Master thesis

An Introduction to the OrbitMethod

J. Maes

Department of Mathematics, Utrecht UniversityInstitute for Theoretical Physics, Utrecht University

Utrecht, April 4, 2011

...from the same principles, I now demonstrate the frame of the System ofthe World.

Isaac Newton ([17])

As time goes on, it becomes increasingly evident that the rules which themathematician finds interesting are the same as those which Nature has

chosen.Paul Dirac at age 36, ([13])

Abstract

The Orbit Method is a method to determine all irreducible unitary represen-tations of a Lie group. It is entangled with its physical counterpart geometricquantization, which is an extension of the canonical quantization scheme togeneral curved manifolds. The main ingredient of the Orbit Method is thenotion of coadjoint orbits, which will be explained. Coadjoint orbits of a Liegroup have the natural structure of a symplectic manifold, as does the phasespace of a classical mechanical system. Naturally, geometric quantizationwill be treated next, since it attempts to provide a geometric interpretationof quantization within an extension of the mathematical framework of classi-cal mechanics (symplectic geometry). In particular, the axioms imposed ona quantization will be discussed. Finally, as an application, coadjoint orbitsand geometric quantization will be brought together by indicating how todetermine the irreducible unitary representations of SU(2) by means of theOrbit Method.

Acknowledgments

This master thesis is not the work of just one person, but the result of thededication of many people. First of all, I would like to thank my supervisors.Dr. Gleb Arutyunov from the Institute for Theoretical Physics, who had thebrilliant ability to give strong intuition for abstract notions in physics andmathematics, without doing deficit to the theory itself, and Dr. Andre Hen-riques from the Department of Mathematics, who amazed me with his deep,fundamental understanding of the concepts we were discussing. He succeededin significantly improving my skills in abstract mathematics the past year,by going back to the very fundamentals. Both of my supervisors were alwayswilling to help me. Some of our appointments even took more then two hours.

Secondly, I want to thank my parents (John and Jolanda Maes), friends(Erwin Engelman, Martine Gennisse, Rolf van Vugt, Tristan van Heijst) andfellow students for their emotional support. At the beginning of my researchI was dissapointed by the rate at which I was able to understand the newabstract theory and the growing consciousness that the final goal, to studythe Virasoro group, could probably not be achieved. My parents and fel-low students were always there to put things in a more realistic perspectiveand encouraged me to look for goals in other things then technical achieve-ment too. This led me to write a thesis which is intended for both physicsstudents with little knowledge of abstract mathematics and mathematicsstudents interested in the physical ideas behind quantum mechanics as well.Only knowledge of calculus, linear algebra, elementary set theory and sometopology is presupposed.

In realizing the, in my opinion, beautiful pictures in this thesis I was tech-nically supported by Bram Jenneskens, who posseses skills in realizing vec-tor pictures. Many thanks go to him as well. Further thanks go Tristanvan Heijst, who corrected some earlier versions of my thesis in a criticaland motivating way, Vivian Jacobs, Maarten Kater and Thessa Fokkemawho helped me with computer tecnnical problems, Marleen Kooiman, whohelped me with Mathematica, Thomas Wasserman, who answered many ofmy questions about representation theory, Daniel Prins, whose presentationon Frobenius theorem was useful in the subsection on polarizations and JulesLamers, whose latex code I could use to give form to my own master thesistalk.

Considering the contents, for the vocabulary section I owe many thanks to thelecture notes of Erik van den Ban ([3]) and Eduard Looijenga ([10]). I usedthe notes of Matthias Blau ([9]) for both the introduction and the sectionon Geometric quantization. I am grateful for using the notes of AlexandreKirillov ([1]) in an intensive manner for both sections about Coadjoint orbits.The section on geometric quantization could not have been written withoutthe book of Nicholas Woodhouse ([7]). The lectures notes of Stefan Vandoren([6]) where useful for both section 5 and section 6. Finally, for the sectionabout the Orbit Method and the representaions of SU(2) the lecture notesof Erik van den Ban were again very important.

I am sure I forget to mention people here which do owe credit to this thesis.This is my final thanks to all of those people.

Contents

1 Conventions and notation 8

2 Introduction 10

3 Vocabulary 143.1 The language of manifolds . . . . . . . . . . . . . . . . . . . . 143.2 Structure-preserving maps (morphisms) . . . . . . . . . . . . . 183.3 Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . . 203.4 Actions and representations . . . . . . . . . . . . . . . . . . . 23

4 Coadjoint orbits 264.1 The coadjoint representation . . . . . . . . . . . . . . . . . . . 264.2 The coadjoint orbits of SU(2) . . . . . . . . . . . . . . . . . . 31

4.2.1 Deriving the coadjoint orbits by finding conjugation-invariant functions on su(2)∗ . . . . . . . . . . . . . . . 31

4.2.2 Deriving the coadjoint orbits by comparing with SO(3) 35

5 Symplectic structure on coadjoint orbits 395.1 The Kirillov form . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 The Kirillov form on the coadjoint orbits of SU(2) . . . . . . . 44

5.2.1 The Kirillov form on the coadjoint orbits of SU(2) inthe spherical coordinate system . . . . . . . . . . . . . 46

5.2.2 The Kirillov form on the coadjoint orbits of SU(2) inthe stereographic coordinate system . . . . . . . . . . . 49

6 Geometric quantization 556.1 The mathematical framework of classical

mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.1.1 A crash course in symplectic geometry . . . . . . . . . 566.1.2 Symplectic geometry and classical mechanics . . . . . . 62

6.2 From classical to quantum mechanics . . . . . . . . . . . . . . 666.3 A heuristic discussion of prequantization . . . . . . . . . . . . 706.4 Formalizing the discussion . . . . . . . . . . . . . . . . . . . . 74

6.4.1 Putting more structure on vector bundles . . . . . . . . 746.4.2 Cech cohomology and the de Rham isomorphism . . . 81

6.5 The integrality condition . . . . . . . . . . . . . . . . . . . . . 866.6 The prequantum Hilbert space . . . . . . . . . . . . . . . . . . 956.7 Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.7.1 Polarizations in general . . . . . . . . . . . . . . . . . . 986.7.2 Kahler polarizations . . . . . . . . . . . . . . . . . . . 101

6.8 The Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . 104

7 The orbit method and the representations of SU(2) 1097.1 The integrality condition for the coadjoint orbits of SU(2) . . 1097.2 Constructing the Hilbert space on the

spheres of half-integer radius . . . . . . . . . . . . . . . . . . . 1127.3 Quantizing observables on the spheres of half-integer radius . . 114

7.3.1 Quantizing observables on the spheres of integer radiusin the spherical coordinate system . . . . . . . . . . . . 115

7.3.2 Quantizing observables on the spheres of half-integerradius in the stereographic coordinate system . . . . . 115

7.4 The irreducible unitary representations of SU(2) . . . . . . . . 1177.4.1 Finding the irreducible representations of SU(2) by

means of the infinitesimal method . . . . . . . . . . . . 1177.4.2 Finding the irreducible representations of SU(2) by

means of the orbit method . . . . . . . . . . . . . . . . 120

8 Conclusions and outlook 1238.1 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 1238.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A An identity at the heart of quantization 126

References 130

Index 133

1 Conventions and notation

-Today’s shocks are tomorrow’s conventions.-Carolyn Heilbrun ([15])

Although sometimes the notation used is already explained in the thesisitself, the total amount of notation introduced is quite big. Therefore, thissection serves as a handy overview of (often) used notation troughout thisthesis. Furthermore, the conventions used will be given.

Notation:K: the field of either the real or complex numbers.U ≡ Uii∈A: an open cover of a manifold. Here A denotes some index set.S2: the two-dimensional sphere.(M,ω): a symplectic manifold M with symplectic 2-form ω.(M,J, ω): a Kahler manifold M with symplectic 2-form ω and complex struc-ture J .TmM : the tangent space of a manifold M at the point m ∈M .TM : the tangent bundle of a manifold M .T ∗M : the cotangent bundle of a manifold M .Ωk(M): the set of k-forms on a manifold M .Vect(M): the set of vector fields on a manifold M .Vect(M ;P ): the set of vector fiels X such that X|m ∈ Pm for every m ∈M .Here P denotes a polarization.End(X): the set of endomorphisms of a mathematical object X.Aut(X): the set of automorphisms of a mathematical object X.Mat(n,K): the group of invertible n× n-matrices with entries in K.GL(V ): the group of invertible, linear transformations on a vector space V .C∞(M,K): the space of either real or complex-valued functions on a mani-fold M .C∞(M,K;P ): the functions in C∞(M,K) whose flow leaves the polarizationinvariant.L2(M,C): the space of square-integrable, smooth, complex-valued functionson a manifold M .P (M): the space of polynomials on an manifold M .Pn(M): the space of homogeneous polynomials of degree n on an manifoldM .ΓX(M): the set of smooth sections over a real manifold M of a line bundleX.h∗: the push-forward corresponding to the diffeomorphism h : M → N withM , N manifolds.

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h∗: the pull-back corresponding to the smooth map h : M → N with M , Nmanifolds.exp: the exponential map.g: the Lie algebra of a Lie group G.Id: the Identity map.π = (π, V ): a representation of a Lie group G in a vector space V .π∗ = (π∗, V ∗): the representation dual to π in the dual space V ∗.V C: the complexification of a vector space V .

TMC: the complexified tangent bundle. TMC ≡∐m∈M

(TmM)C.

V ∗: the dual of a vector space V .Ad: the adjoint representation.Ad∗: the coadjoint representation.ad: ad ≡ TeAd.ad∗: ad∗(X) ≡ (−ad(X))∗ for X ∈ g.Xf : a Hamiltonian vector field corresponding to a real scalar function f .Lv: the Lie derivative along a vector field v.∇: the connection.[, ]: the Lie bracket., : the Poisson bracket.〈, 〉: evaluated on a linear functional f in the first slot and a vector v in thesecond slot it denotes the value of the linear functional f on the vector v.Otherwise, it denotes the inner product.Cp(U,R): the set of p-cochains corresponding to real, locally constant func-tions. Here U denotes an open covering of a manifold M .Zp(U,R): the set of p-cocycles corresponding to real, locally constant func-tions. Here U denotes an open covering of a manifold M .Hp(U,R): the pth Cech cohomology group with coefficients in R. Here Udenotes an open covering of a manifold M .Hp(M,R): the pth de Rham cohomology group of a manifold M with coef-ficients in R.

Conventions:Skew symmetrization: T [ab...c] = 1

p!

∑σ sign(σ)T σ(a)σ(b)...σ(c),

where T is a p-index tensor and the summations are over all permutationsof a, b, ..., c.Interior product: i(X)ω = kω(X, ...) for a vector field X and ω ∈ Ωk(M),where M denotes a manifold.

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2 Introduction

- When one looks back over the development of physics, one sees that it canbe pictured as a rather steady development with many small steps and su-perposed on that a number of big jumps. These big jumps usually consist inovercoming a prejudice... And then a physicist has to replace this prejudiceby something more precise, and leading to some entirely new conception ofnature.-Paul Dirac on Quantum Theory ([13])

At the macroscopic level our world seems to be pretty much gouverned bythe laws of classical physics, that is, by Newtonian mechanics on the onehand and by Maxwell theory on the other. Newtonian mechanics describesthe motion of particles under the influence of forces acting on them. Maxwelltheory covers almost the entire spectrum of phenomena occuring in electro-magnetism and optics. The in these two theories is governed by deterministicequations of motion and it is possible, given the initial conditions, to predictthe results of measurements on the system at any later time.

At first sight, classical physics could thus provide us with a very satisfactorydescription of the world we live in. However, it fails to give an explanationfor a number of phenomena observed at the microscopic level and, moreover,is in plain contradiction with experimental evidence.

As an illustration of what these phenomena are and what they may be try-ing to tell us about a theory which will have to replace classical physics, wemention two examples. The first is regarding the stability of atoms. Fromscattering and other experiments it had been deduced that atoms consist of atiny positively charged nucleus orbited at quite some distance by negativelycharged point-like particles, the electrons. Classical physics would predictsuch structures to collapse within fractions of a second, in contradiction withthe stability of the world around us: orbital motion is accelerated motion,and according to Maxwell theory accelerated charges emit radiation; conse-quently, the electron would radiate away energy and spiral into the nucleusof the atom. It was also observed that simple atoms (like hydrogen) wereable to adsorb and emit energy only in certain discrete quantities. Com-bining these two observations it thus appeared to be necsesary to postulatethe existence of stable orbits for electrons at certain discrete radii (energylevels). This suggests that at the microscopic level, as opposed to in classicalphysics, nature allows for a discrete (or quantized) structure.

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The second example is the set of experiments related to the nature of light.On the one hand, there is the famous double-slit experiment which inves-tigated the diffraction and interference patterns of beams of particles likeelectrons. The experiment is depicted in figure 1 (from [11]). It indicated

Figure 1: The double-slit experiment. A single photon (being a probability wave) departsfrom slit a in plate S1. When the wavefront reaches slits b and c in plate S2 it creates twonew probability waves, one emerging from slit b and one from slit c. The single photonwill show up on the detector screen F (at say d) according to the net probability valuesresulting from the co-incidence of the probability waves coming by way of the two slits band c.

that under certain circumstances particles (like electrons) can show inter-ference patterns and thus wave-like nature. On the other hand, Einstein’sinterpretations of the photo-electric effect made clear that light showed be-haviour characteristic of particles and not of waves. In the words of the greatDirac (see [12, p.3]),

‘We have here a very striking and general example of the breakdown of clas-sical mechanics - not merely an inaccuracy of its laws of motion, but aninadequacy of its concepts to supply us with a description of atomic events.’

Furthermore, the outcome of these experiments appeared to depend on themeasuring process itself, i.e. whether or not one was checking through whichslit the electron went. This, for the first time, implied that in order to de-scribe a system the effects of observation of that system had to be taken into

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account. In particular, it implied that one cannot make any observations ona suitably ‘small’ system without perturbing the system itself (expressed inthe uncertainty principle).

Heisenberg and Schrodinger provided two mathematical models, later shownto be equivalent, which were able to reproduce the above results and makemany other succesfully tested predictions. These models, collectively knownas quantum mechanics , describe the quantum behaviour of particles inflat space under the influence of external forces. Supplemented by someinterpretation of, roughly speaking, the role of the measuring process or ob-server, they constitute a major step forward in the understanding of quantumphysics in general.

However, at the conceptual level the situation was not satisfactory. In partic-ular, it was not clear how general the proposed models were, which featureswere to be regarded as fundamental to any quantum version of a classical the-ory and which were to be attributed to particular properties of the systemsconsidered so far. To gain some insight into this question, it was in particularDirac who emphasized the formal similarities between classical and quantummechanics and the necessity of properly understanding these. Abstractingfrom the analogy found between classical mechanics and Schrodinger andHeisenberg quantum mechanics, Dirac formulated a general quantum condi-tion, a guideline for passing from a given classical system to the correspond-ing quantum theory. This process in general is known as quantization .Roughly speaking, quantization consists of replacing the classical algebra ofobservables (smooth functions on phase space) by an algebra of operators act-ing on some Hilbert space, the quantum condition relating the commutatorof two operators to the Poisson bracket of their classical counterparts:

Poisson Brackets→ Commutators.

At first sight, the very concept of quantization appears to be ill-foundedsince it attempts to construct a ‘correct’ theory from a theory which is onlyapproximately correct. After all, our world is quantum, and while it seemsa legitimate task to try to extract classical mechanics in some limit fromquantum mechanics, there seems too little reason to believe that the inverseconstruction can always be performed:

Classical limit: Quantum mechanics =⇒ Classical mechanics

Quantization: Quantum mechanics?⇐= Classical mechanics.

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Furthermore, there is no reason to believe that such a construction wouldbe unique as there could well be (and, in fact, are) lots of different quantumtheories which have the same classical limit. Unfortunately, however, it isconceptually very difficult to describe a quantum theory from scratch, with-out the help of a reference classical theory. Moreover, there is enough to theanalogy between classical and quantum mechanics to make quantization aworthwile approach. Perhaps, ultimately, the study of quantization will tellus enough about quantum theory itself to do without the very concept ofquantization.

Unfortunately, Dirac’s quantum condition is not as general as one mighthave hoped it to be, or, at least, not sufficiently unambiguous (see, [12]).Thus, it is desirable to find a more intrinsic and constructive description ofthis quantization procedure. This is the aim of geometric quantizationwhich attempts to provide a geometric interpretation of quantization withinan extension of the mathematical framework of classical mechanics (symplec-tic geometry).

Geometric Quantization is the physical counterpart of the Orbit Method. The Orbit Method is a method to determine all irreducible unitary repre-sentations of a Lie group. This is useful, since representation theory remainsthe method of choice for simplifying the physical analysis of systems poss-esing symmetry. The main ingredient of the Orbit Method is the notion ofcoadjoint orbits. What is so special about coadjoint orbits is that they havethe natural structure of a symplectic manifold, as does the phase space ofa classical mechanical system. Indeed, Geometric Quantization is anotheringredient of the Orbit Method. That is why the Orbit Method brings theingredients Geometric Quantization and the notion of coadjoint orbits to-gether in a beautiful way. The idea behind it is to unite harmonic analysiswith symplectic geometry. This idea can be considered as a part of the moregeneral idea of the unification of mathematics and physics.

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3 Vocabulary

-The golden age of mathematics, that was not the age of Euclid, it is ours.-C.J. Keyser ([16])

To study the orbit method, a good knowledge and understanding of thelanguage of manifolds, Lie groups and representation theory is essential. Forthis reason this section states the most important definitions and resultsrelated to these subjects. In this way the exposition of the orbit method ismade as self-contained as possible. The reader can therefore view this sectionas a dictionary and look back at important definitions whenever necessary.It begins with manifolds, continues with listing some morphisms, then treatsLie groups and Lie algebras and ends with some basic theory about actionsand representations. This section would not have been possible without thelecture notes [3] and [10].

3.1 The language of manifolds

Definition 3.1.1. A k-smooth n-dimensional manifold is a topologicalspace M that admits a covering by open sets Uα, α ∈ A, endowed with one-to-one continuous maps φα : Uα → Vα ⊂ Rn so that all maps φα,β ≡ φα φ−1

β

are k-smooth (i.e. have continuous partial derivatives of order ≤ k) wheneverdefined.

We call the manifold M smooth when all the maps φα,β are infinitely differ-entiable.

The following terminology is used:

• the sets Uα are called charts;

• the functions xiα = xi φα are called local coordinates. Here xi1≤i≤nare the standard coordinates in Rn;

• the functions φα,β are called the transition functions;

• the collection Uα, φαα∈A is called the atlas on M .

A given topological space M can be endowed with several different atlasessatisfying all the requirements of the definition. Should the correspondingobjects be considered as different or as the same manifold? To clear this up,we introduce

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Definition 3.1.2. Two atlases Uαα∈A and U ′ββ∈B are called equivalentif the transition functions from any chart of the first atlas to any chart of thesecond one are k-smooth (whenever defined). The structure of a smoothmanifold on M is an equivalence class of atlases.

Remark 3.1.1. When dealing with a smooth manifold, we usually follow acommon practice and each time use only one, the most appropriate, atlas,while keeping in mind that we can always replace it by an equivalent one. Inpractical computations people usually prefer minimal atlases. For example,for the n-dimensional sphere Sn and for the n-dimensional torus T n thereexist atlases with only two charts.

The following definition tells us when a subset of a (Ck-)manifold is a(Ck-)manifold itself.

Definition 3.1.3. Given a Ck−manifold N of dimension n, then we say thata subset M ⊂ N is a Ck-submanifold of dimension m if M can be coveredby charts (U, φ) of N with the property that M ∩ U = φ−1(Rm × 0). (Ifk =∞ we will also call this a smooth submanifold.)

Hence we must have m ≤ n and Rn is written as Rm×Rn−m. Coming to thegeometry of manifolds, the following notion is essential:

Definition 3.1.4. A vector bundle over the manifold M (called the basemanifold) is a real manifold X (called the total space) together with asmooth map pr : X →M (called the projection) such that

(VB1) For each m ∈M , Xm = pr−1(m) is a vector space over K of constantdimension r (Xm is called the fibre over m and r is called the fibredimension),

(VB2) For each m0 ∈ M , there is a neighbourhood U ∈ m0 in M togetherwith a diffeomorphism ψ : U ×Kr → pr−1(U) such thatψ|m : m × Kr → Xm is a linear isomorphism for each m ∈ U (thepair (U, ψ) is called a local trivialization).

In the definition above M is a n-dimensional real manifold and K denotesthe field of either the real or complex numbers. A map s : U ⊂ B → X iscalled a section of a vector bundle X if pr s = Id.

The most important example of a vector bundle is the tangent bundle TM . Itwas the source of the whole theory of vector bundles. In order to understandwhat the space TM is we first introduce an auxiliary definition. A Curve

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at m ∈ M is a smooth map γ from an open neighbourhood of 0 in R to Mwith γ(0) = m. On the collection of curves at m we define an equivalencerelation: γ1 ≡ γ2 if for a chart U at m and the corresponding diffeomorphismα, α γ1 and α γ2 have the same derivative in 0. The equivalence class ofγ relative to this equivalence relation will be called the derivative of γ at0 and will be denoted by γ(0). This relation is independent of the choice ofU . This leads us to the following definition:

Definition 3.1.5. A tangent vector of M at m is the derivative at 0 of acurve atm. The collection of tangent vectors ofM atm is the tangent spaceof M at m and will be denoted by TmM . The union TM ≡

⋃m∈M TmM is

called the tangent bundle .

The union TM forms a vector bundle over M with the projection pr thatmaps all elements of TmM to m: Indeed, let (U, φ), with U ⊂M , be a localchart with coordinates (x1, ..., xn). First of all, TmM is a vector space. Howis addition and scalar multiplication defined on this vector space? If we taketwo curves γ(t) and γ(t) trough m with γ(0) = γ(0) = m, then adding themor multiplying them by a scalar need not produce a curve in M . However,we can add these curves, using the chart φ, as follows:(1) Addition: (γ + γ)(t) = φ−1((φ(γ(t))) + φ(γ(t)))(2) Scalar multiplication: (λγ)(t) = φ−1(λφ(γ(t))) (λ ∈ R)Taking the derivatives of γ + γ and λγ at the point 0 ∈ R will, by the chainrule, produce the sum and scalar multiples of the corresponding tangent vec-tors (see also [30]). Furthermore, any tangent vector a ∈ TmM can be writtenas a = ak∂k where ak = ak(x) are numerical coefficients and ∂k ≡ ∂/∂xk de-notes the partial derivative. So the set pr−1(U) is identified with U ×Rn. Itis by definition a local chart on TM with coordinates (x1, ..., xn; a1, ..., an).This proves the claim.

The smooth sections of TM are called smooth vector fields on M . Wedenote the space of all smooth vector fields on M by Vect(M).

The second important example of a vector bundle is the so-called cotangentbundle.

Definition 3.1.6. Given a manifold M , then the cotangent space of Mat m ∈ M is the dual T ∗mM of the tangent space TmM . The cotangentbundle of M is the dual of the tangent bundle and is denoted by T ∗M . Adifferential 1-form on M is a section of its cotangent bundle.

We denote the space of all smooth differential 1-forms on M by Ω1(M), which

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is dual to Vect(M). Locally, we can write a differential 1-form as the expres-sion ω = ωidx

i for some chart U , where dxi1≤i≤n forms the basis of Ω1(U)dual to the basis ∂i1≤i≤n of Vect(U).

The space Ωk(M) of smooth differential k-forms on M can be definedcan be defined as the k-th exterior power of Ω1(M). In a local coordinatesystem a k-form ω looks like

ω =∑

i1<...<ik

ωi1,...,ik(x)dxi1 ∧ ... ∧ dxik , (1)

where all indices is, 1 ≤ s ≤ k, run from 1 to n and the wedge product isbilinear, associative, and anti-symmetric.

We will now define the exterior derivative . The exterior derivative dis the unique R-linear mapping from k-forms to (k + 1)-forms satisfying thefollowing properties:

(1) df is the differential 1-form of f for smooth functions f .

(2) Flatness: d d = 0.

(3) Leibniz rule: d(α∧ β) =dα∧ β + (−1)p(α∧ dβ) where α is a differentialp-form.

We will end this subsection with a very important object in differential geom-etry, which is the Lie derivative L. Let M be a manifold. Three particularcases of the Lie derivative are:

(a) The ordinary directional derivative v · f of a real smooth scalar functionf : M → R along the vector field v is just Lvf ;

(b) The so-called Lie bracket of two vector fields v, w ∈ Vect(M) is[v, w] = Lvw = −Lwv;

(c) For a general differential k-form ω the Cartan formula holds:Lvω = ivdω + d(ivω) where iv is the interior multiplication, that is,it is the map iv : Ωk(M) → Ωk−1(M) which sends a k-form ω to the(k − 1)-form ivω defined by the property that(ivω)(v1, ..., vk−1) = ω(v, v1, ..., vk−1) for any vector fields v1, ..., vk−1.

The Lie derivative acting on differential k-forms (case (c)) has some particularproperties. In order to quote them we first need to define an auxiliary notion.

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Definition 3.1.7. A flow on a manifold M is a smooth map H : R×M →Mwith the property that if Ht : M → M is defined by Ht(p) = H(t, p), thenH0 is the identity map and HsHt = Hs+t for all s, t ∈ R.

You may think of a flow in the definition above as a fluid in motion whichdoes not change in time (the map in the definition above then describes howa given fluid particle moves in time). The orbit of a point p ∈M is the imageof the map γp : R→M γp(t) ≡ H(t, p). The orbit map γp defines a tangentvector γp(0) ∈ TpM . In the physical situation of a fluid in motion this canbe interpreted as the velocity of the particle at p. By letting p vary, we geta vector field on M . We call this vector field the infinitesimal generatorof the flow and denote it by ∂H

∂t

∣∣t=0

.

We now quote without proof three properties satisfied by the Lie derivativeacting on differential k-forms.

Theorem 3.1.1. The Lie derivative L obeys the following three properties.

(1) commutation with d: dLV = LV d,

(2) Leibniz rule: if α, β ∈ Ωk(M), thenLV (α ∧ β) = LV (α) ∧ β + α ∧ LV (β),

(3) Infinitesimal pull-back: if V is the infinitesimal generator of a flowH on M , then for every k-form α on M ,∂∂t

∣∣t=0

H∗t α = LV α.

3.2 Structure-preserving maps (morphisms)

The notion of morphism (structure-preserving map) plays a central role inmany modern branches of mathematics. In this thesis, treating a subject inmathematics and physics, this notion is dominantly present. In particular, inthe process of quantization we want to preserve as much structure as possible,when going from classical to quantum physics. The general study of mor-phisms and of the structures (called objects ) over which they are defined istreated in category theory . Much of the terminology of morphisms, as wellas the intuition underlying them, comes from concrete categories, where theobjects are simply sets with some additional structure, and morphisms arefunctions preserving the structure. For the purposes of this thesis it sufficesto define some particular, often used, types of morphisms. From the contextit should always be clear what the mathematical objects under considerationare. Giving the abstract definition of morphism from category theory is un-necessary here and would only distract the reader from the main goal. This

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subsection is therefore an enumeration of the concrete types of morphismsused in this thesis and their definitions.

We start with:

Definition 3.2.1. (Homomorphism)Given are two groups (G, ∗), (H, ·). A Homomorphism from (G, ∗) to (H, ·)is a function h : G→ H such that for u, v ∈ G it holds thath(u ∗ v) = h(u) · h(v).

From this definition follows:

• h maps the unit element eG of G to the unit element eH of H. Namely:h(eG) = h(eG ∗ eG) = h(eG) · h(eG)⇒ h(eG) = eH .

• h maps inverses to inverses. Namely:h(u) · h(u−1) = h(u ∗ u−1) = h(eG) = eH ⇒ h(u−1) = h(u)−1.

We say that h is compatible with the group structure. In general, the def-inition of homomorphism depends on the type of algebraic structure underconsideration. The common theme is that a homomorphism is a functionbetween two algebraic objects that respects the algebraic structure. Let usnow define:

Definition 3.2.2. (Isomorphism)An Isomorphism is a bijective map f such that both f and its inverse f−1

are homomorphisms.

Definition 3.2.3. (Endomorphism)An endomorphism is a morphism from a mathematical object to itself.An endomorphism of a vector space V is a linear map f : V → V . Anendomorphism of a group G is a homomorphism f : G→ G.

Definition 3.2.4. (Automorphism)An automorphism is an endomorphism which is also an isomorphism. Incase the mathematical objects are groups an automorphism is simply anisomorphism from the group to itself. In case the mathematical objects arevector spaces an automorphism is an invertible linear map from the vectorspace to itself.

The set of all automorphisms is a subset of End(X), which denotes the set ofall endomorphisms of a mathematical objectX, with a group structure, called

19

the automorphism group of X and denoted by Aut(X). The definitionsand their relations are summarized in the diagram below:

automorphism −−−→ isomorphismy yendomorphism −−−→ (homo)morphism.

Here an arrow denotes implication.

We finally define:

Definition 3.2.5. (Diffeomorphism)Given two manifolds M and N , a bijective map f from M to N is called adiffeomorphism if both f : M → N and its inverse f−1 : N →M are differ-entiable (if these functions are r times continuously differentiable f is calleda Cr-diffeomorphism). A diffeomorphism is an isomorphism in the categoryof smooth manifolds. Two manifolds M and N are (Cr-)diffeomorphic ifthere is an (r times) continuously differentiable bijective map f from M toN with an (r times) continuously differentiable inverse.

Two useful notions related to morphisms are the push-forward and pull-back .Push-forward: given a diffeomorphism h : M → N , then objects on M (suchas a function f : M → R or a vector field V : M → TM) can be transferedto N to give a corresponding object on N . This is called push-forward andthe transfering map is often denoted by h∗. For instance, h∗f ≡ f h−1 andh∗V ≡ Dh V h−1 : N → TN .Pull-back: similarly, the pull-back refers to transfering objects on N via h toobjects on M and is denoted by h∗. This is in fact the push-forward of h−1.The difference is however that a pull-back often makes sense when h only isa C∞-map (without insisting that it is a diffeomorphism). For instance, thepull-back of a function g on N is simply h∗g ≡ g h.

3.3 Lie groups and Lie algebras

Definition 3.3.1. A Lie group is a smooth (C∞) manifold G equippedwith a group structure so that the maps µ : G × G → G (x, y) 7→ xy andι : G→ G x 7→ x−1 are smooth.

We quote without proof (for the proof, see [3, p.6]):

Lemma 3.3.1. Let G be a Lie group, and let H ⊂ G be both a subgroupand a smooth submanifold. Then H is a Lie group.

20

Next we come to the concept of isomorphic Lie groups. Let G and H be Liegroups.

(a) A Lie group homomomorphism from G to H is a smooth mapφ : G→ H that is a homomorphism of groups.

(b) A Lie group isomorphism from G to H is a bijective Lie group homo-morphism φ : G→ H whose inverse is also a Lie group homomorphism.

(c) A Lie group automorphism of G is a Lie group isomorphism of Gonto itself.

If φ : G → H is a Lie group isomorphism, then φ is smooth and bijectiveand its inverse is smooth as well. Hence φ is a diffeomorphism.

Before we define the exponential map we first need to define what left in-variant vector fields are. A vector field v ∈ Vect(G) is called left invariant if(lx)∗v = v for all x ∈ G, where lx is the translation map lx : G→ G y 7→ xy,which is a diffeomorphism. Equivalently, v is left invariant, when

v(xy) = Ty(lx)v(y) (x, y ∈ G). (2)

From the above equation with y = e we see that a left invariant vector fieldis completely determined by its value v(e) ∈ TeG. Now

Definition 3.3.2. The exponential map exp = expG : TeG → G is definedby exp(X) = αX(1) where αX is the maximal integral curve with initial pointe of the left invariant vector field vX on G determined by vX(e) = X, i.e.

d

dtαX(t) = vX(αX(t)). (3)

The following lemma, of which the proof can be found in for example[3, p.16], states some properties of the exponential map and at the sametimes justifies its name.

Lemma 3.3.2. For all s, t ∈ R, X ∈ TeG we have

(i) exp(sX) = αX(s)

(ii) exp((s+ t)X) = exp(sX)exp(tX).

Moreover, the map exp: TeG→ G is smooth and a local diffeormorphism at0. Its tangent map at the origin is given by T0exp= ITeG.

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We now come to a very important application of the exponential map. Theproof of the lemma can be found in [3, p.17].

Lemma 3.3.3. Let φ : G → H be a homomorphism of Lie groups. Thenthe following diagram commutes:

Gφ−−−→ HxexpG xexpH

TeGTeφ−−−→ TeH

We now define the notion of Lie algebra. Note that in general this definitiondoes not need to have any connection with Lie groups. However, the factthat there is such a connection in specific cases will become clear in the nextsection.

Definition 3.3.3. A real Lie algebra is a real linear space a equipped witha bilinear map [·, ·] : a× a→ a such that for all X, Y, Z ∈ a we have

(1) [X, Y ] = −[Y,X] (anti-symmetry)

(2) [X, [Y, Z]] + [Y, [Z,X]] + [Z, [X, Y ]] = 0 (Jacobi-identity)

Note that:

• condition (1) may be replaced by the equivalent condition(1’) [Z,Z] = 0 for all Z ∈ a: without loss of generality we can assumeZ = X + Y . Then, using (1),[Z,Z] = [X + Y,X + Y ] = [X,X] + [Y,X] + [Y,X] + [Y, Y ] =[X, Y ] + [Y,X] = [X, Y ]− [X, Y ] = 0,and the claim follows.

• In view of antisymmetry (1), condition (2) may be replaced by theequivalent condition (2’) [[X, Y ], Z] = [X, [Y, Z]]− [Y, [X,Z]]. Anotherequivalent form of the Jacobi identity is given by the Leibniz type rule[X, [Y, Z]] = [[X, Y ], Z] + [Y, [X,Z]].

