UNIVERSITÄT LEIPZIGFakultät für Physik und Geowissens haftenInstitut für Theoretis he Physik

Diplomarbeit im Studiengang PhysikThe Costratied Hilbert Spa e Stru tureof an SU(3) Latti e Gauge Model

Leipzig, den 03.10.2008 vorgelegt von Jörn Boehnkegeboren am 12.02.1984

Betreuer: Prof. Dr. Gerd RudolphDr. Matthias S hmidtGuta hter: Prof. Dr. Gerd RudolphProf. Dr. Hans-Bert Radema her

A knowledgementsI thank Prof. Gerd Rudolph for giving me the opportunity to work in his group on thisboth interesting and hallenging proje t. It ombined both of my passions, mathemati s andtheoreti al physi s, in a very well-balan ed way and inspired me to venture further into theworld of mathemati al physi s in the future.Furthermore, I owe many thanks to Dr. Matthias S hmidt. With his tireless support andguidan e he has ontributed substantially to making this diploma thesis as su essful aspossible. In ountless dis ussions he helped me nd answers to various questions appearingduring the ourse of this work.I also thank all the people who supported and en ouraged me over the last one and a halfyears. In parti ular, I thank the people with whom I shared my o e for their very enjoyable ompany, their suggestions towards this thesis and many fruitful dis ussions; Dr. ElmarBittner for his patien e in dealing with all the omputer rashes I aused; Dr. CorneliaWoitek for her supportive letters; and Reinhard Höll and Kaj Bernhardt for their lingualsupport.Last but denitely not least, I dearly thank my whole family who always supported me andbelieved in me in the strongest possible terms DANKE.

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Abstra tThe weak and strong intera tions are modelled by non-abelian gauge eld theory. Pertur-bation methods are the usual approa h in dealing with the orresponding gauge elds. Forsome fundamental phenomena of gauge theory, however, only non-perturbative methods areappli able. In this work, we [i analyse a toy model of lassi al SU(3) latti e gauge theory ona single plaquette.At rst, we give an introdu tion of the mathemati al foundations and the model is ana-lysed. In order to redu e the gauge symmetry, we subsequently apply the singular Marsden-Weinstein redu tion to the phase spa e of our system, a Hamiltonian G-manifold. Thispro edure yields the redu ed phase spa e, a symple ti stratied spa e.The redu ed phase spa e is a singular obje t whi h is omposed of seven dierent on-ne ted omponents: three zero-dimensional strata, three two-dimensional strata and onefour-dimensional stratum. The physi al Hilbert spa e of our system arises by geometri quantisation. We hara terise and analyse the Hilbert spa es asso iated with the singularstrata of the SU(3) toy model. The set of these single Hilbert spa es amounts to the stru tureof a ostratied Hilbert spa e. The results obtained may be regarded as a ontribution tonon-perturbative quantisation pro edures in latti e gauge theory.

[i throughout this thesis, we refers to the author and you the readerv

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Contents1 Introdu tion and overview 12 Fundamental mathemati al on epts 52.1 Lie groups, Lie algebras and representations 52.2 G-manifolds 112.3 Momentum mappings 132.4 Singular Marsden-Weinstein redu tion 192.4.1 Redu tion: the regular ase 202.4.2 Redu tion: the singular ase 213 Gauge theory 253.1 Mathemati al obje ts of gauge theory 253.2 Classi al gauge theory 323.3 Hamiltonian formulation 333.4 Latti e gauge model 343.5 Tree gauge 363.6 Classi al redu tion in latti e gauge theory 374 Representation theory - SU(3) and su(3) 414.1 Chara terisation of representations 414.1.1 General aspe ts 414.1.2 Chara ters, S hur's orthogonality relations and the Peter-Weyl theorem 454.1.3 Maximal tori and Weyl's integral formula 494.2 The stru ture of semi-simple Lie algebras 534.2.1 The roots of a Cartan subalgebra 53vii

viii CONTENTS4.2.2 Simple roots 554.2.3 The Weyl ree tions 574.3 The weights of a representation 614.4 The hara ters of SU(3) 665 The ostratied Hilbert spa e for SU(3) 755.1 Kähler stru ture and symmetry redu tion 755.2 The ostratied Hilbert spa e stru ture 785.3 The Hilbert spa es H1m asso iated with the strata P1m 835.4 The Hilbert spa e H2 asso iated with P2 885.4.1 A fun tional des ription of z 915.4.2 The generating fun tion of V2 935.4.3 The basis of V2 975.4.4 Orthogonal fun tions on V2 1026 Outlook 117A SU(3) hara ter values 119B Proof of the spe ial ases of proposition 5.4.5 123C Expli it al ulation of the 15 types of fun tions spanning V2 129D Mathemati a 6.0 ode 133Referen es 139Index 143

1 Introdu tion and overviewEs kann zutreend sein, dass Personen, die reine Mathematiker sind, be-stimmte spezis he Dezite besitzen, aber dies ist ni ht der Mathematikges huldet, denn es gilt für jede andere exklusive Berufung glei herma-ÿen. It may be true, that men, who are mere mathemati ians, have ertain spe i short omings, but that is not the fault of mathemati s,for it is equally true of every other ex lusive o upation.Carl Friedri h GauÿThere are four fundamental intera tions in nature: ele tromagnetism, weak and strong in-tera tion and gravity. Ele tromagnetism was the rst to be phrased in a onsistent theoryby J.C. Maxwell in 1864. The beauty of his approa h was that it revealed invarian es inthe stru tures of its equations. Additionally, in a later formulation, it allowed to des ribephysi s in a oordinate-free way. In luding symmetry into a onsistent des ription of natureis the basi idea of gauge theory. In its onsequen es, the symmetry yields inner degrees offreedom. To x them in a ertain way is alled a gauge.Ele tromagnetism as well as the weak and the strong nu lear for e are onsistently des ribedby the Standard Model. Attempts to unify all four fundamental for es, i.e. to in lude grav-ity, have not su eeded yet. The Standard Model hara terises the intera tions using gaugeelds, i.e. equations losely modelled after Maxwell's equations. This mathemati al formal-ism su eeded, as it provided a urate experimental predi tions regarding the three for es.A quantisation of the gauge elds des ribes the intera tion by parti les: the ele tromagneti for e is arried by the photon, the weak for e by the W and Z parti les and the strong for eby gluons. They intera t with so- alled harged parti les, e.g. quarks (whi h feel the strongfor e) and leptons (whi h do not). All of these harged parti les have orresponding antipar-ti les of the same mass and opposite harge.Inner symmetries are des ribed by stru ture groups. For ele tromagnetism the stru turegroup is the ommutative group U(1). Yang C.N. and R. Mills generalised this theory tonon-abelian stru ture groups in 1954. Thereby, weak intera tion is modelled by the stru -ture group SU(2) and strong intera tion by the stru ture group SU(3); the latter is the1

2 1 INTRODUCTION AND OVERVIEWobje t of quantum hromodynami s. In the Standard Model the stru ture groups ombineto SU(3) × SU(2) × U(1).Gauge theory is modeled by obje ts of dierential geometry. The elements of the ongura-tion spa e are onne tions in a prin ipal bre bundle over the spa e-time manifold. Gaugetransformations smoothly map the onguration spa e onto itself. They are des ribed byverti al automorphisms on this prin ipal bre bundle. This approa h has been initiated byworks of E. Lubkin, R. Hermann and espe ially A. Trautman in the late 1960s and early1970s and was rst summarised in [DV80 and [EGH80.As in every eld theory, the onguration spa e of the gauge eld is innite-dimensional.Avoiding this problem, the system is simplied by repla ing the ontinuous spa e by a dis- rete latti e. This idea is based on the assumption that the limit of an innitely extendedand innitely ne latti e re overs the ontinuum theory. It marks the transition to latti egauge theory introdu ed by K.G. Wilson in 1974, f. [Wil74. In this thesis we use theansatz of Hamiltonian latti e gauge theory, vide [KS75. This approa h does not dis retisethe whole spa e-time but only spa e itself, leaving time ontinuous and thus permitting thetime development hara terised by the Hamiltonian. This transformation yields a ongura-tion spa e of nite dimensions. The redu ed phase spa e repla es the gauge orbit spa e inthe Hamiltonian formulation.Quantum hromodynami s has proven to be a potent approa h on the level of perturbativemethods. For some fundamental phenomena of gauge theory, however, only non-perturbativemethods are appli able. E.g. the question of physi al relevan e of the stratied stru turein the lassi al onguration spa e of non-abelian gauge theory requires a non-perturbativeansatz. Generally speaking, there are two strategies to onstru t a non-perturbative quan-tum gauge theory: (i) The observable algebra is rstly onstru ted for the quantised system.Subsequently, this algebra is redu ed with respe t to gauge symmetries. Compare [KRT97,[KRS98 and [JKR05 for this approa h. (ii) The observable algebra is rstly onstru ted ona lassi al level. Subsequently, the system is redu ed and amounts to a symple ti stratiedspa e. Finally, with the aid of geometri quantisation, this stratied spa e is quantised. Com-pare [RSV02 and [CKRS05 for the stru ture of the gauge orbit spa e and [S h03, [Hal02,[Hüb01, [Hüb04 and [Hüb06 for the strati ation its quantisation.This diploma thesis fo uses on lassi al SU(3) gauge theory. We study a toy model des rib-ing strong intera tion on a simple latti e onsisting of a single plaquette. The phase spa eis des ribed by a Hamiltonian G-manifold. We apply singular Marsden-Weinstein redu tion,[CB97, in order to redu e the system's gauge symmetry. Hen e, the redu ed ongurationspa e of our system amounts to a two-simplex, whi h de omposes into three subsimpli es:the verti es, the edges and the inner area, the three strata. The redu ed phase spa e de- omposes analogously. Its onne ted omponents amount to three zero-dimensional strata,

3one two-dimensional stratum and one four-dimensional stratum and onstitute a symple -ti stratied spa e. We perform geometri quantisation in order to onstru t the Hilbertspa e on the stratied phase spa e. The physi al Hilbert spa e arises by Kähler quantisa-tion. Using the holomorphi Peter-Weyl theorem, [Hüb07, this spa e is set in relation to theHilbert spa e obtained by ordinary S hrödinger quantisation. We hara terise and analysethe Hilbert spa es asso iated with the singular strata of our SU(3) toy model. These Hilbertspa es nally amount to the stru ture of a ostratied Hilbert spa e. For a toy model withstru ture group SU(2), the ostratied quantisation is arried out in [HRS07.Chapter 2 ontains an outlines of the basi mathemati al on epts used in the ourse ofthis thesis. Notions like Lie groups, Lie algebras, their orresponding exponential map andrepresentations, adjoint representations, Killing forms and Killing ve tor elds, G-manifolds,symple ti manifolds, Hamiltonian G-manifolds, momentum mappings and stratied sym-ple ti spa es are dened and some basi results are mentioned. Nevertheless, this shortmathemati al introdu tion does not attempt to be omplete.Chapter 3 starts with a qui k introdu tion of the mathemati s of gauge theory, namely prin- iple bre bundles. After lassi al gauge theory is introdu ed in a dierential geometri way,we perform the temporal gauge by separating time from spa e, yielding the Hamiltoniandes ription. Subsequently, we perform the lassi al redu tion in latti e gauge theory. Then,the phase spa e of our model amounts to a Hamiltonian G-manifold. In order to redu egauge symmetries, we apply singular Marsden-Weinstein redu tion and obtain a quotientspa e whi h amounts to a symple ti stratied spa e.Chapter 4 provides a detailed introdu tion to representation theory, as it is essential for thisthesis. This in ludes integration over groups, irredu ible representations, hara ter and roottheory. We furthermore derive on rete formulae for SU(3), the symmetry group of the toymodel analysed.Chapter 5. To quantise our phase spa e, we rstly apply Kähler quantisation to the unre-du ed phase spa e in this hapter. As des ribed above, we identify the obtained quantisedspa e with the Hilbert spa e obtained by ordinary S hrödinger quantisation. The redu tionto the subspa e of lass fun tions yields pre isely the Hilbert spa e orresponding to thestratied phase spa e.Sin e the zero-dimensional and two-dimensional strata are of zero measure, we hoose an in-dire t approa h to onstru t the asso iated Hilbert spa es. First, we determine the vanishingfun tions of ea h su h stratum and then, we ompute the fun tions orthogonal to the vanish-ing fun tions. These fun tions orrespond to our on eption of lo alised wave fun tions on agiven stratum. The set of all these single Hilbert spa es arries the stru ture of a ostratiedHilbert spa e.

4 1 INTRODUCTION AND OVERVIEWChapter 6. This diploma thesis is on luded by an outlook.Appendix. The al ulations in hapter 5 result in a large amount of spe ial ases. To keepthe fo us on the main ideas, a few exemplary tables are presented and a number of spe ial ases are onsidered here. Additionally, the Mathemati a 6.0 ode for plots and al ulationsappearing in this thesis is presented in appendix D.

2 Fundamental mathemati al on eptsThis hapter provides the basi mathemati al on epts leading to the singular Marsden-Weinstein redu tion. A proof of the laims mentioned here will be omitted as they an befound in standard literature.Knowledge of obje ts like smooth manifolds, tangent manifolds, ve tor elds and their push-forward, pull-ba k and integral urves, dierential forms and their integration, tensor algebraand exterior algebra are assumed.2.1 Lie groups, Lie algebras and representationsDenition 2.1.1: An r-parametri Lie group G is an r-dimensional smooth manifold,in whi h the group operation(·, ·) : G×G→ G , (g, h) 7→ g · h−1 (2.1)is of lass C∞.In the sequel, we identify the left and right multipli ation of a ∈ G on G with La and

Ra, respe tively. Above's denition assures that these mappings are dieomorphisms.Denition 2.1.2: Let G and H be Lie groups.(i) A Lie group homomorphism from G to H is a smooth map φ : G → H that is ahomomorphism of groups.(ii) A Lie group isomorphism from G onto H is a bije tive Lie group homomorphismφ : G→ H whose inverse is also a Lie group homomorphism.(iii) An automorphism of G is an isomorphism of G onto itself.Denition 2.1.3: Let L be a nite-dimensional ve tor spa e over the eld K = R (K =C). L is alled a real ( omplex) Lie algebra if there is a rule of omposition (A,B) →

[A,B] in L (the Lie bra ket) whi h satises the following axioms:(i) [αA+ βB,C] = α[A,C] + β[B,C] for all A,B,C ∈ L and α, β ∈ K (linearity),5

6 2 FUNDAMENTAL MATHEMATICAL CONCEPTS(ii) [A,B] = −[B,A] for all A,B ∈ L (antisymmetry),(iii) [A, [B,C]] + [B, [C,A]] + [C, [A,B]] = 0 for all A,B,C ∈ L (Ja obi identity).Note: The ommon Lie bra ket of ve tor elds in X(G) is given by[A,B](f) := A(B(f)) −B(A(f)) for A,B ∈ X(G) and f ∈ C∞(G).Denition 2.1.4: A smooth ve tor eld A on G is left-invariant if

(La)∗A = Afor all a ∈ G.The notation (·)∗ denotes the push-forward of ve tor elds (here X(G) → X(G)) indu edby dieomorphisms (here La). The left-invariant ve tor elds on G ompose a ve tor spa eand (La)∗ ommutes with the ommon Lie bra ket sin e(La)∗[A,B](f) = [A,B] (f La) L−1

a

= A(B(f La) L−1a La) L−1

a −B(A(f La) L−1a La) L−1

a

= ((La)∗A)(((La)∗B)f

)− ((La)∗B)

(((La)∗A)f

)

= [(La)∗A, (La)∗B](f) .Thus, the obje t g = A ∈ X(G) | A left-invariant is a Lie algebra.Denition 2.1.5: The Lie algebra g is alled Lie algebra of the Lie group G.The denition of (La)∗ is leading to Aa = L′aAe. To ompletely hara terise a left-invariantve tor eld it therefore su es to know its value at one point. The anoni al mapping

g → TeG , A 7→ Aedenes an isomorphismg ∼= TeG (see [Rud05, Satz 5.2.3). (2.2)Note: The spa e of all ve tor elds X(M) of an arbitrary manifold M forms a (innitedimensional) Lie algebra with the ommon Lie bra ket. From equation (2.2), one sees thatthe Lie algebra g of the Lie group G is a subalgebra of X(G).In the following, we will not distinguish between both spa es. Now, let φA

t : G→ G, t ∈ R,be the global one-parameter group of dieomorphisms on G generated by A ∈ g,i.e. φAt ≡ φAt′( d

dt t) = A(φAt ) (its existen e for ea h A ∈ g is ensured by [Rud05, Satz 5.2.6).Then, we an give the following denition.

2.1 LIE GROUPS, LIE ALGEBRAS AND REPRESENTATIONS 7Denition 2.1.6: The exponential map between the Lie algebra g and the Lie group Gis dened byexp : g → G , A 7→ φA1 (e) , (2.3)where φAt is the ow of A ∈ g.Remark 2.1.7: For A ∈ g and t ∈ R there holds(i) φAt (g) = Lgφ

At (e) (by left-invarian e and [Rud05, Satz 3.2.11),(ii) R→ G : t 7→ φAt (e) = exp(tA) is a group homomorphism and(iii) φAt (g) = g exp(tA).In parti ular, for any K-valued n× n-matri es B and C we have(iv) expB =

∑∞k=0

1k!B

k ( f. [Rud05, Fml. 5.14),(v) exp(BT ) = (expB)T , exp(B†) = (expB)† and exp(CBC−1) = C(expB)C−1 (dire t onsequen es of (iv)) and(vi) det(expB) = etrB ( f. [Rud05, Lem. 5.1.8).Note: B† denotes the onjugate of the transpose of B.Example 2.1.8 (SU(n), the spe ial unitary group) : We are going to illustrate the om-putation of the Lie algebra of a Lie group here. Sin e it will be the main fo us of this thesis,we take the spe ial unitary group SU(n) = B ∈ U(n) | detB = 1, where U(n) = B ∈

GL(n,C) | BB† = 1 denotes the unitary group and GL(n,C) is the group of all invertiblen× n matri es with omplex entries.If t 7→ B(t) is a urve in U(n) with B(0) = 1, then we have

0 =d

dt t=0

(1) =d

dt t=0

(B(t)B(t)†) = B′(0)B(0)† +B(0)B′(0)† = B′(0) +B′(0)† .Thus, for s(n) := C ∈ GL(n,C) | C + C† = 0, we have s(n) ⊇ u(n) (u(n) denotes theLie algebra of U(n)). Conversely if C ∈ s(n), then (expC)(expC)† = (expC)(expC†) =

exp(C) exp(−C) = 1 and so expC ∈ U(n). It isC =

d

dt t=0

exp tC ∈ u(n) ,when e u(n) = s(n).The Lie algebra su(n) of SU(n) is the subalgebra of u(n) onsisting of matri es with tra e0 (i.e. su(n) = C ∈ u(n) | trC = 0). This follows immediately from the formuladet(expC) = etrC .

8 2 FUNDAMENTAL MATHEMATICAL CONCEPTSThe mathemati al on ept of representing a Lie group in a (nite-dimensional) ve tor spa eis fundamental for this thesis. As we will see in hapter 3 and 4, the inner automorphisms ofG are of spe ial interest.Denition 2.1.9: A representation (V,Γ) of a Lie groupG on the (nite-dimensional)ve tor spa e V the representation spa e is a group homomorphism

Γ : G→ Aut(V )from G to the automorphism group Aut(V ).Denition 2.1.10: A representation (V , Γ) of a Lie algebra g over the eld K onthe (nite-dimensional) ve tor spa e V is a homomorphismΓ : g → Aut(V )from g to the automorphism group Aut(V ) su h that(i) ΓαA+βB = αΓA + βΓB for all A,B ∈ g and α, β ∈ K and(ii) Γ[A,B] = [ΓA, ΓB ] ≡ ΓAΓB − ΓBΓA for all A,B ∈ g.Remark 2.1.11: (i) V and V are also referred to as G-module and g-module, re-spe tively.(ii) The Lie group representation (V,Γ) des ribes a ontinuous a tion Γ : G × V → V of

G on V su h that for ea h g ∈ G the translation v 7→ Γgv is a linear map.(iii) The Lie algebra representation (V , Γ) des ribes a Lie algebra homomorphism Γ : g →gl(V ) from g to the Lie algebra of endomorphisms on a ve tor spa e V .(iv) (V,Γ) is alled a real (respe tively omplex) representation if and only if V is a real(respe tively omplex) ve tor spa e.

It is mostly lear from ontext whether a Lie group or Lie algebra representation is given.For the sake of simpli ity, we will then just say let (V,Γ) be a representation.Remark 2.1.12: Two representations (V,ΓV ) and (W,ΓW ) ofG are said to be equivalentif V and W are G-isomorphi , i.e. if there exists an isomorphism η : V → W su h that thefollowing diagrams ommute

2.1 LIE GROUPS, LIE ALGEBRAS AND REPRESENTATIONS 9G× V

ΓV - V

G×W

idG×η

?

ΓW- W

η

?

GΓV- Aut(V )

Aut(W ) ,ση

?ΓW -where ση(ΓV ·) = ηΓV ·η

−1.For n-dimensional matrix representations (V,ΓV ) and (W,ΓW ) we have Aut(V ) = Aut(W ) =

GL(n). The last equation then reads σA(B) = ABA−1 for B ∈ GL(n). For α ∈ T∗G, A ∈ TG and g ∈ G dene〈α,A〉g = 〈αg, Ag 〉 := αg(Ag) . (2.4)Denition 2.1.13: The Adjoint representation of G on g is the mapping

Ad : G→ Aut(g) , a 7→ Ada = (LaRa−1)′e . (2.5)The dual representation of Ad, Ad∗ : G → Aut(g∗) is alled oAdjoint representationand is given by〈Ad∗

a α,A〉 := 〈α,Ada−1 A〉 , α ∈ T∗eG

∼= g∗ .Note: For matrix groups there holds AdA(B) = ddt t=0

A−1(exp tB)A = A−1BA.These representations are obviously smooth and indu e representations d Ad : g → Aut(g)and d Ad∗ of g on itself and on g∗, respe tively. They are alled adjoint representationand oadjoint representation and denoted ad and ad∗, respe tively. We nd adA(B) =ddt t=0

Adexp tAB ([Rud05, Bem. 5.3.2), whi h for matrix groups isadA(B) =

d

dt t=0

Adexp tA(B) =d

dt t=0

((exp tA)−1B exp tA) = [A,B] . (2.6)Note: Sin e it is ne essary to distinguish between the dierent adjoint representations, the ase sensitive spelling will be maintained in the following.Denition 2.1.14: A Lie algebra g is said to be(i) simple ⇔ g is non-abelian and only 0 and g itself are its ideals [i and(ii) semi-simple ⇔ g is a dire t sum of simple Lie algebras.[i a subspa e I of a Lie algebra g that is losed under the Lie bra ket and satises [g, I ] ⊆ I is alled anideal of the Lie algebra g

10 2 FUNDAMENTAL MATHEMATICAL CONCEPTSA Lie group is said to be(i) simple ⇔ its Lie algebra is simple and(ii) semi-simple ⇔ its Lie algebra is semi-simple.Note: Sin e all one-dimensional Lie algebras are abelian, simple and semi-simple Lie algebrasmust have dimension greater than one.Denition 2.1.15: The Killing form of g is dened byK(·, ·) : g × g → R , (A,B) 7→ K(A,B) := tr(adA adB) . (2.7)Properties 2.1.16: For the Killing form there holds(i) K(A,B) = K(B,A) for all A,B ∈ g (symmetri ),(ii) K(αA, β(B+C)) = αβ(K(A,B)+K(A,C)) for all A,B,C ∈ g and α, β ∈ K (bilinear),(iii) if ψ is any automorphism of g, K(ψ(A), ψ(B)) = K(A,B) for all A,B ∈ g,(iv) K([A,B], C) = K(A, [B,C]) for all A,B,C ∈ g and(v) K is non-degenerate, if and only if g is semi-simple [ii.Proof : Cf. [Cor95, page 486 . If the Lie group G is ompa t and semi-simple, K is negative denite ( f. [Rud05, Satz 5.3.8).Hen e, K indu es an identi ation of g with g∗ through A 7→ −K(A, ·). We dene

F : g → g∗ , A 7→ 〈F (A), ·〉 := −K(A, ·) . (2.8)Furthermore, the Killing form denes an Ad-invariant Riemannian metri ε on G byεa(Aa, Ba) := −K(A,B) , A,B ∈ g .Example 2.1.17 (The Killing form of the su(2)) : The generators [iii of su(2) are

A1 =1

2

(0 i

i 0

) , A2 =1

2

(0 1

−1 0

) , A3 =1

2

(i 0

0 −i

)(Aj = − i2σj , where σj denote the Pauli matri es, j ∈ 1, 2, 3). Hen e, the basi ommuta-tion relations are

[A1, A2] = −A3 , [A2, A3] = −A1 , [A3, A1] = −A2 .[ii f. denition 2.1.14[iii A1, .., An generate g ⇔ the only subalgebra of g ontaining A1, .., An is g itself

2.2 G-MANIFOLDS 11Using (2.6), these equations implyadA1 =

0 0 0

0 0 1

0 −1 0

, adA2 =

0 0 −1

0 0 0

1 0 0

, adA3 =

0 1 0

−1 0 0

0 0 0

,from whi h it follows by (2.7) that

K(Ai, Aj) = −2δij , i, j ∈ 1, 2, 3 .2.2 G-manifoldsDenition 2.2.1: Let M be a smooth manifold and G a Lie group. A (left) G-a tion on

M is a smooth mappingΦ : G×M →M , (g,m) 7→ Φgmwhi h satises(i) Φe = idM(ii) Φa Φb = Φab for all a, b ∈ G.The triplet (M,Φ) is alled G-manifold.Note: If M is a (nite-dimensional) ve tor spa e, the G-manifold (M,Φ) is a representationof G.Two points m1,m2 ∈ M are onjugate under the G-a tion Φ if there exists an a ∈ G with

m2 = Φa(m1). This property is an equivalen e relation on M .Denition 2.2.2: Equivalen e lasses of the G-a tion Φ are alled orbits and are givenbyG ·m = Φa(m) | a ∈ G ⊆M .The isotropy group (or stabiliser subgroup) of m is the subsetGm = a ∈ G | Φa(m) = m ⊆ G .Denote the orbit spa e by M/G := M/ ∼. For m1,m2 ∈M we havem1 ∼ m2 ⇔ G ·m1 = G ·m2 .

12 2 FUNDAMENTAL MATHEMATICAL CONCEPTSDenition 2.2.3: The G-a tion Φ of G on M is said to be(i) faithful (or ee tive) ⇔ Φg = idM ⇒ g = e(ii) free ⇔ the only element of G that stabilises m is the identity, i.e. Gm = e for allm ∈M(iii) (simply) transitive ⇔ for every m1,m2 ∈ M , there exists a (exa tly one) groupelement g ∈ G su h that g ·m1 = m2, i.e. G ·m = M for all m ∈M(iv) proper ⇔ the mapping M ×G→M ×M , (m, g) 7→ (m,Φg(m)) is proper [iv.Proposition 2.2.4: If Φ is a free proper G-a tion, then the orbit spa e M/G is a smoothmanifold and the orbit mapping π : M →M/G is a submersion.Proof : See [AM85, Pro. 4.1.23. Denition 2.2.5: Let (M,ΦM ) and (N,ΦN ) be G-manifolds. A map f : M → N is saidto be equivariant if it is ompatible with both mappings ΦM and ΦN . I.e. the diagram

Mf - N

M

ΦMg

?

f- N

ΦNg

? ommutes for all g ∈ G.For any A ∈ g the group a tion and the exponential mapping on a G-manifold (M,Φ) denea mapΨA : R×M →M , (t,m) 7→ Φexp tA(m) , (2.9)whi h is an a tion of R on M and therefore a one-parameter group of dieomorphisms.Consequently, ΨA indu es a ve tor eld on M .Denition 2.2.6: Let A ∈ g. The ve tor eld on M indu ed by the one-parameter groupof dieomorphisms given in (2.9) is alled Killing ve tor eld of A and is denoted by A∗.The linear map g → X(M) , A 7→ A∗ is an antihomomorphism from the Lie algebra g to theLie algebra X(M) of ve tor elds on M (see [CB97, Claim B.3.1). For m ∈M we nd

A∗m =d

dt t=0

ΨAt (m) =

d

dt t=0

Φm(exp tA) = Φ′mAe .[iv map between Hausdor spa es is proper ⇔ map is losed and the preimage of every ompa t set underthis map is ompa t

2.3 MOMENTUM MAPPINGS 13Sin e A∗m ∈ TmM , the subspa e of TmM spanned by the Killing ve tor elds A∗, A ∈ g,mat hes the tangent spa e of the orbit ontaining m, i.e. for m ∈ M there is Tm(G ·m) =

A∗(m) | A ∈ g.Denition 2.2.7: The orbit type of m ∈M is the onjuga y lass of the isotropy groupGm, i.e. gGmg−1 | g ∈ G.Note: Conjuga y is an equivalen e relation and thus partitions G.Studying the orbit spa eM/G, we often fa e tri ky situations. We try to resolve it by dividingM into subsets of elements with the same orbit type, analysing them separately (fa torisewith respe t to the group a tion) and trying to merge these obje ts together afterwards.Therefore we need the following sets.Denition 2.2.8: Let (M,Φ) be a G-manifold and H ⊆ G subgroup of G. We dene(i) MH := m ∈M | Gm = H and(ii) M(H) := m ∈M | ∃ g ∈ G : gGmg

−1 = H.Hen e, MH ⊆M(H), M(H) = G ·MH and M(H) onsists of all elements of orbit type H.2.3 Momentum mappingsIn this se tion we dene the on ept of a momentum mapping of a Hamiltonian group a tion.In addition, with Q denoting the onguration spa e of a me hani al system, we subsequentlyderive that the physi al phase spa e T∗ Q is a Hamiltonian G-manifold in a natural way.Proposition 2.3.1 (Darboux theorem) : Suppose ω is a non-degenerate [v two-form on a2n-manifold M . Then, dω = 0 if and only if there is a hart (U,ϕ) at ea h m ∈M su h thatϕ(m) = 0, and with ϕ(u) = (x1(u), .., xn(u), y1(u), .., yn(u)) we have

ωU=

n∑

i=1

dxi ∧ dyi . (2.10)Proof : Cf. [AM85, Thm. 3.2.2. Denition 2.3.2: A symple ti form ω on a smooth manifold M is a non-degenerate, losed two-form ω ∈ Ω2(M) operating on X(M). The pair (M,ω) is alled symple ti manifold.[v ω non-degenerate ⇔ detω(ei, ej) 6= 0, ei basis of M⇔ dim M = 2n and ωn := ω ∧ .. ∧ ω is a volume form on M

14 2 FUNDAMENTAL MATHEMATICAL CONCEPTSLet (M,ω) be a symple ti manifold and let E be a linear subspa e of TmM for somem ∈M .Dene the symple ti omplement of E to be the subspa eE⊥ := X ∈ TmM | ω(X,Y ) = 0 ∀Y ∈ E . (2.11)It satises (E⊥)⊥ = E and dimE + dimE⊥ = dim TmM . However, unlike orthogonal omplements, E ∩ E⊥ does not need to be 0 (vide [Rud05, 6).A ording to proposition 2.3.1, it is always possible to determine lo al oordinates in asymple ti manifold, in whi h the symple ti form is given by equation (2.10). They are alled Darboux oordinates (xi, yi) (in a physi al ontext we usually use (qi, pi)).Remark 2.3.3: Let the manifold Q be the onguration spa e of a me hani al system.Therefore, T∗Q is the orresponding phase spa e. It a quires a symple ti stru ture in anatural way.Let π : TQ → Q be the anoni al proje tion in the tangent bundle. Setting

θp(X) := p(π′X) , p ∈ T∗ Q , X ∈ Tp(T∗Q) and

ω := −dθ .Then, in parti ular, dω = 0.One an show (see [AM85, Thm. 3.2.10), that ω is non-degenerate andθ = pidq

i and ω = dpi ∧ dqi ,where qi are the omponents of the lo al hart's oordinate fun tions on Q and pi arethe bre oordinates on T∗Q indu ed by the lo al hart. The form ω is alled anoni alsymple ti form on T∗Q. The Darboux theorem 2.3.1 already gives great insight into symple ti manifolds. Theylo ally appear like the lassi al phase spa e T∗Rn, where the q- omponents of ϕ representposition and the p- omponents of ϕ represent momentum.These ir umstan es stress the importan e of symple ti geometry in physi s. Phase spa esare symple ti manifolds in a anoni al way.We subsequently dene Hamiltonian ve tor elds.Denition 2.3.4: Let f be a smooth fun tion on a symple ti manifold (M,ω). The ve toreld Xf dened byXf y ω = −df [vi (2.12)is alled Hamiltonian ve tor eld.[vi (· y ·) is dened by (X y ω)(Y ) ≡ 〈ω, X ∧ Y 〉

2.3 MOMENTUM MAPPINGS 15This denition is equivalent to〈−df, Y 〉 = 〈ω,Xf ∧ Y 〉 ≡ ω(Xf , Y ) .In Darboux oordinates we nd (see [Rud05, Bem. 7.1.2)

