Geometric quantization and the orbit method

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Geometric Quantization and The

Orbit Method

July 12, 2019

Author:

Khallil Berrekkal

Supervisors:

prof. dr. Eric Opdam

dhr. dr. Vladimir Gritsev

Bachelor Thesis

Mathematics

and

Physics

Institute of Physics

Korteweg-de Vries Instituut voor Wiskunde

Faculteit der Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam

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Abstract

In this thesis the process of geometric quantization is described. Using geometry, a quantum system is constructed from a classical system, meaning a Hilbert space and a quantum observable are constructed, given a phase space and classical observable. We also look at the orbit method, where unitary irreducible representations of Lie groups are found, using geometrical objects called coadjoint orbits. The orbit method is the mathematical counterpart of geometrical quantization. Here we have looked at two examples, namely the Heisenberg Lie group and SU (2).

Title: Geometric Quantization and The Orbit Method

Author: Khallil Berrekkal, khallil1997@gmail.com, 11293780 Supervisors: prof. dr. Eric Opdam, dhr. dr. Vladimir Gritsev Second reviewers: dhr. dr. H.B. Posthuma and dr. W.J. Waalewijn End date: July 12, 2019

Korteweg-de Vries Instituut voor Wiskunde Universiteit van Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.science.uva.nl/math Institute of Physics

Universiteit van Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.iop.uva.nl

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1

Introduction . . . 5

I

Part One: Mathematical Framework

7

2

Symplectic Geometry and Classical Mechanics . . . 9

2.1 Hamilton Formalism 9 2.2 Symmetries and Lie Groups 12

3

Mathematical Notions . . . 15

3.1 Connections and Bundles 15 3.1.1 Bundles . . . 15

3.1.2 Connections . . . 16

3.2 Integrality 19 3.2.1 Parallel Transport . . . 19

3.2.2 The Condition . . . 20

II

Part Two: Quantization

23

4

Geometric Quantization . . . 25

4.1 Prequantization 25 4.2 Polarization 27 4.2.1 The real case . . . 28

4.2.2 The complex case . . . 28

4.2.3 The reduced Hilbert space . . . 29

4.2.4 Kähler . . . 29

4.3 _{Quantization of R}2 ₃₀ 4.3.1 Integrality condition and the prequantum Hilbert space . . . 30

4.3.2 The correct Hilbert space - Real polarization . . . 30

4.3.3 Quantizing observables . . . 31

4.4 Quantization of S2 31 4.4.1 Integrality condition . . . 31

4.4.2 Constructing the prequantum Hilbert space of S2 _{. . . 31}

4.4.3 The correct Hilbert space of S2 . . . 32

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4

III

Part Three: Orbits

35

5

The Orbit Method . . . 37

5.1 Coadjoint Orbits 37 5.1.1 The symplectic structure on coadjoint orbits . . . 38

5.2 Quantization and Representations 39 5.2.1 The Heisenberg algebra and group . . . 39

5.2.2 Quantizing the coadjoint orbits of the Heisenberg Lie group . . . 40

5.2.3 Heisenberg Lie group/algebra representations . . . 41

5.2.4 SU(2) and S2 . . . 42 5.2.5 Quantization . . . 43 Conclusion 45 Popular Summary 46 Bibliography . . . 47 Books 47 Articles 47 Index . . . 47

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1. Introduction

On the scale of everything around us that we encounter in our daily lives, everything can be described pretty well with the laws of Newton. The corresponding model is also known as classical mechanics, which fits well into our intuition. Classical mechanics has multiple formulations, of which the Hamilton formulation. Hamiltonian mechanics is simply a reformulation of classical mechanics using a geometrical language. This formalism provides a more abstract understanding of the theory. A classical state is described as being an element of a symplectic manifold and the physical quantities that can be measured are smooth real-valued functions from the manifold. Since the beginning of the 20th century it is well known that when we zoom into the world of atoms, electrons and other microscopical particles, that the laws of Newton clearly fall apart. Quantum mechanics was a new theory that described the anomalies that were visible on a fundamental scale. A state is now described as a wave function, an element of a Hilbert space, and the physical quantities that can be observed are operators which do not necessarily commute with each other. Quantization is a way to try to understand quantum mechanics through classical mechanics. The aim of geometric quantization is to construct a quantum system out of a classical system, meaning that from a symplectic manifold we construct a Hilbert space and from a classical observable (physical quantity that can be measured) we construct a quantum observable, using a geometric language. At least, this is the aim for a theoretical physicist.

In mathematics representation theory is very useful to reduce abstract problems to linear algebra. Finding representations of a group is therefore of great significance. However, finding the represen-tations of a group might be difficult. Kirillov focused on finding the unitary represenrepresen-tations of Lie groups through the orbit method. The orbit method constructs a correspondence between unitary representations and geometric objects called coadjoint orbits. This is more a method than a theory, since it has only worked perfectly for certain Lie groups. Moreover, geometric quantization can be thought of as the physical counterpart of the orbit method. So for a mathematician, the aim of geometric quantization could be finding the representations of Lie groups, using a physics intuition as guideline.

***

I want to thank my supervisors prof. dr. Eric Opdam and dhr. dr. Vladimir Gritsev for providing me with this amazing subject and for guiding me through my thesis. Whenever I had questions, they were always easily available through email and open to answer my questions. I also want to thank Sliem el Ela for helping me in finding mistakes and for his support. A special thanks to thank David García Zelada who was prepared to help me even when I got stuck around 4 am, when pulling all-nighters.

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I

2

Symplectic Geometry and Classical Me-chanics . . . 9

2.1 Hamilton Formalism

2.2 Symmetries and Lie Groups

3

Mathematical Notions . . . 15

3.1 Connections and Bundles 3.2 Integrality

Part One: Mathematical

Framework

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2. Symplectic Geometry and Classical Mechanics

S

ymplectic geometry is the geometrical language of classical mechanics. It is a strong tool to give a coordinate free formulation of the equations of motion of some system, considering phase spaces as symplectic manifolds. First we need to define a symplectic manifold.

Definition 2.0.1 — Symplectic manifold. A symplectic manifold (M, ω) is a smooth manifold Mendowed with a 2-form ω such that

1. ω is closed, i.e. dω = 0,

2. ω is non-degenerate, i.e. ωp(v, w) = 0 ∀w ∈ TpM =⇒ v = 0.

In local coordinates we write ω = ∑ni, j=1ωi jdxi∧ dxj, where ωi jis an anti-symmetric, invertible

matrix. From this it follows that a symplectic manifold should always be even dimensional, since det(ωi j) = det(ωi j)T = det(−(ωi j)) = (−1)ndet(ωi j). The determinant is only non-zero if n is

even.

Example 2.0.2 Let R2nhave global coordinates (q1, . . . , qn, p1, . . . , pn). We can equip R2nwith

the canonical symplectic form ω0= ∑ni=1dqi∧ dpi.

Definition 2.0.3 — Morphism of symplectic manifolds . Let (M, ω1) and (N, ω2) be

sym-plectic manifolds with F : M → N a smooth map. F is called symsym-plectic or a morhphism of symplectic manifoldsif

F∗ω2= ω1.

Moreover, if F is a diffeomorphism and F−1is symplectic as well, we call F a symplectomor-phism.

These basic definitions give us a very strong tool to formulate classical mechanis as we will see in the next paragraph. We need one more important result from symplectic geometry which we will not prove.

Theorem 2.0.4— Darboux. Let (M, ω) be a 2n−dimensional symplectic manifold. For every m∈ M, there exists an open neighbourhood U, such that (U, ω|U) is symplectomorphic to an

open neighbourhood in R2nendowed with the canonical symplectic form ω0.

This is a nice result, since it means that every symplectic manifold has locally the same symplectic structure as R2n.

2.1 Hamilton Formalism

First we will formulate the well known coordinate-dependent Hamilton formalism, where we don’t really need a symplectic structure yet. Let M = R2n_{with global coordinates (q}

i, pi), which we call

the phase space of some classical system and the smooth functions on M are called the observables of the system. We endow M with an observable H ∈ C∞_{(M), which we call the Hamiltonian , and}

we endow C∞_{(M) with the Poisson bracket {·, ·}}

pwhich is (for the coordinate dependent version)

defined as { f , g}P= n

∑

i=1 ∂ f ∂ qi ∂ g ∂ pi − ∂ f ∂ pi ∂ g ∂ qi . For the Hamilton formalism we postulate that:

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10 Chapter 2. Symplectic Geometry and Classical Mechanics

1. a classical state is given by an element of the phase space M; 2. the dynamics of the classical system is given by

d f

dt = {H, f }P, with f ∈ C∞_(M).

Symplectic geometry comes in handy if we want to give a coordinate free formulation of the Hamilton formalism. The postulates itself are not really different, but we can define the Poisson bracket using the symplectic structure. The main idea is that we associate a Hamiltonian vector field to the Hamiltonian and the dynamics will be interpreted by how an observable changes along the trajectories of the Hamiltonian vector field flow.

