+) −→ End(R)

first. Notice that

φ(exp(r))(s) = exp(−r)(s) = exp(ϕ(r))(s) ⇒ ϕ(r)(t) = −rt ∀t ∈ R,

so the commutation relation is given by

[(b1, r1), (b2, r2)] = (ϕ(r1)(b2)− ϕ(r2)(b1), [r1, r2]) = (b1r2 − b2r1, 0), (4.4.5)

where we have used the fact that L(R+) = R is an Abelian Lie algebra.

Now we need to consider the one-parameter subgroups of R oϕ R+.

t 7→ exp(tb, 0) = (tb, 1) t 7→ exp(0, tr) = (0, exp(tr)) (4.4.6)

47


to compute the vector field associated with (b, 0) ∈ L(R oϕ R+).

γ(b,0)(x, p) =d

dt

∣∣∣∣t=0

`(tb,1)(x, p)

=d

dt

∣∣∣∣t=0

(x, p− tb)

= b∂

∂p,

and with (0, r) ∈ L(R oϕ R+).

γ(0,r)(x, p) =d

dt

∣∣∣∣t=0

`(0,exp(tr))(x, p)

=d

dt

∣∣∣∣t=0

(exp(tr)x, exp(−tr)p)

= −rx ∂∂x

+ rp∂

∂p.

Hence,

γ(b,r) = −rx ∂∂x

+ (b+ rp)∂

∂p. (4.4.7)

We need to check that this vector field is globally Hamiltonian, however since

H1dR(T ∗R+) = 0 this is always the case. Moreover, if we define f : T ∗R+ −→ R

by f(s) = bx(s) + rx(s)p(s) and compute

iγ(b,r)(dx ∧ dp) = (dx ∧ dp)(−rx ∂

∂x+ (b+ rp)

∂

∂p

)= −rxdp− (b+ rp)dx,

= −df

we ensure the desired condition. As we saw in our first example, it is natural to

define

P (b,r)(s) = bx(s) + rx(s)p(s). (4.4.8)

To compute the cohomology class defined by the action we compute the Poisson

bracket first

{P (r1,b1), P (r2,b2)} = {b1x+ r1xp, b2x+ r2xp}

= (dx ∧ dp)(r1x

∂

∂x− (b1 + r1p)

∂

∂p, r2x

∂

∂x− (b2 + r2p)

∂

∂p

)= −r1x(b2 + r2p) + r2x(b1 + r1p)

= (b1r2 − b2r1)x.

48

4.4 Q = R+

And on the other hand

P [(b1,r1),(b2,r2)] = P (b1r2−b2r1,0)

= (b1r2 − b2r1)x,

therefore, for this action, the cohomology class vanishes.

Now we should focus on the representations of R oϕ R+). If we consider a weakly

continuous, irreducible and unitary representation (v, λ) 7→ U(v, λ) it is convenient to

define operators

U(λ) = U(0, λ) V (b) = U(exp(b), 1) (4.4.9)

Proposition 4.4.1. The operators defined above satisfy the following relations

1. U(λ1)U(λ2) = U(λ1λ2).

2. V (b1)V (b2) = V (b1 + b2).

3. U(λ)V (b) = V (λ−1b)U(λ).

Proof. The first two relations are easy to see. We will only discuss the last relation.One hand we have

U(λ)V (b) = U(0, λ)U(exp(b), 1)

= U((0, λ)(exp(b), 1))

= U(λ−1 exp(b), λ)

= U(λ−1b, λ),

and on the other

V (λ−1b)U(λ) = U(exp(λ−1b), 1)U(0, λ)

= U(exp((λ−1b), 1)(0, λ))

= U(exp(λ−1b), λ)

= U(λ−1b, λ).

Now we will work out the commutation relations of the associated self-adjoint op-

erators. Define x and π by

U(exp r) = exp(−irπ) V (exp b) = exp(−ibx) (4.4.10)

49


Proposition 4.4.2. The commutation relation of the self-adjoint operators defined byequation (4.4.10) is

[x, π] = ix. (4.4.11)

Proof. We proceed as we did before

AdU(λ)(−ix) =d

dt

∣∣∣∣t=0

U(λ)V (t)U(λ)−1

=d

dt

∣∣∣∣t=0

V (λ−1t)

= −iλ−1x.

Hence,

[−iπ,−ix] =d

dt

∣∣∣∣t=0

AdU(exp t)(−ix)

=d

dt

∣∣∣∣t=0

(−i exp(−t)x)

= ix−

Again, this commutation relations is compatible with the quantizing map

P (b,r) = bx+ rxp ˆ7−→1i(bx+ rπ). (4.4.12)

There is a very rich theory on representations of semidirect products developed by

Mackey in [Mac68] and in [Mac78], that we will try to explore later. For now, as Isham

suggests, we will guess a representation in terms of wave functions defined on the con-

figuration space.

We might try a similar approach inspired on equation(4.2.14) defining for x ∈ R+

(U(λ)ψ)(x) = ψ(λ−1x). (4.4.13)

However, this operator in not unitary since the Lebesgue measure is not scale invariant,

but on L2(R+, dx/x) this operator is unitary. For the other operator, we should take

(V (b)ψ)(x) = exp(−ibx)ψ(x). (4.4.14)

50

4.4 Q = R+

These operators define a unitary, weakly continuous irreducible representations of Rnϕ

R+. For instance,

(U(λ)V (b)ψ)(x) = U(λ) exp(−ibx)ψ(x)

= exp(−ibλ−1b)ψ(λ−1x)

= V (λ−1b)ψ(λ−1x)

= (V (λ−1b)U(λ)ψ)(x).

Now we will derive the action of the self-adjoint operators

(xψ)(x) =d

dt

∣∣∣∣t=0

iV (exp t)ψ(x)

=d

dt

∣∣∣∣t=0

i exp(−itx)ψ(x)

= xψ(x)

and

(πψ)(x) =d

dt

∣∣∣∣t=0

iU(exp t)ψ(x)

=d

dt

∣∣∣∣t=0

iψ(exp(−t)x)

= −ixdψdx

(x).

Remark 4.4.1. When Mackey’s theory is applied to R oϕ R+ it shows that thereare three irreducible representations that are described as follows: Notice that thecommutation relation of Proposition 4.4.2 is preserved by the dilation x 7−→ tx fort ∈ R. So we can define a one-parameter of representations of R nϕ R+ by

(U (t)(λ)ψ)(x) = ψ(λ−1x) (V (t)(b)ψ)(x) = exp(itb)ψ(x) (4.4.15)

However, for t > 0 al these representations are unitarily equivalent since

U(t)−1V (b)U(t) = V (tb).

By the same argument, for t < 0 all these representations are unitarily equivalent, buta type t > 0 can not be equivalent to a type t < 0 since the spectrum of x differs ineach case, for t > 0 the spectrum is R− whereas for t < 0 is R+.The third case is the trivial case where t = 0, and since R+ is an Abelian group weknow that all its irreducible represenatations are one-dimensional, son we obtain

(U (0)(λ)ψ)(x) = λψ(x) (V (0)(b)ψ)(x) = ψ(x) (4.4.16)

51


Finally we are going to discuss how does Planck’s constant makes it appearance.

The main idea is that locally, we should recover the Q = R theory. We can consider

the canonical group R × R acting on a neighborhood of the point (x0, 0) ∈ T ∗R+ and

therefore, we could find some preferred observables that will be common to both ac-

tions. For these observables we require that the same operator is assigned for both

cases. This consistency requirement will serve to fix dimensions in the global theory.

For a point (x0, 0) ∈ T ∗R+ consider the exponential map and its inverse i : R 7−→ R+

and j : R+ 7−→ R defined by

i(q) = x0 exp(q/q0) j(x) = q0 log(x/x0) (4.4.17)

These maps induce diffeomorphisms i∗ : T ∗R+ 7−→ T ∗R and j∗ : T ∗R 7−→ T ∗R+. To

find explicit expressions for these maps we proceed as follows: We can write

i∗

(∂

∂q

)= f

∂

∂x,

from which we can determine f by

f = i∗

(∂

∂q

)x =

∂

∂qx0 exp(q/q0) =

x0

q0exp(q/q0),

and on the other hand

j∗

(∂

∂x

)= h

∂

∂q,

from which we can determine h by

h = j∗

(∂

∂x

)q =

∂

∂xq0 log(x/x0) = q0

1x.

So we can compute

i∗(pdp)(∂

∂q

)= pdp

(i∗

(∂

∂q

))= pdp

(i∗

(x

q0j∗

(∂

∂x

)))=xp

q0

52

4.4 Q = R+

and

j∗(pdp)(∂

∂x

)= pdp

(j∗

(∂

∂x

))= pdp

(i∗

(q0x0

exp(−q/q0)j∗

(∂

∂q

)))=q0x0

exp(−q/q0)p,

which in coordinates can be written as

i∗(x, p) = (q0 log(x/x0), xp/q0) j∗(q, p) = (x0 exp(q/q0), q0 exp(−q/q0)p/x0)

Since i∗ and j∗ are symplectic diffeomorphisms, we can define an action ˜ on T ∗R+

induced by the group R× R by

˜(u,v)(x, p) = (j∗ ◦ `(u,v) ◦ i∗)(x, p)

= (j∗ ◦ `(u,v))(q0 log(x/x0), xp/q0)

= j∗(q0 log(x/x0) + u, xp/q0 − v)

= (x exp(u/q0), exp(−u/q0)(p− q0v/x0)).

The associated vector field of (a, b) ∈ L(R× R) is given by

γ(a,b)(x, p) = − d

dt

∣∣∣∣t=0

˜(ta,tb)(x, p)

= − d

dt

∣∣∣∣t=0

(x exp(ta/q0), exp(−ta/q0)(p− q0tb/x0))

= − aq0x∂

∂x+(a

q0p+

q0b

x

)∂

∂p.

We already know that this vector field is globally Hamiltonian since H1dR(T ∗R+) is

trivial, but moreover, if we define f : T ∗R+ −→ R by

f(s) =a

q0x(s)p(s) + q0b log

(x(s)x0

),

we have explicitly that iγ(a,b)(dx ∧ dp) = −df .