Xf = (∂pif)∂qi − (∂qif)∂pi

.A symple ti manifold (M,ω) together with a Hamiltonian ve tor eld H ∈ C∞(M) is alledHamiltonian system and denoted by the triplet (M,ω,H). Hamiltonian ve tor eldshave the following physi al interpretation: Let H ∈ C∞(T∗Q) be a Hamilton fun tion onthe physi al system (T∗ Q, ω). The ow of the Hamiltonian ve tor eld asso iated with Hthen denes the time evolution of the system.Later on, we often nd the following stru ture.Denition 2.3.5: A Poisson algebra A is a ve tor spa e over a eld K equipped withtwo bilinear produ ts · and , satisfying the following onditions(i) the produ t · forms an asso iative K-algebra with unit element,(ii) the produ t , , alled the Poisson bra ket, is a Lie bra ket, i.e. it is anti-symmetri and obeys the Ja obi identity [vii and(iii) the Poisson bra ket a ts as a derivation of the asso iative produ t · su h that for anythree elements f, g, h ∈ A , one has f, g · h = f, g · h+ g · f, h (Leibniz' law).We use the notation (A , , , ·).The linear mapadf : A → A , g 7→ f, g (2.13)is alled Hamiltonian derivation asso iated with f . Its formal ow is dened by

φft = exp(t · adf ) =

∞∑

n=0

1

n!tn adnf . (2.14)The Hamiltonian ow φft of a Hamiltonian derivation adf exists and is unique (see [CB97,page 354). The Poisson algebra is non-degenerate if adf ≡ 0 implies that f is aK-multipleof the unit.A Poisson ideal is a subset I ⊆ A , whi h is an ideal of A with regard to the Poissonbra ket and the multipli ation.[vii f, g, h + h, f, g + g, h, f = 0 , f, g, h ∈ A

16 2 FUNDAMENTAL MATHEMATICAL CONCEPTSA Poisson map ϕ : M → N of two Poisson algebras (M, , M , ·) and (N, , N , ·) is a mapwhi h satisesf, gN ϕ = f ϕ, g ϕM (2.15)for all f, g ∈ C∞(N).Leibniz' law assures that adf is indeed a derivation on A .Remark 2.3.6: Let (M,ω) be a symple ti manifold and f, h ∈ C∞(M). The mappingdened by

f, h := ω(Xf ,Xh) (2.16)is a Poisson bra ket. C∞(M) together with (2.16) is a Poisson algebra (see [Rud05, Satz7.1.13).Furthermore, we ndω(Xf , Y ) = (Xf y ω)(Y )

(2.12)= (−df)(Y )

Def. of d= −Y (f) and

ω(Xf ,Xg) = Xg y (Xf y ω) = −Xg y (df) = −Xg(f) .Hen ef, g = −Xg(f) = Xf (g)

[Xf ,Xg] = Xf,g . (2.17)Therefore we an identify adf with the Hamiltonian ve tor eld Xf .Denition 2.3.7: Let (M,ω) be a symple ti manifold and G a Lie group with an a tion

Φ on M . Φ is symple ti ifΦ∗gω = ω (here, (·)∗ denotes the pull-ba k operation)for all g ∈ G. In this ase the triple (M,ω,Φ) is alled symple ti G-manifold.Note: If the Lie group G is onne ted, Φ is symple ti in a natural way ( f. [CB97, Lem.B.3.2).So there are two distinguished types of ve tor elds on symple ti manifolds: Killing ve -tor elds and Hamiltonian ve tor elds. Linking those two ve tor elds, we introdu e themomentum mapping.Denition 2.3.8: Let (M,ω,Φ) be a symple ti G-manifold. We say that Φ is a Hamil-tonian G-a tion if for every A ∈ g the Killing ve tor eld A∗ is also a Hamiltonian ve tor

2.3 MOMENTUM MAPPINGS 17eld on (M,ω). In other words, for every A ∈ g, there is a smooth fun tion JA : M → Rsu h thatA∗ = XJA

. (2.18)The map g → R : A 7→ JA(m) is linear in A for all m ∈ M . Hen e, bringing together theinformation given in (2.18) we dene a mapping J through the relationJ : M → g∗ , X〈J(·),A 〉 = A∗ ⇔ JA(·) =: 〈J(·), A〉 , A ∈ g . (2.19)Denition 2.3.9: The map J given in (2.19) is alled momentum map of the Hamil-tonian G-a tion Φ.Together with the symple ti G-manifold (M,ω,Φ) on whi h it operates, the system is alledHamiltonian G-manifold and denoted by (M,ω, J,Φ).Proposition 2.3.10: If (M,ω) is onne ted, then the momentum mapping is determinedup to an additive onstant µ0 ∈ g∗.Proof : Suppose J is another momentum map for the Hamiltonian a tion Φ and let J :=

J − J . For every A ∈ g and every Xm ∈ TmM with m ∈M xed there holds〈dJA,X 〉m Def. of d

= Xm(JA)(2.19)= Xm〈J (·), A〉 = (J ′

m(Xm))(A)and thus(J ′

m(Xm))(A) = 〈dJA,X 〉m= (dJA)mXm − (dJA)mXm

= ωm((XJA)m − (XJA

)m,Xm)

= ωm((A∗)m − (A∗)m,Xm) = 0 .Sin e J ′m vanishes for every m ∈ M and M is onne ted, it follows that J = µ0 for somexed µ0 ∈ g∗. Proposition 2.3.11: Let (M,ω, J,Φ) be a Hamiltonian G-manifold and H ∈ C∞(M)Ga G-invariant [viii smooth fun tion on M . Then, the fun tion JA is a onstant of motion inthe Hamiltonian system (M,ω,H) for all A ∈ g, i.e.

d

dtJA(γ(t)) = 0 ,for all γ integral urves of XH (thus γ(t) remains within one level set of JA).[viii H ∈ C∞(M) G-invariant ⇔ H Φg ≡ Φ∗

gH = H ∀ g ∈ G,the set of G-invariant smooth fun tions on M is denoted by C∞(M)G

18 2 FUNDAMENTAL MATHEMATICAL CONCEPTSNote: If H is G-invariant, then Φ is alled a symmetry of the Hamiltonian system. Symme-tries give rise to onserved quantities.Proof : We start by verifying that JA,H = 0.JA,H(m)

(2.17)= (XJA

)mH = A∗mH =d

dt t=0

H Φexp tA(m) = 0 .Using f ∈ C∞(M) onstant of motion ⇔ f,H = 0 andd

dt t

f γ(t) = XH(f) γ(t) (2.17)= H, f γ(t) ,the assertion is shown. Remark 2.3.12: Let (Q,Φ) be a G-manifold with the a tion ϕ and π : T∗ Q → Q thenatural proje tion introdu ed in remark 2.3.3. The dieomorphism ϕg indu es a dieomor-phism Φg : T∗Q → T∗Q for all g ∈ G, alled point transformation, given by the onditions(i) the diagram

T∗ Q Φg- T∗Q

Q

π

?

ϕg- Q

π

? ommutes and(ii) for all p ∈ T∗Q and X ∈ Tϕgπ(p) Q there is〈Φg(p),X 〉 = 〈p, ϕ′

g−1X 〉 .So, for k-forms α ∈ Ωk(Q) there isΦg α = (ϕ∗

g−1α) ϕg .The assignment g 7→ Φg denes a mapΦ : G× T∗ Q → T∗ Qwhi h is a G-a tion on T∗ Q (see [Rud05, Bem. 7.8.7). As one an easily see

π Φg = ϕg π , (2.20)i.e. the natural proje tion π is equivariant. An a tion with the property (2.20) is alled liftof the a tion ϕ to T∗Q.Resulting from these fa ts, we nd (see [Rud05, Satz 6.4.5)Φ∗gθ = θ .Therefore, Φg is symple ti and (T∗ Q, ω = dθ,Φ) is a symple ti G-manifold.

2.4 SINGULAR MARSDEN-WEINSTEIN REDUCTION 19Proposition 2.3.13: Dene the map J : T∗Q → g∗ by〈J(p), A〉 := 〈p,Aϕ∗ (π(p))〉 , (2.21)where Aϕ∗ is the Killing ve tor eld of A on Q indu ed by ϕ. Then, J is a momentummapping for the lifted a tion Φ whi h is equivariant with respe t to the oAdjoint a tion of

G on g∗, i.e.J Φg(p) = Ad∗

g J(p) , g ∈ G , p ∈ T∗Q .Proof : See [CB97, Claim B.3.8. 2.4 Singular Marsden-Weinstein redu tionThe symmetry of a system is equivalent to inner degrees of freedom. In order to fo uson physi al questions, we want to simplify the system and remove these degrees of freedom.Redu tion is the basi te hnique in symple ti geometry for removing symmetry from Hamil-tonian systems. In this subse tion we will rst introdu e the regular redu tion theorem andthen state and motivate the singular ase.Let Φ : G×M →M be a free proper Hamiltonian a tion of a Lie group G on a symple ti manifold (M,ω) with the equivariant momentum map J : M → g∗, i.e. (M,ω, J,Φ) is aHamiltonian G-manifold. For µ ∈ g∗ let m be a point in the level set J−1(µ) and Oµ the oAdjoint orbit through µ, i.e.Oµ = Ad∗

g µ | g ∈ G ⊆ g∗ .Let Gm be the isotropy group of m under the G-a tion Φ and let Gµ be the isotropy groupof µ under the oAdjoint a tion of G on g∗ (i.e. Gm := g ∈ G | Φg(m) = m and Gµ :=

g ∈ G | Ad∗g µ = µ).The following lemma relates redu tion at an arbitrary oAdjoint orbit Oµ with redu tion at

O0 = 0.Lemma 2.4.1 (shifting tri k) : Dene a G-a tion on M ×Oµ byG× (M ×Oµ) →M ×Oµ , (g, (m, ν)) 7→ (Φg(m),Ad∗

g ν) .This a tion is proper and has a oAdjoint equivariant momentum mapping J : M ×Oµ → g∗given by Jm,ν(·) = J·(m) − ν(·).The level sets J−1(µ) ×Oµ and J−1(0) are lo ally dieomorphi .Proof : The fa t that (Φg(·),Ad∗g ·) is a proper a tion and that J is its equivariant momentummapping is a dire t onsequen e of the fa t that they are ompositions of su h maps.For the shifting tri k see [CB97, Fa t B.3.19.

20 2 FUNDAMENTAL MATHEMATICAL CONCEPTSCorollary 2.4.2: For every µ ∈ g∗ the level set J−1(µ) is lo ally ar wise onne ted.Proof : Cf. [CB97, page 332. 2.4.1 Redu tion: the regular aseWe now formulate the regular redu tion theorem.Theorem 2.4.3 (Regular redu tion theorem) : Let (M,ω, J,Φ) be a HamiltonianG-manifold.Suppose that µ ∈ g∗ is a regular value [ix of J .Then, the redu ed spa eMµ = J−1(µ)/Gµ (2.22)is a smooth symple ti manifold with symple ti form ωµ, uniquely dened by

π∗µωµ = ι∗ω . (2.23)Here πµ : J−1(µ) → Mµ is the orbit map (= redu tion map) of the Gµ-a tion ΦGµ×J−1(µ)and ι : M ⊇ J−1(µ) →M is the anoni al embedding.Proof : The redu ed spa e Mµ is a smooth manifold, be ause Gµ a ts freely and properly onthe smooth manifold J−1(µ) ( f. proposition 2.2.4). Thus, we only need to onstru t thesymple ti form ωµ on Mµ. For that purpose vide [CB97, Thm. B.4.1.

HJ−1(µ)

indu es a smooth fun tion Hµ on the redu ed spa eMµ, alled the redu ed Hamil-tonian, whi h satises the equationπ∗µHµ = i∗H .Thus, we onstru ted a redu ed Hamiltonian system (Mµ, ωµ,Hµ).Its importan e lies in the fa t that the ve tor eld XH on J−1(µ) is πµ-related to theredu ed ve tor eld XHµ , i.e. the diagramM

XH- TM

Mµ

πµ

?

XHµ

- TMµ

π′µ

? ommutes. By use of symmetry, we redu e the number of variables in the me hani al system.[ix n ∈ N is a regular value of the smooth map between manifolds f : M → N ⇔ for all m ∈ f−1(n) themap dfm : TmM → TnN is surje tive

2.4 SINGULAR MARSDEN-WEINSTEIN REDUCTION 212.4.2 Redu tion: the singular aseWe start by introdu ing the on ept of stratied symple ti spa es.Denition 2.4.4: Let X be a para ompa t Hausdor spa e and (D,≤) a partially orderedset. A D-de omposition of X is a lo ally nite partition of X into lo ally losed [x, disjointmanifolds Sd ⊆ X , d ∈ D, satisfying the following onditions:(i) X =⋃d Sd,(ii) Sd ∩ Se 6= ∅ ⇔ Sd ⊂ Se ⇔ d ≤ e (frontier ondition).

Sd are alled pie es of X.A simple example of su h a D-de omposition is an n-simplex. Its de omposition is S0 =edges, S1 = sides, S2 = inner area et . (Sn = n-dimensional fa es).A stratied spa e is a D-de omposed spa e X, whose pie es satisfy further onditions ( f.[P01, Def. 1.2.2). Sin e those will not be of interest in the following, it is adequate to referto it as a D-de omposed spa e. The pie es of a stratied spa e are the so- alled strata.Denition 2.4.5: A stratied symple ti spa e X is a stratied spa e together with aC∞(X)-stru ture [xi whi h satises:(i) ea h stratum S is a symple ti manifold,(ii) C∞(X) is a Poisson algebra and(iii) the embeddings S → X are Poisson maps.Compared to the regular ase, we relax the hypothesis. In parti ular, we only assume thatthe Hamiltonian a tion is proper (and not ne essarily free). There are no other assumptionson the value of the momentum map µ. Hen e the redu ed spa e Mµ is not ne essarily amanifold. The fa t that it still has a manageable stru ture is ensured byTheorem 2.4.6 (Singular redu tion theorem) : Let (M,ω, J,Φ) be a Hamiltonian G-manifold and µ ∈ g∗. Then,

Mµ = J−1(Oµ)/G = π(J−1(Oµ)) (2.24)together with the de omposition given by the orbit types of Φ is a stratied symple ti spa e.[x Y ⊆ X is lo ally losed in X ⇔ ea h point y ∈ Y has an open neighbourhood Uy ⊆ X su h that Y ∩Uyis losed in Uy (with the subspa e topology)[xi subalgebra of the algebra of ontinuous fun tions from X to R

22 2 FUNDAMENTAL MATHEMATICAL CONCEPTSIn the sequel, we are going to study the quotient Mµ, onstru t the geometri obje ts de-s ribed above and thereby motivate the singular redu tion theorem. For the fa t that thede omposition mentioned above forms a stratied spa e, ompare [P01, Thm. 4.3.7.Constru tion of the strataUsing lemma 2.4.1 (shifting tri k), we are able to set µ = 0 without loss of generality. Fur-thermore, we partition J−1(0) by orbit types.Let H ⊆ G be a ompa t subgroup. We setJH := J−1(0) ∩MH and J(H) := J−1(0) ∩M(H)

[xii . (2.25)Consequently, JH ⊆ J(H) and the orbit type of subset J(H) is the onjuga y lass of H.The following propositions are key results in onstru ting the strata of M0.Proposition 2.4.7: There holds(i) JH and J(H) are submanifolds of (M,ω) and(ii) Tm J(H) ∩ (Tm J(H))⊥ = TmOm

[xiii for all m ∈M .Proof : See [CB97, page 349 . Let N(H) be the normaliser of H in G [xiv. Then, JH and J(H) are G-manifolds with respe tto the proper Lie group a tionsΦ1 : N(H)/H × JH → JH , ([n], x) 7→ Φn(x) andΦ2 : G× J(H) → J(H) , (g, x) 7→ Φg(x) .

N(H)/H is a Lie group, as N(H) is losed in G and H is losed and normal in N(H).Obviously, Φ1 is a free a tion on JH . Therefore, JH/(N(H)/H) is a smooth manifold (propo-sition 2.2.4).We now prove that J(H)/G and JH/N(H) are homeomorphi (JH/N(H) = JH/(N(H)/H)sin e the isotropy group of JH is dened to be H).Proposition 2.4.8: Let ιH be the anoni al embedding of JH in J(H) and h the map,[xii MH and M(H), f. denition 2.2.8[xiii (·)⊥ is the symple ti omplement, f. (2.11)[xiv dened by g ∈ G | gHg−1 = H; subgroup of G

2.4 SINGULAR MARSDEN-WEINSTEIN REDUCTION 23whi h is dened by the following ommutative diagramJH

ιH - J(H)

JH/N(H)

π1

?

h- J(H)/G .π2

?Then, h is a homeomorphism.Proof : Surje tivity : Let x be a representative of [x](H) ∈ J(H). Then, there exists an elementg ∈ G su h that Φg(x) ∈ JH .Inje tivity : Let [x1]H , [x2]H ∈ JH/N(H) su h that h([x1]H) = h([x2]H) and let x1, x2 ∈ JHbe their representatives.Sin e h([x1]H) = h([x2]H), there is a g ∈ G su h that gιH(x1) = ιH(x2). Then, gx1 = x2 andhen e there is Gx1 = Gx2 = Ggx1 = H. On the other hand, with Ggx = gGxg

−1, we havegHg−1 = H ⇒ g ∈ N(H) and therefore, [x1]H = [x2]H .Bi ontinuity : h is ontinuous as ιH and π2 are ontinuous and π1 is open. h is open be auseh is surje tive and ιH and π2 are losed. Due to the regular redu tion theorem, the spa e JH/N(H) arries the stru ture of a dier-entiable manifold. Hen e J(H)/G is a manifold.Proposition 2.4.7 (ii) assures that ω(H) ∈ Ω2(J(H)/G), dened by

π∗2ω(H) = ι∗ω ,is a symple ti form. Here, ι : J(H) →M denotes the anoni al embedding.Denition 2.4.9: A stratum of Mµ is dened by

(M(H)µ , ω(H)

µ ) = (J(H)/G,ω(H)) . (2.26)Constru tion of the Poisson algebraAt rst, we dene C∞(Mµ).Denition 2.4.10: A fun tion fµ on Mµ, fµ : Mµ → R is said to be smooth if thereexists a smooth G-invariant fun tion f : M → R , f ∈ C∞(M)G su h that

π∗fµ = fJ−1(0)

.

24 2 FUNDAMENTAL MATHEMATICAL CONCEPTSConsequently, C∞(Mµ) equals C∞(M)GJ−1(0). Thus, let I G be the vanishing ideal of J−1(0),

I G := f ∈ C∞(M)G | fJ−1(0)

= 0 .Therefore we formulateC∞(Mµ) = C∞(M)G/I G .As J−1(0) is losed, I G is a losed subspa e of C∞(M)G and we may give C∞(Mµ) thequotient topology.Sin e ω is G-invariant, C∞(M)G is a Poisson subalgebra of C∞(M). Due to the fa t, thatHamiltonian ve tor elds of fun tions in C∞(M)G are tangential to J−1(0), I G is a Poissonideal of C∞(M)G.Therefore, C∞(Mµ) itself is a Poisson algebra sin e it is the quotient of a Poisson algebraand a Poisson ideal.The indu ed Poisson bra ket on C∞(Mµ) has the following stru ture:Denition 2.4.11: Let fµ, gµ ∈ C∞(Mµ) and f, h ∈ C∞(M)G with

π∗fµ = fJ−1(0)

and π∗hµ = hJ−1(0)

.The Poisson bra ket of fµ and hµ is dened byπ∗fµ, hµµ = f, h

J−1(0).Moreover, one an prove that this Poisson bra ket is non-degenerate (see [CB97, page 348).The Hamiltonian ow ( f. (2.13)) of adfµ an be obtained by pulling down the Hamiltonianow φft of a fun tion f ∈ C∞(M)G with π∗fµ = f

J−1(0).

3 Gauge theoryIn this hapter we are going to outline the physi al model of gauge theory. In the framework ofthis theory fundamental intera tions in physi s an be modelled by onne tions on prin ipalbre bundles. An illustrative introdu tion to gauge theory and its physi al relevan e an befound in [BM94.LetM be the spa e-time ontinuum. In semi lassi al physi s, a parti le is des ribed in termsof a wave fun tion ψ : M → V , where V is some ve tor spa e (typi ally over the omplexnumbers). Some xed basis of V is hosen with the basis elements orresponding to ertainstates of the parti le. Impli it in the determination of ψ is the hoi e of a referen e frame inV at m (e.g. C3 for strong intera tion). Let Pm denote the spa e of all possible referen eframes at m. Any two referen e frames at m are uniquely related by an element of somegroup G of transformations of V (e.g. rotations).If ψ(p) is the value of ψ relative to p ∈ Pm, then ψ(pg) = g−1ψ(p) is the value relative to pg.A smooth on atenation P of the various Pm as m ranges over M is alled a prin ipal brebundle with group G and Pm is alled bre over m.If U is a subregion of M , then a smooth map s : U → P , su h that s(m) ∈ Pm for all m ∈ Uis alled a lo al se tion (i.e. a ontinuous hoi e of referen e frames).The main goal is to dene physi al laws and quantities independent of the experimental onditions, thus independent of the hoi e of gauge (unlike the lo al wave fun tions). Hen eit is attempted to onstru t real-valued fun tions on M that depend on the wave fun tion(and its dierential) but are independent of the hoi e of gauge. Su h a fun tion is alledan a tion density. For this onstru tion an additional obje t on P is needed, alled a gaugepotential or onne tion.3.1 Mathemati al obje ts of gauge theoryPrin ipal bre bundleSin e all measurements are made relative to a xed frame and the measurement pro ess annever be ompletely divor ed from the aspe t of the universe being measured, we are led tothe on lusion that the bundle of referen e frames should play a role in the very stru ture ofour universe. 25

26 3 GAUGE THEORYThis bundle of referen e frames is the entral obje t of every gauge theory model. It is alledprin ipal bre bundle P over a spa e-time manifold M with a stru ture group G.Denition 3.1.1: A prin ipal bre bundle (P,M,π,G, χi) onsists of a manifoldP ( alled the total spa e), a Lie group G ( alled the stru ture group), a base manifoldM and a proje tion map π : P →M su h that(i) G a ts freely and dierentiably on P to the right (i.e. P ×G→ P , (p, g) 7→ pg).(ii) The map π : P → M is onto and π−1(π(p)) = pg | g ∈ G (whi h is the orbit of Gthrough p). If m ∈M , then π−1(m) is alled the bre over m (see gure 3.1).Note that for ea h p ∈ π−1(m) there is a map G → π−1(m) given by g 7→ pg. Thismap is a dieomorphism by (i), but depends on p. Thus, all the bres π−1(m) aredieomorphi to G, but there is no anoni al identi ation of it and hen e no naturalgroup stru ture on π−1(m).(iii) For ea h m ∈ M there is an open set Ui with m ∈ Ui and a dieomorphism χi :

π−1(Ui) → Ui × G of the form χi(p) = (π(p), κi(p)), where κi(p) : π−1(Ui) → G hasthe property κi(pg) = κi(p)g for all g ∈ G, p ∈ π−1(Ui). The map χi is alled a lo altrivialisation. χi is alled global trivialisation if Ui = M .Remark 3.1.2: Condition (iii) is equivalent to: For a overing (Ui)i∈I of M there is afamily of dieomorphisms χi : π−1(Ui) → Ui ×G su h that the diagramπ−1(Ui)

χi - Ui ×G

Ui

π

?pr1

ommutes and κi := pr2 χi satises κi(pg) = κi(p)g.Furthermore, we abbreviate the foregoing by saying that π : P → M is a prin ipal brebundle with group G. Example 3.1.3: Let (P, ν) be a right G-manifold and let ν be proper and free. A ordingto proposition 2.2.4, P is a prin ipal bre bundle over M := P/G with stru ture group G. Denition 3.1.4: Let χi : π−1(Ui) → Ui × G and χj : π−1(Uj) → Uj × G be twolo al trivialisations of a prin ipal bre bundle π : P → M with group G. The transitionfun tion from χi to χj is the map gij : Ui ∩ Uj → G, dened for m = π(p) ∈ Ui ∩ Uj bygij(m) = κi(p)(κj(p))

−1.

3.1 MATHEMATICAL OBJECTS OF GAUGE THEORY 27PSfrag repla ements

x

p

M

P

TxM

π

π−1(x)

Fig. 3.1: Visualisation of the prin ipal bre bundle above M .Note: gij(m) is independent of the hoi e of p ∈ π−1(m) be auseκi(pg)(κj(pg))

−1 = κi(p)g(κj(p)g)−1 = κi(p)gg

−1(κj(p))−1 = κi(p)(κj(p))

−1 .Remark 3.1.5: We an nd(i) gii(m) = e, for all m ∈ U ,(ii) gji(m) = (gij(m))−1, for all m ∈ Ui ∩ V ,(iii) gij(m)gjk(m)gki(m) = e, for all m ∈ Ui ∩ Uj ∩W .Thus, the transition fun tions des ribe how the various produ ts U × G, V × G, ... gluetogether to form the total spa e P . Denition 3.1.6: We dene a lo al se tion of a prin ipal bre bundle (P,M, π,G, χi)to be a map s : Ui → P with Ui ⊆M open su h that π s = idi.Note: If U = M , then s is a global se tion. A prin ipal bre bundle with a global se tion is alled trivial.Example 3.1.7: Let M be a smooth manifold and G a Lie group. Then, P := M × Gwith the proje tion π(m, g) := m and the right group a tion (m, g)a := (m, ga) is a trivialprin ipal bre bundle. Conne tionsConne tions dene the mathemati al model for gauge potentials. They an be dened in twodierent ways. In the sequel, we show that both are equivalent. We hen eforth assume that(P,M, π,G, χi) is a prin ipal bre bundle, dimM = n and g is the Lie algebra of G.

28 3 GAUGE THEORYPSfrag repla ements

x

p

M

P

TxM0TxM

TpP

Hp

Hp

Vp

Vp

ππ∗

Fig. 3.2: De omposition of the tangent spa e of P .Denition 3.1.8: A onne tion assigns to ea h p ∈ P a subspa e Hp ⊆ TpP (the hor-izontal subspa e) su h that for Vp = X ∈ TpP | π∗(X) = 0 (the verti al subspa e)we have TpP = Hp ⊕ Vp. Furthermore, Rg ′(Hp) = Hpg is required.Denition 3.1.9: A onne tion one-form is a g-valued one-form ω dened on P su hthat the following properties hold:(i) ωp(A∗p) = A for all A ∈ g [i and(ii) ωpg(Rg′X) = Adg−1 ω(X) for all g ∈ G, p ∈ P and X ∈ TpP . In other words,

R∗gω = Adg−1 ω.Proposition 3.1.10: Denitions 3.1.8 and 3.1.9 are equivalent geometri des riptions.Proof : The general idea is to use ω, the onne tion one-form of denition 3.1.9, and then set

Hp = X ∈ TpP | ωp(X) = 0. Thus, Hp⊕Vp = TpP . Conversely, we set ωp(A∗p+Xp) =

A with A∗p dened as in 3.1.9 (i) and Xp ∈ Hp.For the detailed proof ompare [Ble05, Thm. 1.2.4. [i A∗p, f. denition 2.2.6

3.1 MATHEMATICAL OBJECTS OF GAUGE THEORY 29Proposition 3.1.11: Any prin ipal bre bundle admits a onne tion.Proof : Cf. [KN96a, Thm. 2.2.1. Remark 3.1.12: In the same way that adjoining the time dimension to our three-dimen-sional spa e permits the geometrisation of the for e of gravity, non-gravitational for es anbe geometrised via onne tions on a prin ipal bre bundle.Let gij(m) be the transition fun tion of the lo al trivialisations χi and χj and let ωi andωj be g-valued one-forms over Ui and Uj , respe tively (dened by ωi = s∗iω, where si(m) =

χ−1i (m, e), see below). The transformation rule from ωi to ωj an be expressed as

ωj = g−1ij dgij + g−1

ij ωigij . (3.1)Compare [Ble05, Def. 1.2.3 for this formula.Physi al interpretation: The one-forms ωi are referred to as gauge potentials. Below,we give an example of how to ompute the eld strength starting with ωi. Therefore we onsider the spe ial ase of ele tromagnetism.Suppose that (M,η) is a Minkowski spa e (hen e, M = R4 and η is a metri , su h thatη(∂0, ∂0) = 1, η(∂ i, ∂ j) = −δij , i, j ∈ 1, 2, 3). Let π : P → M be a prin ipal brebundle with group U(1) = eiθ | θ ∈ R. The Lie algebra is u(1) = iR. Suppose that ω is a onne tion one-form on P and let si : Ui → P be a lo al se tion.Then, ωi = s∗

iω = −iAi (following physi al onvention, the result in ludes −i), whereAi ∈ Λ1(Ui,R) is alled the potential one-form (or ve tor potential). The ele tromag-neti eld strength relative to si : Ui → P is then Fi = −dAi ∈ Λ2(Ui,R). If sj : Uj → Pis another lo al se tion, then Fj = Fi: Using (3.1) and the fa t that U(1) is Abelian weobtain ωj = g−1

ij dgij + ωi. The relation g−1ij gij = 1 implies that d(g−1

ij )gij + g−1ij dgij = 0 or

d(g−1ij ) = −g−2

ij dgij. Thus, −d(g−1ij dgij) = g−2

ij dgij ∧ dgij = 0 and it follows that dωj = dωior Fj = Fi. Hen e the Fi (for various s : Ui → P ) pie e together to yield a well-denedF ∈ Λ2(M,R).While the lo al gauge potentials transform in a quite ompli ated way under gauge transfor-mation ( ompare with (3.1)), the lo al eld strengths obey a simpler transformation rule. Inthe ase of an Abelian group, like we have in ele tromagnetism, the lo al eld strengths areinvariant under a hange of gauge and pie e together to yield a well-dened eld strength onthe base.For non-abelian groups the eld strength is a well-dened g-valued two-form on the totalspa e P . It is omputed through the urvature of the onne tion.

30 3 GAUGE THEORYCurvatureGiven a onne tion one-form ω on a prin ipal bre bundle (P,M, π,G, χi), we an writeany X ∈ TpP as X = XV +XH, where XV is verti al (i.e. π∗(XV ) = 0) and XH ishorizontal (i.e. ω(XH) = 0).Let V be a nite dimensional ve tor spa e and N be a manifold. We will denote the set ofall V -valued k-forms on N by Λk(N,V ).Denition 3.1.13: The exterior ovariant derivative of α ∈ Λk(P, g) isDωα(X1, ..,Xk+1) := (dα)H(X1, ..,Xk+1) ≡ dα(XH

1 , ..,XHk+1) ∈ Λk+1(P, g) . (3.2)Denition 3.1.14: The urvature of the onne tion ω ∈ Λ1(P, g) is Ωω ≡ Dω ω ∈

Λ2(P, g). When ω is regarded as a potential, Ωω is alled the eld strength of ω.The following stru tural equation will permit us to write an expression for the eld strengththat looks more familiar.Proposition 3.1.15 (Stru tural equation) : The urvature form is given by Dω ω = dω+12 [ω, ω] (= Ωω).Proof : Cf. [Ble05, Thm. 2.2.4 Con lusion 3.1.16: In physi al appli ations the gauge potential shall be expressed byobje ts on spa etime. In general, this an be a hieved only lo ally: Let s : U → π−1(U) bea lo al se tion, then

A := s∗ω , F := s∗Ωω . (3.3)Remark 3.1.17: From the stru tural equation we an derive

F = dA+1

2[A,A] .Let s be a lo al se tion and κ : π−1(U) → G the asso iated equivariant map. For a givenlo al gauge potential A and a given lo al eld strength F we are able to obtain the onne tionform and the urvature by

ωp = Ad(κ(p)−1)(π∗(A))p + (κ∗Θ)p andΩp = Ad(κ(p)−1)(π∗(F ))p , (3.4)where Θ denotes the Maurer-Cartan form [ii on G ( f. [Rud06, Satz 2.1.3). [ii g-valued one-form Θ on G with 〈Θ, A〉(a) = A, where A ∈ g and a ∈ G, ompare [Rud05, Satz 5.3.11

3.1 MATHEMATICAL OBJECTS OF GAUGE THEORY 31PSfrag repla ements

x M

FPFB P VB× ·/G−→

P ×G F

π

π−1(x)

Ψ

Fig. 3.3: Diagram of the parti le eld Ψ.Asso iated bre bundles and parti le eldsLet (P,M, π,G, χi) be a prin ipal bre bundle with right a tion denoted by ν and (F, σ)be a left G-manifold. Let σ := σa−1 . Sin e ν a ts freely, ν× σ is a right free a tion on P ×F .WithP ×G F := (P × F )/G ,we obtain the orbit spa e of this a tion. There is a unique map ρ : P ×G F →M , su h thatthe diagramP × F

ι- P ×G F

P

pr1

?

π- M

ρ

? ommutes, where ι : P × F → P ×G F is the natural proje tion (vide [Rud06, Satz 4.1.1). Therefore, P ×G F itself denes a ve tor bundle with typi al bre F . It is alled theasso iated bre bundle.A parti le eld Ψ an be regarded as a se tion of the asso iated ve tor bundle P ×G F ,Ψ ∈ Γ∞(P ×G F ). Equivalently, it an also be onsidered as an equivariant ve tor-valuedfun tion on P . Examples in lude the S hrödinger wave fun tion and the Dira ele tron eld.Typi ally, a real-valued fun tion ( alled a tion density) on the base is assigned to the parti leelds. These elds obey a dierential equation, the Lagrange's equation, obtained by settingthe rst variation of the integral of the a tion density to zero. However, sin e this thesis hasno dire t relation to these onstru tions, we will not spe ify them any further at this point.

32 3 GAUGE THEORY3.2 Classi al gauge theoryThe spa e of elds of a pure gauge theory isΦ = ω onne tions on P . (3.5)Not all elements of Φ an be distinguished. Equivalent elds are obtained by gauge trans-formation. The group of gauge transformations G is realised mathemati ally by verti alautomorphisms in P .Denition 3.2.1: An automorphism of a prin ipal bre bundle (P,M, π,G, χi) is adieomorphism ϑ : P → P su h that ϑ(pg) = ϑ(p)g for all g ∈ G, p ∈ P .