Definition 3.3.4. Let a, b be Lie algebras. A Lie algebra homomorphismfrom a to b is a linear map φ : a→ b such that

φ([X, Y ]a) = [φ(X), φ(Y )]b (4)

for all X, Y ∈ a.

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3.4 Actions and representations

To understand what a representation is we first need some theory aboutgroup actions.

Definition 3.4.1. Let M be a set and G a group. A (left) action of G onM is a map α : G×M →M such that

(a) α(g1, α(g2,m)) = α(g1g2,m) (m ∈M, g1, g2 ∈ G);

(b) α(e,m) = m (m ∈M).

Instead of the cumbersome notation α we usually exploit the notation g ·mor gm for α(g,m). Then the above rules (a) and (b) become:g1 · (g2 ·m) = (g1g2) ·m, and e ·m = m.

If g ∈ G, then we sometimes use the notation αg for the mapm 7→ α(g,m) = gm, M →M . From (a) and (b) we see that αg is a bijectionwith inverse map equal to αg−1 . Let Bij(M) denote the set of bijections fromM onto itself. Then Bij(M), equipped with the composition of maps, is agroup. According to (a) and (b), the map α : g 7→ αg is a group homomor-phism of G into Bij(M).

Remark 3.4.1. Similarly, a right action of a group G on a set M is de-fined to be a map β : M ×G→ M, (m, g) 7→ mg, such that meG = m and(mg1)g2 = m(g1g2) for all m ∈ M and g1, g2 ∈ G. Notice that these require-ments on β are equivalent to the requirement that the map β∨ : G×M →Mdefined by β∨(g,m) = mg−1 is a left action. Thus, all results for left actionshave natural counterparts for right actions.

As a next step we concentrate on continuous actions. This is most naturallydone for topological groups.

Definition 3.4.2. A topological group is a group G equipped with atopology such that the multiplication map µ : G×G→ G, (x, y) 7→ xy andthe inversion map ι : G×G, x 7→ x−1 are continuous.

Hence it is important to note that a Lie group is in particular a topologicalgroup.

Definition 3.4.3. Let G be a topological group. By a continuous leftaction of G on a topological space M we mean an action α : G ×M → Mthat is continuous as a map between topological spaces. A (left) G-space isa topological space equipped with a continuous (left) G-action.

23

Sets of the form mG (m ∈ M) are called orbits of the action α. Note thatfor two orbits m1G,m2G either m1G = m2G or m1G ∩ m2G = ∅. Thus,the orbits constitute a partition of M . The set of all orbits, called the or-bit space , is denoted by M/G. Let us denote the canonical projectionM →M/G, m 7→ mG by π.

The orbit space X = M/G is equipped with the quotient topology. Thisis the finest topology for which the map π : M →M/G is continuous. Thus,a subset O of X is open if and only if its preimage π−1(O) is open in M .

The action of G on M is called transitive if it has only one orbit, thefull manifold M . In this case the G-space M is said to be a homogeneousspace for G.

We are now ready to define a representation, a notion that is essential duringthis thesis. In the rest of this subsection G will always be a Lie group.

Definition 3.4.4. Let V be a locally convex space. A continuous repre-sentation π = (π, V ) of G in V is a continuous left action π : G× V → V ,such that π(x) : v 7→ π(x)v = π(x, v) is a linear endomorphism of V , forevery x ∈ G. The representation is called finite dimensional if dimV <∞.

Remark 3.4.2. If G is a group, and V a linear space, one defines a repre-sentation of G in V similarly, but without the requirement of continuity.

We quote without proof (for the proof the reader is referred to [3, p.69]):

Lemma 3.4.1. Let (π, V ) be a finite dimensional representation of G. If πis continuous, then π is smooth.

The next definition contains an important property of continuous represen-tations.

Definition 3.4.5. Let π be a representation of G in a linear space V . Byan invariant subspace we mean a linear subspace W ⊂ V such thatπ(x)W ⊂ W for every x ∈ G. A continuous representation π of G in acomplete locally convex space V is called irreducible , if 0 and V are theonly closed invariant subspaces of V .

Besides, the notion of a representation of a Lie group there also exists thenotion of a representation of a Lie algebra.

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Definition 3.4.6. Let g be a Lie algebra. A representation of g in acomplex linear space V is a bilinear map ι × V → V, (X, v) 7→ Xv, suchthat

[X, Y ]v = XY v − Y Xv

for all X, Y ∈ g and v ∈ V . In other words, the map X 7→ X· is a Lie algebrahomomorphism from g into End(V ).

Earlier in this section we already defined what it means for two atlases to beequivalent. We end this section by stating what it means for two represen-tations to be equivalent.

Definition 3.4.7. If (πj, vj)(j = 1, 2) are continuous representations of G incomplete locally convex spaces, then a continuous linear map T : V1 → V2 issaid to be equivariant, or intertwining if the following diagram commutesfor every x ∈ G:

V1T−−−→ V2xπ1(x)

xπ2(x)

V1T−−−→ V2

The representations π1 and π2 are said to be equivalent if there exists alinear isomorphism T from V1 onto V2 which is equivariant.

If the above representations are finite dimensional, then one does not need torequire T to be continuous, since every linear map V1 → V2 has this property.

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4 Coadjoint orbits

-Isaac Newton encrypted his discoveries in analysis in the form of an anagramthat deciphers to the sentence, ‘It is worthwhile to solve differential equa-tions’. Accordingly, one can express the main idea behind the orbit methodby saying ‘It is worthwhile to study coadjoint orbits’.-A.A. Kirillov ([1])

The notion of coadjoint orbits is the main ingredient of the orbit method. Itis also the most important new mathematical object that has been broughtinto consideration in connection with the orbit method.

4.1 The coadjoint representation

Let G be a Lie group. Unless stated otherwise, we will assume G to be amatrix group, that is, a subgroup and at the same time a smooth submanifoldof GL(n,R). Let g = Lie(G) be the associated Lie algebra. Let x ∈ G. Thenthe translation maps lx : G → G y 7→ xy and rx : G → G y 7→ yxare diffeomorphisms from G onto itself, hence they are automorphisms (thenotions of diffeomorphism and isomorphism are identical for Lie groups). Asa consequence the conjugation map Cx = lx r−1

x : G → G y 7→ xyx−1 isan automorphism too, also called an inner automorphism. This map fixesthe neutral element e. So we can define its tangent map at e which is alinear automorphism of TeG = Lie(G) (the last identification will be justifiedlater in this section). Thus, because the automorphisms form a group undercomposition of morphisms, we have TeCx ∈ GL(TeG).

Definition 4.1.1. If x ∈ G we define Ad(x) ∈ GL(TeG) by Ad(x) ≡ TeCx.The map Ad: G → GL(TeG) is called the adjoint representation of G inTeG.

Because here G is a matrix group the Lie algebra g is a subspace of Mat(n,R).Hence the adjoint representation is simply matrix conjugation:

Ad(g)X = g ·X · g−1, X ∈ g, g ∈ G. (5)

Indeed Ad is a representation: for g1, g2 ∈ G,X ∈ g we haveAd(g1)Ad(g2)X =Ad(g1)(g2Xg

−12 ) = g1g2Xg

−12 g−1

1 =g1g2X(g1g2)−1 =Ad(g1g2)X.

Now consider g∗ ≡ f : g → R|f linear, the vector space dual to g. For

26

any linear representation (π, V ) of a group G one can define a dual represen-tation (π∗, V ∗) in the dual space V ∗:

π∗(g) : V ∗ → V ∗ π∗(g) ≡ π(g−1)∗ (6)

where the asterisk in the right-hand side means the dual operator in V ∗

defined by:

〈π(g−1)∗f, v〉 ≡ 〈f, π(g−1)v〉 for any v ∈ V, f ∈ V ∗ (7)

and by 〈f, v〉 we denote the value of the linear functional f on a vector v. Inparticular:

Definition 4.1.2. Let X ∈ g, F ∈ g∗. Then the coadjoint representationAd∗ : G×g∗ → g∗ ofG in g∗ is defined by Ad∗(g) : g∗ → g∗ Ad∗(g) ≡Ad(g−1)∗.So by definition, 〈Ad(g−1)∗F,X〉 = 〈F,Ad(g−1)X〉.

Indeed Ad∗ is a representation:〈Ad∗(g1)Ad∗(g2)F,X〉 = 〈Ad(g−1

1 )∗Ad(g−12 )∗F,X〉

= 〈Ad(g−12 )∗F,Ad(g−1

1 )X〉 = 〈F,Ad(g−12 )Ad(g−1

1 )X〉= 〈F,Ad(g−1

2 g−11 )X〉 = 〈F,Ad((g1g2)−1)X〉

= 〈Ad((g1g2)−1)∗F,X〉 = 〈Ad∗(g1g2)F,X〉.

Now the groups under consideration are matrix groups. What specific formdoes the coadjoint representation take for matrix groups? The space Mat(n,R)has a bilinear form

S : Mat(n,R)×Mat(n,R)→ R (A,B) 7→ tr(AB), (8)

where tr(-) is the operation of taking the trace of a matrix. It is:

• non-degenerate (meaning S(A,B) = 0 ∀B ∈ Mat(n,R) ⇒ A = 0):first we will show tr(ATA) = 0 ⇔ A = 0. Namely, tr

(ATA

)=∑n

i,j=1ATijAji =

∑ni,j=1(Aji)

2 = 0 ⇔ (Aji)2 = 0 ∀i, j ⇔ A = 0. Hence

S(A,B) = tr(AB) = tr(BA) = 0 ∀B ⇒tr(ATA) = 0⇒ A = 0 (matrixwith all entries zero).

• invariant under conjugation: let C ∈Mat(n,R). Then tr(CABC−1) =tr(C−1CAB) =tr(AB). Hence S is invariant under conjugation.

The bilinear form S gives an identification between Mat(n,R) and Mat(n,R)∗

via the map β : A→ S(A,−) ∈Mat(n,R)∗ defined by β(A)(B) = S(A,B).

27

Now we want to show that the space g∗, dual to te subspace g ⊂Mat(n,R)can be identified with the quotient space Mat(n,R)/g⊥, where the sign ⊥means the orthogonal complement with respect to the the form S:

g⊥ = A ∈ Mat(n,R)|S(A, ι(B)) = 0 ∀B ∈ g (9)

where ι is the inclusion map from g to Mat(n,R).

To start with the following table comparing properties of a map and itsdual is good to have in mind:

f f ∗

isomorphism isomorphismzero image zero imageinjective surjectivesurjective injective

Now consider the following short exact sequence of vector spaces (π ι = 0):

Aι→ B

π→ C

with ι an injective and π a surjective map. Then C ∼= B/ι(A). Further-more π ι = 0 ⇒ (π ι)∗ = ι∗ π∗ = 0 (see the second property quoted inthe table above). Hence

A∗ι∗← B∗

π∗← C∗

is a short exact sequence of vector spaces with ι∗ a surjective and π∗ aninjective map (see the third and fourth property quoted in the table above).And A∗ ∼= B∗/π∗(C∗) (where the first property quoted in the table is used).Now define:

A⊥ ≡ φ ∈ B∗|φ(ι(a)) = 0 ∀a ∈ A. (10)

We will show that π∗(C∗) = A⊥ from which follows A∗ ∼= B∗/A⊥. Indeed:

• π∗(C∗) ⊂ A⊥: take f ∈ C∗, i.e. f : C → R, then π∗(f) : B → R. Sofor a ∈ A we have (π∗(f))(ι(a)) ≡ (f π)(ι(a)) = (f π ι)(a) = 0,because π ι = 0. Hence f ∈ A⊥.

• A⊥ ⊂ π∗(C∗): take g ∈ A⊥. Then g ∈ B∗ and g(ι(a)) = 0 ∀a ∈ A.Now for f ∈ C∗ and b ∈ B define f([b]) ≡ g(b), where[b] ≡ (b + ι(a)) ∈ C(a ∈ A), is the equivalence class of an elementb ∈ B. This definition is independent of the choice of b ∈ [b]. Namely,

28

g(b + ι(a)) = g(b) + g(ι(a)) = g(b), because g ∈ A⊥. It is left to checkthat g = π∗(f), because then g ∈ π∗(C∗). Indeed g(b) = f([b]) =f(π(b)) = (f π)(b) ≡ π∗(f)(b).

From π∗(C∗) ⊂ A⊥ and A⊥ ⊂ π∗(C∗) follows π∗(C∗) = A⊥.

Now take A = g, B = Mat(n,R), ι the inclusion map from g to Mat(n,R) andπ the projection map from Mat(n,R) to Mat(n,R)/ι(g). Indeed π ι = 0for this choice, by definition of the quotient space Mat(n,R)/ι(g). It fol-lows by the previous argument that g∗ ∼= Mat(n,R)∗/g⊥. Here is usedS(A, ι(B)) = β(A)(ι(B)) and β(A) ∈ Mat(n,R)∗. Finally, because thebilinear form S is non-degenerate, the map β is an isomorphism. HenceMat(n,R) ∼=Mat(n,R)∗. We arrive at g∗ ∼=Mat(n,R)∗/g⊥ ∼=Mat(n,R)/g⊥.

In practice this quotient space Mat(n,R)/g⊥ is often identified with a sub-space V ⊂Mat(n,R) that is transversal to g⊥ and has complementary di-mension. So we can write Mat(n,R) = V ⊕ g⊥. Let pV be the projectionof Mat(n,R) onto V parallel to g⊥. Hence for matrix groups the coadjointrepresentation takes the form:

Ad∗(g) : g∗ ∼= V → g∗ ∼= V F 7→ pV (gFg−1), g ∈ G = Mat(n,R). (11)

Remark 4.1.1. If we could choose V ∼= g∗ invariant under Ad(g), g ∈ G,then gFg−1 ∈ V ∼= g∗, F ∈ V ∼= g∗, hence we can drop the projection inthe above definition of the coadjoint representation. This can be done if g issemisimple.

We can now define the notion of a coadjoint orbit :

Definition 4.1.3. Given F ∈ g∗. The coadjoint orbit O(F ) is the imageof the map κ : G→ g∗ g 7→Ad∗(g)F .

This section shall be concluded with the introduction of the so-called in-finitesimal version of the coadjoint representation. From the definition of Cxit follows that Ce = IG. Hence Ad(e) = ITeG. Hence, using TIGL(TeG) =End(TeG), we can define the map tangent to Ad:

Definition 4.1.4. The linear map ad: TeG →End(TeG) is defined by ad≡ TeAd

Furthermore,

Definition 4.1.5. We define the Lie bracket [, ] : TeG × TeG → TeG by[X, Y ] ≡ (adX)Y .

29

We will again derive what specific form the Lie bracket takes for matrixgroups. First of all, note that by the chain rule

ad(X) =d

dt

∣∣∣∣t=0

Ad(exp tX), t ∈ R (12)

where is used the exponential map defined in section 3. We already knewthat the adjoint representation takes the form:

Ad(g)X = g ·X · g−1, X ∈ g, g ∈ G

Substituting g = exp tY, Y ∈ g, and differentiating the resulting expressionwith respect to t at t = 0 we obtain, using (8),

[Y,X] = (adY )X =d

dt

∣∣∣∣t=0

exptY X exp−tY = Y X −XY, (13)

the commutator bracket of X and Y .

Hence the Lie bracket:

• is a bilinear map,

• is anti-symmetric: [X, Y ] = XY − Y X = −(Y X −XY ) = −[Y,X],

• satisfies the Jacobi identity:[X, [Y, Z]] + [Y, [Z,X]] + [Z, [X, Y ]] =[X, Y Z − ZY ] + [Y, ZX −XZ] + [Z,XY − Y X] =[X, Y Z]− [X,ZY ] + [Y, ZX]− [Y,XZ] + [Z,XY ]− [Z, Y X] =XY Z−Y ZX−XZY +ZY X+Y ZX−ZXY −Y XZ+XZY +ZXY −XY Z − ZY X + Y XZ = 0, Z ∈ g.

It follows that TeG together with the Lie bracket is a Lie algebra. This jus-tifies the identification of TeG with g, the Lie algebra of G, already usedearlier in this subsection. The Lie bracket is a Lie algebra representation(also called the infitesimal representation associated with the representa-tion of the Lie group) of g in End(g), which follows from the definitionof a Lie algebra representation given in section 3. Namely, for Z ∈ g,[X, Y ]ad(Z) = (XY − Y X)ad(Z) = XY ad(Z)− Y Xad(Z).

We are now ready to define the infinitesimal version of the coadjoint rep-resentation:

30

Definition 4.1.6. Let X, Y ∈ g, F ∈ g∗. Then ad∗ : g × g∗ → g∗ isdefined by ad∗(X) : g∗ → g∗ ad∗(X) ≡ (−ad(X))∗. So by definition,〈(−ad(X))∗F, Y 〉 = 〈F,−ad(X)Y 〉 = 〈F,−[X, Y ]〉 = 〈F, [Y,X]〉.

For matrix groups it takes the form

ad∗(X)F = pV ([X,F ]) for X ∈ g, F ∈ V ∼= g∗. (14)

Again, if g is semisimple, the projection can be dropped in the equationabove.

4.2 The coadjoint orbits of SU(2)

In this subsection we will apply the theory of the coadjoint orbits to theLie group SU(2), that is, we will argue what the coadjoint orbits of SU(2)are. The approach in this subsection will be less formal than in previoussubsections. Because SU(2) is a relatively simple application of the theory itwill give a good intuition for the notion of coadjoint orbits.

4.2.1 Deriving the coadjoint orbits by finding conjugation-invariantfunctions on su(2)∗

First of all, SU(2) is the matrix group defined by

SU(2) ≡ x ∈ Mat(2,C)|x†x = I, det(x) = 1. (15)

From the conditions on x contained in the definition we can derive the formof a matrix in SU(2). Let us first take a general matrix in Mat(2,C). It hasthe form(

α βγ δ

)with α, β, γ, δ ∈ C. By the definition of SU(2) we have that x† = x−1. Atthe level of matrices this implies, using det(x) = αδ − βγ = 1, that(

α∗ γ∗

β∗ δ∗

)=

(δ −β−γ α

)using Cramer’s rule to invert the matrix. Hence it follows thatα∗ = δ, γ∗ = −β, β∗ = −γ and δ∗ = α. So at the level of matrices we canwrite

SU(2) =

(α −β∗β α∗

)∣∣∣∣ (α, β) ∈ C2, |α|2 + |β|2 = 1

.

(16)

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The next step is to calculate the Lie algebra of SU(2), which we will denoteby su(2). We can write x = expεX for X ∈ g, ε ∈ R small and where exp isthe exponential map defined in subsection 3.3. Taylor expanding this expres-sion around ε and keeping terms up to first order in ε we have x = I + εX,so the condition x†x = I translates into

(I+εX)†(I+εX) = I⇔ I+ε(X+X†)+O(ε2) = I⇔ X+X† = 0⇔ X = −X†.

Furthermore the condition det(x) = I translates into

det(x) = explog(det(x)) = exptr log(x) = exptr log(expεX) = exptr(εX) =

expεtr(X) ∼= I + εtrX = I⇔ trX = 0.

Hence,

su(2) = X ∈ Mat(2,C)|X = −X†, trX = 0. (17)

Again, from the conditions on X we can derive the form of a matrix in su(2).Take the same general matrix in Mat(2,C) as before. At the level of matricesthe condition X = −X† becomes(

α βγ δ

)=

(−α∗ −γ∗−β∗ −δ∗

).

Hence it follows that(i) α∗ = −α, (ii) δ∗ = −δ, (iii) − β = γ∗ and (iv) − β∗ = γ.If we write α = a + ib, β = c + id, γ = e + if, δ = g + ih and note that theconditions (iii) and (iv) are equivalent we deduce thata = 0, g = 0, d = f, c = −e.Finally tr(X) = 0 implies that b = −h. We conclude that at the level ofmatrices

su(2) =

(ib c+ id

−c+ id −ib

)∣∣∣∣ b, c, d ∈ R.

(18)

It follows that su(2) is determined in terms of three free real parametersb, c, d and hence is isomorphic to R3.

The Lie algebra su(2) is semisimple. To show this, it suffices to show thatsu(2) has an invariant inner product, since SU(2) is a compact Lie group(see [31]). An inner product 〈, 〉 is invariant under the action of su(2) if, forX, Y, Z ∈ su(2) ([32]),

〈[X, Y ], Z〉 = −〈Y, [X,Z]〉, (19)

32

where [, ] denotes the Lie bracket. Since su(2) is a matrix group, the Liebracket [, ] is just a commutator bracket. Consequently, an invariant innerproduct is given by

〈X, Y 〉 = tr(XY ), (20)

for X, Y ∈ su(2). Indeed,

〈[X, Y ], Z〉 = tr([X, Y ]Z) = tr(Z[X, Y ]) = tr([Z,X]Y ) =tr(Y [Z,X]) = −tr(Y [X,Z]) = −〈Y, [X,Z]〉,

for X, Y, Z ∈ su(2).

By remark 4.1.1. we can now drop the projection pV in equation (7). Further-more, we know by the previous subsection that on matrix groups Mat(n,R)there exists a bilinear form S and the corresponding earlier defined isomor-phism β. This isomorphism β tells us that for SU(2) we have g∗ ∼= g. TakeF ∈ g∗ ∼= g ∼= R3. Now by definition 4.1.3. in order to find the coadjointorbits of SU(2) we look for the points F ′ ∈ g∗ ∼= g ∼= R3 for which

F ′ = gFg−1 for a certain g ∈ SU(2). (21)

Let Q be a funtion Q: su(2)→ C which is conjugation invariant. Then (21)implies that Q(F ′) = Q(gFg−1) = Q(F ) with F ′ ∈ Ω, a coadjoint orbit ofSU(2). Hence Q(Ω) = constant ∈ C. That is, different coadjoint orbits arecharacterized by different constants in the equation above. We have to checkall conjugation invariant functions on su(2) to find all coadjoint orbits ofSU(2).

Take for Q the function Q: su(2) → C, F 7→ tr(F ). Then Q is indeedconjugation invariant: tr(gFg−1) = tr(F ), by cyclicity of the trace. Butfor SU(2) we already know that tr(F ) = 0 for all F ∈ su(2), so in particu-lar for points in Ω, so this gives no information about the specific points in Ω.

Now take for Q the function Q : su(2) → C F 7→ tr((F )2). Indeed thisQ is conjugation invariant:

tr(gFg−1gFg−1) = tr(gF 2g−1) = tr(g−1gF 2) = tr((F )2).

Let us calculate tr((F ′)2) for F ′ ∈ Ω,using the matrix representation wehave for elements in su(2):

tr((F ′)2) = tr

(ib c+ id

−c+ id −ib

)·(

ib c+ id−c+ id −ib

)=

33

tr

(−b2 − c2 − d2 ...

... −b2 − c2 − d2

)= -2(b2 + c2 + d2) = constant.

Note that we can write without loss of generality, constant/(−2) = r2 for acertain other constant r ∈ C. So the condition on b, c, d ∈ R becomes:

b2 + c2 + d2 = r2. (22)

Because the constants b, c, d are real, the expression b2 + c2 + d2 must bereal. Hence r2 must be real too. Therefore, expression (21) tells us thatat least some coadjoint orbits of SU(2) are spheres S2 ⊂ R3 and we have apoint as coadjoint orbit in case b = c = d = 0. We will now make clear whyall coadjoint orbits of SU(2) are two-dimensional spheres or a point (one-dimensional).

Take for Q the function Q : su(2) → C F 7→ tr(FF †). Because F ∈ su(2)it holds that F = −(F )†. Hence tr(FF †) = −tr((F )2). It follows that Qis conjugation invariant and the condition on b, c, d ∈ R corresponding withpoints on the coadjoint orbits becomes the same and we again get a pointand two-dimensional spheres as coadjoint orbits.

Take for Q the function Q: su(2) → C F 7→ tr(FN) for N ∈ Z. Wealready proved tr(F 2) = −2(b2 + c2 + d2) for b, c, d ∈ R. With induc-tion it follows that tr(FN) = (−b2 − c2 − d2)(N−1)/2tr(F ) for N odd andtr((F )N) = (−2)N/2(b2 + c2 + d2)N/2 for N even. So for N odd we get no in-formation about the specific points in Ω, because tr(F ) = 0 for all F ∈ su(2).For N even we can write

tr(FN) = (−2)N/2(b2 + c2 + d2)N/2 = (−2(b2 + c2 + d2))N/2 = (tr(F 2))N/2.

Hence Q is conjugation invariant and the condition on b, c, d ∈ R againbecomes b2 + c2 + d2 = r2 for r ∈ R and we get a point and two-dimensionalspheres as coadjoint orbits.

The reader can convince himself/herself that all conjugation invariant func-tions on su(2) are similar to the ones described above and hence lead tothe same coadjoint orbits: two-dimensional spheres or points. The formal,rigourous proof of the fact that the two-dimensional spheres and a pointexhaust the coadjoint orbits of SU(2) will not be given in this thesis.

34

4.2.2 Deriving the coadjoint orbits by comparing with SO(3)

Another way to see that the coadjoint orbits of SU(2) are two-dimensionalspheres or a point is to establish an isomorphism between the Lie algebrassu(2) and so(3) and then prove from this that the coadjoint orbits of both Liegroups are equivalent. Because SO(3) is the group of rotations in R3 it is thenclear from visual point of view that its coadjoint orbits are two-dimensionalspheres or a point.

Let a Lie group homomorphism φ : G → H be given. Then we know fromthe theory given in subsection 3.3 that the map induced on the Lie algebrasφ : g → h is a (Lie algebra) homomorphism. This induces the followingcommutative diagram:

G× gAd−−−→ gyφ×φ yφ

H × hAd−−−→ h

with Ad the adjoint representation. This diagram is equivalent to the fol-lowing commutative diagram

G× g Ad //

IdG×φ

gφ

''PPPPPPPPPPPPPPPPP

G× hφ×Idg

// H × hAd

// h

with IdG and Idh the identity mappings on G and g respectively. The factthat these diagrams are equivalent is obvious from the fact thatφ× φ = (φ× Id) (Id× φ).

In general, if forG a group and V , W vector spaces the maps α1 : G×V → V ,α2 : G ×W → W are continuous G-representations and a homomorphismφ : V → W is given this induces the commutative diagram:

G× V α1−−−→ VyIdG×φyφ

G×W α2−−−→ W

We then say that φ is a homomorphism of G-representations . In thissituation it is generally true that G-orbits in V are sent to G-orbits in W .

35

So in the particular case considered above we have V = g, W = h, φ = φ,α1 = Ad and α2 = Ad (φ × Id). Indeed α2 is a continuous representationin this case, because the composition of a continuous representation with ahomomorphism is again a continuous representation. It follows that φ is a ho-momorphism of G-representations and G-orbits in g are sent to G-orbits in h.

However, we are interested in the coadjoint orbits, not in the adjoint ones.If φ : g→ h is a homomorphism, then φ∗ : h∗ → g∗ is a homomorphism. Thefollowing commutative diagram is induced:

G× g∗ Ad∗ // g∗

G× h∗φ∗×Idh∗

//

IdG×φ∗OO

H × h∗Ad∗

// h∗

φ∗hhQQQQQQQQQQQQQQQQ

We conclude that φ∗ is a homomorphism of G-representations and G-orbitsin h∗ are sent to G-orbits in g∗.

Now take G = SU(2) and H = SO(3). We will now show that su(2) ∼= so(3).Because su(2) is the algebra of X ∈ Mat(2,C) with X† = −X and tr(X) = 0one can see from this that as a real linear space su(2) is generated by theelements

r1 =

(i 00 −i

), r2 =

(0 1−1 0

), r3 =

(0 ii 0

).

Note that rj = iσj, where σ1, σ2, σ3 are the famous Pauli spin matrices.One readily verifies that r2

1 = r22 = r2

3 = −I and r1r2 = −r2r1‘ = r3 andr2r3 = −r3r2 = r1.

It follows from the above product rules that the commutator brackets aregiven by

• [r1, r1] ≡ ad(r1)r1 = 0, [r1, r2] ≡ ad(r1)r2 = r1r2 − r2r1 = 2r3,[r1, r3] ≡ ad(r1)r3 = r1r3 − r3r1 = r2

1r2 + r23r2 = −Ir2 − Ir2 = −2r2.

• [r2, r1] = −[r1, r2] = −2r3, [r2, r2] = 0, [r2, r3] = r2r3 − r3r2 = 2r1.

• [r3, r1] = −[r1, r3] = 2r2, [r3, r2] = −[r2, r3] = −2r1, [r3, r3] = 0.

36

From this it follows that the endomorphisms ad(rj) ∈ End(su(2)) have thefollowing matrices with respect to the basis r1, r2, r3:

mat(ad r1) =

0 0 00 0 −20 2 0

, mat(ad r2) =

0 0 20 0 0−2 0 0

, mat(ad r3) =

0 −2 02 0 00 0 0

.

The above elements belong to

s0(3) = X ∈ Mat(3,R)|X = −XT,

the Lie algebra of the group SO(3).

If a ∈ R3, then the exterior product map X 7→ a×X, R3 → R3 has matrix

Ra =

0 −a3 a2

a3 0 −a1

−a2 a1 0

with respect to the standard basis e1, e2, e3 of R3. Clearly Ra ∈ so(3).

Write Rj = Rej for j = 1, 2, 3. Then by the above formulas for mat ad (rj)we have

mat(ad rj) = 2Rj (j = 1, 2, 3).

We now define the map φ: SU(2) → GL(3,R) by φ(x) = mat Ad (x),the matrix being taken with respect to the basis r1, r2, r3. Then φ isa homomorphism of Lie groups. The theory in subsection 3.3 gives thatφ(expX) = mat ead X = emat ad X from which we see that φ maps SU(2)einto SO(3). Since SU(2) is connected, we have SU(2) = SU(2)e, so that φis a Lie group homomorphism from SU(2) to SO(3). The tangent map ofφ is given by φ : su(2) → so(3) X 7→ mat ad X. It maps the basis rjof su(2) onto the basis 2Rj of s0(3), hence is a linear isomorphism. Sosu(2) ∼= s0(3).

It follows that φ is a local diffeomorphism at I, hence its image im(φ) con-tains an open neighbourhood of I in SO(3). By homogeneity, im(φ) is anopen connected subgroup of SO(3), and we see that im(φ) = SO(3)e. Thelatter subgroup equals SO(3), by connectedness of the latter group. Fromthis we conclude that φ : SU(2) → SO(3) is a surjective group homomor-phism. Hence SO(3) ∼= SU(2)/ker(φ). The kernel of φ may be computed asfollows. If x ∈ ker(φ), then Ad (x) = I. Hence xrj = rjx for j = 1, 2, 3.

37

From this one sees that x ∈ −I, I. Hence ker(φ) = −I, I. SoSU(2)/±I ∼= SO(3).

So what do we have so far? We have that φ : su(2) → so(3) X 7→ ad(X)is a linear isomorphism, hence φ∗ is a linear isomorphism, hence in the com-mutative diagram above we can replace h∗ = so(3)∗ by g∗ = su(2)∗. So wehave that φ∗ is a homomorphism of SU(2)-representations and SU(2)-orbitsin so(3)∗ ∼= su(2)∗ are send to SU(2)-orbits in su(2)∗ ∼= so(3)∗.

Moreover, we can now show that the SU(2)-orbits in so(3)∗ ∼= su(2)∗ areequivalent to the SO(3)-orbits in so(3)∗. Two points x, y ∈ so(3)∗ lie bothon a SU(2)-orbit if g · x = y, where g ∈ SU(2) and ‘·’ means ‘acting’. Wedenote this by the equivalence relation x ∼SU(2) y. Because φ is a homomor-phism g · x = y ⇒ φ(g) · x = y. The fact that φ(g) ∈ SO(3) thus givesx ∼SU(2) y ⇒ x ∼SO(3) y. Now suppose that the points x, y ∈ so(3)∗ lie bothon a SO(3)-orbit, that is, h · x = y for h ∈ SO(3). As proved earlier, φ issurjective, meaning ∃ g such that φ(g) = h. This implies that g · x = y.Hence x ∼SO(3) y ⇒ x ∼SU(2) y and we are done.

In other words, the coadjoint orbits of SO(3) and SU(2) are equivalent.

38

5 Symplectic structure on coadjoint orbits

-One feature of coadjoint orbits is eye-catching when you consider a few ex-amples: they always have an even dimension. This is not accidental, but hasa deep geometric reason.-A.A. Kirillov ([1, p.4])

5.1 The Kirillov form

Definition 5.1.1. A symplectic manifold (M,ω) is a smooth manifold Mon which there exists a closed nondegenerate differential 2-form ω (called thesymplectic form). In local coordinates:

ω = ωµν(x)dxµ ∧ dxν , dω = 0,

where ωµν is an anti-symmetric, invertible matrix.

In the appendix it is proven that the closedness dω = 0 of the symplecticform ω at the level of the matrix ωµν translates to the condition

∂κωµν + ∂µωνκ + ∂νωκµ = 0,

which is known as the Jacobi identity for symplectic forms .

From the definition it follows that symplectic manifolds should have evendimension. Indeed,

det(ωµν) = det(−(ωµν)T ) = (−1)ndet((ωµν)

T ) = (−1)ndet(ωµν),

with ωµν an n × n-matrix. In the first identity the anti-symmetry of ωµνis used. In the second and third identity properties of the determinant areused. Consequently, if n is odd, det(ωµν) = 0 and ωµν does not satisfy theinvertibility property anymore. Hence a symplectic manifold should be even-dimensional.

The important and beautiful fact that the coadjoint orbits of Lie groups (inour case matrix groups) are symplectic manifolds is encoded in the followingtheorem:

Theorem 5.1.1. On every coadjoint orbit Ω of a matrix group G, thereexists a nondegenerate closed G-invariant differential 2-form BΩ (called theKirillov form) defined by BΩ(F )(X, Y ) ≡ 〈F, [X, Y ]〉 for X, Y ∈ g, F ∈ g∗.