Now we need to take care about the obstruction cocycle:

Define a map P : L(R× R) −→ C∞(T ∗R)+,R by

P (a,b)(s) =a


(x(s)x0

). (4.4.18)

53


Now we compute

{P (a1,b1), P (a2,b2)} =(a1

q0p+

q0b1x

)(a2

q0x

)−(a2

q0p+

q0b2x

)(a1

q0x

)= −(b2a1 − b1a2)

and

P [(a1,b1),(a2,b2)] = P (b2a1−b1a2,0) = (b2a1 − b1a2)xp

q0.

So in this case, the obstruction cocycle does not vanish and wee need to do a central

extension with moment map

P (a,b,t)(s) =a


(x(s)x0

)+ t. (4.4.19)

Notice that the operator xp appears in the set of preferred observables of both

actions. By comparing with the momentum map for Q = R given by equation (4.2.5)

we can read that

q = q0 log(x

x0

)p =

1q0π (4.4.20)

If we impose that [q, p] = i~ and realizing that for n > 0 we have [qn, p] = ni~qn−1

we conclude that

[x, π] = x0q0[exp(q/q0), p]

= i~x.

4.5 Q = S1

This is our first topologically non-trivial configuration space, for instance its fundamen-

tal group is π1(S1) = Z and its first de Rham cohomology group is HdR(S1,R) = R.

Nevertheless, the cotangent bundle is still a product T ∗S1 = S1 × R. On a local coor-

dinate system we can write the symplectic form as ω = dφ ∧ dl, where 0 < φ < 2π is

the usual angle in S1 and l is the projection on the second component of S1 × R.

A natural choice for the canonical group would be R× SO(2) acting by

`(n,ϕ)(φ, l) = ((φ+ ϕ)mod2π, l − n),

which would generate vector fields

γ(n,0) = n∂

∂lγ(0,t) = −t ∂

∂φ.

54

4.5 Q = S1

However, γ(n,0) is not globally Hamiltonian: even since iγ(n,0)ω = −d(nφ), we need to

keep in mind that φ is not a continuous function from T ∗S1 to R.

The problem arises mainly because S1 needs at least two charts to be covered, hence

we could try to consider a group with three parameters. Our candidate will be the Eu-

clidean group E2 = R2oSO(2), which is a semidirect product of a vector space and a Lie

group. As we discussed before, the group product is given by (m2, n2, ϕ2)(m1, n1, ϕ1) =

(m′, n′, (ϕ2 + ϕ1)mod2π), where(m′

n′

)=(m2

n2

)+(

cosϕ2 − sinϕ2

sinϕ2 cosϕ2

)(m1

n1

)(4.5.1)

We are going to study now its Lie algebra. We can identify L(R2 o SO(2)) with R3,

and the commutator induced by the semidirect product can be found using equation

(4.3.6) and using the fact that

so(2) ={(

0 −ττ 0

)∈M2(R)

∣∣∣∣ τ ∈ R},

one can verify that

[(b1, d1, τ1), (b2, d2, τ2)] =((

0 −τ1τ1 0

)(b2d2

)−(

0 −τ2τ2 0

)(b1d1

), 0)

= (τ2d1 − τ1d2, τ1b2 − τ2b1, 0).

Now we will define the action of E2 on T ∗S1 by

`(m,n,ϕ)(φ, l) = ((φ+ ϕ)mod2π, l +m sin(φ+ ϕ)− n cos(φ+ ϕ)). (4.5.2)

An easy computation shows that this definition defines an action which is effective

and transitive, however is instructive to show that the action is symplectic:

`∗(m,n,ϕ)(dφ ∧ dl) = d(`∗(m,n,ϕ)φ) ∧ d(`∗(m,n,ϕ)l)

= d(φ ◦ `(m,n,ϕ)) ∧ d(l ◦ `(m,n,ϕ))

= d((φ+ ϕ)mod2π) ∧ d(l +m sin(φ+ ϕ)− n cos(φ+ ϕ))

= dφ ∧ (dl +m cos(φ+ ϕ)dφ+ n sin(φ+ ϕ)dφ)

= dφ ∧ dl.

55


To find the infinitesimal generator associated with (b, d, τ) ∈ L(E2) we compute, as

usual,

γ(b,d,τ)(φ, l) = − d

dt

∣∣∣∣t=0

`(tb,td,tτmod2π)(φ, l)

= − d

dt

∣∣∣∣t=0

((φ+ tτ)mod2π, l + tb sin(φ+ tτ)− td cos(φ+ tτ))

= −τ ∂∂φ− (b sinφ− d cosφ)

∂

∂l.

Now we will verify that this vector field is indeed Hamiltonian. Define f ∈ C∞(T ∗S1,R)

by

f(s) = τ l(s) + b cosφ(s) + d sinφ(s),

then

iγ(b,d,τ)(dφ ∧ dl) = −τdl + (b sinφ− d cosφ)dφ = −df.

So our natural candidate to be the moment map P : L(E2) −→ C∞(T ∗S1,R) is defined

by

P (b,d,τ)(s) = τ l(s) + b cosφ(s) + d sinφ(s). (4.5.3)

The next step is to compute the obstruction cocycle. The Poisson bracket is given by

{P (b1,d1,τ1), P (b2,d2,τ2)}

= (dφ ∧ dl)(τ1∂

∂φ+ (b1 sinφ− d1 cosφ)

∂

∂l, τ2

∂

∂φ+ (b2 sinφ− d2 cosφ)

∂

∂l

)= τ1(b2 sinφ− d2 cosφ)− (b1 sinφ− d1 cosφ)τ2

= (τ2d1 − τ1d2) cosφ− (τ2b1 − τ1b2) sinφ.

And on the other hand

P [(b1,d1,τ1),(b2,d2,τ2)](φ, l) = P (τ2d1−τ1d2,τ1b2−τ2b1,0)(φ, l)

= (τ2d1 − τ1d2) cosφ+ (τ1b2 − τ2b1) sinφ,

so there in no obstruction cocycle.

Now we are going to work out the representations. Denote by (m,n, ϕ) 7−→U ′(m,n, ϕ) a unitary, weakly continuous, irreducible representation of E2 and consider

in particular the operators

V (b, d) = U ′(exp b, exp d, 0) U(φ) = U ′(0, 0, φ) (4.5.4)

56

4.5 Q = S1

Proposition 4.5.1. The operators defined by equation (4.5.4) satisfy the followingrelations

1. U(φ1)U(φ2) = U((φ1 + φ2)mod2π).

2. V (b1, d1)V (b2, d2) = V (b1 + b2, d1 + d2).

3. U(φ)V (b, d) = V (b′, d′)U(φ).

where (b′

d

)=

(cosφ − sinφsinφ cosφ

)(b

d

).

Proof. Au usual, the two first two relations are easy to prove, the last relation is followsimmediately from the multiplication law of E2.

Now we are going to work out the commutation relations of the self-adjoint operators

defined by

V (b, d) = exp(−i(bc+ ds)) U(τmod2π) = exp(−iτ J) (4.5.5)

Proposition 4.5.2. Tho commutator relations of the self-adjoint operators defined byequation (4.5.5) are given by

1. [c, s] = 0.

2. [s, J ] = ic.

3. [c, J ] = −is.

Proof. For the first relation we compute, as usual,

AdV(b,0)(−is) =

d

dt

∣∣∣∣t=0

V (b, 0)V (0, t)V (b, 0)−1

=d

dt

∣∣∣∣t=0

V (0, t)

= −is,

which implies that [c, s] = 0 since AdV(t,0)(−is) does not depend on t.

For the second relation we compute

AdU(φ)(−is) =d

dt

∣∣∣∣t=0

U(φ)V (0, t)U(φ)−1

=d

dt

∣∣∣∣t=0

V (−t sinφ, t cosφ)

= i sinφc− i cosφs,

57


therefore

[−iJ ,−is] =d

dt

∣∣∣∣t=0

AdU(t)(−is)

=d

dt

∣∣∣∣t=0

i sinφt− i cosφt

= ic.

The third case follows from an analogue calculation.

Remark 4.5.1. In this example the quantizing map is given by

P (b,d,τ) = τ l + b cosφ+ d sinφ ˆ7−→1i(τ J + bc+ ds).

Remark 4.5.2 (Helicity). The commutation relations obtained in Proposition 4.5.2are the same ones obtained in the context of the classification of massless particlesmade by Wigner, studying the irreducible representation of the Poincare group:Particles are classified by eigenvalues of mass and spin, i.e. by the eigenvalues ofe2 = eie

i (see Example 4.3.2) and of W (spin vector of Pauli and Lubanski), thespin taking only integer or half-integer values. The dynamical states of free particlescan be characterized by the eigenvalues of the four operators P of the energy and themomentum, and by one component of spin. (see [Sch07]).

Since E2 is a semidirect product, it representations can be studied using Mackey’s

theory. In this case we will also try to guess them. Our candidate for the representation

space will be L2(S1, dφ) and the action of E2 on functions will be given by

(U(ϕ)ψ)(φ) = ψ((φ− ϕ)mod2π) (4.5.6)

(V (b, d)ψ)(φ) = exp(−i(b cosφ+ d sinφ))ψ(ψ) (4.5.7)

which induce the following action at the level of the self-adjoint operators

(Jψ)(φ) =d

dt

∣∣∣∣t=0

i(U(t)ψ)(φ)

=d

dt

∣∣∣∣t=0

iψ((φ− t)mod2π)

= −idψdφ

(φ)

and

(cψ)(φ) = cosφψ(φ) (sψ)(φ) = sinφψ(φ)

58

4.5 Q = S1

Remark 4.5.3. The commutation relations of Proposition 4.5.2 are unchanged byc, s 7−→ λc, λs, so actually we have a one-parameter family of representations

(U (λ)(ϕ)ψ)(φ) = ψ((φ− ϕ)mod2π) (4.5.8)

(V (λ)(b, d)ψ)(φ) = exp(−iλ(b cosφ+ d sinφ))ψ(ψ) (4.5.9)

However notice that U (λ)(π)V (λ)(b, d)U (λ)(π)−1 = V (λ)(−b,−d) = V (−λ)(b, d), there-fore the λ and (−λ) representations are unitarily equivalent. For the case were λ, λ′ ≥ 0and λ 6= λ′ the associated representations are inequivalent.In the next chapter, we will discuss in a little detail the origin of these expressions formMackey’s theory.