ϑ indu es a unique dieomorphism ϑ : M →M by the ommutative diagramP

ϑ - P

M

π

?

ϑ- M .π?A verti al automorphism (or gauge transformation) is an automorphism with ϑ = idM .The group of all verti al automorphisms is denoted by AutM(P ).Denote: G = AutM (P ).Remark 3.2.2: ϑ respe ts the bres, i.e. it indu es dieomorphisms ϑ

π−1(m): π−1(m) →

π−1(m) for any m ∈M .In general, the maps Rg : P → P , p 7→ pg are not gauge transformations be ause Rg(pg) =

pgg, while Rg(p)g = pgg. Thus, unless G is Abelian (or g is in the entre of G), we do nothave Rg(pg) = Rg(p)g. Gauge transformations a t through pull-ba k on gauge elds, i.e.G × Φ → Φ , (ϑ, ω) 7→ ϑ∗ω .The dynami s in gauge elds is represented by the Lagrangian and the prin iple of leasta tion. The lassi ally possible eld ongurations satisfy the eld equation. Thisequation is derived from the G-invariant Lagrange density

L : Φ → C∞(P )by variation of the fun tionalS : Φ → R , ω 7→

∫

PL(ω) (prin iple of least a tion). (3.6)

3.3 HAMILTONIAN FORMULATION 33In the following, we will denote all elds whi h minimise (3.6) by Φ.Summarising, we have: In lassi al gauge theory, for a given prin ipal bre bundle P and agiven Lagrange density L, one wants to nd onne tions ω whi h minimise the a tion integralS. Furthermore, one does not distinguish between solutions whi h an be obtained by gaugetransforming one into another. Thus, the main fo us in lassi al gauge theory is on thequotient

Φ/G . (3.7)A gauge is a hoi e of a representative in this spa e. To redu e the omplexity of (3.7), weare going to map Φ on a temporary quotient Φ → Φ → Φ/G = Φ/G.3.3 Hamiltonian formulationWe simplify the given problem by dis retising the spatial degrees of freedom. To do so, the ontinuous spa e is substituted by latti e points. Let ST denote our spa etime and G be thegauge group, hen eforth.Sin e only spa e is dis retised, we onstrain gauge elds on spa etime to spa e only using thetemporal gauge. To perform this gauge, we have to make further simplifying assumptions:(i) P (ST,G) is trivial and(ii) spa etime ST an be de omposed ST = M ×R, where M is spa e and R is time.This separation of time allows the hange from Lagrangian to Hamiltonian formulation.Sin e P is trivial, Φ = Ω1(ST ) ⊗ g. The tangent bres an be written as T(m,t)(M ×R) = TmM ⊕ TtR. Hen e, for any X = XM + XR and α ∈ T∗(m,t)(M × R) there are

αM , αR ∈ T∗(m,t)(M ×R) whi h obey

αM (X) = α(XM ) and αR(X) = α(XR) .Therefore we are able to de ompose the dual spa eT ∗

(m,t)(M ×R) = Ann(TtR) + Ann(TmM) (Ann is the annihilator)and we obtainω = ωM + ω0 with ωM ∈ Ann(TtR) , ω0 ∈ Ann(TmM) . (3.8)We an show, that for any given ω ∈ Φ there is a gauge transformation ϑ ∈ G su h that

(ϑ∗ω)0 = 0. Hen e, ea h gauge orbit ontains an ω with ω0 = 0, i.e.ΦM := ω ∈ Φ | ω0 = 0

34 3 GAUGE THEORYis an admissible gauge (the temporal gauge).Subsequently, we turn to the question of symmetries in ΦM , i.e. we are looking for a subgroupGM ⊆ G satisfying ΦM/GM = Φ/G.The following holds: If there exists a ϑ ∈ G for given ξ, η ∈ ΦM with ξ = ϑ∗η, then ϑ is onstant in time. Contrariwise, for ea h ϑ ∈ G that is onstant in time and ω ∈ ΦM , we haveϑ∗ω ∈ ΦM , too. Hen e GM := ϑ ∈ G | ϑ onstant in time. Due to the triviality of theprin ipal bre bundle and of spa etime, we an substitute GM by AutM (M ×G). Sin e forea h ϑ there is an equivariant u : M → G with ϑ(p) = p · u(m), we an further substituteAutM (M ×G) by HomG(M,G):= C∞

G (M,G) (equivariant maps with respe t to the a tionof G on itself by onjugation on G). Therefore,GM = HomG(M,G) ≡ C∞

G (M,G) . (3.9)Subsequently, we outline that gauge elds are des ribed as obje ts over M by virtue of ΦM .Let ιt : M → M × t ⊆ ST be the anoni al embedding. Via pull-ba k with ιt, we anidentify ω ∈ ΦM with the traje tory t 7→ ι∗tω in Ω1(M) ⊗ g. Let ξ(t), η(t) ∈ Ω1(M) ⊗ g bethe traje tories orresponding to ξ, η ∈ ΦM . Sin e the elements GM are onstant in time, we on lude: If there exists a t0 ∈ R and ϑ ∈ GM with ϑ∗ξ(t0) = η(t0), this holds for all t ∈ R.In the Hamiltonian formulation the onguration spa e of our system is the G-spa e (Ω1(M)⊗g,GM ). It in ludes the whole gauge theory.3.4 Latti e gauge modelIn the last se tion, we separated time from spa e. Denote P = M × G, G = HomG(M,G)and Φ = onne tions in P.We assume that G ⊆ GL(n,K) is a linear Lie group. This assumption an be made sin e thisthesis will fo us on the gauge group of strong intera tion, the linear stru ture group SU(3).Due to the matrix representations of G and g we an substitute Adu−1 · by u−1 · u and u∗Θby u−1u′ in (3.4). Hen e, for gauge transformation of gauge elds ω we have the followingequation:

ϑ∗ω = u−1ωu+ u−1u′ , (3.10)where u : M → G with ϑ(p) = pu(m) as introdu ed earlier.We hoose a nite ubi [iii latti e Λ in M and denote the subspa e of k-dimensional obje tsof Λ by Λk,Λ0 = nodes, Λ1 = edges, Λ2 = plaquettes, Λ3 = ubes.[iii ea h node has six edges

3.4 LATTICE GAUGE MODEL 35Latti es des ription of gauge eldsIn latti e gauge theory it is not of importan e what happens in between adja ent nodes.We are just interested in the point of arrival in the bre of the se ond node. Therefore, a onne tion assigns a group element to ea h edge, whi h denotes the hange in bres.This happens in a anoni al way:Let x, y ∈ Λ0 be two neighbouring nodes and let xy ∈ Λ1 denote the onne ting edges. Edges orrespond to urves γ : [0, 1] ⊂ R→M in M and onne tions ω ∈ Φ indi ate ω-horizontallifts to (x, e) ∈ P of these urves, γ.(x, e) ∈ P

γ[0,1]- (y, axy) ∈ P

x ∈M

π

?

γ[0,1]

-

6

y ∈M

π

?

[0, 1] ⊂ R γ - M

[0, 1] ⊂ R γ - P

T[0, 1] ∼= [0, 1] ×R γ′- TP

ondition:ω(γ′(t)) = 0g ,

∀ t ∈ [0, 1]

Hen e, ea h edge is assigned a group element axy by these lifts. (y, axy) ∈ P is the pointobtained by the parallel transport of (x, e) ∈ P along the edge xy, regarding ω.Thus, onne tions assign an element of G to ea h edge, i.e. gauge elds are given by mappingsfrom edges to the symmetry group of the prin ipal bre bundleΦ = a : Λ1 → G .For the path from x via y to z we obtain the group element ayzaxy sin e parallel transportsare equivariant.Latti e des ription of gauge transformationsThe parallel transport of ϑ∗ω along xy is omputed here. We start with a proof of thefollowing fa t.Proposition 3.4.1: Let ϑ ∈ AutM (P ), ω ∈ Φ, γ be a urve inM and p ∈ π−1(γ(0)) ⊆ P .The ϑ∗ω-horizontal lift of γ to ϑ−1(p) ∈ P is ϑ−1 γ.Proof : For all t there is (ϑ∗ω)ϑ−1γ(t)((ϑ

−1γ)′(t)) = ωγ(t)((ϑϑ−1)′ ˙γ(t)) = ωγ(t)( ˙γ(t)) = 0.Thus, ϑ−1 γ is ϑ∗ω-horizontal. Sin e ϑ−1 γ starts in ϑ−1(p) the assertion follows. The parallel transports of ϑ∗ω yields a map ϑ−1(p) 7→ ϑ−1(γ(1)).Let u ∈ HomG(M,G) as introdu ed in (3.9), i.e. ϑ(p) = p · u(π(p)). By the fa t that parallel

36 3 GAUGE THEORYtransports are equivariant, the parallel transport of ϑ∗ω along γ maps p = ϑ−1(p)u(π(p)) onϑ−1(γ(1))u(π(p)). If p = (x, e) and γ is the edge xy, we obtain the following formula for theparallel transport of p:

ϑ−1((xy)(1))u(x) = ϑ−1(y, axy)u(x)

= ϑ−1(y, e)axy)u(x)

= (u(y)−1(y, e)axy)u(x)

= (y, u(y)−1axyu(x)) .Thus, gauge transformations are maps from the verti es into the stru ture group of theprin ipal bre bundleG = g : Λ0 → G . (3.11)Their a tion is given byagxy = g(y)−1axyg(x) . (3.12)Con lusion 3.4.2: The onguration spa e of a latti e gauge system is the G-manifold

(GΛ1, GΛ0

). The a tion of GΛ0 on GΛ1 is given by (3.12) and represents the gauge trans-formation ϑ. The phase spa e is given by the otangent bundle T∗GΛ1 of the ongurationspa e. Note: Elements of GΛ1 and GΛ0 orrespond to the onne tion ω and the verti al automor-phism ϑ, respe tively.3.5 Tree gaugeChoose a maximal simply onne ted subset T ⊆ Λ1, a so alled maximal tree, and setC := a ∈ GΛ1 | aT = id .Thus, for xy ∈ T there is axy = e ∈ G. The following proposition will spe ify our phasespa e further.Proposition 3.5.1: (i) For ea h a ∈ GΛ1 there is a g ∈ GΛ0 with the transformed gaugeeld ag ∈ C.(ii) Let a, b ∈ C be gauge-equivalent elds and g ∈ GΛ0 its transformation, i.e. b = ag.Then, g is onstant.Proof : (i) Let x ∈ Λ0. This hoi e denes a starting point. Sin e T is maximal and simply onne ted, for ea h y ∈ Λ0 there is a uniquely dened path xiy .. x1x2 xx1 with elementsin T onne ting x with y.

3.6 CLASSICAL REDUCTION IN LATTICE GAUGE THEORY 37We onstru t g ∈ GΛ0 su h that ag ∈ C as follows: Starting in x we hoose a g(x) = e.Inserting it into equation (3.12), we obtain g(xj) = axj−1xj..ax1x2axx1 for all subsequentnodes xj. Then, agxjxj+1 = g(xj+1)

−1axjxj+1g(xj) = e.(ii) For xjxj+1 ∈ T there holdse = bxjxj+1 = agxjxj+1

= g(xj+1)−1axjxj+1g(xj) = g(xj+1)

−1g(xj) .Hen e g(xj) = g(xj+1). Let y ∈ GΛ0 and xiy .. x1x2 xx1 be the uniquely dened pathfrom x to y in T . By the same pro edure as in (i) we obtain g(x) = g(y) for ea h node. The gaugeΦ = C , G = g : Λ0 → G | g is onstant = Gis alled tree-gauge. Thus, our system is simplied essentially: Values of a an only be hosenon Λ1 \ T freely and gauge transformations are onstant maps from Λ0 to G, i.e. given by Gitself.Denoting N = ♯(Λ1 \ T ), we set Φ = GN . G is still given by G sin e there is diagonal onjugation a tion

ag = (a1, .., aN )g = (g−1a1g, .., g−1aNg) .Our new onguration spa e is given by the G-manifold (GN , G).3.6 Classi al redu tion in latti e gauge theorySet K := GN and hen e the onguration spa e now reads (K,G). Classi al redu tion onsists of the onstru tions of the redu ed onguration and the redu ed phase spa e andof the dis ussion of the dynami s given by a G-invariant Hamiltonian.The redu ed onguration spa e QThe redu ed onguration spa e is given by the quotient K/G =: Q (with respe t to Ga ting on K via onjugation). Due to the fa t that in physi al models we always have a ompa t stru ture group G, Q is a stratied spa e with dierentiable stru ture C∞(Q) =

C∞(K/G) = C∞(K)G.The redu ed phase spa e PThe onstru tion of the redu ed phase spa e is ondu ted in the following two steps.(i) We onstru t the asso iated Hamiltonian G-spa e T∗K ( f. remark 2.3.3) with itslifted a tion and the anoni al momentum map J : T∗K → g∗.(ii) Now, we arry out singular Marsden-Weinstein redu tion ( f. theorem 2.4.6). Theresulting quotient J−1(0)/G is a stratied symple ti spa e, denoted by P.

38 3 GAUGE THEORYTogether with the dierentiable stru ture C∞(P) = C∞(J−1(0)/G) = C∞(T∗K)G, P is aPoisson spa e. The ow of a G-invariant Hamilton fun tion is stratum preserving, i.e. if thestate of the system of a spe i time is given by an element in S then the state will neverleave this stratum. In general this is wrong for the strata of the onguration spa e.The proje tion T∗K → K of the otangent bundle on the base manifold indu es a proje tionπ : P → Q . (3.13)This redu ed otangent bundle provides the physi al des ription of lassi al gauge theory.Case N = 1, K ≡ GLet G be ompa t and onne ted in the sequel. We identify the otangent bundle T∗G with

G× g via(i) g ∼= g∗ by F : g → g∗ ( f. equation (2.8))(ii) T∗G =⋃

a∈GT∗aG =

⋃

a∈Gαa | αa = F (A) L′

a−1 , A ∈ TaG (∼= g)

⇒ T∗G ∼= G× TaG ∼= G× g . (3.14)This ongruen e is given by the dieomorphismς : G× g → T∗G , (a,A) 7→ F (A) L′

a−1 . (3.15)Let ϕ be the a tion of G×G on G× g dened byϕx,y : (a,A) 7→ (xay,Ady−1 A) .

ς is equivariant under G×G-a tion ϕ sin e:F (Ady−1 A) L′

(xay)−1 = (Ad∗y−1 F (A)) L′

y−1L′a−1L

′x−1Def.

= F (A) Ady−1 L′y−1L

′a−1L

′x−1

Def.= F (A) R′

y−1L′yL

′y−1L

′a−1L

′x−1

= ς(a,A) R′y−1 L′

x−1 (as R and L ommute).L′x−1 and R′

y−1 are pre isely the point transformations indu ed by Lx and Ry, respe tively.The lifted a tion of G in the variables (a,A) ∈ G× g reads(a,A) 7→ (xax−1,AdxA)and the symple ti potential θ : T T∗G→ R ( f. remark 2.3.3) is given by

θ(a,A)(L′aX,Y ) = 〈A,X 〉 , X, Y ∈ g .

3.6 CLASSICAL REDUCTION IN LATTICE GAUGE THEORY 39A ordingly, the symple ti form ω = −dθ an be determined.As shown in [Fis04, page 22 [iv, the momentum mapping J of our model is given byJ(a,A) = AdaA−A (3.16)and the Hamiltonian fun tion reads

H(a,A) = −1

2||A ||2 +

ν

2(3 − Re tr(a)) (3.17)for all (a,A) ∈ G × g, where || · || denotes the norm dened by the inner produ t on g and

ν := 1g2

with g denoting the oupling onstant.Case G = SU(3) (and N = 1, K ≡ G)In physi s, strong intera tion is modelled by the symmetry group G = SU(3). We now analyseit on one plaquette, i.e. ♯Λ0 = ♯Λ1 = 4. After hoosing a tree we obtain ♯(Λ1 \T ) = 1. Thus,(i) the onguration spa e of our system is given by the G-manifold SU(3), where SU(3)a ts on itself by onjugation and(ii) the phase spa e is given by T∗ SU(3) and an be trivialised T∗ SU(3) = SU(3) × su(3)using the Killing metri , as des ribed above.Applying the pre edingly introdu ed redu tion pro edure to our onguration spa e, weobtain the redu ed otangent bundleπ : P = J−1(0)/SU(3) → SU(3)/SU(3) = Q . (3.18)In the following hapters, we will quantise the phase spa e and subsequently hara terisethe Hilbert spa e of this SU(3) toy model. We refer to [Fis04 and [Spi06 for the study ofdierent aspe ts of our model.

[iv sin e T∗ G is there trivialised by right translation the sign onventions ne essarily dier

40 4 REPRESENTATION THEORY - SU(3) AND su(3)

4 Representation theory - SU(3) and su(3)The general on ept of nite-dimensional Lie group / Lie algebra representation and Adjoint/ adjoint / oAdjoint / oadjoint representation has already been introdu ed in se tion 2.1.The Killing form K(·, ·), dened in 2.1.15 will also play a major role in the following.In this hapter we will fo us on the theory of roots, hara ters and weights with spe ialinterest in the onsequen es for SU(3) and su(3). We will provide a general overview of theresults needed in this thesis. For an in-depth introdu tion to Lie groups and representationtheory we refer to [Hsi00, [Ban03, [Cor94 and [Cor95. All aspe ts dis ussed in this hapterare restri ted to the nite-dimensional ase.4.1 Chara terisation of representationsThis se tion starts with a general introdu tion to representation theory. This in ludes inte-gration over ompa t Lie groups, the on ept of irredu ible representations and the de om-position of representation spa es. Later on we will see that there is a one-to-one relationbetween representations and their hara ters and with the Peter-Weyl theorem we will provethat the representative fun tions are dense in L2(G). Finally, we will learn that hara tersform a basis in the lass fun tions of G, i.e. the onjugation invariant ones.4.1.1 General aspe tsOne of the most important stru tures in group theory is the n-torus. An n-torus is a Liegroup isomorphi to Rn/Zn, i.e.T n ∼= Rn/Zn ∼= S1 × ..× S1

︸ ︷︷ ︸n

.An element t ∈ T n is alled a generator (or generating element) if the group tk | k ∈ Z,algebrai ally generated by t, is dense in T n.The next proposition hara terises ompa t abelian Lie groups.Proposition 4.1.1: A ompa t abelian Lie group is isomorphi to the produ t of a torusand a nite abelian group.Proof : Cf. [BD85, Cor. 1.3.7. 41

42 4 REPRESENTATION THEORY - SU(3) AND su(3)The subsequent assertion is used in the proof of Cartan's theorem 4.1.25. It determines theproperties of generating elements of a torus T n.Proposition 4.1.2 (Krone ker's theorem) : A ve tor v ∈ Rn is a generator of T n if andonly if 1 and the ve tor omponents v1, .., vn are linearly independent over the rational numbersQ.Thus, almost every element of T n is a generating element and they form a dense subspa e ofthe torus.Proof : See [BD85, Thm. 1.4.13. Using standard methods of dierential geometry, we introdu e the integration over om-pa t Lie groups. Let α ∈ Ωn(M) be a dierential n-form on an arbitrary m-dimensionalorientable manifold M . To integrate α on M , we rstly sele t an orientation (a spe ial overing) on M and se ondly hoose a partition of unity subordinate to the overing. Thispro edure allows to as ribe the integration of M to a standard integration on Rm via pull-ba k of the inverse overing maps. We refer to standard literature here, e.g. [Rud05, 4.2.Thus, there is a unique integral∫

M: Ωn(M) → R , α 7→

∫

Mαmapping α on the real numbers. In most ases we prefer to integrate fun tions, whi h isa hieved in the following way. If M is endowed with a volume form ω and f : M → R a ontinuous fun tion with ompa t support. Then,

∫

Mf :=

∫

Mf · ω .Hen e, the integral of f is dened to be the integral of the form f · ω on M with orientationdetermined by ω.Proposition 4.1.3: Let G be a ompa t Lie group. There exists an invariant integral of ontinuous fun tions on G

∫

G: C0(G) → R , f 7→

∫

Gf dg , (4.1)whi h is uniquely determined by the following properties:(i) ∫

G is linear, monotone [i and normalised (∫G dg = 1) and(ii) ∫G is left-invariant: ∫G f Lh dg =

∫G f dg for any h ∈ G (Lh as in denition 2.1.4).Proof : Cf. [Ban03, Lem. 17.12 and [BD85, Thm. 1.5.13. [i if f(g) ≤ h(g) for all g ∈ G, then R

Gf dg ≤

R

Gh dg

4.1 CHARACTERISATION OF REPRESENTATIONS 43Denition 4.1.4: The well-dened integral of proposition 4.1.3 is alled invariant Haarintegral and the orresponding volume form dg the Haar measure of G.Subsequently, we dene the essential obje ts of representation theory and state some basi theorems. Let G be a Lie group and g its Lie algebra.Denition 4.1.5: Let V and W be G-modules. An intertwining operator (or mor-phism) f : V → W between representation spa es is a linear map whi h is equivariant,i.e. whi h satises f(gv) = gf(v) for all g ∈ G (g ∈ g) and v ∈ V . The set of all su hintertwining operator is denoted by HomG(V,W ).Note: G an a t on V and W dierently.Denition 4.1.6: Let V be a omplex G-module. An (Hermitian) inner produ t V ×V →C , (u, v) 7→ 〈u, v 〉 is alled G-invariant if 〈gu, gv 〉 = 〈u, v 〉 for all g ∈ G and u, v ∈ V .A representation whose representation spa e is equipped with a G-invariant inner produ t is alled a unitary representation.Proposition 4.1.7: Let (V,Γ) be a real ( omplex) representation of the ompa t group G.Then, V possesses a G-invariant inner (Hermitian) produ t.Note: This shows the importan e of dealing with ompa t groups.Proof : Let 〈·, ·〉∗ be an arbitrary inner (respe tively Hermitian) produ t on V . Set〈v,w 〉 :=

∫

G〈Γgv,Γgw 〉∗ dgwith the integral being normalised and left-invariant. It is straightforward al ulation toverify that 〈·, ·〉 is again an inner (respe tively Hermitian) produ t on V and moreover

〈Γav,Γaw 〉 =

∫

G〈Γgav,Γgaw 〉∗ dg .Dening g′ := ga and using dg′ = dg (be ause the Haar measure is left-invariant), we obtain

〈Γav,Γaw 〉 =

∫

G〈Γg′v,Γg′w 〉∗ dg′ .

Denition 4.1.8: Let V be a G-module. A subspa e U ⊆ V whi h is G-invariant (i.e.gu ∈ U for all g ∈ G and u ∈ U) is alled a submodule of V .A non-zero representation (V,Γ) of a group G (Lie algebra g) is alled irredu ible if V hasno submodules other than 0 and V . A representation whi h is not irredu ible is alledredu ible.

44 4 REPRESENTATION THEORY - SU(3) AND su(3)Proposition 4.1.9 (S hur's lemma) : Let G be any group and let V and W be irredu ible omplex G-modules. Then,(i) an intertwining operator V →W is either zero or an isomorphism,(ii) every intertwining operator f : V → V has the form f(v) = λv for some λ ∈ C and(iii) dim HomG(V,W ) =

1 if V ∼= W ,0 if V ≇ W .The same holds for V and W being irredu ible g-modules of a Lie algebra representation.Proof : Sin e V is irredu ible, the kernel of f is either 0 or V : Suppose ker f =: U ( V ,

U 6= V and let u ∈ U . There exists a g ∈ G with gu /∈ U and f(gu) = gf(u) = 0. This is a ontradi tion to ker f = U . Thus, ker f = 0 or ker f = V .In the latter ase f is zero and in the former f is inje tive. If f is inje tive, its image is anon-zero submodule of the irredu ible G-module W and hen e is identi al W . We on ludethat f is an isomorphism, proving (i).To prove (ii), assume that f is non-trivial and let λ be any eigenvalue of f and Uλ the orresponding eigenspa e. Thus, Uλ = v ∈ V | f(v) = λv and this is again a G-submodule.Hen e, Uλ = V , whi h proves (ii).The third part follows from (ii) and (iii). The proof for irredu ible g-modules of a Lie algebrag is obvious as all arguments given above also apply for intertwining operators between Liealgebra representations. Corollary 4.1.10: An irredu ible representation of an abelian Lie group G is one-dimen-sional. The same holds for irredu ible representations of abelian Lie algebras.Proof : Sin e G is abelian, the translation Γg : V → V is an intertwining operator of rep-resentations for ea h g ∈ G. By 4.1.9 (ii), every Γg is a multipli ation by λ(g) ∈ C. Thisimplies that any subspa e of V is G-invariant.The result follows sin e for dimV > 1, V has one-dimensional subspa es. As all subspa esare submodules, this ontradi ts the irredu ibility of V .The proof for irredu ible representations of abelian Lie algebras is analogous. Theorem 4.1.11 (Weyl theorem full redu ibility of representations) : Let G be a ompa tLie group and let (Γ, V ) be any nite-dimensional representation of G. Then,

V = V1 ⊕ .. ⊕ Vn , (4.2)

4.1 CHARACTERISATION OF REPRESENTATIONS 45where ea h submodule Vi is irredu ible (i.e. (Γ, Vi) is an irredu ible representation).The same holds for representations of onne ted semi-simple ( f. denition 2.1.14) Lie groupsand semi-simple Lie algebras.Note: Representations whose representation spa e de omposes into a dire t sum of irredu iblesubmodules ( ompare (4.2)) are alled ompletely redu ible.Proof : As G is a ompa t Lie group, a G-invariant inner produ t on V an be hosen ( f.proposition 4.1.7). Hen e, orthogonal omplements are again G-submodules. By indu tionon the dimension of V : If V 6= 0 is redu ible, then V = U ⊕W with 0 < dimU < dimV .For onne ted semi-simple Lie groups and semi-simple Lie algebras see [Cor95, Thm. 15.1.1and Thm. 15.1.2. Remark 4.1.12: Let (W,ΓW ) be an irredu ible representation of a ompa t Lie groupG. We all the dimension dim HomG(W,V ) the multipli ity of W in V . It is of thefollowing signi an e: Suppose we have a de omposition V =

⊕j Vj of V into its irredu iblesubmodules Vj . Then HomG(W,V ) =

⊕j HomG(W,Vj) and by S hur's lemma, proposition4.1.9 (iii), dim HomG(W,V ) is the number of Vj that are isomorphi to W . In parti ular, themultipli ity is non-zero if and only if W is ontained in V . 4.1.2 Chara ters, S hur's orthogonality relations and the Peter-Weyl theoremLet G hen eforth denote the set of equivalen e lasses of nite dimensional irredu ible rep-resentations of G, ompare remark 2.1.12. In this subse tion, we introdu e a unique identi- ation for equivalen e lasses of representations [Γ] ∈ G, the hara ters. Furthermore, thePeter-Weyl theorem proves that hara ters of [Γ] provide a basis in the spa e of square inte-grable fun tions whi h are invariant under onjugation, L2(G)G. For the sake of simpli ity,we will write Γ ∈ G hen eforth, indi ating that Γ is a representative for an element of G.Let G be a ompa t Lie group and let (V,ΓV ) and (W,ΓW ) be omplex representations of

G. G operates on Hom(V,W ) by (g · f)v = ΓWgf(ΓV−1g v), where f ∈ Hom(V,W ). Withthis a tion Hom(V,W )G = HomG(V,W ), the spa e of G-maps V →W . Using the invarian e∫

G hg dg =∫G g dg we obtain proje tion operators

pr1 : V → V G , v 7→∫

GΓV gv dg and

pr2 : Hom(V,W ) → HomG(V,W ) , f 7→∫

G(g · f) dg . (4.3)For an irredu ible representation (V,Γ) we have Γ = λf idV with λf ∈ C, HomG(V, V ) ∼= C,by S hur's lemma (proposition 4.1.9). Applying the linear map tr : Hom(V, V ) → C we

46 4 REPRESENTATION THEORY - SU(3) AND su(3)furthermore haveλf dimV = tr(λf idV ) = tr

∫

G(g · f) dg = tr

∫

GΓgfΓ−1

g dg =

∫

Gtr(f) dg = tr f .Hen e, for irredu ible representations (V,Γ) there holds

∫

G(g · f) dg =

1

dimVtr(f) idV . (4.4)Denition 4.1.13: The hara ter χ of a representation (V,Γ) is the fun tion

χ(V,Γ) : G→ C , g 7→ tr(Γg) ,where tr(Γg) is the tra e of the linear map Γg : V → V , v 7→ Γgv. The hara ter ofan irredu ible representation is alled an irredu ible hara ter and the hara ter of the omplex Lie group GC is alled omplex hara ter, denoted χC.Properties 4.1.14: Let (V,ΓV ) and (W,ΓW ) be representations of G. There holds(i) χV is of lass C∞,(ii) χV (ghg−1) = χV (h),(iii) χV⊕W = χV + χW ,(iv) χV ⊗W = χV · χW ,(v) χV ∗(g) = χV (g−1),(vi) χV (g) = χV (g) = χV (g−1),(vii) χV (e) = dimC V .Proof : Cf. [BD85, Pro. 2.4.10. Proposition 4.1.15 (S hur's orthogonality relations) : There holds(i) ∫

GχV (g) dg = dimV G,(ii) 〈χV , χW 〉 :=

∫

GχV χW dg = dim HomG(W,V ) (〈·, ·〉 inner produ t) and(iii) if V and W are irredu ible, ∫

GχV χW dg =

1 if V ∼= W ,0 if V ≇ W .

4.1 CHARACTERISATION OF REPRESENTATIONS 47Proof : (i) Using the proje tion operator pr1 from (4.3) we obtaindimV G = tr(pr1) = tr

∫

GΓg dg =

∫

Gtr(Γg) dg =

∫

GχV (g) dg .(ii) With Hom(W,V )G = HomG(W,V ) and (i) we have dimHomG(W,V ) =∫

G χHomG(W,V )(g) dg and χHomG(W,V ) = χW ∗⊗V = χWχV .(iii) This assertion follows dire tly from (ii) and S hur's lemma, 4.1.9. Proposition 4.1.16: A representation of a ompa t Lie group is determined up to equiv-alen e by its hara ter.Proof : Let G be a ompa t Lie group, (V,ΓV ) a ( omplex) representation of G andΓV ∼=

⊕

ΓW∈G

〈χV , χW 〉 ΓW (4.5)be a de omposition of (V,ΓV ) into irredu ible representations (Weyl theorem 4.1.11 andremark 4.1.12). Then, χV =∑

ΓW ∈G〈χV , χW 〉χW using property 4.1.14 (iii) and the assertionis shown. Corollary 4.1.17: Let (V,ΓV ) and (W,ΓW ) be two representations of the ompa t Liegroup G. Then,ΓV ∼= ΓW ⇔ χV = χW .Proposition 4.1.18: A representation of a ompa t Lie group is irredu ible if and only if

〈χV , χV 〉 = 1.Proof : For the de omposition of V into irredu ible representations ΓV ∼=⊕

ΓW ∈G〈χV , χW 〉ΓWthere holds 〈χV , χV 〉 =∑

ΓW∈G〈χV , χW 〉2. With proposition 4.1.15 (iii) we see that thereexists exa tly one 〈χV , χV 〉 = 1 and the assertion is proven. Let now G be a ompa t abelian group. Then, G is isomorphi to a produ tG ∼= S1 × ..× S1 × Z/m1 × ..× Z/mkof ir le groups and a nite abelian group ( f. proposition 4.1.1). By orollary 4.1.10 theirredu ible omplex G-modules are one-dimensional and hen e given by homomorphisms G→

S1. Thus, su h homomorphisms are hara ters of G. In order to determine all representationsof G, it su es to determine the irredu ible representations of S1 and Z/m.Example 4.1.19: The hara ters of the torus T n = Rn/Zn all have the formχ : [x] → e2πiα(x) , x ∈ Rn ,

48 4 REPRESENTATION THEORY - SU(3) AND su(3)where α(x) = 〈a, x〉 =∑

i aixi with a = (a1, .., an) ∈ Zn. Cf. [BD85, Pro. 2.8.1 here.Note: Consequently, for irredu ible omplex representations of S1 (respe tively U(1)) the hara ters are given by z 7→ zm, m ∈ Z. We now introdu e the on ept of dire t sum ompletion sin e it will be used in the Peter-Weyltheorem. Let Hι be a olle tion of Hilbert spa es, indexed by a set A. Then, the algebrai dire t sum⊕ι∈AHι is a pre-Hilbert spa e when equipped with the dire t sum inner produ t

〈∑ι vι,∑

ι wι 〉 =∑

ι〈vι, wι 〉. Its ompletion is alled the Hilbert dire t sum of the spa esHι and denoted by

⊕

ι∈AHι . (4.6)The ompletion may be realised as the spa e of sequen es v = (vι)ι∈A with vι ∈ Hι and

|| v ||2 =∑

ι∈A|| vι ||2 <∞ .Its inner produ t is given by

〈v,w 〉 =∑

ι∈A〈vι, wι 〉 .If Γι is a unitary representation of G in Hι for ι ∈ A then the dire t sum of the Γιs extendsto a unitary representation of G in (4.6).Theorem 4.1.20 (Peter-Weyl theorem) : (i) The representative fun tions are dense inboth C0(G) and L2(G). There holds

L2(G) =⊕

Γι∈G

Hι (4.7)with Hι being the linear span of the irredu ible unitary representation Γι ∈ G.(ii) The irredu ible hara ters generate a dense subspa e of the spa e of ontinuous lassfun tions, i.e. for f ∈ L2(G)G = f ∈ L2(G) | LgRg−1f = f ∀ g ∈ G there holdsf =

∑

Γι∈G

〈f, χι 〉 χι (4.8)with onvergen e in the L2-norm.Proof : Cf. [BD85, Thm. 3.3.1. Corollary 4.1.21: Let G be a ompa t Lie group. Then, the hara ters χι, Γι ∈ G forma basis in L2(G)G.