Definition 2.1.1 — Hamiltonian vector field. Let (M, ω) be a symplectic manifold and H ∈ C∞_{(M). We call a vector field X}

Hon M a Hamiltonian vector field when we have that

ω (XH, −) =dH(−).

We call (M, ω, H) a Hamiltonian system.

Now for f , g ∈ C∞_{(M) we define the Poisson bracket coordinate free as}

{ f , g}PB ω (XF, Xg).

Where you should note that ω(Xf, Xg) ∈ C∞(M). Lemma 2.1.2 For f , g ∈ C∞_{(M) we have { f , g}}

P= Xg( f ).

Proof. We simply write it out starting from the right hand side. Xg( f ) =d f (Xg) = ω(Xf, Xg) = { f , g}P.

By Darboux we can always choose local coordinates for any symplectic manifold (M, ω) such that ω has the form of ω0= ∑ni=1dqi∧ dpi. This is very useful for the following theorem.

Theorem 2.1.3 For a chart (qi, pi) on a symplectic manifold (M, ω), where the symplectic form

ω is of the form ω0= ∑ni=1dqi∧ dpi, we have, for smooth functions f , g ∈ C∞(M),

{ f , g}P= n

∑

i=1 ∂ f ∂ qi ∂ g ∂ pi −∂ f ∂ pi ∂ g ∂ qi .

Proof. From the previous lemma we know in particular that { f , g}P=d f (Xg).d f can be written

in local coordinates as d f = n

∑

i=1 ∂ f ∂ qi dqi+ ∂ f ∂ pi dpi. Similarly, dg = n

∑

i=1 ∂ g ∂ qi dqi+ ∂ g ∂ pi dpi.

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2.1 Hamilton Formalism 11 Xgis a vector field and can be written in local coordinates as

Xg= n

∑

i=1 ai ∂ ∂ qi + bi ∂ ∂ pi ,

where ai and biare smooth functions. We’ll try to find those coefficient functions aiand bi, using

that ω (Xg, −) = n

∑

i=1 dqi∧ dpj(ai ∂ ∂ qi + bi ∂ ∂ pi , −) = n

∑

i=1 aidpi− bidqi.

By definition this is equal to dg. By comparing the terms we find our coefficient functions. Therefore Xg= n

∑

i=1 ∂ g ∂ pi ∂ ∂ qi −∂ g ∂ qi ∂ ∂ pi . (2.1)

By writing out the Poisson bracket, we get { f , g}P= ω(Xf, Xg) =d f (Xg) = n

∑

i=1 ∂ f ∂ qi dqi+ ∂ f ∂ pi dpi ∂ g ∂ pi ∂ ∂ qi −∂ g ∂ qi ∂ ∂ pi = n

∑

i=1 ∂ f ∂ qi ∂ g ∂ pi − ∂ f ∂ pi ∂ g ∂ qi . Therefore because of Darboux, for any symplectic manifold (M, ω), we’re allowed to choose local coordinates such that

d f = n

∑

i=1 ∂ f ∂ qi dqi+ ∂ f ∂ pi dpi,

for f , g ∈ C∞_{(M). The postulates for the Hamilton formalism state the same, but now it has been}

extended to all symplectic manifolds.

Theorem 2.1.4 In local coordinates (xi), we can write for a symplectic manifold (M, ω) and

f ∈ C∞_(M) Xf = ωk j ∂ f ∂ xj ∂ ∂ xk .

Proof. Throughout the theorem and the proof, the summation convention is used. In a local frame (∂i), we can write for the vector field Xf and Y

Xf = ξi ∂ ∂ xi , Y = ηi ∂ ∂ xi ,

for smooth real valued functions ξiand ηi. Applying the symplectic form on these vector fields,

gives us in local coordinates ω (Xf,Y ) = ωi jdxi∧ dxj ξi ∂ ∂ xi ηi ∂ ∂ xi = ωi j(ξiηj− ξjηi) = ωi jξiηj.

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12 Chapter 2. Symplectic Geometry and Classical Mechanics

We made use of the anti-symmetry of ωi j in our last step. Now we write outd f (Y ) and get ∂ f

∂ xidxi(ηj

∂ ∂ xj) =

∂ f

∂ xiηi. Using the definition of Xf, we obtain

ω (Xf,Y ) =d f (Y ) =⇒ ωi jξiηj= ∂ f ∂ xi ηi =⇒ (ωi jξi− ∂ f ∂ xi )ηi= 0.

Since this holds for every vector field Y , we can conclude ωijξi− ∂ f ∂ xi = 0 =⇒ ωijξi= ∂ f ∂ xi .

By ω being a symplectic form, ωi j is invertible. Meaning we have ωkiωi j= δkj, the identity matrix.

By applying ωki_{to the left and right of ω}

ijξi=_{∂ x}∂ f

i, we get ξk= ω

ki ∂ f

∂ xi. By change of index, we get

Xf = ξk ∂ ∂ xk = ωk j∂ f ∂ xj ξk ∂ ∂ xk . Usually we consider the phase space M as a cotangent bundle of a configuration space Q. A coordinate system (qi) on Q, which represents the position, can be extended to a coordinate system

(qi, pi) on M = T∗Q, with pithe momentum. This induces a symplectic form ∑n_i=1dqi∧ dpiin the

local coordinates. The Hamiltonian is in general the total energy (kinetic + potential energy).

Example 2.1.5 — One dimensional free particle. Consider a free particle of mass m in one

dimension. The classical system is given by (M, ω, H), with the phase space M simply being R2 with coordinates (q, p), ω the canonical symplectic formdq ∧ dp and H = _2mp2. Using the Hamilton formalism we can describe the dynamics of the system.

dq dt = {q, H}P =∂ q ∂ q ∂ H ∂ p − ∂ q ∂ p ∂ H ∂ q = p m. And dp dt = {p, H}P =∂ p ∂ q ∂ H ∂ p− ∂ p ∂ p ∂ H ∂ q = 0.

This means that the particle moves with a speed of p/m and the momentum does not change.

2.2 Symmetries and Lie Groups

A symmetry in physics is some transformation of a system that leaves the system unchanged, mean-ing that the equations of motion stay the same. In classical mechanics, all the equations of motion are encoded in a Hamiltonian system (M, ω, H). Studying symmetries is important, since, from the famous Noether’s theorem, we know it is the same as studying conserved quantities. However, We shall see that studying symmetry from a physical perspective might help us better understanding symmetry from a mathematical perspective, in particular representation theory.

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2.2 Symmetries and Lie Groups 13

Definition 2.2.1 — Symmetry. Let (M, ω, H) be a Hamiltonian system. A symmetry is a diffeomorphism S : M → M, such that

S∗H= H and S∗ω = ω .

A symmetry S is therefore in particular a symplectomorphism.

Symmetries form a group with composition as operator. As we shall understand through some examples, we want the group of symmetries to have some smoothness to it. Therefore we like to think of symmetries as Lie groups.

Definition 2.2.2 — Symmetry group. Let (M, ω, H) be a Hamiltonian system and G a Lie group. Let

S: G × M → M

be an action such that Sg: M → M is a symmetry for all g ∈ G. We call G a symmetry group of

the Hamiltonian system.

Example 2.2.3 Consider the one dimensional free particle system like in Example 2.1.5. Take

the Lie group (R, +), acting on M = R2by translation: St : R2→ R2, (q, p) 7→ (q + t, p),

for t ∈ R. We claim that St is a symmetry for all t ∈ R. We see that

S_t∗ω = S∗t(dq ∧ dp) = (dS ∗ tq) ∧ (dS ∗ t p) = ( ∂ ∂ q(q + t)dq) ∧ ( ∂ ∂ p(p)dp) = dq ∧ dp = ω, and S_t∗H= H ◦ St = H(q + t, p) = p 2m= H.

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3. Mathematical Notions

T

his chapter is meant to develop the necessary machinery, in order to understand the processof prequantization. We introduce some notions like covariant derivatives of sections and the curvature 2-form which we all need for prequantization. Therefore this chapter will mainly contain definitions.

3.1 Connections and Bundles

3.1.1 Bundles

Definition 3.1.1 — Complex line bundle . Let M be a smooth manifold. A smooth manifold Lis called a complex line bundle over M, if it is a complex vector bundle bundle over M of rank 1.

From basic differential geometry we already know what a vector bundle is. One could argue whether it was necessary to even define complex line bundles. The main difference between this and bundles from Lee [6], is that we consider complex vector spaces. Moreover, complex line bundles will be used often in the process of prequantization. We will therefore also simply refer to them as line bundles.