Remark 4.5.4. The way in which ~ will come out in this case will be by ensuring thatlocally, we recover the commutation relations of the case Q = R, or as we will see later,when the radius of the circle is made infinitely large we should recover this case too.Notice that c2+ s2 = λ2, therefore for consistency, if we fix λ2 = µ = ~, we can convinceourselves that the physical operators should be

cphys = c(c2 + s2)−1 sphys = s(c2 + s2)−1 (4.5.10)

Remark 4.5.5 (Contraction of a Lie Algebra). For each ε > 0 define a Lie bracket onR3 by

[(b1, d1, τ1), (b2, d2, τ2)]ε =(τ2d1 − τ1d2,

τ1b2 − τ2b1ε

, 0)

(4.5.11)

(ε will represent the radius of the circle), and a map Aε : L(E2) −→ L(E2) by

Aε(b, dτ) = (b, εd, τ/ε). (4.5.12)

A straight computation shows that [X,Y ]ε = A−1ε ([Aε(X), Aε(Y )]) for X,Y ∈ L(E2).

Now we can take the limit ε→∞ and obtain the contraction of the Lie algebra L(E2)whose bracket is given by

[(b1, d1, τ1), (b2, d2, τ2)]∞ = (τ2d1 − τ1d2, 0, 0), (4.5.13)

which should be familiar to us, since it is the Heisenberg Lie algebra.We can also contract the self-adjoint operators if we define qε = εs and pε = J/ε, whichon the limit ε→∞ satisfy

[c, q∞] = 0 [q∞, p∞] = ic [c, p∞] = 0 (4.5.14)

At this point we should give a more clear interpretation to what we have done. Define foreach ε > 0, the a Wε : L2(S1, dφ/2π) −→ L2([0, 2πε], dx/2πε) by Wε(ψ)(x) = ψ(x/ε).

59


We will explore concrete the representations of the contracted operators trough nextproposition.

Proposition 4.5.3. From the representations discussed in Remark 4.5.3 we have thefollowing relations

1. (WεqεW−1ε ψ)(x) = λε sin(x/ε)ψ(x).

2. (WεcW−1ε ψ)(x) = λ cos(x/ε)ψ(x).

3. (WεJW−1ε ψ)(x) = −idψ

dx(x).

Proof. Just compute

1. (WεsεW−1ε ψ)(x) = Wεsεψ(εx) = Wελε sin(x)ψ(εx) = λε sin(x/ε)ψ(x).

2. (WεcεW−1ε ψ)(x) = Wεcεψ(εx) = Wελ cos(x)ψ(εx) = λ cos(x/ε)ψ(x).

3. (WεJεW−1ε ψ)(x) = WεJεψ(εx) = Wε(−i)

dψ

d(εx)(εx) = −idψ

dx(x).

So here we see explicitly that, as ε→∞, c tends to a multiple of the identity map.

Moreover, we can define the dimensioned operators by

q = q0q∞ p = p0p∞ (4.5.15)

and conclude that [q, p] = iλ2 = i~. Notice also that, for consistency, the operators p

and pε should agree on the limit ε→∞, this forces that Jphys = ~J .

4.6 Uncertainty Relations

In order to study experimental evidence of the commutation relations it is useful to

work out the uncertainty relations. The main purpose of this section is to find a general

formula for these expressions. Our main reference for this section is [Sha94].

We are going to start with two Hermitian operators A and B and a general form of

a commutation relation [A,B] = iC to derive the associated uncertainty relation.

Remark 4.6.1. Notice that C is Hermitian too since

−iC† = [A,B]† = −[A,B] = −iC.

60

4.6 Uncertainty Relations

Recall that we define the expectation value by

〈A〉 = 〈ψ|A|ψ〉

and the variance

(∆A)2 = 〈A2〉 − 〈A〉2.

We want to compute

(∆A)2(∆B)2 = 〈ψ|A2 − 〈A〉2|ψ〉〈ψ|B2 − 〈B〉2|ψ〉

= 〈ψ|A2|ψ〉〈ψ|B2|ψ〉

= 〈Aψ|Aψ〉〈Bψ|Bψ〉,

where we have defined A = A2 − 〈A〉2 (and clearly [A, B] = iC). Now we can apply

Schwartz inequality to obtain

(∆A)2(∆B)2 ≥ |〈Aψ|Bψ〉|2

≥ |〈ψ|AB|ψ〉|2

≥ |〈ψ|12

[A, B]+ +12

[A, B]|ψ〉|2,

where [A, B]+ = AB + B, A is the anti-commutator, which is clearly Hermitian.

Remark 4.6.2. Since C is Hermitian then 〈[A, B]〉 is purely imaginary, therefore

|〈ψ|[A, B]+|ψ〉+ i〈ψ|C|ψ〉|2 = 〈ψ|[A, B]+|ψ〉2 + 〈ψ|C|ψ〉2.

So finally we obtain

(∆A)2(∆B)2 ≥ 14〈ψ|[A, B]+|ψ〉2 +

14〈ψ|C|ψ〉2. (4.6.1)

Example 4.6.1. In the case Q = R we have that [q, p] = i~, so in this case equation(4.6.1) becomes

(∆q)2(∆p)2 ≥ 14〈ψ|[q, p]+|ψ〉2 +

~2

4.

However, since the first term is positive definite, we conclude that

∆q∆p ≥ ~2. (4.6.2)

which is Heisenberg uncertainty principle.

61


Example 4.6.2. Now we are going to explore the uncertainty relations for Q = S1.Recall that [s, J ] = ic and [c, J ] = −is, therefore the associated uncertainty relationsare

(∆c)(∆J) ≥ 14〈s〉2 (∆s)(∆J) ≥ 1

4〈c〉2 (4.6.3)

We want to finish this chapter with a strong recommendation: to refer to the work

done by J. Rehacek, Z. Bouchal, R. Celechovsky Z. Hradil, and L. L. Sanchez-Soto

[RBC+08] where they find experimental evidence supporting equation (4.6.3).

62

5

The General Case: Through

Examples

The goal of this section is to study the quantization method of the general case. How-

ever, since we have worked some explicit examples, the arguments and constructions

given in this chapter should not be “strange”. In particular we are going to explore the

construction of the representations of the canonical group. At this point it should be

mentioned that we only are looking for the motivation and spirit of the method, since

the mathematical and technical details (that we do not need to take for granted) are

beyond the scope of this work.

5.1 Q = T 2

In this first section we are going to give a brief description of a work done by Z.

Hishamuddin [His89] where he works out the canonical group quantization method for

the 2-torus T 2 = S1 × S1. As we shall see, this case is simply an extension of the case

Q = S1, that is why, in addition, Hishamuddin works the case where there is a presence

of a magnetic field.

The cotangent space of the configuration space Q = T 2 is trivial, T ∗T 2 = T 2 ×R2.

The symplectic form is given by ω = dϕ1∧dl1 +dϕ2∧dl2, where (φ1, φ2) are the angles

on T 2 and (l1, l2) are the coordinates in R2. Form our discussion in the case Q = S1, a

natural choice for the canonical group is E2×E2 which acts on T ∗T 2 in a natural way

63

5. THE GENERAL CASE: THROUGH EXAMPLES

(like equation (4.5.2) in each component). This action is clearly symplectic, transitive

and effective. Moreover, since we can reproduce what we did for Q = S1 on each

component it is not surprising to find commutation relations of the form

[ca, sb] = 0 = [Ja, Jb] [sa, Jb] = icaδab [ca, Jb] = −isaδab (5.1.1)

To make this problem more interesting, what Hishamuddin proposed was to consider

the same configuration space with a constant magnetic field. To model this situation

he modify the symplectic form adding a new term

ωF = ω + F, (5.1.2)

where F is the field strength 2-form

F12dφ1 ∧ dφ2. (5.1.3)

If we denote an element (m1, n1, ϕ1,m2, n2, ϕ2) ∈ E2 ×E2 by (~m,~n, ~ϕ) we see that

the action is still symplectic:

`∗(~m,~n,~ϕ)F12dφ1 ∧ dφ2 =12F12d((φ1 + ϕ1)mod2π) ∧ d((φ2 + ϕ2)mod2π) = ωF .

The infinitesimal generator with respect to (b, d, t, β, δ, τ) ∈ L(E2 × E2) is

γ(b,d,t,β,δ,τ) = −t ∂

∂φ1− (b sinφ1 − d cosφ1)

∂

∂l1− τ ∂

∂φ2− (β sinφ2 − δ cosφ2)

∂

∂l2.

We need to check that these vector fields are Hamiltonian. We compute

iγ(b,d,t,β,δ,τ)(dϕ1 ∧ dl1 + dϕ2 ∧ dl2) =− tdl1 + (b sinφ1 − d cosφ1)dφ1

− τdl2 + (β sinφ2 − δ cosφ2)dφ2

and

iγ(b,d,t,β,δ,τ)(F12dφ1 ∧ dφ2) = F12(−tdφ2 + τdφ1),

so the natural candidate to be the moment map would be

P (b,d,t,β,δ,τ)(s) =tl1(s) + b cosφ1(s) + d sinφ1(s)

+ τ l2(s) + β cosφ2(s) + δ sinφ2(s) + F12(tφ2(s)− τφ1(s)).

64

5.1 Q = T 2

Be careful! Recall that φ1(s) and φ2(s) are not smooth functions on T 2, so what we

need to do is to modify the infinitesimal generator:

γ(b,d,t,β,δ,τ) =− t ∂

∂φ1− (b sinφ1 − d cosφ1 + F12τ)

∂

∂l1

− τ ∂

∂φ2− (β sinφ2 − δ cosφ2 − F12t)

∂

∂l2,

and leave the moment map as

P (b,d,t,β,δ,τ)(s) =tl1(s) + b cosφ1(s) + d sinφ1(s) + τ l2(s) + β cosφ2(s) + δ sinφ2(s).