4.1 CHARACTERISATION OF REPRESENTATIONS 494.1.3 Maximal tori and Weyl's integral formulaIn the sequel we will introdu e the on ept of maximal tori for onne ted ompa t Lie groups.We will see that every onne ted ompa t Lie group G ontains a maximal torus T , whi h isunique up to onjugation. Furthermore, the union of onjugates of T overs all of G.Denition 4.1.22: A subgroup T ⊆ G is a maximal torus if T is a torus and there isno other torus T with T ( T ⊆ G.Sin e tori are ompa t and onne ted if T ( T , then dimT < dim T . Hen e, maximal toriexist. A maximal torus is the same thing as a maximal onne ted abelian subgroup.Denition 4.1.23: Let T be a maximal torus in G and N the normaliser of T in G, i.e.N = g ∈ G | gTg−1 = T .The group W = N/T is alled the Weyl group of G.The normaliser N of T operates on T by onjugationN × T → T , (n, t) 7→ ntn−1 .Sin e the operation of T on T is trivial, we obtain an indu ed a tion of the Weyl groupW × T → T , (nT, t) 7→ ntn−1 . (4.9)Proposition 4.1.24: The Weyl group is nite.Proof : As N is a losed subgroup of G, it is ompa t. W is the image of a ompa t groupunder a ontinuous map and hen e W is ompa t. As T is the onne ted omponent of theidentity in N , N/T is dis rete. Thus, W is nite. Theorem 4.1.25 (Cartan's theorem) : Any two maximal tori in a ompa t onne ted Liegroup G are onjugate and every element of G is ontained in a maximal torus.Note: Hen e, speaking of the maximal torus in G and the Weyl group of G in the sequel,we mean that we expli itly hoose one. Any other one hosen is onjugate by the pre edingtheorem and hen e its Weyl groups are isomorphi .To prove Cartan's theorem we use the following lemma.Lemma 4.1.26: Let G be a ompa t onne ted Lie group and T a maximal torus in G.The map

q : G/T × T → G , (gT, t) 7→ gtg−1 (4.10)is of degree deg(q) = |W | ( |W | denotes the order of W). In parti ular, q is surje tive.

50 4 REPRESENTATION THEORY - SU(3) AND su(3)Proof : To ompute the mapping degree of q we need to analyse the transformation of the anoni al volume form dq. We hoose a metri on the Lie algebra g ≡ TeG of G whi h isinvariant under the Adjoint representation.Using an arbitrary metri , the Lie algebra TeG de omposes into the tangent spa e to T at e,namely t ≡ Te T and its orthogonal omplement t⊥ whi h we denote by Te(G/T ) hen eforth,i.e.g = TeG = Te(G/T ) ⊕ Te T = t⊥ ⊕ t .This de omposition is invariant under the operation AdT

. Sin e T is maximal, T a tstrivially on Te T and non-trivially on every non-zero ve tor Te(G/T ). The indu ed a tionon the rst summand is given byAdt⊥

T: T → Aut t⊥ ≡ AutTe(G/T ) , t 7→ (Lt)

′e . (4.11)The proje tion π : G → G/T indu es a map between TeG = Te(G/T ) ⊕ Te T and thetangent spa e of G/T at the point eT . This maps Te(G/T ) isomorphi ally onto the tangentspa e of G/T at the point eT and we identify both spa es by Te(G/T ), hen eforth.From proposition 4.1.7 we know that there exist left-invariant volume forms d(gT ), dt and

dg on G/T , T and G, respe tively. The assumption follows analogously to [BD85, ProofLem. 4.1.7 by analysing the pull-ba k π∗d(gT ) ∈ ΩdimG−dimT (G). Thus, we ndπ∗d(gT ) ∧ dτ = dg (4.12)with dτ ∈ ΩdimT (G) being a left-invariant dierential form satisfying dτT

= dt, i.e. theJa obian of the proje tion map π is one. We now on lude the proof of Cartan's theorem 4.1.25. Let T and T be maximal tori and ta generator of T , see Krone ker's theorem 4.1.2. By lemma 4.1.26, there exists a g ∈ G witht ∈ gTg−1 and sin e gTg−1 is a torus we have T = gTg−1. As q is surje tive every elementof G is ontained in some onjugate of T . Hen e, equation (4.12) determines the volume form on G and

α = pr∗1 d(gT ) ∧ pr∗2 dt (4.13)is the one on G/T ×T (pr1 and pr2 denote the proje tions onto the rst, respe tively se ondsummand of TeG = Te(G/T ) ⊕ Te T ). The Ja obian det(q′) : G/T × T → R of the onjugation map q, ompare equation (4.10), is dened byq∗ dg = det(q′) α . (4.14)Proposition 4.1.27: The determinant of the onjugation map q : G/T ×T → G is givenby

det(q′)(gT, t) = det((Adt⊥

T)t−1 − 1G/T) (4.15)

4.1 CHARACTERISATION OF REPRESENTATIONS 51with Adt⊥T

dened in (4.11).Proof : Sin e the forms dg, d(gT ) and dt are left-invariant under the a tions of G and T ,respe tively, we analyse the dierential of a transformed map whi h sends (T, e) to e insteadof al ulating the determinant of q′ at (gT, t) dire tly. Considering a map G/T × T → Gindu ed by the ompositionG× T

La- G× Tq - G

Lb - G

(x, y)La- (gx, ty)

q- (gx)(ty)(gx)−1 Lb- gt−1xtyx−1g−1with a = (g, t), b = gt−1g−1 and q : G×T → G , (g, t) 7→ gtg−1. Here, c(g) : G→ G , x 7→gxg−1 denotes the onjugation map and thus gt−1xtyx−1g−1 = c(g)(c(t−1)(x) · y · x−1). Thedeterminant we are looking for is the determinant of this omposition's tangent map at thepoint (e, e) restri ted to t⊥ ⊕ t = Te(G/T ) ⊕ Te T ⊆ TeG⊕ Te T = g ⊕ t.Now, c′g = Adg has determinant 1 and the dierential of a produ t is the sum of the dier-entials. Hen e, the determinant of q equals the determinant of

(X,Y ) 7→ (Adt⊥T

)t−1X + Y −X .In matrix form this linear operator reads(

(Adt⊥T

)t−1 − 1G/T 0

0 1T)and thus the assertion follows. The following formula is an important tool in representation theory and will be used in hapter 5.Theorem 4.1.28 (Weyl's integral formula) : Let G be a ompa t onne ted Lie group, Tthe maximal torus and f a ontinuous fun tion on G. Then,|W |

∫

Gf(g) dg =

∫

T

(det((Adt⊥

T)t−1 − 1G/T )

∫

Gf(gtg−1) dg

)dt . (4.16)Proof : From equation (4.12) we know that the proje tion map π : G→ G/T has no inuen eon the volume form, i.e. the Ja obian equals one. Thus, for ft : G/T → R , g 7→ f(gtg−1) =

ftπ(g) there holds ∫G f(gtg−1) dg =∫G/T ft d(gT ). Using the theorem on mapping degrees, ompare [BD85, Thm. 1.5.19, we ompute

∫

T

(det((Adt⊥

T)t−1 − 1G/T )

∫

Gf(gtg−1) dg

)dt

=

∫

T

(det((Adt⊥

T)t−1 − 1G/T )

∫

G/Tft(g) d(gT )

)dt

52 4 REPRESENTATION THEORY - SU(3) AND su(3)(4.15)=

∫

G/T

∫

Tf q(gT, t) det(q′)(gT, t) dt d(gT )

=

∫

(G/T )×Tf q(gT, t) q∗dg

= deg(q)

∫

Gf(g) dg

= |W |∫

Gf(g) dg .

Example 4.1.29 (The maximal tori in U(n) and SU(n)) : Let D(n) ⊆ GL(n,C) denotethe diagonal n×n matri es. We dene UD(n) := U(n)∩D(n) and SUD(n) := SU(n)∩D(n),hen e (S) UD(n) =

z1 0. . .0 zn

, zj ∈ S1 (z1..zn = 1).Claim: The groups UD(n) and SUD(n) are maximal tori in U(n) and SU(n).Proof : Let T be a torus with U D(n) ⊆ T and SUD(n) ⊆ T , respe tively. The in lusions

T ⊆ U(n) and T ⊆ SU(n), respe tively, for any torus T are unitary representations of abeliangroups. By orollary 4.1.10 they are homomorphisms with image in D(n). Claim: The Weyl group of the unitary group U(n) is the full symmetri group Sn.Proof : Any element of UD(n) an be represented by the n-tuple (ϑ1, .., ϑn) ∈ (R/Z)n withzj = ei2πϑj . Sin e ϑj are eigenvalues and eigenvalues are invariant under onjugation, ompareequation (4.9), the Weyl group of UD(n) permutes the diagonal entries. Thus, W is ontainedin Sn.By (

0 −1

1 0

)(z1 0

0 z2

)(0 1

−1 0

)=

(z2 0

0 z1

)we see that every two- y le and hen e every permutation of (ϑ1, .., ϑn) is indu ed by anelement of the Weyl group and thus the assertion is shown. A diagonal matrix aD = (ϑ1, .., ϑn) is in SUD(n) if and only if ϑ1 + ..+ ϑn−1 = −ϑn modZ.Thus, the torus SUD(n) has dimension n− 1. Sin e the eigenvalue argument remains validand the transformation used in this proof has determinant one, we nd that the Weyl groupof SU(n) is also the full symmetri group Sn.

4.2 THE STRUCTURE OF SEMI-SIMPLE LIE ALGEBRAS 534.2 The stru ture of semi-simple Lie algebrasThe on ept of simple and semi-simple Lie algebras and groups was introdu ed in denition2.1.14. If not stated otherwise, let g be a semi-simple omplex Lie algebra. When analysingreal Lie algebras the omplexi ation of it should be onsidered: The omplexi ation ofa real Lie algebra g is given by g = C⊗R g with the operations(i) (a, b) + (a′, b′) = (a+ a′, b+ b′) with (a, b), (a′, b′) ∈ g × g ∼= C⊗R g and(ii) (µ+ iν)(a, b) = (µa− νb, µb+ νa) with (a, b) ∈ g × g , (µ+ iν) ∈ C.In the ase that the hosen basis a1, .., an of a real Lie algebra g is also linearly independentover the eld of omplex numbers, its omplexi ation is simply g := Ca1, .., an.Denition 4.2.1: A Cartan subalgebra h of a semi-simple omplex Lie algebra g is asubalgebra of g with the following two properties:(i) h is a maximal abelian [ii subalgebra of g,(ii) adh is ompletely redu ible for every h ∈ h.The dimension of a Cartan subalgebra h is alled rank of g.Note: The existen e of the Cartan subalgebra for every semi-simple omplex Lie algebrag 6= 0 is ensured by the Satz von Engel. It proves that there have to be diagonalisableelements in g. Compare [VRS97, 2.1 and [HN91, Kor. 2.2.6 here.4.2.1 The roots of a Cartan subalgebraLet h1, .., hl be a basis of a Cartan subalgebra h of a semi-simple omplex Lie algebra gof rank l and dimension n. As h is an abelian subalgebra and its adjoint representationis ompletely redu ible (denition 4.2.1 (ii)) it is evident that adh , h ∈ h, de omposes intoone-dimensional irredu ible representations ( orollary 4.1.10). Consequently, the matri esadhi

are simultaneously diagonalisable for i = 1, .., l and adhia ts trivially. By a similaritytransformation we an determine the remaining n − l basis elements of g, a1, .., an−l, su hthat

[hi, aj ] = αj(hi)aj , i ∈ 1, .., l , j ∈ 1, .., n − l , (4.17)where αj(hi) are a set of omplex numbers. As h is abelian,[hi, hj ] = 0 for i, j ∈ 1, .., l.The αj dene linear fun tionals on h whi h are alled roots of g. The property maximalabelian assures that for ea h j = 1, .., n−l there must exist at least one i su h that αj(hi) 6= 0,[ii h is abelian, but every subalgebra of g ontaining h as a proper subalgebra is not abelian

54 4 REPRESENTATION THEORY - SU(3) AND su(3)hen e αj 6= 0.The set of elements aα ∈ g su h that[h , aα] = α(h)aα for all h ∈ hform a subspa e of g whi h will be denoted by gα, alled the root subspa e orrespondingto α. Sometimes h is regarded to be the subspa e of g orresponding to zero-roots denoted

g0.The set of distin t roots will hen eforth be denoted by ∆. Therefore,g = h ⊕

⊕

α∈∆

gα . (4.18)Properties 4.2.2: For roots and root subspa es there holds(i) aα ∈ gα, aβ ∈ gβ ⇒ if α+ β ∈ ∆: [aα, aβ ] ∈ gα+β or if α+ β /∈ ∆: [aα, aβ] = 0,(ii) aα ∈ gα, aβ ∈ gβ and α+ β 6= 0 ⇒ K(aα, aβ) = 0,(iii) α ∈ ∆ ⇒ −α ∈ ∆ and(iv) the Killing form of g provides a non-degenerate symmetri bilinear form on h, i.e.K(h , h ′) 6= 0 for all h 6= h ′, h , h ′ ∈ h.Proof : Cf. [Cor95, page 499 . Note: A dire t onsequen e of (ii) is that K(h, aα) = 0 for all h ∈ h and any aα ∈ gα and that

K(aα, aα) = 0 for all α ∈ ∆.With property 4.2.2 (iv) it is possible to asso iate a unique element hα of h with every linearfun tional α(h) on h by the denitionK(hα, h) := α(h) for all h ∈ h ( f. [Cor94, Thm. B.6.1). (4.19)The following properties hold:(i) hα+β = hα + hβ and(ii) α(hβ) = β(hα) = K(hα, hβ).For onvenien e we denote 〈α, β〉 = K(hα, hβ). We an ndProperties 4.2.3:(i) For ea h α ∈ ∆ and every aα ∈ gα, there exists an element a−α of g−α su h that

K(aα, a−α) 6= 0,

4.2 THE STRUCTURE OF SEMI-SIMPLE LIE ALGEBRAS 55(ii) for every α, β ∈ ∆, there is 〈α, β 〉 ∈ Q and 〈α,α〉 > 0 ( f. [Cor95, E.7),(iii) h ∈ h ⇒ h =∑

α∈∆ µαhα with µα ∈ C⇒ there are l roots β1, .., βl su h that hβ1 , .., hβl

⊆ hα | α ∈ ∆ form a basis of h,(iv) every root α of ∆ an be written in the form α =∑l

j=1 κjβj with κj ∈ Q (hβ1, .., hβlbasis of h, ompare (iii)),(v) α ∈ ∆ ⇒ dim gα = 1 and kα ∈ ∆ only if k = 1 or −1 and(vi) 〈α, β 〉 =∑

γ∈∆〈α, γ 〉〈β, γ 〉.Let hR denote the real ve tor spa e spanned by hβ1, .., hβl.(vii) For any α ∈ ∆, α(h) is real for all h ∈ hR.Proof : Cf. [Cor95, page 505 . Proposition 4.2.4: For any α, β ∈ ∆, 2〈β,α 〉

〈α,α 〉 an only take the integer values 0,±1,±2,

±3. The quantities 2〈β,α〉〈α,α〉 are alled the Cartan integers.Proof : See [Cor95, Thm. 13.5.10. The following proposition xes a basis for the subspa es gα, α ∈ ∆.Proposition 4.2.5: There are basis elements eα of gα satisfying

K(eα, e−α) = −1 and [eα, eβ ] = Nα,βeα+β ,where N−α,−β = Nα,β for all α, β ∈ ∆.Note: There are dierent sets of onventions used in literature whi h an be hosen from. E.g.[BR80 denes K(eα, e−α) = 1 and N−α,−β = −Nα,β. The set introdu ed above, however, isthe one used by Weyl (e.g. [Wey25).Proof : See [Wey25, 2. 4.2.2 Simple rootsAs property 4.2.3 (iv) holds, we are able to dene positive and simple roots.Denition 4.2.6: Let a basis hβ1, .., hβlof h be hosen. A root α ∈ ∆ is said to bepositive (with respe t to the basis hβ1 , .., hβl) if the rst non-vanishing oe ient of κ1, .., κlin 4.2.3 (iv) is positive (α > 0). The set of all positive roots is denoted by ∆+.A root α ∈ ∆ is said to be simple (with respe t to the basis of h again) if α is positive and

56 4 REPRESENTATION THEORY - SU(3) AND su(3)

α annot be expressed in the form α = β + γ with β, γ ∈ ∆+. The set of all simple roots isdenoted by ∆S.Note: With the denition of positive roots we an dene a lexi ographi ordering on ∆.Let α, β ∈ ∆ then α > β if (α− β) > 0.Properties 4.2.7: Let α, β ∈ ∆S and α 6= β. For the simple roots there holds(i) α− β /∈ ∆ and(ii) 〈α, β 〉 ≤ 0.Proof : Suppose α − β ∈ ∆ ⇒ α − β ∈ ∆+ or β − α ∈ ∆+ ⇒ α = (α − β) + β /∈ ∆S orβ = (β − α) + α /∈ ∆S , respe tively. This obviously ontradi ts the assumptions made and(i) is proven.(ii) follows by proposition 4.2.4 and property 4.2.2 (iv). Theorem 4.2.8: If g has rank l then g possesses pre isely l simple roots α1, .., αl. Theyform a basis for the dual spa e h∗.Moreover if α ∈ ∆+ then

α =

l∑

j=1

kjαj , k1, .., kl ∈ N .Proof : See [Cor95, E.9. Denition 4.2.9: The Cartan matrix A of g is the l × l matrix dened byAij :=

2〈αi, αj 〉〈αi, αi 〉

, αi, αj ∈ ∆S , i, j ∈ 1, .., l . (4.20)Properties 4.2.10: For the Cartan matrix there holds(i) Ajj = 2 for all j ∈ 1, .., l and(ii) Aij = 0 ⇔ Aji = 0.Proof : (i) is obvious.(ii) follows be ause 〈αi, αj 〉 = 12 Aij〈αi, αi 〉 and 〈αi, αj 〉 = 〈αj , αi 〉 = 1

2 Aji〈αj , αj 〉. As〈αi, αi 〉 and 〈αj , αj 〉 both are non-zero (K non-degenerate on h), Aij = 0 ⇔ 〈αi, αj 〉 ⇔Aji = 0.

4.2 THE STRUCTURE OF SEMI-SIMPLE LIE ALGEBRAS 574.2.3 The Weyl ree tionsLet β ∈ h∗. For any root α ∈ ∆ we dene the linear fun tional Sα β on h bySα β := β − 2〈β, α〉

〈α,α〉 α . (4.21)Properties 4.2.11: It is obvious that for the operators Sα on h∗, α ∈ ∆, β, γ ∈ h∗ thereholds(i) Sα α = −α,(ii) Sα µ = µ for µ ⊥ α,(iii) Sα(Sα β) = β ⇒ S2α = 1,(iv) Sαj

αi = αi − Aji αj for all αi, αj ∈ ∆S ,(v) 〈Sα β,Sα γ 〉 = 〈β, γ 〉 for all s, t ∈ R,(vi) Sα(sβ + tγ) = s Sα β + t Sα γ for all s, t ∈ R.Hen e, Sα dene linear operators on h∗. By virtue of properties (i) and (ii) the map α 7→ Sαis a ree tion with respe t to the hyperplane perpendi ular to the ve tor α. Thus, the S are alled Weyl ree tions. They are very useful to determine the omplete set of weights (seese tion 4.3, example 4.3.11).Proposition 4.2.12: The linear group generated by the Weyl ree tions Sα, α ∈ ∆S isisomorphi to the Weyl group W, dened in 4.1.23.Thus, we will not distinguish between these two groups in the sequel and refer to both of themas Weyl group W.Note: If S ∈ W is represented by an even produ t of ree tions relative to the hyperplanesperpendi ular to roots, then det S = 1. Otherwise det S = −1.Proof : See [BD85, Thm. 5.2.12 Lemma 4.2.13: Let αi ∈ ∆S. If α ∈ ∆+ and α 6= αi then Sαiα ∈ ∆+. Let α, β ∈ ∆. If

Sαiα = Sαi

β for some i = 1, .., l then α = β.Proof : See [Cor94, Thm. 13.9.1 and Thm. 13.9.2. Proposition 4.2.14: For ζ := 12

∑α∈∆+

α there holds〈ζ, αi 〉 =

1

2〈αi, αi 〉 (4.22)for all i = 1, .., l.

58 4 REPRESENTATION THEORY - SU(3) AND su(3)Proof : Lemma 4.2.13 implies that Sαimaps the set of positive roots ∆+ \ αi into the setof positive roots for any i = 1, .., l. Thus

Sαiζ =

1

2

∑

α∈∆+\αiSαi

α +1

2Sαi

αi4.2.11 (i)=

1

2

∑

α∈∆+\αiα − 1

2αi

= ζ − αi .Using 4.2.11 (v), we ompute −〈ζ, αi 〉+ 〈αi, αi 〉 = 〈ζ−αi,−αi 〉 = 〈Sαi(ζ−αi),Sαi

(−αi)〉 =

〈ζ − αi + αi, αi 〉 and hen e 2〈ζ, αi 〉 = 〈αi, αi 〉. We now analyse the root system of the omplexi ation of the real Lie algebra su(3).Example 4.2.15 (The Cartan subalgebra and roots of suC(3)) : As shown in example2.1.8, su(n) = A ∈ GL(n,C) | A+A† = 0 , trA = 0. The Gell-Mann tra eless Hermitianmatri esλ1 =

0 1 0

1 0 0

0 0 0

, λ2 =

0 −i 0

i 0 0

0 0 0

, λ3 =

1 0 0

0 −1 0

0 0 0

, λ4 =

0 0 1

0 0 0

1 0 0

,

λ5 =

0 0 −i0 0 0

i 0 0

, λ6 =

0 0 0

0 0 1

0 1 0

, λ7 =

0 0 0

0 0 −i0 i 0

, λ8 =

1√3

1 0 0

0 1 0

0 0 −2

form a basis for suC(3), the omplexi ation of su(3) [iii. They satisfy the ommutationrelations

[λp, λq] =

8∑

r=1

2ifpqrλr (4.23)with the non-vanishing stru ture onstants (fpqr is antisymmetri in all three indi es)f123 = 1, f147 = f246 = f257 = f345 = −f156 = −f367 =

1

2and f458 = f678 =

√3

2.As a onvenient basis for the real Lie algebra su(3) we hoose the tra eless anti-Hermitianmatri es ap = iλp, p = 1, .., 8. By dire t al ulation, analogous to example 2.1.17 (here,ea h linear operator adλp , p ∈ 1, .., 8 equals an 8 × 8 matrix represented in the basis of all

λ1, .., λ8), we obtainK(λp, λq) = 12δpq and K(ap, aq) = −12δpq , respe tively. (4.24)Consequently, by property 2.1.16 (v), su(3) and its omplexi ation suC(3) are semi-simple(detK = (−12)8).[iii therefore, λp

2are the generators of SU(3)

4.2 THE STRUCTURE OF SEMI-SIMPLE LIE ALGEBRAS 59Lemma 4.2.16: Let again D(n) denote the diagonal n×n matri es. suCD(n) := suC(n)∩D(n) is a Cartan subalgebra of suC(n).Proof : We have suCD(n) ⊆ su(n) and it is obvious that suCD(n) is abelian. Let d(n) ⊆su(n) be an abelian subalgebra. Then, any given representation of d(n) de omposes into itsirredu ible subspa es (theorem 4.1.11) and the obtained irredu ible representations are allone-dimensional ( orollary 4.1.10). Hen e, with an appropriate basis transformation d(n) ⊆suCD(n).Therefore, suCD(n) is the maximal abelian subalgebra and the assertion is proved. The Cartan subalgebra suCD(3) is the subspa e spanned by h1 := λ3 and h2 := λ8, whi himplies that the rank l is 2. It follows that

[h1, (λ1 + iλ2)] = 2(λ1 + iλ2) ,[h1, (λ6 + iλ7)] = −(λ6 + iλ7) ,[h1, (λ4 + iλ5)] = (λ4 + iλ5) ,[h1, (λ1 − iλ2)] = −2(λ1 − iλ2) ,[h1, (λ6 − iλ7)] = (λ6 − iλ7) ,[h1, (λ4 − iλ5)] = −(λ4 − iλ5) ,

[h2, (λ1 + iλ2)] = 0 ,[h2, (λ6 + iλ7)] =

√3(λ6 + iλ7) ,

[h2, (λ4 + iλ5)] =√

3(λ4 + iλ5) ,[h2, (λ1 − iλ2)] = 0 ,[h2, (λ6 − iλ7)] = −

√3(λ6 − iλ7) ,

[h2, (λ4 − iλ5)] = −√

3(λ4 − iλ5) .Thus, there are six roots: α1, α2, α3 and −α1, −α2, −α3 withα1(h1) = 2 ,α2(h1) = −1 ,α3(h1) = 1 , α1(h2) = 0 ,

α2(h2) =√

3 ,α3(h2) =

√3 . (4.25)

gα1 , gα2 , gα3 , g−α1 , g−α2 and g−α3 are all one-dimensional, with basis elements (λ1 + iλ2),(λ6 + iλ7), (λ4 + iλ5), (λ1 − iλ2), (λ6 − iλ7) and (λ4 − iλ5), respe tively. Furthermore,α3 = α1 + α2.We now al ulate the matri es hα1 , hα2 and hα3 . Suppose hα1 = κ1h1 + κ2h2. Then, with thedenition of hα and the properties 2.1.16 of K(·, ·) we nd that κ1 K(h1, h)+κ2 K(h2, h) = α1(h)for h = h1, h2. With equation (4.24), we see that κ1 = 1

6 and κ2 = 0 and therefore,hα1 =

1

6h1 =

1

6

1 0 0

0 −1 0

0 0 0

. (4.26)Similarly,

hα2 = − 1

12h1 +

√3

12h2 =

1

6

0 0 0

0 1 0

0 0 −1

and hα3 = hα1 + hα2 =

1

6

1 0 0

0 0 0

0 0 −1

.

60 4 REPRESENTATION THEORY - SU(3) AND su(3)Furthermore, with equation (4.24) and h1 ≡ λ3, h2 ≡ λ8, we ompute〈α1, α1 〉 =

1

3,

〈α2, α1 〉 = −1

6, 〈α1, α2 〉 = −1

6,

〈α2, α2 〉 =1

3. (4.27)Orthonormal basis of the Cartan subalgebra h : By dire t al ulation using equation(4.27), we nd the orthonormal basis elements (alternatively, apply the Gram-S hmidt or-thogonalisation pro ess to hα1 and hα2 , see [Cor94, B.2)

h1 =√

3hα1 =

√3

6λ3 ,

h2 = hα1 + 2hα2 =

√3

6λ8 . (4.28)Thus, using equation (4.27) with αi dened by hαi

:= hi and 〈αi, αj 〉 = δij , we omputeα1 =

√3

3α1 ,

α2 = −√

3

6α1 +

1

2α2 . (4.29)Positive roots of suC(3) : With respe t to the basis β1 := α1, β2 := α2 ( ompare property4.2.3 (iii)) the positive roots are α1, α2 and α1+α2. If another basis in h∗ is hosen, e.g. β1 :=

α1, β2 := α1 +α2, the positive roots are α1, −α2 and α1 +α2 (here, −α2 = α1 − (α1 +α2)).Simple roots of suC(3) : With β1 and β2 it is obvious that α1 and α2 are the only simpleroots. However, with respe t to β1 and β2, the simple roots are −α2 and α1 + α2 sin eα1 = (α1 + α2) + (−α2).Cartan matrix of suC(3) : With (4.20) and (4.27) it is obvious that

AsuC(3) =

(2 −1

−1 2

) . (4.30)Weyl groupWsuC(3) : Using the Cartan matrix AsuC(3) from above and property 4.2.11 (iv)we omputeSα1 α1 = −α1 ,Sα1 α2 = α1 + α2 , Sα2 α1 = α1 + α2 ,

Sα2 α2 = −α2 .It then follows thatSα2 Sα1 α1 = −(α1 + α2) ,Sα2 Sα1 α2 = α1 , Sα1 Sα2 α1 = α2 ,

Sα1 Sα2 α2 = −(α1 + α2)(4.31)

4.3 THE WEIGHTS OF A REPRESENTATION 61andSα1 Sα2 Sα1 α1 = Sα2 Sα1 Sα2 α1

(4.20)= Sα1+α2 α1 = −α2 ,

Sα1 Sα2 Sα1 α2 = Sα2 Sα1 Sα2 α2(4.20)= Sα1+α2 α2 = −α1 .With S2

α1= S2

α2= 1 (property 4.2.11 (iii)) and Sα1 Sα2 Sα1 = Sα2 Sα1 Sα2 we on lude thatthe six distin t elements of the non-abelian group WsuC(3) are

WsuC(3) = 1,Sα1 ,Sα2 ,Sα1 Sα2 ,Sα2 Sα1 ,Sα1 Sα2 Sα1 . (4.32)4.3 The weights of a representationMost of representation theory of semi-simple omplex Lie algebras rests on the notion ofweights. It turns out that ea h irredu ible representation is uniquely spe ied by its highestweight. All other weights and further properties of the irredu ible representation (su h asits dimension) an be dedu ed from it. With the knowledge of weights we are then able toinvestigate the representations of the orresponding semi-simple Lie group. Due to the Weyl hara ter formula we an infer the irredu ible hara ters of a Lie group representation fromthe highest weight and the simple roots of its Lie algebra (see se tion 4.4).Consider a representation (V,Γ) with dimension d of a semi-simple omplex Lie algebra gand let h be its Cartan subalgebra. As Γ provides a representation of the ompa t form

g ompa t of g on g ompa t ( ompare [Cor95, 14.2), it is equivalent to a representation byanti-Hermitian matri es. Hen e, Γh are diagonalisable for all h ∈ h by the same similaritytransformation, i.e. it is simultaneously diagonalisable.Let ψ1, .., ψd form a basis of the representation spa e V in whi h the Γh are diagonal. Hen e,for ea h a ∈ g the operator Γa is dened by(Γaψ)j :=

d∑

k=1

(Γa)kjψk , j ∈ 1, .., d ,where (Γa)kj denotes the matrix of Γa with respe t to the basis in V . Then, for ea h h ∈ h,(Γhψ)j = (Γh)jjψj , j ∈ 1, .., d .Let h , h ′ ∈ h and s, t ∈ R. As Γsh+th ′ = sΓh +tΓh ′ , for the diagonal elements (Γh )jj, j = 1, .., dthere holds

(Γsh+th ′)jj = s(Γh)jj + t(Γh ′)jj .Hen e, the (Γ·)jj dene linear fun tionals on h.

62 4 REPRESENTATION THEORY - SU(3) AND su(3)Denition 4.3.1: Let (V,Γ) be a representation of g and let h be its Cartan subalgebra.As shown above, V de omposes into eigenspa es Vλ indu ed by the eigenvalue fun tionalsλ ∈ h∗ given by

Vλ := v ∈ V | Γhv = λ(h)v ∀ h ∈ h .The linear fun tionals λ are alled weights of the representation (V,Γ) of g if Vλ 6= 0(λ(·) = (Γ·)jj for any j = 1, .., d). In this ase Vλ is alled the weight spa e of λ. Theintegerm(λ) := dimVλis alled multipli ity of the weight λ. The system of weights of the representation Γ is denotedby ΞΓ.Note: If the multipli ity equals 1, the weight is alled simple.Note 2: V de omposes into V =

⊕λ∈ΞΓ

Vλ.As weights an also be dened for representations of tori and for the sake of ompleteness,we will present this denition next. In the ongoing dis ussion, however, we will just use thepre eding denition.Denition 4.3.2: Let T be a torus and V a omplex T -module. An irredu ible hara terχ : T → S1 is alled weight of V if the orresponding spa e

Vχ = v ∈ V | x · v = χ(x)v ∀x ∈ Tis non-zero. In this ase Vχ is alled the weight spa e of χ.Remark 4.3.3: Every omplex T -module is the dire t sum of its weight spa es be auseof orollary 4.1.10 and theorem 4.1.11. By a weight of a real T n-module V we mean a weightof its omplexi ation V = C⊗R V .For a representation (V,Γ) of G the derivative of Γ in e (identity element in G) readsΓ′Av =

d

dt t=0

Γexp tAv , v ∈ V , A ∈ g .Hen e, the diagramg ∼= TeG

Γ′·- Te Aut(V ) ∼= End(V )

G

exp

?

Γ·- Aut(V )

exp

?

A - Γ′A

eA?

- ΓeA

?