We follow [4] on constructing bundles. Let (E, π, M) be a vector bundle of rank n, and let {Uα}

be an open cover of M that gives the trivializations of the bundle, together with the trivialization maps φα: π−1(Uα) → Uα× Cn. This gives us the transition maps on the non-empty intersections

Uα∩Uβ: φβ α: Uα∩Uβ→ Gl(n, C), which is defined by φβ◦ φ 1 α(x, v) = (x, φβ α(x)v), where x ∈ Uα∩ Uβ and v ∈ R

n_{. The following theorem tells us how the construction bundles of}

bundles is related to transition maps.

Theorem 3.1.2 Let (E, π, M) be a vector bundle with trivialization chart {Uα} and

correspond-ing maps {φα}.

E=

∏

α

Uα× R n_{/ ∼,}

where the equivalence relation is defined by (x, v) ∼ (y, w) ⇔ x = v and w = φβ α(x)v,

for x ∈ Uα, y ∈ Uβand v, w ∈ R n_.

Proof. See [4] and [3] for a proof of this and for more information on constructing bundles. This theorem is interesting since it helps us constructing vector bundles using transition maps. If we find an open cover {Uα} of M, such that each Uα is contractible, then we know for sure that

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16 Chapter 3. Mathematical Notions

the bundle is trivial on Uα and we can therefore use Theorem 3.1.2 to construct a bundle using

transition maps. [4]

3.1.2 Connections

In euclidean spaces the tangent spaces are all canonically identified with the space itself. It is not hard to talk about second derivatives of curves or how a vector field changes in a certain direction. All we need to do, is to compare the different vectors, which is possible, since they basically live in the same tangent space. Each element of a vector field on a manifold in general, lives in a different tangent space (see Figure 3.1). This imposes a problem to define second derivatives coordinate free. One way to solve this, is by ‘connecting‘ the different tangent spaces, through something we call a connection. This new definition of a connection is compatible with the definition of second derivatives in Euclidean spaces. To read more about this, we refer you to Lee’s book on Riemannian geometry [7].

Figure 3.1:γ (t˙0) and ˙γ (t) lie in different vector spaces. [7]

Definition 3.1.3 — Connection. Let π : E → M be a vector bundle over M, and let X(M) and Γ (E) respectively denote the space of all vector fields on M and the space of all smooth sections of E. A connection is a map

∇ : X(M) × Γ (E) → Γ (E), (X ,Y ) 7→ ∇X(Y ),

that satisfies the following properties: 1. ∇XY is lineair over C∞(M) in X :

∇f X1+X2Y = f ∇X1Y+ ∇X2Y, for f ∈ C

∞_(M);

2. ∇XY is linear over R in Y :

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3.1 Connections and Bundles 17 3. ∇ satisfies the leibniz rule:

∇X( f Y ) = f ∇XY+ (X f )Y.

R The symbol ∇ is read ‘nabla’ or ‘del’ and ∇XY is called the covariant derivative of Y in the

direction X.

Definition 3.1.4 A Hermitian structure on a line bundle L over M, is a choice of an inner product h·, ·i : π−1({x}) × π−1({x}) C × C → R on each fiber π−1({x}), in the sense that if s is a smooth section, then

H_{: M × M → R, (m, n) 7→ hs(m), s(n)i} is smooth.

The line bundle equipped with the Hermitian structure will be called a Hermitian line bundle. A connection ∇ on a Hermitian line bundle is called Hermitian or compatible with the Hermitian structure, if for every vector field X we have

h∇X(s1), s2i + hs1, ∇X(s2)i = X hs1.s2i,

for all smooth sections s1 and s2 of L. When we talk about a Hermitian line bundle L with

connection, we will always refer to a Hermitian connection.

First of all, note that hs1.s2i is a smooth function. Moreover, by local trivialization of the line

bundle, we can choose a chart such that we locally define a section s0on it that does not vanish. The

fibers of the vector spaces are all isomorphic to C, but not canonical. Since S0does not vanish there,

we could have chosen it such that hs0, s0i = 1, and induce an isomorphism of the fiber with C, with

as result that any section s can locally be written as s = f s0for a unique complex-valued smooth

function f . We call s0a local isometric trivialization of L, and we have that s0(m) is identified with

‘1’ ∈ C, for m ∈ M. Given s0as local isometric trivialization, we can associate a 1-form θ to s0.

Given a vector field X , θ (X ) is the unique smooth function, such that

∇Xs0= −iθ (X )s0.

Proposition 3.1.5 θ is a well-defined 1-form.

Proof. Being well defined in this case, would mean that the value of θ (X ) in p only depends on the value of X in p. Suppose X1(p) = X2(p), for two different vector fields X1, X2, then we must

have θ (X1)(p) = θ (X2)(p). By definition we have

θ (X )s0(p) = i∇Xs0(p),

for some p ∈ M. We can locally write X1= ∑ j

fj∂jand X2= ∑ j

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X1(p) = X2(p), we have fj(p) = gj(p) for all j. Now we compute

θ (X1)(p) = i∇X1s0(p) = i∇_∑ j fj∂js0(p) = i

_∑

j fj(p)∇∂js0(p) = i

_∑

j gj(p)∇∂js0(p) = i∇_∑ j gj∂js0(p) = i∇X2s0(p) = θ (X2)(p).

With this we conclude the prove.

R Since every section can be written as f s0for some smooth function f , we can write ∇X

locally as X − iθ (X ).

Definition 3.1.6 Curvature 2-form Let (L, ∇) be a Hermitian line bundle with connection. The curvature 2-formis the complex differential 2-form ω of ∇ that is determined by requiring that

ω (X ,Y )s = i(∇X∇Y− ∇Y∇X− ∇[X ,Y ])(s)

for all X ,Y ∈ X(L) and S ∈ Γ (L).

We state the following proposition without proof. But the idea of the proof is again, that you show that the value of ω(X ,Y )s(p) depends on X (p) and Y (p) only. [2]

Proposition 3.1.7 ω is a well-defined 2-form.

Lemma 3.1.8 Let s0 be a local isometric trivialization and θ the associated 1-form. Then the

curvature 2-form ω of ∇ is expressed locally as ω = dθ .

Proof. Let θ be the 1-form associated to the local isometric trivialization s0. Writing i(∇X∇Y−

∇Y∇X− ∇[X ,Y ]) out, gives us

i((X − iθ (X ))(Y − iθ (Y )) − (Y − iθ (Y ))(X − iθ (X )) − (XY −Y X − iθ (XY −Y X ))

= i (−iX (θ (Y )) + XY −Y X − θ (X )θ (Y ) +Yi(θ (X )) + θ (X )θ (Y ) − XY +Y X + iθ (XY −Y X )) = (X (θ (Y )) −Y (θ (X )) − θ ([X ,Y ])

=dθ (X,Y ).

In the last step we used the identity dη(X ,Y ) = X (η(Y )) −Y (η(X )) − η([X ,Y ]) [2]. With this we

conclude the proof.

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3.2 Integrality 19

3.2 Integrality

In this section we try to receive some geometrical insight into the meaning of what we call the integrality condition, which is something that necessarily applies to the introduced curvature 2-form, following the first edition of Woodhouse. [13]

3.2.1 Parallel Transport

Definition 3.2.1 — Parallel . Suppose a Hermitian line bundle π : L → M with connection and curvature 2-form ω =dθ exists. A smooth curve Γ : [a, b] → L∗= L − {0}, with tangent vector field Ξ is said to be parallel if there exists a section s and a vector field ξ ∈ X(M) satisfying,

1. ξ = π∗(Ξ) on π (Γ ([a, b]));

2. Γ ([a, b]) ⊂ s(M);

3. ∇_ξs= 0 on π (Γ ([a, b])).

The intuition behind this, is that the curve Γ is parallel to the base space. The curve does not move ‘vertically’ at some point, nor does it cross itself. See Figure 3.2 for a visualization.

So if γ : [a, b] → M is a smooth curve with p ∈ L∗_{γ (a)}, then there is a unique parallel curve Γ : [a, b] → L∗through p such that

π ◦ Γ = γ .

We say that Γ is a curve, covering γ. Moreover, we say that Γ (b) is reached from Γ (a) through parallel transportalong γ.

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For simplicity, it is safe to assume that γ : [0, 1] → M, with ξ = γ∗(_{∂ t}∂), the tangent vector field. Let

the s = φ s0be the section, satisfying the requirement for a parallel curve Γ . It follows that

∇_ξ(φ s0) = 0

=⇒ φ ∇ξs0+ (ξ φ )s0= 0

=⇒ − iφ θ (ξ )s0+ ˙φ s0= 0

=⇒ ∂ φ

∂ t = iθ (ξ )φ .

In our last step, we used the assumption, that s0does not vanish. We have an ODE (with boundary

condition φ (0)), which gives us the unique solution [10] φ (t) = exp i Z γ (t) γ (0) θ φ (0). 3.2.2 The Condition

Now suppose that γ is bounded by some 2-surfaces Σ1 and Σ2, such that Σ = Σ1∪ Σ2 forms a

closed orientable surface (so ∂ Σ1= γ and ∂ Σ2= −γ, taking the orientation into account). Then by

Stokes‘ theorem, as dθ = ω, we have I γ θ = Z Σ1 ω = − Z Σ2 ω .