To study the obstruction cocycle we only need to compute the term

(F12dφ1 ∧ dφ2)(γ(b1,d1,t1,β1,δ1,τ1), γ(b2,d2,t2,β2,δ2,τ2)) = F12(t1τ2 − t2τ1),

so we see that the obstruction cocycle does not vanish, therefore we need to make a

central extension of the Lie algebra and define a new bracket by

[(b1, d1, t1, β1,δ1, τ1, r1), (b2, d2, t2, β2, δ2, τ2, r2)]

= (t2d1 − t1d2, t1b2 − t2b1, 0, τ2δ1 − τ1δ2, τ1β2 − τ2β1, 0, F12(t1τ2 − t2τ1)).

We are not going to go further in this example, however we will point out two

important remarks.

Remark 5.1.1. In this article they give arguments to show that the appropriate newcanonical group will be E2 o (E2 × U(1)), where E2 is the universal cover R2 o R.

Remark 5.1.2. We can “read” from the bracket of the central extension that theonly commutator that is going to change from equation (5.1.1) will be the commutatorbetween J operators. This can be seen since the new term only contains ti and τi,hence is not surprising to find out that [J1, J2] = −iF12

It should be mentioned that this article works out the representations, so we en-

courage to refer to it.

We will stop here working out explicit examples to start describing the general case,

however we will develop a final example (the sphere S2) to expose the techniques that

we are going to study in the following sections.

65


5.2 Brief Discussion: The General Case

Since we have explored various examples, the spirit of this quantization method should

be clear by now, even if we have not been totaly explicit in some aspects (for instance

the study of the representations of the canonical group). Having these examples in

mind, we will discuss how to proceed in the general case.

Let φ ∈ Diff(Q), then we can define an action of Diff(Q) on T ∗Q by `φα = φ∗α, for

each α ∈ T ∗Q. This action is not transitive since each zero element in each fibre remains

unaffected. To obtain a transitive action we are going to define diffeomorphisms up to

fibres as follows: Let ϑ ∈ Γ(T ∗Q) and define for α ∈ T ∗Q

ψ(α) = α− ϑπ(α),

where π : T ∗Q −→ Q is the natural projection. Notice that ψ is a well defined

diffeomorphism on each fibre, that is π ◦ ψ = π, so we would like to find conditions on

this action to be symplectic. For v ∈ Tα(T ∗Q) we compute ψ∗θ, where θ is the Liouville

1-form,

〈(ψ∗θ)α, v〉α = 〈θψ(α), ψ∗v〉ψ(α) = 〈ψ(α), π∗ψ∗v〉π(ψ(α)) = 〈(α− ϑπ(α)), π∗v〉π(α)

= 〈α, π∗v〉π(α) − 〈ϑπ(α), π∗v〉π(α) = 〈θα, v〉α − 〈(π∗ϑ)α, v〉α.

Thus, we can conclude that ψ∗θ = θ − π∗ϑ, and if we apply the exterior derivative we

obtain ψ∗ω = ω + dϑ, where ω = −dθ is the natural symplectic form on T ∗Q. There-

fore, the action is symplectic if, and only if, ϑ is a closed form, in particular if ϑ = dh

for h ∈ C∞(Q,R).

We would like to define an action of C∞(Q,R)/R×Diff(Q) on T ∗Q by

`(h,φ)α = (φ−1)∗α− (dh)φ(q). (5.2.1)

Remark 5.2.1. To avoid a heavy notation it should be noticed that h ∈ C∞(Q,R)/Ractually denotes an equivalence class.

66

5.3 Q = G/H

The group operation on C∞(Q,R)/R×Diff(Q) should satisfy, for α ∈ T ∗qQ,

`(h2,α2)`(h1,φ1)α = `(h2,α2)((φ−11 )∗α− (dh1)φ1(q))

= (φ−12 )∗((φ−1

1 )∗α− (dh1)φ1(q))− (dh2)φ2(φ1(q))

= ((φ2 ◦ φ1)−1)∗α− d(h2 + h1 ◦ φ−12 )(φ2◦φ1)(q)

= `(h2 + h1 ◦ φ−12 , φ2 ◦ φ1)α

to define an action. Therefore, we see that we need to work with the semidirect product

C∞(Q,R)/RoDiff(Q), where the product rule, which should be familiar for us, is given

by

(h2, φ2)(h1, φ1) = (h2 + h1 ◦ φ−12 , φ2 ◦ φ1). (5.2.2)

Remark 5.2.2. We already know that this action is symplectic, transitive and effec-tive, and it can be shown that the action is Hamiltonian. Moreover, the obstructioncocycle vanishes if we restrict ourselves to Diff(M), but on the C∞(Q,R)/R this it notguaranteed in general. For the discussion on these aspects please refer to [Ish84].

Remark 5.2.3. We need to point out that both, C∞(Q,R)/R and Diff(Q) are infinite-dimensional Lie groups. The study of these objects is beyond the scope of this work.However, for a wide family of examples, it is enough to consider a finite-dimensionalsubgroup of C∞(Q,R)/R o Diff(Q) of the form V nG, where V is a finite-dimensionalvector space and G is a Lie group acting on Q.

5.3 Q = G/H

In this section we are going to discuss the case where the configuration space is a ho-

mogeneous space. However, as you will see, we will describe the procedure “backwards”.

To start suppose that we are given a finite dimensional vector space V and a finite

dimensional Lie group G endowed with a representation R : G −→ Aut(V ). Recall

that the dual representation R∗ : G −→ Aut(V ∗) is defined by 〈R∗(g)ξ, v〉 = 〈ξ,R(g)v〉for v ∈ V and ξ ∈ V ∗. Since there is a natural identification between T ∗V and V ×V ∗,we can define a left action of the semidirect product V ∗ oG on T ∗V by

`(ξ,g)(w, η) = (R(g)w,R∗(g−1)η − ξ), (5.3.1)

where the group product on V ∗oG is given by (ξ2, g2)(ξ1, g1) = (ξ2 +R∗(g−12 )ξ1, g2g1).

67


Remark 5.3.1. Notice that 〈R∗(g2)R∗(g1)ξ, v〉 = 〈R∗(g1)ξ,R(g2)v〉 = 〈ξ,R(g1g2)v〉,therefore R∗(g2)R∗(g1) = R∗(g1g2), which is why we have used R∗(g−1) instead ofR∗(g) in the definition of the action.

If A ∈ g, then R(A) ∈ End(V ) will be defined by R(exp(A)) = exp(R(A)) (No-

tice that R∗(A) = −R(A)). We can also define a moment map P : L(V ∗ o G) −→C∞(T ∗V,R) by

P (ξ,A)(p, q) = p(R(A)q) + p(q). (5.3.2)

Remark 5.3.2. From our study in semidirect products we have the following Liebracket on L(V ∗ oG)

[(ξ1, A1), (ξ2, A2)] = (R∗(A1)ξ2 −R∗(A2)ξ1, [A1, A2]). (5.3.3)

It can be shown that this action is symplectic and that the cocycle obstruction

vanishes for the map defined above. Instead of proving this for the general case, we

are going to work out a concrete example. Consider V = Rn and G = SO(n) and let

R : SO(n) −→ Aut(Rn) be the identity map. We will treat vectors in Rn as column

vectors and covectors in (Rn)∗ treated as row vectors, so that the pairing between Rn

and (Rn)∗ is just matrix multiplication. In this context, the dual representation is given

by R∗(g)η = ηR(g) for η ∈ (Rn)∗, therefore the action given by equation (5.3.1) is then

`(ξ,g)(w, η) = (R(g)w, ηR(g)T − ξ). (5.3.4)

If we give coordinates to V by {q1, · · · qn} and associate its dual basis {p1, · · · pn}then the symplectic form can be written as ω = dqα ∧ dpα, where, in order to avoid

heavy notation, we omit the sum over repeated indices. To show that the action is

symplectic we compute

`∗(ξ,g)(q, p)(dqα ∧ dpα) = d(qα ◦ `(ξ,g)) ∧ d(pα ◦ `(ξ,g))

= d(R(g)αβqβ) ∧ d(pσ(R(g)T )σα − ξα)

= (R(g)αβdqβ) ∧ (dpσ(R(g)T )σα)

= (R(g)T )σαR(g)αβdqβ ∧ dpσ

= (R(g)TRg)σβdqβ ∧ dpσ

= δσβdqβ ∧ dpσ

= dqβ ∧ dpβ.

68

5.3 Q = G/H

Now we will compute the obstruction cocycle. If we compute the Poisson bracket

{P (ξ1,A1), P (ξ2,A2)}(p, q)

=∂P (ξ1,A1)

∂qj∂P (ξ2,A2)

∂pj− ∂P (ξ1,A1)

∂pj− ∂P (ξ2,A2)

∂qj

=(pαR(A1)αj + (ξ1)j)(R(A2)jβqβ)− (R(A1)jβq

β)(pαR(A2)αj + (ξ2)j)

=pα(R(A1)R(A2))αβqβ + (ξ1)jR(A2)jβq

β − pα(R(A2)R(A1))αβqβ − (ξ2)jR(A1)jβq

β

=p(R([A1, A2])q) + (ξ1(R(A2))− ξ2(R(A1)))q

=p(R([A1, A2])q) + (R∗(A1)ξ2 −R∗(A2)ξ1)q

=P [(ξ1,A1),(ξ2,A2)](p, q).

Hence, the obstruction cocycle vanishes.

Remark 5.3.3. For ξ ∈ V ∗ we can associate a function f (ξ) ∈ C∞(V,R) by f (ξ)v =ξ(v). In this case the restriction of action defined by equation (5.3.1) to V ∗can bewritten as

`(ξ,e)(v, η) = (u, η − ξ) = (u, η)− (0, ξ) = (u, η)− (df (ξ))u (5.3.5)

and therefore we see that the action if of the form equation (5.2.1).

The action defined by (5.3.1) satisfies almost all the conditions that we want. How-

ever the action is not transitive. Recall that if we fix v ∈ V , then its orbit is de-

fined by Ov = {R(g)v | g ∈ G}, so clearly G acts transitively on Ov. Moreover,

there is a smooth group isomorphism Ov ∼= G/Hv given by R(g)v 7−→ [g]Hv , where

Hv = {g ∈ G |R(g)v = v} is the isotropy group or the little group of v. This means

that we can regard G/Hv as a submanifold of V with embedding

ιv : G/Hv// V

[g]Hv� // R(g)v

We would like to see if there exists a well-behaved action of V ∗oG on T ∗(G/Hv) keeping

in mind our previous construction. The natural action of G is given by g′[g] = [g′g].