39

Remark 5.1.1. Notice that although this theorem applies only to matrixgroups, the result can also be proven for general Lie groups. The proof of thismore general theorem can be found in for example [2, p.227-230]. Becausein applications in (mathematical) physics the Lie groups are almost alwaysmatrix groups the above theorem suffices for the purposes of this thesis.

Proof. In this proof we will proceed as follows. We will first prove the ex-istence of this differential form. Then we will prove that the differentialform is nondegenerate, G-invariant and closed, respectively. This togetherestablishes the claim of the theorem.

• Existence: Note that Ω is by definition a transitive G-space (beingone orbit). So it follows from the Orbit-Stabilizer theorem (for theproof, see for example [5]) that Ω ∼= G/StabG(F ) for F ∈ Ω. HereStabG(F ) ≡ g ∈ G| Ad∗(g)F = F.

Take X ∈ g ≡ Lie(G). Then we know from the theory in subsection 3.1that X is an equivalence class of curves γ : [−ε, ε]→ G, 0 7→ e, whichwe will denote as [γ]. Let an action of G on a manifold M be given. Wecan now construct a vector field on M . To achieve this we somehowneed to map the interval [−ε, ε] to the manifold M . A natural way to

do this is as follows: [−ε, ε] γ→ G ×M α→ M 0 7→ (e,m) 7→ m whereγ ≡ ιm γ with ιm : G → G×M, g 7→ (g,m), the inclusion map andα the action map. So for a given X ∈ g we can define ξX ∈ Vect(M)by (ξX)m ≡ [t 7→ γ(t) ·m] where ‘·’ means ‘acting on’.

Now take for M the orbit space Ω, for the curve the specific realisationt 7→ exptX with t ∈ R and for the action the coadjoint representation.Then for F ∈ Ω

ξX(F ) =d

dt

∣∣∣∣t=0

Ad∗(etX)F =

d

dt

∣∣∣∣t=0

pV(etXFe−tX

)= pV ([X,F ]). (23)

We will construct a differential two-form on Ω. At the point F ∈ Ωa differential two form gives an alternating bilinear map B : TFΩ ×TFΩ → R. Take a, b ∈ TFΩ. We can write a = ξX(F ), b = ξY (F ) forX, Y ∈ Lie(Ω) = g/Lie(StabG(F )). This follows from the fact that thesurjective map G→ Ω given by the coadjoint representation induces asurjective map on the tangent spaces

40

TeG ≡ g→ TeΩ ≡ g/Lie(StabG(F )) and the fact that these choices fora and b do not depend on the choices for X, Y ∈ Lie(Ω). Namely, forZ ∈ Lie(StabG(F )) we have

ξX+Z(F ) = pV ([X + Z, F ]) = pV ([X,F ]) + pV ([Z, F ]) = pV ([X,F ]) =ξX(F ).

That [Z, F ] = 0, and hence pV ([Z, F ]) = 0, follows from restrictingad∗(Z) to Ω ∼= G/StabG(F ).

We are now in the positon to defineB(a, b) ≡ BΩ(F )(X, Y ) ≡ 〈F, [X, Y ]〉.Indeed this definition does not depend on our choice for X, Y ∈ Lie(Ω):take Z ∈ Lie(StabG(F )). Then

BΩ(F )(Z, Y ) = 〈F, [Z, Y ]〉 = 〈F,ad(Z)Y 〉= 〈ad∗(Z)F, Y 〉 = 〈pV [Z, F ], Y 〉 = 0,

so it follows that,

BΩ(F )(X + Z, Y ) = 〈F, [X + Z, Y ]〉 = 〈F, [X, Y ]〉+ 〈F, [Z, Y ]〉 =〈F, [X, Y ]〉 = BΩ(F )(X, Y ).

Indeed BΩ(F )(X, Y ) defines a differential two-form:

BΩ(F )(X, Y ) = 〈F, [X, Y ]〉 = −〈F, [Y,X]〉 = −BΩ(F )(Y,X).

Hence we have established the existence of the differential two-formBΩ(F )(X, Y ) on the coadjoint orbits of the matrix group G.

• Nondegeneracy: To prove that B(a, b) ≡ BΩ(F )(X, Y ) is nondegen-erate is now easy. We want to prove that for a 6= 0 there exists ab such that B(a, b) 6= 0. This is equivalent to proving that for X /∈Lie(StabG(F )) there exists a Y such that BΩ(F )(X, Y ) 6= 0. TakeY ∈ g. Then

BΩ(F )(X, Y ) = 〈F, [X, Y ]〉 = 〈F,ad(X)Y 〉 = 〈(ad(X))∗F, Y 〉 =〈−(ad∗(X))F, Y 〉 = 〈−(pV [X,F ]), Y 〉.

We already had that [X,F ] = 0 ⇔ X ∈ Lie(StabG(F )). Hence[X,F ] 6= 0⇔ X /∈ Lie(StabG(F )). It follows that 〈−(pV [X,F ]), Y 〉 6= 0and nondegeneracy is established.

41

• G-invariance: BΩ(F )(X, Y ) being G-invariant means thatl∗gBΩ(F )(X, Y ) = BΩ(F )(X, Y ) where l∗g is the pull-back of lg as definedin section 3. In general, a mathematical object α has a natural groupaction defined on it and we call the mathematical object G-invariantif l∗gα = α. If α is a function f we get l∗g(f) : M → R m 7→ f(g ·M)and l∗gf = f ⇔ f(g ·m) = f(m). Here · means ‘acting on’. This gives,using BΩ(F )(X, Y ) ∈ C∞(Ω), that,

l∗gBΩ(F )(X, Y ) = BΩ(F )(X, Y ) = 〈F, [X, Y ]〉,

implying that 〈F, [X, Y ]〉 = 〈Ad∗(g)F, [Ad(g)X,Ad(g)Y ]〉.

So this last equation is the one that needs to be checked. Indeed,

〈Ad∗(g)F, [Ad(g)X,Ad(g)Y ]〉 = 〈Ad∗(g)F, [gXg−1, gY g−1]〉.

Furthermore,

[gXg−1, gY g−1] ≡ gXg−1gY g−1 − gY g−1gXg−1

= gXY g−1 − gY Xg−1 = g[X, Y ]g−1.

Hence,

〈Ad∗(g)F, [gXg−1, gY g−1]〉 = 〈Ad∗(g)F, g[X, Y ]g−1〉 =〈(Ad(g−1))∗F, g[X, Y ]g−1〉 = 〈F,Ad(g−1)(g[X, Y ]g−1)〉 =〈F, g−1g[X, Y ]g−1g〉 = 〈F, [X, Y ]〉.

This gives G-invariance of BΩ(F )(X, Y ).

• Closedness: We have to prove that dBΩ = 0, where the definition ofd is used as given in section 3. For notational convenience we dropthe subscript Ω in BΩ for a moment. The notation c.p. denotes cyclicpermutation in X, Y, Z of the expression on the left hand side of it.

Using the shape of B in local coordinates and taking X, Y, Z ∈ Lie(Ω)gives

dB(X, Y, Z) =

d(∑

i<j Bijdxi ∧ dxj

)(X, Y, Z) =(∑

k

∑i<j ∂kBijdx

k ∧ dxi ∧ dxj)

(X, Y, Z) =∑k

∑i<j ∂kBij(X

kY iZj −XkY jZi + c.p.).

42

We claim that dB(X, Y, Z) = X · B(Y, Z)− B([X, Y ], Z) + c.p.. Hereis used that for a function f and and a vector field v the expres-sion v · f denotes the directional derivative of f in the direction v,that is v · f ≡ vi ∂f

∂xi. Let us check this claim by first calculating

X ·B(Y, Z) + c.p.. We have

X ·B(Y, Z) + c.p. =∑k

Xk ∂

∂Xk

∑i<j

Bij(YiZj − Y jZi) + c.p. =∑

i<j

∑k

Xk(∂kBij)YiZj +XkBij∂kY

iZj +XkBijYi∂kZ

j

−∑i<j

∑k

Xk(∂kBij)YjZi +XkBij∂kY

jZi +XkBijYj∂kZ

i+ c.p. (24)

It follows that the terms containing ∂kBij match with the expressionfor dB(X, Y, Z) found at the beginning. So it is only left to check thatthe terms not containing ∂kBij cancel against −B([X, Y ], Z)+c.p.. Wehave

−B([X, Y ], Z) + c.p. =−∑

i<j Bij([X, Y ]iZj − [X, Y ]jZi) + c.p.

Using [X, Y ] = X i∂iYj − Y i∂iX

j this equals∑k

∑i<j

Bij(−Xk∂kYiZj + Y k∂kX

iZj

+Xk∂kYjZi − Y k∂kX

jZi) + c.p. (25)

Hence the terms not containing ∂kBij in (24) indeed cancel against theterms in (25). This verifies the claim.

Notice that because B(Y, Z) is a C∞-function, its directional derivativealong the vector field X is equal to the Lie derivative LXB(Y, Z) (seesubsection 3.1). The same holds for the cyclic permutations of B(Y, Z).Using the Leibniz rule for the Lie derivative,

X ·B(Y, Z)−B([X, Y ], Z) =(LXB)(Y, Z) +B(LXY, Z) +B(Y, LXZ)−B([X, Y ], Z) =B([X, Y ], Z) +B(Y, [X,Z])−B([X, Y ], Z) = B(Y, [X,Z]).

43

Here is used that LXB = 0: earlier in this proof we showed that Bis G-invariant. Write g = exptA for t ∈ R, A ∈ g, g ∈ G. Then Bbeing G-invariant means l∗g(B) = B with lg the left translation map.

So l∗exp(tA)(B) = B ⇔ ddt

∣∣t=0

l∗exp(tA)(B) ≡ LXA(B) = 0 and we are done.

One can apply similar mathematics on the cyclic permutations of theexpression X ·B(Y, Z)−B([X, Y ], Z). We end up with the three termsB(Y, [X,Z]), B(X, [Z, Y ]), B(Z, [Y,X]). Using the anti-symmetry of Band the Lie bracket, rewriting of these three terms gives:

(1) B(Y, [X,Z]) = −B([X,Z], Y )

(2) B(X, [Z, Y ]) = −B(X, [Y, Z]) = B([Y, Z], X)

(3) B(Z, [Y,X]) = −B(Z, [X, Y ]) = B([X, Y ], Z).

It follows that for X, Y, Z ∈ Lie(Ω) we have (inserting the subscript Ωin BΩ again):

dBΩ(X, Y, Z) = BΩ([X, Y ], Z)−BΩ([X,Z], Y ) +BΩ([Y, Z], X) =〈F, [[X, Y ], Z]〉 − 〈F, [[X,Z], Y ]〉+ 〈F, [[Y, Z], X]〉 =〈F, [[X, Y ], Z]− [[X,Z], Y ] + [[Y, Z], X]〉 =〈F,−[Z, [X, Y ]] + [Y, [X,Z]]− [X, [Y, Z]]〉 =〈F,−[Z, [X, Y ]]− [Y, [Z,X]]− [X, [Y, Z]]〉,

where the anti-symmetry of the Lie bracket is used in the last twoidentities. The last expression is indeed equal to zero by the Jacobiidentity. Hence we have established that BΩ is closed.

It follows that the coadjoint orbits of matrix groups are always even-dimensional.

5.2 The Kirillov form on the coadjoint orbits of SU(2)

This second subsection of section 5 will again be an application of the theorydescribed in the foregoing subsection (as was the case in section 4) and willhopefully give more insight into this theory. Here the explicit ‘local coordi-nates shape’ of the Kirillov form on the physically interesting coajoint orbitsof SU(2) (being the two-dimensional spheres) will be derived.

Before we come to this calculation we make a small excursion to the the-ory of Poisson manifolds and Poisson brackets.

44

Definition 5.2.1. A Poisson manifold is a differentiable mainifold M suchthat the the algebra C∞(M) of smooth functions over M is equipped with abilinear map , : C∞(M)× C∞(M)→ C∞(M) such that ,

(1) is skew-symmetric, i.e. f, g = −g, f,

(2) obeys the Jacobi identity: f, g, h+ g, h, f+ h, f, g = 0,

(3) is a derivation of C∞(M) in its first argument:fg, h = fg, h+ gf, h for all f, g, h ∈ C∞(M).

The map , is called the Poisson bracket.

In local coordinates the Poisson bracket can be written as

f, g ≡ −1

2

∂f

∂xµωµν(x)

∂g

∂xν, (26)

where x ∈M and ωµν(x) is a matrix such that f, g satisfies properties (1)and (2) in the above definition (so in particular it must be an anti-symmetricmatrix by virtue of property 1). Expression (26) also satisfies property (3):

fg, h = −12∂(fg)∂xµ

ωµν(x) ∂h∂xν

=

−f 12∂g∂xµ

ωµν(x) ∂h∂xν− g 1

2∂f∂xµ

ωµν(x) ∂h∂xν

=fg, h+ gf, h,for all f, g, h ∈ C∞(M).

Given a symplectic manifold (M,ω) and f, g ∈ C∞(M). We can definethe Poisson brackets in local coordinates by

f, g ≡ −1

2

∂f

∂xµωµν(x)

∂g

∂xν, (27)

where in this case ωµν is the inverse of the matrix ωµν corresponding to thesymplectic form ω = ωµν(x)dxµ ∧ dxν . The proof that this definition of , does indeed define a Poisson bracket is postponed untill subsection 6.1.

As a consequence, all symplectic manifolds are Poisson manifolds. The op-posite need not be true. This is because on a Poisson manifold, the matrixωµν , defining the Poisson bracket, need not be invertible.

Our procedure to calculate the explicit ‘local coordinate’ shape of the Kir-illov form on the two-dimensional spheres will be to calculate the matrix ωµν

in (26) locally and show that it is invertible. Hence in the SU(2)-case we

45

can calculate the inverse matrix ωµν and hence the corresponding symplecticform ω. By showing that ωµν is closed and SU(2)-invariant we thus find theKirillov form on the two-dimensional spheres, the physically relevant coad-joint orbits of SU(2). We will do the calculation in two often used coordinatesystems: the spherical coordinate system and the stereographic coordinatesystem.

5.2.1 The Kirillov form on the coadjoint orbits of SU(2) in thespherical coordinate system

Let x1, x2, x3 be the standard coordinates in R3. The spheres of radius Rin R3 are given by the equation x2

1 + x22 + x2

3 = R2.

We can cover the sphere by two charts. One chart being the whole sphereminus the northpole (θ = 0), which we denote by U+, the other chart beingthe whole sphere minus the southpole (θ = π), which we denote by U−. Be-cause we calculate the Kirillov form on the two-dimensional sphere in localcoordinates, we have to perform the calculation on both charts U+ and U−.Let us first consider U−.

The well known parametrization of U− is in terms of the parameters φ, θand is given by

x1 = R sin(θ) cos(φ)x2 = R sin(θ) sin(φ)x3 = R cos(θ)0 ≤ φ < 2π, 0 ≤ θ < π.

Indeed,

x21 + x2

2 + x23 =

(R sin(θ) cos(φ))2 + (R sin(θ) sin(φ))2 + (R cos(θ))2 =R2(sin2(θ) cos2(φ) + sin2(θ) sin2(φ) + cos2(θ)) =R2(sin2(θ) + cos2(θ)) = R2.

As a first step towards finding the Kirillov form on U− in the spherical coor-dinate system the Poisson brackets θ, φ, φ, θ, θ, θ and φ, φ need tobe calculated. By formula (26) and in particular the antisymmetry of ωθθ,ωθφ and ωφφ it immediately follows that

θ, θ = 0

φ, φ = 0

θ, φ = −φ, θ.

46

So it is left to calculate φ, θ. The most convenient way to do this is to

first calculate the bracketx1

x2, x3

in terms of the parameters θ, φ and sub-

sequently, to express this bracket in terms of φ, θ.

In subsubsection 4.2.2 we already proved the commutation relations forthe generators of su(2) ∼= R3. For the choice of generators x1, x2, x3 ofR3 ∼= su(2) here we thus have:

[x1, x2] = 2x3, [x1, x3] = −2x2, [x2, x3] = 2x1.

We claim that a bilinear map , : C∞(R3)×C∞(R3)→ C∞(R3) which is aderivation in its first argument and which satisfies the bracket relations

xi, xj = cεijkxk,

for c ∈ R and εijk the Levi-Civita tensor, defines a Poisson bracket:

• Jacobi-identity: For f, g ∈ C∞(R3) we have

f, g =∂f

∂xi

∂g

∂xjxi, xj with i, j ∈ 1, 2, 3, (28)

which easily follows from equation (26) and the chain rule. To check theJacobi identity for general functions in C∞(R3) it is by (28) sufficientto check it for the coordinates x1, x2, x3 on R3. Indeed,

x1, x2, x3+ x2, x3, x1+ x3, x1, x2 =x1, cx1+ x2, cx2+ x3, cx3 = 0.

• Anti-symmetry: follows directly from the definition of the Levi-Civitatensor and (28).

This turns R3 into a Poisson manifold. Moreover, for the Poisson bracketinduced by the Kirillov form the constant c in the bracket relations aboveequals c = 2, since in that case the structure constants of the Poisson bracketrelations and the Lie bracket relations between the generators of su(2) arethe same (without proof). Consequently, for these induced Poisson bracketsit holds that,

x1, x2 = 2x3, x1, x3 = −2x2, x2, x3 = 2x1.

Furthermore, in order to calculatex1

x2, x3

we use that

f(x1, x2), x3 = ∂f∂x1x1, x3+ ∂f

∂x2x2, x3.

47

Consequently,x1

x2, x3

= 1

x2x1, x3 − x1

x22x2, x3 = 1

x2(−2x2)− x1

x22(2x1) =

2(−1− x2

1

x22

)= −2(1 + tan2(φ)) = −2

cos2(φ).

Again, from (26) and the chain rule follows f(φ), g(θ) = ∂f∂φ

∂g∂θφ, θ. So,

on the other hand,x1

x2, x3

= tan(φ), R cos(θ) =

(−1

cos2(φ)

)R sin(θ)φ, θ.

It thus follows that

φ, θ =2

R sin(θ), (29)

which has a singularity at the point θ = 0.

Hence, the matrix ωµν in (26) is given by

ωµν =

(0 2

R sin(θ)

− 2R sin(θ)

0

).

It clearly has a nonvanishing determinant, implying that it is invertible. Theinverse is given by

ωµν =

(0 −1

2R sin(θ)

12R sin(θ) 0

).

Consequently, the symplectic form on U− equals

ω = 0 dφ ∧ dφ+ 0 dθ ∧ dθ − 1

2R sin(θ) dφ ∧ dθ +

1

2R sin(θ) dθ ∧ dφ

= R sin(θ) dθ ∧ dφ. (30)

Indeed, ω is closed: dω = R cos(θ) dθ ∧ dθ ∧ dφ = 0. Hence, in order to showthat ω defines the Kirillov form on U+ in the spherical coordinate system wehave to prove it is SU(2)-invariant.

Since we have characterized our points on the sphere by the coordinatesx1, x2, x3 satisfying x2

1 + x22 + x2

3 = R2 we cannot act on these vectors byelements of SU(2), being 2 × 2 matrices. However, since we proved earlier

48

that SU(2)/±I ∼= SO(3), proving S0(3)-invariance is equivalent to prov-ing SU(2)-invariance of ω. We can of course let SO(3), the rotation groupin 3 dimensions, act on the vectors corresponding to points on the sphere.Since our choice of axes is arbitrary it suffices to prove rotation invarianceby choosing a specific rotation, being the rotation around the x3−axis by aconstant angle φ. The matrix representing this rotation is given by: cos(φ) sin(φ) 0

− sin(φ) cos(φ) 00 0 1

.

Letting this matrix act on the vector (x1, x2, x3)T gives cos(φ) sin(φ) 0

− sin(φ) cos(φ) 00 0 1

R sin(θ) sin(φ)R sin(θ) cos(φ)

R cos(θ)

=

R sin(θ) sin(φ) cos(φ) +R sin(θ) cos(φ) sin(φ)

−R sin(θ) sin(φ) sin(φ) +R sin(θ) cos(φ) cos(φ)R cos(θ)

=

R sin(θ) sin(φ+ φ)

R sin(θ) cos(φ+ φ)R cos(θ)

.

So the transformed coordinates are φ′ = φ+ φ and θ′ = θ. We conclude

ω′ = R sin(θ′) dθ′ ∧ d(φ+ φ) = R sin(θ) dθ ∧ dφ = ω,

showing SU(2)-invariance of ω.

Finally, notice that the calculation on U+ is fully analogous, except thatin this case 0 < θ ≤ π and φ, θ has a singularity at θ = π.

5.2.2 The Kirillov form on the coadjoint orbits of SU(2) in thestereographic coordinate system

Another often used coordinate system to calculate the ‘local coordinate’shape of the Kirillov form on the the two-dimensional spheres is the stereo-graphic coordinate system. To this end, consider the stereographic projection

49

that projects the two-dimensional sphere to the complex plane. This projec-tion is depicted in figure 2.

Figure 2: The stereographic projection. For any point P on the sphere (minus thesouthpole), which we denoted by U−, there is a unique line trough the northpole and P ,and this line intersects the complex plane (which runs trough the southpole of the sphere)in exactly one point z.

The plane x3 = 0 runs trough the southpole of the sphere, which coincideswith the origin of R3. Let M be the sphere minus the southpole (U−). Forany point P on M , there is a unique line trough the northpole and P , andthis line intersects the complex plane x3 = 0 in exactly one point z. Thestereographic projection of P is defined to be this point z. We denote thedistance from the southpole of the sphere to the point z by ρ. The stereo-graphic coordinates z can thus be expressed in terms of the variables φ, ρ.We already know that the points on the sphere can be expressed in terms ofthe variables θ,φ. So if we find an expression for ρ in terms of θ (and the

50

constant radius of the sphere R), this enables us to write z as a function of θand φ (and R). As we shall see, this will be helpful in calculating the Poissonbrackets between the independent coordinates, Re(z) and Im(z), that spanthe complex plane.

To express ρ in terms of θ, consider the triangle in figure 3. We denote

Figure 3: The triangle A lifted from figure 1 and the small rectangular triangle B at the‘top’ of A.

the triangle as a whole by A and the rectangular triangle at the ‘top’ of Aby B. Because both triangles A and B have angle γ and a right angle incommon, their third angle should be equal too (the sum of the angles being180 for every triangle). For triangle B we already know that this third angleequals θ/2. Hence, the third angle of triangle A equals θ/2 too (see figure).It follows that

tan(θ/2) =2R

ρ⇔ ρ = 2R cot(θ/2). (31)

51

Consequently, Re(z) = 2R cot(θ/2) cos(φ) and Im(z) = 2R cot(θ/2) sin(φ).Using Euler’s formula, cos(φ) + i sin(φ) = eiφ, we conclude that

z = 2R cot(θ/2)eiφ, (32)

being the superposition of Re(z) and Im(z).

The next step is to calculate the Poisson brackets Re(z),Re(z), Re(z), Im(z),Im(z),Re(z) and Im(z), Im(z). By formula (26) and in particular theantisymmetry of ωRe(z),Re(z), ωIm(z),Im(z) and ωRe(z),Im(z) it immediatelyfollows that

Re(z),Re(z) = 0

Im(z), Im(z) = 0

Re(z), Im(z) = −Im(z),Re(z).

So it is left to calculate Re(z), Im(z). The most convenient way to do thisis expressing the bracket firstly in terms of z, z and secondly to expressz, z in terms of the brackets θ, φ, φ, θ, θ, θ and φ, φ. The lastfour brackets we already calculated and hence Re(z), Im(z) can be writ-ten as a function of the variables θ, φ. By using the expression of z in termsof θ, φ (and R) we can then write Re(z), Im(z) as a function of z, z (and R).

From the identities Re(z) = 12(z + z) and Im(z) = 1

2i(z − z) it follows

Re(z), Im(z) = 14iz + z, z − z =

14iz, z − 1

4iz, z+ 1

4iz, z − 1

4iz, z = 1

2iz, z.

To find the expression for z, z, we in particular use the fact that the Pois-son bracket is a derivation in its first argument, giving,

z, z = −z, z =− 4R2cot(θ/2)eiφ, cot(θ/2)e−iφ =− 4R2 cot(θ/2)eiφ, cot(θ/2)e−iφ − 4R2eiφcot(θ/2), cot(θ/2)e−iφ =4R2 cot(θ/2)cot(θ/2)e−iφ, eiφ+ 4R2eiφcot(θ/2)e−iφ, cot(θ/2) =4R2 cot2(θ/2)e−iφ, eiφ+ 4R2 cot(θ/2)e−iφcot(θ/2), eiφ+ 4R2eiφ cot(θ/2)e−iφ, cot(θ/2)+ 4R2cot(θ/2), cot(θ/2).

We can now work out the four remaining brackets, usingf(θ, φ), g = ∂f

∂θθ, g+ ∂f

∂φφ, g and equation (28):

• cot(θ/2), cot(θ/2) = 0

52

• e−iφ, eiφ = −ie−iφφ, eiφ = ie−iφeiφ, φ = −e−iφeiφφ, φ = 0

• cot(θ/2), eiφ = −12(1 + cot2(θ/2))θ, eiφ =

12(1 + cot2(θ/2))ieiφφ, θ = (1+cot2(θ/2))ieiφ

R sin(θ)

• e−iφ, cot(θ/2) = 12(1 + cot2(θ/2))ie−iφφ, θ = (1+cot2(θ/2))ie−iφ

R sin(θ).

As a consequence, the equation for z, z becomes

z, z =

0 + 4R2 cot(θ/2)e−iφ( (1+cot2(θ/2))ieiφ

R sin(θ)) + 4R2 cot(θ/2)eiφ( (1+cot2(θ/2))ie−iφ

R sin(θ)) + 0 =

8iR cot(θ/2)sin(θ)

(1 + cot2(θ/2)) = 4iR (1 + cot2(θ/2))2.

Using cot2(θ/2) = zz4R2 it follows that z, z = 4iR

(1 + zz

4R2

)2. Hence,

Re(z), Im(z) = 2R(

1 +zz

4R2

)2

. (33)

Hence, the matrix ωµν in (26) is given by

ωµν =

(0 2R

(1 + zz

4R2

)2

−2R(1 + zz

4R2

)20

).

It clearly has a nonvanishing determinant, implying that it is invertible. Theinverse is given by

ωµν =

(0 − 1

2R

(1 + zz

4R2

)−2

12R

(1 + zz

4R2

)−20

).

Consequently, the symplectic form on U− equals

ω = − 1

2R

(1 +

zz

4R2

)−2

dRe(z) ∧ dIm(z) +1

2R

(1 +

zz

4R2

)−2

dIm(z) ∧ dRe(z)

=1

R

(1 +

zz

4R2

)−2

dIm(z) ∧ dRe(z)

= i1

2R

(1 +

zz

4R2

)−2

dz ∧ dz. (34)

In subsubsection 7.3.2 will be discussed what the shape of the symplecticform on U+ is in terms of the coordinates z and z and what this implies forthe singularity structure of the symplectic form.

53

Remark 5.2.2.1. Although sometimes z and z are treated as independent,they are in fact not. The variables Re(z) and Im(z) are independent. Thisis why we calculated the matrix ωµν with respect to the basis Re(z), Im(z)of the complex plane and only at the very end substituted z and z in theexpression for ω. Notice, however, that the complexifications of z and z areindependent.

Indeed, ω is closed:

dω = i 12R

∂∂z

(1 + zz

4R2

)−2dz ∧ dz ∧ dz + i 1

2R∂∂z

(1 + zz

4R2

)−2dz ∧ dz ∧ dz = 0.

Of course, because the form of ω is independent of the choice of coordi-nates, if ω is closed in the spherical coordinate system, it is closed in anycoordinate system. Similarly, when ω is SU(2)-invariant in the spherical co-ordinate system, it is SU(2)-invariant in any coordinate system. We concludethat equation (33) defines the Kirillov form on the two-dimensional spheresin the stereographic coordinate system.

54

6 Geometric quantization

-The correspondence between a classical theory and a quantum theory shouldbe based not so much on a coincidence between their predictions in the limit~→ 0, as on an analogy between their mathematical structures: the primaryrole of the classical theory is not in approximating the quantum theory, butin providing a framework for its interpretation.-P.A.M. Dirac ([14])

Geometric quantization (abbreviated to GQ) refers to a body of ideas pi-oneered independently by Souriau ([18]) and Kostant ([19]) in the late 60’sand early 70’s. The aim of the GQ programme is to find a way of formulat-ing the relationship between classical and quantum mechanics in a geometriclanguage, as a relationship between symplectic manifolds (classical phasespaces) and Hilbert spaces (quantum phase spaces). The passage from clas-sical to quantum mechanics depends on the introduction into the classicalphase space of an additional geometric structure called a polarization.

I will start this section with a formulation of classical mechanics in termsof symplectic geometry, then discuss some criteria (axioms) which a quan-tization has to satisfy, after which I give a heuristic discussion how to con-struct a prequantum Hilbert space. This is the space corresponding to theprequantization construction. Prequantization is a ‘weak’ version of quan-tization, satisfying only a part of the axioms mentioned above. After this,prequantization is formalized by using the ‘natural’ mathematical objects onwhich one has to apply prequantization. After applying this prequantizationprocedure the Hilbert space is still ‘too big’ and we need to reduce it. Wedo this by means of polarizations, which are explained in the subsequentsubsection. Finally, equipped with all the necessary mathematical tools, weconstruct the Hilbert space.

6.1 The mathematical framework of classicalmechanics

Symplectic geometry is the adequate mathematical framework for describingthe Hamiltonian version of classical mechanics. As such it is also the mostsuitable starting point for a geometrization of the canonical (Dirac) quanti-zation procedure.

The purpose of subsubsection 6.1.1 is to introduce the formalism of sym-plectic geometry. Subsubsection 6.1.2 serves to establish the relation of this

55

formalism with that of classical Hamiltonian mechanics.

6.1.1 A crash course in symplectic geometry

To start with, let us repeat definition 5.1,

Definition 5.1. A symplectic manifold (M,ω) is a smooth manifold Mon which there exists a closed nondegenerate differential 2-form ω (called thesymplectic form). In local coordinates:

ω = ωµν(x)dxµ ∧ dxν , dω = 0,

where ωµν is an anti-symmetric, invertible matrix.

Furthermore, this implies that symplectic manifolds always have even dimen-sion (see section 5.1).

The most important example of a symplectic manifold is the cotangent bun-dle M = T ∗Q of an n-dimensional manifold Q. This is the set of pairs (p, q)where q ∈ Q and p is a differential 1-form at q. It is made into a vector bun-dle (and hence into a manifold) by using as coordinates the set of 2n funtionspa, qb where the qbs are coordinates on Q and the pas are the correspondingcomponents of the differential 1-forms p. pa, qb is called the extension toT ∗Q of the coordinate system qb.

The symplectic structure on M is the differential 2-form ω defined by

ω = dpa ∧ dqa. (35)

Indeed ω defines a symplectic form:

(1) ω is a differential 2-form, because it is of the form (1) for k = 2.

(2) ω is closed. Indeed, define the 1-form θ by

θ = padqa. (36)

From this, one sees that ω = dθ, so it is globally exact and hence closedsince d d = 0 by definition of the exterior derivative.

From this we can now show that expression (35) is invariant underchange of coordinates. Denote the transformed coordinates by pa, qb.

56

Then the transformed and untransformed coordinates are related, usingthe Einstein summation convention, via, ([20, p.181])

pa =∂qb

∂qapb. (37)

It follows that

ω = dθ = d(padqa) = d(pa

∂qa

∂qbdqb) =

d(pbdqb) = dθ = ω.

Hence (35) is invariant under change of coordinates.

(3) ω is nondegenerate. We write ω = ωµν(x)dxµ ∧ dxν = aijdpi ∧ dpj +

bijdqi ∧ dqj + cijdp

i ∧ dqj + dijdqi ∧ dpj. Since aij = bij = 0 ∀ i, j and

cij = 12∀i, j, dij = −1

2∀i, j we get

ωµν =

(0 1

2I

−12I 0

),

where I is the n × n identity matrix. Hence the determinant of ωµνis nonzero. We conclude that ωµν is an invertible and anti-symmetricmatrix.

In typical physical applications, Q is the configuration space of some me-chanical system and T ∗Q is the phase space of the system. The cotangentbundle is a fundamental example since all symplectic manifolds have thisform locally. This follows from Darboux’s theorem, which we state withoutproof (for the proof, see [7, p.7-9]).

Theorem 6.1.1.1. Let (M,ω) be a 2n-dimensional symplectic manifold andlet m ∈M . Then there is a neighbourhood U of m and a coordinate systempa, qb(a, b = 1, 2, ..., n) on U such that ω|U = dpa ∧ dqa.

Consequently, in any symplectic manifold (M,ω), it is possible to find a 1-form θ in some neighbourhood of each point such that ω = dθ; such a 1-formis called a symplectic potential .

The symplectic form ω gives, at each m ∈ M , an isomorphism betweenthe tangent and cotangent spaces of M at m, ωm : TmM → T ∗mM , expressedin local coordinates as

X i 7→ X iωik. (38)

Indeed,

57

• ωm is a homomorphism. Indeed,ωm(X i + Y i) =(X i + Y i)ωik = X iωik + Y iωik = ωm(X i) + ωm(Y i).

• ωm is surjective. Take Yk ∈ T ∗mM . Then Yk = X iωik = ωm(X i) for acertain X i.

• ωm is injective. Suppose X iωik = Y jωjk for certain X i, Y j. Then−ωkiX i = −ωkjY j ⇔ ωlkωkiX

i = ωlkωkjYj ⇔ δliX

i = δljXj

⇔ X l = Y l.

This extends to an isomorphism between TM and T ∗M and between vectorfields and one-forms on M ,

X 7→ i(X)ω = 2ω(X, .),

where i(X) denotes the contraction of a differential form with the vectorfield X, as in (38), i.e. the insertion of X into the first ‘slot’ of a differentialform. The conventions of section 2 are used in applying this contraction. Itis sometimes helpful to think of ω as an antisymmetric metric on M ; then(38) corresponds to ‘lowering the index’ on X.