Σ

γ

1

2

Figure 3.3: The integrality condition. Notice thatH

γθ =

Rγ (1)

γ (0)θ . By the uniqueness of of the ODE, we obtain the same solutions for φ (1),

which gives us φ (1) = exp i I γ θ φ (0) = exp i Z Σ1 ω φ (0) = exp −i Z Σ2 ω φ (0).

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3.2 Integrality 21 From this we derive

exp i I γ θ = exp −i Z Σ2 ω =⇒ exp i Z Σ2 ω + i Z Σ2 ω = 1 =⇒ exp i I Σ ω = 1 =⇒ i I Σω ∈ 2π iZ.

For every orientable closed 2-surface Σ, we can split it into two 2-surfaces, using some closed curve γ. We therefore see that in order for the 2-form ω to be a curvature form on a Hermitian line bundle, it must satisfy the Weil’s integrality condition, that is

1 2π

I

Σω ∈ Z,

(3.1) for every closed orientable 2-surface Σ. The following theorem tells us that beside the integrality condition (IC) being necessary, it is also a sufficient condition for the existence of a Hermitian line bundle with connection with a curvature 2-form.

Theorem 3.2.2 Suppose ω is a closed 2-form on M, such that ω/2π is integral in the sense of 3.1. Then there exists a Hermitian line bundle L over M with connection ∇, such that ω is equal the curvature 2-form of ∇.

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II

4

Geometric Quantization . . . 25 4.1 Prequantization 4.2 Polarization 4.3 Quantization of R2 4.4 Quantization of S2

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4. Geometric Quantization

I

n classical mechanics a state is an element of a symplectic manifold, where observables are smooth functions and the time evolution is given by Poisson brackets. In the quantum mechanis a state is given by an element of a Hilbert space (which is a vector space with some conditions). An observable isn’t a smooth function, but a self-adjoint operator. Given a Hamiltonian, which is a self-adjoint operator, ˆH, the time evolution of an observable ˆf is given by

d ˆf

dt =

1 i¯h[ ˆf, ˆH],

where [A, B] = AB − BA are the commutator brackets.

The idea of quantization is that we want move from classical mechanics to quantum mechanis. We try to quantize a classical system. This means that of we’re given a symplectic manifold (M, ω) and an observable f , we want to construct some corresponding Hilbert space H and a corresponding quantum observable ˆf. Moreover, we see that the commutator brackets are the quantum equivalent of the Poisson brackets, so we also want to have some kind of Lie algebra morphism involved in quantization. Dirac suggestested some conditions, and we define therefore his idea of quantization as follows:

Definition 4.0.1 — Dirac quantization. Denote L(V ) for the set of self-adjoint operators on a vector space V . Let (M, ω) be a symplectic manifold and D ⊂ C∞_{(M) a subalgebra. We define}

Dirac quantizationof (M, ω, D) as the construction of a Hilbert space H and a map Q: D → L(H), f 7→ ˆf,

where Q satisfies the following properties: 1. Q({ f , g}P) =_i¯h1[Q( f ), Q(g)];

2. Q(1) = I;

3. Q(λ f + g) = λ Q( f ) + Q(g), ∀λ ∈ R;

4. If { fα} is a complete set of classical observables, then {Q( fα)} is also complete.

For a complete set of classical obervables { fα} it means that { fα, g}P = 0 ∀α =⇒ g is a

constant function. Likewise for a completse set of quantum observables { ˆfα}, we have [ ˆfα, g] =

0 ∀α =⇒ g = λ I for some constant λ . This last condition is called the irreducibility postulate, but unfortunately, the theorem of Groenewold-van Hove [11] tells us that Dirac quantization does not exist. Dirac quantization without the irreducibility postulate is called prequantization.

4.1 Prequantization

We shall demonstrate the general process of prequantization. What we will notice, is that the Hilbert space we obtain is ‘too big’.

Definition 4.1.1 — Quantizable. A symplectic manifold (M, ω) is quantizable if 1

2π ¯h I

Sω ∈ Z

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26 Chapter 4. Geometric Quantization

For prequantization we want to construct a Hermitian line bundle with a connection having curvature 2-form which will be equal to the symplectic form ω divided by ¯h. Therefore it is important that ω /¯h satisfies the IC. From now on, we will assume that (M, ω) is a symplectic manifold and that (L, ∇) is a Hermitian line bundle with connection over M with curvature Ω = ω/¯h. We say that a section s ∈ Γ (L) is square integrable if

Z

M

hs(x), s(x)iλ (x)

is finite, where λ (x) = ∏idpidqi is the Liouville volume form. Two sections s1and s2 are said

to be equivalent, if they are almost everywhere the same with respect to the Liouville volume measure.

Definition 4.1.2 — Prequantum Hilbert space. The space of all equivalence classes of square-integrable sections of the Hermitian line bundle L over M is called the prequantum Hilbert space of M, denoted by Hpre(M).

The innerproduct on Hpre(M) is given by

hs1|s2i =

Z

M

hs1(x), s2(x)iλ (x),

which induces a norm and a metric. For prequantization, we map a smooth function f ∈ C∞_{(M) to}

the prequantum operator Qpre( f ), which acts on Hpre(M).

Qpre( f ) : Hpre(M) → Hpre(M), s 7→ (−i¯h∇Xf+ f )(s).

With other words, (pre)quantizing observables, gives us f7→ −i¯h∇Xf + f = Qpre( f ).

It is important to note here that Qprereally depends on the choice of your connection ∇. If we know

how a connection ∇ acts on a local isometric trivialization s0, then by the leibniz rule we know how

it acts on all the other sections. Moreover, the choice of a 1-form θ such that ω is locally equal to dθ , is equal to the choice of the connection ∇. In particular, we have

∇X = X − iθ (X ).

Example 4.1.3 Canonical quantization] Let M = R2nbe the euclidean space with the symplectic

form ω =dq ∧ dp and let H =1₂(q2+ p2) be the n-dimensional harmonic oscillator. As symplectic potential we choose θ =1₂(qdp − p dq). We get

I Σ ω = I ∂ Σ θ = 0.

So ω (and ω/¯h) satisfies the integrality condition. One could check that the trivial line bundle L_{= M × C is a hermitian line bundle such that it has connection ∇ with curvature ω/¯h. The} prequantum hilbert space equals the space of all square integrable complex valued smooth functions. Quantizing q and p gives us, keeping in mind that Xq= −_{∂ p}∂ and Xp=_{∂ q}∂ (see Equation 2.1), the

following: Qpre(q) = −i¯h(Xq− i ¯hθ (Xq)) + q = −i¯h(− ∂ ∂ p− i 1 2¯h(qdp − p dq)(− ∂ ∂ p) + q = −i¯h(− ∂ ∂ p − i · 1 2¯h· −q) + q = i¯h ∂ ∂ p+ 1 2q,

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4.2 Polarization 27

Figure 4.1: A polarization splits the coordinates of a manifold into two direction.

and, Qpre(p) = −i¯h(Xp− i ¯hθ (Xp)) + p = −i¯h( ∂ ∂ q − i1 2¯h(qdp − p dq)( ∂ ∂ q) + p = −i¯h( ∂ ∂ q − i · 1 2¯h· −p) + p = −i¯h ∂ ∂ q+ 1 2p. And in the same way, we get

Qpre(H) = i¯h(q ∂ ∂ q− p ∂ ∂ q). [2]

This particular example is able to show us that prequantization brings some trouble with it. When solving the prequantized harmonic oscillator, you will find that the eigenstates have eigenvalues n¯h, where n ∈ Z. This is highly problematic, since it means that there exists no ground state, which is physically incorrect. Our Hilbert space is too large and we need to reduce it. This is to be expected, since elements of the Hilbert space should either depend on the position coordinates, or the momentum coordinates only. However, the sections that live in our prequantum Hilbert space depend on 2n coordinates, instead of n coordinates. We need to get rid of the extra coordinates by adding some structure which we call a polarization.

4.2 Polarization

A way to get rid of our extra variables, is to endow our manifold with some extra structure through a choice of a sub-bundle P ⊂ T M that essentially polarizes the coordinates (see Figure 4.1). Let us as usually introduce some definitions first and understand this concept, following Litsgård.[12]

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Definition 4.2.1 Lagrangian submanifold Let (M, ω) be a 2n−dimensional symplectic mani-fold. A submanifold N ⊂ M is called Langrangian if for all p ∈ M, its tangent space Lp= TpN

satisfies the following properties:

Lpis isotropic, i.e. ωp(v, w) = 0 for all v, w ∈ Lp;

•• Lpis maximally isotropic, i.e. for any isotropic subspace LP⊂ K ⊂ TpMwe have K = Lp.