On the other hand, motivated on the construction above we define, for each ξ ∈ V ∗, a

smooth function h(ξ) ∈ C∞(G/Hv,R) by

h(ξ)([g]) = f (ξ)(ιv([g])) = ξ(ιv([g])) = ξ(R(g)v),

69


which we will use to define the V ∗ oG-action on T ∗(G/Hv) by

`(ξ,g′)α = (g−1)∗α− (dh(ξ))[g′g] for α ∈ T ∗[g]G/Hv. (5.3.6)

What is important is that it can be shown than for this action the obstruction

cocycle vanishes too, and therefore it satisfies all the conditions that we require for the

quantization programme.

Remark 5.3.4. As Isham recalls, in practice we are given a configuration space of theform G/H, therefore the first step of this procedure is to find the smallest-dimensionalreal vector space V that carries a linear representation of G with the property that thereis a G-orbit in V that is diffeomorphic to G/H. This leads to the canonical group beingthe semidirect product V ∗ oG.

5.4 Induced Representations of Semidirect Products

We are going to discuss in a little detail the problem of the representations of the

canonical group. The first part is devoted to study representations induced by an action,

and in particular to describe the representation space as sections on some Hermitian

vector bundle.

To start suppose we are given a (connected) Lie group G acting on manifold M , we

would like to find unitary representations of G derived from the action. Our first candi-

date for being the representation space would be some Hilbert space H = L2(M,C, dµ)

where µ is a measure on X. The inner product would be

〈ψ, φ〉 =∫Mψ∗(x)ψ(x)dµ(x). (5.4.1)

A natural representation of G on H is (U(g)ψ)(x) = ψ(g−1x), in this case, since we

want unitary representations, we require that

〈U(g)ψ,U(g)φ〉 =∫Mψ∗(g−1x)ψ(g−1x)dµ(x) =

∫Mψ∗(x)ψ(x)dµg(x),

where roughly speaking, dµg(x) = dµ(g−1x). This actually means that we define

µg(B) = µ(g−1B), where B is a Borel subset on M (this measure is known as the

push-forward of µ by f). We will call a measure G-invariant if µg = µ for all g ∈ G.

Now we state the following important result:

70


Theorem 5.4.1. Suppose that the measure µ is G-quasi-invariant, that is, for all g ∈G, both µ and µg have the same sets of measure zero, then there exists a positive (µ-a.e)continuous function on M , denoted by dµg/dµ and known as the Radon-Nikodymderivative of µg with respect to µ, such that

µg(B) =∫B

(dµgdµ

(x))dµ(x)

for all Borel sets B.

With this result in mind, we can define a unitary representation of G on H by

(U(g)ψ)(x) =(dµgdµ

(x))1/2

ψ(g−1x). (5.4.2)

We can extend our framework of the representations space to consider, instead of

H = L2(M,C, dµ), the completion of the space of sections of some vector bundle E on

M endowed with a Hermitian metric 〈, 〉.

Remark 5.4.1. We will not discuss here the theory of fiber bundles, you can refer toAppendix (A) for the main definitions. However, we highly recommend to go to thereferences proposed, since this appendix is far away from a comprehensive treatmenton this theory, and instead its purpose is to establish the main definitions, conventionsand notation.

We can define for ψ1, ψ2 ∈ Γ(E) an inner product by

〈ψ1, ψ2〉 =∫M〈ψ1(x), ψ2(x)〉xdµ(x)

and complete this space to get the Hilbert space L2(Γ(E), dµ). A first approach to

define a representation would be as equation (5.4.2), however in this new context this

equation does not make sense, since each side of the equation lies in a different fibre.

This motivates or following definition.

Definition 5.4.1. Let ` : G ×M −→ M be a left right action on M , we say that anaction `↑ : G×E −→ E covers ` if for each g ∈ G we have the following commutativediagram:

E

=π

��

`↑g // E

π

��M

`g // M

(5.4.3)

71


Under this scenario, the representation should be given by

(U(g)ψ)(x) =(dµgdµ

(x))1/2

`↑gψ(g−1x). (5.4.4)

Notice that if we want unitary operators we require the following condition for lift

action: 〈`↑gu, `↑gv〉x = 〈u, v〉x for all u, v ∈ Ex, where Ex denotes the fibre over x ∈M .

Remark 5.4.2. There are important comments and remarks in our main reference,[Ish84], about the irreducibility of this representation and other technical aspects onthe existence and classification of lifts on a vector bundle. We will not go deeper intothose aspects, however it is important to comment that they exist and should not betreated lightly.

There is a common way of constructing Hermitian vector bundles: Suppose we are

given P a principal K-bundle and a unitary representation U : K −→ Aut(Cm), then

we can construct the associated bundle P ×U Cm, which is a vector bundle with fiber

Cm. On each fiber we can define an inner product by

〈[p, u], [pk, v]〉x = 〈u,U(k)v〉Cm for p ∈ π−1(x) (5.4.5)

Notice that 〈[pk,U(k−1)u], [(pk)(k−1k), v]〉x = 〈U(k−1)u,U(k−1k)v〉Cm = 〈u,U(k)v〉Cm ,

therefore the inner product is well defined.

There is an other advantage of working with the associated bundle: we can define

the inner product on equivariant functions. Recall that the space of sections can be

seen as (Appendix (A)),

Γ(P ×U Cm) ∼= {ψ ∈ C∞(P,Cm) | ψ(pk) = U(k−1)ψ(p) ∀k ∈ K}. (5.4.6)

Under this identification the inner product becomes∫M〈ψ1(px), ψ2(px)〉Cmdµ(x) for px ∈ π−1(x), x ∈M (5.4.7)

which is well defined since the representation U is unitary:∫M〈ψ1(pxk), ψ2(pxk)〉Cmdµ(x) =

∫M〈U(k−1)ψ1(px),U(k−1)ψ2(px)〉Cmdµ(x)

=∫M〈ψ1(px), ψ2(px)〉Cmdµ(x).

Now we are going to describe how to construct form a lift in the associated bundle

form a lift in the principal bundle.

72


Definition 5.4.2. A family of diffeomorphisms {tg | g ∈ G} of a principal K-bundle issaid to be a G-lift of the G-action if, for each g ∈ G, the following conditions hold:

1. The following diagram commutes

P

=π

��

tg // P

π

��M

`g // M

(5.4.8)

2. tg2 ◦ tg1 = tg2g1 for all g1, g2 ∈ G.

3. tg is a fibre preserving map such that tg(pk) = tg(p)k for all k ∈ K.

Proposition 5.4.1. Given such a G-lift, a lift `↑g in the associated bundle P ×U Cm

can be defined by `↑g([p, u]) = [tg(p), u]

Proof. It is easy to see that the lift is well defined and that π ◦ `↑g = `g ◦ π. We remainto check the G acts invariantly on the inner product, that is,

〈`↑g([p, u]), `↑g([pk, v])〉x = 〈[tg(p), u], [tg(pk), v]〉x= 〈[tg(p), u], [tg(p)k, v]〉x= 〈[p, u], [pk, v]〉x.

Consider a section Ψ ∈ Γ(G×U Cm) of the form Ψ(π(p)) = [p, ψ(p)], where ψ(pk) =

k−1ψ(p) for all k ∈ K, then if we compute

(U(g)Ψ)(π(p)) =(dµgdµ

(π(p)))1/2

`↑gΨ(g−1π(p))

=(dµgdµ

(π(p)))1/2

`↑gΨ(π(tg−1p))

=(dµgdµ

(π(p)))1/2

`↑g[tg−1p, ψ(tg−1p)]

=(dµgdµ

(π(p)))1/2

[p, ψ(tg−1p)]

73


we see that, in this context, the representation given by equation (5.4.4) can be written

in terms of equivariant functions ψ ∈ C∞(P,Cm) as

(U(g)ψ)(p) =(dµgdµ

(π(p)))1/2

ψ(tg−1g). (5.4.9)

Using what we have done in this section, we are now ready to describe (briefly)

Mackey’s theory on induced representations of semidirect products for the particular

case V oϕ G, which is the general form of the canonical groups that we have studied.

Definition 5.4.3. Let V be a finite dimensional vector space, we define the set ofcharacters of V to be the set of all continuous homomorphisms from W into thecomplex numbers. There is a one-to-one correspondence between Char(V ) and V ∗

which associates to each covector ξ ∈ V ∗ the character χξ(w) = exp(−i(ξ(w)).

The homomorphism ϕ : G −→ Aut(V ) induces a left action of G on the character

group given by (`gχ)(ξ) = χ(R∗(g−1)ξ). The main idea of this formalism is to associate,

to each orbit Og, a family of irreducible representations of V oϕ G: Choose χ0 ∈ Og,

denote its isotropy group by Hχ0 and let U be an irreducible unitary representation of

Hχ0 on Cm (for simplicity). We can consider the vector bundle π : G×UCm −→ G/Hχ0

and define a representation of G on L2(Γ(G ×U Cm), dµ) ∼= {ψ : G −→ Cn | ψ(gh) =

U(h−1)ψ(g), ∀h ∈ Hχ0} by

(U(g)ψ)(χ) =(dµgdµ

(x))`↑gψ(`g−1χ). (5.4.10)

Remark 5.4.3. Notice that we are evaluating the section in a character: we are usingthe fact that Oχ0

∼= G/Hχ0 .

For the V part we define for each v ∈ V ,

(V (v)ψ)(χ) = χ(v)ψ(χ). (5.4.11)

At first sight this procedure seems complicated, therefore we are going to study an

explicit example to see this method in action:

Example 5.4.1. We are going to work out the representations obtained in Section 4.5for the configuration space Q = S1, using this formalism. The canonical group in thisexample is E2 = R2 o SO(2). We can represent each element ϕ ∈ SO(2) as

Rϕ =

(cosϕ − sinϕsinϕ cosϕ

).

74


Notice that Char(R2) ∼= (R2) ∼= R2, where the last identification is induced by thecanonical inner product 〈, 〉 in R2. For an element ~α ∈ (R2)∗ and ~v ∈ R2 we compute

(`ϕχ~α)(~v) = exp(−i〈~α,R−1ϕ ~v〉) = exp(−i〈Rϕ~α,~v〉) = χRϕ~α(~v),

so we conclude that `ϕχ~α = χRϕ~α. This shows that the orbits of the SO(2)-action inChar(R2) are labeled by λ ≥ 0, and are given by

Oλ = {(α1, α2) ∈ R2 | α21 + α2

2 = λ2}.