Therefore, the existence of ω on a symplectic manifold allows us to asso-ciate a vector field Xf to every function f ∈ C∞(M,R), the space of smoothreal-valued functions, via

Xf∼= i(Xf )ω = 2ω(Xf , .) = −df. (39)

In order to justify this statement we have to prove that for every functionf ∈ C∞(M,R) we can establish the identity 2ω(Xf , .) = −df . This meansthat if we write Xf = ξk ∂

∂xkwe can find an explicit expression for the coeffi-

cients ξk. This is the case, as follows from

2ω(Xf , .) = 2ωijdxi ∧ dxj(ξk ∂

∂xk, .) = df(.)⇔

2ξkωkjdxj = ∂jfdx

j ⇔∂jf = 2ξkωkj ⇔ξk = 1

2∂jfω

jk,

where in the last identity the invertibility of ωkj is used. We call a vec-tor field Xf satisfying (39) a Hamiltonian vector field of f .

The Lie derivative of ω along Xf is zero:

58

L(Xf )ω ≡ d(i(Xf )ω) + i(Xf )dω = −ddf = 0,

where the definition of the Hamiltonian vector field (39), the closedness ofthe symplectic form ω and the identity d d = 0 are used respectively. Thisimplies by property (3) of the Lie derivative (see section 3.1) that

∂∂t

∣∣t=0

(φf )∗tω = L(Xf )ω = 0⇔

(φf )∗tω = constant,

where φf : dom(Xf ) ⊂ M × R → M is the flow generated by Xf andt ∈ R. It follows that the flow φf leaves ω invariant.

If (M,ω) is a symplectic manifold and f, g ∈ C∞(M,R) we can globallydefine the Poisson bracket by

f, g ≡ 2ω(Xf , Xg) ∈ C∞(M). (40)

It describes the change of g ∈ C∞(M,R) along Xf (or vice versa), since

f, g = −2ω(Xg, Xf ) = −i(Xf )i(Xg)ω = i(Xf )dg = dg(Xf ) =Xf (g) = L(Xf )g = −L(Xg)f .

In local coordinates expression (40) matches with expression (27). Indeed,

f, g ≡ 2ω(Xf , Xg) = 2ωijdxi ∧ dxj(1

2ωkl ∂f

∂xl∂∂xk

, 12ωmn ∂g

∂xn∂

∂xm) =

2ωij12ωil ∂f

∂xl12ωjn ∂g

∂xn= −1

2∂f∂xjωjn ∂g

∂xn .

Here is used the expression for the Hamiltonian vector field in local coor-dinates (A.1), which is derived in the appendix. As promised, we will nowprove that , in (27) defines a Poisson bracket:

(1) , is a derivation: already proved at the beginning of subsection 5.2.

(2) , is anti-symmetric: follows immediately from anti-symmetry of ωµν ,since ω is a symplectic 2-form.

(3) , satisfies the Jacobi identity: let f, g, h ∈ C∞(M,R) and let xµ bethe local coordinates on M . Using the notation ∂µg ≡ ∂

∂xµg it follows

that

f, g, h = f,−12∂µgω

µν∂νh =−1

2∂σfω

σπ∂π(−12∂µgω

µν∂νh) =

59

14∂σfω

σπ∂π∂µgωµν∂νh+ 1

4∂σfω

σπ∂µg∂πωµν∂νh+ 1

4∂σfω

σπ∂µgωµν∂π∂νh.

Consequently,

f, g, h+ f, g, h+ f, g, h =14∂σfω

σπ∂π∂µgωµν∂νh+ 1

4∂σfω

σπ∂µg∂πωµν∂νh+ 1

4∂σfω

σπ∂µgωµν∂π∂νh+

14∂σgω

σπ∂π∂µhωµν∂νf+ 1

4∂σgω

σπ∂µh∂πωµν∂νf+ 1

4∂σgω

σπ∂µhωµν∂π∂νf+

14∂σhω

σπ∂π∂µfωµν∂νg+1

4∂σhω

σπ∂µf∂πωµν∂νg+1

4∂σhω

σπ∂µfωµν∂π∂νg =

14∂σfω

σπ∂µg∂πωµν∂νh+1

4∂σgω

σπ∂µh∂πωµν∂νf+1

4∂σhω

σπ∂µf∂πωµν∂νg =

14∂µfω

µπ∂νg∂πωνσ∂σh+1

4∂νgω

νπ∂σh∂πωσµ∂νf+1

4∂σhω

σπ∂µf∂πωµν∂νg =

14∂µf∂νg∂σh(ωµπ∂πω

νσ + ωνπ∂πωσµ + ωσπ∂πω

µν) = 0,

where in the second identity is used that the terms containing sec-ond order derivatives of f , g or h cancel each other out by applyingsuitable interchanges of indices (which is allowed since all indices aresummed over). In the third identity a similar interchange of indices isapplied. In the last identity the Jacobi identity for Poisson forms isused, (A.3), proved in the appendix.

We have now established that the pair (C∞(M,R), , ) forms a Lie algebra,since the space of smooth real-valued functions is a linear space and , is abilinear map satisfying anti-symmetry and the Jacobi identity.

A very important identity, relating the Lie algebras of Hamiltonian vectorfields and smooth real-valued functions on M , is

[Xf , Xg] = Xf,g, (41)

where f, g ∈ C∞(M,R) and Xf , Xg, Xf,g are Hamiltonian vector fields onM . On the left hand side, [, ] denotes the Lie bracket of vector fields, whichis defined by,

[X, Y ]b = Xa∂aYb − Y a∂aX

b, (42)

for X, Y ∈ Vect(M). In the appendix identity (41) is proved and from thisthe fact is proved that the space of Hamiltonian vector fields together withcommutator bracket forms a Lie algebra.

Finally, we consider certain submanifolds of symplectic manifolds, which willbecome important in the study of polarizations later in this thesis. Beforeintroducing these submanifolds, let us begin with the following definition.

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Definition 6.1.1.1. A symplectic vector space is a pair (V, ω) in whichV is a 2n (n ∈ Z) dimensional real vector space and ω : V × V → C is abilinear form on V satisfying

• ω(X, Y ) = −ω(Y,X) X, Y ∈ V (anti-symmetry),

• ω(X, .) = 0⇔ X = 0 X ∈ V (non-degeneracy).

With this information we can now define,

Definition 6.1.1.2. A Lagrangian subspace H ⊂ V is a subspace H of asymplectic vector space V with the properties

• dim(H) = 12dim(V ) = n,

• ω(X, Y ) = 0 X, Y ∈ H.

We are now ready to define the certain submanifolds of symplectic manifoldsmentioned previously. Analogously to the linear case,

Definition 6.1.1.3. A Lagrangian submanifoldN ⊂M is an n-dimensionalsubmanifold N of a 2n-dimensional (n ∈ Z) symplectic manifold (M,ω) sat-isfying ω|TN = 0.

To give some examples of Lagrangian submanifolds, let us again considerthe cotangent bundle T ∗Q of an n-dimensional manifold Q. Then Q is aLagrangian submanifold of T ∗Q. Indeed,

(1) Q is a n-dimensional submanifold of T ∗Q: if U are local charts on Q,then U × Rn are local charts on T ∗Q (the argument is analogous tothe one given in section 3 for the tangent bundle). Let φ denote thefunctions φ : U ×Rn → R2n corresponding to the local charts on T ∗Q.It follows that Q∩ (U ×Rn) = U ×0 = φ−1(Rn×0), which provesthe claim.

(2) ω|TQ = 0: the symplectic structure on T ∗Q is given by the differential2-form ω = dpa ∧ dqa. A vector field on the tangent space of T ∗Q canbe written as v = v1,a

∂∂pa

+ v2,a∂∂qa

. On TQ it holds that v1,a = 0 (since

pa = 0 on Q), so a vector field on TQ takes the form v = va∂∂qa

. Hence,

for v, w ∈ TQ, ω(v, w) = (dpa ∧ dqa)(vb ∂∂qb

)(vc∂∂qc

) = 0 − 0 = 0. This

implies ω|TQ = 0.

61

Furthermore, for fixed q ∈ Q, the fibres T ∗qQ are Lagrangian submanifolds ofT ∗Q. In this case v2,a = 0 in the above form for v and T ∗qQ

∼= Rn gives then-dimensionality of T ∗qQ.

Locally, any Lagrangian submanifold N is given by the vanishing of n smoothreal-valued functions fk on M which are in involution, i.e. satisfying,

fk, fl = 0 ∀ k, l. (43)

Indeed, the vanishing of n real-valued functions on M gives n restrictionson M , reducing its dimension by n, leaving a dimension 2n − n = n for N .Besides, fk, fl = 0 ⇔ Xfk(fl) = ωqs ∂fk

∂xq∂fl∂xs

= 0. Since the last equationholds for all k, l this implies that the Hamiltonian vector fields Xfk are tan-gent to

⋂lFl = 0, so that they locally span the tangent bundle TN . Then,

noticing that (43) holds if and only if ω(Xfk , Xfl) = 0, this implies ω|TN = 0.

This concludes our crash course in symplectic geometry. Many results ofmodern symplectic geometry are not mentioned here, and the adventurousreader is referred to the bible of symplectic geometry and classical mechanics,[20], for a detailed account.

6.1.2 Symplectic geometry and classical mechanics

To establish the relation between symplectic geometry and classical mechan-ics, let us start by recalling the formalism of classical Hamiltonian mechanics.In the Hamiltonian formalism, the arena for classical mechanics in the sim-plest mechanical systems is the phase space . This is a 2n-dimensional(n ∈ Z) real linear space, and hence isomorphic to R2n, with coordinatesq1, ..., qn, p1, ..., pn describing the position and the momentum (velocity) ofthe n particles involved. These coordinates are called canonical coordi-nates . The dynamics (or time evolution) of the system is gouverned byHamilton’s equations , given by

d

dtqk =

∂H

∂pkd

dtpk = −∂H

∂qk. (44)

Here H ≡ H(qk, pk), the Hamiltonian , is a real smooth function on phasespace describing the energy of the system.

Typically, H is of the form H = T + V with T ∝ p2 ≡ pkpk (Einstein

62

summation convention used) the kinetic energy and V = V (qk) the poten-tial energy, minus whose gradient describes the forces acting on the particles.

As an example, let us consider the harmonic oscillator in one dimension(n = 1). It is described by the Hamiltonian H = 1

2(p2 + q2). Hamilton’s

equations become

d

dtq = q =

∂H

∂p= p

d

dtp = p = −∂H

∂q= −q.

By substituting q = p into p = −q it follows that q = −q. The solution tothis second order ordinary differential equation is a fimiliar one:

q(t) = A cos(t) +B sin(t) A,B ∈ R, (45)

leading to the characteristic oscillating behaviour. Sinceq(0) = A cos(0) +B sin(0) = A and p(0) = q(0) = −A sin(0) +B cos(0) = Bwe can write (45) as q(t) = q(0) cos(t) + p(0) sin(t).

If H does not depend on time explicitly, the equations of motion (44) implythat H is conserved along every trajectory in phase space. Indeed,

d

dtH =

∂H

∂qkqk +

∂H

∂pkpk

=∂H

∂qk∂H

∂pk− ∂H

∂pk

∂H

∂qk= 0, (46)

where again the Einstein summation convention is used. The evolution ofany other smooth real-valued function on phase space, called an observable,is given by

d

dtf =

∂f

∂qkqk +

∂f

∂pkpk

=∂f

∂qk∂H

∂pk− ∂f

∂pk

∂H

∂qk .(47)

In the example of the harmonic oscillator in one dimension, the conserva-tion of the Hamiltonian along any trajectory in phase space implies thatddtH = d

dt

(12(p2 + q2)

)= 0 ⇔ p2 + q2 = Constant ∈ R. Since Constant ∈ R,

we can write Constant = r2 for some constant r ∈ R. Hence the phase spacetrajectories of the one-dimensional harmonic oscillator are circles of radius

63

r. This is in agreement with the explicit solution of the equations of motion(45), since

q(t)2 + p(t)2 = (q(0) cos(t) + p(0) sin(t))2 + (p(0) cos(t)− q(0) sin(t))2 =q(0)2 cos2(t) + p(0)2 sin2(t) + 2q(0)p(0) cos(t) sin(t)−2q(0)p(0) cos(t) sin(t) + p(0)2 cos2(t) + q(0)2 sin2(t) =q(0)2(cos2(t) + sin2(t)) + p(0)2(cos2(t) + sin2(t)) =q(0)2 + p(0)2 = Constant.

The equations (44), (46), (47) characterize Hamiltonian mechanics. Theyarise naturally in the formalism of symplectic geometry if we look at R2n

as the cotangent bundle T ∗Rn ∼= Rn × Rn of the configuration space Rn,equipped with the symplectic form (35). If we write the symplectic form ωfrom (35) as ωµνdx

µ ∧ dxν , then we already derived in subsubsection 6.1.1that

ωµν =

(0 1

2I

−12I 0

).

By Cramer’s rule, the inverse matrix is given by

ωµν =

(0 −2I2I 0

).

Hence, using the form of a Hamiltonian vector field in local coordinates,(A.1), derived in the appendix, the Hamiltonian vector field Xf of a functionf(q1, ..., qn, p1, ..., pn) equals

Xf =1

2ωµν

∂f

∂xµ∂

∂xν

=0 · ∂f∂pk

∂

∂pk− ∂f

∂qk∂

∂pk+∂f

∂pk

∂

∂qk+ 0 · ∂f

∂qk∂

∂qk

=∂f

∂pk

∂

∂qk− ∂f

∂qk∂

∂pk(48)

. Therefore the Poisson bracket equals

f, g = Xf (g) =∂f

∂pk

∂g

∂qk− ∂f

∂qk∂g

∂pk(49)

. It follows, that the Poisson brackets between the coordinates and momenta(called the canonical Poisson brackets) are

64

qm, ql = ∂qm

∂pk

∂ql

∂qk− ∂qm

∂qk∂ql

∂pk= 0− 0 = 0

pm, pl = ∂pm∂pk

∂pl∂qk− ∂pm

∂qk∂pl∂pk

= 0− 0 = 0

pm, ql = ∂pm∂pk

∂ql

∂qk− ∂pm

∂qk∂ql

∂pk= δkmδ

lk − 0 = δlm.

The functions qk and pl form a so-called complete set of observables. Thismeans that any observable which Poisson commutes (has vanishing Pois-son brackets) with all of them is a constant (which means constant in allcoordinates and momenta). Indeed, suppose that an observable f Poissoncommutes with all coordinates and all momenta. Then

f, ql = ∂f∂pk

∂ql

∂qk− ∂f

∂qk∂ql

∂pk= ∂f

∂pl= 0 ∀l,

f, pl = ∂f∂pk

∂pl∂qk− ∂f

∂qk∂pl∂pk

= − ∂f∂qm

= 0 ∀l,which implies f(q1, ..., qn, p1, ..., pn) = Constant.

Since H, ql = ∂H∂pk

∂ql

∂qk− ∂H

∂qk∂ql

∂pk= ∂H

∂pland H, pl = ∂H

∂pk

∂pl∂qk− ∂H

∂qk∂pl∂pk

= −∂H∂ql

in the formalism of symplectic geometry (44) becomes

(44)⇔ d

dtql = H, ql, d

dtpl = H, pl. (50)

Similarly, since H,H = ∂H∂pk

∂H∂qk− ∂H

∂qk∂H∂pk

, (46) becomes

(46)⇔ H,H = 0. (51)

Finally, since H, f = ∂H∂pk

∂f∂qk− ∂H

∂qk∂f∂pk

, (47) becomes

(47)⇔ d

dtf = H, f = XH(f), (52)

so time evolution in Hamiltonian mechanics is determined by the Hamilto-nian vector field XH of the Hamiltonian H.

A first advantage of this new formulation, is that it makes manifest the forminvariance of the equations of classical mechanics under symplectomorphisms(diffeomorphisms leaving the symplectic form invariant), since the Poissonbracket is defined in terms of the symplectic form. Because canonical trans-formations are a particular type of symplectomorphisms, the form-invarianceof the equations of classical mechanics under canonical transformations (inparticular coordinate transformations) is also manifest.

A second advantage of this formulation is that it generalizes immediatelyto more complicated systems where the configuration space is some curved

65

or compact symplectic manifold. The necessity to consider such more exoticsystems in physics has arisen in recent years in a number of different con-texts, e.g. for the description of internal degrees of freedom and in conformalfield theory.

6.2 From classical to quantum mechanics

As mentioned earlier, the aim of the geometric quantization (GQ) programmeis to formulate the relationship between classical and quantum mechanics ina geometric language. At the classical level the state space is a symplecticmanifold (M,ω) and the observables are smooth real-valued functions on M .At the quantum level, the states of a physical system are represented by raysin a Hilbert space H and the observables by a collection Q of Hermitianoperators on H. Let us explain all this terminology. To start with,

Definition 6.2.1. A Hilbert space H is a complex inner product spacethat is also a complete metric space with respect to the distance functioninduced by the inner product.

Here,

Definition 6.2.2. A complex inner product space is a vector space Vover the field C together with an inner product , i.e. with a map,〈·, ·〉 : V × V → C that, for all vectors x, y, z ∈ V and all scalars a ∈ C,satisfies

• Conjugate symmetrie: 〈x, y〉 = 〈y, x〉,

• Antilinearity in the first argument:〈ax, y〉 = a〈x, y〉,〈x+ y, z〉 = 〈x, z〉+ 〈y, z〉,

• Positive-definiteness: 〈x, x〉 ≥ 0 with equality only for x = 0.

The inner product in this Hilbert space is necessary for the probabilisticinterpretation of the theory and is thus of fundamental importance. Theexistence of this inner product allows us to define a norm , which is the real-valued function ‖x‖ =

√〈x, x〉. This function is real-valued since the inner

product is conjugate symmetric, from which follows 〈x, x〉 = 〈x, x〉. Thedistance function between two points x, y ∈ V is defined in terms of thenorm by d(x, y) = ‖x− y‖. The terminology ‘distance function’ is justifiedby the fact that d(x, y) is symmetric (d(x, y) = d(y, x)), the distance betweenpoints x and y is non-negative (d(x, x) ≥ 0), only zero if x = y, and d(x, y)satisfies the triangle equality (d(x, z) ≤ d(x, y) + d(y, z)). Now,

66

Definition 6.2.3. A complete metric space is a metric space M , i.e. anon-empty set equipped with a distance function (called a metric), that iscomplete, which means that every Cauchy sequence of points in M has alimit that is also an element of M .

Since it is now clear what a Hilbert space is, we can say what rays in a Hilbertspace are, which were the quantum states. Each point (vector) in the Hilbertspace (apart from the origin) corresponds by definition to a so-called purequantum state . Since multiplying this vectors by non-zero complex scalarsleaves the Schrodinger equation invariant, two vectors that differ only by anon-zero complex scalar are considered to correspond to the same state. Inthis sense, each pure state is a ray in the Hilbert space.

Finally,

Definition 6.2.4. A Hermitian operator on a Hilbert space H is a linearmap Q : H → H for which 〈Qx, y〉 = 〈x, Qy〉, for x, y ∈ H and 〈·, ·〉 theinner product on H.

The first and simplest problem of quantization concerns the kinematic rela-tionship between the classical and quantum domains. The kinematic problemis: given M and ω, is it possible to construct H and Q? In the rest of section6, we will shed more light onto this problem. A more subtle problem concernsthe dynamical relationship between the classical and quantum domains. Thisproblem will not be treated in this thesis.

Classically, the space C∞(M,R) of observables has in addition to the Liealgebra structure provided by the Poisson bracket, the structure of commu-tative algebra under pointwise multiplication, since then for f, g ∈ C∞(M,R)and x ∈M , we have:

(fg)(x) = f(x)g(x) = (gf)(x). (53)

It is this property which has to be sacrificed when moving from the classicalto the quantum theory. As already discussed in the introduction, the non-commutative nature of observables is at the heart of phenomena in quantummechnanics.

Denote by S a subalgebra of C∞(M,R) (that is, a subspace of C∞(M,R)which is closed under pointwise multiplication of smooth real-valued func-tions on M). Let us look at quantization as an assignment

Q : S → operators on H f 7→ Q(f) (54)

67

of operators Q(f) on some Hilbert space to classical observables f . Ab-stracting from the analogy between classical mechanics and Schrodinger andHeisenberg quantum mechanics (for example, by using the observations fromthe experiments described in the introduction) the assignment Q has to sat-isfy some requirements. Firstly,

(Q1) R-linearity:

Q(rf + g) = rQ(f) +Q(g) ∀r ∈ R, f, g ∈ S. (55)

Secondly,

(Q2)

[Q(f), Q(g)] = −i~Q(f, g). (56)

Here [·, ·] denotes the Lie bracket of vector fields and ~ is Planck’s constant, aconstant of nature characteristic of quantum effects. It is a very small num-ber (~ ≈ 1.05457162853 × 10−34Joule · second), and for most macroscopicconsiderations the fact that it is not zero can be neglected. Treating ~ as aparameter, and taking the limit ~ → 0 (which is considered to be the limitcorresponding with classical mechanics), one recovers from (Q2) the commu-tative structure of classical mechanics. At the microscopic level, however,effects of order ~ can no longer be neglected and here classical mechanicsneeds to be replaced by quantum mechanics.

In quantum mechanics, the eigenvalues of operators on the Hilbert spaceH correspond to the possible results of measurements, and are hence real.This justifies that the operators Q on Hilbert space are taken to be Hermi-tian, since Hermitian operators have real eigenvalues. Indeed, suppose Q haseigenvalue q ∈ C, that is Qx = qx ∀x ∈ H and let again 〈·, ·〉 denote theinner product on H. Then for x, y ∈ H,

〈Qx, y〉 = 〈x, Qy〉 ⇔ 〈qx, y〉 = 〈x, qy〉 ⇔ q〈x, y〉 = 〈qy, x〉 = q〈y, x〉 = q〈x, y〉⇔ q = q ⇔ q ∈ R.

So thirdly,

(Q3)

〈Q(f)x, y〉 = 〈x,Q(f)y〉 f ∈ S, x, y ∈ H. (57)

68

The three requirements above can be summarized in one single statement,using definition 3.4.6, with H the complex vector space and (S, ·, ·) the Liealgebra, where ·, · denotes the Poisson bracket. That is, the operators inQ form a representation of a subalgebra S of the classical observables. Thisstatement is Dirac’s quantum condition (see [14]) and Dirac formulated itby saying that the Poisson bracket is the classical analogue of the quantumcommutator.

However, the quantum condition does not, in itself, uniquely determine theunderlying quantum system. Furthermore, not every Hilbert space and op-erators that satisfy it represent a ‘physically reasonable’ quantization. Thereare some guiding principles about when a quantization is ‘physically reason-able’. First of all, S must contain the constant functions in C∞(M,R) andthese must be represented in Q by the corresponding multiples of the identityoperator. This is equivalent to:

(Q4)

Q(1) = I, (58)

with 1 the constant function (1 : M → R m 7→ 1) and I the identity oper-ator (I : H → H x 7→ x). This ensures by (Q2) that pa, qa = 1 at theclassical level implies [Q(q), Q(p)] = i~I at the quantum level, required bythe uncertainty principle (see, for example, [29]).

Secondly, requiring only Dirac’s quantum condition and (Q4), the Hilbertspace is still ‘too big’ and we need some irreducibility condition to makeit smaller, which appears to be a natural requirement. In general, we can-not have a one-to-one correspondence between Q and the whole of C∞(M)without making H ‘too large’ and the selection of a subalgebra S for which(Q1), (Q2), (Q3) and (Q4) should hold involves picking out some additionalstructure in M .

One way to phrase this irreducibility condition makes use of the concept ofa complete set of observables introduced in subsubsection 6.1.2. In completeanalogy, a complete set of operators is defined such that the only operatorswhich commute with all the operators from that set are multiples of theidentity operator. The condition is then formulated as:

(Q5) If f1, ..., fk is a complete set of observables, Q(f1), ..., Q(fk) is acomplete set of operators.

69

However, GQ uses another approach to make the Hilbert space irreducible,using so-called polarizations. Let us for a moment put aside the problemof the ‘size’ of H and look at a construction for H and Q where only thesymplectic structure on M is used and where there is an operator in Q forevery classical observable in C∞(M,R) (so not only for observables in S). SoQ works on the whole of C∞(M,R) and satisfies an extension of (Q1)-(Q4)to all f ∈ C∞(M,R). Such an assignment is called a prequantization andshall be treated next.

6.3 A heuristic discussion of prequantization

The prequantization construction is already sophisticated, since the pre-quantization Hilbert space consisting of ‘ordinary’ (square-integrable) scalarfunctions associated with the classical symplectic manifold M , appears notto be the correct prequantum Hilbert space to consider. The (square inte-grable) scalar functions do not have the correct transformation properties.The objects which do have the correct transformation behaviour are calledsections (see section 3). However, since most of the readers of this thesis willbe more familiar with scalar functions than with sections, to gain intuitionfor what prequantization actually is about, it pays to first naively considerthe wrong prequantum Hilbert space consisting of (square-integrable) scalarfunctions. Immediately diving into new formalism would only distract usfrom the main idea of this section.

In the quantization process an important role is played by the identity provedin the appendix,

[Xf , Xg] = Xf,g, (39)

where f, g ∈ C∞(M,R) and Xf , Xg, Xf,g are Hamiltonian vector fields onM . Using definition 3.4.6, with C∞(M,R) the linear space and(C∞(M,R), ·, ·) the Lie algebra, this identity shows that Hamiltonian vec-tor fields give a representation of the classical Poisson bracket algebra byfirst order differential operators on M . To see this, notice that a Hamilto-nian vector field is an element of End(C∞(M,R)) and that the mapφ : C∞(M,R) → End(C∞(M,R)) f 7→ Xf is a Lie algebra homomorphismfrom (C∞(M,R), ·, ·) into End(C∞(M,R)). Indeed,

φ(f, g) = Xf,g = [Xf , Xg] = [φ(f), φ(g)].

The classical phase space M carries a natural measure,

70

εω ≡ (−1)12n(n−1) 1

n!ωn ≡ (−1)

12n(n−1) 1

n!ω ∧ ... ∧ ω︸ ︷︷ ︸n times

, (59)

for n ∈ Z and ω = dp ∧ dq, the symplectic form on T ∗Q with Q a1-dimensional manifold. εω is called the Liouville form .

Let the Hilbert space associated with M be the space L2(M,C) of square-integrable, smooth, complex-valued functions on M , with the inner product

〈φ, ψ〉 =

(1

2π~

)n ∫M

φψεω; φ, ψ ∈ L2(M,C). (60)

Recall that a φ ∈ L2(M,C) is called square-integrable if∫M‖φ‖2 εω is finite.

Here ‖·‖ is the norm on the Hilbert space L2(M,C) induced by the innerproduct 〈·, ·〉.

Let us try to find the assignment Q in (54). Using definition 3.4.1 (and theremark beneath), each classical observable f ∈ C∞(M,R) acts on the setL2(M,C) according to

φ 7→ −i~Xfφ; φ ∈ L2(M,C). (61)

To see this, note that C∞(M,R) is a group under pointwise addition andthat the map Q : f 7→ −i~Xf is a group homomorphism. Indeed, using(A.1), it follows that Q(f + g) = −i~Xf+g = −i~Xf − i~Xg = Q(f) +Q(g).Furthermore, since the map in (61) is clearly linear, it follows that Q is arepresentation of C∞(M,R) (with pointwise addition as group operation) inL2(M,C).

The assignment Q in (54) corresponding with this representation is indeedthe map Q defined above. For simplicity, let us denote the extension of (Q1)-(Q4) to all f ∈ C∞(M,R) just by the same (Q1)-(Q4). Unfortunately, Qcannot be a correct prequantization, since condition (Q4) is not satisfied.Indeed, Q(1) = −i~X1 = 0 6= I. Here 1 and I are the constant function,mapping every element in M to 1 ∈ R and the identity operator definedpreviously.

Trying to remedy this, we propose a second attempt for the assignment Q,

Q : C∞(M,R)→ End(L2(M,C)) f 7→ −i~Xf + fI (62)

71

In this case, we indeed have Q(1) = −i~X1 + 1I = 1I = I. But unfortu-nately, in this case (Q2) is not satisfied, since

[Q(f), Q(g)] = [−i~Xf + fI,−i~Xg + gI] =−~2[Xf , Xg]− i~[Xf , gI]− i~[fI,Xg] + [fI, gI] =−~2Xf,g − i~Xf (g)I + i~Xg(f)I + [f, g]I =−~2Xf,g − 2i~f, gI = −i~(−i~Xf,g + 2f, gI) 6= −i~Q(f, g).

In the third identity, we used the identity (41) and the fact that for φ ∈L2(M,C) the identity [Xf , g](φ) = Xf (gφ) − gXf (φ) = Xf (g)φ + gXf (φ) −gXf (φ) = Xf (g)φ holds. In the fourth identity we used equation (49) andthe commutatitivity of real-valued smooth scalar function under pointwisemultiplication.

Suppose, ω = dθ for some 1-form θ, the symplectic potential. Adding anotherterm, the assignment

Q : C∞(M,R)→ End(L2(M,C)) f 7→ −i~Xf + fI − θ(Xf )I, (63)

does satisfy all conditions (Q1)-(Q4). Indeed,

• (Q1): Let r ∈ R, f, g ∈ C∞(M,R). Then

Q(rf + g) = −i~Xrf+g + (rf + g)I − θ(Xrf+g)I =rXf +Xg + rfI + gI − θ(rXf +Xg)I =rXf +Xg + rfI + gI − rθ(Xf )I − θ(Xg)I = rQ(f) +Q(g).

• (Q2): Noting that real-valued smooth scalar functions commute underpointwise multiplication, for f, g ∈ C∞(M,R),

[Q(f), Q(g)] = [−i~Xf + fI − θ(Xf )I,−i~Xg + gI − θ(Xg)I] =−~2Xf,g − 2i~f, gI + i~[Xf , θ(Xg)I] + i~[θ(Xf )I,Xg] =−~2Xf,g − 2i~f, gI + i~[Xf , θ(Xg)I] + i~[θ(Xf )I,Xg] =−~2Xf,g − 2i~f, gI + i~(Xf (θ(Xg))I −Xg(θ(Xf ))I.

Simplifying further, using

θ(Xf ) = θidxi(Xj

f∂∂xj

) = θiXjfdx

i( ∂∂xj

) = θiXif , it follows,

Xf (θ(Xg))−Xg(θ(Xf ) = X if∂i(θjX

jg)−X i

g∂i(θjXjf ) =

X if (∂iθj)X

jg +X i

fθj(∂iXjg) +X i

g(∂iθj)Xjf +X i

gθj(∂iXjf ) =

2ωijXfi X

gj + θj(X

if∂iX

jg −X i

g∂iXjf = f, g+ θj[Xf , Xg]

j =

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f, g+ θjXjf,g = f, g+ θ(Xf,g).

This indeed leads to

[Q(f), Q(g)] = −~2Xf,g − 2i~f, gI + i~I(f, g+ θ(Xf,g) =−i~(−i~Xf,g − θ(Xf,g) + f, g) = −i~Q(f, g).

• (Q3): Using the expression for the inner product, (60), on L2(M,C),for f ∈ C∞(M,R), φ, ψ ∈ L2(M,C), we have

〈Q(f)φ, ψ〉 =(

12π~

)n ∫M

(−i~Xf (φ) + fφ− θ(Xf )φ)ψεω =(1

2π~

)n ∫M

(φi~X if (

∂∂xiψ) + fφψ − θ(Xf )φψ)εω =(

12π~

)n ∫M

(φ−i~X if (

∂∂xiψ) + φfψ − φθ(Xf )ψ)εω =(

12π~

)n ∫MφQ(f)ψεω = 〈φ,Q(f)ψ〉.

Here, in the second equality we used partial integration in the firstterm of Q(f) (choosing the boundary conditions on φ, ψ ∈ C∞(M,R)such that the ‘boundary term’ appearing after partial integration van-ishes), the identity Xf = X i

f∂∂xi

and the invariance of εω under Xf .In the third equality is used that f , X i

f and θ(Xf ) are real-valued.Furthermore, the hermiticity of the derivative operator is used here.

• (Q4): Q(1) = −i~X1 +1I−θ(X1)I = 1I−0I = I, with 0 the zero mapdifined previously. Indeed, by linearity of θ it follows that θ(X1) = 0,since θ(0) = θ(r0) = rθ(0)⇒ θ(X1) = θ(0) = 0 (r ∈ R).

However, there is still a difficulty with the last assignment Q. It depends onthe choice for the symplectic potential θ. So Q cannot be defined globallyunless ω is exact (see Darboux’ theorem). At first sight, a way to escapethis problem, is by means of so-called gauge-transformations : replace θby θ′ = θ + du for some u ∈ C∞(M,R). Then, ω is invariant under thistransformation, since ω′ = dθ′ = d(θ + du) = dθ + ddu = dθ = ω. Replaceφ ∈ L2(M,C) by φ′ = exp(iu/~)φ. Then the inner-product (and hence thephysics involved, since the inner product is the probability measure) is in-variant under this transformation. Indeed, for φ, ψ ∈ L2(M,C),

〈exp(iu/~)φ, exp(iu/~)ψ〉 =(

12π~

)n ∫M

exp(iu/~)φ exp(−iu/~)ψεω =(1

2π~

)n ∫M

exp(iu/~)φ exp(−iu/~)ψεω = 〈φ, ψ〉.

The transformed quantization assignment equals

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−i~Xfφ′ − θ′(Xf )φ

′ + fφ′ =−i~Xf (exp(iu/~)φ)− (θ + du)(Xf ) exp(iu/~)φ+ f exp(iu/~)φ =−i~ exp(iu/~)Xf (φ)− i~(i/~) exp(iu/~)Xf (u)φ− θ(Xf ) exp(iu/~)φ−Xf (u) exp(iu/~)φ+ f exp(iu/~)φ =−i~ exp(iu/~)Xf (φ)− θ(Xf ) exp(iu/~)φ+ f exp(iu/~)φ =exp(iu/~)(−i~Xf (φ)− θ(Xf )φ+ fφ).