We call Lpa Lagrangian subspace of TPM. It appears to be that N always has dimension1₂dim M =

n.

4.2.1 The real case

Definition 4.2.2 A real polarization on a symplectic manifold (M, ω) is a choice of a sub-bundle P ⊂ T M, such that

1. each fiber Ppis a Lagrangian subspace of TpM;

2. for all X ,Y ∈ Γ (P), w have [X ,Y ] ∈ Γ (P).

Example 4.2.3 Consider S1 as the configuration space, giving the phase space M = T∗S1 S1× R. Our phase space admits local coordinates (θ , p) and therefore also a symplectic form ω =dθ ∧ dp. We let π be the canonical projection map

π : M → S1. Note that ker(dπ) = a ∂ ∂ θ|m+ b ∂ ∂ p|m∈ T M | a = 0, b ∈ R .

We can endow M with the polarization P B ker(dπ). For a fixed point m = (θ , p), we have that Pmis a Lagrangian subspace of TmM. For Xm,Ym∈ Pmwe can write Xm= a_{∂ p}∂ |mand Ym= b_{∂ p}∂ |m.

It follows that ω (Xm,Ym) =dθ ∧ dp(a ∂ ∂ p|m, b ∂ ∂ p|m).

By this we have that Pmis isotropic. Moreover it is maximally isotropic, since the only larger space,

containing Pm, is TmMitself. A section X ∈ Γ (P) is of the form

X = α ∂

∂ p,

where α is a smooth real valued function. It follows that for X ,Y ∈ Γ (P) that [X ,Y ] = 0, which

lies again in Γ (P).

4.2.2 The complex case

Some phase spaces, like S2do not admit real polarizations. We therefore need to generalize this concept to complex polarizations. For the basic theory on complex manifolds, we refer you to Nakahara [8], where basic concepts like differential forms and connections are extended e.g. to complexified vector fields as well.

Definition 4.2.4 — Complex polarization . Let (M, ω) be a symplectic manifold and P a n-ranked sub-bundle of TC_M_{(where T}C

pM= TpM⊗ C is the complexification of TpM). P is

called a complex polarization if

1. each fiber Ppis a Lagrangian subspace of TpCM;

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4.2 Polarization 29 3. for all X ,Y ∈ Γ (P), w have [X ,Y ] ∈ Γ (P).

4.2.3 The reduced Hilbert space

Now we have enough tools to construct the correct Hilbert space.

Definition 4.2.5 Polarized sections Let (L, ∇) be a hermitian line bundle over a symplectic manifold (M, ω) with some polarization P. A section s ∈ Γ (L) is said to be polarized (w.r.t. P) if

∇_Xs= 0,

for X ∈ Γ (P) (meaning X ∈ X(M), such that Xp∈ Pp).

These are all technical definitions that we need for our quantization program to work. Of course there is more geometrical intuition behind this, however this is beyond the scope of the thesis and for this we refer you to Woodhouse. [9] Now for our actual Hilbert space; we take (the closure of) the polarized sections of the prequantum Hilbert space. So this means that quantization is prequantization and a choice of a (complex) polarization. However, it is important that the Hilbert space is closed under action of the observables, meaning that the prequantized observables map polarized sections to polarized sections. We therefore say a smooth complex-valued function f on (M, ω) is quantizable with respect to P if Qpre( f ) preserves polarized sections.

4.2.4 Kähler

We dedicate a subsection to a special class of manifolds which are well suited for geometric quantization, namely Kähler manifolds.

Definition 4.2.6 — Complex structure . A complex structure on a vector space V is a linear transformation

J: V → V,

such that J2= −I. Moreover, if for a manifold M, Jp is a complex structure on each TpM,

varying smoothly with p ∈, then we say J is an almost complex structure on a manifold M. Complex structures are important, since they are a part of Kähler manifolds. We shall show in the next example that every Kähler manifold admits a Kähler polarization , i.e. a complex polarization P, such that P ∩ P = 0.

Definition 4.2.7 — Kähler manifold . (M, ω, J) is called a Kähler manifold if • (M, ω) is symplectic;

• (M, J) is a complex manifold, i.e. J is an almost complex structure such that for all X,Y ∈ X(M)

[JX , JY ] − J[JX ,Y ] − J[x, JY ] − [X ,Y ] = 0; • ω(JX , JY ) = ω(X ,Y );

• g(X ,Y ) B ω(JX,Y ) is a Riemannian metric on M.

Example 4.2.8 Let (M, ω, J) be a Kähler manifold. Define the the sub-bundle

P = {v ∈ TC_M_{| Jv = −iv} C T M}+_{⊂ T}C_M.

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1. P is Lagrangian: Suppose X ,YinΓ (P), then

ω (X ,Y ) = ω (JX , JY ) = i2ω (X ,Y ) = −ω (X ,Y ) =⇒ ω (X ,Y ) = 0;

2. Γ (P) is closed under Lie brackets: Suppose X ,Y ∈ Γ (P), then

0 = [JX , JY ]−J[JX ,Y ]−J[x, JY ]−[X ,Y ] = −[X ,Y ]+iJ[X ,Y ]+iJ[X ,Y ]−[X ,Y ] =⇒ J[X ,Y ] = −i[X ,Y ]; 3. Pp∩ Ppis obviously always 0, since

P = {v ∈ TC_M_{| Jv = iv} C T M}−_{⊂ T}C_M

In particular its dimension is constant for every p ∈ M.

Since the dimension of the intersection is 0, we have that P is a Kähler polarization. P and P are called the holomorphic and anti-holomorphic polarization respectively. They’re respectively

generated by ∂

∂ ¯zk and

∂

∂ zk. Once again, to read more about this in detail, see the 8th chapter of

Nakahara. [8]

If (M, ω, J) is a Kähler manifold, ω is usually called the Kähler form. We use a result from Woodhouse [9] that there exists a local smooth function f (z, ¯z), such that we can write the Kähler form in local coordinates as

ω = i¯h−∂

2_f

∂ ¯zj∂ zkd¯z

j_{∧ z}k

= i¯h∂ ∂ f , with symplectic potentials

θ = i¯h∂ f = i¯h∂ f ∂ zjdz

j_,

where θ and ¯θ are respectively adapted to P and P .

4.3 Quantization of R2

We have all our tools for the geometric quantization program, and we shall apply it in this section to quantize the 2-dimensional plane. For simplicity we shall assume ¯h = 1. The first part shall resemble Example 4.1.3.

4.3.1 Integrality condition and the prequantum Hilbert space

Let M = R2be the 2-dimensional plane with coordinates (q, p). Since R2= T ∗ R, it comes with the symplectic form ω =dq ∧ dp. R2_{doesn’t have closed 2-surface and satisfies the integrality}

condition trivially. Every bundle over R2is trivial and therefore the hermitian line bundle over M is simple R2× C, with curvature ω. The connection ∇ is given by the choice of the symplectic potential θ . Just like in Example 4.1.3, the prequantum Hilbert space consists of square integrable complex-valued smooth functions.

4.3.2 The correct Hilbert space - Real polarization

We admit R2with the real polarization P = ∂

∂ p and choose θ = −pdq as symplectic potential. Now

by the requirement for ψ, an element of the prequantum Hilbert space, that it satisfies ∇∂ ∂ p ψ = 0, we get ∇∂ ∂ p ψ = ( ∂ ∂ p− ipdq( ∂ ∂ p))ψ = 0 =⇒ ∂ ψ ∂ p = 0.

Therefore, the Hilbert space consists of square integrable complex-valued smooth functions that only depend on q.

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4.4 Quantization of S2 31

4.3.3 Quantizing observables

We quantize the observables by mapping a classical observable f to ¯f, where ¯

f= −i∇Xf + f .

By quantizing the coordinates q and p, we quantize all the observables. We have that Xq= −_{∂ p}∂

and Xp=_{∂ q}∂ . We compute

−i∇Xq+ q = −i(Xq− iθ (Xq) + q

= −i(− ∂ ∂ p− i · −pdq(− ∂ ∂ p)) + q = i ∂ ∂ p+ q, and

−i∇Xp+ p = −i(Xp− iθ (Xp) + p

= −i( ∂ ∂ q− i · −pdq( ∂ ∂ q)) + p = −i ∂ ∂ q+ 2p.

However, since these quantum observables will only act on (square integrable smooth) functions that only depend on q and need to preserve the polarization, we get

q7→ q, p 7→ −i ∂ ∂ q.

4.4 Quantization of S2

We have all our tools for the geometric quantization program, and we shall apply it in this section to quantize the 2-sphere.