For the case λ > 0 equations (5.4.10) and (5.4.1) become

(Uλ(ϕ)ψ)(α1, α2)ψ(α1 cosϕ+ α2 sinϕ,−α1 sinϕ+ α2 cosϕ)

(V (λ)(~v)ψ) = exp(−iλ(v1α1 + v2α2))ψ(α1, α2)

where ~v = (v1, v2). The Radon-Nikodym is one since the Lebesgue measure is rotationalinvariant.We can parametrize each ~α ∈ Oλ with an angle φ as

α1 = λ cosφ α2 = λ sinφ

Notice that

α1 cosϕ+ α2 sinϕ = λ(cosφ cosϕ+ sinφ sinϕ)

= λ cos(φ− ϕ)

and

−α1 sinϕ+ α2 cosϕ = −λ(cosφ sinϕ+ sinφ cosϕ)

= λ sin(φ− ϕ).

Thus, we recover the form of the representation discussed in Section 4.5 (equations(4.5.8) and (4.5.9))

(U (λ)(ϕ)ψ)(φ) = ψ((φ− ϕ)mod2π) (5.4.12)

(V (λ)(~v)ψ)(φ) = exp(−iλ(v1 cosφ+ v2 sinφ))ψ(φ) (5.4.13)

75


5.5 Q = S2

In this section we are going to give a brief look to the work done by C. Benavides and

A. Reyes in [BR08] where they apply this formalism to the sphere S2. If we consider the

U(1)-bundle S3 −→ S2 (which known as the Hopf bundle and is discussed in Appendix

(A) as SU(2) −→ SU(2)/U(1), a natural choice for the canonical group would be

(R3)∗ o SU(2).

Remark 5.5.1. Recall that SU(2) is the universal (2-sheeted) cover of the rotationgroup SO(3), therefore the orbits of the natural action of SO(3) on R3 are concentricspheres centered at the origin and the origin itself. So we see that the form of thecanonical group is of the form discussed in section 5.3.

In this work they are particularly interested in the construction of the U operators,

in this case the Radon-Nikodym factor is one since the Lebesgue measure induced on S2

is rotationally invariant. The first step is to construct unitary irreducible representation

of U(1). This can be easily done: For each integer n we define

Un : U(1) // GL(C)

eiφ� // einφ

(5.5.1)

Then we can consider the associated bundle Ln = SU(2)×Un C.

Remark 5.5.2. There is a useful way to regard U(1) as a subgroup of SU(2): Foreach λ ∈ U(1) define a matrix in SU(2) by(

λ 00 λ

)

Therefore, the right action of U(1) on SU(2) can bee seen just as matrix multiplication.Moreover, since each element of SU(2) can be written of the form(

z0 z1

−z1 z0

)

for z0, z1 ∈ C such that |z0|2+ |z1|2 = 1, then the right action can be seen as (z0, z1) 7−→(λz0, λz1)

Remark 5.5.3. Recall that S2 ∼= CP 1, so we can see the projection map of theprincipal U(1)-bundle SU(2) −→ CP 1 as π(z0, z1) = [z0 : z1]. In this context, the

76

5.5 Q = S2

action of SU(2) on CP 1 can be written (under the appropriate identifications discussedabove) as

`(α,β)([z0 : z1]) =

(α β

−β α

)(z0 z1

−z1 z0

)= [αz0 − βz1 : βz0 + αz1]. (5.5.2)

This action can be lifted to the line bundle Ln by

`↑(α,β)([(z0, z1), v]) = ([(α, β)(z0, z1), v]), (5.5.3)

for g = (α, β), (z0, z1) ∈ SU(2) and v ∈ C, as seen by Proposition 5.4.1.

SU(2)×Un C

π

��

`↑g // SU(2)×Un C

π

��S2

`g // S2

(5.5.4)

The nice characteristic of this article is that they do explicit calculations. For this

purpose lets construct local charts on S2 by defining open sets

UN = S2/{north pole} US = S2/{south pole}

and the stereographic projection onto C

ζN : UN −→ C ζS : US −→ C

It is easy tho show that, in spherical coordinates, these projections take the form

ζN (ϕ, θ) =eiϕ sin θ1− cos θ

ζS(ϕ, θ) =eiϕ sin θ1 + cos θ

(5.5.5)

Remark 5.5.4. Under the identification S2 ∼= CP 1 we can also define local charts bydefining open sets U0 = {[z0 : z1] | z1 6= 0} and U1 = {[z0 : z1] | z0 6= 0} and functionsζ0 : U0 −→ C and ζ1 : U1 −→ C by

ζ0([z0 : z1]) =z0z1

ζ1([z0 : z1]) =z1z0

(5.5.6)

In this case we can identify (UN , ζN ) and (US , ζS) with (U0, ζ0) and (U1, ζ

1) respectively.

77


Now we are going to construct local trivializations of the line bundle Ln using these

open sets. Define maps

ϕ0 : π−1(U0) −→ U0 × C ϕ1 : π−1(U1) −→ U1 × C

by

ϕ0([(z0, z1), v]) =(

[z0 : z1],(z1|z1|

)nv

)(5.5.7)

ϕ1([(z0, z1), v]) =(

[z0 : z1],(z0|z0|

)nv

)(5.5.8)

Hence, the composite map is given by

(ϕ1 ◦ ϕ−10 )([z0 : z1], v) = ϕ1

([(z0, z1),

(z1|z1|

)−nv

])

=

([z0 : z1],

(z0|z0|

)n( z1|z1|

)−nv

)

=(

[z0 : z1],(ζ0

|ζ0|

)nv

).

Remark 5.5.5. Notice that the transition function of Ln is given by g10 = (ζ0/|ζ0|)n.This means that its first Chern number of is n, therefore the label of each unitaryirreducible representation of U(1) determines the topology of the line bundle. Hence,if n 6= n′, then Ln and Ln′ are inequivalent. (See Appendix A)

Without loss of generality we are going to work on the local chart U0. Consider the

following diagram:

SU(2)×Un C

ϕ0

��

π

��

`↑g // SU(2)×Un C

π

�� ϕ0

��

π

��U0 ⊆ S2

`g // U0 ⊆ S2

U0 × Cσg // U0 × C

(5.5.9)

We would like to find an explicit expression for σg = ϕ0 ◦ `↑g ◦ ϕ−10 , in an proper

domain where this expression makes sense. If we let g = (α, β), we have

78

5.5 Q = S2

(ϕ0 ◦ `↑g ◦ ϕ−10 )([z0 : z1], v) = (ϕ0 ◦ `↑g)

([(z0, z1),

(z1|z1|

)−nv

])

= ϕ0

([(α, β)(z0, z1),

(z1|z1|

)−nv

])

= ϕ0

([(αz0 − βz1, βz0 − αz1),

(z1|z1|

)−nv

])

=(

[αz0 − βz1 : βz0 − αz1],(|z1|(βz0 + αz1)z1|βz0 + αz1|

)nv

)=(

[αz0 − βz1 : βz0 − αz1],(βζ0 + α

|βζ0 + α|

)nv

).

Notice that any local section s0 : U0 −→ Ln can be written in the form s0(x) =

(x, ψ0(x)) where ψ0 ∈ C∞(U0,C). Using one this local section we can find a local

expression for the U operator,

(Uloc(g)s0)(x) = σg(s0(g−1x)) = (g−1x, ψ0(g−1x)) = (x, ωn(x, g)ψ0(g−1x)),

where we define

ωn(x, g) =(βζ0(x) + α

|βζ0(x) + α|

)n. (5.5.10)

As usual, to find the expressions for the infinitesimal generators, we take a curve that

passes trough the identity and take the derivative with respect to the curve parameter

and evaluate at t = 0,

(Js0) = id

dt

∣∣∣∣t=0

(Uloc(g(t))s0)(x). (5.5.11)

This generator can always be written as J = ω + L, where ω depends on x and L

is a differential operator, this can be seen explicitly since,

id

dt

∣∣∣∣t=0

ωn(x, g(t))ψ0(g−1(t)x) =(id

dtωn(x, g(t))

)︸︷︷︸

=ωn(x)

ψ0(x) + id

dt

∣∣∣∣t=0

ψ0(g−1(t)x)︸︷︷︸=Lψ0(x)

.

The generator of rotations around the z-axis can be obtained if we consider the

curve t 7→ (α(t), β(t)) = (eit/2, 0). In this case L = L(z), the orbital angular momentum

operator. For the other summand we have

ω(z)n = i

d

dt

∣∣∣∣t=0

ωn(x, g(t)) = id

dt

∣∣∣∣t=0

e−int/2 =n

2.

79


For the generator of rotations around the y-axis we consider the curve t 7→ (α(t), β(t)) =

(cos t/2, sin t/2), therefore

ω(y)n = i

d

dt

∣∣∣∣t=0

ωn(x, g(t)) = id

dt

∣∣∣∣t=0

(cos t/2 + ζ0(x) sin t/2cos t/2 + ζ0(x) sin t/2

)n/2= −in

4(ζ0(x)− ζ0(x)) = −in

4

((eiϕ − e−iϕ) sin θ

1− cos θ

)=n

2

(sin θ sinϕ1− cos θ

)=n

2

(y

1− z

).

Remark 5.5.6 (Magnetic Monopole). In this article they explore the relation betweenthese results and the quantization of the electric charge via the existence of the magneticmonopole. Classically, one can show that the conserved quantity associated to therotational symmetry of the magnetic monopole system can be written as a vector

~J = ~L− eg

2~KN , (5.5.12)

where

• ~L = ~q × ~p, is the orbital angular momentum.

• ~B = gr/r2, and we define r2 = |~q|2 and r = ~q/r.

• ~KN = g−1(~q× ~AN ) + r = (~q− rz)/(r− z), where ~AN is a (local) vector potentialdefined by ~AN (~q) = (−q2, q1, 0)/(r(r + q3)).