Hence Q(f) becomes independent of the choice of symplectic potential.

Another problem remains: u is determined by θ and θ′ only up to a constant,since d(u + Constant) = du + dConstant = du. So given θ, θ′, and φ, thereis an ambiguity in the overall ‘phase’ of φ′:

φ′ = exp(iu/~)φ = exp(i(u+Constant)/~)φ = exp(iu/~) exp(iConstant/~)φ.

To take this into account, and to put the prequantization construction ona more rigorous foundation, Q(f) must not act on functions in L2(M,C)(subject to gauge-transformations), but on the sections of a Hermitian linebundle-with-connection.

6.4 Formalizing the discussion

A natural way to continue is to explain what a Hermitian line bundle withconnection is. Furthermore, for a Hermitian line bundle with connectionto exist some condition needs to be satisfied. In order to understand thiscondition one needs to have some knowledge of Cech cohomology. Thissubsection serves to explain the new mathematical luggage needed in orderto put the prequantization construction on a more rigorous foundation.

6.4.1 Putting more structure on vector bundles

To start with, noting that C is a vector space,

Definition 6.4.1.1. A line bundle is a vector bundle with K = C and fibredimension 1.

Let X be a line bundle over the real manifold M . Recall (see section 3) thata section over U of a line bundle X is a map s : U ⊂ M → X satisfyingpr(s(m)) = m ∀m ∈ U . We denote the set of all smooth sections over Uof a line bundle X with ΓX(U). It is a vector space, where the scalars areC-valued functions on U .

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Denote by Vect(M,C) the space of smooth complex vector fields (so multi-plication by complex numbers is included in the scalar multiplication) on Mand by C∞(M,C) the set of smooth complex-valued funtions on M . Nowdefine,

Definition 6.4.1.2. A connection ∇ on X is a map ∇ : Vect(M,C) →End(ΓX(M)) that assigns to each X ∈ Vect(M,C) an operator ∇X onΓX(M) such that

(a) ∇fX+Y = f∇X +∇Y ; f ∈ C∞(M,C), X, Y ∈ Vect(M,C),

(b) ∇X(fs) = X(f)s+ f∇Xs; X ∈ Vect(M,C), s ∈ ΓX(M),

(c) ∇X(s+ s′) = ∇X(s) +∇X(s′); X ∈ Vect(M,C), s, s′ ∈ ΓX(M).

A line bundle equipped with a connection is called a line bundle-with-connection. We still need more structure on the line bundle. Let m ∈ M ,and let s, s′ ∈ ΓX(M).

Definition 6.4.1.3. A Hermitian structure on a line bundle is an innerproduct〈·, ·〉 : (m × C)× (m × C) ∼= C× C→ Rs(m)× s′(m) 7→ 〈s, s′〉(m) ≡ 〈s(m), s′(m)〉in each fibre that is smooth in the sense thatH : X → R (m, s(m)) 7→ 〈s, s〉(m) is smooth.

A connection ∇ and a Hermitian structure 〈·, ·〉 are called compatible iffor every X ∈ Vect(M,C) and for every s, s′ ∈ ΓX(M) we have

X〈s, s′〉 = 〈∇Xs, s′〉+ 〈s,∇Xs

′〉. (64)

Here X denotes the formal complex conjugate of the complex vector fieldX. The formal complex conjugate of X satisfies by definition the followingrules,

• X + Y = X + Y with X,Y vector fields.

• λX = λX with X a vector field, λ ∈ C and λ the complex conjugateof λ.

A line bundle equipped with a connection and a Hermitian structure whichare compatible is called a Hermitian line bundle-with-connection .

Let X be a line bundle-with-connection ∇ over M .

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Definition 6.4.1.4. The curvature 2-form of ∇ is the complex differential2-form Ω on M determined by

Ω(X, Y )s =i

2([∇X ,∇Y ]−∇[X,Y ])s; X, Y ∈ Vect(M), s ∈ ΓX(M). (65)

Here [∇X ,∇Y ] = ∇X∇Y −∇Y∇X is a commutator bracket. Indeed Ω definesa differential 2-form:

(1) Ω is anti-symmetric in X, Y ∈ Vect(M), since

Ω(X, Y ) = i2([∇X ,∇Y ]−∇[X,Y ]) = i

2(−[∇Y ,∇X ]−∇−[Y,X]) =

i2(−[∇Y ,∇X ] +∇[Y,X]) = −Ω(Y,X).

In the third equality property (a) of the connection is used.

(2) Ω is bilinear in X, Y ∈ Vect(M). Suppose X = f1X1 +X2 forf1 ∈ C∞(M,R), X1, X2 ∈ Vect(M). Then, using properties (a) and (b)of the connection,

[∇X ,∇Y ]s = [∇f1X1+X2 ,∇Y ]s = [f1∇X1 +∇X2 ,∇Y ]s =[f1∇X1 ,∇Y ]s+[∇X2 ,∇Y ]s = f1∇X1(∇Y s)−∇Y (f1∇X1s)+[∇X2 ,∇Y ]s =f1∇X1(∇Y s)− f1∇Y (∇X1s)− Y (f1)∇X1s+ [∇X2 ,∇Y ]s =f1[∇X1 ,∇Y ]s+ [∇X2 ,∇Y ]s− Y (f1)∇X1s.

And,

∇[X,Y ]s = ∇[f1X1+X2,Y ]s = ∇[f1X1,Y ]s+∇[X2,Y ]s =∇f1[X1,Y ]s+∇−Y (f1)X1s+∇[X2,Y ]s = f1∇[X1,Y ]s−Y (f1)∇X1s+∇[X2,Y ]s.

In the third equality is used that [f1X1, Y ] = f1[X1, Y ] − Y (f1)X1

is satisfied by the Lie bracket for vector fields. Consequently,

Ω(X, Y )s =i2(f1∇[X1,Y ]s−Y (f1)∇X1s+∇[X2,Y ]s−f1∇[X1,Y ]s+Y (f1)∇X1s−∇[X2,Y ]s) =f1Ω(X1, Y ) + Ω(X2, Y ).

Hence, Ω is linear in its first argument. That it is linear in its sec-ond argument is proved fully analogously.

Furthermore, Ω is also linear in s over C∞(M,C). Firstly, using property (c)of the connection, for f1 ∈ C∞(M,C), s1, s2 ∈ ΓX(M),

76

[∇X ,∇Y ](f1s1 + s2) = ∇X(∇Y (f1s1))−∇Y (∇X(f1s1)) + [∇X ,∇Y ]s2 =∇X(Y (f1)s1 + f1∇Y s1)−∇Y (X(f1)s1 + f1∇Xs1 + [∇X ,∇Y ]s2 =X(Y (f1))s1 + Y (f1)∇Xs1 +X(f1)∇Y s1 + f1∇X(∇Y s1)− Y (X(f1))s1

−X(f1)∇Y s1 − Y (f1)∇Xs1 − f1∇Y (∇Xs1) + [∇X ,∇Y ]s2 =f1[∇X ,∇Y ]s1 + [∇X ,∇Y ]s2 + [X, Y ](f1)s1.

And,

∇[X,Y ](f1s1 + s2) = [X, Y ](f1)s1 + f1∇[X,Y ]s1 +∇[X,Y ]s2.

It follows that,

Ω(X, Y )(f1s1 + s2) =i2(f1[∇X ,∇Y ]s1 + [∇X ,∇Y ]s2 + [X, Y ](f1)s1

− [X, Y ](f1)s1 − f1∇[X,Y ]s1 −∇[X,Y ]s2) =f1Ω(X, Y )s1 + Ω(X, Y )s2.

Let (U, ψ) be a local trivialization of X. Define the map

1 : U ⊂M → X m 7→ ψ(m, 1(m)). (66)

where 1 ∈ C∞(U,C) denotes the identity map 1 : U → C m 7→ 1. The map(68) is clearly a section on U ⊂ M , since the image of ψ|m is pr−1(m). Itfollows that pr(s(m)) = m, as was to be shown. It is called the unit section.Let X ∈ Vect(U). Let β be the complex 1-form defined on U by

β(X)1 = i∇X 1. (67)

Notice that one can always define such a 1-form β in this way, since, firstof all, both sides of the equation are linear in X. On the one hand, β is a1-form and so by definition linear in X. On the other hand, the connectionis by definition linear in X by property (a) of the connection. Furthermore,every function can be written as the composition of a 1-form and a vector field(by definition) and every section can be written as a complex-valued functiontimes some other section (since ΓX(M) is a vector space with complex-valuedfunctions as scalars). β is called the potential 1-form for ∇. We claim thata general section s on U ⊂ M can be written as a complex-valued functiontimes the unit section. Indeed a general section s on U ⊂M can be writtenas

s : U ⊂M → X m 7→ ψ(m, f(m)), (68)

77

where f ∈ C∞(U,C) is a general smooth complex-valued function. Since1(m) = 1 ∈ C ∀m ∈ U is a complex basis vector for the one-dimensional vec-tor space C, it follows that 1 is a basis vector for the complex one-dimensionalvector space ΓX(M). The claim follows. It follows that we can write the con-nection of a general section s as,

∇Xs = ∇X

(f 1)

= X(f)1 + f∇X 1 =

X(f)1 + f(−iβ(X)1

)= X(f)1− iβ(X)s. (69)

With expression (69) we are able to show that the curvature 2-form of ∇is a closed form. Writing out the right-hand side of definition (65) forX, Y ∈ Vect(U), s ∈ ΓX(U), using (69) and (67), gives

i2([∇X ,∇Y ]−∇[X,Y ])s = i

2(∇X∇Y −∇Y∇X −∇[X,Y ])s =

i2∇X(Y (f)1−iβ(Y )s)− i

2∇Y (X(f)1−iβ(X)s)− i

2([X, Y ](f)1−iβ([X, Y ])s) =

i2X(Y (f))1 + i

2Y (f)∇X 1 + i

2(−iX(β(Y ))s) + i

2(−iβ(Y )∇Xs)− i

2Y (X(f))1

− i2X(f)∇Y 1− i

2(−iY (β(X))s)− i

2(−iβ(X)∇Y s)

− i2([X, Y ](f)1− iβ([X, Y ])s) =

i2Y (f)∇X 1 + i

2(−iX(β(Y ))s) + i

2(−iβ(Y )∇Xs)− i

2X(f)∇Y 1

− i2(−iY (β(X))s)− i

2(−iβ(X)∇Y s)− i

2(−iβ([X, Y ])s) =

12X(β(Y ))s− 1

2Y (β(X))s− 1

2β([X, Y ])s+ i

2Y (f)(−iβ(X)1)− i

2X(f)(−iβ(Y )1)

+12β(Y )X(f)1− 1

2β(X)Y (f)1− i

2β(Y )β(X)s+ i

2β(X)β(Y )s =

12X(β(Y ))s− 1

2Y (β(X))s− 1

2β([X, Y ])s.

In the fourth equality is used that X(Y (f)) − Y (X(f)) = [X, Y ](f) forX, Y ∈ Vect(U), f ∈ C∞(U,C). As a next step to establish the closedness ofthe curvature 2-form Ω, we prove that

dβ(X, Y )s = 12X(β(Y ))s− 1

2Y (β(X))s− 1

2β([X, Y ])s,

the expression with which the derivation above ended. Denote the set oflocal coordinates on the n-dimensional manifold B by xi1≤i≤n . WritingX = X i ∂

∂xifor X ∈ Vect(U) and β = βjdx

j for the potential 1-form β, itindeed follows that

dβ(X, Y ) = d(βjdxj)(X, Y ) = (dβj ∧ dxj)(X, Y ) =

(∂βj∂xidxi ∧ dxj)(X i ∂

∂xk, Y m ∂

∂xm) = 1

2

∂βj∂xi

(X iY j − Y iXj) =12(X i ∂βj

∂xiY j − Y i ∂βj

∂xiXj) =

12(X i ∂

∂xi(βjY

j)− βjX i ∂∂xiY j − Y i ∂

∂xi(βjX

j) + βjYi ∂∂xiXj) =

12(Xβ(Y )− Y β(X)− βj[X, Y ]j) = 1

2X(β(Y ))− 1

2Y (β(X))− 1

2β([X, Y ]).

78

It follows that on every neighbourhood U ⊂ M we have Ω = dβ. Hence onevery U ⊂M the identity dΩ = (d d)β = 0 holds, hence Ω is closed.

Another ‘tool’ we will need in the next section are so-called transition func-tions. Let (U1, ψ1) and (U2, ψ2) be two local trivializations of M and takethe intersection U1 ∩ U2 non-empty.

Definition 6.4.1.5. A transition function between (U1, ψ1) and (U2, ψ2)is the function c12 : U1 ∩ U2 → C defined by

ψ2|(U1∩U2)×C(m, z) = c12(m)ψ1|(U1∩U2)×C(m, z). (70)

Notice that this function is well defined, since ψ1(m, z) and ψ2(m, z) bothtake values in the fibre over m. Indeed, by definitionψ1,2|m : m × C → Xm, so that ψ1,2(m, z) = ψ1,2|m(z) are values in Xm.Since the Xm

∼= C are by definition vector spaces over C, it follows that itselements can be multiplied by complex numbers, such as c12(m).

We quote without proof, that it is always possible to find a collection (Uj, ψj)of local trivializations of X such that Uj is a contractible open cover of M(which means that each Uj and each finite intersection of Ujs is contractibleto a point). Denote the unit section determined by (Uj, ψj) by 1j. Thenwe know from previous discussions that any s ∈ Γ(M) can be written assj = f 1j on Uj for some f ∈ C∞(Uj,C). In particular, on each nonemptyUj ∩ Ui ≡ Uji,

1i|Uji(m) = cji(m)1j|Uji(m), (71)

where cji is the transition function between (Uj, ψj) and (Ui, ψi). Indeed, cjibeing a transition function means by definition that it satisfies the equationψi|Uji×C(m, z) = cji(m)ψj|Uji×C(m, z). It follows that

1i|Uji = ψi|Uji×C(m, 1) = cji(m)ψj|Uji×C(m, 1) = cji(m)1j|Uji(m),

as stated by (71).

Whenever Uk ∩ Uj ∩ Ui ≡ Ukji is nonempty, the transitions functions satisfythe relations

cij = (cji)−1, (72)

ckj|Uikj(m)cji|Uikj(m)cik|Uikj(m) = 1. (73)

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Notice that (cji)−1 denotes the inverse of (cji) with respect to pointwise multi-

plication. First we prove property (72). Note that cji satisfies ψi|Uji×C(m, z) =cji(m)ψj|Uji×C(m, z) and cij satisfies ψj|Uij×C(m, z) = cij(m)ψi|Uij×C(m, z).Consequently, cij = (cji)

−1 is smooth (since ψi, ψj are smooth) and

ψi|Uji×C(m, z)ψj|Uij×C(m, z) = cji(m)cij(m)ψi|Uij×C(m, z)ψj|Uji×C(m, z)⇔cji(m)cij(m) = 1⇔ cij = (cji)

−1.

Here is used that Uij = Uji. To prove relation (73) we first construct func-tions cji : Uji×C→ Uji×C from the original complex-valued scalar functionscji : Uji → C. This can be done by means of the map

cji : Uji × C→ Uji × C, (m, z) 7→ (m, cji(m)z) (74)

which is well-defined, since m ∈ Uji and cji(m)z ∈ C. Notice that this mapis a diffeomorphism,

• cji is smooth: This follows directly from the definition, since cji issmooth.

• The inverse cji−1 exists with respect to composition and is smooth: the

inverse map is given bycji−1 : Uji × C→ Uji × C (m, z) 7→ (m, c−1

ji (m)z). Indeed,

(cji cji−1)(m, z) = cji(m, c−1ji (m)z) = (m, cji(m)c−1

ji (m)z) = (m, z).

Smoothness follows from the fact that c−1ji = cij is smooth.

• cji is bijective: it is surjective, since the expression cji(m)z spans C. Forinjectivity take cji(m

′, z′) = cji(m, z)⇔ (m′, cji(m′)z′) = (m, cji(m)z).

This implies that m = m′ and so cji(m)z′ = cji(m)z ⇔ z = z′. Injec-tivity follows.

Furthermore, the restrictioncji|m×C : m × C ∼= C → m × C ∼= C z 7→ cji(m)z is a linear isomor-phism,

• cji|m×C is linear: all linear maps from C to C are classified by mapsof the form f : C → C z 7→ cz where c ∈ C. Indeed cji(m) ∈ C, solinearity follows.

• cji|m×C is bijective: This follows directly from the discussion above.

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• cji|m×C is a homomorphism under addition, sincecji|m×C(z1 + z2) = cji(z1 + z2) = cjiz1 + cjiz2 =cji|m×C(z1) + cji|m×C(z2) for z1, z2 ∈ C.

From this discussion it follows that we can write

cji = (ψ−1i ψj)|Uij×C : Uij × C

ψj |Uij×C−−−−−→ pr−1(Uij)

ψ−1i |pr−1(Uij)

−−−−−−−−→ Uij × C.Consequently,

(ckj cji cik)|Uijk×C = (ψ−1j ψk)|Uijk×C(ψ−1

i ψj)|Uijk×C(ψ−1k ψi)|Uijk×C =

(ψ−1k ψi ψ

−1i ψj ψ−1

j ψk)|Uijk×C = 1ij|Uijk×C,

where 1ij : Uij × C→ Uij × C (m, z) 7→ (m, z). Using (74) we derive

(ckjcjicik)|Uijk×C(m, z) = (m, ckj|Uijk(m)cji|Uijk(m)cik|Uijk(m)z) = (m, z)⇔ckj|Uijk(m)cji|Uijk(m)cik|Uijk(m) = 1,

which proves property (73).

6.4.2 Cech cohomology and the de Rham isomorphism

As stated before, for a Hermitian line bundle-with-connection to exist a con-dition needs to be satisfied, the so-called integrality condition. This is for-mulated in terms of elements of the Cech cohomology group. In this subsub-section it is therefore explained what Cech cohomology is and the related deRham isomorphism will be discussed.

Let M be a manifold and let U = Ui be an open cover of M , where i ∈ I,some index set.

Definition 6.4.2.1. A p-cochain Cochain relative to U is a collection f =fi1i2...ip+1 of functions, of some specified type (i.e. smooth, locally constant,holomorphic) with the properties

(CO1) fi1i2...ip+1 is defined on Ui1 ∩ Ui2 ∩ ... ∩ Uip+1,

(CO2) f contains just one function fi1i2...ip+1 for each ordered set of p + 1indices for which Ui1 ∩ Ui2 ∩ ... ∩ Uip+1 is nonempty,

(CO3) fi1i2...ip+1 = f[i1i2...ip+1] ≡ 1(p+1)!

∑σ fσ(i1)σ(i2)...σ(ip+1). Here σ denotes a

permutation of the set of indices i1i2...ip+1.

Furthermore,

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Definition 6.4.2.2. Let Ui0 ∈ U and let Ui0 ∩Ui1 ∩ ...∩Uip+1 be nonempty.Then the coboundary operator Coboundary operator δ is defined oncochains by

δf = (δf)i0...ip+1

(δf)i0...ip+1 = (p+ 2)

p+1∑l=0

(−1)l(fi0...il...ip+1)|Ui0∩...∩Uip+1

(75)

Here il denotes omitting the l-th index of fi0...il...ip+1 .

The coboundary operator δ takes p-cochains into (p+ 1)-cochains. Indeed,

• (δf)i0i1...ip+1 is defined on Ui0 ∩ Ui1 ∩ ... ∩ Uip+1 : follows directly fromthe definition (75),

• δf contains just one function (δf)i0i1...ip+1 for each ordered set of p+ 2indices for which Ui0∩Ui1∩...∩Uip+1 is nonempty: follows from definition(75) and the fact that f contains just one function fi1i2...ip+1 wheneverUi1 ∩ Ui2 ∩ ... ∩ Uip+1 is nonempty,

• (δf)i0i1...ip+1 = (δf)[i0i1...ip+1]: Indeed,(δf)[i0i1...ip+1] ≡ 1

(p+2)!

∑σ sign(σ)(δf)σ(i0)σ(i1)...σ(ip+1) =

1(p+2)!

(p+ 2)!(δf)i0i1...ip+1 = (δf)i0i1...ip+1 .

Definition 6.4.2.3. A p-cochain is called a p-cocycle if δf = 0 and ap-coboundary if f = δg for some (p− 1)-cochain g.

One can prove that δ2 = 0, which implies that every p-coboundary is also ap-cocycle. The only important case we consider later on in this thesis willbe when p = 2 (i.e. that 2-coboundaries are also 2-cocycles). So for thepurposes of this thesis it suffices to show δ2 = 0 for p = 2. A 2-cochain is acollection of functions fi1i2i3 of some specified type satisfying (CO1),(CO2)and (CO3). Let Ui0 ∩ Ui1 ∩ ... ∩ Ui4 be nonempty. Indeed,

(δ(δf))i0i1i2i3i4 ≡5((δf)i1i2i3i4 − (δf)i0i2i3i4 + (δf)i0i1i3i4 − (δf)i0i1i2i4 + (δf)i0i1i2i3)|Ui0i1...i4 =54(fi2i3i4 − fi1i3i4 + fi1i2i4 − fi1i2i3)− 4(fi2i3i4 − fi0i3i4 + fi0i2i4 − fi0i2i3) +4(fi1i3i4 − fi0i3i4 + fi0i1i4 − fi0i1i3)− 4(fi1i2i4 − fi0i2i4 + fi0i1i4 − fi0i1i2) +4(fi1i2i3 − fi0i2i3 + fi0i1i3 − fi0i1i2)|Ui0i1...i4 = 0.

Consequently, δ2 = 0 when p = 2.

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In the rest of this subsubsection let us consider a specified type of functionswhich is referred to in the definition for p-cocycles and that will also be usedlater on in this thesis. These are the locally constant functions, functionswhich are constant in a neighbourhood of each point and take values in R.For these functions we denote the set of p-cochains by Cp(U,R) and the setof p-cocycles by Zp(U,R). We can now define,

Definition 6.4.2.4. The Cech cohomology group with coefficients in R,denoted by Hp(U,R), is the quotient group Zp(U,R)/δCp−1(U,R).

The equivalence class of f ∈ Zp(U,R) in Hp(U,R) is denoted by [f ]. Two co-cycles f in Hp(U,R) that are equivalent (that is, differ by a coboundary) aresaid to be (Cech) cohomologous. Indeed the above described equivalencedefines an equivalence relation ∼,

• f ∼ f : follows from f = f + δh for h = 0. Indeed when h = 0 we havethat δh = 0 is a p-coboundary.

• f ∼ g ⇒ g ∼ f : since f ∼ g ⇔ f = g + δh for some coboundary δh, itfollows that g = f − δh = f + δ(−h)⇒ g ∼ f .

• f ∼ g, g ∼ h⇒ f ∼ h: if f ∼ g ⇔ f = g + δi andg ∼ h⇔ g = h+ δj thenf = h+ δj + δi = h+ δ(i+ j)⇒ f ∼ h.

Let U = Ui be a locally finite contractible cover of M . That is, eachpoint in M has a neighbourhood intersecting only a finite number of Uis ∈ U(locally finite cover of M) and each Ui ∈ U and each finite intersection ofUis ∈ U is contractible to a point (contractible cover of M). Now define,

Definition 6.4.2.5. A partition of unity subordinate to U is a collectionof functions hi : M → R with the properties

(PU1) 0 ≤ hi(m) ≤ 1,

(PU2) supp(hi) ⊂ Ui,

(PU3)∑

i hi(m) = 1.

We quote without proof that for every manifold it is always possible to find alocally finite contractible cover U of M and a partition of unity subordinateto U (for the proof, see for example [21]).

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For f = fi1i2...ip+1 ∈ Cp(U,R), define fi1i2...ip+1 : M → R by

fi1i2...ip+1 ≡fi1i2...ip+1 x ∈ Ui1i2...ip+1

0 x /∈ Ui1i2...ip+1

Subsequently, define α : Cp(U,R)→ p-forms on M by

α(f) ≡ αf =∑

i1,i2,...,ip+1

fi1i2...ip+1hi1dhi2 ∧ ... ∧ dhip+1 . (76)

From (76) it follows that

d(αf ) = αδf . (77)

Again, since the only important case we consider later on in this thesis willbe when p = 2, we prove (77) for this case only. In this case,αf =

∑i1,i2,i3

fi1i2i3hi1dhi2 ∧ dhi3 . The left-hand side of (77) gives

dαf = d(∑

i1,i2,i3fi1i2i3hi1dhi2 ∧ dhi3

)=∑

i1,i2,i3hi1dfi1i2i3 ∧ dhi2 ∧ dhi3 +

∑i1,i2,i3

fi1i2i3dhi1 ∧ dhi2 ∧ dhi3 =∑i1,i2,i3

fi1i2i3dhi1 ∧ dhi2 ∧ dhi3

In the last identity is used that dfi1i2i3 = 0, since fi1i2i3 is locally constant.Furthermore, it follows from

∑i hi = 1 that d(

∑i hi) =

∑i dhi = 0. Using

this, the right-hand side of (77) gives

αδf =∑

i0,i1,i2,i3(δf)i0i1i2i3hi0dhi1 ∧ dhi2 ∧ dhi3 =∑

i0,i1,i2,i3(fi1i2i3 − fi0i2i3 + fi0i1i3 − fi0i1i2)|Ui0i1i2i3hi0dhi1 ∧ dhi2 ∧ dhi3 =∑

i0,i1,i2,i3(fi1i2i3 − fi0i2i3 + fi0i1i3 − fi0i1i2)hi0dhi1 ∧ dhi2 ∧ dhi3 =∑

i0,i1,i2,i3fi1i2i3hi0dhi1 ∧ dhi2 ∧ dhi3 =

∑i1,i2,i3

fi1i2i3dhi1 ∧ dhi2 ∧ dhi3 .

In the third identity is used that hi0dhi1 ∧ dhi2 ∧ dhi3 equals zero outsideUi0i1i2i3 ≡ Ui0 ∩ Ui1 ∩ Ui2 ∩ Ui3 . In the fourth identity is used that∑

i0hi0 = 1,

∑i1dhi1 =

∑i2dhi2 =

∑i3dhi3 = 0. Identity (77) now follows.

The map (76) and the identity (77) tell us that associated with any[f ] ∈ Hp(U,R) we have a set of closed p-forms, any two of which differ byan exact p-form. Indeed, take [f ] ∈ Hp(U,R). Then, using (76) and (77)respectively, it follows that δf = 0⇒ αδf = dαf = 0.

So αf is indeed a closed p-form. The fact that any two of these differ by

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an exact p-form follows from the following argumentation. In the deductionabove the only freedom we have is in choosing the cocycle f satisfying δf = 0.We could also have chosen the cocycle f+δh, where δh is a coboundary, sinceδ(f + δh) = δf + δ2h = δf . So another closed p-form can be given byαf+δh = αf + αδh = αf + dαh,which indeed differs from the original αf by an exact p-form dαh.

Now define,

Definition 6.4.2.6. The pth de Rham cohomology group ofM , denotedby H(M,R), is the space of equivalence classes of p-forms, αf , with two formsregarded as equivalent if they differ by an exact p-form.

Indeed, the equivalence described in the definition above defines an equiva-lence relation ∼, since,

• αf ∼ αf : follows from αf = αf + dαh for αh = 0. Indeed when αh = 0we have that dαh = 0 is an exact p-form.

• αf ∼ αg ⇒ αg ∼ αf : since αf ∼ αg ⇔ αf = αg + dαh for some exactform dαh, it follows that αg = αf − dαh = αf + dα−h ⇒ αg ∼ αf .

• αf ∼ αg, αg ∼ αh ⇒ αf ∼ αh: if αf ∼ αg ⇔ αf = αg + dαi andαg ∼ αh ⇔ αg = αh + dαj thenαf = αh + dαj + dαi = αh + dαi+j ⇒ αf ∼ αh.

In the rest of this subsection we prove that the map α in (76) determines anisomorphism between the pth Cech cohomology group and the pth de Rhamcohomology group. This isomorphism is called the de Rham isomorphism.. First define ι : Hp(U,R)→ Zp(U,R) [f ] 7→ f andπ : closed p-forms on M → Hp(U,R) αf 7→ 〈αf〉. Subsequently, defineα = π α|Zp(U,R) ι :

Hp(U,R)ι→ Zp(U,R)

α|Zp(U,R)−−−−−→ closed p-forms on M π→ Hp(U,R).To prove the claim we have to show that α is an isomorphism. Indeed,

(1) α is a homomorphism: Indeed,α([f ] + [g]) = (π α|Zp(U,R) ι)([f ] + [g])= (π α|Zp(U,R) ι)([f + g]) = (π αZp(U,R))(f + g) = π αZp(U,R)(f + g)= π (αZp(U,R)(f) + αZp(U,R)(g)) = 〈αZp(U,R)(f) + αZp(U,R)(g)〉 =〈αZp(U,R)(f)〉+ 〈αZp(U,R)(g)〉 = α([f ]) + α([g]).

(2) α is bijective: clearly ι and π are bijective. Furthermore, α|Zp(U,R) isbijective. Surjectivity of α|Zp(U,R) follows directly from (77), whereasinjectivity follows directly from (76). This implies that the compositionα = π α|Zp(U,R) ι is bijective too.

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6.5 The integrality condition

To make the prequantization construction considered in subsection 6.3 rigourousone needs a Hermitian line bundle-with-connection over the classical phasespace (the symplectic manifold M). Existence of such a line bundle-with-connection is however not guaranteed. For such a line bundle-with-connectionto exist a certain condition should be satisfied by the curvature 2-form Ω de-fined on it. This is the so-called integrality condition (see [22],[19]). Let usformulate it:

(IC) Integrality Condition: Let M be a symplectic manifold and let Ωbe a closed real 2-form on M (the symplectic form). There exists anopen cover U = Uj of M such that the equivalence class defined by1

2πΩ in H2(U,R) contains a cocycle z in which all the zijk are integers.

In the sequel of this section I shall prove that this integrality condition isboth a necessary and sufficient condition for a Hermitian line bundle-with-connection to exist. The construction presented below is called the Cechconstruction .

Theorem 6.5.1. The Integrality Condition is both a necessary and sufficientcondition for a Hermitian line bundle-with-connection to exist.

Proof. We proceed by showing seperately sufficience and necessarity of ICfor a Hermitian line bundle-with-connection to exist.

• The Integrality Condition is a sufficient condition for a Hermitian linebundle-with-connection to exist:

Let a closed real 2-form Ω on M (the symplectic form) and a con-tractible open cover U = Uj of M satisfying (IC) be given. ByDarboux’s theorem there exist real 1-forms βj on Uj (the symplecticpotentials) such that on Uj,

Ω = dβj. (78)

Furthermore, on each non-empty intersection Ui ∩ Uj, there exists asmooth real function fij = −fji such that

dfij = βi − βj. (79)

Indeed βi − βj is only well-defined on Ui ∩ Uj, as is fij. Furthermore,since dfij = βi − βj = −(βj − βi) = −dfji we can choose the functionssuch that they satisfy fij = −fji. Subsequently, on each non-empty

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triple intersection Ui ∩ Uj ∩ Uk there exists a locally constant smoothreal function aijk defined by

aijk ≡ fij + fjk + fki. (80)

Indeed aijk is a locally constant function, sincedaijk = dfij + dfjk + dfki = βi − βj + βj − βk + βk − βi = 0.Moreover, the constant aijk define a 2-cochain: properties (CO1) and(CO2) follow directly from its definition. Property (CO3) follows froma[ijk] ≡ 1

6aijk − aikj + ajki − ajik + akij − akji =

16fij + fjk + fki − fik − fkj − fji + fjk + fki + fij− fji − fik − fkj + fki + fij + fjk − fkj − fji − fik =fij + fjk + fki = aijk,where the anti-symmetry of fij, fjk and fki is used in the third identity.The constant aijk’s even define a cocycle a ∈ Z2(U,R), since, using (75),(δa)ijkl = ajkl − aikl + aijl − aijk =fjk + fkl + flj − fik − fkl − fli + fij + fjl + fli − fij − fjk − fki = 0.In the last identity the anti-symmetry of the functions flj and fik isused. The equivalence class [a] ∈ H2(U,R) determined by a ∈ Z2(U,R)depends only on Ω and not on the choices made for the βi, fij and aijk.

To see this, choose a general βi = βi + dhi for some exact 1-formdhi. Then βi and βi give rise to the same closed real 2-form Ω, sincedβi = d(βi + dhi) = dβi = Ω.For this new βi we get the newdfij = βi − βj = βi + dhi − βj − dhj = dfij + dhi − dhj⇔ d(fij − fij − hi + hj) = 0

⇔ fij = fij + hi − hj + dij,where dij = −dji are constant functions on nonempty Ui ∩ Uj. Hencethe new aijk become

aijk = fij + fjk + fki =fij + hi − hj + dij + fjk + hj − hk + djk + fki + hk − hi + dki =aijk + dij + djk + dki.Now the eijk ≡ dij + djk + dki define a 2-coboundary e ≡ eijk. It isclear that the eijk form a 2-cochain, since the properties (CO1),(CO2)and (CO3) can be checked completely analogous as in the aijk case.The fact that they in particular form a 2-coboundary follows from(δd)ijk = dij − dik + djk = dij + djk + dki = eijk.But this implies that the cocycles a ≡ aijk and a ≡ aijk only differby the coboundary e. This in turn implies that a and a define the sameequivalence class [a] ∈ H2(U,R). As a consequence, the equivalenceclass [a] only depends on the choice for Ω, not on the choices for the

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βi, fij and aijk.Next, define zijk ≡ 1

2π(aijk + dij + djk + dki) on nonempty Ui ∩Uj ∩Uk.