4.4.1 Integrality condition

Let S2⊂ R3_{be the 2-dimensional sphere, with radius R > 0, equipped with the local coordinates}

(θ , φ ). The sphere comes with the symplectic (scaled volume) form ω = R sin(θ )dθ ∧ dφ . [9] We compute I S2ω = Z 2π 0 Z π 0 Rsin(θ )dθ dφ = 2πR Z π 0 sin(θ )dθ = 4πR. Therefore _{2π ¯h}1 H

S2ω ∈ Z, precisely when R =n¯h₂, for n ∈ R. For simplicity we set ¯h = 1, and notice

that the sphere S2is only quantizable for half-integer radii.

4.4.2 Constructing the prequantum Hilbert space of S2

Consider S2⊂ R3_{, the 2-dimensional sphere with a half integer radius R =}1

2k> 0 and symplectic

form ω = R sin(θ )dθ ∧ dφ . By the integrality condition we can construct a Hermitian line bundle with connection with curvature ω. From basic geometry we know that S2 CP1, the Riemann-sphere. Through the stereographic projection we can equip S2locally with complex coordinates.

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We can cover M = S2with U+, which is S2minus the north pole (S2− {n}) and with U− which

is the sphere minus the south pole (S2− {s}). We have through the stereographic projection that U± C. For a cover {Ui} of M, where each Uiis contractible, the line bundle is constructed as

L=

∏

i

Ui× C/ ∼ .

Where two points are equivalent depending on the transition functions. On A B U−∩U+we have a

homeomorphism given by

C − {0} U+− {s} → U−− {n} C − {0}, z 7→

1 z.

The bundle for a sphere S2with radius R is constructed by the transition functions ci, j: Ui→ Ujon

Abelow. c++= 1 c−−= 1 c+−= 1 z2R.

This gives us in this case

L=(U+× C)

∏

(U−× C) / ∼ (C × C)

∏

(C × C)/ ∼,

where the equivalence relation ∼ is defined by (p, z) ∼ (p0, z0)

if and only if p = p0, z = z0 for p, p ∈ U±, and p0=1_p, z0=_z12R for p ∈ U+, p0∈ U−. See [12] for

more details on this. The square integrable sections of this space form the prequantum Hilbert space. We will however determine directly how the sections look like after applying a polarization, since that’s more convenient.

4.4.3 The correct Hilbert space of S2

The symplectic form ω can locally be written on U−(on the sphere with radius R) as

ω = i

R∂ ∂ log 1 + |z|

2₌ i

R(1 + |z|2₎2dzd¯z,

and S2is therefore a Kähler manifold. The connection on the line bundle is given by the symplectic potential

θ = i¯z

R(1 + |z|2₎dz.

Since S2 is Kähler, it admits the anti-holomorphic polarization ¯P = ∂

∂ z. Polarized sections in

this case are sections ψ ∈ Γ (L) such that for X ∈ Γ (P), we have ∇_Xψ = 0. Furthermore, on U+

we can simply write ψ as f s0, with f some complex valued function, ans s0the local isometric

trivialization. So the polarized sections will require that ∇∂

∂ ¯zψ = 0 =⇒ ∇∂ ¯∂z( f s0) = 0 =⇒

∂ ∂ ¯zf = 0.

Here we assumed that the potential 1-form corresponding to ∇ vanishes along ¯P. [12] Since f is a holomorphic function, from complex analysis we know that the condition implies that f can

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4.4 Quantization of S2 33 be written as ∑∞

k=0akpkand hence we notice the following. On the intersection A we have by the

transition map that (p, f (p)) 7→ (1 p, f ( 1 p)) = ( 1 p, ∞

∑

k=0 akp−k) ∼ (p, p2R ∞

∑

k=0 akp−k).

Since p2R∑∞k=0akp−kshould be holomorphic, it requires that 2R − k ≥ 0. Therefore 2R > k. So

f must be a polynomial with degree ≤ 2R. Therefore the Hilbert space consists of all square integrable polynomials with degree ≤ 2R.

4.4.4 Quantizing the observables

Using the previous subsection and 2.1.4, we can find Xf, for some f ∈ C∞(M), by a straightforward

computation. This gives us Xf = iR(1 + |z|2)2(

∂ f ∂ ¯zdz −

∂ f ∂ zd¯z). And we also get

θ (Xf) = −¯z(1 + |z|2)

∂ f ∂ ¯z.

Therefore we get in stereographic coordinates ˆ f= −i∇Xf + f = R(1 + |z| 2₎2₍∂ f ∂ ¯zdz − ∂ f ∂ zd¯z) + i¯z(1 + |z| 2₎∂ f ∂ ¯z + f .

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III

5

The Orbit Method . . . 37

5.1 Coadjoint Orbits

5.2 Quantization and Representations

Conclusion Popular Summary Bibliography . . . 47 Books Articles Index . . . 47

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5. The Orbit Method

T

his chapter will discuss some concepts that have been introduced by Kirillov in [5]. These concepts revolve around some main objects here that we call coadjoint orbits. These abstract, yet easy to visualize, objects provide interesting tools to study representations of Lie groups. In fact, the method using this objects to find representations (the orbit method) is the mathematical counterpart of quantization! Roughly, the idea is that every Lie group has a representation that we call the coadjoint representation. Its orbits are symplectic manifolds, and quantizing those symplectic manifolds corresponds to finding a unitary irreducible representation (unirrep ) of the Lie group. However, this is in general not an easy method. Moreover, it only works for specific groups.

5.1 Coadjoint Orbits

Let G be a Lie group. For every g ∈ G we have a diffeomorphism Cg: G → G, that maps h to

ghg−1. This induces an isomorphism on the tangent spaces TeGthat has been identified with the

Lie algebra of G, g TeG. [6]

Adg: g → g, X 7→ TeCg(X ).

Adgis the differential of Cgat the identity element e ∈ G. Since this is an isomorphism, it means

we have a Lie group representation Ad: G → Gl(g), g 7→ Adg.

We call this the adjoint representation. Indeed, by the chain rule, this is a representation, since Ad(g1)Ad(g2)X = TeCg1(X )TeCg2(X ) = TeCg1g2X= Ad(g1g2)X .

Recall that for a representation π : G → Gl(v), the dual of the representation (π∗,V∗) is defined as

π∗: G → Gl(V∗), πg∗(ξ )(v) = ξ (πg−1(v)), (5.1)

where ξ ∈ V∗, the dual of V and v ∈ V . When V∗is the dual of V , we call the canonical duality pairing

h·, ·i : V∗×V → R, (ξ , v) 7→ ξ (v)

the natural pairing. Using this notation, we can define a dual representation (π∗,V∗) as π∗(g) B (π(g−1))∗,

where the asterisk on the right side means the dual of an operator on V∗which is defined by hA∗ξ , vi = hξ , Avi,

for A ∈ End(V ), ξ ∈ V∗and v ∈ V. The use of the natural pairing is simply for the notation. Now the coadjoint representation, is the dual representation of the adjoint representation, denoted by K. So we have a homomorphism

K: G → Gl(g∗), such that

hK(g)ξ , vi = hξ , Ad(g−1)vi.

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38 Chapter 5. The Orbit Method

Definition 5.1.1 — Coadjoint Orbit. Let G be a Lie group and K: G → Gl(g∗)

the coadjoint representation. The coadjoint orbit through ξ ∈ g∗is defined as O_ξ _{B {K(g)ξ | g ∈ G} ⊂ g}∗.

In this thesis we identify g with g∗through the map below. This identification is not necessarily always possible, but it is for the examples we shall consider. [12]

φ : g∗→ g, F 7→ XF,

where Xf is defined by

F(Y ) = hF, [Xf,Y ]i,

for all Y ∈ g. By this identification, we will use that

O_ξ = {gξ g−1: g ∈ G}. (5.2)

In general this is not true, but it is for the examples we will consider in this thesis.[5] [12]

5.1.1 The symplectic structure on coadjoint orbits

It is well known that orbits of a Lie group are embedded submanifold because of the smoothness of the group itself. However, coadjoint orbits even admit a natural symplectic structure. Given the adjoint representation of a Lie group

Ad: G → Gl(g),

we can obtain a Lie algebra representation by deriving it at the identity element, getting ad: g → Lie(Gl(g)) ⊂ End(g), X 7→ adX = Te(Ad)X .

The adjoint representation satisfies adX(G) = [X ,Y ]. We define K∗(X ) B −ad(X)∗, the obtained

Lie algebra representation from the coadjoint representation. This implies that hK∗(X )ξ ,Y i = hξ , −ad(X )Y i,

for X ,Y ∈ g and ξ ∈ g∗.

Theorem 5.1.2 Let O be a coadjoint orbit in g∗. There exists a symplectic G-invariant 2-form ω on O that is defined by

ωO(F)(K∗(X )ξ , K∗(Y )ξ ) = hξ , [X ,Y ]i.