4.3 THE WEIGHTS OF A REPRESENTATION 63 ommutes and the map g → Aut(V ), A 7→ Γ′Av is the innitesimal version of the represen-tation G→ Aut(V ). It is a representation of the Lie algebra g sin e [Γ′

A,Γ′B ]v = Γ′

[A,B]v ( f.[Rud05, Satz 3.3.5).Let V be a omplex T -module of a torus T . We an dene an a tion of the algebra t of T byt × V → V , (X, v) 7→ Γ′

Xv .Using this innitesimal version of the T -module V we an dene the innitesimal weight ofthe T -module V . An R-linear form Θ : t ∼= Te T → Te U(1) = iR is alled an innitesimalweight of the T -module V if the orresponding weight spa eVΘ := v ∈ V | Γ′

Xv = Θ(X)v ∀X ∈ tis non-zero.For the maximal torus T of G there holds t ∼= h. Therefore, the weights of the representation(V,Γ) of g (∼= TeG) dened in 4.3.1 and the innitesimal weights of V with T being themaximal torus of G dened above are the same obje ts. Depending on the preferen es of theauthor both obje ts an be found in literature. Remark 4.3.4 (Weights of the adjoint representation of g) : The basis of g onsists of (i)the basis of h = g0, hα1 , .., hαl

, and (ii) the basis of gα, eα with α ∈ ∆ dened in 4.2.5.In this basis adh is diagonal for ea h h ∈ h. The diagonal element orresponding to hαjis 0and the diagonal element orresponding to eα is α(h). Thus, ea h root α of g is a weight ofthe adjoint representation and, as dim gα = 1, ea h su h weight is simple. The other weight

λ(h) equals 0 for all h ∈ h (with multipli ity l). Properties 4.3.5: Let λ be a weight of a representation (V,Γ) of the semi-simple omplexLie algebra g. Then the following statements are true:(i) for α ∈ ∆, λ+ α is also a weight of the same representation,(ii) for α ∈ ∆, 2〈λ,α 〉〈α,α 〉 ∈ Z,(iii) λ an be written as λ =

∑lj=1 κjαj with κj ∈ Q and α ∈ ∆S

⇒ λ(h) is real for ea h h ∈ hR (hR denotes again the real ve tor spa e spanned byhβ1, .., hβl

) and(iv) tr Γa = 0 for all a ∈ g

⇒∑

λ∈ΞΓm(λ)λ(a) = 0 (the sum ranges over all weights λ and m(λ) denotes themultipli ity of λ).Proof : Cf. [Cor95, page 563 .

64 4 REPRESENTATION THEORY - SU(3) AND su(3)In order to dene the highest weight of a representation we need to dene a lexi ographi ordering for weights. Analogously to 4.2.6, we dene the weight λ to be positive (withrespe t to the hosen basis hβ1, .., hβlof h ⇒ elements of ∆S) if the rst non-vanishing oe ient of κ1, .., κl in 4.3.5 (iii) is positive (λ > 0).Then, λ > λ′ :⇔ (λ− λ′) > 0.Denition 4.3.6: Let Λ be a weight of a representation (V,Γ) of g, su h that Λ > λ forevery other weight λ of this representation. Then, Λ is said to be the highest weight of

(V,Γ).Properties 4.3.7: For the highest weight Λ of an irredu ible representation there holds(i) Λ is a simple weight (i.e. m(Λ) = 1) and(ii) every other weight λ of the representation has the formλ = Λ −

l∑

j=1

kjαj , α1, .., αl ∈ ∆S and k1, .., kl ∈ N . (4.33)Proof : Cf. [Wey25, Satz 1 and Satz 2. In omputing the highest weights the following obje ts are used.Denition 4.3.8: The fundamental weights Λ1, ..,Λl of g are the l linear fun tionalson h dened byΛj =

l∑

k=1

(A−1)kjαk , α1, .., αl ∈ ∆S . (4.34)Sin e2〈Λj , αk 〉〈αk, αk 〉

=

l∑

p=1

2(A−1)pj〈αp, αk 〉〈αk, αk 〉

=

l∑

p=1

2(A−1)pj Akp = δjk ,the fundamental weights are uniquely determined by2〈Λj , αk 〉〈αk, αk 〉

= δjk . (4.35)Theorem 4.3.9: For every irredu ible representation of a semi-simple omplex Lie algebrag the highest weight Λ an be expressed by

Λ =l∑

j=1

njΛj , (n1, .., nl) ∈ Nl and Λ1, ..,Λl fundamental weights. (4.36)Furthermore, for every l-tuple (n1, .., nl) ∈ Nl there exists an irredu ible representation of gwith the highest weight Λ given by (4.36).

4.3 THE WEIGHTS OF A REPRESENTATION 65The dimension d of an irredu ible representation of g with highest weight Λ is given bydΛ =

∏

α∈∆+

〈Λ + ζ, α〉〈ζ, α〉 with ζ =

1

2

∑

α∈∆+

α. (4.37)Proof : See [Cor95, Thm. 15.3.2 and Thm. 15.3.3. Remark 4.3.10: The theorem states that every irredu ible representation an be uniquelyidentied by the l-tuple (n1, .., nl) ∈ Nl.For any simple omplex Lie algebra g equation (4.37) an be rewritten asdΛ =

∏

α∈∆+

(∑lj=1 njk

αj ωj∑l

j=1 kαj ωj

+ 1

) , (4.38)where α =∑l

j=1 kαj αj , Λ =

∑lj=1 njΛj and ωj = 〈αj , αj 〉. This follows by the observations

〈Λ, α〉 =l∑

i,j=1

nikαj 〈Λi, αj 〉(4.35)

=l∑

i,j=1

nikαj

1

2δij〈αj , αj 〉

=1

2

l∑

i=1

nikαi ωi ,

〈ζ, α〉 =

l∑

j=1

kαj 〈ζ, αj 〉(4.22)=

1

2

l∑

j=1

kαj 〈αj , αj 〉

=1

2

l∑

j=1

kαj ωj .We hen eforth denote the irredu ible representation of g indu ed by (n1, .., nl) ∈ Nl by

Γg

n1,..,nl.Example 4.3.11 (The irredu ible representations of suC(3)) : We start by analysing theweights of the adjoint representation of suC(3) ( ompare example 4.2.15). Either by dire t al ulation using equation (4.26) or by using remark 4.3.4 about the weights of the adjointrepresentation we nd

λ1(hα1) =1

6,

λ2(hα1) = −1

6,

λ3(hα1) = 0 , λ1(hα2) = 0 ,λ2(hα2) =

1

6,

λ3(hα2) = −1

6.To determine the fundamental weights of suC(3), we invert (4.30) and obtain

A−1suC(3) =

(23

13

13

23

) . (4.39)

66 4 REPRESENTATION THEORY - SU(3) AND su(3)Hen e, using (4.34), the fundamental weights of suC(3) areΛ1 =

2

3α1 +

1

3α2 ,

Λ2 =1

3α1 +

2

3α2 . (4.40)Therefore, by (4.36), the highest weight of every irredu ible representation of suC(3) is givenby

Λn1n2 = n1Λ1 + n2Λ2 , n1, n2 ∈ N . (4.41)As proven in 4.3.9, for every pair of integers (n1, n2) ∈ N2 there exists an irredu ible repre-sentation of suC(3) with highest weight Λ, denoted by Γn1n2.Sin e ∆S = α1, α2, ω1 and ω2 in (4.38) both equal 1

3 and the dimension d of Γn1n2 is givenbydn1n2 = (n1 + 1)(n2 + 1)

(1

2(n1 + n2) + 1

) . (4.42)In order to on lude the studies on weights of representations, we subsequently ompute theweights of Γ00, Γ10 and Γ01 and visualise the last two in gure 4.1. Therefore we onsiderboth the natural basis α1, α2 and the orthonormal basis α1, α2 of h∗, ompare equation(4.25) and (4.29).Γ00 : With (4.42) we nd d = 1. Consequently this irredu ible representation has only oneweight, namely the highest weight Λ = 0.Γ10 : With (4.42) we nd d = 3. The highest weight is Λ = Λ1 = 2

3α1+13α2. Now, by equation(4.33), we al ulate the remaining two weights: λ2 = Λ−α1 = −1

3α1 + 13α2 = Sα1 Λ and λ2 =

Λ− (α1 +α2) = −13α1− 2

3α2 = Sα2 Sα1 Λ. Thus, the weights of the irredu ible representationin the orthonormal basis read √3

6 α1 + 16 α2, −√

36 α1 + 1

6 α2 and −13 α2, respe tively. They aresimple.

Γ01 : Here we nd again d = 3. The highest weight is Λ = Λ2 = 13α1 + 2

3α2 = 13 α2 and theremaining two (simple) weights are 1

3α1 − 13α2 =

√3

6 α1 − 16 α2 = Sα2 Λ and −2

3α1 − 13α2 =

−√

36 α1 − 1

6 α2 = Sα1 Sα2 Λ. 4.4 The hara ters of SU(3)In this se tion we use the hara ter formula introdu ed by H. Weyl in [Wey25 to al ulatethe omplex hara ters of SU(3), i.e. the hara ters of the omplexied Lie group SUC(3) =

SL(3,C). As we will see in se tion 5.2, the hara ters of K = SU(3) form an orthogonal basisin the Hilbert spa e of K-invariant square-integrable holomorphi fun tions on T∗K ∼= KC,HL2(KC, e−κ

~ ηε)K .

4.4 THE CHARACTERS OF SU(3) 67PSfrag repla ements16

16

16

16

16

16

16

16

Sα1Λ = 1

6(−

√3, 1) Λ = 1

6(√

3, 1)

Sα2Sα1

Λ = 16(0,−2)

Λ = 16(0, 2)

Sα1Sα2

Λ = 16(−

√3,−1) Sα2

Λ = 16(√

3,−1)

α1 α1

α2 α2

Fig. 4.1: Weight diagrams of Γ10 and Γ01.Theorem 4.4.1 (Weyl's hara ter formula) : Let (V,Γ) be an irredu ible representationof the semi-simple Lie group G determined by the highest weight Λ =∑l

j=1 njΛj (equation(4.36)) and let ζ = 12

∑α∈∆+

α (as dened in proposition 4.2.14). SetξΛ(a = eA) :=

∑

S∈Wdet(S) exp

((S(Λ + ζ)

)A), A ∈ g , (4.43)where W is the Weyl group dened in 4.1.23. Then, the hara ter of a = eA ∈ G, A ∈ g isgiven by

χCΛ (a) =ξΛ(a)

ξ0(a). (4.44)Note: We obtain the real hara ters by restri ting χC to the real Lie group.Proof : Cf. [Ban03, Thm. 35.4. For SU(3) with its omplexied Lie algebra suC(3) = sl(3,C) the highest weight Λ is denedby the two integers n1 and n2 ( ompare example 4.3.11). Taking the Weyl group WsuC(3)dened in equation (4.32) of example 4.2.15, the omplex hara ters of SU(3) are given by

χCn1n2(eA) =

∑S∈WsuC(3)

det(S)e(S(Λn1n2+ζsuC(3)))A

∑S∈WsuC(3)

det(S)e(S ζsuC(3))A≡ ξCn1n2

ξC00 , A ∈ suC(3) . (4.45)In the sequel we will ompute this formula expli itly. For SU(3) it su es to al ulate (4.45)for the diagonal matri es SUD(3) sin e every element of SU(3) is onjugate to an element ofSUD(3) and a basis transformation has no impa t on the hara ter. For the sake of simpli ity,we hen eforth denote Λn1n2 by Λ and ζsuC(3) by ζ.

68 4 REPRESENTATION THEORY - SU(3) AND su(3)We start by analysing the exponent of the numerator of (4.45). With equation (4.40) we ndΛ + ζ = (n1Λ1 + n2Λ2) +

1

2(α1 + α2 + (α1 + α2))

=

(2

3n1 +

1

3n2 + 1

)α1 +

(1

3n1 +

2

3n2 + 1

)α2 . (4.46)The elements S ∈ WsuC(3) a t on Λ + ζ via (note Sα1α2α1 ≡ Sα2α1α2)1 (Λ + ζ) = Λ + ζ ,

Sα1(Λ + ζ) = −(

2

3n1 +

1

3n2 + 1

)α1 +

(1

3n1 +

2

3n2 + 1

)(α1 + α2)

=

(−1

3n1 +

1

3n2

)α1 +

(1

3n1 +

2

3n2 + 1

)α2 ,

Sα2(Λ + ζ) =

(2

3n1 +

1

3n2 + 1

)α1 +

(1

3n1 −

1

3n2

)α2 ,

Sα1α2(Λ + ζ) =

(−1

3n1 −

2

3n2 − 1

)α1 +

(1

3n1 −

1

3n2

)α2 ,

Sα2α1(Λ + ζ) =

(−1

3n1 +

1

3n2

)α1 +

(−2

3n1 −

1

3n2 − 1

)α2 and

Sα1α2α1(Λ + ζ) =

(−1

3n1 −

2

3n2 − 1

)α1 +

(−2

3n1 −

1

3n2 − 1

)α2 .

(4.47)Ea h A ∈ suCD(3) an be written in the basis h1 ≡ λ3, h2 ≡ λ8 ( ompare example 4.2.15)by

A = w1h1 + w2h2 , w1, w2 ∈ C .Hen e, we dedu eα1(A) = w1α1(h1) + w2α1(h2)

(4.25)= 2w1 and

α2(A)(4.25)= −w1 +

√3w2Thus, inserting A in (4.47), we obtain1 (Λ + ζ)(A) = 2w1

(2

3n1 +

1

3n2 + 1

)+ (−w1 +

√3w2)

(1

3n1 +

2

3n2 + 1

)

= w1(n1 + 1) + w2

(1√3n1 +

2√3n2 +

√3

) ,Sα1(Λ + ζ)(A) = w1(−n1 + 1) + w2

(1√3n1 +

2√3n2 +

√3

) ,Sα2(Λ + ζ)(A) = w1(n1 + n2 + 1) + w2

(1√3n1 −

1√3n2

) ,Sα1α2(Λ + ζ)(A) = w1(−n1 − n2 − 1) + w2

(1√3n1 −

1√3n2

) ,Sα2α1(Λ + ζ)(A) = w1(n2 + 1) + w2

(− 2√

3n1 −

1√3n2 −

√3

) andSα1α2α1(Λ + ζ)(A) = w1(−n2 − 1) + w2

(− 2√

3n1 −

1√3n2 −

√3

) .(4.48)

4.4 THE CHARACTERS OF SU(3) 69We therefore omputeξCn1n2

(A) = ew1(n1+1)+

w2√3(n1+2n2+3) − e

w1(−n1+1)+w2√

3(n1+2n2+3) − e

w1(n1+n2+1)+w2√

3(n1−n2)

+ ew1(−n1−n2−1)+

w2√3(n1−n2) + e

w1(n2+1)+w2√

3(−2n1−n2−3) − e

w1(−n2−1)+w2√

3(−2n1−n2−3) .Denoting x1 := ew1 and x2 := e

w2√3 , a ∈ SUCD(3) reads

a = eA =

x1x2 0 0

0 x−11 x2 0

0 0 x−22

and we get

ξCn1n2(a) = xn1+1

1 xn1+2n2+32 − x−n1+1

1 xn1+2n2+32 − xn1+n2+1

1 xn1−n22

+ x−n1−n2−11 xn1−n2

2 + xn2+11 x−2n1−n2−3

2 − x−n2−11 x−2n1−n2−3

2

= xn1+2n2+32 (xn1+1

1 − x−(n1+1)1 ) + x

−(2n1+n2+3)2 (xn2+1

1 − x−(n2+1)1 )

− xn1−n22 (xn1+n2+2

1 − x−(n1+n2+2)1 ) .We on lude

ξC00(a) = x32(x1 − x−1

1 ) + x−32 (x1 − x−1

1 ) − (x21 − x−2

1 )

= (x1 − x−11 )(x3

2 + x−32 − x1 − x−1

1 )and therefore,χCn1n2

x1x2 0 0

0 x−11 x2 0

0 0 x−22

=

xn1+2n2+32 (xn1+1

1 − x−(n1+1)1 )

(x1 − x−11 )(x3

2 + x−32 − x1 − x−1

1 )+

+x−(2n1+n2+3)2 (xn2+1

1 − x−(n2+1)1 ) − xn1−n2

2 (xn1+n2+21 − x

−(n1+n2+2)1 )

(x1 − x−11 )(x3

2 + x−32 − x1 − x−1

1 ). (4.49)Another parameterisation of suCD(3) leads to a more simplied version of this formula. Let

B =

y1 0 0

0 y2 0

0 0 −(y1 + y2)

=

1

2(y1 − y2)h1 +

√3

2(y1 + y2)h2and hen e

b = eB =

z1 0 0

0 z2 0

0 0 z−11 z−1

2

with z1 := ey1 , z2 := ey2 .

70 4 REPRESENTATION THEORY - SU(3) AND su(3)Analogously to the above approa h (using example 4.2.15 and equation (4.47)) we omputeα1(B)

(4.25)= y1 − y2 ,

α2(B)(4.25)= y1 + 2y2and 1 (Λ + ζ)(B) = (y1 − y2)

(2

3n1 +

1

3n2 + 1

)+ (y1 + 2y2)

(1

3n1 +

2

3n2 + 1

)

= y1(n1 + n2 + 2) + y2(n2 + 1) ,Sα1(Λ + ζ)(B) = y1(n2 + 1) + y2(n1 + n2 + 2) ,Sα2(Λ + ζ)(B) = y1(n1 + 1) + y2(−n2 − 1) ,

Sα1α2(Λ + ζ)(B) = y1(−n2 − 1) + y2(n1 + 1) ,Sα2α1(Λ + ζ)(B) = y1(−n1 − 1) + y2(−n1 − n2 − 2) and

Sα1α2α1(Λ + ζ)(B) = y1(−n1 − n2 − 2) + y2(−n1 − 1) .(4.50)

We an therefore writeξCn1n2

(b) = zn1+n2+21 zn2+1

2 − zn2+11 zn1+n2+2

2 − zn1+11 z

−(n2+1)2

+ z−(n2+1)1 zn1+1

2 + z−(n1+1)1 z

−(n1+n2+2)2 − z

−(n1+n2+2)1 z

−(n1+1)2 .For the denominator (n1 = n2 = 0) we nd

ξC00(b) = z21z2 − z1z

22 − z1z

−12 + z−1

1 z2 + z−11 z−2

2 − z−21 z−1

2

= (z1 − z2)

(z1 −

1

z1z2

)(z2 −

1

z1z2

) .Therefore,χCn1n2

z1 0 0

0 z2 0

0 0 z−11 z−1

2

=

zn1+n2+21 zn2+1

2 − zn2+11 zn1+n2+2

2 − zn1+11 z

−(n2+1)2

(z1 − z2)(z1 − 1z1z2

)(z2 − 1z1z2

)

+z−(n2+1)1 zn1+1

2 + z−(n1+1)1 z

−(n1+n2+2)2 − z

−(n1+n2+2)1 z

−(n1+1)2

(z1 − z2)(z1 − 1z1z2

)(z2 − 1z1z2

). (4.51)Proposition 4.4.2: Equation (4.51) is equivalent to

χCn1n2

z1 0 0

0 z2 0

0 0 z−11 z−1

2

=

n1∑

i=0

n2∑

j=0

n2+i−j∑

k=0

zn1+n2−i−2j−k1 zn2+i−j−2k

2 . (4.52)

4.4 THE CHARACTERS OF SU(3) 71To prove this, we will use the following three equations.Lemma 4.4.3: For k, l, r ∈ N and s, t ∈ C there holdssk+1 − tk+1 = (s− t)(sk + sk−1t+ ..+ stk−1 + tk) (4.53)

= (s− t)

k∑

i=0

sk−iti ,sktl − sk−2rtl−r =

(s− 1

st

)(sk−1tl + sk−3tl−1 + ..+ sk−2r+1tl−r+1) (4.54)

=

(s− 1

st

) r−1∑

i=0

sk−(2i+1)tl−i for all s, t 6= 0 andsktl − sk−rtl−2r =

(t− 1

st

)(sktl−1 + sk−1tl−3 + ..+ sk−r+1tl−2r+1) (4.55)

=

(t− 1

st

) r−1∑

i=0

sk−itl−(2i+1) for all s, t 6= 0Proof : Only the rst and last term remain after expansion for all three equations. Applying (4.53) to the numerator of equation (4.51) (ξCn1n2(b)) leads to

zn1+n2+21 zn2+1

2 − zn2+11 zn1+n2+2

2 = zn2+11 zn2+1

2 (zn1+11 − zn1+1

2 )

(k=n1)= zn1+1

1 zn1+12 (z1 − z2)(z

n11 + zn1−1

1 z2 + ..+ zn12 ) ,

−zn1+11 z

−(n2+1)2 + z

−(n2+1)1 zn1+1

2 = −z−n2−11 z−n2−1

2 (zn1+n2+21 − zn1+n2+2

2 )

(k=n1+n2+1)= −z−n2−1

1 z−n2−12 (z1 − z2)(z

n1+n2+11 + zn1+n2

1 z2 + ..+ zn1+n2+12 ) ,

z−(n1+1)1 z

−(n1+n2+2)2 − z

−(n1+n2+2)1 z

−(n1+1)2 = z−n1−n2−2

1 z−n1−n2−22 (zn2+1

1 − zn2+12 )

(k=n2)= z−n1−n2−2

1 z−n1−n2−22 (z1 − z2)(z

n21 + zn2−1

1 z2 + ..+ zn22 ) .Consequently, ξCn1n2

(b) readsξCn1n2

(b) = (z1 − z2)((zn1+n2+1

1 zn2+12 + zn1+n2

1 zn2+22 + ..+ zn2+1

1 zn1+n2+12 )

− (zn11 z−n2−1

2 + zn1−11 z−n2

2 + ..+ z−n2−11 zn1

2 )

+ (z−n1−21 z−n1−n2−2

2 + z−n1−31 z−n1−n2−1

2 + ..+ z−n1−n2−21 z−n1−2

2 ))

(4.56)To apply equation (4.54), we pair the summands of (4.56) in the following wayzn1+n2+11 zn2+1

2 with − zn1−n2−11 z0

2 , ... , zn2+11 zn1+n2+1

2 with − z−n2−11 zn1

2 andz−n1−21 z−n1−n2−2

2 with − zn11 z−n2−1

2 , ... , z−n1−n2−21 z−n2−2

2 with − zn1−n21 z−1

2 .

72 4 REPRESENTATION THEORY - SU(3) AND su(3)For i = 0, .., n1 and j = 0, .., n2 and using equation (4.54), we ndzn1+n2+1−i1 zn2+1+i

2 − zn1−n2−1−i1 zi2︸ ︷︷ ︸

(k=n1+n2+1−i , l=n2+1+i , r=n2+1)

=

(z1 −

1

z1z2

)(zn1+n2−i

1 zn2+1+i2

−2/−1+..+ zn1−n2−i

1 z1+i2 ) and

z−n1−2−j1 z−n1−n2−2+j

2 − zn1−j1 z−n2−1+j

2 = − (zn1−j1 z−n2−1+j

2 − z−n1−2−j1 z−n1−n2−2+j

2 )︸ ︷︷ ︸(k=n1−j , l=−n2−1+j , r=n1+1)

= −(z1 −

1

z1z2

)(zn1−1−j

1 z−n2−1+j2

−2/−1+..+ z−n1−1−j

1 z−n1−n2−1+j2 ) .Therefore, ξCn1n2

(b) readsξCn1n2

(b) = (z1 − z2)

(z1 −

1

z1z2

)((zn1+n2

1 zn2+12

−2/−1+..+ zn1−n2

1 z12)︸ ︷︷ ︸

n2 terms−1/+1 , n1 terms

+ ... + (zn21 zn1+n2+1

2

−2/−1+..+ z−n2

1 zn1+12 )︸ ︷︷ ︸

n2 terms− (zn1−1

1 z−n2−12

−2/−1+..+ z−n1−1

1 z−n1−n2−12 )︸ ︷︷ ︸

n1 terms−1/+1 , n2 terms

− ... − (zn1−n2−11 z−1

2

−2/−1+..+ z−n1−n2−1

1 z−n1−12 )︸ ︷︷ ︸

n1 terms )and simplies toξCn1n2

(b) = (z1 − z2)

(z1 −

1

z1z2

) n1∑

i=0

n2∑

j=0

(zn1+n2−2j−i1 zn2−j+i

2 − zn1−1−2i−j1 z−n2−1−i+j

2

) .(4.57)Applying equation (4.55), we obtainzn1+n2−2j−i1 zn2+1−j+i

2 − zn1−1−2i−j1 z−n2−1−i+j

2︸ ︷︷ ︸(n1 + 1)(n2 + 1) terms with (k=n1+n2−2j−i , l=n2+1−j+i , r=n2+1−j+i , r=1,..,n1+n2+1)

=

(z2 −

1

z1z2

)(zn1+n2−2j−i

1 zn2−j+i2 + zn1+n2−2j−i−1

1 zn2−j+i−22 + ..+ zn1−2i−j

1 z−n2−i+j2 )

=

(z2 −

1

z1z2

) r−1∑

m=0

zn1+n2−2j−i−m1 zn2−j+i−2m

2 . (4.58)We on lude the proof of proposition 4.4.2 by ombining equations (4.58) and (4.57), insert-ing them into (4.51) and an elling the denominator.

4.4 THE CHARACTERS OF SU(3) 73As we will see in hapter 5, the hara ters of the entre elements of SU(3) are needed todetermine the vanishing fun tions on subspa es asso iated with ertain strata of our model'sredu ed phase spa e P, ompare se tion 3.6. The entre of SU(3) is given by the three rootsof unityZ = ei 2m

3π1 | m = 1, 2, 3 .Using equation (4.52) and e 1

3i2πm = e−

23i2πm, we subsequently ompute the irredu ible har-a ters at these points.

χCn1n2

ei2m3π 0 0

0 ei2m3π 0

0 0 ei2m3π

=

n1∑

l=0

n2∑

j=0

n2+l−j∑

k=0

ei2m3π(n1+n2−l−2j−k+n2+l−j−2k)

= ei2m3π(n1+2n2)

n1∑

l=0

n2∑

j=0

n2+l−j∑

k=0

ei2m3π(−3j−3k)

= ei2m3π(n1+2n2)

n1∑

l=0

n2∑

j=0

n2+l−j∑

k=0

e−i2mπ(j+k)︸ ︷︷ ︸

= 1︸ ︷︷ ︸= n2+l−j+1

= ei2m3π(n1+2n2)(n1 + 1)(n2 + 1)2 + (n2 + 1)

n1∑

l=0

l

︸ ︷︷ ︸=

n1(n1+1)2

−(n1 + 1)

n2∑

j=0

j

︸ ︷︷ ︸=

n2(n2+1)2

= ei2m3π(n1+2n2)(n1 + 1)(n2 + 1)

((n2 + 1) +

n1

2− n2

2

)

= ei2m3π(n1+2n2) (n1 + 1)(n2 + 1)

(1

2(n1 + n2) + 1

)

︸ ︷︷ ︸= dn1n2 , ompare (4.42) . (4.59)An other simple argument to verify this equation is the following. Let (V,Γ) be an irredu iblerepresentation of SU(3).It is obvious that

(ei2m3π1)3 = 1 ⇒ Γ

(ei 2m3 π1)3 = 1 4.1.9 (ii)⇒ Γ

ei2m3 π1 = µ 1 for some µ ∈ C with µ3 = 1.To determine µ it su es to al ulate Γ

ei2m3 π1vΛ with vΛ being the highest weight ve tor.Sin e vΛ is an eigenve tor of Γ′

sh2with eigenvalue Λ(sh2) (s ∈ C and h2 ≡ λ8),

µvΛ = Γei2m

3 π1vΛRem. 4.3.3= exp

(Γ′i 2m

3π√

3h2

)vΛ

= exp

(Λ

(i2m

3π√

3h2

))vΛ

= exp

((n1Λ1 + n2Λ2

)i 2m

3π√

3h2

)vΛ

74 4 REPRESENTATION THEORY - SU(3) AND su(3)(4.40)= exp

(n1

(2

3α1 +

1

3α2

)

i 2m3π√

3h2

+ n2

(1

3α1 +

2

3α2

)

i 2m3π√

3h2

)vΛ(4.25)

= exp

(n1

(2

30 +

1

3

√3

(i2m

3π√

3

))1+ n2

(1

30 +

2

3

√3

(i2m

3π√

3

))1)vΛ= exp

(i2m

3π(n1 + 2n2)

)vΛ .Using property 4.1.14 (vii), we obtain

χCn1n2(Γei 2m

3 π1) = µχCn1n2(1)︸ ︷︷ ︸

= dimC V = ei2m3π(n1+2n2)dn1n2 ,whi h is identi al to equation (4.59).

5 The ostratied Hilbert spa e for SU(3)In this hapter we will dedu e the ostratied Hilbert spa e stru ture, ontinuing the dis us-sions of se tion 3.6.Re all: Let K = GN be a ompa t onne ted Lie group with G a ting on K via onjuga-tion (g 7→ LgRg−1). Let Q = K/G be the redu ed onguration spa e and let J be the anoni al momentum map on the unredu ed phase spa e T∗K, J : T∗K → g∗. For ourmodel we assume N = 1 and hen e K ≡ G. The isomorphism T∗G ∼= G × g is given byς : G × g → T∗G , (a,A) 7→ F (A) L′

a−1 (F : g → g∗, f. (2.8)) and the a tion of G ×Gon G× g is dened by ϕx,y : (a,A) 7→ (xay,Ady−1 A). We have shown that ς is equivariantunder this a tion.For the following dis ussions we suppose G to be realised as a losed subgroup of some unitarygroup U(n).5.1 Kähler stru ture and symmetry redu tionDenition 5.1.1: A Kähler manifold is a omplex manifold endowed with a positivedenite Hermitian form whose imaginary part, ne essarily an ordinary real two-form, is losedand non-degenerate and hen e a symple ti stru ture.Note: Consequently, a Kähler manifold is a smooth manifold equipped with a ompatible omplex and symple ti stru tureThe unredu ed phase spa e T∗G a quires a Kähler stru ture in the following manner: Usingthe polar de ompositionU(n) × u(n) → GL(n,C) , (a,A) 7→ aeiAwith G realised as a losed subgroup of some unitary group U(n), we obtain a dieomorphism(see [Hal94, Proof Lem. 12)

G× g → GC , (a,A) 7→ aeiA (5.1) alled the polar de omposition of GC.Note: The polar de omposition is equivariant with respe t to the a tion ϕ sin e ϕx,y(a,A) 7→75

76 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)

xayeiAdy−1 A = xayeiy

−1Ay 2.1.7 (v)= xaeiAy = ϕx,y(ae

iA).Thus, there holdsT∗G ∼= G× g ∼= GC . (5.2)Combining the omplex stru ture on GC and the symple ti stru ture on the otangentbundle T∗G we obtain the ϕ-equivariant Kähler stru ture. The global Kähler potential isthe real analyti fun tion κ : T∗G→ R given byκ(aeiA) = ||A ||2 [i . (5.3)An expli it proof for this assertion is given in [Hal02, Fml. 2.5.The redu ed onguration spa e of our model is the onjuga y quotient X := G/G (withrespe t to the G-a tion on G given by onjugation). We an de ompose X into a disjointunion X =

⋃τ,j Xτj , where τ ranges over the orbit types of the G-a tion, Xτ is the subset onsisting of orbits of type τ and j labels the onne ted omponents. This de omposition isknown as the orbit type strati ation of X . As dis ussed in [P01, Thm. 4.3.2, the Xτjare manifolds and the frontier ondition holds ( f. denition 2.4.4 (ii)).Let now T ⊆ G be a maximal torus and let W be the Weyl group of G. With the naturalin lusion T → G there holdsProposition 5.1.2: There exists a anoni al homeomorphism

T/W → X = G/G , Wt 7→ G · t = gtg−1 | g ∈ Gtaking the orbit of t under the operation of W to the onjuga y lass of t.Note: Hen e, there is a one-to-one relation whi h identies the orbit type strata in X viaT/W.Proof : Cf. [BD85, Pro. 4.2.6. The redu ed phase spa e of our model is the redu ed otangent bundle P = J−1(0)/Gobtained by singular Marsden-Weinstein redu tion, ompare with se tions 2.4.2 and 3.6. Itsstrati ation is given by the onne ted omponents of the orbit type subsets, P =

⋃τ,j Pτj ,where τ ranges over the orbit types of the G-a tion, Pτ is the subset onsisting of orbits oftype τ and j labels the onne ted omponents again.For an expli it des ription of P let t ⊆ g be the Lie algebra of T . A ording to the mo-mentum map stated in equation (3.16), the vanishing of J(a,A) for given (a,A) ∈ G × gimplies that a and A ommute. Hen e, the pair (a,A) is onjugate to an element of T × t( f. Main Theorem on Tori [BD85, Thm. 4.1.6) and the inje tion T × t → G × g indu es[i || · || denotes the norm dened by the inner produ t on g

5.1 KÄHLER STRUCTURE AND SYMMETRY REDUCTION 770 1

323 1 ϑ1

0

13

23

1

ϑ2

SUD(3)

·/W

[0,0,0]

[ 13, 13, 13]

[ 23, 23, 23]

SUD(3)/S3Fig. 5.1: Illustration of the redu tion pro ess for SUD(3). Identi ations are indi ated by the thingray lines and foldings by the thi k gray arrows.a homeomorphism of the quotient (T × t)/W onto the redu ed phase spa e P (analogous toproposition 5.1.2 above). This homeomorphism identies the orbit type strata.Using the results of example 4.1.29 we know that for G = SU(n), T an be hosen as thesubgroup of diagonal matri es in G, i.e. SUD(n), and W then equals the symmetri groupSn permuting the entries in T . Then, t is the subalgebra of diagonal matri es in g and Where also operates by permutation.As shown in example 4.1.29, the elements of T are represented by n-tuples (ϑ1, .., ϑn) ∈(R/Z)n with zj = ei2πϑj and ϑ1 + .. + ϑn−1 = −ϑnmodZ. Hen e, T/W is equivalent toan (n − 1)-simplex and the orbit type strata of X ∼= T/W orrespond to its (open) subsim-pli es. In gure 5.1 the redu tion pro ess and the resulting identi ations are illustrated forSUD(3). The orbit type strata of the redu ed onguration spa e are labelled by partitionsn = n1 + ..+nk of n with nj being positive integer ree ting the multipli ity of the entries ofelements of T . For the redu ed phase spa e P the partitions of n label the orbit types of thea tion of W on T × t again. The nj's amount to the dimensions of the ommon eigenspa esof pairs in T × t.For our model, G = SU(3), the maximal torus T equals SUD(3) and its Weyl group W is thesymmetri group S3. The orbit strata of X ∼= T/W are labelled by the three partitions 3 = 3,3 = 2 + 1 and 3 = 1 + 1 + 1, ompare gure 5.2. The subsets a quired by de omposition ofX are therefore denoted X1, X2 and X3, respe tively. X1 onsists of the lasses of the entrelabelled by the three roots of 1, i.e. ei 2m

3π, m = 1, 2, 3, denoted by X1m.The redu ed phase spa e P ∼= (T × t)/W de omposes in the following manner: The orbittype subsets P1m onsist of the lasses ([ei

2m3π, ei

2m3π, ei

2m3π], [0]) and have dimension zero.