Then it follows from the argument given above that z ≡ zijk is anelement of the class defined by 1

2πΩ in H2(U,R). Let m ∈ Uijk. By

(IC), when Ui ∩ Uj is nonempty, we can define the constant dij suchthat all zijk(m) are integers when Ui∩Uj∩Uk is nonempty. Notice thatwe wouldn’t be able to choose the dij such that the zijk(m) are integerswithout (IC), since in general the aijk are only locally constant and theintersections Ui ∩ Uj ∩ Uk do not need to be connected.We are now ready to define on each nonempty Ui ∩ Uj,

cij ≡ exp(−i(fij + dij)) (81)

Then, on each nonempty Uijk it holds thatcij(m)cjk(m)cki(m) =exp(−i(fij(m) + fjk(m) + fki(m) + dij(m) + djk(m) + dki(m))) =exp(−2πizijk) = cos(−2πzijk) + i sin(−2πzijk) = 1.Furthermore, (cij)

−1 = cji, sincecji(m)cij(m) = exp(−i(fji(m) + dji(m) + fij(m) + dij(m))) =exp(0) = 1.In other words, the complex functions cij defined above satisfy thesame properties (72) and (73) as transitions functions. We claim thatthe smooth complex functions cij defined on nonempty Uij allow usto construct a line bundle X over M of which the cij are transition

functions. This is the line bundle whose total space is∐i

Ui×C/ ∼ with

∼ the equivalence relation defined by: (j,m, z) ∼ (k, m, z) whenever(j,m, z) ∈ I×Uj×C, (k, m, z) ∈ I×Uk×C andm = m and z = cjk(m)z,

and with the projection map pr :∐i

Ui × C/ ∼→ M [(j,m, z)] 7→ m.

Here the points in∐i

Ui×C are denoted as triples (j,m, z) ∈ I×Uj×C

with I some index set. Notice that the projection map pr is well-definedsince all points (j,m, z) ∼ (k, m, z) are mapped to the same pointm = m ∈ M by pr. Furthermore, notice that ∼ does indeed define anequivalence relation, since

– (j,m, z) ∼ (j,m, z): follows from m = m and z = cjj(m)z. Thelast identity holds, because cjj(m) = exp(−i(fjj(m) + djj(m))) =exp(0) = 1, using the anti-symmetry of fjj and djj.

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– (j,m, z) ∼ (k, m, z)⇒ (k, m, z) ∼ (j,m, z): suppose(j,m, z) ∼ (k, m, z), then m = m and z = cjk(m)z. This impliesm = m and z = ckj(m)z using (72), i.e. (k, m, z) ∼ (j,m, z).

– (j,m, z) ∼ (k, m, z), (k, m, z) ∼ (l, m, z) ⇒ (j,m, z) ∼ (l, m, z):suppose (j,m, z) ∼ (k, m, z) and (k, m, z) ∼ (l, m, z), that is,m = m = m and z = cjkz = cjkcklz. This implies m = m andz = cjlz using (73) and (72) respectively (cjkcklclj = 1⇒ cjkckl =cjl). It thus follows that (j,m, z) ∼ (l, m, z).

We define addition and scalar multiplication on the fibers (m ∈ M

fixed) of X ≡∐i

Ui×C/ ∼ by [(j,m, z1)]+[(j,m, z2)] = [(j,m, z1 +z2)]

and λ[(j,m, z1)] = [(j,m, λz1)] (λ ∈ C) respectively. Let us now provethe claim:

(1) X satisfies (VB1) for K = C: First of all,pr−1(m) = (j,m, z) ∈ I×Uj×C/ ∼. Since m is fixed and j ∈ Ionly takes those values such that m ∈ Uj (let us say that thereare a such values), we can view (j,m, z) ∈ I×Uj ×C as the setconsisting of a copies of C. Then pr−1(m) can be viewed as justone copy of C, since in this case points (k,m, z) and (l,m, z) (mfixed) are ‘identified’ if z = ckl(m)z and the map f : C→ C z 7→ckl(m)z is clearly an isomorphism (with respect to addition onC). Our goal is indeed to show that pr−1(m) is isomorphic to C.To this end, define αk : C → pr−1(m) z 7→ [(k,m, z)]. That αkis surjective follows directly from the discussion above. It is alsoinjective. Indeed,αk(z) = αk(z)⇔ [(k,m, z)] = [(k,m, z)]⇔ (k,m, z) ∼ (k,m, z)⇔ z = ckk(m)z = z.The last identity indeed holds, since ckk(m) = 1. Finally, it is ahomomorphism, because for z1, z2 ∈ C it holds thatαk(z1 + z2) = [(k,m, z1 + z2)] =[(k,m, z1)] + [(k,m, z2)] = αk(z1) + αk(z2).

(2) X satisfies (VB2) for K = C: take m ∈M . If we pick a neighbour-hood U of m we can always choose it such that it is contained inthe intersection of the Ui which contain m, since this intersectionis an open set. With this in mind,pr−1(U) = (j,m, z) ∈ I × Ui × C|m ∈ U ⊂ Ui/ ∼. Now definethe map Ψk : U × C → pr−1(U) (m, z) 7→ [(k,m, z)]. Its surjec-tivity follows directly from surjectivity of αk and the fact that Ψk

sends m to m = m (Ψk is the ‘identity map in m’). Injectivity

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follows from,Ψk(m, z) = Ψk(m, z)⇔ [(k,m, z)] = [(k, m, z)]⇔ (k,m, z) ∼ (k, m, z)⇔ m = m, z = ckk(m)z = z.It follows that (m, z) = (m, z). Furthermore Ψk is smooth: itsends (m, z) to [(k,m, z)] where (k,m, z) ∼ (k, m, z) if m = mand z = ckk(m)z = z, so it sends m to m = m in a smooth fashion(‘identity map in m’) and it sends z to z = z in a smooth fashion(‘identity map in z). It follows that Ψk is smooth. The smooth-ness of Ψk on Uk ∩ Uj should not depend on the choice k. Indeedit does not, since (k,m, z) ∼ (j, m, z) if m = m, z = ckj(m)z andckj is smooth. For the inverse map, define(Ψk)

−1 : pr−1(U) → U × C [(k,m, z)] 7→ (m, z). Indeed it isan inverse, since (Ψ−1

k Ψk)(m, z) = Ψ−1k ([(k,m, z)]) = (m, z).

That the inverse map is well-defined follows from ckk(m) = 1. Itssmoothness is proved in the same way the smoothness of Ψk isproved.

The fact that the restriction map,Ψk|m : m×C→ pr−1(m) (m, z) 7→ [(k,m, z)] is an isomorphismfollows directly from m × C ∼= C and the fact that αk is anisomorphism. Linearity follows fromΨk|m(λz1 + z2) = [(k,m, λz1 + z2)] = [(k,m, λz1)] + [(k,m, z2)] =λ[(k,m, z1)] + [(k,m, z2)] = λΨk|m(z1) + Ψk|m(z2).

(3) The cij are the transition functions of the line bundle X: in orderto verify this, we have to check if the cijs satisfy (70). Indeed, on(Ui ∩ Uj)× C, we havecjk(m)Ψj(m, z) = cjk(m)[(j,m, z)] = [(j,m, cjk(m)z)] = [(k,m, z)] =Ψk(m, z).

What is left to check is that this line bundle X is equipped with aHermitian structure and a connection which are compatible. For theHermitian structure, define, for m ∈ Ui the inner product in each fibreby〈·, ·〉 : pr−1(m) ∼= C× pr−1(m) ∼= C→ R([(i,m, z1)], [(i,m, z2)]) 7→ 〈[(i,m, z1)], [(i,m, z2)]〉 = z1z2.In order to show that this inner product is well-defined for m ∈ Ui∩Uj,note that the fij, dij ∈ R. It follows that|cij|2 = cij cij = exp(−i(fij + dij)) exp(i(fij + dij)) = exp(0) = 1.Hence, on Ui ∩ Uj we have〈[(i,m, z1)], [(i,m, z2)]〉 = 〈[(i,m, cijz1)], [(i,m, cijz2)]〉 =

¯cijz1cijz2 = cij z1cijz2 = |cij|2z1z2 = z1z2,

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as should be the case. Furthermore, for m ∈ Ui the mapH : X → R (m, z) 7→ 〈[(i,m, z)], [(i,m, z)]〉 = zz is smooth and onUi ∩Uj (where (i,m, z) ∼ (j, m, z) if m = m and z = cij z) the smooth-ness does not depend on the choice i, since cij is smooth. Hence 〈·, ·〉defines a Hermitian structure on X.

The line bundle X is also equipped with a connection. From the realβi we can construct complex βi ≡ iβi. Pick a collection (Ui,Ψi) oflocal trivializations of X, a unit section 1i on Ui and a vector fieldX ∈ Vect(Ui). From the complex βi we can define a connection ∇X via

βi1i = i∇X 1i (82)

Since a general section on Ui can be written as si = f 1i forf ∈ C∞(Ui,C) a connection of a general section on Ui can be writtenas

∇Xsi = X(f)1i − iβi(X)si, (83)

as derived previously on a general neighbourhood U . From (71) itfollows that on Uij in general sections si,sj are related by

si(m) = cji(m)sj(m). (84)

Since ∇Xsi should be a section, in order for ∇X to be a well-definedconnection the relation (∇Xsj)(m) = cij(m)(∇Xsi)(m) should be sat-isfied on Uij. Indeed, on Uij,(∇Xsj)(m) = (∇Xcijsi)(m) = (∇Xcijf 1i)(m) =(X(cijf)1i)(m)− i(βj(X)cijsi)(m) =(X(cij)si)(m) + (cijX(f)1i)(m)− i(βj(X)cijsi)(m) =− i(cij(βi(X)− βj(X))si)(m) + (cijX(f)1i)(m)− i(βj(X)cijsi)(m) =(cij(X(f)1i − icijβi(X)si))(m) = cij(m)(∇Xsi)(m).In the fifth identity we usedX(cij) = dcij(X) = d(exp(−i(fij + dij)))(X) =(−id(fij + dij) exp(−i(fij + dij)))(X) = (−idfijcij)(X) =(−i(βi − βj)cij)(X).Finally, for X to be a Hermitian line bundle with connection we haveto check that the constructed connection and Hermitian structure arecompatible. Let X ∈ Vect(Ui), si, s

′i ∈ ΓX(Ui). Then we know that

si = f 1i for f ∈ C∞(Ui,C) and s′i = g1i for another g ∈ C∞(Ui,C).Hence, on the one hand,X〈si, s′i〉 ≡ X〈f 1i, g1i〉 ≡ X(f g)〈1i, 1i〉 =X(f)g〈1i, 1i〉+ fX(g)〈1i, 1i〉.

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On the other hand,〈∇Xsi, s

′i〉+ 〈si,∇Xs

′i〉 =

〈X(f)1i − iβi(X)si, s′i〉+ 〈s,X(g)1i − iβi(X)s′i〉 =

X(f)〈1i, s′i〉+ iβi(X)〈si, s′i〉+X(g)〈si, 1i〉 − iβi(X)〈si, s′i〉 =X(f)〈1i, s′i〉+X(g)〈si, 1i〉 = X(f)g〈1i, 1i〉+X(g)f〈1i, 1i〉.This establishes the compatability condition (64) on some Ui. Thefact that the compatibility condition continues to hold on the overlapsUi ∩ Uj in a well-defined manner follows from the well-definedness ofthe inner product on the overlaps (since |cij|2 = 1) and from∇Xsj = ∇X(cijsi) = cij∇Xsi, so we have proved the ‘sufficient’-part ofthe theorem (since the open cover of M in (IC) can always be chosensuch that it is contractible for a collection Ui,Ψi of local trivializa-tions of X).

Remark 6.5.1. Note that we started with a symplectic 2-form Ω onM . As we constructed the Hermitian line bundle-with-connection wein particular constructed the connection ∇ defined by (83) on generalsections si = f 1i for f ∈ C∞(Ui,C). It now follows from the argumentpresented in section 6.4.1 that the curvature 2-form of ∇ on M equalsthis symplectic form on M , once M is the base manifold of a linebundle-with-connection.

• The Integrality Condition is a necessary condition for a Hermitian linebundle-with-connection to exist:

Suppose we are given a line bundle X over M with connection ∇. Picka selection of local trivializations Ui,Ψi of B such that U = Ui isa contractible open cover of M . Pick the diffeomorphismsΨi : Ui × C→ pr−1(Ui) such that the restriction mapsΨi|m : m×C ∼= C→ pr−1(m) are compatible with the inner product,meaning that for z1, z2 ∈ C we have 〈Ψi|m(z1),Ψi|m(z2)〉 ≡ 〈z1, z2〉.We can always choose the Ψi|m this way. To see this, take z1 ∈ Cand suppose that 〈Ψi|m(z1),Ψi|m(z1)〉 6= 〈z1, z1〉. Then, since both〈Ψi|m(z1),Ψi|m(z1)〉 and 〈z1, z1〉 belong to R they should be related bysome real positive function λ : Ui → R>0 where R>0 is the set of posi-tive real numbers. So we can choose a rescaled

Ψi|m(z1) ≡(√

λ(m))−1

Ψi|m(z1) which then gives

〈Ψi|m(z1), Ψi|m(z1)〉 =(√

λ(m))−1 (√

λ(m))−1

〈Ψi|m(z1),Ψi|m(z1)〉 =(√λ(m)

)−1 (√λ(m)

)−1

λ(m)〈z1, z1〉 = 〈z1, z1〉.

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Since Ψi|m is linear it follows that this map does define a linear iso-morphism compatible with the inner product. Hence we can alwaysconstruct such a map, because the Ψi|m we started with was arbitrary.Since we have a well known inner product on C, namely 〈z1, z2〉 ≡ z1z2

we now have an inner product 〈Ψi|m(z1),Ψi|m(z2)〉 = 〈z1, z2〉 ≡ z1z2 onpr−1(m) for m ∈ Ui too. Subsequently, consider the functions

cji = (ψ−1i ψj)|Uij×C : Uij × C

ψj |Uij×C−−−−−→ pr−1(Uij)

ψ−1i |pr−1(Uij)

−−−−−−−−→ Uij × Cintroduced in subsection 6.4.1 for m ∈ Uij. Since Ψj|m is compatiblewith the inner product, so is Ψ−1

i |m×C and so iscij|m×C ≡ Ψj|m Ψ−1

i |m×C. Using (74) this implies for the norm of thecorresponding transition functions cij that〈cij|m×C(z), cij|m×C(z)〉 = 〈z, z〉 ⇔ 〈cij(m)z, cij(m)z〉 = 〈z, z〉⇔ |cij(m)|2〈z, z〉 ⇔ |cij(m)| = 1.From this it follows that the inner product 〈·, ·〉 on pr−1(m) is well-defined for m ∈ Ui ∩ Uj as well. Indeed, take z1, z2 ∈ pr−1(m) ∼= C.We know that the sections si ∈ ΓX(Ui), sj ∈ ΓX(Uj) are related bysj = cijsi on Uij. Hence, writing z1 ≡ si(m), z2 ≡ si(m), on Uij wehave〈sj, sj〉(m) = 〈cij si, cij si〉(m) = |cij(m)|2〈si, si〉(m) = 〈z1, z2〉,as should be the case. The line bundle X is now provided globally witha Hermitian structure, since for m ∈ Ui the mapH : X → R (m, z) 7→ 〈z, z〉 = zz is smooth and this smoothness doesnot depend on the choice i, since cij is smooth.

The existence of the connection ∇ allows us to define a complex 1-form βi on Ui via 1iβi(X) = ∇X(1i) for X ∈ Vect(Ui), and for generalsections si = f 1i on Ui (f ∈ C∞(Ui,C)) via the relation ∇Xsi =X(f)1i − iβi(X)si. The compatibility of the connection ∇ and theHermitian structure 〈·, ·〉 follows now by the same arguments as in the‘sufficient’-part, since |cij(m)|2 = 1 for m ∈ Uij in this case too. Hencewe have a Hermitian line bundle-with-connection.

Next, define fij ≡ −1i

log(cij) on Uij where log is the complex logarithmwhose values are only determined upto integer multiples n ∈ Z of 2πi.And from this define, zijk ≡ fij + fjk + fki on Uijk. In subsubsection6.4.1 we proved that the transition functions cij satisfycij(m)cjk(m)cki(m) = 1 on Uijk. It follows thatzijk(m) ≡ fij(m) + fjk(m) + fki(m) =−1i

log(cij(m)) + −1i

log(cjk(m)) + −1i

log(cjk(m)) =−1i

log(cij(m)cjk(m)cki(m)) = − log(1)i

+nijk2πi

i= 2πnijk ∈ 2πZ

for nijk : Uijk → Z a constant integer. Hence the zijks are constants.

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The fact that these zijk form a cochain follows from the same argumentsas for the aijk in the ‘sufficient’-part. In particular, the zijk determinea cocycle z ≡ zijk. Indeed,(δz)ijkl = zjkl − zikl + zijl − zijk =fjk + fkl + flj − fik − fkl − fli + fij + fjl + fli − fij − fjk − fki = 0.Here is used the anti-symmetry of the fij. Indeed, since we proved insubsubsection 6.4.1 that the transition functions cij satisfycij(m) = (cji(m))−1, it follows thatfij = −1

ilog(cij) = −1

ilog((cji)

−1) = 1i

log(cji) = −fji.Our independently constructed βi and fij satisfy the relationdfij = βi− βj. Indeed, since sections transform as sj = cijsi on Ui ∩Ujwe know that ∇Xsj for X ∈ Vect(Uj) satisfies ∇Xsj = cij∇Xsi. Itfollows, writing si = f 1i for f ∈ C∞(Ui,C), that∇X(cijsi) = cij∇Xsi ⇔X(cijf)1i − iβj(X)cijsi = cij(X(f)1i − iβi(X)si)⇔X(cij)f 1i + cijX(f)1i − iβj(X)cijsi = cij(X(f)1i − iβi(X)si)⇔X(cij)si − icijsi(βj(X)− βi(X)) = 0⇔ X(cij) = icij(βj − βi)(X)⇔dcij(X) = icij(βj − βi)(X)⇔ dcij

cij= i(βj − βi)⇔

−1id log(cij) = βi − βj ⇔ dfij = βi − βj.

Subsequently, define Ω = dβi on Ui. Since for general sectionssi = f 1i (f ∈ C∞(Ui,C)) the βi are defined via (83) it follows bythe argument given in subsubsection 6.4.1 that Ω equals the curvature2-form corresponding to X. Hence, by remark 6.5.1 it equals the sym-plectic 2-form on M (since M is the base manifold of the (Hermitian)line bundle-with-connection X).

Let [z] ∈ H2(U, 2πZ) be the equivalence class determined by the cocyclez ∈ Z2(U, 2πZ). This equivalence class depends only on Ω and not onthe choices made for the βi, fij (in particlar not on the ambiguityin defining them) and zijks. The argument is fully equivalent to theone presented in the ‘sufficient’-part, except that in this case the factthat the values of fij(m) (m ∈ Uij) are only determined up to integermultiples of 2πi has to be taken into account. This ambiguity is absentfor the equivalence classes [z] ∈ H2(U, 2πZ). Indeed, takefij = fij − 2πinij on Uij for nij : Ui ∩ Uj → Z an integer. These newfij give rise to the same 2-form Ω, since the nij are constant functions.Furthermore,zijk ≡ fij + fjk + fki = fij − 2πinij + fjk − 2πinjk + fki − 2πinki =zijk − 2πi(nij + njk + nki) = zijk − 2π(δn)ijkon Uijk. So the z differ at most by a coboundary and hence give riseto the same equivalence class [z].

94

As a consequence we have found an open cover U of M such that theclass [z] defined by Ω in H2(U,R) contains a cocycle z in which all thezijk are 2π multiples of integers. The integrality condition is clearly anequivalent statement and the proof of the ‘necessary’-part is complete.

6.6 The prequantum Hilbert space

In subsection 6.3 we already mentioned that a prequantum Hilbert spaceconsisting of ‘ordinary’ (square-integrable) complex-valued scalar functionsf : M → C is not the correct Hilbert space to consider. Now that we havetreated the right objects to consider, sections of a Hermitian line bundle-with-connection, and formulated the condition for the classical phase spaceM to be quantizable we are ready to define the correct prequantum Hilbertspace and the prequantum operators acting on it. This will be the purposeof this subsection.

Suppose that the symplectic manifold (M,ω) is quantizable in the sense that1~ω satisfies the integrality condition (IC). Then we know that it is possible toconstruct a Hermitian line bundle-with-connection X with curvature 2-form1~ω, called the prequantum line bundle Prequantum line bundle. Now letm ∈M and s1, s2 ∈ ΓX(M). Define

(s1, s2)(m) ≡ (1

2π~)n∫M

〈s1, s2〉(m)εω. (85)

Here εω denotes the Liouville form as defined in (59) and 〈·, ·〉 denotes theHermitian structure on X as defined in the previous subsection by〈s1, s2〉(m) ≡ 〈s1(m), s2(m)〉 ≡ 〈z1, z2〉 ≡ z1z2 using the notationz1 ≡ s1(m) ∈ C, z2 ≡ s1(m) ∈ C. We claim that (·, ·) defines an innerproduct. Note that the space ΓX(M) is a vector space with complex-valuedfunctions as scalars. Let us now verify the claim,

• (·, ·) is conjugate symmetric: first of all, 〈·, ·〉 is conjugate symmetric,since〈s1, s2〉(m) ≡ z1z2 = ¯z1z2 = ¯〈s1, s2〉(m).Conjugate symmetry of (·, ·) now follows from(s1, s2)(m) ≡ ( 1

2π~)n∫M〈s1, s2〉(m)εω = ( 1

2π~)n∫M

¯〈s2, s1〉(m)εω =¯( 1

2π~)n∫M〈s1, s2〉(m)εω,

where in the last identity εω = εω and the fact that integration is alinear operation are used.

95

• (·, ·) is antilinear in the first argument: Let f be a complex-valuedfunction on M . To begin with, 〈·, ·〉 is antilinear in its first argument,since〈fs1, s2〉(m) = 〈f(m)s1(m), s2(m)〉 = ¯f(m)z1z2 =

¯f(m)z1z2 = ¯f(m)〈s1, s2〉(m)and for s3 ∈ ΓX(M) we have〈s1 + s2, s3〉(m) = ¯z1 + z2z3 = z1z3 + z2z3 = 〈s1, s3〉(m) + 〈s2, s3〉(m),denoting s3(m) ≡ z3.Antilinearity of (·, ·) in its first argument now follows directly fromintegration being a linear operation.

• (·, ·) is positive definite: denote z1 ≡ a+ ib for a, b ∈ R. Firstly, 〈·, ·〉 ispositive definite, because〈s1, s1〉(m) = ¯z1z1 = (a+ ib)(a− ib) = a2 + b2 ≥ 0,and〈s1, s1〉(m) = 0⇔ a2 + b2 = 0⇔ a = 0 and b = 0⇔z1 = 0⇔ s1(m) = 0.Positive-definiteness of (·, ·) now follows directly from the properties ofintegration and εω and the positive value of the prefactor 1

2π~ .

We are now ready to define the (correct) prequantum Hilbert space.

Definition 6.6.1. The prequantum Hilbert space H is the space of alls ∈ ΓX(M) equipped with the inner product (·, ·) defined in (85) for whichthe integral of (s, s)εω over M exists and is finite.

Furthermore, each classical observable f ∈ S ⊂ C∞(M,R) acts on an appro-priate subset of H according to

s 7→ −i~∇Xf s+ fs, (86)

where Xf denotes the Hamiltonian vector field corresponding to the classicalobservable f ∈ S ⊂ C∞(M,R). The corresponding quantization assignmentQ defined in (54) is

Q(f) = −i~∇Xf + f. (87)

Our goal is now to show that this assigment satisfies the conditions (Q1)-(Q4)introduced in subsection 6.2 and hence is a suitable prequantum assignmentof operators on the prequantum Hilbert space (actually, some subset of it,which is argued below) to classical observables f . Indeed Q satisfies theseconditions,

96

(1) Q satisfies (Q1): let f, g ∈ S ⊂ C∞(M,R) and λ ∈ R. Then,Q(λf + g) = −i~∇Xλf+g

+ (λf + g) = −i~∇λXf+Xg + λf + g =− i~λ∇Xf − i~∇Xg + λf + g = λQ(f) +Q(g),which establishes the claim.

(2) Q satisfies (Q2): let f, g ∈ S ⊂ C∞(M,R) and let 1~ω be the symplectic

form on M (as mentioned before). Then[Q(f), Q(g)] = [−i~∇Xf + f,−i~∇Xg + g] =− i~[∇Xf , g]− i~[f,∇Xg ]− ~2[∇Xf ,∇Xg ] =

− i~[∇Xf , g]− i~[f,∇Xg ]− ~2(∇[Xf ,Xg ] +

2ω(Xf ,Xg)

i~

)=

− i~[∇Xf ,g − i~[f,∇Xg ]− ~2(∇Xf,g + f,g

i~

)=

− i~[∇Xf , g]− i~[f,∇Xg ]− ~2∇Xf,g + i~f, g.In the second identity the commutativity of scalar functions underpointwise multiplication is used. In the third identity we used thefact that the symplectic form 1

~ω (which equals the curvature 2-formas a base manifold of X) satisfies by definition the equalityω(Xf ,Xg)

~ ≡ i2

([∇Xf ,∇Xg ]−∇[Xf ,Xg ]

). In the fourth identity the defini-

tion of the Poisson bracket (40) and the fundamental identity (41) areused. Furthermore,taking s ∈ H we have[∇Xf , g]s = ∇Xf (gs)− g∇Xf s = g∇Xf s+Xf (g)s− g∇Xf s =Xf (g)s = f, gs,and hence [∇Xf , g] = f, g. And[f,∇Xg ] = −[∇Xg , f ] = −g, f = f, g.It thus follows that,[Q(f), Q(g)] = −2i~f, g+ i~f, g − ~2∇Xf,g =− i~(f, g − i~∇Xf,g = −i~Q(f, g),as was to be shown.

(3) Q satisfies (Q3): let s1, s2 ∈ H and f ∈ S ⊂ C∞(M,R). Then(Q(f)s1, s2) = (−i~∇Xf s1 + fs1, s2) =i~(∇Xf s1, s2) + f(s1, s2) = i~( 1

2π~)n∫M〈∇Xf s1, s2〉εω + f(s1, s2) =

− i~( 12π~)n

∫M〈s1,∇Xf s2〉εω + i~( 1

2π~)n∫M

(Xf (s1, s2))εω + f(s1, s2) =− i~( 1

2π~)n∫M〈s1,∇Xf s2〉εω + (s1, fs2) = (s1, Q(f)s2).

In the fourth identity the compatibility of the Hermitian structure 〈·, ·〉on X and the connection ∇ on X is used. In the fifth identity weused that on the appropriate subset of the prequantum Hilbert spacewe are considering the term ( 1

2π~)n∫M

(Xf (s1, s2))εω vanishes. For thetechnical proof of this fact we refer the reader to ([7, p.129]).

(4) Q satisfies (Q4): let 1 denote the map 1 : M → R m 7→ 1 and 0 thezero vector field. Then

97

Q(1) = −i~∇X1 + 1 = −i~∇0 + 1 = 1,where linearity of ∇ is used in the sense that ∇0 = 0 with 0 the zerooperator on ΓX(M) and so we are done.

6.7 Polarizations

As mentioned in the introduction of this section the prequantum Hilbertspace is considered ‘too large’, in the sense that it cannot represent thephase space of a physically reasonable quantum system. One way to reduceit is by putting an additional geometric structure on the classical phase space(as a base manifold of the prequantum line bundle), called a polarization.As metioned by Woodhouse (see [7, p.133]), it should be stressed that thejustification is not based on general mathematical results (such as the irre-ducibility of representations) but on the examination of particular examples:the construction unifies and generalizes a number of familiar techniques that,in the past, have not appeared to have had any obvious connection with eachother and that have sometimes seemed to be overspecialized with applica-tions only to particular physical systems.

In this subsection we will first treat polarizations in general and then lookat a specific kind of polarization, the so-called Kahler polarization.

6.7.1 Polarizations in general

The prequantum Hilbert space consists of sections s of a prequantum linebundle X which depend on all the 2n coordinates of the symplectic manifold(M,ω) on which they are defined. We will abstract the idea of a polarizedsection by first considering complex scalar functions depending on 2n co-ordinates. In subsubsection 6.1.1 we already proved that the configurationspace Q, depending on the n position coordinates q1, ..., qn, is a Lagrangiansubmanifold of the cotangent bundle T ∗Q, depending on the the 2n coordi-nates q1, ..., qn, p1, ..., pn, so complex scalar functions on T ∗Q depend on2n coordinates, whereas complex scalar functions on Q depend only on ncoordinates. More generally, if we consider complex scalar functions f on ageneral symplectic manifold M , a way of eliminating half of the 2n coordi-nates they depend on is to demand that the functions f are constant along nindependent vector fields on M . That is, for k ∈ 1, ..., n, Xk ∈ Vect(M,R),we have

Xk(f) = 0. (88)

98

Indeed, this gives n restrictions on M , reducing its dimension by n. Nowdefine,

Definition 6.7.1.1. Let M be a manifold. For all m ∈ M consider thesubspaces Pm ⊂ TmM . If m has an open neighborhood U ⊂M such that forall u ∈ U a set of r independent vector fields Xj exists with SpanXj =Pu, then

P ≡∐m∈M

Pm ⊂ TM (89)

is called a distribution of dimension r on M .

If we denote the space of vector fields X ∈ Vect(M,R) such thatX|m ∈ Pm for every m ∈ M by Vect(M,R;P ) it follows that the n vectorfields Xk considered above definition 6.7.1.1 are the base elements of the (inthis case) n-dimensional space of vector fields Vect(M,R;P ).

Let us generalize this story to sections s on M . Since the directional deriva-tive does not map sections to sections it has no invariant meaning for sectionsof X. However, the connection ∇ does map sections to sections. Considerthe n-dimensional distribution P of the tangent bundle TM of M . A naturalgeneralization of (88) to sections is

∇Xs = 0, (90)

for all X ∈ Vect(M,R;P ). Let X, Y ∈ Vect(M,R;P ) be arbitrary ands ∈ ΓX(M). Then (90) implies

[∇X ,∇Y ]s = ∇X(∇Y s)−∇Y (∇Xs) = 0, (91)

for all X, Y ∈ Vect(M,R;P ). Since the curvature 2-form ω~ of ∇ on X

satisfies ω(X,Y )~ ≡ i

2

([∇X ,∇Y ]−∇[X,Y ]

)equation (91) implies

∇[X,Y ]s−2i

~ω(X, Y )s = 0 ∀X, Y ∈ V ect(M,R;P ). (92)

Condition (92) is automatically satisfied, provided that

(i) [X, Y ] = 0 ∀X, Y ∈ Vect(M,R;P ),

(ii) ω(X, Y ) = 0 ∀X, Y ∈ Vect(M,R;P ).

Let us first consider condition (i). To this end,

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Definition 6.7.1.2. Let P be a distribution on a manifold M . If[X, Y ] ∈ V ect(M,R;P ) ∀X, Y ∈ Vect(M,R;P ), the distribution is calledinvolutive .

In other words, condition (i) translates to the statement that P on M mustbe an involutive distribution. To continue,

Definition 6.7.1.3. Let M be a manifold. If for every point on M , a coor-dinate chart U can be found such that a distribution P on M is spanned byjust the derivatives with respect to the coordinates on U , the distribution iscalled completely integrable .

The following important theorem, known as Frobenius theorem, is quotedwithout proof. For the proof, the interested reader is referred to ([23, p.170]).

Theorem 6.7.1.1. For any distribution P , P is completely integrable if andonly if P is involutive.

It follows that condition (i) is equivalent to saying that P on M must becompletely integrable. Furthermore,

Definition 6.7.1.4. Let M be a manifold and P a distribution on M . IfN ⊂ M is an n-dimensional submanifold of M with TmN ⊂ Pm ∀m ∈ M ,N is called an integral manifold of the distribution P .

We state without proof, that given a rank r completely integrable distribu-tion P on M , there exists an integral submanifold N ⊂M of dimension n = rfor all m ∈M such that TmN = Pm (for the proof, see ([23])). Consequently,from condition (i) it follows that for all m ∈M there exist integral manifoldsN ⊂ M through Pm = TmN of dimension n, since the rank of P on thesymplectic manifold M is n(= dim(P )).

Now looking at condition (ii) and definition 6.1.1.3, we see that condition(ii) translates to the statement that the integral manifolds N ⊂ M are La-grangian submanifolds of M . So, in first instance, one would define a polar-ization of (M,ω) to be a maximally integrable distribution P of TM suchthat Pm is a Lagrangian subspace of TmM for every m ∈ M . However, thisdefinition is far too restrictive. For instance, on a two-dimensional surface apolarization corresponds to a nowhere-vanishing vector field. But the two-dimensional sphere, S2, does not have a globally defined nowhere-vanishingvector field and hence a polarization in the above sense does not exist forS2. The solution to this problem is to complexify the tangent bundle of M .

That is, TM will be replaced by TMC ≡∐m∈M

TmMC ≡

∐m∈M

(TmM ⊗ C).

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Maximally integrable distributions P of TMC, such that Pm = (TmN)C forN ⊂ M a Lagrangian submanifold of M , are more likely to exist. We thusarrive at

Definition 6.7.1.5. Let (M,ω) be a symplectic manifold. A polarizationP of (M,ω) is a maximally integrable distribution of TMC such thatPm is a Lagrangian subspace of TmM

C for all m ∈M .

To conclude this subsubsection, we define the corresponding polarized sec-tions as follows,

Definition 6.7.1.6. A polarized section of the prequantum line bundleX is a section s over M satisfying

∇Xs = 0 ∀X ∈ V ect(M,C;P ). (93)

6.7.2 Kahler polarizations

A particular class of symplectic manifolds for which geometric quantizationis fairly well understood and works with comparative ease are Kahler man-ifolds. These manifolds have natural and well-behaved polarizations. Theyare the subject of this subsubsection.

To understand what a Kahler manifold is we first need to define what analmost complex manifold and complex manifold are respectively.