Proof. For the proof, we refer the reader to [5]. We only prove G-invariance, by showing the following:

hAd∗(g)F, [Ad(g)X , Ad(g)Y ]i = hAd∗(g)F, [gX g−1, gY g−1]i

= hAd∗(g)F, gX g−1gY g−1− gY g−1gX g−1]i = hAd(g−1)∗, g[X ,Y ]g−1i

= hF, Ad(g−1)g[X ,Y ]g−1i = hF, [X ,Y ]i.

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5.2 Quantization and Representations 39

5.2 Quantization and Representations

The coadjoint orbits are symplectic manifolds. The first thing that comes to mind is, that we can realize these manifolds as phase spaces. Moreover, from the previous section, we see that a Lie group G acts as a symmetry on its own coadjoint orbits. The idea behind the orbit method, is that when quantizing a system that has a certain symmetry, there will be the same kind of symmetry in the quantized system. However, a symmetry on a Hilbert space, is simply a unitary representation. Moreover, G acts ’irreducibly’ on its coadjoint orbits, therefore it is natural to think that the induced unitary representation on the Hilbert space obtained from that coadjoint orbit will be irreducible as well. The orbit method is based on the philosophy of this. It is not something that has a rigorous explanation, but Kirilov showed that using this philosophy we can use quantization to find unirreps of a Lie group (see Figure 5.1).

G

1 M

H

2

quan�za�on

Lie(G)

C (M)

∞ quan�za�on

3 _Hermi�an

Operators

H

Lie algebra representa�on

4

Figure 5.1: Philosophy: Lie group G acts irreducibly on coadjoint orbits (1), where one orbit, M, is quantized (2). This gives rise to a Lie algebra representation (3), which in turn can be lifted to a unirrep of G on H.

5.2.1 The Heisenberg algebra and group

The Heisenberg Lie algebra and Lie groups are interesting groups in the quantum mechanics and they provide as a good example for the orbit method. The Heisenberg group is defined as

G B    (x, y, z) =   1 x z 0 1 y 0 0 1  : x, y, z ∈ R   

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with Lie algebra

g B    pP+ qQ + cC =   0 p c 0 1 q 0 0 1  : p, q, c ∈ R    .

Equivalently , we could have defined the Heisenberg Lie algebra as the 3 dimensional vector space with basis elements

{P, Q,C}

and the Lie bracket defined by

[P, P] = [Q, Q] = [P,C] = [Q,C] = 0, [P, Q] = C,

and the Heisenberg Group as the Lie group obtained by exponentiation. We want to find the coadjoint orbits of the Heisenberg group. We already know what g is. Through the dual pair

h·, ·i : A × g → R, hX,Y i = Tr(XY ), where A B    (a1, a2, a3) =   0 0 0 a1 0 0 a₃ a₂ 0  : ai∈ R    ,

we can identify A with g∗. Since if for all X ∈ A and Y ∈ g, we have Tr(XY ) = 0 =⇒ a1p+ a2q+

a₃c= 0. If this is 0 for a fixed (p, q, c) and for all ai, then obviously p, q, c = 0. So the Heisenberg

group G acts on strictly lower triangular matrices, g∗through the coadjoint representation. For g= (x, y, z) ∈ G and (a1, a2, a3) we get Kg(a1, a2, a3) = (a1− xa3, a2+ ya3, a3) =   0 0 0 a1− xa3 0 0 a₃ a₂+ ya3 0  .

This is a straightforward computation using (5.2). We notice that for a3= 0, G is the stabalizer of

(a1, a2, 0). So the coadjoint orbit simply consists of the point (a1, a2, 0), which are trivial symplectic

manifolds. However, for the coadjoint orbit of a point (a, 1, a2, a3), where a3, 0, we get for every pair (x, y) ∈ R2a point (a1− xa3, a2+ ya3, a3) in the orbit of (a, 1, a2, a3). Therefore, we see that for

every a3, 0, the coadjoint orbit O(a1,a2,a3)is a copy of R

2_{⊂ R}3_{. The corresponding Kirilov-Kostant}

form should be a multiple of the canonical symplectic form. In [5] we see that

ω = 1

a₃dx∧ dy.

5.2.2 Quantizing the coadjoint orbits of the Heisenberg Lie group

Let us consider the coadjoint orbits of the Heisenberg group as our phase spaces. Let us start with the one point orbits (a1, a2, 0). This is not a very interesting case, since the line bundle is

L= (a1, a2, 0) × C C. The sections are simply functions from (a1, a2, 0) to C. Moreover, the

tangent bundle is 0-dimensional, so the polarization is trivial. Therefore the Hilbert space space is isomorphic to C. Observables are functions from λ : (a1, a2, 0) → R. Because of this we can think

of λ as a real number. The quantization of λ is simply given by λ 7→ λ . Since it is real, we have that λ is self-adjoint.

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Figure 5.2: A visualization of the coadjoint orbits of the Heisenberg group and constructing a line bundle.

The other orbits are more interesting. For each a3= α , 0 we have a sheet in R3, a copy of

M_{= R}2with symplectic form ω = _α1dx∧ dy. In Section 4.3 we saw how we can quantize R2_{. The}

prequantum Hilbert space Hpre(M) consist of all smooth complex-valued functions ψ : M → C.

However, because the symplectic form is scaled, it will change how the observables are quantized. Let θ = −_α1pdq be the symplectic potential. Simple calculation gives us that Xq= α −_{∂ p}∂ , and

Xp= α_{∂ q}∂ . Using the polarization P = _{∂ p}∂ , we can repeat the steps as we did before. Once again,

the Hilbert space consist of square-integrable, smooth complex valued functions ψ that only depend on q. However now we have

q7→ q, p 7→ −iα ∂

∂ q. (5.3)

5.2.3 Heisenberg Lie group/algebra representations

Let us consider the vector space A, spanned by {q, p, α}. This gives us a sub Lie algebra of C∞_(M),

with the Poisson brackets as Lie brackets. Writing out the commutation relations with the Poisson bracket of the basis vectors, gives us

{q, q}P= {p, p}P= {q, α}P= {p, α}P= 0, {p, q}P= α.

Therefore A is the Heisenberg Lie algebra. The quantization map iQ: A → End(HP(M)), f 7→ iQ( f )

given by (5.3) is therefore a Lie algebra representation of A. A arbitrary element (x, y, z) ∈ G, the Heisenberg group, can be written as exp(xq) exp(yp) exp(zα). Through deriving a Lie group representation π one gets a Lie algebra representation π∗, and from a Lie algebra representation π∗

one gets a Lie group representation π through exponentiation. From Appendix V2 in Kirillov’s book [5] we know that exp(iπ∗(X )) = π(exp(X )). Here π∗= Q, the quantization map. Therefore

π : G → Gl(HP(M)),

π (g) = π (x, y, z)

= π(exp(xq) exp(yp) exp(zα)) = exp(iπ∗(xq) + iπ∗(yp) + iπ∗(zα))

= eixq+αy∂ q∂ +izα

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Writing eα y∂ q∂ out using taylor expansion, gives us

1 + αy ∂ ∂ q+ 1 2!α 2_y2₍ ∂ ∂ q) 2_{+ ....}

Letting the representation act on some ψ ∈ HP(M), gives us

π (x, y, z)ψ (q) = ei(xq+zα)eα y ∂ ∂ q_{ψ (q)} = ei(xq+zα)· (1 + αy ∂ ∂ q+ 1 2!α 2_y2₍ ∂ ∂ q) 2_{+ ...)ψ(q)} = ei(xq+zα)· (ψ(q) + αyψ‘(q) +1 2α 2_y2 ψ “(q) + ...) = ei(xq+zα)ψ (q + α y).

For each α , 0, we found a representation of the Heisenberg group. For α = 0 it is easier. The classical observables of the 0-dimensional orbits F = (a1, a2, 0) simply form R. We can correspond

a Lie group representation of G with F, by

πF : G → Gl(C) C, πF(g) = π(exp(X )) = eihF,Xi.

One should notice that using a different polarization will give us different representations. These representations are however equivalent. The different polarization correspond to the different realizations of the canonical commutation relations in [5]. By the famous Stone-von Neumann theorem, these are indeed all the unirreps. [2]

5.2.4 SU(2)and S2

Even though the representations of SU (2) are already well known, it is a good example to show the power of the orbit method by finding the representations through this method. However, we will go less into details than we did for the Heisenberg Lie group. We can realize SU (2) as the matrix group G= SU (2) = a b −¯b a¯ : a, b ∈ C, |a|2+ |b|2= 1 .

From basic differential geometry, we know the corresponding Lie algebra is g = su(2) = {X ∈ Mat(2, C) : X = −X†, TrX = 0} = ib c+ id −c + id −ib : b, c, d ∈ R . [1]

We can identify g∗with g, which is again identified with R3, since the real parameters b, c and d fix the element. Define

Q_{: su(2) → C, F 7→ Tr(F}2).