P2 orresponds to the lasses ([x, x, x−2], [iz, iz,−i2z]) with x ∈ S1 \ ei 23π, ei 43π, 1 for z = 0and x ∈ S1 for z ∈ R \ 0. It amounts to a ylinder over the edges of the triangle, missing

78 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)PSfrag repla ements[1, 1, 1] h

ei 2

3π, ei 2

3π, ei 2

3π

i

hei 4

3π, ei 4

3π, ei 4

3π

i

[x,x, x−2] , x = ei2πϑx , ϑx ∈ (0, 13)

[x,x, x−2] , ϑx ∈ ( 13, 23)

[x, x, x−2] , ϑx ∈ ( 23, 1)

[y1, y2, (y1y2)−1]

Fig. 5.2: Visualisation of X ∼= T/W (x ∈ S1 \ ei 2

3π, ei 4

3π, 1 and y1, y2 ∈ S1, y1 6= y2).the three points of the triangle's vertexes, and has dimension two. P3 is of dimension fourand onsists of the lasses ([y1, y2, y

−11 y−1

2 ], [iz1, iz2,−iz1 − iz2]) with y1, y2 ∈ S1, y1 6= y2 forz1 = z2 and y1, y2 ∈ S1 for z1 6= z2, z1, z2 ∈ R.The Weyl group W a ts on T∗ T by pull-ba k and on TC via permutation of entries. TheW-equivariant dieomorphisms T∗ T → T × t → TC given in (3.15) and (5.1) indu e a home-omorphism between P and the quotients T∗ T/W ∼= TC/W. As explained in the pre edingse tion the symple ti stru ture of T∗ T and the omplex stru ture of TC ombine to theKähler stru ture on T∗ T ∼= TC. This spa e is simply denoted by TC in the sequel.P regarded as T∗ T/W inherits a stratied symple ti stru ture by singular Marsden-Weinsteinredu tion. Hen e,(i) the algebra C∞(TC)W of ordinary smooth W-invariant fun tions on TC inherits aPoisson bra ket and thus furnishes a Poisson algebra of ontinuous fun tions on P ∼=

TC/W,(ii) the Poisson stru ture yields an ordinary symple ti Poisson stru ture on ea h stratumand(iii) the restri tion map from C∞(TC)W to the algebra of ordinary smooth fun tions onthat stratum is a Poisson map.P a quires a omplex analyti stru ture by TC/W in a natural way. Both, the Poissonstru ture and the omplex stru ture satisfying the ompatibility requirement that for ea hstratum the symple ti and omplex stru ture on that stratum ombine to give an ordinaryKähler stru ture ( f. denition 5.1.1 and beneath) merge to form a stratied Kählerstru ture on P.5.2 The ostratied Hilbert spa e stru tureOur physi al Hilbert spa e H is given by a ertain spa e of square-integrable holomorphi fun tions on GC whi h arises by Kähler quantisation on GC, ompare [Woo92, 9. This geo-metri quantisation for ompa t Lie groups was a hieved by Hall's work about the generalised

5.2 THE COSTRATIFIED HILBERT SPACE STRUCTURE 79Segal-Bargmann transformation, see [Hal02. Analogously to publi ations of Hübs hmann([Hüb07), we then put our physi al Hilbert spa e in relation with the Hilbert spa e arisingby ordinary S hrödinger quantisation on G through an analogon of the Peter-Weyl theorem,the holomorphi Peter-Weyl theorem. In the Hilbert spa e yielded by Kähler quantisation,we sear h for wave fun tions lo alised on the strata Pτj of P. The subspa es of lo alisedwave fun tions arry an additional ostrati ation stru ture.S hrödinger quantisation of T∗G yields the Hilbert spa e L2(G, da) of ordinary square-integrable fun tions on G with s alar produ t ( f. [Hal02)〈ψ1, ψ2 〉 =

1

volG

∫

Gψ1ψ2 da .

da is the normalised Haar measure whi h oin ides with the Riemannian volume measure onG. The Hilbert spa e asso iated with P by redu tion after (S hrödinger) quantisation is thesubspa e L2(G, da)G of G- onjugation-invariant fun tions.Kähler quantisation on GC yields the Hilbert spa e HL2(GC, e−κ

~ ηε) of holomorphi square-integrable fun tions on GC relative to the measure e−κ~ ηε ( f. [Hal02). The sin-gle omponents of this measure are:(i) κ: ϕ-equivariant global Kähler potential dened in (5.3), i.e. the real analyti fun tionon GC, κ(aeiA) = ||A ||2.(ii) η: ϕ-equivariant analyti fun tion on GC dened by

η(aeiA) =

√det

(sin(ad(A))

ad(A)

), a ∈ G , A ∈ gderived from the Ja obian of the map G× g → GC, onrm [Hal96, Proof Lem. 5. Ifwe hoose a Cartan subalgebra t of g, we an express η in terms of a root system ∆.Hen e, for T × t ∼= TC, η is given by

η(teiH) =∏

α∈∆+

sinhα(H)

α(H), t ∈ T , H ∈ t , ompare [Hal96, Lem. 5 (∆+ denotes the set of positive roots).(iii) ε: Using the polar de omposition (5.1) we obtain the identity ε = da dA, where dais the normalised Haar measure on G (see above) and dA the Lebesgue measure on gindu ed by the inner produ t on g.Thus, the s alar produ t in HL2(GC, e−κ

~ ηε) is given by〈ψ1, ψ2 〉 =

1

volG

∫

GC ψ1ψ2e−κ

~ ηε .

80 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)Left and right multipli ation dene a unitary representation of G×G on HL2(GC, e−κ~ ηε).A basis of HL2(GC, e−κ

~ ηε) is obtained by the following pro edure. Let again χCΛ denotethe irredu ible omplex hara ter of GC for a highest weight Λ. From the holomorphi Peter-Weyl theorem, ompare [Hüb07, Thm. 1.14, we know that the total Hilbert spa eHL2(GC, e−κ

~ ηε) ontains the omplex ve tor spa e C[GC] of representative fun tions on GCas a dense subspa e. This is a hieved in the following way: Let Λ be a highest weight and(ΓΛ, VΛ) the orresponding irredu ible representation of GC. For q ∈ GC, ϕ ∈ V ∗

Λ and v ∈ VΛthe map q 7→ ϕ((ΓΛ)qv) indu es an inje tive morphismιΛ : V ∗

Λ ⊗ VΛ → C[GC] , ϕ ⊗ v 7→ ϕ((ΓΛ)·v)of (GC ×GC)-representations. HL2(GC, e−κ~ ηε) de omposes as a dire t sum

HL2(GC, e−κ~ ηε) =

⊕

ΓΛ∈dGC ι(V ∗Λ ⊗ VΛ) . (5.4)Furthermore, the holomorphi Peter-Weyl theorem implies that HL2(GC, e−κ

~ ηε)G, the Hil-bert spa e of G- onjugation-invariant fun tions, ontains the omplex ve tor spa e C[GC]Gof G-invariant polynomials on GC as a dense subspa e. The irredu ible hara ters χCΛ ofGC, where Λ runs through the highest weights of the irredu ible representations, form anorthogonal basis of HL2(GC, e−κ

~ ηε)G.Hen e, the Hilbert spa e asso iated with P by redu tion after (Kähler) quantisation is thesubspa e HL2(GC, e−κ~ ηε)G with orthogonal basis χCΛ . Sin e the χΛ's form an orthonormalbasis of L2(G, da)G, we now put L2(G, da)G and HL2(GC, e−κ

~ ηε)G in relation. ThereforedeneCΛ := (~π)

12

dim(G)e~ || Λ+ζ ||2 , (5.5)where ζ = 12

∑α∈∆+

α is the half sum of positive roots (see proposition 4.2.14) and || · || refersto the inner produ t on g ( ompare dis ussion pre eding properties 4.2.3).Theorem 5.2.1: The assignment χΛ 7→ χCΛ :=√CΛχΛ, where Λ ranges over the highestweights, denes a unitary isomorphism

L2(G, da)G → HL2(GC, e−κ~ ηε)G (5.6)of Hilbert spa es.Proof : Dire t onsequen e of [Hüb07, Thm. 5.3. Thus, the omplex hara ters χCΛ satisfy the orthogonality relations

〈χCΛ , χCΛ′ 〉 = CΛδΛΛ′ (5.7)

5.2 THE COSTRATIFIED HILBERT SPACE STRUCTURE 81and 1√CΛ

χCΛ , with Λ ranging over the highest weights, form an orthonormal basis ofHL2(GC, e−κ

~ ηε)G. Thus, the transition between the realisations of our Hilbert spa e H inS hrödinger (L2(G, da)) and holomorphi representation (HL2(GC, e−κ~ ηε)G) is a hieved bytheorem 5.2.1. In the sequel, we denote the orthonormal basis fun tions ofHL2(GC, e−κ

~ ηε)G,1√CΛχCΛ = χΛ, by |Λ〉, labelled by the highest weights.Remark 5.2.2: Using Weyl's integral formula, ompare theorem 4.1.28, we des ribe theHilbert spa es L2(G, da)G and HL2(GC, e−κ

~ ηε)G of G- onjugation-invariant fun tions asHilbert spa es of W-invariant fun tions on the omplexied maximal torus TC that aresquare-integrable relative to measures of the kind det(q′)dt and e−κ~ γεT , respe tively. Here

det(q′) = det((AdG/T )t−1 − 1G/T ) with onjugation map q is the density fun tion on T( f. equations (4.10) and (4.15)), γ is a suitable density fun tion on TC and εT denes theLiouville volume form on TC ∼= T∗ T . These Hilbert spa es are equivalent to those obtainedby quantisation after redu tion. In a natural way, we want the Hilbert spa e asso iated with the stratum Pτj for ertain τand j to be the set of fun tions lo alised on this stratum. But we annot dene lo alisedwave fun tions to have support on a ertain stratum Pτj sin e the strata are zero measuresets. However, the fun tions in HL2(GC, e−κ~ ηε)G are pointwise well-dened whi h allows usto hoose a dierent approa h: Firstly identify the vanishing fun tions on the hosen stratum

Pτj and se ondly ompute the orthogonal set of fun tions, i.e. the set of fun tionals mappingvanishing fun tions onto zero. This orthogonal set of fun tions will be our Hilbert spa easso iated with the stratum Pτj . The system of Hilbert spa es arising by the set of strata isa ostratied Hilbert spa e related to the redu ed phase spa e P.Denition 5.2.3: Let X be a stratied spa e and CX be the ategory whose obje ts arethe strata of X and whose morphisms are the in lusions S1 ⊆ S2 where S1 and S2 are strata.A ostratied Hilbert spa e relative to X assigns a Hilbert spa e CS to ea h stratum S,together with a bounded linear map CS2 → CS1 for ea h in lusion S1 ⊆ S2 su h that forS1 ⊆ S2 and S2 ⊆ S3 the omposite of CS3 → CS2 and CS2 → CS1 oin ides with the boundedlinear map CS3 → CS1 asso iated with the in lusion S1 ⊆ S3.The following pro edure des ribes how to single out the subspa es Hτj of the Hilbert spa eHL2(GC, e−κ

~ ηε)G asso iated with the strata Pτj. Thus, the elements of HL2(GC, e−κ~ ηε)Gdene fun tions on P and we identify the subspa e Vτj of fun tions vanishing on the stratum

Pτj of P viaVτj := f ∈ HL2(GC, e−κ

~ ηε)G | fPτj= 0 .

82 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)The Hilbert spa e Hτj asso iated with Pτj is determined to be the orthogonal omplementof Vτj. Therefore,HL2(GC, e−κ

~ ηε)G = Hτj ⊕ Vτj for all τ and j. (5.8)By onstru tion there holdsHτ1j1 ⊆ Hτ2j2 and Vτ2j2 ⊆ Vτ1j1 for Pτ1j1 ⊆ Pτ2j2 . Let prτ1j1,τ2j2 :

Hτ2j2 → Hτ1j1 denote the orthogonal proje tion operator (thus, prτ1j1,τ2j2 is self-adjoint,idempotent and positive). The resulting system of subspa es Hτj and the orthogonalproje tors prτ1j1,τ2j2 for Pτ1j1 ⊆ Pτ2j2 form the ostratied Hilbert spa e asso iatedwith P .Now, we expli itly ompute the Ja obian det(q′). For this, letδ : t ∼= Te T → C , H 7→

∏

α∈∆+

(e

12α(H) − e−

12α(H)

) . (5.9)With ζ being the half sum of positive roots and −i 2 sin(iα) = eα − e−α, δ an be writtenδ = eζ

∏

α∈∆+

(1 − e−α

) (5.10a)=∏

α∈∆+

(−i 2 sin

iα

2

) . (5.10b)It an be shown (e.g. [Hel84, Pro. 1.5.15) that analogously to Weyl's hara ter formula(theorem 4.4.1) there holdseζ∏

α∈∆+

(1 − e−α

)=∑

S∈Wdet(S)eS ζ . (5.11)Proposition 5.2.4: There holds δδ = det((AdG/T )t−1 − 1G/T ).Proof : The eigenvalues of (AdG/T )t−1 with t = eH , H ∈ t are the values of the global roots

e±α(H), α ∈ ∆+. Thus,det((AdG/T )t−1 − 1G/T ) =

∏

α∈∆+

(eα(H) − 1

)(e−α(H) − 1

) (∗)= δδ .(∗) uses equation (5.10a) and the fa t that the value of the roots on t is purely imaginary, i.e.

e12α(H) − e−

12α(H) = e−is − eis = e−

12α(H) − e

12α(H) with is := α(H)

2 , s ∈ R. Thus, using (5.10b) and iα(H) ∈ R, the Ja obian det(q′) equalsdet(q′) = δδ =

∏

α∈∆+

4 sin2 iα

2(5.12)and the Hilbert spa es L2(G, da)G and HL2(GC, e−κ

~ ηε)G an be des ribed as the spa es ofW-invariant square-integrable fun tions L2(T, δδdt)W and HL2(TC, e−κ

~ γεT ).

5.3 THE HILBERT SPACES H1m ASOCIATED WITH THE STRATA P1m 835.3 The Hilbert spa es H1m asso iated with the strata P1mThe purpose of this se tion is to determine the basis elements of the subspa es H11, H12 andH13 of fun tions orthogonal to V11, V12 and V13, the spa e of vanishing fun tions on the orbittype strata P11, P12 and P13, respe tively.By virtue of the polar de omposition map (5.1), G× g → GC, the three points (diag(ei

2m3π,

ei2m3π, ei

2m3π), 0), m = 1, 2, 3 are mapped onto diag(ei

2m3π, ei

2m3π, ei

2m3π). Hen e, V1m on-sists of fun tions ψ ∈ H = HL2(GC, e−κ

~ ηε)G whi h satisfy ψ(diag(ei2m3π, ei

2m3π, ei

2m3π)) = 0for m = 1, 2, 3. Using equation (4.59) we nd that V1m is spanned by

χCn1n2− ei

2m3π(n1+2n2)dn1n2χ

C00 , n1, n2 = 0, 1, 2, .. and (n1, n2) 6= (0, 0) . (5.13)A ording to the holomorphi Peter-Weyl theorem ( f. equation (5.4)), the χCn1n2

form anorthogonal basis in H. Consequently we have: (i) (5.13) is linearly independent and (ii)arbitrary ψ ∈ V1m for a xed m an be expressed in the basis of H using the elements in(5.13) together with χC00. We see that the vanishing of ψ(diag(ei2m3π, ei

2m3π, ei

2m3π)) impliesthat the oe ient of χC00 is zero and ompleteness of (5.13) is shown for ea h m. Therefore,the elements in (5.13) onstitute a basis of the subspa e V1m.By omputing the onstant CΛ for the irredu ible representations of SU(3) we will expressthe basis of V1m in terms of the orthonormal |Λ〉. Thus, we are able to dedu e the basisof H1m from that of V1m. Let Λ be a highest weight of an irredu ible representation of G.There holds

||Λ + ζ ||2 ≡ 〈Λ + ζ,Λ + ζ 〉= 〈Λ,Λ〉 + 2〈Λ, ζ 〉 + 〈ζ, ζ 〉 .Subsequently, we determine the three terms of this equation. As any invariant inner produ t

〈·, ·〉 on g = su(3) is proportional to the (negative denite) tra e form (vide [VRS97, Satz3.5), there exists β ∈ R, β > 0 with 〈·, ·〉 = − 12β2 tr(· ·). Thus, for the Killing form wend β = 1√

12as a dire t onsequen e of equation (4.24) in example 4.2.15. With αj ∈ ∆S,

ωj = 〈αj , αj 〉 and Λ =∑l

i=1 niΛi, Λi fundamental weights ( f. equation (4.36)) we ompute〈Λ,Λ〉 =

l∑

i,j=1

ninj〈Λi,Λj 〉

=l∑

i,j,k=1

ninj(A−1)kj〈Λi, αk 〉 , A Cartan matrix , αk ∈ ∆S(4.35)

=

l∑

i,j,k=1

ninj(A−1)kj

1

2δik〈αk, αk 〉

=1

2

l∑

i,j=1

ninj(A−1)ijωi ,

84 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)

〈Λ, ζ 〉 =

l∑

j=1

nj〈Λj , ζ 〉

=

l∑

i,j=1

nj(A−1)ij〈αi, ζ 〉(4.22)

=1

2

l∑

i,j=1

nj(A−1)ij 〈αi, αi 〉︸ ︷︷ ︸

= ωi

,〈ζ, ζ 〉 =

1

2

∑

α∈∆+

〈ζ, α〉

=1

2

∑

α∈∆+

l∑

j=1

kαj 〈ζ, αj 〉 , α =

l∑

j=1

kαj αj dened in rem. 4.3.10(4.22)=

1

4

∑

α∈∆+

l∑

j=1

kαj 〈αj , αj 〉︸ ︷︷ ︸= ωj

.For G = SU(3) there holds ωj ≡ ||αj ||2 = 1

3 = 4β2 =: ω for j = 1, 2, A−1suC(3) =

(23

13

13

23

) and∆

suC(3)+ = α1, α2, (α1 + α2) (see equation (4.27) and (4.39) in example 4.2.15). Hen e,

〈Λn1n2,Λn1n2 〉 =1

2

2∑

i,j=1

ninj(A−1)ijω

= 2β2∑

i

ni(n1(A−1)i1 + n2(A

−1)i2)

= 2β2(2

3n2

1 + 21

3n1n2 +

2

3n2

2)

=4

3β2(n2

1 + n22 + n1n2) ,

〈Λn1n2 , ζ 〉 =1

2

2∑

i,j=1

nj(A−1)ijωi

= 2β2(n1

∑

i

(A−1)i1

︸ ︷︷ ︸= 1

+n2

∑

i

(A−1)i2

︸ ︷︷ ︸= 1

)

= 2β2(n1 + n2) and〈ζ, ζ 〉 =

1

4

∑

α∈∆+

2∑

j=1

kαj ω

= β2(1 + 1 + (1 + 1))

= 4β2 .

5.3 THE HILBERT SPACES H1m ASOCIATED WITH THE STRATA P1m 85Thus,||Λn1n2 + ζ ||2 =

4

3β2(n2

1 + n22 + n1n2) + 4β2(n1 + n2) + 4β2

= 2β2 2

3

(n2

1 + n22 + n1n2 + 3(n1 + n2 + 1)

) (5.14)=: 2β2n1n2and using equation (5.5) we have

Cn1n2 = (~π)4e2~β2n1n2 . (5.15)Hen e, using equation (5.13) with χCn1n2=√Cn1n2χn1n2 =

√Cn1n2 |n1n2〉, we obtain thebasis ve tors of V1m in terms of |n1n2〉:

(~π)2e~β2n1n2 |n1n2〉 − (~π)2e2~β2ei

2m3π(n1+2n2)dn1n2 |00〉 . (5.16)Proposition 5.3.1: The subspa es H1m, m = 1, 2, 3 have dimension one and are spannedby the normalised ve tors

ψ1m =1

N

∞∑

n1,n2=0

e−~β2n1n2e−i2m3π(n1+2n2)dn1n2 |n1n2〉 (5.17)with the normalisation fa tor N2 =

∑∞n1,n2=0 e

−2~β2n1n2d2n1n2

.Proof : The ve tor ψ1m provides a normalised basis of H1m for all m: (i) As∣∣∣Re(e−~β2n1n2e−i

2m3π(n1+2n2)dn1n2

)∣∣∣

=∣∣∣(e−~β2n1n2 cos

(−2m

3π(n1 + 2n2)

)

︸ ︷︷ ︸∈ [−1,1]

dn1n2

)∣∣∣

∃N : ∀ n1,n2≥N≤ n3

1n32e

−(n21+n2

2) and∣∣∣Im(e−~β2n1n2e−i

2m3π(n1+2n2)dn1n2

)∣∣∣

=∣∣∣(e−~β2n1n2 sin

(−2m

3π(n1 + 2n2)

)

︸ ︷︷ ︸∈ [−1,1]

dn1n2

)∣∣∣

∃N : ∀ n1,n2≥N≤ n3

1n32e

−(n21+n2

2)the sum (5.17) onverges for all m sin e n31n

32e

−(n21+n2

2) is a onverging majorant for the realand the imaginary part. (ii) ψ1m is normalised as〈ψ1m |ψ1m〉 =

1

N2

∞∑

n1,n2=0

e−2~β2n1n2d2n1n2

= 1 .

86 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)

0.5 1 1.5 2 2.5

0.2

0.4

0.6

0.8

1.0

PSfrag repla ements~β2

| 〈ψ1m, ψ1m′ 〉 |2

Fig. 5.3: Tunnelling probabilities | 〈ψ1m |ψ1m′ 〉 |2 between the point strata P1m and P1m′ , m 6= m′.(iii) ψ1m is orthogonal to the basis ve tors of the orresponding subsets V1m in (5.16) sin eψ†

1m · (5.16) =1

Ne−~β2n1n2ei

2m3π(n1+2n2)dn1n2 · (~π)2e~β2n1n2

− 1

Ne−2~β2 · (~π)2e2~β2

ei2m3π(n1+2n2)dn1n2

= 0 .As a onsequen e of proposition 5.3.1 we obtain the orthogonal proje tors prj,1m : Hj → H1mfor j = 2, 3 and m = 1, 2, 3 dened by

prj,1m := |ψ1m 〉 〈ψ1m | . (5.18)Tunnelling between the point strata P1mBy straightforward al ulation we derive the tunnelling probabilities between the onstituentsH11, H12 and H13 of the ostratied Hilbert spa e H relative to the orbit type strati ationof P. With m,m′ = 1, 2, 3, we have

〈ψ1m |ψ1m′ 〉 =1

N2

∞∑

n1,n2=0

e−~β2n1n2 e−i23π(n1+2n2)(m′−m)d2

n1n2. (5.19)The tunnelling probability between the strata P1m and P1m′ , m 6= m′, is given by the abso-lute square | 〈ψ1m |ψ1m′ 〉 |2, i.e. the probability for a state prepared at P1m to be measuredat P1m′ and vi e versa.As we see in gure 5.3 the tunnelling probability depends strongly on the onstant ~β2. Forlarge values of ~β2, | 〈ψ1m |ψ1m′ 〉 |2 is almost one. This is an evident result: The distin-guishing oe ient between ψ1m and ψ1m′ in equation (5.19) is e−i 23π(n1+2n2)(m′−m). Sin ethe pre eding fa tor e−~β2n1n2 = (e−~β2

)n1n2 drasti ally de lines for in reasing values of

5.3 THE HILBERT SPACES H1m ASOCIATED WITH THE STRATA P1m 87n1 and n2, the most inuential ontribution is obtained for small values of n1 and n2 whi hsatisfy n1 + 2n2 6= 3k, k ∈ N (hen e e−i 23π(n1+2n2)(m′−m) 6= 1), e.g. (n1, n2) = (1, 0) and(n1, n2) = (0, 1). Thus, the inuen e of the distinguishing oe ient vanishes for small e−~β2 ,i.e. for large ~β2 and for ~β2 → ∞ there holds ψ1m → ψ1m′ .Contrariwise, for ~β2 → 0 we have | 〈ψ1m |ψ1m′ 〉 |2 → 0, i.e. in the semi lassi al limit thetunnelling probability vanishes.Representation in L2([−π, π]2, dx1dx2)In order to produ e plots of a wave fun tion ψ ∈ H, we hoose a suitable parametrisation ofX . Hen e, we realise H in the S hrödinger representation as L2(T, δδdt)W , ompare remark5.2.2. In our model, T = SUD(3) is the maximal torus, t = suCD(3) is the omplexied Liealgebra and ∆

suC(3)+ = α1, α2, (α1 +α2) are the positive roots of suC(3). Analogous to theintrodu tion pre eding proposition 4.4.2, we hoose the following parametrisation of T :

H = diag(−ix1,−ix2, i(x1 + x2)) and t = eH = diag(e−ix1 , e−ix2 , ei(x1+x2)) , (5.20)yieldingα1(H) = −i(x1 − x2) and α2(H) = −i(x1 + 2x2) .We determine the measure δδdt in this parametrisation. Using equation (5.12), we nd

δδ(t) = 64 sin2 x1−x22 sin2 x1+2x2

2 sin2 2x1+x22 . (5.21)Therefore, the assignment

H → L2([−π, π]2, δδdx1dx2) : ψ 7→ ψ(diag(e−ix1 , e−ix2 , ei(x1+x2))) (5.22)denes a Hilbert spa e isomorphism onto a losed subspa e.Furthermore, ∫ π−π ∫ π−π sin2 x1−x22 sin2 x1+2x2

2 sin2 2x1+x22 dx1 dx2 = 3π2

8 yields the normalisedintegration Ja obian δδ := 124π2 δδ. Hen e, multipli ation by √δδ denes a se ond Hilbertspa e isomorphism

L2([−π, π]2, δδdx1dx2) → L2([−π, π]2) . (5.23)Let Υ : H → L2([−π, π]2) denote the Hilbert spa e isomorphism (onto a losed subspa e) ombining rstly the parametrisation given in (5.22) and se ondly the multipli ation givenin (5.23). By virtue of Weyl's hara ter formula for SU(3), see equation (4.49), we omputethe transformed basis ve tor of H1m, Υψ1m. Dening e−ix1 =: z1z2 and e−ix2 =: z−11 z2, were eive z1 = e−

i2(x1−x2) and z2 = e−

i2(x1+x2). Thus,

Υψ1m(x1, x2)

=sin x1−x2

2 sin x1+2x22 sin 2x1+x2

2√24π

√∑∞n1,n2=0 e

−2~β2n1n2d2n1n2

· (5.24)

88 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)

·( ∞∑

n1,n2=0

e−~β2n1n2e−i2m3π(n1+2n2)dn1n2(

e−i2(x1−x2) − e

i2(x1−x2)

)(e−i

32(x1+x2) + ei

32(x1+x2) − e−

i2(x1−x2) − e

i2(x1−x2)

)

·(e−

i2(x1+x2)(n1+2n2+3)

(e−

i2(x1−x2)(n1+1) − e

i2(x1−x2)(n1+1)

)

+ ei2(x1+x2)(2n1+n2+3)

(e−

i2(x1−x2)(n2+1) − e

i2(x1−x2)(n2+1)

)

− e−i2(x1+x2)(n1−n2)

(e−

i2(x1−x2)(n1+n2+2) − e

i2(x1−x2)(n1+n2+2)

)))with m = 0, 1, 2, n1n2 = 23

(n2

1 + n22 + n1n2 + 3(n1 + n2 + 1)

) and dn1n2 = (n1 + 1)(n2 +

1)(

12(n1 +n2)+1

). Using this expli it formula, the distribution |Υψ10 | for x1, x2 ∈ [−π, π]2and ~β2 = 1, 14 and 1

16 is visualised in gure 5.4. We see that there is a strong dependen eon the onstant ~β2. In the semi lassi al limit, for ~β2 → 0, |ψ10 | is highly lo alised aboutits asso iated point stratum P10, respe tively (0, 0). Contrariwise, for large values of ~β2,|ψ10 | extends to whole X , respe tively [−π, π]2. The vanishing of |Υψ10 | on the symmetryboundaries (and espe ially at the point strata) is be ause of the sine-terms in the densityfun tion (5.21).Due to the symmetry in equation (5.24) with respe t to m, |Υψ11 | and |Υψ12 | are ofidenti al shape the symmetry entre just moves to the asso iated point stratum P11 andP12, respe tively. This shift is demonstrated in gure 5.5. The reason why the m = 1 plotappears in the left and the m = 2 plot in the right of the m = 0 plot is that we hose anegative parametrisation of T , H = diag(−ix1,−ix2, i(x1 + x2)), f. (5.20).We remark that the value ~β2 = 1

12 appears when we hoose ~ = 1 and the negative of theKilling form as the invariant s alar produ t on g.5.4 The Hilbert spa e H2 asso iated with P2We study the Hilbert spa e H2 in this se tion. The elements of H2 are those fun tion-als orthogonal to V2, the spa e of vanishing fun tions on the phase spa e stratum P2∼=

([x, x, x−2], [y, y,−2y]) with x ∈ S1 \ ei 23π, ei 43π, 1 for y = 0 and x ∈ S1 for y ∈ R \ 0, ompare the introdu tion at page 77. Therefore we begin by omputing the basis ve tors ofV2.Analogously to se tion 5.3, we use the polar de omposition map (5.1) to map the two matri es(diag(x, x, x−2),diag(iy, iy,−i2y)) ∈ G × g onto diag(xe−y, xe−y, x−2ei2y). With z := xe−yand the domains written above, we on lude z ∈ w ∈ C | w 6= 0, 1, ei

23π, ei

43π =: Z. Let P2be parametrised by t = diag(z, z, z−2), with z ∈ Z, hara terising a ylinder over the edgesof the triangle in gure 5.2 without the three vertexes. Applying Weyl's hara ter formula

5.4 THE HILBERT SPACE H2 ASSOCIATED WITH P2 89PSfrag repla ements PSfrag repla ements

~β2 = 1

PSfrag repla ements PSfrag repla ements~β2 = 1

4

PSfrag repla ements PSfrag repla ements~β2 = 1

16Fig. 5.4: Distribution of |Υψ10 | for x1, x2 ∈ [−π, π]2 and ~β2 = 1, 1

4, 1

16. The thin lines in thedensity plots on the right-hand side represent the symmetry boundaries of the maximaltorus SU D(3).