On the other hand, the quantum generators discussed above can be also written in theform of a vector

~J = ~L− n

2~KN . (5.5.13)

Therefore, taking into account equations (5.5.12) and (5.5.13), we need to impose that2eg = n~c (recall n ∈ Z) in order to obtain a consistent quantum theory.

To end this chapter we will mention an important comment about the origin of the

construction introduced here:

The power of Mackey’s work lies not so much in the actual construction of the

representations (which is a direct extension of Wigner’s work on the Poincare group)

but in the subtle measure theoretic arguments that are needed to prove their

irreducibility and uniqueness.[Ish84]

80

6

The Riemann Hypothesis:

Spectral Aproach

The objective of this chapter is to explore some possible applications of the quantization

method studied in this work.

6.1 The Riemann Zeta Function

This section is devoted to a brief discussion on the definition and its appearance in

physics of the Riemann zeta function. The main reference of this part of the work is

[Edw74].

Definition 6.1.1. The complex variable function ζ which is the analytic continuationof the series

ζ(s) =∞∑n=1

1ns, (6.1.1)

which converges for Re(s) > 1, is known as the Riemann Zeta Function.

Remark 6.1.1. The origins of the Rieman zeta function lie in the study of the distri-bution of prime numbers. In 1737 Euler proved that the sum of the reciprocals of theprime number diverges. Later on, he also proved the product formula

∞∑n=1

1ns

=∏

p prime

11− p−s

.

There is a functional expression for the Riemann zeta function:

ζ(s) = 2sπs−1 sin(πs

2

)Γ(1− s)ζ(1− s).

81

6. THE RIEMANN HYPOTHESIS: SPECTRAL APROACH

It is a known fact that the Euler gamma function has simple poles in each non-positive

integer, so the last equation shows that, for Re(s) < 0, the Riemmann zeta function

has zeros at each s = −2n for n ∈ N+, which are known as the trivial zeros of ζ(s).

The Riemann hypothesis is a conjeture about the other zeros of ζ(s): the non-

trivial zeros of ζ(s) have real part 1/2. The Riemann hypothesis is still an open problem

and is part of Hilbert’s list of unsolved problems.

Now we are going to give some examples where the Riemann zeta function appears

in physics:

• In a Bose-Einstein gas, the pressure can be written in the form [Rie98]:

P = −kBTV

ln(1− z) +kBT

λ3T

g5/2(z)− IP(z,λT√π

L

),

where z is the fugacity. The function g5/2(z) is given by

g5/2(z) =∞∑n=1

zn

n5/2.

Therefore g5/2(1) = ζ(5/2).

• The Riemann zeta function appears in the context of random matrices: more

specifically, in the distribution of the values taken by the characteristic polyno-

mials of random unitary matrices.

• In the context of quantum field theory: The Casimir effect.

At this point it might not be clear what is the relation of the quantization method

we have studied and the Riemann zeta function. To begin we should mention some

works on the Riemann hypothesis from a spectral point of view: Hilbert and Polya

conjectured that the imaginary part of the Riemann zeta function are the eigenvalues

of a self-adjoint operator. Important work has been done in this direction by M. Berry

and J. Keating [BK99]: they worked on a semiclassical model of one particle in one

dimension with classical Hamiltonian H = xp. The idea is to find the number of states

N(E) with energy less than E, however there is a problem: the orbits on the phase

82

6.2 Landau Levels

space of this system are unbounded hyperbolae so N(E) is not finite. To make this

number finite they work with a regularized model where they restrict the particle in a

finite region. In this work Berry and Keating found that

N(E) ≈ E

2πlog(E

2π

)− E

2π. (6.1.2)

What is the relation of this result with the Riemann hypothesis? If we denote by N(E)

the number of complex zeros of the Riemann zeta function with imaginary part less

than E, the Riemann-van Mangoldt formula states that

N(E) = N(E) +1π

log ζ(

12

+ iE

)where N(E) has an asymptotic expression given by equation (6.1.2).

Remark 6.1.2. Notice that the classic Hamiltonian H = xp studied in these worksquantizes to π in the example where the configuration space is Q = R+.

In the next section we are going to study an approach developed by German Sierra

[ST08] in which similar expressions are obtained studying Landau levels of a particular

physical system.

6.2 Landau Levels

First of all we will describe the idea behind Landau levels. Consider a particle of mass

µ and charge q moving in the xy-plane under the influence of a constant magnetic field

along the z-axis. The Hamiltonian of the system is given by

H =1

2µ(px + qBy/2c)2 +

12µ

(py − qBx/2c)2, (6.2.1)

where we have used the vector potential

~A =B

2(−yx+ xy).

We can perform a canonical transformation

Q =cpx + qBy/2

qBP = py − qBx/2c

83


P ′ =cpx − qBy/2

qBQ′ = py + qBx/2c

and obtain the transformed Hamiltonian

H =1

2µP 2 +

12µω2

cQ2, (6.2.2)

where ωc = qB/µc is known as the cyclotron frequency. We can see that the

Hamiltonian can be written as

H = [a†a+ 1/2]~ωc, (6.2.3)

where

a =(µωc

2~

)1/2Q+ i

(1

2µωc~

)1/2

P. (6.2.4)

Hence, the energy levels are given by En = (n+ 1/2)~ωc, which are known as Landau

Levels.

Remark 6.2.1. Notice that the variables Q′ and P ′ are cyclic, we will see that thisis reflected by the fact that the Landau levels are infinitely degenerated. Let |0〉 bethe Lowest Landau Level (LLL), hence it satisfies a|0〉 = 0. If we take px = i~∂x andpy = i~∂y and work in complex coordinates z = x + iy and z = x − iy, we have incoordinate representation (

∂

∂z+qB

4~cz

)ψ0(z, z) = 0. (6.2.5)

If we suppose ψ0 of the form

ψ0(z, z) = exp(− qB

4~czz

)u(z, z)

we have

∂

∂zψ0(z, z) = − qB

4~czφ0(z, z) + exp

(− qB

4~czz

)∂

∂zu(z, z).

Equation(6.2.5) implies that∂

∂zu(z, z) = 0. (6.2.6)

So u belongs to the vector space of analytic functions, which has the set of monomials{zm |m = 0, 1, 2, · · · } as a basis.

84

6.2 Landau Levels

Now we are going to consider the case studied by [ST08]. This time we have an

external electric potential of the form ϕ(x, y) = λxy, where λ > 0. Working with a

particle of charge q = −e and in the Landau gauge, ~A = Bxy, the Lagrangian of the

system can be written as

L =µ

2(x2 + y2)− eB

cxy − eλxy.

This Lagrangian has two normal modes given by

ωc =eB

µccoshϑ ωh = i

eB

µcsinhϑ,

where sinh(2ϑ) = (2λµc2)/(eB2) and the Landau levels are also given by En = (n +

1/2)~ωc. In the limit where ωc � |ωh| they approximate to

ωc ≈eB

µcωh ≈ i

λc

B.

To vanish the high energy levels, since the gap is given by ~eB/µc, we can take the

limit µ→ 0. In this context, we have an effective Lagrangian given by

LLLL =eB

cxy − eλxy = px− |ωh|xp, (6.2.7)

where we have defined

p =~y`2

and `2 =~ceB

Remark 6.2.2. Each level has an associated quantum magnetic flux given by Φ0 =2π~c/e, that occupies an area 2π`2 in the xy-plane.The quantum magnetic flux Φ0 appears naturally in the context of the Aharonov-Bohmeffect treated with the path-integral formalism: an integral multiple of Φ0 will not makeany observable difference to the quantum mechanics of the particle, i.e. the interferencepattern (relative to the two paths) will not be affected. (see [Sha94]).

Remark 6.2.3. In the limit we are working on, the zero point energy of the systemdiverges, therefore we need subtract this term form the Lagrangian, which explains thechange of sign of the first term in equation (6.2.7). (for details see [Sha94]).

Therefore we see that the Hamiltonian H = xp follows naturally from this model

in the appropriate approximations.

85


As we discussed above, the number of states in the lowest Landau level is infinite

(in this case too). One way of making the number of states finite at each level is to

restrict the particle motion to the square |x| < L and |y| < L. Under this restriction

there is a maximum magnitude for the energy

|E| = |ωh||xp| ≤ ~|ωh|L2

`2.

Remark 6.2.4. We will use units such that ~|ωh| = 1, this will fix the value of λ.

If we take large values of L/`, in the semiclasical limit the number of quantum

states allowed is less than L2/2π`2. For a fixed value of admitted energy E the classical

trajectory for the particle will be the hyperbola xy = E`2, so the number of semiclasical

quantum states Nsc(E) with energy less that E will be the area between the two

branches of the hyperbola divided by 2π`2. However, since the Lagrangian is invariant

under (x, y) 7→ (−x,−y), the double degeneracy of the energy levels implies that we

only need to consider the region 0 < x < L. The area A(E) of this region is calculated

as follows

A(E) = x0L+∫ L

x0

E`2

xdx

= E`2 + E`2 log(L2

2π`2

)− E`2 log

(E

2π

),

where x0 satisfies L = E`2/x0. Hence, Nsc(E) is given by

Nsc(E) =E

2π+E

2πlog(L2

2π`2

)− E

2πlog(E

2π

). (6.2.8)

This equation should be compared with equation (6.1.2). Notice that the second term

is actually the regularization term since it diverges as we take L → ∞. Moreover, as

Sierra itself explains, this term may be interpreted as a regularization of the continuum

of states in the lowest Landau level for a particle free to move on the entire xy-plane.

6.3 Conclusions and Perspectives

The main motivation behind the canonical group quantization method developed by

Isham was the understanding of the canonical structure of classical gravity and its

possible implications for a quantum theory of gravity (this original program has been

86

6.3 Conclusions and Perspectives

superseded by e.g. loop quantum gravity). In the context of the canonical group ap-

proach, Isham gives some arguments -which are beyond the scope of this work- to state

that the configuration space that should be studied is of the form C∞(Σ,GL+(3,R))

where space-time is modeled as Σ × R with Σ an oriented compact 3-manifold. The

complications that arise in this context are mainly due to the difficulty of the repre-

sentation theory of the corresponding canonical group. In any case, in order to attack

complicated problems like this one it is always a good strategy to start from the begin-

ning: studying simple models!