We notice that Q(Kg(F)) = Q(gFg−1) = Tr(gFg−1gFg−1) = Tr(gF2g−1) = Tr(F2) = Q(F). So

Qis invariant under the coadjoint action. This means that if you apply Q on two elements of the same coadjoint orbit, you’ll get the same complex value. Moreover, computing Q(F) gives us

Q(F) = Tr F2 = Tr ib c+ id −c + id −ib · ib c+ id −c + id −ib = Tr−b 2_{− c}2_{− d}2 _... ... −b2_{− c}2_{− d}2 = −2(b2+ c2+ d2) = constant.

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So we can write b2+ c2_{+ d}2_{= constant/ − 2 = R}2_{, for some positive real number R. So, if we}

have 2 points lying in a different sphere with 0 as the origin of the sphere, then Q would have a different value for those points and they would be therefore be in different coadjoint orbits. We can conclude that 2 points of the some orbit, will always lie in the same sphere. It is obvious that the origin, the point (0, 0, 0) forms a 0-dimensional coadjoint orbit on itself and we state without proof that all the spheres exactly form all the coadjoint orbits. [12] So all the coadjoint orbits of SU (2) are 2-dimensional spheres in R3, centered around the origin and one point.

5.2.5 Quantization

To find the representations of SU (2), we need to quantize the coadjoint orbit. Since the 0-dimensional orbit is analogue to the 0-0-dimensional orbits of the Heisenberg Lie group and simply gives us the trivial representation, we shall first look at the 2-dimensional spheres and take a particular sphere S2_R, with radius R as an example. In Section 4.4 we saw that only spheres with half integer radius (R = 1/2k, k ∈ Z) are quantizable, and the Hilbert space consisted of square integrable polynomials with degree ≤ 2R. This will form the representation space

As we know from Section 4.4, we have f7→ −i∇Xf+ f = R(1 + |z| 2₎2₍∂ f ∂ ¯zdz − ∂ f ∂ zd¯z) + i¯z(1 + |z| 2₎∂ f ∂ ¯z + f .

Just like in the case of the Heisenberg Lie algebra, we are only interested in quantizing g. We note that su(2) is generated by

u1= 0 i i 0 , u2= 0 −1 1 0 , u3= i 0 0 −i .

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We have [u1, u2] = 2u3, [u2, u3] = 2u1and [u3, u1] = 2u2. Therefore we need f1, f2and f3, such that

[ ˆf1, ˆf2] = 2 ˆf3, [ ˆf2, ˆf3]2 ˆf1and [ ˆf3, ˆf1] = 2 ˆf2. Using the fact that [ ˆf, ˆg] = −i{ f , g}ˆ p, the reader can

convince himself that f1= i(z + ¯z R(1 + |z|2₎, f2= z− ¯z R(1 + |z|2₎, f3= −i(|z|2_{− 1)} R(1 + |z|2₎

satisfy the desired property. [12] Just like the Heisenberg Lie group case, ¯fiinduce a Lie algebra

representation, and it can be lifted to an unirrep. By [1] these are indeed all irreducible unitary representations.

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45 Conclusion

We saw that the orbit method is a good method to find the representations of certain Lie groups. In the thesis there is more of an emphasis on the philosophy of the method, meaning the link between theoretical physics and mathematics. It is beautiful to see how strong the language of geometry might be. Geometry helps us in obtaining a more abstract understanding of classical mechanics, it provides a strong tool to translate classical mechanics into quantum mechanics and by the philisophy of the orbit method, the same geometry provides a tool of finding and understanding representations of Lie groups. The philosophy, obtained from geometrical quantization, of finding representations using coadjoint orbits in general for any Lie group, however it does not always work. There exists a correspondence between representations and coadjoint orbits, but it is not always easy to find the correspondence. Here we looked at the Heisenberg Lie group and SU (2), which are respectively a nilpotent and a compact group. The Kirillov orbit method always works for these kind of groups, but for some groups, mathematicians still do not know how to find the correspondence between the orbits and the representations. This subject is therefore also interesting, since it is an active research area.

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Popular Summary

When we gave a geometrical object, we can describe its symmetries. A symmetry is some kind of operation that leaves the object invariant. For example, an equilateral triangle can be rotated 120 degrees, or you could mirror flip it through certain axes (see Figure 5.4). All the information of symmetries is encoded in an abstract way in some algebraic object we call a group.

Figure 5.4: Visualisation of D3. https://mattheusic.wordpress.com/2013/09/07/ group-theory-in-music-2-generators-and-cyclic-groups/

A group is a set with a binary operation that satisfies some conditions and it obtains information about symmetries. However, it appears that a lot of times it is very useful to translate this abstract kind of symmetries to symmetries on vector spaces, which are spaces that we understand better. There are multiple ways of translating those symmetries and this is a whole area in mathematics that we call representation theory. Sometimes it might be hard to find the ways to translate the symmetries. In this thesis we focus on special kind of symmetries, that have a smooth nature. In physics we have classical mechanics, which is the physics that is intuitive to us. A state of a particle is simply described by its position and velocity in a certain direction. The state of a particle in quantum mechanics is stranger, since probability plays a role. A state of a particle is a wave function, that specifies the probability of finding the particle at a certain point in space. The physics goals of this thesis is to translate classical mechanics to quantum mechanics using a geometrical language. It is very interesting to note that by doing this, we automatically find ways to translate some kind of symmetries. Translating classical mechanics to quantum mechanics is inherently connected with translating abstract symmetries in a group to symmetries on vector spaces.

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Bibliography

Books

[1] Brian Hall. Lie groups, Lie algebras, and representations: an elementary introduction. Volume 222. Springer, 2015 (cited on pages 42, 44).

[2] Brian C Hall. Quantum theory for mathematicians. Volume 267. Springer, 2013 (cited on pages 18, 27, 42).

[3] Dale Husemoller. Fibre bundles. Volume 5. Springer, 1966 (cited on page 15).

[4] Jürgen Jost and Jèurgen Jost. Riemannian geometry and geometric analysis. Volume 42005. Springer, 2008 (cited on pages 15, 16).

[5] Aleksandr Aleksandrovich Kirillov. Lectures on the orbit method. Volume 64. American Mathematical Soc., 2004 (cited on pages 37, 38, 40–42).

[6] John M Lee. Introduction to smooth manifolds. Springer, 2001 (cited on pages 15, 37). [7] John M Lee. Riemannian manifolds: an introduction to curvature. Volume 176. Springer

Science & Business Media, 2006 (cited on page 16).

[8] Mikio Nakahara. Geometry, topology and physics. CRC Press, 2003 (cited on pages 28, 30). [9] Nicholas Michael John Woodhouse. Geometric quantization. 2nd edition. Oxford University

Press, 1997 (cited on pages 21, 29–31). Articles

[10] Andrea Carosso. “Geometric Quantization”. In: arXiv preprint arXiv:1801.02307 (2018)

(cited on page 20).

[11] Viktor L Ginzburg and Richard Montgomery. “Geometric quantization and no go theorems”. In: arXiv preprint dg-ga/9703010 (1997) (cited on page 25).

[12] Malte Litsgård. “The Orbit Method and Geometric Quantisation”. In: (2018) (cited on

pages 27, 32, 38, 43, 44).

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Index

Symbols Γ (E) . . . 16 X(M) . . . 16 A Adjoint Representation . . . 37 Anti-holomorphic Polarization . . . 30 C Coadjoint Orbit . . . 38 Coadjoint Representation . . . 37 Complete Set . . . 25

Complex Line Bundle . . . 15

Complex Structure . . . 29 Connection . . . 16 Covariant Derivative . . . 17 Curvature . . . 18 D Dirac Quantization . . . 25 Dual Representation . . . 37 H Hamilton Formalism . . . 9

Hamiltonian (Classical Mechanics) . . . 9

Hamiltonian System . . . 10

Hamiltonian Vector Field . . . 10

Heisenberg Group . . . 39

Heisenberg Lie Algebra . . . 40

Hermitian Line Bundle . . . 17

Hermitian Structure . . . 17 Holomorphic Polarization . . . 30 I Integrality Condition . . . 21 K Kähler Form . . . 30 Kähler Manifold . . . 29 Kähler Polarization . . . 29 Kirillov-Kostant Form . . . 38 L Langrangian Submanifold . . . 28 Line Bundle . . . 15 M Morphism of symplectic manifolds . . . 9

N Natural Pairing . . . 37 Non-degenerate . . . 9 O Observables . . . 9 P Parallel . . . 19 Phase Space . . . 9

Poisson Bracket (Coordinate Dependent) . . . 9

Polarization (complex) . . . 28

Polarization (real) . . . 28

Prequantization . . . 25

Prequantum Hilbert Space . . . 26

Prequantum Operator . . . 26 S Symmetry . . . 13 Symmetry Group . . . 13 Symplectic Manifold . . . 9 Symplectic Potential . . . 18 Symplectomorphism . . . 9 U Unirrep . . . 37

Geometric quantization and the orbit method