Definition 6.7.2.1. An almost complex manifold (M,J) is a real man-ifold M that is equipped with a smooth real tensor field J such that atevery point m ∈ M the linear endomorphisms Jm : TmM → TmM satisfyJm Jm = −Im with Im : TmM → TmM v 7→ v the identity operator onTmM .

We need one more ingredient before we come to the definition of a complexmanifold,

Definition 6.7.2.2. Let (M,J) be an almost complex manifold. The smoothreal tensor field J is called integrable if the Nijenhuis tensorN(X, Y ) ≡ [X, Y ] + J [JX, Y ] + J [X, JY ]− [JX, JY ] vanishes for all X, Y ∈Vect(M,R).

Subsequently,

Definition 6.7.2.3. A complex manifold (M,J) is an almost complexmanifold such that J is integrable. In this case, the smooth tensor field J iscalled the complex structure .

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We are now ready to define what a Kahler manifold is.

Definition 6.7.2.4. A Kahler manifold (M,J, ω) is a complex manifold(M,J) which at the same time is a symplectic manifold (M,ω) such that thesymplectic form ω and complex structure J are compatible, that is,ω(JX, JY ) = ω(X, Y ) ∀X, Y ∈ Vect(M,R).

Now suppose we are given a Kahler manifold (M,J, ω). Let m ∈ M be ar-bitrary. Consider the complexified tangent space (TmM)C ≡ TmM ⊗ C. Wecan diagonalize the linear endomorphisms Jm : TmM → TmM in (TmM)C.To this end, let v be a vector in (TmM)C and set Jmv = λv with λ ∈ Can eigenvalue of Jm. Since we know that Jm Jm = −Im, it follows for theeigenvalues λ of Jm that(Jm Jm)(v) = −v ⇔ Jm(λv) = λ2v = −v ⇔ λ2 = −1⇔ λ = ±i.The eigenspace (TmM)(1,0) corresponding to the eigenvalue i of Jm is spannedby vectors of the form v−iJmv for v ∈ TmM . Indeed, denote such an elementin Span(v − iJmv) by v ≡

∑i ci(vi − iJmvi) for vi ∈ TmM and ci ∈ C. Then

Jmv = Jm∑

i ci(vi − iJmvi) =∑

i ci(Jmvi)−∑

i i(ciJm(Jmvi) =∑i ci(Jmvi) + i

∑i civi = i

∑i ci(vi − iJmvi) = iv.

Similarly, the eigenspace (TmM)(0,1) corresponding to the eigenvalue −i ofJm is spanned by vectors of the form v + iJmv for v ∈ TmM .

Furthermore, since (M,J, ω) is in particular a symplectic manifold it has realdimension 2n and so does its tangent space TmM for m ∈M . It follows that(TmM)C has complex dimension 2n. The spaces (TmM)(1,0) and (TmM)(0,1)

are formal complex conjugates of each other, since the formal complex con-jugate of an element in (TmM)(1,0) is in an element in (TmM)(0,1). Indeed,∑

i ci(vi − iJmvi) =∑

i ci(vi − iJmvi) =∑i civi + i

∑i ciJmvi =

∑i ci(vi + iJmvi).

Hence, the spaces (TmM)(1,0) and (TmM)(0,1) have the same dimension n,since (TmM)(1,0) ∪ (TmM)(0,1) = TmM

C. Furthermore,(TmM)(1,0) ∩ (TmM)(0,1) = 0 with 0 the zero vector in TmM

C.

As a next step we show that (TmM)(0,1) is a Lagrangian subspace of (TmM)C.We see that (TmM)(0,1) is a linear subspace of (TmM)C, since the zero vectoris contained in it (take vi = 0) and it is clearly closed under vector additionand scalar multiplication. That it is even a Lagrangian subspace of (TmM)C

follows fromωm (

∑i ci(vi − iJmvi),

∑i di(vi − iJmvi)) =

ωm (Jm∑

i ci(vi − iJmvi), Jm∑

i di(vi − iJmvi)) =ωm (i

∑i ci(vi − iJmvi), i

∑i di(vi − iJmvi)) =

− ωm (∑

i ci(vi − iJmvi),∑

i di(vi − iJmvi))⇔

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ωm(v, w) = 0 ∀v, w ∈ (TmM)(0,1),where ωm : (TmM)C × (TmM)C → C is the alternating multilinear map de-fined by the symplectic 2-form ω for m ∈M .

Now construct from this the distribution TM (0,1) ≡∐m

(TmM)(1,0) on TMC.

The distribution TM (0,1) is maximally integrable. To see this, first note thatthis is equivalent to showing that the distribution is involutive. Secondly,because M is a complex manifold the complex structure J defined on it isintegrable. It follows that for any X, Y ∈ Vect(M,R) it holds thatN(X, Y ) ≡ [X, Y ] + J [JX, Y ] + J [X, JY ]− [JX, JY ] = 0⇔i[JX, Y ] + i[X, JY ] = iJ [X, Y ]− iJ [JX, JY ].Take two vector fiels X ≡

∑i ci(Xi − iJXi) ∈ TM (0,1) and

Y ≡∑

j dj(Yj − iJYj) ∈ TM (0,1) for Xi, Yj ∈ Vect(M,R) and ci, dj ∈ C.

That TM (0,1) is involutive now follows from[∑i ci(Xi − iJXi),

∑j dj(Yj − iJYj)

]=∑

i,j cidj [Xi − iJXi, Yj − iJYj] =∑i,j cidj [Xi, Yj]− i[JXi, Yj]− i[Xi, JYj]− [JXi, JYj] =∑i,j cidj [Xi, Yj]− iJ [Xi, Yj] + iJ [JXi, JYj]− [JXi, JYj] =∑i,j cidj (I − iJ)[Xi, Yj]− (I − iJ)[JXi, JYj] =∑i,j cidj(I − iJ)([Xi, Yj]− [JXi, JYj]) ∈ TM (0,1).

In the third identity equation the vanishing of the Nijenhuis tensor is used.As a consequence we have shown that the space TM (0,1) is a polarizationof (M,J, ω)! We call this polarization a Kahler polarization since it isnaturally constructed on a Kahler manifold. Similarly, one can check thatTM (1,0) is a Kahler polarization of (M,J, ω).

Is there an easy way we can recognize whether a symplectic manifold isa Kahler manifold, so that we can construct the Kahler polarization on it?That there indeed is such a way can be seen by going to local coordinates. Ifwe introduce local coordinates z and z on the patches of a Kahler manifold(M,J, ω), in first instance the Kahler form (the symplectic form compatiblewith the complex structure) can locally be written asω = iωij(z, z)dzi ∧ dzj + iωij(z, z)dzi ∧ dz j +

iωij(z, z)dz i ∧ dzj + iωij(z, z)dz i ∧ dz j.From the additional structure on a Kahler manifold one can deduce that theωij-component and ωij-component of ω vanish and more particularly, that ωtakes the form

ω = 2iωij(z, z)dzi ∧ dz j. (94)

The proof of this will not be provided in this thesis and we refer the reader

103

to ([6]) for a much more detailed derivation. The fact that the Kahler formis closed has important consequences. Indeed,dω = 0⇔ 2id(ωij(z, z)dzi ∧ dz j) = 0⇔2i∂ωij(z,z)

∂zkdzk ∧ dzi ∧ dz j + 2i

∂ωij(z,z)

∂zkdzk ∧ dzi ∧ dz j = 0.

Both parts of the expression above must separately vanish. Denoting ∂i = ∂∂zi

and ∂i = ∂∂zi

, this implies that

• ∂kωijdzk ∧ dzi ∧ dz j = 0⇔∂kωijdz

k ∧ dzi ∧ dz j + ∂iωkjdzi ∧ dzk ∧ dz j = 0⇔

∂kωij − ∂iωkj = 0,

• ∂kωijdzk ∧ dzi ∧ dz j = 0⇔∂kωijdz

k ∧ dzi ∧ dz j + ∂jωikdzj ∧ dzi ∧ dzk = 0⇔

∂kωij − ∂jωik = 0.

This implies that locally in a patch Ua of the Kahler manifold M thereexists a function Ki(z, z), called the Kahler potential , such that ωij =∂i∂jKa(z, z). Hence the Kahler form can locally be written as

ω = 2i∂i∂jKa(z, z)dzi ∧ dz j = 2i∂∂Ka, (95)

with ∂ ≡ dzk∧∂k and ∂ ≡ dzk∧∂k such that d = ∂+∂. As a consequence, twonatural symplectic potentials on a Kahler manifold are 2i∂Ka and −2i∂Ka.Indeed,d(i∂Ka(z, z)) = 2i ∂

∂zi

(∂∂zjKa(z, z)dzi ∧ dzj

)+2i ∂

∂zi

(∂∂zjKa(z, z)dzi ∧ dzj

)=

2i ∂∂zi

(∂∂zjKa(z, z)dzi ∧ dzj

)= 2i ∂

∂zi∂∂zjKa(z, z)dzi ∧ dzj = ω

andd(−i∂Ka(z, z)) = −2i ∂

∂zi

(∂∂zjKa(z, z)dzi ∧ dzj

)−2i ∂

∂zi

(∂∂zjKa(z, z)dzi ∧ dzj

)= −2i ∂

∂zi

(∂∂zjKa(z, z)dzi ∧ dzj

)= −2i ∂

∂zi∂∂zjKa(z, z)dzi ∧ dzj = ω.

This concludes the subsection on polarizations.

6.8 The Hilbert space

Equipped with all the necessary mathematical tools, we are now ready to con-struct the quantum phase space (Hilbert space). At this point it is importantto notice that the general theory about quantization is far from completelyunderstood at the moment. We restrict our attention to quantum phasespaces which arise from symplectic manifolds equipped with a Kahler polar-ization, because here the least difficulties arise. In this case, there are noobstructions in constructing the Hilbert space and defining a natural mea-sure on it. We will determine the quantum Hilbert space corresponding to a

104

Kahler polarization and discuss the operators acting on it.

Let us denote the space of polarized sections of the prequantum line bundleX over the Kahler manifold (M,J, ω) with respect to a Kahler polarizationP by ΓX(M ;P ). As previously, denote the prequantum Hilbert space by H.

Definition 6.8.1. The Hilbert space HP is the space defined byHP ≡ H ∩ ΓX(M ;P ). In other words, it consists of all square-integrablepolarized sections of X.

Since a Kahler polarization just picks out a particular subspace of the pre-quantum Hilbert space, the Hilbert space is well-defined, since the intersec-tion H ∩ ΓX(M ;P ) is nonempty. The technical proof that HP does indeeddefine a Hilbert space (so in particular has a natural inner product definedon it) will not be provided in this thesis. For a detailed proof we refer thereader to ([7, p.136-137]).

The next question to ask is which classical observables define operators onHP . The requirement is that the prequantization assignment Q(f) for f ∈C∞(M,R) maps polarized (square-integrable) sections to polarized (squareintegrable) sections. Let f ∈ C∞(M,R), X ∈ Vect(M,C;P ) and s ∈ HP .Then∇X(Q(f)s) = ∇X(−i~∇Xf s + fs) = −i~∇X∇Xf s + f∇Xs + X(f)s =−i~∇Xf∇Xs− i~[∇X ,∇Xf ]s+ f∇Xs+ X(f)s =− i~∇Xf∇Xs− 2ω(X,Xf )− i~∇[X,Xf ]s+ f∇Xs+ X(f)s =− i~∇Xf∇Xs− i~∇[X,Xf ]s+ f∇Xs = Q(f)(∇Xs)− i~∇[X,Xf ]s.In the fourth identity we used that the curvature 2-form ω/~ of ∇ on Msatisfies ω(X,Xf )/~ = i

2([∇X ,∇Xf ]−∇[X,Xf ]). In the fifth identity we used

that ω is non-degenerate and vanishes on the polarization P and X(f) = 0on P . That X = 0 follows from the fact that the polarized sections satisfy∇Xs = 0. Writing s = g1 for g ∈ C∞(M,C) and β for the potential 1-formcorresponding to ∇ this indeed implies thatX(g)1− iβ(X)s = 0⇔ X(g)1 = 0⇔ X = 0.The first equivalence follows from the fact that under the regularity condi-tions assumed in this subsection one can locally always find potential 1-formscorresponding to∇ which vanish on P (which means β(X) = 0). Such poten-tial 1-forms are called locally adapted to P . As a consequence of the abovediscussion we see that polarized sections are indeed mapped to polarizedsections by Q(f) provided that

[Xf , X] ∈ Vect(M,C;P ) ∀X ∈ Vect(M,C;P ). (96)

105

This condition (96) tells us that a classical observable f on M defines anoperator on HP via the assignment Q(f) provided that its flow leaves thepolarization invariant. We denote the functions in C∞(M,R) whose flowleaves the polarization invariant by C∞(M,R;P ).

Now take the specific Kahler polarization P = TM (0,1), as defined in subsub-section 6.7.1, for a 2-dimensional Kahler manifold M . Locally, introducingthe complex coordinates z and z on M , it is spanned by the vector ∂

∂z. The

observables preserving this polarization are of the form

f(z, z) = f0 + f1z + f1z + f2zz (97)

with f0, f2 ∈ R and f1 ∈ C. Indeed, using (A.1) we see that the Hamiltonianvector field on M is given by

Xf =1

2ωzz

∂f

∂z

∂

∂z+

1

2ωzz

∂f

∂z

∂

∂z. (98)

Hence condition (96) givesf ∈ C∞(M,R;P )⇔ [Xf ,

∂∂z

] ∈ Vect(M,C;P )⇔− 1

2ωzz ∂

2f∂z∂z

∂∂z− 1

2ωzz ∂

2f∂z∂z

∂∂z∈ Vect(M,C;P )⇔ ∂2f

∂z∂z= 0⇔

f(z, z) = g(z) + h(z)z,with g(z) and h(z) smooth complex valued functions only depending on z.But since observables are real and hence satisfy f(z, z) = f(z, z), equation(97) follows.

We will conclude this section by showing that for the harmonic oscillator,a physically relevant and easy application, the quantization construction de-scribed above already fails to give correct results. The required modificationwhich is needed to give the correct results is called metaplectic quantization.We will not discuss metaplectic quantization in this thesis, but will show thatit gives the correct results for the harmonic oscillator. The Hamiltonian ofthe harmonic oscillator in 1 dimension is given by

H(q, p) =1

2(p2 + q2) = zz, (99)

where q, p are the position and momentum coordinate on the phase space ofthe harmonic oscillator, which is T ∗R2, and z ≡ 1√

2(p+ iq). Note that T ∗R2

is a 2-dimensional Kahler manifold, with complex structure defined byJ( ∂

∂q) = − ∂

∂p, J( ∂

∂p) = ∂

∂q

and symplectic form ω = idz ∧ dz. Hence H(q, p) is an observable whichpreserves the polarization P = TM (0,1), since it is a specific case of (97). In

106

terms of z and z the symplectic form ω equalsω = dp ∧ dq = −i

2(dz + dz) ∧ (dz − dz) = i

2(dz ∧ dz)− i

2(dz ∧ dz).

Hence the Hamiltonian vector field associated with the observable H equals,using (A.1),XH = i∂H

∂z∂∂z− i∂H

∂z∂∂z

= iz ∂∂z− iz ∂

∂z.

Furthermore, the polarized sections with respect to P = TM (0,1) are holo-morphic scalar functions (scalar functions depending on the z coordinateonly). Indeed, the connection along directions in P takes the form of an‘ordinary’ directional derivative, since the potential 1-form β correspondingto ∇ vanishes on P . This implies, for s ∈ ΓX(M ;P ), X ∈ Vect(M,C;P )and ψ the scalar function associated with s, that∇Xs = 0⇔ d

dzψ(z, z) = 0⇔ ψ(z, z) = ψ(z).

As a consequence, the assignment Q(H) acting on holomorphic functions,equals,Q(H)ψ(z) = −i~(∇XH − iβ(XH))ψ(z) +Hψ(z) =− i~XH(ψ(z))− i~(−iβ(XH)) +Hψ(z) =~z ∂

∂zψ(z)− i~(−i)

(i zdz~

) (iz ∂

∂z− iz ∂

∂z

)ψ(z) + zzψ(z) =

~z ∂∂zψ(z)− zz + zz = ~z ∂

∂zψ(z).

Hence the quantized Hamiltonian corresponds with the operator ~z ∂∂z

. Theeigenfunctions of this operator are the monomials zn (n ≥ 0) with eigenval-ues n~. Indeed,~z ∂

∂zzn = ~znzn−1 = n~zn.

But as the reader might know from introductory quantum mechanics thisspectrum of the Hamiltonian is incorrect and should be shifted by the groundstate energy 1

2~. The problem arises from the fact that viewed as an operator

the Hamiltonian H = z ˆz is not Hermitian, whereas the symmetrized Hamil-tonian H = 1

2(z ˆz + ˆzz) is. This symmetrization process can be incorporated

by a quantization procedure called metaplectic quantization.

A general rule of thumb for including the metaplectic correction (in thiscase the term 1

2~) to the operator corresponding to a polarization preserving

observable is the following (see [24]). Let the polarization P be spanned bythe n complex vector fields Xk on a Kahler manifold M , k ∈ 1, ..., n. If fpreserves the polarization P , there is a matrix a(f) ≡ (akl(f)) of functionson M satisfying

[Xf , Xk] = alk(f)Xl, (100)

as follows from (96). In terms of this matrix, the corrected quantum operatoris (see for example [9]),

Q(f) = Q(f)− 1

2i~ tr(a(f)). (101)

107

Furthermore,[XH ,

∂∂z

] = −i(∂∂zz)∂∂z

+ i(∂∂zz)∂∂z

= i ∂∂z

.It follows that the matrix a(f) in this case equals

a(f) =

(0 00 i

).

Hence tr(a(f)) = i. Thus the corrected quantum operator becomesQ(H) = ~z ∂

∂z− 1

2i~(i) = ~

(z ∂∂z

+ 12

).

The eigenfunctions of this operators are again the monomials zn (n ≥ 0)with the correct eigenvalues (n+ 1

2)~. Indeed,

~(z ∂∂zzn + 1

2zn)

= ((n+ 12)~)zn.

108

7 The orbit method and the representations

of SU(2)

-It is the harmony of the diverse parts, their symmetry, their happy balance;in a word it is all that introduces order, all that gives unity, that permits usto see clearly and to comprehend at once both the ensemble and the details.-Henri Poincare about symmetry ([25])

The goal of the orbit method is to establish a correspondence between ir-reducible unitary reprensentations of a Lie group and its coadjoint orbits. Itthus provides a general framework for finding all irreducible unitary represen-tations of a Lie group by finding its coadjoint orbits. Representation theoryremains the method of choice for symplifying the physical analysis of systemspossesing (a high degree of) symmetry and its appearance is ubiquitous in,for example, high energy physics.

In this section the representation theory of SU(2) will be treated by meansof the orbit method. The method of investigation is less formal then theone used in sections 4, 5 and 6. In order to determine the irreducible rep-resentations of SU(2) by means of the orbit method we proceed as follows.We construct the quantum phase space on the coadjoint orbits of SU(2) de-termined in section 4 (the spheres) which satisfy the integrality condition.To this end, we first have to check what the integrality means in the caseof a sphere. Subsequently, we quantize the observables on these spheres bygeometric quantization, as described in section 6. Since SU(2) is a compactmanifold, its representations were already well-known to mathematicians longbefore the orbit method was invented. In order to finally understand the rep-resentation theory of SU(2) by means of the orbit method, it is worthwile tofirst determine the representations of SU(2) by the old-fashioned method.

7.1 The integrality condition for the coadjoint orbitsof SU(2)

Denote by S2 the sphere of some arbitrary radius R ∈ R and by ω thesymplectic form defined on it. In order to understand what the integralitycondition means in the case of a sphere of radius R we proceed by provingthat the statements beneath are equivalent.

(I1) There exists an open cover U = Uj of S2 such that the class definedby ω in H2(U,R) contains a cocycle a in which all the aijk are integralmultiples of 2π.

109

(I2) The integral of ω over S2 is an integral multiple of 2π.

Theorem 7.1.1. The formulations (I1) and (I2) of the Integrality Conditionfor S2 are equivalent.

Proof. We proceed by showing the implications (⇐) and (⇒) seperately.

• (I1) ⇐ (I2): suppose that∫S2 ω ∈ 2πZ is given. Now consider the

pyramid cover U = Ui i ∈ 1, 2, 3, 4 of the sphere depicted in figure 4below.

Figure 4: The pyramid cover of the sphere. The sphere is covered by the four patchesU1 (blue), U2 (violet), U3 (green) and U4 (red) as indicated in the figure. Considering theintersections U123, U124, U134 and U234 and picking a point in the center of each of them,drawing all the geodesics between these points gives the edges of a regular tetrahedron.

In subsection 6.5 it is already proved that the existence of the closedreal 2-form ω on a symplectic manifold M (so in particular a closedreal 2-form on the sphere) gives the existence of real 1-form βj on Ujsuch that ω = dβj on Uj, the existence of anti-symmetric smooth realfunctions fij = −fji on Uij such that dfij = βi − βj on Uij and the

110

existence of locally constant smooth real functions aijk ≡ fij +fjk +fkion Uijk. When considering the pyramid cover of the sphere this givesthe globally constant functions (since the Uijk are connected)a123 ≡ f12 + f23 + f31 on U123

a124 ≡ f12 + f24 + f41 on U124

a134 ≡ f13 + f34 + f41 on U134

a234 ≡ f23 + f34 + f42 on U234.We can define new fij from the old fij byf12 ≡ f12 − a124 on U124

f13 ≡ f13 − a134 on U134

f23 ≡ f23 − a234 on U234

and leaving the other fij unchanged, since daijk = 0 and so ω remainsthe same. In terms of the new fij the new aijk ≡ fij + fjk + fki becomea124 ≡ f12 + f24 + f41 = f12 − a124 + f24 + f41 =f12 − (f12 + f24 + f41) + f24 + f41 = 0a134 ≡ f13 + f34 + f41 = f13 − a134 + f34 + f41 =f13 − (f13 + f34 + f41) + f34 + f41 = 0a234 ≡ f23 + f34 + f42 = f23 − a234 + f34 + f42 =f23 − (f23 + f34 + f42) + f34 + f42 = 0a123 6= 0.Hence a124, a134, a234 ∈ 2πZ. If we can prove that

∫S2 ω = −a123 it fol-

lows that a123 ∈ 2πZ and consequently, using the results of subsection6.5, we have found an open cover U of S2 such that the class definedby ω in H2(U,R) contains a cocycle a in which all the aijk are integralmultiples of 2π. Indeed, for a consistent choice of orientations on theboundaries of the Ui,∫S2 ω =

∫U1ω +

∫U2ω +

∫U3ω +

∫U4ω =∫

U1dβ1 +

∫U2dβ2 +

∫U3dβ3 +

∫U4dβ4 =∫

∂U1β1 +

∫∂U2

β2 +∫∂U3

β3 +∫∂U4

β4 =

−∫U12

β1 +∫U13

β1 −∫U14

β1 +∫U21

β2 −∫U23

β2 +∫U24

β2 −∫U31

β3 +∫U32

β3 −∫U34

β3 +∫U41

β4 −∫U42

β4 +∫U43

β4 =∫U12

(−β1 + β2) +∫U13

(β1 − β3) +∫U14

(−β1 + β4) +∫U23

(−β2 + β3) +∫U24

(β2 − β4) +∫U34

(−β3 + β4) =∫U12

df21 +∫U13

df13 +∫U14

df41 +∫U23

df32 +∫U24

df24 +∫U34

df43 =∫∂U12

f21 +∫∂U13

f13 +∫∂U14

f41 +∫∂U23

f32 +∫∂U24

f24 +∫∂U34

f43 =

(f21(u123)− f21(u124))+(f13(u123)− f13(u134))+(f41(u124)− f41(u134))+(f32(u123)−f32(u234))+(f24(u124)−f24(u234))+(f43(u134)−f43(u234)) =−a123(u123)+a124(u124)+a134(u134)+a234(u234) = −a123(u123) = −a123(·).In the second identity we used Stokes’ theorem. In the third iden-

111

tity we used that ∂U1 = U12 ∪ U13 ∪ U14, et cetera. In the sixthidentity again Stokes’ theorem is used. In the seventh identity theu123, u124, u234, u134 denote points in U123, U124, U234, U134 respectivelyand is used that ∂V12 = V123 ∪ V124, et cetera. In the last identity isused that a123 is constant.

• (I1) ⇒ (I2): this implication is now easy to prove. Suppose we aregiven the pyramid cover considered above such that the class definedby ω in H2(U,R) contains a cocycle a in which all the aijks are integralmultiples of 2π. Such aijk always exist, since the intersections Uijk areall connected for the pyramid cover, from which it follows that the aijkare globally constant. We thus know that a123, a124, a134 and a124 areintegral multiples of 2π. It follows that−a123(u123) + a124(u124) + a134(u134) + a234(u234) =

∫S2 ω

and hence that∫S2 ω ∈ 2πZ, since a123, a124, a134 and a124 are integral

multiples of 2π. The implication follows.

As a consequence, for a Hermitian line bundle-with-connection over S2 toexist (the prequantum line bundle over S2) condition (I2) should be satisfied.We already know from subsubsection 5.2.1 that ω = R sin(θ)dθ ∧ dφ. Hencecondition (I2) gives∫S2 ω =

∫ π0

∫ 2π

0R sin(θ)dθdφ = 2πR

∫ π0

sin(θ)dθ = 4πR ∈ 2πZ⇔ R ∈ Z/2.In other words, we can construct the prequantum line bundle for all spheresof half-integer radius. We are now ready to construct the quantum phasespace on the spheres of half-integer radius. This shall be done in the nextsubsection.

7.2 Constructing the Hilbert space on thespheres of half-integer radius

Consider a sphere S2 of half-integer radius k ∈ Z/2. We cover the sphereby the two charts already introduced in subsubsection 5.2.1, i.e. the coverU ≡ U±. As explained in subsubsection 5.2.2 these charts are isomorphicto C. We can writeS2 = C

∐C/ ∼ with ∼ the equivalence relation defined by (1, z1) ∼ (1, z2)

when z1 = z2, (2, z1) ∼ (2, z2) when z1 = z2 and (1, z1) ∼ (2, z2) when z2 = 1z1

and z1 6= 0. Here (1, z) for z ∈ C denotes an element in the first copy of Cin the definition for S2 and (2, z) for z ∈ C denotes an element in the secondcopy of C in the definition for S2. Indeed ∼ defines an equivalence relation,since

112

• (i, z) ∼ (i, z): we have (i, z) ∼ (i, z)⇔ z = z for both i = 1 and i = 2,which is indeed true.

• (i, z1) ∼ (j, z2) ⇒ (j, z2) ∼ (i, z1): we will prove this relation for thecase i = 1, j = 2. Suppose (1, z1) ∼ (2, z2), which means z2 = 1

z1and

z1 6= 0. Then z1 = 1z2

and z2 6= 0. Hence (2, z2) ∼ (1, z1).

• (i, z1) ∼ (j, z2), (j, z2) ∼ (k, z3) ⇒ (i, z1) ∼ (k, z3): we will prove thisrelation for the case i = 1, j = 2, k = 1. Suppose (1, z1) ∼ (2, z2)and (2, z2) ∼ (1, z3). Then, by definition, z2 = 1

z1, z1 6= 0, z3 = 1

z2and

z2 6= 0. Hence z3 = 1(1z1

) = z1. It follows that (1, z1) ∼ (1, z3).

On the overlap of the first copy of C and the second copy of C in the def-inition for S2 points are indeed identified by ∼ if one views the first copyof C (minus the point (1, 0), but the equivalence relation does act triviallyon this point anyway) as the image under the map α : C/0 → C z 7→ zand the second copy of C (minus the point (2, 0)) as the image under themap β : C/0 → C z 7→ 1

z. The fact that one can let the north pole of S2

correspond to the point (1, 0) and the southpole to the point (2, 0) followsfrom the fact that (1, ε) ∼ (2, ε) when ε = 1

εfor ε 6= 0 infinitesimally small.

In subsection 6.5 we have seen that on a symplectic manifold M (with coverU ≡ Ui) one can construct a prequantum line bundleX whose total space is∐i

Ui×C/ ∼ with ∼ the equivalence relation defined by: (j,m, z) ∼ (k, m, z)

whenever (j,m, z) ∈ I × Uj × C, (k, m, z) ∈ I × Uk × C and m = m andz = cjk(m)z. In case M = S2 with cover U ≡ U±, the total space becomes

X =∐i

Ui ×C/ ∼= (U1 ×C)∐

(U1 ×C)/ ∼= (C×C)∐

(C×C)/ ∼ with

∼ the equivalence relation defined by (1,m, z) ∼ (1, m, z) when m = m andz = z, (2,m, z) ∼ (2, m, z) when m = m and z = z, (1,m, z) ∼ (2, m, z)when m = 1

mand m 6= 0 and z = 1

zkz. Here is used that the transition

functions on S2 of radius k ∈ Z with cover U satisfy c11(m) = 1, c22(m) = 1and c12(m) = 1

zk.

At any point m ∈ S2 the tangent space of S2 equals the plane R2 ∼= C.It follows that TU+

∼= U1 × C ∼= C× C and similarly TU− ∼= C× C. Conse-

quently, (TU+)C ≡∐m∈U+

TmU+∼= C×C2 and (TU−)C ∼= C×C2. Together the

(TU+)C and (TU+)C determine (TS2)C. A point in (TU+)C can be denoted

by(z, a ∂

∂x+ b ∂

∂y

)for z ∈ C, (x, y) ∈ R2 ∼= C and a, b ∈ C.

113

As a next step we show that S2 is a Kahler manifold. We know from sub-section 5.2.2 that the symplectic form ω on S2 equals

ω = i2R

(1 + zz

4R2

)−2dz ∧ dz. We claim that ω can be written as ω = 2i∂∂K

for K = R log(1 + zz/4R2), which shows that S2 is a Kahler manifold. Here∂ ≡ dz ∧ ∂

∂zand ∂ ≡ dz ∧ ∂

∂z. Indeed,

∂K = ∂∂zK ∧ dz = ∂

∂z(R log(1 + zz/4R2)) ∧ dz = z/4R

1+zz/4R2 ∧ dz.Subsequently,

2i∂∂K = i∂ z/2R1+zz/4R2 ∧ dz = i(dz ∧ ∂

∂z)(

z/2R1+zz/4R2 ∧ dz

)=

i(

1/2R1+zz/4R2 − zz/4R2

(1+zz/4R2)2

)dz ∧ dz = i

2R

(1 + zz

4R2

)−2dz ∧ dz,

which proves the claim.

Now consider the specific Kahler polarization P ≡ (TS2)(0,1). Locally, tak-ing z ≡ x + iy and z ≡ x − iy as complex coordinates on the patchesof S2, it is spanned by the vector ∂

∂z. Hence a point in (TU+)(0,1) can be

denoted by (z, a ∂∂z

) for a ∈ C. Furthermore, on the patch U+, the polar-ized sections with respect to P are holomorphic complex-valued scalar func-tions, as already shown in subsection 6.8. In other words, scalar functionsf which satisfy ∂

∂zf = 0. Since the f are holomorphic they can be written

as f(z) =∑∞

i=0 aizi with ai ∈ C. From (68) it follows that a section on

U+ evaluated in a point m ∈ U+ can be written as ψ(m, f(m)) with ψ thediffeomorphism ψ : pr−1(C)→ C×C. Since ψ(m, f(m)) ∈ pr−1(C) ∼= C×Cwe might as well denote the points ψ(m, f(m)) by (m, f(m). Using that

S2 = C∐

C/ ∼ it follows that on the intersections U+− of a sphere of radius

k ∈ Z/2 with cover U we have for z ∈ C,(z, f(z)) ∼

(1z, f(1

z))

=(

1z,∑∞

i=0 aiz−i) ∼ (z, zk∑∞i=0 aiz

−i).Since zk

∑∞i=0 aizi should be a holomorphic function, this can only be the

case if k − i ≥ 0. Hence k ≥ i. Hence f is a polynomial of degree ≤ k on S2

with radius k ∈ Z/2. Notice that the space of polynomials of degree ≤ k onS2 (with radius k ∈ Z/2) is indeed an irreducible subspace of the space ofall smooth complex-valued scalar functions on S2 (with radius k ∈ Z/2. Asa consequence, the Hilbert space on S2 is the space of all square-integrablepolynomials of degree ≤ k on S2 (with radius k ∈ Z/2).

7.3 Quantizing observables on the spheres of half-integerradius

In this subsection we present an explicit description of the quantization as-signment in case the symplectic manifold is a sphere of half-integer radius.

114

We will describe this quantization assignment both in spherical and stereo-graphic coordinates.

7.3.1 Quantizing observables on the spheres of integer radius inthe spherical coordinate system

From subsubsection 5.2.1 we know that in spherical coordinates the symplec-tic form on a patch of S2 with covering U ≡ U± equalsω = R sin(θ)dθ ∧ dφ. Hence on spheres S2 of half-integer radius R ∈ Z/2the curvature 2-form locally equals Ω = ω

~ = R~ sin(θ)dθ ∧ dφ. Hence on a

patch Ui of such a sphere of radius R ∈ Z/2 a potential 1-form is given byβ = −R

~ cos(θ)dφ. Indeed,dβ = −d(R~ cos(θ)dφ) = − ∂

∂θ(R~ cos(θ))dθ ∧ dφ − ∂

∂φ(R~ cos(θ))dφ ∧ dφ =

− ∂∂θ

(R~ cos(θ))dθ ∧ dφ = R~ sin(θ)dθ ∧ dφ.

Since the spheres S2 of half-integer radius R ∈ Z/2 are Kahler manifolds,they admit the natural Kahler polarization P ≡ (TS2)(0,1). From subsection6.8 it follows that the only quantizable observables are the ones preservingthe polarizations. Let f ∈ C∞(S2,R;P ). The corresponding Hamiltonianvector field in spherical coordinates equals, using (A.1),

Xf ≡ 12ωkj ∂f

∂xj∂∂xk

= 12ωθφ ∂f

∂θ∂∂φ

+ 12ωφθ ∂f

∂φ∂∂θ

= 1R2 sin(θ)

(∂f∂θ

∂∂φ− ∂f

∂φ∂∂θ

).