90 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)

PSfrag repla ements PSfrag repla ementsFig. 5.5: Distributions of |Υψ1m | for x1, x2 ∈ [−π, π]2, ~β2 = 1

16and m = 1 (left at (− 2

3π,− 2

3π)),

m = 0 (middle at (0, 0)) and m = 2 (right at (2

3π, 2

3π)).for SUC(3) (proposition 4.4.2) to the elements of P2, we obtain

χCn1n2

z 0 0

0 z 0

0 0 z−2

=

n1∑

i=0

n2∑

j=0

n2+i−j∑

k=0

zn1+2n2−3j−3k

= zn1+2n2

n1∑

i=0

n2∑

j=0

z−3jn2+i−j∑

k=0

(z−3)k

︸ ︷︷ ︸= z−3n2−3i+3j−z3

1−z3

=zn1+2n2

1 − z3

n1∑

i=0

n2∑

j=0

z−3j((z−3)n2+i − z3z−3j

)

=zn1+2n2

1 − z3

((n2 + 1)z−3n2

n1∑

i=0

(z−3)i

︸ ︷︷ ︸= z−3n1−z3

1−z3

− (n1 + 1)z3n2∑

j=0

(z−3)j

︸ ︷︷ ︸= z−3n2−z3

1−z3

)

=zn1+2n2

((n2 + 1)z−3n2(z−3n1 − z3) − (n1 + 1)z3(z−3n2 − z3)

)

(z3 − 1)2

=(n1 + 1)z3n1+3n2+6 − (n1 + n2 + 2)z3n1+3 + n2 + 1

z2n1+n2 (z3 − 1)2. (5.25)Our main goal is to determine vanishing fun tions on P2, hen e, ombinations of χCn1n2identi al to the zero fun tion for all values of P2. As the expression in equation (5.25) israther di ult to handle, we rstly seek for ongurations of χCn1n2

(diag(z, z, z−2)) yielding asimpler term. For the rst 28 tuples (n1, n2) the values of χCn1n2(diag(z, z, z−2)) are expli itly

5.4 THE HILBERT SPACE H2 ASSOCIATED WITH P2 91 omputed in table A.1, appendix A. Analysing the stru ture in these values, we see thatdenominators are reo urring for ertain tuples (n1, n2), e.g. all pairs (j+1, 0) and (j, 2) andall pairs (j + 1, 1) and (j, 3), j ∈ N. Combining these pairings in a way that they an el outall powers of z in the numerator whi h are smaller then these of the denominator, we ndthe two denominator-vanishing forms(3χCj+1,0 − χCj,2)(diag(z, z, z−2))

=z−2j−2

(z3 − 1)2

(3(j + 2)z3j+9 − 3(j + 3)z3j+6 + 3 − (j + 1)z3j+12 + (j + 4)z3j+3 − 3

)

=z−2j−2

(z3 − 1)2z3j+3

(−(j + 1)z9 + 3(j + 2)z6 − 3(j + 3)z3 + (j + 4)

)

︸ ︷︷ ︸= −(z3−1)2

((j+1)z3−(j+4)

)

= − (j + 1)zj+4 + (j + 4)zj+1 (5.26)and(2χCj+1,1 − χCj,3)(diag(z, z, z−2))

=z−2j−3

(z3 − 1)2

(2(j + 2)z3j+12 − 2(j + 4)z3j+6 + 4 − (j + 1)z3j+15 + (j + 5)z3j+3 − 4

)

=z−2j−3

(z3 − 1)2z3j+3

(−(j + 1)z12 + 2(j + 2)z9 − 2(j + 4)z3 + (j + 5)

)

︸ ︷︷ ︸= −(z3−1)2

((j+1)z6−2z3−(j+5)

)

= − (j + 1)zj+6 + 2zj+3 + (j + 5)zj . (5.27)The rst twelve values of the two ombinations above are displayed in table A.2 and A.3,respe tively.5.4.1 A fun tional des ription of zThe approa h presented in this subse tion is driven by the following idea: Analogous tose tion 5.3, we want to use equation (5.25) to formulate an expression for the vanishing wavefun tions on P2. A fun tional des ription of z ∈ Z (= w ∈ C | w 6= 0, 1, ei23π, ei

43π, f. page88) using the omplex hara ters an repla e the matrix parameter z in the nal formula ofequation (5.25). We now stateLemma 5.4.1: In the parametrisation t = diag(z, z, z−2), z ∈ B2 ≡ w ∈ C | |w | < 1,the two-dimensional losed ball, we nd

z =

∞∑

j=0

1

(3j + 1)(3j + 4)(3χC3j+1,0 − χC3j,2) (diag(z, z, z−2)) . (5.28)

92 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)Proof : Using (5.26) we have (3χC3j+1,0 −χC3j,2)(diag(z, z, z−2)) = −(j + 1)zj+4 + (j + 4)zj+1.Thus, there holds

m∑

j=0

1

(3j + 1)(3j + 4)(3χC3j+1,0 − χC3j,2) (diag(z, z, z−2))

=1

4(4z − z4) +

1

4 · 7(7z4 − 4z7) +1

7 · 10(10z7 − 7z10) + ...+

+1

(3m− 2)(3m + 1)

((3m+ 1)z3m−2 − (3m− 2)z3m+1

)

+1

(3m+ 1)(3m + 4)

((3m+ 4)z3m+1 − (3m+ 1)z3m+4

)

= z +1

3m+ 4z3m+4 .Sin e z ∈ B2, z + 1

3m+4z3m+4 → z for m→ ∞ and the assertion is shown. Thus, (∑∞

j=01

(3j+1)(3j+4) (3χC3j+1,0 − χC3j,2)) exists pointwise on P2 \ diag(z, z, z−2) | z ∈C \B2 and we dene the fun tional

µz :=

∞∑

j=0

1

(3j + 1)(3j + 4)(3χC3j+1,0 − χC3j,2) . (5.29)I.e. µz : P2 \ diag(z, z, z−2) | z ∈ C \B2 → B2 ⊆ C , diag(z, z, z−2) 7→ z = µz(diag(z, z,

z−2)) pointwise. Using the on ept of analyti ontinuation, we know that this progressionis uniquely ontinuable to whole C (we refer to [BS72, 11.1, µz(diag(z, z, z−2)) and z areholomorphi and identi al on B2). The vanishing fun tions on P2 \ diag(z, z, z−2) | z ∈C \B2 are spanned byχCn1n2

− (n1 + 1)µ3n1+3n2+6z − (n1 + n2 + 2)µ3n1+3

z + n2 + 1

µ2n1+n2z (µ3

z − 1)2χC00

⇔ µ2n1+n2z

(µ3z − 1

)2χCn1n2

−((n1 + 1)µ3n1+3n2+6

z − (n1 + n2 + 2)µ3n1+3z + n2 + 1

)χC00 .(5.30)As (∑∞

j=01

(3j+1)(3j+4) (3χC3j+1,0 − χC3j,2)) only exists pointwise, equations (5.29) and (5.30)are both meant for some diag(z, z, z−2) ∈ P2 \ diag(z, z, z−2) | z ∈ C \B2.But trying to transform µz into the |n1n2〉-representation by virtue of equation (5.15) and

χCn1n2=√Cn1n2 |n1n2〉, leads to (µz)m :=

∑mj=0

1(3j+1)(3j+4)

(3(~π)2e

12

~β23j+1,0 |3j + 1, 0〉 −(~π)2e

12

~β23j,2 |3j, 2〉) with n1n2 = 2

3(2n21 + 2n1n2 + 2n2

2 + 3n1 + 3n2 + 6). Thus, (µz)m isnot onverging in L2 and we therefore abandon this approa h here.In the following, we perform a more indire t approa h: In [Hüb05, the generating fun tionof V2 is exhibited. Multiplying this fun tion by arbitrary basis elements of H (= HL2(GC,e−

κ~ ηε)G), we obtain all elements of V2.

5.4 THE HILBERT SPACE H2 ASSOCIATED WITH P2 935.4.2 The generating fun tion of V2We develop an expli it des ription for V2, the spa e of vanishing fun tions on the stratumP2, in this subse tion. Sin e the pro edure des ribed above is based on the multipli ationof irredu ible fun tions, we need to de ompose the underlying dire t produ t of irredu iblerepresentations of the SU(3) into their irredu ible omponents. This spe i problem is overed in the arti le [Col64 of S. Coleman. Given (n1, n2) and (m1,m2), two irredu iblerepresentations of SU(3), the redu tion takes pla e in the following two steps:(i) The produ t (n1, n2) ⊗ (m1,m2) de omposes

(n1, n2) ⊗ (m1,m2) =

min(n1,m2)⊕

i=0

min(n2,m1)⊕

j=0

(n1 − i,m1 − j;n2 − j,m2 − i)with (l1, l2; l3, l4) ∈ SU(3)i1·..·in1+m1j1·..·jn2+m2

≡ L(SU(3), ..,SU(3)︸ ︷︷ ︸j1·..·jn2+m2

,SU(3)∗, ..,SU(3)∗︸ ︷︷ ︸i1·..·in1+m1

) and l1, .., l4∈ N.(ii) The de omposition of arbitrary tensors in the basis (n1,m1;n2,m2) into a sum oflinear ombinations of irredu ible representations of SU(3), i.e. ompletely symmetri tra eless tensors, is obtained by

(n1,m1;n2,m2) = (n1 +m1, n2 +m2)

⊕min(n1,m1)⊕

i=1

(n1 +m1 − 2i, n2 +m2 + i)

⊕min(n2,m2)⊕

j=1

(n1 +m1 + j, n2 +m2 − 2j) .By help of [Hüb05, we stateProposition 5.4.2: The fun tionχCG := 15χC00 + 3χC30 − χC22 − 6χC11 + 3χC03 (5.31)generates V2, i.e. V2 is obtained by multipli ation of χCG with elements of H.Note: For the purpose of larity, we label some dire t sum signs with integers (e.g. 1

⊕) in thefollowing set of equationsProof : Adapting the notations of [Hüb05, 5, we have: The stratum orresponding to thepartition (2, 1) is given by the variety DSU(3)(2,1)

in A3,1coef , i.e. the vanishing fun tions on thestratum (2, 1) are hara terised by the equation

a21a

22 − 4a3

1 − 4a32 − 27 + 18a1a2 = 0

94 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)with (a1, a2) oordinates on A3,1coef . As explained in the introdu tion of [Hüb05, 5, a1 and

a2 are identied with the fundamental hara ters restri ted to the maximal torus. Usingthe two three-dimensional fundamental representations of SU(3), (1, 0) and (0, 1), we havea1 = χC10 and a2 = χC01. Thus, the generator of V2 is given by

χC102χC012 − 4χC103 − 4χC013 − 27χC00 + 18χC10χC01 .We de ompose the produ t of hara ters into their irredu ible omponents by applying theredu tion te hnique introdu ed above. There holds

(1, 0) ⊗ (1, 0) = (1, 1; 0, 0) = (2, 0) ⊕ (0, 1) ,(0, 1) ⊗ (0, 1) = (0, 0; 1, 1) = (0, 2) ⊕ (1, 0) ,(2, 0) ⊗ (0, 2) = (2, 0; 0, 2)

1

⊕ (1, 0; 0, 1)2

⊕ (0, 0; 0, 0)

= (2, 2)1

⊕ (1, 1)2

⊕ (0, 0) ,(2, 0) ⊗ (1, 0) = (2, 1; 0, 0) = (3, 0) ⊕ (1, 1) ,(0, 1) ⊗ (0, 2) = (0, 0; 1, 2) = (0, 3) ⊕ (1, 1) ,(0, 1) ⊗ (1, 0) = (0, 1; 1, 0)

1

⊕ (0, 0; 0, 0) = (1, 1)1

⊕ (0, 0) .Thus,(1, 0)2 ⊗ (0, 1)2 = (3, 0) ⊕ (2, 2) ⊕ 4(1, 1) ⊕ (0, 3) ⊕ 2(0, 0) ,

(1, 0)3 = (3, 0) ⊕ 2(1, 1) ⊕ (0, 0) ,(0, 1)3 = (0, 3) ⊕ 2(1, 1) ⊕ (0, 0) ,

(1, 0) ⊗ (0, 1) = (1, 1) ⊕ (0, 0) .Colle ting all oe ients and multiplying the equation by (−1), we on lude that V2 isgenerated by15χC00 + 3χC30 − χC22 − 6χC11 + 3χC03and thus, the assertion is shown. Hen e, V2 is des ribed by

V2 = spann1,n2∈N χCn1n2

· χCG . (5.32)Using Coleman's redu tion te hnique again, we now expli itly al ulate the elements of V2given in (5.32) starting with χCn1n2· χCG, n1, n2 ∈ N.

χCn1n2· χCG = 15χCn1n2

· χC00 + 3χCn1n2· χC30 − 3χCn1n2

· χC22 − 6χCn1n2· χC11 + 3χCn1n2

· χC03We therefore ompute(n1, n2) ⊗ (0, 0) = (n1, 0;n2, 0) = (n1, n2) ,

5.4 THE HILBERT SPACE H2 ASSOCIATED WITH P2 95(n1, n2) ⊗ (3, 0) = (n1, 3;n2, 0)

1

⊕ (n1, 2;n2 − 1, 0)︸ ︷︷ ︸n2≥1

2

⊕ (n1, 1;n2 − 2, 0)︸ ︷︷ ︸n2≥2

3

⊕ (n1, 0;n2 − 3, 0)︸ ︷︷ ︸n2≥3

= (n1 + 3, n2) ⊕ (n1 + 3 − 2, n2 + 1)︸ ︷︷ ︸n1≥1

⊕ (n1 + 3 − 4, n2 + 2)︸ ︷︷ ︸n1≥2

⊕ (n1 + 3 − 6, n2 + 3)︸ ︷︷ ︸n1≥3

1

⊕ (n1 + 2, n2 − 1)︸ ︷︷ ︸n2≥1

⊕ (n1 + 2 − 2, n2 − 1 + 1)︸ ︷︷ ︸n1≥1 , n2≥1

⊕ (n1 + 2 − 4, n2 − 1 + 2)︸ ︷︷ ︸n1≥2 , n2≥1

2

⊕ (n1 + 1, n2 − 2)︸ ︷︷ ︸n2≥2

⊕ (n1 + 1 − 2, n2 − 2 + 1)︸ ︷︷ ︸n1≥1 , n2≥2

3

⊕ (n1, n2 − 3)︸ ︷︷ ︸n2≥3

,(n1, n2) ⊗ (2, 2) = (n1, 2;n2, 2)

1

⊕ (n1 − 1, 2;n2, 1)︸ ︷︷ ︸n1≥1

2

⊕ (n1, 1;n2 − 1, 2)︸ ︷︷ ︸n2≥1

3

⊕ (n1 − 1, 1;n2 − 1, 1)︸ ︷︷ ︸n1≥1 , n2≥1

4

⊕ (n1 − 2, 2;n2, 0)︸ ︷︷ ︸n1≥2

5

⊕ (n1, 0;n2 − 2, 2)︸ ︷︷ ︸n2≥2

6

⊕ (n1 − 2, 1;n2 − 1, 0)︸ ︷︷ ︸n1≥2 , n2≥1

7

⊕ (n1 − 1, 0;n2 − 2, 1)︸ ︷︷ ︸n1≥1 , n2≥2

8

⊕ (n1 − 2, 0;n2 − 2, 0)︸ ︷︷ ︸n1≥2 , n2≥2

= (n1 + 2, n2 + 2) ⊕ (n1 + 2 − 2, n2 + 2 + 1)︸ ︷︷ ︸n1≥1

⊕ (n1 + 2 − 4, n2 + 2 + 2)︸ ︷︷ ︸n1≥2

⊕ (n1 + 2 + 1, n2 + 2 − 2)︸ ︷︷ ︸n2≥1

⊕ (n1 + 2 + 2, n2 + 2 − 4)︸ ︷︷ ︸n2≥2

1

⊕ (n1 + 1, n2 + 1)︸ ︷︷ ︸n1≥1

⊕ (n1 + 1 − 2, n2 + 1 + 1)︸ ︷︷ ︸n1≥2

⊕ (n1 + 1 − 4, n2 + 1 + 2)︸ ︷︷ ︸n1≥3

⊕ (n1 + 1 + 1, n2 + 1 − 2)︸ ︷︷ ︸n1≥1 , n2≥1

2

⊕ (n1 + 1, n2 + 1)︸ ︷︷ ︸n2≥1

⊕ (n1 + 1 − 2, n2 + 1 + 1)︸ ︷︷ ︸n1≥1 , n2≥1

⊕ (n1 + 1 + 1, n2 + 1 − 2)︸ ︷︷ ︸n2≥2

⊕ (n1 + 1 + 2, n2 + 1 − 4)︸ ︷︷ ︸n2≥3

3

⊕ (n1, n2)︸ ︷︷ ︸n1≥1 , n2≥1

⊕ (n1 − 2, n2 + 1)︸ ︷︷ ︸n1≥2 , n2≥1

⊕ (n1 + 1, n2 − 2)︸ ︷︷ ︸n1≥1 , n2≥2

4

⊕ (n1, n2)︸ ︷︷ ︸n1≥2

⊕ (n1 − 2, n2 + 1)︸ ︷︷ ︸n1≥3

⊕ (n1 − 4, n2 + 2)︸ ︷︷ ︸n1≥4

5

⊕ (n1, n2)︸ ︷︷ ︸n2≥2

⊕ (n1 + 1, n2 − 2)︸ ︷︷ ︸n2≥3

⊕ (n1 + 2, n2 − 1)︸ ︷︷ ︸n2≥4

6

⊕ (n1 − 1, n2 − 1)︸ ︷︷ ︸n1≥2 , n2≥1

⊕ (n1 − 1 − 2, n2 − 1 + 1)︸ ︷︷ ︸n1≥3 , n2≥1

7

⊕ (n1 − 1, n2 − 1)︸ ︷︷ ︸n1≥1 , n2≥2

⊕ (n1 − 1 + 1, n2 − 1 − 2)︸ ︷︷ ︸n1≥1 , n2≥3

96 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)

8

⊕ (n1 − 2, n2 − 2)︸ ︷︷ ︸n1≥2 , n2≥2

,(n1, n2) ⊗ (1, 1) = (n1, 1;n2, 1)

1

⊕ (n1 − 1, 1;n2, 0)︸ ︷︷ ︸n1≥1

2

⊕ (n1, 0;n2 − 1, 1)︸ ︷︷ ︸n2≥1

3

⊕ (n1 − 1, 0;n2 − 1, 0)︸ ︷︷ ︸n1≥1 , n2≥1

= (n1 + 1, n2 + 1) ⊕ (n1 + 1 − 2, n2 + 1 + 1)︸ ︷︷ ︸n1≥1

⊕ (n1 + 1 + 1, n2 + 1 − 2)︸ ︷︷ ︸n2≥1

1

⊕ (n1, n2)︸ ︷︷ ︸n1≥1

⊕ (n1 − 2, n2 + 1)︸ ︷︷ ︸n1≥2

2

⊕ (n1, n2)︸ ︷︷ ︸n2≥1

⊕ (n1 + 1, n2 − 2)︸ ︷︷ ︸n2≥2

3

⊕ (n1 − 1, n2 − 1)︸ ︷︷ ︸n1≥1 , n2≥1

,(n1, n2) ⊗ (0, 3) = (n1, 0;n2, 3)

1

⊕ (n1 − 1, 0;n2, 2)︸ ︷︷ ︸n1≥1

2

⊕ (n1 − 2, 0;n2, 1)︸ ︷︷ ︸n1≥2

3

⊕ (n1 − 3, 0;n2, 0)︸ ︷︷ ︸n1≥3

= (n1, n2 + 3) ⊕ (n1 + 1, n2 + 3 − 2)︸ ︷︷ ︸n2≥1

⊕ (n1 + 2, n2 + 3 − 4)︸ ︷︷ ︸n2≥2

⊕ (n1 + 3, n2 + 3 − 6)︸ ︷︷ ︸n2≥3

1

⊕ (n1 − 1, n2 + 2)︸ ︷︷ ︸n1≥1

⊕ (n1 − 1 + 1, n2 + 2 − 2)︸ ︷︷ ︸n1≥1 , n2≥1

⊕ (n1 − 1 + 2, n2 + 2 − 4)︸ ︷︷ ︸n1≥1 , n2≥2

2

⊕ (n1 − 2, n2 + 1)︸ ︷︷ ︸n1≥2

⊕ (n1 − 2 + 1, n2 + 1 − 2)︸ ︷︷ ︸n1≥2 , n2≥1

3

⊕ (n1 − 3, n2)︸ ︷︷ ︸n1≥3

.Thus, the produ t χCn1,n2· χCG with n1, n2 ∈ N redu es to

χCn1,n2· χCG = 15χCn1,n2

+ 3χCn1+3,n2− χCn1+2,n2+2 − 6χCn1+1,n2+1 + 3χCn1,n2+3

n1 ≥ 1 + 2χCn1+1,n2+1 − χCn1,n2+3 − 3χCn1−1,n2+2 − 6χCn1,n2

n1 ≥ 2 + 2χCn1−1,n2+2 − χCn1−2,n2+4 − χCn1,n2− 3χCn1−2,n2+1

n1 ≥ 3 + 2χCn1−3,n2+3 − χCn1−2,n2+1 + 3χCn1−3,n2

n1 ≥ 4 − χCn1−4,n2+2

n2 ≥ 1 − 3χCn1,n2− χCn1+3,n2

+ 2χCn1+1,n2+1 − 6χCn1,n2

n2 ≥ 2 − 3χCn1+1,n2−2 − χCn1+4,n2−2 + 2χCn1+2,n2−1 − χCn1,n2

n2 ≥ 3 + 3χCn1,n2−3 + 2χCn1+3,n2−3 − χCn1+1,n2−2

n2 ≥ 4 − χCn1+2,n2−4

n1 ≥ 1, n2 ≥ 1 + 5χCn1,n2− χCn1+2,n2−1 − χCn1−1,n2+2 − 6χCn1−1,n2−1

n1 ≥ 2, n2 ≥ 1 + 2χCn1−2,n2+1 + 2χCn1−1,n2−1

5.4 THE HILBERT SPACE H2 ASSOCIATED WITH P2 97n1 ≥ 1, n2 ≥ 2 + 2χCn1−1,n2−1 + 2χCn1+1,n2−2

n1 ≥ 2, n2 ≥ 2 − χCn1−2,n2−2

n1 ≥ 3, n2 ≥ 1 − χCn1−3,n2

n1 ≥ 1, n2 ≥ 3 − χCn1,n2−3 .This table ontains all summands emerging from the redu tion pro ess. It reads in thefollowing way: The sum of terms satisfying the onditions for n1 and n2 are put together toone formula. Afterwards, we simplify the obtained sum by olle ting all oe ients of equal hara ters. For instan e, the formula of the general ase n1, n2 ≥ 4 redu e from 40 down to19 summands.From equation (5.32) we know that V2, the spa e of vanishing fun tions on P2, is ompletely hara terised by the table above. Sin e it is a rather unhandy des ription of V2, we subse-quently sear h for more pra ti able expressions.5.4.3 The basis of V2Examining the formulae for (3χCj+1,0−χCj,2)(diag(z, z, z−2)) and (2χCj+1,1−χCj,3)(diag(z, z, z−2))again (see table A.2 and A.3), we note that ombining three of them in a spe i way reatesvanishing fun tions, ompare

(6(3χC10 − χC02) + 2(3χC40 − χC32) − 4(2χC21 − χC13))(diag(z, z, z−2)) = 0 .This observation nally leads to proposition 5.4.5. In order to motivate this approa h, thefollowing two examples show how equation (5.32) and the ansatz above boil down to oneidentity.Example 5.4.3: For the parametrisation t = diag(z, z, z−2) ∈ TC of P2 there holds

((j + 5)(3χCj,0 − χCj−1,2) + (j + 1)(3χCj+3,0 − χCj+2,2)

− (j + 3)(2χCj+1,1 − χCj,3))︸ ︷︷ ︸

=: γC0,j+2

(diag(z, z, z−2)) = 0 (5.33)with j ∈ N and χC−1,2 := 0 (analogous to the dire t al ulations in equation (5.25)).For j = 0, γC0,2 equals the generating fun tion of V2 (χCG, f. equation (5.31)) and equation(5.33) trivially holds.

98 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)Furthermore, using equations (5.26) and (5.27) for j ≥ 1, we nd((j + 5)(3χCj,0 − χCj−1,2) + (j + 1)(3χCj+3,0 − χCj+2,2)

− (j + 3)(2χCj+1,1 − χCj,3))(diag(z, z, z−2))

= (j + 5)(−jzj+3 + (j + 3)zj

)+ (j + 1)

(−(j + 3)zj+6 + (j + 6)zj+3

)

− (j + 3)(−(j + 1)zj+6 + 2zj+3 + (j + 5)zj

)

= − (j + 1)(j + 3)zj+6 + (j + 3)(j + 1)zj+6

− (j + 5)jzj+3 + (j + 1)(j + 6)zj+3 − (j + 3)2zj+3

+ (j + 5)(j + 3)zj − (j + 3)(j + 5)zj

= 0 .The question arises wheather or not there are more series like γC0,j+2 in example 5.4.3 spanningthe whole of V2, i.e. if all χCn1,n2

·χCG an be represented in terms of γC0,j+2 and other series. Inpursuit of that idea we start by analysing all fun tions χCj,0 ·χCG. Trying to represent χCj,0 ·χCGby γC0,j+2 for j ∈ N yields a new series whi h equals the dieren e γC0,j+2 − (j + 1)χCj,0 · χCG,i.e.Example 5.4.4:((j + 1)χCj−2,4 − (j − 1)χCj,3 − (2j + 2)χCj−3,3 − 4χCj−1,2 + (j + 1)χCj−4,2

+ (2j − 2)χCj+1,1 + (4j + 4)χCj−2,1 − (5j − 7)χCj,0 − (3j + 3)χCj−3,0

)

︸ ︷︷ ︸=: γC1,j

(diag(z, z, z−2)) = 0(5.34)with diag(z, z, z−2) ∈ TC, j ∈ N and χC−3,k := −χC1,k−2, χC−2,k := −χC0,k−1, χC−1,k := 0 fork ∈ N (analogous to the dire t al ulations in equation (5.25)).Furthermore, there holds

(j + 1) χCj,0 · χCG = γC0,j+2 − γC1,j . (5.35)To see that, refer to proposition 5.4.5 and its proof. The proposition is not using any resultsof this example. Continuing with χCj,1 · χCG, χCj,2 · χCG and so on we nd a reo urring stru ture whi h issummarised by the following proposition.

5.4 THE HILBERT SPACE H2 ASSOCIATED WITH P2 99Proposition 5.4.5: LetγCi,j := (3j + 5)χCj−2,i − (3j − 5)χCj,i−1 − (2j + 2)(χCj−3,i−1 + χCj−3,i+2)

+ (2j − 2)(χCj+1,i + χCj+1,i−3) + (j + 1)(χCj−2,i+3 + χCj−4,i+1 + χCj−2,i−3)

− (j − 1)(χCj,i+2 + χCj+2,i−2 + χCj,i−4) − 4(χCj−1,i+1 + χCj−1,i−2)

= (j + 1)χCj−2,i+3 − (j − 1)χCj,i+2 − (2j + 2)χCj−3,i+2 − 4χCj−1,i+1 (5.36)+ (j + 1)χCj−4,i+1 + (2j − 2)χCj+1,i + (3j + 5)χCj−2,i − (3j − 5)χCj,i−1

− (2j + 2)χCj−3,i−1 − (j − 1)χCj+2,i−2 − 4χCj−1,i−2 + (2j − 2)χCj+1,i−3

+ (j + 1)χCj−2,i−3 − (j − 1)χCj,i−4(sorted by the se ond index in des ending order) with i, j ∈ N andχCk,−1 := 0 , χC−1,k := 0 ,χCk,−2 := −χCk−1,0 , χC−2,k := −χC0,k−1 ,χCk,−3 := −χCk−2,1 , χC−3,k := −χC1,k−2 andχCk,−4 := −χCk−3,2 , χC−4,k := −χC2,k−3 .for k ∈ N (analogously to dire t al ulations in equation (5.25)). There holds

(j + 1) χCj,i · χCG = γCi,j+2 − γCi+1,j (5.37)and γCi,j(diag(z, z, z−2)) = 0 for all i, j ∈ N and z ∈ Z.Remark 5.4.6: (i) The series γC0,j+2 dened in example 5.4.3 is equivalent with the onedened above.(ii) The γCi,j, i, j ∈ N form a generating system of V2 due to proposition 5.4.2 and equation(5.37).(iii) With the given substitution rules, we observeγCi,0 = χC−2,i+3 + χC0,i+2 − 2χC−3,i+2 + χC−4,i+1 − 2χC1,i + 5χC−2,i

+ 5χC0,i−1 − 2χC−3,i−1 + χC2,i−2 − 2χC1,i−3 + χC−2,i−3 + χC0,i−4

= − χC0,i+2 + χC0,i+2 + 2χC1,i − χC2,i−2 − 2χC1,i − 5χC0,i−1

+ 5χC0,i−1 + 2χC1,i−3 + χC2,i−2 − 2χC1,i−3 − χC0,i−4 + χC0,i−4

= 0 andγCi,1 = − 4χC−2,i+2 − 4χC0,i+1 + 2χC−3,i+1 + 2χC1,i−1 − 4χC−2,i−1 − 4χC0,i−2

= 4χC0,i+1 − 4χC0,i+1 − 2χC1,i−1 + 2χC1,i−1 + 4χC0,i−2 − 4χC0,i−2

= 0 .

100 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)Proof of proposition 5.4.5 : Due to irregularities in the produ t χCj,i · χCG and the series γCi,jfor indi es i < 4 and/or j < 4, ompare subse tion 5.4.2, we only demonstrate the proof ofthe rst assertion in the most general ase i, j ≥ 4 here. The remaining 24 spe ial ases areperformed in appendix B.For i, j ≥ 4 there holds(j + 1) χCj,i · χCG = (j + 1)

(−χCj−2,i+4 + 2χCj,i+3 + 2χCj−3,i+3 − χCj+2,i+2 − 2χCj−1,i+2

− χCj−4,i+2 − 2χCj+1,i+1 − 2χCj−2,i+1 + 2χCj+3,i + 6χCj,i + 2χCj−3,i

− 2χCj+2,i−1 − 2χCj−1,i−1 − χCj+4,i−2 − 2χCj+1,i−2 − χCj−2,i−2

+ 2χCj+3,i−3 + 2χCj,i−3 − χCj+2,i−4

) .Furthermore, we haveγCi,j+2 − γCi+1,j = (j + 3)χCj,i+3 − (j + 1)χCj+2,i+2 − (2j + 6)χCj−1,i+2 − 4χCj+1,i+1

+ (j + 3)χCj−2,i+1 + (2j + 2)χCj+3,i + (3j + 11)χCj,i − (3j + 1)χCj+2,i−1

− (2j + 6)χCj−1,i−1 − (j + 1)χCj+4,i−2 − 4χCj+1,i−2 + (2j + 2)χCj+3,i−3

+ (j + 3)χCj,i−3 − (j + 1)χCj+2,i−4

− (j + 1)χCj−2,i+4 + (j − 1)χCj,i+3 + (2j + 2)χCj−3,i+3 + 4χCj−1,i+2

− (j + 1)χCj−4,i+2 − (2j − 2)χCj+1,i+1 − (3j + 5)χCj−2,i+1 + (3j − 5)χCj,i+ (2j + 2)χCj−3,i + (j − 1)χCj+2,i−1 + 4χCj−1,i−1 − (2j − 2)χCj+1,i−2

− (j + 1)χCj−2,i−2 + (j − 1)χCj,i−3

= − (j + 1)χCj−2,i+4 + (j + 3)χCj,i+3 + (j − 1)χCj,i+3 + (2j + 2)χCj−3,i+3

− (j + 1)χCj+2,i+2 − (2j + 6)χCj−1,i+2 + 4χCj−1,i+2 − (j + 1)χCj−4,i+2

− 4χCj+1,i+1 − (2j − 2)χCj+1,i+1 + (j + 3)χCj−2,i+1 − (3j + 5)χCj−2,i+1

+ (2j + 2)χCj+3,i + (3j + 11)χCj,i + (3j − 5)χCj,i + (2j + 2)χCj−3,i

− (3j + 1)χCj+2,i−1 + (j − 1)χCj+2,i−1 − (2j + 6)χCj−1,i−1 + 4χCj−1,i−1

− (j + 1)χCj+4,i−2 − 4χCj+1,i−2 − (2j − 2)χCj+1,i−2 − (j + 1)χCj−2,i−2

+ (2j + 2)χCj+3,i−3 + (j + 3)χCj,i−3 + (j − 1)χCj,i−3 − (j + 1)χCj+2,i−4

= (j + 1)(−χCj−2,i+4 + 2χCj,i+3 + 2χCj−3,i+3 − χCj+2,i+2 − 2χCj−1,i+2 (5.38)

− χCj−4,i+2 − 2χCj+1,i+1 − 2χCj−2,i+1 + 2χCj+3,i + 6χCj,i + 2χCj−3,i

− 2χCj+2,i−1 − 2χCj−1,i−1 − χCj+4,i−2 − 2χCj+1,i−2 − χCj−2,i−2

+ 2χCj+3,i−3 + 2χCj,i−3 − χCj+2,i−4

)

= (j + 1) χCj,i · χCG .Thus, the rst assertion is shown for i, j ≥ 4. As all spe ial relations for negative indi es

5.4 THE HILBERT SPACE H2 ASSOCIATED WITH P2 101of χCj,i go onform with equation (5.25), the following omputation holds for all i, j ∈ N.γCi,j(diag(z, z, z−2)) =

j2z3j+9−iz3j+9+ijz3j+9+jz3j+9−2z3j+9−j2z3i+3j+6−2jz3i+3j+6+3z3i+3j+6+i−ij+j−1

zi+2j+2(z3−1)2

+ −2j2z3j+6+2iz3j+6−2ijz3j+6−4jz3j+6+6z3j+6+2j2z3i+3j+9+2jz3i+3j+9−4z3i+3j+9−2i+2ij+2j−2

zi+2j+2(z3−1)2

+ j2z3j+3−iz3j+3+ijz3j+3+3jz3j+3−4z3j+3−j2z3i+3j+12+z3i+3j+12+i−ij−3j+3

zi+2j+2(z3−1)2

1

+ j2z3j+7−iz3j+7+ijz3j+7−3jz3j+7+2z3j+7−j2z3i+3j−2+z3i+3j−2+iz4−ijz4+3jz4−3z4

zi+2j(z3−1)2

+ −j2z3j+4−iz3j+4−ijz3j+4+2jz3j+4+3z3j+4+j2z3i+3j−2−z3i+3j−2+iz7+ijz7−2jz7−2z7

zi+2j(z3−1)2

+ 4iz3j+4+4jz3j+4−4z3j+4−4jz3i+3j+1−4iz4+4z4

zi+2j(z3−1)2

2

+ 2j2z3j+1+2iz3j+1+2ijz3j+1−2jz3j+1−4z3j+1−2j2z3i+3j+1+2jz3i+3j+1+4z3i+3j+1−2iz7−2ijz7

zi+2j(z3−1)2

+ −2j2z3j+7+2iz3j+7−2ijz3j+7+2jz3j+7+2j2z3i+3j+1+2jz3i+3j+1−4z3i+3j+1−2iz+2ijz−4jz+4z

zi+2j(z3−1)2

3

+ 3j2z3j+4−5iz3j+4+3ijz3j+4−2jz3j+4−5z3j+4−3j2z3i+3j+4+2jz3i+3j+4+5z3i+3j+4+5iz−3ijz

zi+2j(z3−1)2

+ −j2z3j−2−iz3j−2−ijz3j−2+z3j−2+j2z3i+3j+4−2jz3i+3j+4−3z3i+3j+4+iz7+ijz7+2jz7+2z7

zi+2j(z3−1)2

4

+ −3j2z3j+1−5iz3j+1−3ijz3j+1−5jz3j+1+3j2z3i+3j+4+2jz3i+3j+4−5z3i+3j+4+5iz4+3ijz4+3jz4+5z4

zi+2j(z3−1)2

+ 4iz3j+1+4jz3j+1+8z3j+1−4jz3i+3j+7−4iz−8z

zi+2j(z3−1)2

5

+ 2j2z3j−2+2iz3j−2+2ijz3j−2+4jz3j−2+2z3j−2−2j2z3i+3j+7+2jz3i+3j+7+4z3i+3j+7−2iz4−2ijz4−6jz4−6z4

zi+2j(z3−1)2

+ −j2z3j−2−iz3j−2−ijz3j−2−4jz3j−2−3z3j−2+j2z3i+3j+10−z3i+3j+10+iz+ijz+4jz+4z

zi+2j(z3−1)2.Combining the fra tions in-between the labelled plus signs, we re eive

γCi,j(diag(z, z, z−2))

= − (j−1)z−i+j+1(−3z3i+3+z3i+6−iz3−2z3+i+j(z3−1)(z3i+3−1)+4)z3−1

1

+−4jz3(i+j)−(j2−6j+i(j−3)+1)z3j+3+(j−1)(i+j−2)z3j+6+(i−2)(j+1)z6+(3j−i(j+3)+1)z3

zi+2j−1(z3−1)2

2

+2

“−j2(z6−1)z3j−2z3j+j(−z3j+2z3(i+j)+z3j+6−2)+i((z6+1)(z3j−1)−j(z6−1)(z3j+1))+2

”

zi+2j−1(z3−1)2

3

− (j+1)((j−1)z3j+2(j−1)z3(i+j+2)+(5−3j)z3j+6−2z9)zi+2j+2(z3−1)2

+i(−(j+1)z3j+(3j−5)z3j+6+(j+1)z9+(5−3j)z3)

zi+2j+2(z3−1)2

4

− (3j2+j−8)z3j+(−3j2−2j+5)z3(i+j+1)+4jz3(i+j+2)−(3j+5)z3+i((3j+1)z3j−(3j+5)z3+4)+8

zi+2j−1(z3−1)2

5

+(j+1)((i+j−1)z3j−2(j−2)z3(i+j+3)+(j−1)z3(i+j+4)−2(i+3)z6+(i+4)z3)

zi+2j+2(z3−1)2.