When discussing about quantization, there are different points of view that can be

taken. From the point of view of high energy physics, it is very important to have a real

understanding of the underlying (if any) classical theory. For instance, the unavailabil-

ity of enough experimental data pertaining to ultra-high energies, forces theoreticians

to try out quantum models based on the only premise that, at low energies, one should

get the classical theory. This is one of the few routes available for those cases! One can

also consider another point of view: quantum theory stands on its own, and there is

no need to seek any ”functorial” properties relating quantum and classical phenomena.

In any case, in most cases both the quantum and the classical description of a given

system is given in terms of the same ”background” (space-time structure, symmetry

groups, etc.). It is this common ground which is, in the end, more important and for

which a rigorous quantization method can be of help. These points are without doubt

interesting, controversial, and deserve further consideration. But that was not our main

concern here. Rather, the objective of this work was to study an interesting proposal

from which we could compute explicit quantities and consider further applications of

the method. As we saw explicitly with some toy models, the soul of this quantization

method lies in the geometric and algebraic skeleton of physical theories: Evidence of

how to pass from a classical theory to its associated quantum model was encoded in

the symmetries of the classical system.

More concretely, we studied three one-dimensional examples: R,R+ and S1 where we

did explicit computations using the geometric tools that we introduced at the beginning

of the work. On each case we computed the commutation relations and guessed their

correct representations. What was also interesting for these case was the analysis of the

appearance of Planck’s constant in each model: we wanted all these theories to agree

87


locally. We also illustrated some works done on two-dimensional configuration spaces:

T 2 and S2. The first one gave us an idea of how to employ this method in presence of

external fields and the second gave us to recover Dirac’s argument for the quantization

of the electric charge via the magnetic monopole, but within a representation-theoretic

context. Finally, we discussed the formalism to find the irreducible representations

of the canonical group, for which we needed to introduce some basic theory on fiber

bundles.

We hope the reader become curious about the possible applications and perspectives

of this quantization method, since there are still a lot of questions to answer. We will

focus on two of them:

1. It is not clear how to employ this formalism for a constrained classical system.

There are some mathematical structures, called Dirac structures, that were born

from the study of constrained systems by Dirac. Today, this is an active research

area and there has been work on prequantization of these structures. It is not

clear, however, if there could an analogue for this quantization method in that

direction.

2. We have included in this chapter some works motivated by Polya’s conjeture

to study the spectrum of certain operators. Motivated by the work done by

Sierra and keeping in mind our work in curved configuration spaces, one could be

tempted to study Landau levels, with appropriate potentials, in these surfaces. If

for example we choose the 2-sphere, since this space is compact the regularization

procedure might be skipped.

88

Appendix A

Principal Bundles

The idea of this appendix is to give a quick review on fiber bundles, and in particular to

the construction of associated bundles. We strongly recommend refer to other sources

to complement the contents of this section. We we are going to follow the text of M.

Nakahara [Nak03] (which we strongly recommend since is full of examples).

Definition A.0.1. A smooth fiber bundle consists of the following elements:

1. Smooth manifolds E,M and F called total space, base space and fiber re-spectively.

2. A smooth surjection π : E −→ M , called the projection, for which the inverseimage of a point p ∈M is diffeomorphic to the fibre, that is, π−1(p) = Fp ∼= F .

3. A Lie group G that acts on F on the left.

4. An open covering {Uj}j∈I of M and diffeomorphisms ϕj : Ui × F −→ π−1(Ui)such that (π ◦ ϕj)(p, f) = p and that the restriction map ϕj |p : {p} × F −→ F isa diffeomorphism too. The pairs {(Uj , ϕj)}j∈I are called local trivializations.

5. On Uj ∩ Uk 6= ∅ we require that gjk(p) = ϕ−1j |p ◦ ϕk|p be an element of G. These

maps gjk : Uj ∩ Uk −→ G are called transition functions.

Remark A.0.1. On Uj ∩Uk ∩Ul the transitions functions satisfy the cocycle condi-tions:

gjj = id gjk = g−1kj gjkgklglj = id (A.0.1)

Definition A.0.2. A smooth map s : M −→ E is called a local section if π ◦ s = id.The space of sections of the fiber bundle is denoted by Γ(E).

89

A. PRINCIPAL BUNDLES

Definition A.0.3. Let (E, π,M,F,G) and (E′, π′,M ′, F ′, G′) be two fiber bundles andh : M −→ M ′ be a smooth map. We say that h : E −→ E′ is a smooth bundle mapover h if h(Fp) = F ′h(p).

Definition A.0.4. A vector bundle is a fiber bundle whose fiber has a vector spacestructure, for instance Ck, and whose structure group in this case is GL(k,C).

Definition A.0.5. A principal bundle is a fiber bundle whose fibre is the samestructure group G. A principal bundle is often called a G-bundle over M and thetotal space is denoted by P .

Remark A.0.2. We can define a right action of G on F as follows: Let {(Uj , ϕj)}j∈Ibe local trivializations and let u ∈ P such that π(u) = p ∈ Uj and ϕ−1

j (u) = (p, bj).Then we define for each a ∈ G, ua = ϕj(p, bja). This definition is independent of thetrivialization since ua = ϕj(p, bja) = ϕj(p, gjk(p)bka) = ϕk(p, bk, a) = ua. It is easilyseen that the action is free and transitive.

Remark A.0.3. We can use local sections to define a canonical trivialization asfollows: Let sj be a smooth section defined on an open subset Uj ⊆ M . For eachelement u ∈ π−1(p) there is a unique element gu ∈ G such that u = sj(p)gu, thereforewe can define ϕ−1

j (u) = (p, gu) and, in particular, sj(p) = ϕj(p, e). Moreover, on anon-empty intersection we have

sj(p) = ϕj(p, e) = ϕk(p, gkl(p)e) = ϕk(p, egkl(p)) = ϕk(p, e)gkj(p) = sk(p)gkj

Example A.0.1. We are going to explore an explicit example: A U(1)-bundle overS2. For ε > 0 define an open covering for the sphere using spherical coordinates by

UN = {(θ, φ) ∈ [0, π/2 + ε)× [0, 2π)}

US = {(θ, φ) ∈ (π/2− ε, π]× [0, 2π)}

so that the intersection is an open band of width 2ε around the equator. Define localtrivializations by

ϕ−1N (u) = (p, exp(iαN ))

ϕ−1S (u) = (p, exp(iαS))

were π(u) = p. The transitions function gNS can be taken of the form gNS(p) exp(inφ(p)),however on the equator we have exp(iαN ) = exp(inφ) exp(iαS), therefore, n should bean integer in order to ensure that the transition function is well defined on the equator.

90

For n = 0 we have a trivial bundle S2 × U(1), whereas for n 6= 0 the bundle is nottrivial. Moreover, for each integer we have inequivalent U(1)-bundles over S2. Thetwisting of these bundles can be understood in terms of the fundamental group, sinceπ1(S1) = Z.

Example A.0.2. Let H be a closed Lie subgroup of a Lie group G. We are goingto see that G is a principal bundle over the homogeneous space M = G/H with fiberH. Define the projection map by π(g) = [g] and the right action by of H on G justby right multiplication. To construct local trivializations we consider a local sectionsj defined on Uj and g ∈ π−1([g]), now define fj : G −→ H by fj(g) = sj([g])−1g.To ensure that this map is well defined notice that s([g]) = ga for some a ∈ H,hence sj([g])−1g = a−1 ∈ H. Also notice that if a ∈ H then fj(ga) = s([ga])−1ga =s([g])−1ga = fj(g)a. Finally, if we define ϕj : Uj × H −→ G by ϕ−1

j (g) = ([g], fj(g))we see that ϕ−1

j (ga) = ([ga], fj(ga)) = ([g], fj(g)a), which is what we wanted.

Example A.0.3. (Hopf Bundle) For instance, it is well known that SO(n)/SO(n−1) ∼=Sn−1, in particular for n = 3 he have a principal U(1)-bundle over S2 with total spaceS3.

Definition A.0.6. Let P be a G-bundle over M and let G acting on a smooth manifoldF on the left. Define a right action of G on P × F by (u, f)g = (ug, g−1f). We definethe associated bundle to be a fiber bundle over M whose total space is P × F/Gand has F as a fiber. The projection map is given by π([(u, f)]) = π(u), which is welldefined since π([(u, f)g]) = π([(ug, g−1f)]) = π(ug) = π(u). We will denote the totalspace of the associated map by P ×G F

Remark A.0.4. Let P be a G-bundle over M and consider the special case in whichF has a vector space structure, for instance let F = Ck. Then, given a representationρ : G −→ GL(k,C) we have a natural left action of G on Ck, namely gv = ρ(g)v forg ∈ G and v ∈ Ck. We will denote the associated bundle by P ×ρ Ck, which is clearlya vector bundle over M.Conversely, let E be a k-dimensional vector bundle over M and let GL(E) the fiberbundle over M whose fiber at p ∈M is GL(k, π−1(p)), then GL(E) is a GL(k,C)-bundleover M and the associated bundle GL(E)×GL(k,C) Ck is naturally equivalent to E. Theprincipal bundle GL(E) is called the frame bundle of E.

Example A.0.4. Let P be a G-bundle over M and consider de adjoint representationAd : G −→ Aut(g). Then we can construct the adjoint associated bundle byP ×Ad g.

91

A. PRINCIPAL BUNDLES

Remark A.0.5. We are going to study the space Γ(P ×G F ). Let ψ ∈ C∞(P, F ) suchthat ψ(ug) = g−1ψ(u) for all g ∈ G, then we can define a section by sψ(p) = [up, ψ(up)]where up ∈ π−1(p). Notice that [upg, ψ(upg)] = [upg, g−1ψ(up)] = [up, ψ(up)], thereforethis section is well defined.On the other hand, given a section s ∈ Γ(P ×G F ), define a function ψs ∈ C∞(P, F )by s(π(u)) = [p, ψs(p)]. If we compute

[p, ψs(u)] = s(π(u)) = s(π(ug)) = [ug, ψs(ug)] = [u, gψs(ug)]

we see that ψ(ug) = g−1ψ(u). Hence, we conclude that

Γ(P ×G F ) ∼= {ψ ∈ C∞(P, F ) | ψ(ug) = g−1ψ(u) ∀g ∈ G} (A.0.2)

92

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