Henceβ(Xf ) = −

(R2

~ cos(θ)dφ)

1R2 sin(θ)

∂f∂θ

∂∂φ− ∂f

∂φ∂∂θ

=

−(R2 cos(θ)R2 sin(θ)

)∂f∂θ

= − cot(θ)∂f∂θ.

Now let s ∈ ΓX(S2) and write s = g1 for g ∈ C∞(S2,C). Then,∇Xf s = ∇Xf

(g1)≡ Xf (g)1 + g∇Xf

(1)

= f, g1− igβ(Xf )1 =1

R2 sin(θ)

(∂f∂θ

∂g∂φ− ∂f

∂φ∂g∂θ

)1 + i cot(θ)∂f

∂θg1.

In the third identity (49) and (67) are used. Finally, the quantization assign-ment Q(f) in spherical coordinates becomes, using (87),

Q(f) = −i~⟨

1

R2 sin(θ)

(∂f

∂θ

∂g

∂φ− ∂f

∂φ

∂g

∂θ

)+ i cot(θ)g

(∂f

∂θ

)⟩+f for f ∈ C∞(S2,R;P ). (102)

7.3.2 Quantizing observables on the spheres of half-integer radiusin the stereographic coordinate system

Notice that in spherical coordinates the symplectic form ω has a singularpoint on both the patches U+ and U−, namely the point θ = π in the first

115

case and the point θ = 0 in the second case. One can get around this prob-lem in case the stereographic coordinates are used as local coordinates onthe sphere. We know from subsubsection 5.2.2 that in projective coordinates

the symplectic form on U− equals ω = i2R

(1 + zz

4R2

)−2dz ∧ dz. Viewing

the south pole of the sphere as the point at ∞, this symplectic form onlyhas a singular point at ∞. One can get rid of this singularity by applyingthe change of coordinates z = 1

ωfor ω ∈ C. As a consequence, when z →∞

then ω → 0. Applying this change of coordinates one gets, taking 2R = 1for notational convenience,ω = 1

(1+|z|2)2dz ∧ dz = 1(1+1/ω2)2d( 1

ω) ∧ d( 1

ω) =

|ω|4(1+|ω|2)2

1ω2ω2dω ∧ dω = 1

(1+|ω|2)2dz ∧ dω.

In this new coordinates ω has no singular point on U− anymore. Noticethat on U+ the symplectic form ω does have a singular point at∞, but thiscan be get rid of by applying the same change of coordinates again. Then ωgets back the original form in which it has no singular points on U+.

Let us now calculate the quantization assignment in stereographic coordi-nates. We choose to do this on the chart U− (the calculation on the chartU+ is similar). On a sphere of half-integer radius R ∈ Z/2 it locally holds

for the curvature 2-form that Ω = ω~ = i

4~R2

(1 + zz

4R2

)−2dz ∧ dz. On U− we

can choose the corresponding 1-form to be β = −i~

(z/2R)dz1+zz/4R2 . Indeed,

dβ = −i~

∂∂z

(z/2R

1+zz/4R2

)dz ∧ dz − i

~∂∂z

(z/2R

1+zz/4R2

)dz ∧ dz =

−i~

∂∂z

(z/2R

1+zz/4R2

)dz ∧ dz = i

~

⟨1/2R

1+zz/4R2 − zz/4R2

(1+zz/4R2)2

⟩dz ∧ dz =

i2R~

(1 + zz

4R2

)−2dz ∧ dz.

Let f ∈ C∞(S2,R;P ). The corresponding Hamiltonian vector fields in pro-jective coordinates are equal too, again using (A.1),

Xf = 12ωzz ∂f

∂z∂∂z

+ 12ωzz ∂f

∂z∂∂z

= R(1 + zz

4R2

)2 (∂f∂z

∂∂z− ∂f

∂z∂∂z

).

Consequently,

β(Xf ) =(−i~

(z/2R)dz1+zz/4R2

)(R)

(1 + zz

4R2

)2 (∂f∂z

∂∂z− ∂f

∂z∂∂z

)=

(−i~z/2R

1+zz/4R2 )(R)(1 + zz

4R2

)2 ∂f∂z

= −iz2~ (1 + zz/4R2) ∂f

∂z.

Moreover,

∇Xf s = R(1 + zz

4R2

)2 (∂f∂z

∂g∂z− ∂f

∂z∂g∂z

)1− z

2~ (1 + zz/4R2) ∂f∂zg1.

Finally, the quantization assingment in stereographic coordinates becomes

116

Q(f) = −i~⟨R(

1 +zz

4R2

)2(∂f

∂z

∂

∂z− ∂f

∂z

∂

∂z

)− z

2~(1 + zz/4R2

) ∂f∂z

⟩+f for f ∈ C∞(S2,R;P ). (103)

7.4 The irreducible unitary representations of SU(2)

In this subsection we first determine the irreducible unitary reprensentationsof SU(2) by means of the so-called infinitesimal method, as describedin ([26]). After this, we conclude by giving a heuristic argument how todetermine the irreducible unitary representations of SU(2) by means of theorbit method.

7.4.1 Finding the irreducible representations of SU(2) by meansof the infinitesimal method

To recall,

SU(2) ≡ x ∈ Mat(2,C)|x†x = I, det(x) = 1.

and

su(2) = X ∈ Mat(2,C)|X = −X†, trX = 0.

Now look at the complexification of su(2), that is su(2)⊗C. Since in su(2)⊗Cthe scalar multiplication is extended to the complex numbers, let us look atwhat this implies for a particular matrix A ∈ su(2). We have, for z ≡ x+iy ∈C, x, y ∈ R, that(zA)† = ((x+ iy)A)† = xA† − iyA† = −xA+ iyA.Hence for a purely imaginary scalar multiplication this implies(iA)† = iA.So if we define B ≡ iA ∈ su(2)⊗C it follows that su(2)⊗C is the extensionof su(2) with Hermitian matrices. Since every matrix C ∈ Mat(2,C) can bewritten as the sum of a Hermitian and anti-Hermitian matrix viaC = 1

2(C + C†) + 1

2(C − C†)

with(

12(C + C†)

)†= 1

2(C + C†)(

12(C − C†)

)†= −1

2(C − C†),

it follows that su(2)⊗ C ∼= sl(2,C) with

sl(2) ≡ C ∈ Mat(2,C)|tr(C) = 0, (104)

117

the Lie algebra of general 2 by 2 complex valued traceless matrices. Sincethis Lie algebra has complex dimension 3 it spanned by the set of matrices:

X =

(0 10 0

), Y =

(0 01 0

), E =

(1 00 −1

).

By ordinary matrix multiplication it can be checked that

[X, Y ] = E, [X,E] = −2X, [Y,E] = 2Y. (105)

The Lie group SU(2) acts on C2 by ordinary matrix multiplication. Theassociated representation π on the space P (C) of polynomial functions p :C→ C is given by the formula

π(g)p(z) ≡ p(g−1z) g ∈ SU(2), z ∈ C. (106)

Indeed π defines a representation, since π(g) is clearly a linear endomorphismand, for g1, g2 ∈ SU(2), we haveπ(g1g2)p(z) = p((g1g2)−1z) = p(g−1

2 g−11 z) = π(g1)p(g−1

2 z) = π(g1)π(g2)p(z).The subspace Pn ≡ Pn(C2) of homogeneous polynomials of degree n is aninvariant subspace for π. The restriction of π to Pn, denoted by πn, is anirreducible representation of SU(2). We denote the associated representationof su(2) in Pn(C2) by πn∗. Then the representation of sl(2,C) in Pn(C2)equals (πn∗)

C, the complexification of πn∗. The basis of Pn(C2) is given by

pj(z) = zj1zn−j2 , (z ∈ C2, j ∈ 0, 1, ..., n). (107)

Let p ∈ Pn(C2), ξ ∈ su(2) and t ∈ R. Then, using (106) it follows that,[πn∗(ξ)p](z) = d

dtp(exp−tξ z)

∣∣t=0

,and hence by the chain rule,[πn∗(ξ)p](z) = ∂p

∂z1(z)(−ξz)1 + ∂p

∂z2(z)(−ξz)2..

The expression on the right-hand side is complex linear in ξ, meaning thatπn∗(iξ) = iπ∗(ξ). This implies that the map πn∗ : su(2) → Pn(C2) can beextended to its complexification πC

n∗. Hence for ξ ∈ sl(2,C) and p ∈ Pn(C2),

(πCn∗)(ξ)p = −

[(ξz)1

∂

∂z1

] + (ξz)2∂

∂z2

]p. (108)

Now consider the base elements X, Y and E of sl(2,C). Substituting themfor ξ in (108) gives

Xp = −z2∂

∂z1

p, Y p = −z1∂

∂z2

p, Ep =

[−z1

∂

∂z1

+ z2∂

∂z2

]p. (109)

118

It follows that the base elements X, Y and E can be associated with theoperators

X = −z2∂

∂z1

, Y = −z1∂

∂z2

, E = −z1∂

∂z1

+ z2∂

∂z2

. (110)

One can indeed check that these operators satisfy the relations (105) andhence form a Lie algebra representation of su(2) ⊗ C ∼= sl(2,C) in Pn(C2).The idea of the infinitesimal method is to reduce the consideration of Liegroups to consideration of the associated (complexified) Lie algebras. Fromthe fact that the operators in (105) form a Lie algebra representation ofsu(2) ⊗ C it indeed follows that every irreducible continuous representationof SU(2) is equivalent to πn for some n ∈ N. For the full rigourous proof seefor example ([3]) or ([26]).

To conclude this subsection, let us consider a specific case of πn. Fromthe discussion above one already concludes that ∀n ∈ N there is a uniquerepresentation of dimension n + 1 (there are n + 1 polynomials pj(z) forj ∈ 0, 1, ..., n). Take n = 3. Then the representation is of the form de-picted in figure 5.

Figure 5: Schematic view of the representation π3 of SU(2) in P3(C2). The pointsrepresent the 4 basis elements of P3(C2). The arrows represent the action of the operatorsX (green), Y (violet) and E (red) on the basis elements. The numbers above the green(violet) arrows denote the coefficients in front of the basis elements on the right (left) ofthe arrow when the corresponding operator has acted on the basis element on the left(right) of the arrow. The numbers above the red arrows denote the coefficients in front ofthe basis elements when the corresponding operator has acted on them.

Here z31 , z2

1z2, z1z22 and z3

2 denote the basis elements of P3(C2). The arrowsrepresent the action of the operators X, Y and E on the basis elements.The numbers above the arrows denote the coefficients in front of the basiselements after the action has taken place. As a check, we apply the operator

119

X to the basis elements of P3(C2). We indeed calculate,−z2

∂∂z1

(z31) = −3z2

1z2,

− z2∂∂z1

(z21z2) = −2z1z

22 ,

− z2∂∂z1

(z1z22) = −z3

2 ,

− z2∂∂z1

(z32) = 0.

7.4.2 Finding the irreducible representations of SU(2) by meansof the orbit method

Consider the coadjoint orbits of SU(2) satisfying the Integrality Condition,i.e. the spheres of half-integer radius. As we have derived in subsubsection5.2.2, the stereographic coordinates z and z on U− are given in terms of thecoordinates θ and φ by

z = 2R cot(θ/2)eiφ,z = 2R cot(θ/2)e−iφ,0 ≤ φ < 2π, 0 ≤ θ < π.

Now consider the rescaled coordinates,

z− =z

2R= cot(θ/2)eiφ, z− =

z

2R= cot(θ/2)e−iφ, (111)

which span the complex plane C too. In terms of z− and z− a parametrizationof U− is given by

x1 = R(z−+z−)1+|z−|2 ,

x2 = −iR(z−−z−)1+|z−|2 ,

x3 = R(|z−|2−1)1+|z−|2 .

,

where x1, x2, x3 are the standard coordinates in R3. Indeed, using (111),

x1 = R(z−+z−)1+|z−|2 = R cot(θ/2)(eiφ+e−iφ)

1+cot2(θ/2)= R 2 cot(θ/2) cos(φ)

1+cot2(θ/2)= R sin(θ) cos(φ),

x2 = −iR(z−−z−)1+|z−|2 = −iR cot(θ/2)(eiφ−e−iφ)

1+cot2(θ/2)= −iR 2 cot(θ/2) sin(φ)

1+cot2(θ/2)= R sin(θ) sin(φ),

x3 = R(|z−|2−1)1+|z−|2 = R(cot2(θ/2)−1)

1+cot2(θ/2)= R cos(θ),

which is the parametrization of U− in spherical coordinates (see subsub-section 5.2.1). Instead of taking x1, x2 and x3 as the coordinates spanningU− ∼= C, one can also consider the linear combinatinions x− = x1 − ix2,x+ = x1 + ix2 and x3 as the coordinates spanning U− ∼= C. In terms of z−and z− these equal,

120

x+ = x1 + ix2 =(R(z−+z−)

1+|z−|2

)+ i(−iR(z−−z−)

1+|z−|2

)= 2Rz−

1+|z−|2 ,

x− = x1 − ix2 =(R(z−+z−)

1+|z−|2

)− i(−iR(z−−z−)

1+|z−|2

)= 2Rz−

1+|z−|2 ,

x3 = R(|z−|2−1)1+|z−|2 .

The quantization assignment (103) can be rewritten in terms of z− and z−,giving,

Q(f) = −i~⟨

1

4R(1 + |z−|2)2

(∂f

∂z−

∂

∂z−− ∂f

∂z−

∂

∂z−

)− z−

2(1 + |z−|2)

∂f

∂z−

⟩+f for f ∈ C∞(S2,R;P ). (112)

Applying the assignment (112) to x+, x− and x3 gives the operators

Q(x+) = (z−)2 ∂∂z−− 2Rz−,

Q(x−) = − ∂∂z−

,

Q(x3) = z−∂∂z−−R.

Note that here is used that Q(f) acts on holomorphic sections, hence the∂∂z−

-terms can be dropped in the calculation. The determined operators

Q(x+), Q(x−), Q(x3) satisfy the same characteristic equations

[r1, r2] = 2r3, [r1, r3] = −2r2, [r2, r3] = 2r1, (113)

as the Lie algebra generators r1, r2, r3 of su(2) (see subsubsection 4.2.2) .Indeed, for s ∈ HP ,

[Q(x+), Q(x−)]s =[(z−)2 ∂

∂z−− 2Rz−,− ∂

∂z−

]s =

−(

(z−)2 ∂∂z−− 2Rz−

)∂s∂z−

+ ∂∂z

((z−)2 ∂

∂z−− 2Rz−

)s =

− z2−∂2s∂z2−

+ 2Rz ∂s∂z−

+ 2z−∂s∂z−

+ z2−∂2s∂z2−− 2Rs− 2Rz−

∂s∂z−

=

2(z−

∂∂z−−R

)s = 2Q(x3)s⇔ [Q(x+), Q(x−)] = 2Q(x3).

Similarly, one can check that [Q(x+), Q(x3)] = −2Q(x−) and[Q(x−), Q(x3)] = 2Q(x+). In subsection 7.2 we showed that for spheres ofradius k ∈ Z

2the Hilbert space consists of polynomials in z ∈ C of degree

≤ k, denoted by P (C). The homogeneous polynomials in z ∈ C of degreen ≤ k, denoted by Pn(C), form an invariant subspace of P (C). Consequently,Q(x+), Q(x−) and Q(x3) form a representation of su(2) in Pn(C) for n ≤ R.Finally, lifting the information from the Lie algebra to the Lie group (see[26]), this gives all the irreducible unitary representations of SU(2).

121

We end this section by giving an alternative method to determine that theHilbert space for spheres of half-integer radius k ∈ Z

2consists of polynomials

of degree ≤ k compared the method described in subsection 7.2.

On U− it holds that

z− =(x1 + ix2)

R− x3

. (114)

Indeed,

x1+ix2

1−x3=⟨(

R(z−+z−)1+|z−|2

)+ i(−iR(z−−z−)

1+|z−|2

)⟩/⟨R− R(|z−|2−1)

1+|z−|2

⟩=⟨

2R1+|z−|2

⟩/⟨2R

1+|z−|2

⟩= z−.

Similarly, on U+ it holds that

z+ =(x1 − ix2)

R + x3

. (115)

On the overlap of the patches U± we have z−z+ = 1, which follows from

z−z+ =(

(x1+ix2)R−x3

)((x1−ix2)R+x3

)=

x21+x2

2

R2−x23

=x2

1+x22

x21+x2

2= 1.

As the patches U± are topologically trivial, any line bundle is trivial whenrestricted to one of these patches. Thus, all we have to do is to give a pre-scription for gluing these trivial bundles together over, say, the equator. Ifone does this with the transition function (z−)−k = (z+)k one obtains theline bundle Lk over a sphere of radius k ∈ Z

2. To determine the global

holomorphic sections of Lk in the trivialization U±, we have to check whichlocal holomorphic functions on U± can be patched together via the transi-tion functions. Since holomorphic functions f(z) can be written as a seriesf(z) =

∑∞i=0 aiz

i with ai ∈ C, it follows that a basis for holomorphic func-tions on U+ is given by the monomials (z+)l and on U− by (z−)m for l andm non-negative integers. We thus have to find the solution to the equation

(z+)l = (z+)k(z−)m ⇔ (z+)l = (z+)k(z+)−m, (116)

where we used z+z− = 1. This equation has non-negative integer solutionsfor l ≤ k and m ≤ k, implying that the Hilbert space consists of polynomialsof degree ≤ k.

122

8 Conclusions and outlook

-Life is infinitely stranger than anything which the mind of man could invent.We would not dare to conceive the things which are really merely common-places of existence.-Sir Arthur Conan Doyle ([28])

In the first subsection of this section we will try to gain insight in the overallstructure of this thesis and recall the important results derived previously.Finally, in the last subsection, we will consider were future research effortsshould be directed.

8.1 Concluding remarks

This thesis consisted of three main parts, being

(1) Coadjoint orbits

(2) Geometric Quantization

(3) The irreducible unitary representations of SU(2).

The parts (1) and (2) are the basic ingredients used in the Orbit Method, themethod to determine all irreducible unitary representations of a Lie group.Part (3) is an application of the Orbit Method to the particular Lie groupSU(2).

(1):In part (1) we introduced the notion of coadjoint orbit, the most importantnew mathematical object that has been brought into consideration in connec-tion with the Orbit Method. As an application, we showed in two alternativeways that the coadjoint orbits of SU(2) are spheres. Subsequently, we provedthe beautiful non-trivial theorem that the coadjoint orbits of a matrix groupposses symplectic structure (and mentioned that this result is still valid forgeneral Lie groups). In particular, it followed that the coadjoint orbits ofa matrix group are always even-dimensional. Then we derived the explicitshape of the Kirillov 2-form for the coadjoint orbits of SU(2) (the spheres)in both spherical and stereographic coordinates. The symplectic structureof the coadjoint orbits provided an important link between coadjoint orbitsand geometric quantization, since quantization was as assignment applied tosmooth real functions on symplectic manifolds (classical observables).

123

(2):This brought us to part (2) of the thesis. Here we showed that GeometricQuantization, formulating a relationship between classical- and quantum me-chanics in a geometric language, is a rigorous quantization scheme applicableto general curved manifolds. In particular, it is showed that the Hilbert spaceover a curved manifold (M,ω) does not consist of square integrable scalarfunctions on (M,ω), but instead of polarized, square integrable sections ofa Hermitian line bundle-with-connection over (M,ω) with curvature Ω = ω

~ .Important roles in the construction of the Hilbert space are played by theIntegrality Condition and the notion of Polarization. We proved the Integral-ity Condition to be both a sufficient and necessary condition for a Hermitianline bundle-with-connection over (M,ω) with curvature Ω = ω

~ to exist. Asan important specific type of polarization (a way of reducing the ‘too big’prequantum Hilbert space) we considered the Kahler polarization, naturallydefined on kahler manifolds. Finally, as an important explicit realization ofgeometric quantization, we showed the assignment (87) acting on sections inthe Hilbert space HP , to satisfy all axioms (Q1)-(Q5), imposed on a quanti-zation assignment.

(3):In the third and last part we applied our geometric quantization construc-tion to the spheres, the coadjoint orbits of SU(2). We find that in thatcase the integrality condition translates into the spheres being of half-integerradius in order for smooth real functions defined on it to be quantizable.Furthermore, we showed that the Hilbert space on these spheres of half-integer radius consists of polynomials of degree n ≤ k, for a sphere of radiusk ∈ Z

2. Subsequently, we determined the irreducible unitary representations

of SU(2) by the Infinitesimal Method. Finally, as an application of the OrbitMethod we brought coadjoint orbits and geometric quantization together bydeterming all irreducible unitary representations of SU(2) by means of theOrbit Method.

8.2 Outlook

Geometric Quantization is a quantization construction applicable to symplec-tic manifolds. Looking at the classical phase space as a symplectic manifoldpresupposes to take point particles as the fundamental building blocks of na-ture. In field theory the fundamental building blocks of nature are consideredto be fields instead of point particles and therefore it is interesting to extendGeometric Quantization to fields, a first step towards making Quantum FieldTheory mathematically rigorous. An introduction to such an approach can

124

be found in, for example, ([7]). In string theory the fundamental build-ing blocks of nature are one-dimensional extensions of point particles, beingstrings. Finding a rigorous quantization scheme corresponding to such a the-ory is another very interesting prospect.

As already indicated at the beginning of the previous section the representa-tion theory of compact manifolds, such as SU(2), was already well-known tomathematicians long before the Orbit Method was invented. In that sense,SU(2) is not the most important application of the Orbit Method to consider.It is indeed shown that the Orbit Method is applicable to non-compact Liegroups as well, in case one utilizes less precise defined mathematics. At themoment one makes the notions ‘mathematically precise’ again, the resultsthe theory predicts can be shown to be incorrect. As a consequence, a lotof research can still be done on the Orbit Method applied to non-compactLie groups. Important examples of non-compact Lie groups to consider areSL(2,R) and the famous Virasoro group (named after the physicist MiguelAngel Virasoro), which has important applications in conformal field theoryand string theory. As a good start to investigate these applications of theOrbit Method, the reader is referred to ([1]) and ([27]).

125

A An identity at the heart of quantization

-Let us try to introduce a quantum Poisson Bracket which shall be the ana-logue of the classical one.-P.A.M. Dirac ([12, p.86])

Let (M,ω) be a symplectic manifold. An essential identity in the quanti-zation procedure is

[Xf , Xg] = Xf,g, (41)

where f, g ∈ C∞(M,R) and Xf , Xg, Xf,g are Hamiltonian vector fieldson M , that is, i(Xf )ω = 2ω(Xf , ·) = −df , i(Xg)ω = 2ω(Xg, ·) = −dg,i(Xf,g)ω = 2ω(Xf,g, ·) = −df, g. This identity will be proved in thisappendix.

First of all, we will derive the form of the Hamiltonian vector fields in localcoordinates. Take Y to be a vectorfield on M . Because Xf and Y are vectorfields, in local coordinates they can be written as: Xf = ξi ∂

∂xi, Y = ηi ∂

∂xi.

Consequently, in order to find the form of Xf in local coordinates we needto determine the coefficients ξi.

We have 2ω(Xf , ·) = −df ⇔ 2ω(Xf , Y ) = −df(Y ). Let us start with writingout the left-hand side of the last equality:

2ω(Xf , Y ) = 2ωijdxi∧dxj

(ξk

∂

∂xk

)(ηl

∂

∂xl

)= ωij(ξ

iηj−ξjηi) = 2ωijξiηj,

where the Einstein summation convention is used and the anti-symmetry ofthe wedge product and ωij are used in deriving the second and last equality,respectively. Writing out the right-hand side gives

−df(Y ) = − ∂f∂xi

dxi(ηj

∂

∂xj

)= − ∂f

∂xiηi.

So 2ω(Xf , Y ) = −df(Y ) ⇔(∂f∂xj

+ 2ωijξi)ηj = 0. Because the ηj are arbi-

trary we thus have ∂f∂xj

+ 2ωijξi = 0 ⇔ ∂f

∂xj= −2ωijξ

i = 2ωjiξi. The fact

that ω is a symplectic form gives that ωji is invertible, that is, ωkjωji = δki .It follows that ∂f

∂xj= 2ωjiξ

i ⇔ ωkj ∂f∂xj

= 2ωkjωjiξi = 2ξk. Hence, in local

coordinates,

Xf =1

2ξk

∂

∂xk=

1

2ωkj

∂f

∂xj∂

∂xk. (A.1)

126

The identity (41) relates the Lie algebras of Hamiltonina vector fields andfunctions on M . We already proved that (C∞(M,R), ., .) has the structureof a Lie algebra. We can use identities (41) and (A.1) to show that Hamilto-nian vector fields together with the Lie bracket for vector fields also form aLie algebra. First of all, notice that the space of Hamiltonian vector fields isa linear space, where smooth real-valued functions play the role of ‘scalars’.Furthermore, the Lie bracket for vector fields satisfies:

• bilinearity: for a ∈ R, h ∈ C∞(M,R), [aXf +Xg, Xh] = [Xaf+g, Xh] =Xaf+g,h = aXf,h + Xg,h = a[Xf , Xh] + [Xg, Xh]. Linearity in thesecond ‘slot’ of the commutator bracket can be derived similarly.

• anti-symmetry: [Xf , Xg] = Xf,g = X−g,f = −Xg,f = −[Xg, Xf ].

• Jacobi identity: for h ∈ C∞(M,R), [Xf , [Xg, Xh]] + [Xg, [Xh, Xf ]]+[Xh, [Xf , Xg]] = Xf,g,h +Xg,h,f +Xh,f,g =Xf,g,h+g,h,f+h,f,g = X0 = 0.

Moreover, regarding the map (Lie algebra homomorphism) f → Xf as anassignment of differential operators to functions, the identity (41) is also anillustration of the quantization paradigm (Dirac’s quantum condition),

Poisson Brackets→ Commutators. (A.2)

To prove identity (41) we will write out both sides of this equality in localcoordinates and check that the corresponding expressions match. The lefthand side equals

[Xf , Xg] = 14ωij∂if∂j(ω

kl∂kg∂l)− 14ωij∂ig∂j(ω

kl∂kf∂l) =14ωij∂if∂j(ω

kl∂kg)∂l + 14ωij∂ifω

kl∂kg∂j∂l − 14ωij∂ig∂j(ω

kl∂kf)∂l−14ωij∂igω

kl∂kf∂j∂l.

By switching the indices i and k in the second term of the last expression(over both indices is summed) one easily sees that the second and fourthterm in this expression cancel. The remaining expression can be written outfurther:

14ωij∂if∂j(ω

kl∂kg)∂l − 14ωij∂ig∂j(ω

kl∂kf)∂l =14ωij∂ifω

kl∂j∂kg∂l−14ωij∂igω

kl∂j∂kf∂l+14ωij∂if∂jω

kl∂kg∂l−14ωij∂ig∂jω

kl∂kf∂l.

Writing out the right-hand side of identity (41) gives, using the definitionof the Poisson bracket in local coordinates, that

127

Xf,g = −12ωij∂if, g∂j = 1

4ωij∂i(ω

kl∂kf∂lg)∂j =14ωij∂iω

kl∂kf∂lg∂j + 14ωijωkl∂i∂kf∂lg∂j + 1

4ωijωkl∂kf∂i∂lg∂j.

If we apply the index interchange j → l, l→ i, i→ k, k → j the second termof the last expression becomes 1

4ωklωji∂k∂jf∂ig∂l = −1

4ωklωij∂k∂jf∂ig∂l, and

if we apply the index interchange j → l, k → i, i → k, l → j the thirdterm of the last expression becomes 1

4ωklωij∂k∂jg∂if∂l. Hence we see that

the terms containing no derivatives of ωkl match on the left- and right-handside of identity (41). What is left to check to establish identity (41), is that

−ωij∂iωkl∂kf∂lg∂j + ωij∂if∂jωkl∂kg∂l − ωij∂ig∂jωkl∂kf∂l = 0.

If we apply the index interchange i → k, k → l, l → j, j → r to thesecond term of this expression, it becomes ωkr∂rω

lj∂kf∂lg∂j. If we apply theindex interchange i→ l, l→ j, j → r to the third term of this expression, itbecomes −ωlr∂rωkj∂kf∂lg∂j. As a whole, the last equality then implies

−ωij∂iωkl + ωkr∂rωlj − ωlr∂rωkj = 0.

Using the anti-symmetrie of ωkr, ωlr and ωkj this is equivalent to

ωrj∂rωkl + ωrk∂rω

lj + ωrl∂rωjk = 0. (A.3)

The identity (A.3) is known as the Jacobi identity for Poisson forms and itwill be proved in the remainder of this appendix.

Since (M,ω) is a symplectic manifold, ω is closed, that is dω = 0. Inlocal coordinates this can be rewritten as

d(ωijdxi ∧ dxj) = ∂kωijdx

k ∧ dxi ∧ dxj = 0⇔13(∂kωijdx

k ∧ dxi ∧ dxj + ∂iωjkdxi ∧ dxj ∧ dxk + ∂jωkidx

j ∧ dxk ∧ dxi) = 0⇔13(∂kωijdx

i ∧ dxj ∧ dxk + ∂iωjkdxi ∧ dxj ∧ dxk + ∂jωkidx

i ∧ dxj ∧ dxk) = 0⇔∂kωij + ∂iωjk + ∂jωki = 0.

The last identity is known as the Jacobi identity for symplectic forms . Wewill derive the Jacobi identity for symplectic forms from the Jacobi identityfor Poisson forms, using that ω is a symplectic form, hence invertible. Letus start by multiplying the Jacobi identity for Poisson forms by ωsl, usingωslω

lk = δks ,

ωkl∂kωmn + ωkn∂kω

lm + ωkm∂kωnl = 0⇔

ωslωkl∂kω

mn + ωslωkn∂kω

lm + ωslωkm∂kω

nl = 0⇔

128

−∂sωmn + ωslωkn∂kω

lm + ωslωkm∂kω

nl = 0

Using differentiation by parts and the fact that the boundary terms,∂k(ωslω

lm) = ∂kδms , vanish, we get

−∂sωmn − ωkn∂kωslωlm − ωkm∂kωslωnl = 0

We can write the first term of this expression as ωmr∂sωrlωln. This follows

from ∂s(ωmnωnl) = 0⇔ ∂sω

mnωnl +ωmn∂sωnl = 0⇔ ∂sωmn +ωmr∂sωrlω

ln =0 ⇔ −∂sωmn = ωmr∂sωrlω

ln. Multiplying the resulting expression by ωpmand ωnq respectively, gives

ωmr∂sωrlωln − ωkn∂kωslωlm − ωkm∂kωslωnl = 0⇔

∂sωplωln − ωnk∂kωsp + ∂pωslω

nl = 0⇔∂sωpq + ∂qωsp + ∂pωqs = 0,

which is indeed the Jacobi identity for symplectic forms! Identity (41) isproved.

129

References

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132

Index

Cech construction, 86Cech cohomology group, 83s0(3), 37su(2), 32

generators of, 36pth de Rham cohomology group, 851-form, 16

Actionleft, 23right, 23transitive, 24

ad, 29ad∗, 31Adapted potential, 105Adjoint representation, 26Automorphism, 19

group, 20

Canonical coordinates, 62Cartan formula, 17Category theory, 18Coadjoint orbit, 29, 123Coadjoint representation, 27Coboundary, 82Cocycle, 82Complete metric space, 67Complex inner product space, 66Complex structure, 101Connection, 75

compatible, 75Cotangent bundle, 16Cotangent space, 16Curvature 2-form, 76Curve, 16

De Rham isomorphism, 85Diffeomorphism, 20Distance function, 66

Distribution, 99completely integrable, 100involutive, 100

Endomorphism, 19Exponential map, 21Exterior derivative, 17

Flow, 18Formal complex conjugate, 75

Gauge-transformations, 73Geometric Quantization, 13, 55, 123

Hamilton’s equations, 62Hamiltonian, 62Hamiltonian vector field, 58Hermitian line bundle with connec-

tion, 75Hermitian operator, 67Hermitian structure, 75

compatible, 75Hilbert space, 66, 105

for coadjoint orbits SU(2), 112Homomorphism, 19

of G-representations, 35

Infinitesimal generator, 18Infinitesimal method, 117Inner product, 66Integrable, 101Integrality condition, 86

for spheres, 109Invariant subspace, 24Isomorphism, 19

Jacobi identity, 22for Poisson forms, 128for symplectic forms, 39, 128

133

k-form, 17Kahler potential, 104Kirillov form, 39

Lagrangian submanifold, 61Lagrangian subspace, 61Lie algebra, 22

homomorphism, 22Lie bracket, 17, 29Lie derivative, 17Lie group, 20

automorphism, 21homomomorphism, 21isomorphism, 21

Line bundle, 74Liouville form, 71

Manifold, 14almost complex, 101complex, 101integral, 100Kahler, 102Poisson, 45symplectic, 39, 56

Metaplectic correction, 107Morphism, 18

Norm, 66

Object, 18Observable, 63

complete set of, 65Orbit, 24

space, 24Orbit Method, 13, 123

for SU(2), 120

Partition of unity, 83Phase space, 62Poisson bracket, 45Polarization, 101

Kahler, 103

Prequantization, 70Prequantum Hilbert space, 96Pull-back, 20Pure quantum state, 67Push-forward, 20Pyramid cover, 110

Quantization, 12, 67for coadjoint orbits SU(2), 114

Quantum condition, 69Quantum mechanics, 12

Ray, 67Representation, 24

equivalent, 25irreducible, 24Lie algebra, 25

Section, 15, 74polarized, 101

Stereographic projection, 50Structure of a smooth manifold, 15SU(2), 31Submanifold, 15Symplectic potential, 57Symplectic vector space, 61

Tangent bundle, 16Tangent space, 16Tangent vector, 16Topological group, 23Transition function, 79

Vector bundle, 15Vector field, 16

left invariant, 21

134