102 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)We sum up the fra tions of same denominator.γCi,j(diag(z, z, z−2))

= − (j−1)z−i+j+1(−3z3i+3+z3i+6−iz3−2z3+i+j(z3−1)(z3i+3−1)+4)z3−1

− j2(−3z3i+3+z6+z3+1)z3j−4z3j+5z3(i+j+1)+z3j+3−2z3j+6+2z6−6z3

zi+2j−1(z3−1)2

+j(3z3j−2z3(i+j+1)+4z3(i+j+2)−6z3j+3+z3j+6+2z6−6z3+4)

zi+2j−1(z3−1)2

+i“−z3j−3z3j+3−z3j+6+z6−2z3+j(z3j+z3j+3+z3j+6+z6−2z3−2)+6

”+4

zi+2j−1(z3−1)2

+(j+1)(−2(j−1)z3(i+j+1)−2(j−2)z3(i+j+2)+(j−1)z3(i+j+3)+(3j−5)z3j+3+2z6−6z3+4)

zi+2j−1(z3−1)2

+i(−5z3j+3+z6−2z3+j(3z3j+3+z6−2z3−2)+6)

zi+2j−1(z3−1)2

= − (j−1)z−i+j+1(−3z3i+3+z3i+6−iz3−2z3+i+j(z3−1)(z3i+3−1)+4)z3−1

+(j−1)z−i+j+1(−3z3i+3+z3i+6−iz3−2z3+i+j(z3−1)(z3i+3−1)+4)

z3−1

= 0 .Therefore, the se ond assertion is shown for all i, j ∈ N. 5.4.4 Orthogonal fun tions on V2The nal step in hara terising the Hilbert spa eH2 lies in the omputation of the fun tionalsorthogonal on V2. By virtue of theorem 5.2.1 and equation (5.15), we transform the fun -tions γClk from the holomorphi into the S hrödinger representation (|kl〉 = χkl = 1√CklχCkl).Consequently, in the S hrödinger representation, γClk reads [ii

(~π)−2 |µk,l〉 = − (k − 1) λ ˜k,l+2 |k, l + 2〉 + (2k − 2) λ ˜k+1,l |k + 1, l〉+ (k + 1) λ ˜k−2,l+3 |k − 2, l + 3〉 − (k − 1) λ ˜k+2,l−2 |k + 2, l − 2〉− 4 λ ˜k−1,l+1 |k − 1, l + 1〉 − (3k − 5) λ ˜k,l−1 |k, l − 1〉− (2k + 2) λ ˜k−3,l+2 |k − 3, l + 2〉 + (2k − 2) λ ˜k+1,l−3 |k + 1, l − 3〉 (5.39)+ (3k + 5) λ ˜k−2,l |k − 2, l〉 − 4 λ ˜k−1,l−2 |k − 1, l − 2〉+ (k + 1) λ ˜k−4,l+1 |k − 4, l + 1〉 − (k − 1) λ ˜k,l−4 |k, l − 4〉− (2k + 2) λ ˜k−3,l−1 |k − 3, l − 1〉 + (k + 1) λ ˜k−2,l−3 |k − 2, l − 3〉[ii as the index tuple of γClk is in inverse order to those of its onstituents χCkl ( f. proposition 5.4.5), weintrodu e a new fun tion µkl instead of ontinuing with γlk here

5.4 THE HILBERT SPACE H2 ASSOCIATED WITH P2 103(sorted by the sum of indi es in des ending order) with k, l ∈ N, λ := e23

~β2 , ˜kl :=(k2 + l2 + kl + 3(k + l + 1)

) (= 32kl from page 85) and

λk,−1 |k,−1〉 = 0 , λ−1,k | − 1, k〉 = 0 ,λk,−2 |k,−2〉 = −λk−1,0 |k − 1, 0〉 , λ−2,k | − 2, k〉 = −λ0,k−1 |0, k − 1〉 ,λk,−3 |k,−3〉 = −λk−2,1 |k − 2, 1〉 , λ−3,k | − 3, k〉 = −λ1,k−2 |1, k − 2〉 andλk,−4 |k,−4〉 = −λk−3,2 |k − 3, 2〉 , λ−4,k | − 4, k〉 = −λ2,k−3 |2, k − 3〉 .Let (qij)ij be a sequen e in C su h that ∑∞

n1,n2=0 qn1n2 exists and setψ :=

∞∑

n1,n2=0

qn1n2 |n1n2〉 . (5.40)If ψ satises〈ψ |µkl 〉 = 0 for all k, l ∈ N, (5.41)it denes an orthogonal fun tion on V2. These fun tions span the Hilbert spa e H2 andwill hen eforth be denoted by ψ2.In the sequel, we determine the oe ients qn1n2 of ψ2 using the ondition formulated in(5.41). We therefore start by analysing formula (5.39). With the given substitution rules and

l ∈ N, there holds(~π)−2 |µ0,l〉 = 0 ( f. remark 5.4.6 (iii)),(~π)−2 |µ1,l〉 = 0 ( f. remark 5.4.6 (iii)),(~π)−2 |µ2,l〉 = − λ ˜2,l+2 |2, l + 2〉 + 2 λ ˜3,l |3, l〉 + 3 λ ˜0,l+3 |0, l + 3〉

− λ ˜4,l−2 |4, l − 2〉 − 4 λ ˜1,l+1 |1, l + 1〉 − λ ˜2,l−1 |2, l − 1〉+ 2 λ ˜3,l−3 |3, l − 3〉 + 11 λ ˜0,l |0, l〉 − 4 λ ˜1,l−2 |1, l − 2〉+ 3 λ ˜−2,l+1 | − 2, l + 1〉 − λ ˜2,l−4 |2, l − 4〉 + 3 λ ˜0,l−3 |0, l − 3〉

= − λ ˜2,l+2 |2, l + 2〉 + 2 λ ˜3,l |3, l〉 + 3 λ ˜0,l+3 |0, l + 3〉− λ ˜4,l−2 |4, l − 2〉 − 4 λ ˜1,l+1 |1, l + 1〉 − λ ˜2,l−1 |2, l − 1〉+ 2 λ ˜3,l−3 |3, l − 3〉 + 8 λ ˜0,l |0, l〉 − 4 λ ˜1,l−2 |1, l − 2〉− λ ˜2,l−4 |2, l − 4〉 + 3 λ ˜0,l−3 |0, l − 3〉 and

(~π)−2 |µ3,l〉 = − 2 λ ˜3,l+2 |3, l + 2〉 + 4 λ ˜4,l |4, l〉 + 4 λ ˜1,l+3 |1, l + 3〉− 2 λ ˜5,l−2 |5, l − 2〉 − 4 λ ˜2,l+1 |2, l + 1〉 − 4 λ ˜3,l−1 |3, l − 1〉− 8 λ ˜0,l+2 |0, l + 2〉 + 4 λ ˜4,l−3 |4, l − 3〉 + 14 λ ˜1,l |1, l〉− 4 λ ˜2,l−2 |2, l − 2〉 − 2 λ ˜3,l−4 |3, l − 4〉 − 8 λ ˜0,l−1 |0, l − 1〉+ 4 λ ˜1,l−3 |1, l − 3〉 .

104 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)Hen e, there are 15 types of non-zero fun tions spanning V2 in the S hrödinger representation,i.e.µ20

µ30 µ21

... µ31 µ22

µk00 ... µ32 µ23

µk11 ... µ33 ...

µk22 ... ... µ2l2

µk33 ... µ3l3

... ...

µklwith k, l, k0, .., k3, l2, l3 ≥ 4.To motivate the next proposition, we expli itly ompute the three fun tions |µ20 〉, |µ21〉 and|µ30 〉 in the sequel. The al ulation of all 15 types of non-zero fun tions is performed inappendix C.

(~π)−2 |µ20〉 = − λ ˜22 |22〉 + 2 λ ˜30 |30〉 + 3 λ ˜03 |03〉 − λ ˜4,−2 |4,−2〉 − 4 λ ˜11 |11〉+ 2 λ ˜3,−3 |3,−3〉 + 8 λ ˜00 |00〉 − 4 λ ˜1,−2 |1,−2〉 − λ ˜2,−4 |2,−4〉+ 3 λ ˜0,−3 |0,−3〉

= − λ ˜22 |22〉 + 3 λ ˜30 |30〉 + 3 λ ˜03 |03〉 − 6 λ ˜11 |11〉 + 15 λ ˜00 |00〉 ,(~π)−2 |µ21〉 = − λ ˜23 |23〉 + 2 λ ˜31 |31〉 + 3 λ ˜04 |04〉 − 4 λ ˜12 |12〉 − λ ˜20 |20〉

+ 2 λ ˜3,−2 |3,−2〉 + 8 λ ˜01 |01〉 − λ ˜2,−3 |2,−3〉 + 3 λ ˜0,−2 |0,−2〉= − λ ˜23 |23〉 + 2 λ ˜31 |31〉 + 3 λ ˜04 |04〉 − 4 λ ˜12 |12〉 − 3 λ ˜20 |20〉

+ λ ˜10 |10〉 + 8 λ ˜01 |01〉 ,(~π)−2 |µ30〉 = − 2 λ ˜32 |32〉 + 4 λ ˜40 |40〉 + 4 λ ˜13 |13〉 − 2 λ ˜5,−2 |5,−2〉

− 4 λ ˜21 |21〉 − 8 λ ˜02 |02〉 + 4 λ ˜4,−3 |4,−3〉 + 14 λ ˜10 |10〉− 4 λ ˜2,−2 |2,−2〉 − 2 λ ˜3,−4 |3,−4〉 + 4 λ ˜1,−3 |1,−3〉

= − 2 λ ˜32 |32〉 + 6 λ ˜40 |40〉 + 4 λ ˜13 |13〉− 8 λ ˜21 |21〉 − 6 λ ˜02 |02〉 + 18 λ ˜10 |10〉 .Using the notation introdu ed in formula (5.40), we are able to ompute the oe ients qn1n2of ψ2 in an iterative way. This pro edure is illustrated in the following example.

5.4 THE HILBERT SPACE H2 ASSOCIATED WITH P2 105Example 5.4.7: The aim of the subsequent al ulations is to have a xed set of arbitrary oe ients qij and then iteratively determine all remaining ones. Therefore we analyse thes alar produ t (5.41) in in reasing order of the sum of indi es k and l with non-vanishingµkl. Thus, we start with 〈ψ2 |µ20 〉 = 0, followed by 〈ψ2 |µ30 〉 = 0 and 〈ψ2 |µ21〉 = 0, then〈ψ2 |µ40〉 = 0, 〈ψ2 |µ31〉 = 0 and 〈ψ2 |µ22 〉 = 0 and so on. Ea h su h pairing yields a unique oe ient qkl, dened by the highest index-sum k + l.Using orthogonality, the rst three relations read

0!= 〈ψ2 |µ20〉 = − λ ˜22q22 + 3 λ ˜30q30 + 3 λ ˜03q03 − 6 λ ˜11q11 + 15 λ ˜00q00 ,

0!= 〈ψ2 |µ30〉 = − λ ˜23q23 + 2 λ ˜31q31 + 3 λ ˜04q04 − 4 λ ˜12q12 − 3 λ ˜20q20 + λ ˜10q10

+ 8 λ ˜01q01 and0

!= 〈ψ2 |µ21〉 = − 2 λ ˜32q32 + 6 λ ˜40q40 + 4 λ ˜13q13 − 8 λ ˜21q21 − 6 λ ˜02q02 + 18 λ ˜10q10 .They transform into

q22 = 3λ ˜30− ˜22q30 + 3λ ˜03− ˜22q03 − 6λ ˜11− ˜22q11 + 15λ ˜00− ˜22q00 ,q32 = 3λ ˜40− ˜32q40 + 2λ ˜13− ˜32q13 − 4λ ˜21− ˜32q21 − 3λ ˜02− ˜32q02 + 9λ ˜10− ˜32q10 ,q23 = 3λ ˜04− ˜23q04 + 2λ ˜31− ˜23q31 − 4λ ˜12− ˜23q12 − 3λ ˜20− ˜23q20 + λ ˜10− ˜23q10

+ 8λ ˜01− ˜23q01with the exponents˜30 − ˜22 = −6 , ˜40 − ˜32 = −6 , ˜04 − ˜23 = −6 ,˜03 − ˜22 = −6 , ˜13 − ˜32 = −9 , ˜31 − ˜23 = −9 ,˜11 − ˜22 = −15 , ˜21 − ˜32 = −18 , ˜12 − ˜23 = −18 ,˜00 − ˜22 = −24 , ˜02 − ˜32 = −24 , ˜20 − ˜23 = −24 ,

˜10 − ˜32 = −30 , ˜10 − ˜23 = −30 ,˜01 − ˜23 = −30 .The in reasing negative exponents motivate the onvergen e if this pro edure is performedin an iterative way. Example 5.4.7 and the 15 types of non-zero fun tions spanning V2 al ulated in appendix Csuggest the subsequent proposition.Proposition 5.4.8: The orthogonal fun tions on V2 are uniquely determined by the fol-lowing pro edure. Set Q := (qi0)i≥0, (qi1)i≥0, (q0j)j≥2, (q1j)j≥2 with omplex entries qi0, qi1,

q0j and q1j. Let Q denote the set of all Q with only a nite number of non-zero members inea h sequen e. For Q ∈ Q dene the oe ients qn1n2 with n1, n2 ≥ 2 as follows:

106 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)(i) Let S(m) dene the set of all tuples (n1, n2) with index sums n1 + n2 = m (level m)satisfying the ondition n1, n2 ≥ 2, e.g. S(4) = (2, 2), S(5) = (3, 2), (2, 3) and soon.(ii) Set m = 4.(iii) For all (n1, n2) ∈ S(m): Use equation ∑(i,j)∈S(n≤m) qij〈ij |µn1,n2−2〉 = 0 and all previ-ously determined qij with i+ j ≤ n1 + n2 to ompute qn1n2.(iv) In rease the value of m by one and ontinue with (iii).There holds∑∞n1,n2=0 | qn1n2 |2 <∞ and the fun tions dened by ψQ2 :=

∑∞n1,n2=0 qn1n2 |n1n2〉are in HL2. Furthermore, spanQ∈QψQ2 ⊆ H2.Remark 5.4.9: Here, we use the notations of the 15 types of non-zero fun tions spanning

V2 on page 104. Denote i := k, i2 := k0, i3 := k1, i4 := k2, i5 := k3 all greater equal 4 andj := l+2, j2 := l2 +2, j3 := l3 +2 all greater equal 6. The pro edure des ribed in proposition5.4.8 is displayed in the following diagram. (Legend: above the solid line: arbitrary hosen oe ients, beneath it: omputed oe ient; above the dotted lines: spe ial ases, beneathit: the general ase)

q00

q10 q01

q20 q11 q02

q30 q21 q12 q03

... q31 q22 q13 q04

q41 q32 q23 q14 q05

... ... q33 q24 q15 ...

qi22 ... q34 q25 q16

qi33 ... q35 ... ...

qi44 ... ... q2j2

qi55 ... q3j3

... ...

qij .

5.4 THE HILBERT SPACE H2 ASSOCIATED WITH P2 107Proof of proposition 5.4.8 : Let i, j ∈ N be given. In the general ase, equation∑

(n1,n2)∈S(n≤m) qn1n2〈n1n2 |µi,j−2〉 = 0 yieldsqi,j = 2λ ˜i+1,j−2− ˜i,jqi+1,j−2 + i+1

i−1λ˜i−2,j+1− ˜i,jqi−2,j+1 − λ ˜i+2,j−4− ˜i,jqi+2,j−4

− 4i−1λ

˜i−1,j−1− ˜i,jqi−1,j−1 − 3i−5i−1 λ

˜i,j−3− ˜i,jqi,j−3 − 2i+2i−1 λ

˜i−3,j− ˜i,jqi−3,j

+ 2λ ˜i+1,j−5− ˜i,jqi+1,j−5 + 3i+5i−1 λ

˜i−2,j−2− ˜i,jqi−2,j−2 − 4i−1λ

˜i−1,j−4− ˜i,jqi−1,j−4

+ i+1i−1λ

˜i−4,j−1− ˜i,jqi−4,j−1 − λ ˜i,j−6− ˜i,jqi,j−6 − 2i+2i−1 λ

˜i−3,j−3− ˜i,jqi−3,j−3

+ i+1i−1λ

˜i−2,j−5− ˜i,jqi−2,j−5 . (5.42)The λ exponents expli itly read

˜i+1,j−2 − ˜i,j = −3j ˜i−2,j−2 − ˜i,j = −6(i+ j)

˜i−2,j+1 − ˜i,j = −3i ˜i−1,j−4 − ˜i,j = 6 − 6i− 9j

˜i+2,j−4 − ˜i,j = 6 − 6j ˜i−4,j−1 − ˜i,j = 6 − 9i− 6j

˜i−1,j−1 − ˜i,j = −3(1 + i+ j) ˜i,j−6 − ˜i,j = −6(−3 + i+ 2j)

˜i,j−3 − ˜i,j = −3(i + 2j) ˜i−3,j−3 − ˜i,j = −9(−1 + i+ j)

˜i−3,j − ˜i,j = −3(2i + j) ˜i−2,j−5 − ˜i,j = −3(−6 + 3i+ 4j)

˜i+1,j−5 − ˜i,j = −3(−3 + i+ 3j)

Sin e all the spe ial ases for negative indi es merely inuen e the oe ients in front ofthe λ∼q∼ expressions and do not add new terms ( ompare the al ulation of the 15 types offun tions spanning V2 in appendix C), we prove all ases by applying adequate estimates tothe general ase.The oe ients whi h dire tly enter qi,j a ording to (5.42) are displayed in the followingdiagram

108 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)

qi−1,j−7 qi−2,j−6 qi−3,j−5 qi−4,j−4 qi−5,j−3 qi−6,j−2 qi−7,j−1

qi−1,j−6 qi−2,j−5 qi−3,j−4 qi−4,j−3 qi−5,j−2 qi−6,j−1

qi,j−6 qi−1,j−5 qi−2,j−4 qi−3,j−3 qi−4,j−2 qi−5,j−1 qi−6,j

qi,j−5 qi−1,j−4 qi−2,j−3 qi−3,j−2 qi−4,j−1 qi−5,j

qi+1,j−5 qi,j−4 qi−1,j−3 qi−2,j−2 qi−3,j−1 qi−4,j qi−5,j+1

qi+1,j−4 qi,j−3 qi−1,j−2 qi−2,j−1 qi−3,j qi−4,j+1

qi+2,j−4 qi+1,j−3 qi,j−2 qi−1,j−1 qi−2,j qi−3,j+1 qi−4,j+2

qi+2,j−3 qi+1,j−2 qi,j−1 qi−1,j qi−2,j+1 qi−3,j+2

qi,jLet m := i+ j denote the iteration level. To obtain an estimate for qij, we need to measurethe number of paths in the latti e of points (a, b) su h that qab ontributes to qij. For laterestimates, we are parti ularly interested in the number of paths onne ting points of thelevel m = 0 with qij. From the diagram above, we see that ea h level m − n, n = 1, .., 6 ontributes with at most two oe ients to qij . Resulting from su essive ontributions,there is a number of indire t inuen es besides the dire t onne tions for ea h n ≥ 2. Thenumber ♯n of paths onne ting points at the level m − n, n = 1, 2, 3, 4 with a xed point(i, j) at m = i + j represents the amount of dire t and indire t ontributions of oe ientsat m− n to qij. There holds

n = 1 ♯1 = 21︸︷︷︸1×1

n = 2 ♯2 = 22︸︷︷︸2×1

+ 21︸︷︷︸1×2

n = 3 ♯3 = 23︸︷︷︸3×1

+ 2 · 22︸ ︷︷ ︸

1×2+1×1

+ 21︸︷︷︸1×3

n = 4 ♯4 = 24︸︷︷︸4×1

+ 3 · 23︸ ︷︷ ︸

1×2+2×1

+ 3 · 22︸ ︷︷ ︸

1×3+1×1 , 2×2

+ 21︸︷︷︸1×4The terms underneath ea h summand denote the number of ertain ba kward-steps made.The rst integer marks the number of ba kward-steps of length of the se ond integer. E.g.1 × 3 denotes that this summand equals the amount of paths using one ba kward-stepof the length three. Sin e there are always two dire t onne tions with a ertain level,

5.4 THE HILBERT SPACE H2 ASSOCIATED WITH P2 109the number of onne tions ontributed by paths with the onguration 1 × 3 equals 21.1 × 2 + 2 × 1 denotes a ombination of paths using one ba kward-step of the length twoand two ba kward-step of the length one. There are again two onne tions of ea h lengthof ba kward-step, whi h results in the number of 23 paths. Sin e the ba kward-steps are ofdierent length, we additionally have to take their order into a ount. Therefore, the totalamount of paths equals 3 · 23.The oe ients in front of the powers of two mat h those of Pas al's triangle. This motivatesthe following general lemma.Lemma 5.4.10: Let i, j ∈ N and m = i+j. Assume that in ea h level m′ < m the are twopoints with dire t onne tions to the point (i, j). The number of paths dire tly and indire tly onne ting arbitrary points at the level m− n (n ∈ N+ and n < m) with (i, j) amounts to♯n =

n−1∑

l=0

(n− 1

l

)2n−l = 2

n−1∑

l=0

(n− 1

l

)2n−1−l = 2 (2 + 1)n−1

= 2 · 3n−1 . (5.43)Proof : We use indu tion to prove the assertion.The base ase was already shown in the examples given above.The indu tive step : Assume that equation (5.43) holds for all integers p ≤ n. There are twodire t onne tions from the xed point qij to the level m−(n+1). Then, there is the numberof onne tions with the level m− 1 multipied by 2 (as there are again two dire t onne tionsfrom the level m− 1 to the level m− (n+1)). ... Finally, we have the number of onne tionswith the level m− n multipied by 2. The sum of all these onne tions equals ♯n+1, hen e,♯n+1 = 2 + (2 · 31) · 2 + (2 · 32) · 2 + ..+ (2 · 3n−2) · 2 + (2 · 3n−1) · 2

= 2 ·(

1 + 2n−1∑

p=0

3p

︸ ︷︷ ︸= 3n−1

)

= 2 · 3n .We used 2(∑n−1

p=0 3p)

= 3(∑n−1

p=0 3p)−(∑n−1

p=0 3p)

= 3 + ..+ 3n − (1 + ..+ 3n−1) = 3n − 1here. Therefore, the assertion is shown for all n ∈ N+, n < m. We now pro eed with the proof of proposition 5.4.8. Lemma 5.4.10 allows us to estimate thenumber of paths onne ting points at level zero with an arbitrary oe ient qij at the levelm = i+ j to be less or equal than 2 · 3m.We analyse the dependen e diagram and the table of λ-exponents of qij again. From thelatter, we learn that with the ex eption of the three dependen es below the dotted line,

110 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)namely qi+2,j−4, qi+1,j−2 and qi−2,j+1, the λ exponents an be estimated by −3m. Weperform a similar estimate for the three ex eptions beneath the dotted line now.Sin e the number of arbitrarily hosen non-zero oe ients qi0, qi1, q0j and q1j in C for i, j ∈N is nite, there exists a minimal highest level m0 su h that all these non-zero oe ientsare on levels m smaller equal m0. Additionally, the three dependen es inspe ted here haveonly a small slope whi h amounts to ±13 if the oordinate system is set to the half-distan ebetween two point horizontally and the distan e of two su essive levels verti ally. Thus,there exists an integer m1 depending on m0 su h that the three dependen es underneath thedotted line an only have non-zero ontributions from paths, whi h itself olle t λ−3m-fa tors;i.e. a hosen path needs to go ba k to the level m0 and therefore has to ross the dottedline at some point. Starting at a level m ≥ m1, all paths have an in reasing number of

λ−3m-fa tors in their iteration steps. Below, we approximate this level-integer m1.Without loss of generality, we analyse the ase where the highest non-zero oe ient's indexis at the left hand position, i.e. qm0,0 or qm0−1,1. The opposite ase is omputed analogously.Set the origin of the oordinate system introdu ed above in the point (m0, 0). The levelm1 is determined by the interse tion point of the two straight lines y1 := −1

3x and y2 :=

−(x− ((m0 − 3) · 2 + 3)

)= −x+ 2m0 − 3. Here, y1 represents the path onsisting only of

qi+2,j−4 and qi+1,j−2 iterations; y2 onsists of the right shift for the width of the interation one m0−3 multiplied by the s ale fa tor 2 and added the left shift of 3 of the point of origin.The des ribed onstru tion is visualised below.m=0

m=1

m=2

m=3

m=4

m=m0

(0,0)

y1 y2

(xs,ys)

m=m1We obtain the interse tion point (xs, ys) =(3m0 − 9

2 ,−m0 + 32

), i.e. any path an haveat most m0 − 1 steps where the λ-fa tor an not be estimated by λ−3m. Thus, m1 :=

(m0−1)+m0 = 2m0−1 denotes the level where there have to be λ−3m-fa tors for some levelm. Therefore, in all paths rea hing oe ients from a level m ≥ m1 the minimum number of

5.4 THE HILBERT SPACE H2 ASSOCIATED WITH P2 111λ−3m-fa tors amounts to (round-up m−m1

7

) sin e the furthest dire t ontribution to qij (i.e.biggest dire t ba kward-step) goes down seven levels, to qi−2,j−5 and the level m0 has to berea hed / rossed.Let m ≥ m1, qm ∈ R denote the highest modulus of the qij- oe ients at the level m =

i + j and let qmax ∈ R denote the highest modulus of the nite number of hosen non-zero oe ients. The highest oe ient in front of the λ∼q∼ term is 15, obtained by therepla ements done for the al ulation of q22. Using ∑m+1k=1 k = 1

2(m+ 1)(m+ 2), we ombineall estimations done toqm ≤ qmax 15m

round-up m−m17∏

k=0

λ−3(k+1)

· 2 · 3m︸ ︷︷ ︸number of paths

= 2 qmax 45mλ−3

“1+2+..+round-up m−m1

7+(round-up m−m1

7+1)

”

= 2 qmax 45mλ−32(round-up m−m1

7+1)(round-up m−m1

7+2)

≤ 2 qmax 45mλ−32(

m−m17

+1)(m−m1

7+2)

≤ 2 qmax 45mλ−398

(m−m1)2

= 2 · 45m1 qmax︸ ︷︷ ︸=: qmax 45m−m1

(λ−

398

(m−m1))m−m1

= qmax(45 λ− 398

(m−m1)︸ ︷︷ ︸(∗) )m−m1 .With λ = e

23

~β2> 1, there exists an m2 ≥ m1 with (∗) < 1 for all m ≥ m2. m2 is given by

m2 := round-up (m1 +98

3logλ 45

) .Let Q ∈ Q. The number of points / oe ients at ea h level m equals m + 1 for m < m0and m− 3 for m ≥ m0. Considering all estimates done, we obtain∞∑

n1,n2=0

| qn1n2 |2 =

∞∑

m=0

∑

i,ji+j=m

| qij |2

≤m1−1∑

m=0

∑

i,ji+j=m

| qij |2 +

m2−1∑

m=m1

∑

i,ji+j=m

| qij |2 +∞∑

m=m2

∑

i,ji+j=m

| qij |2

≤m1−1∑

m=0

(m+ 1) q2m +

m2−1∑

m=m1

(m− 3) q2m +

∞∑

m=m2

(m− 3) q2m

= c+∞∑

m=m2

(m− 3) q2m with a suitable c ∈ R (5.44)

112 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)

≤ c+

∞∑

m=m2

(m− 3) q2max (45 λ− 398

(m−m1))2(m−m1)

≤ c+ q2max ∞∑

m=m2

m(452λ−

398

(m−m1)︸ ︷︷ ︸

=: r

)m−m1

<∞ , as ∑∞m=m2+1m rm−m1 = rm2−m1 (m2+r−m2r)

(r−1)2with r < 1.Therefore, ψQ2 :=

∑∞n1,n2=0 qn1n2 |n1n2〉 with qn1n2 obtained by the given iteration pro edure onverges in HL2. By onstru tion, it is obvious that all ψQ2 with Q ∈ Q solve equation(5.41) and hen e dene fun tions orthogonal on V2. Consequently, spanQ∈QψQ2 ⊆ H2 andproposition 5.4.8 holds. Remark 5.4.11: From an intuitive point of view, we suppose that spanQ∈QψQ2 is a densesubspa e of H2. Then, the onstru tion pro edure given by proposition 5.4.8 hara terisesthe Hilbert spa e asso iated with the stratum P2 as the losure of spanQ∈QψQ2 . However,we have not su eeded in proving this assertion yet.In the following, we present the approa hes whi h have been made and dis uss the prob-lems o urring. From proposition 5.4.8, we already know that spanQ∈QψQ2 ⊆ H2 holds.Contrariwise, let ψ ∈ H2 arbitrary but xed,

ψ =∞∑

i,j=0

qij |ij〉 .As H2 = V⊥2 and V2 is spanned by µkl, there holds

〈ψ |µkl 〉 != 0 for all k, l ∈ N, k ≥ 2. (5.45)Dene Qψ := (qi0)i, (qi1)i, (q0j)j , (q1j)j | qi0, qi1, q0j , q1j ∈ C ∀ i, j ∈ N, j ≥ 2 ∈ Q. Thenumber of non-zero members in ea h sequen e is not ne essarily nite now. For the nite ase, equation (5.45) su essively yields onditions (for in reasing levels) whi h identi allymat h those given by the iteration pro edure of proposition 5.4.8. Thus, ψ ∈ spanQ∈QψQ2 .The innite ase. For some n ∈ N, let Q(n)

ψ ∈ Q denote the oe ients' sequen e-four-tupleobtained by utting o ea h sequen e in Qψ from the level n+1, i.e. Q(n)ψ = (q00, .., qn0, 0, ..),

(q01, .., qn−1,1, 0, ..), (q02, .., q0n, 0, ..), (q12, .., q1,n−1, 0, ..). By virtue of proposition 5.4.8, ea hsu h Q(n)ψ yields a fun tion ψ(n)

2 ∈ spanQ∈QψQ2 ⊆ H2,ψ

(n)2 =:

∞∑

i,j=0

q(n)ij |ij〉 . (5.46)We suppose those fun tions ψ(n)

2 to onverge against ψ2 in a natural way. To show so, wemade the following attempts:

5.4 THE HILBERT SPACE H2 ASSOCIATED WITH P2 113(i) We want to show that ψ(n)2 is a Cau hy sequen e and thus, ψ(n)

2 → ψ2 for n→ ∞. Withoutloss of generality, let n ≥ n′. Therefore,∣∣∣∣∣∣ψ(n)

2 − ψ(n′)2

∣∣∣∣∣∣2

=

∞∑

i,j=0

∣∣∣ q(n)ij − q

(n′)ij

∣∣∣2

=

∞∑

m=0

∑

i,ji+j=m

∣∣∣ q(n)ij − q

(n′)ij

∣∣∣2 .Set q(n,n′)

ij := q(n)ij − q

(n′)ij . Hen e, q(n,n′)

ij = 0 for all (i, j) ∈ S(m), m ≤ n′. At this point, wewant to pro eed with estimates for q(n,n′)ij analogously to the proof of proposition 5.4.8. Buthere it is not possible to determine a xed level m2 from whi h we an apply the previousestimate. The reason for this problem is that the dieren e n − n′ has no upper bound andthus, there is an indenite number of inuential oe ients (qi0)i, (qi1)i, (q0j)j, (q1j)j, with

qi0, qi1, q0j , q1j ∈ C for all i, j ∈ N, i, j ≥ n′ + 1. Attempts to estimate the inuen e of anarbitrary number of these inuential oe ients by the original qijs (whi h onverge in thesense of ∑ | · |2) were not su essful. This is due to the fa t that the parts of qij, j ≤ 1 ori ≤ 1 whi h were ut o (i.e. those of levels m ≥ n′ + 1) an be of positive inuen e on the onvergen e of the hosen ψ. The onstru ted sequen e q(n,n′)

ij is visualised below.m=n

m=n′

0 0 0

6=0 6=0 6=0

0 6=0 0

qij with i≤1 or j≤1

qij with i≥2 or j≤2

(ii) Denoted :=

∞∑

i,j=0

| qij |2 and (5.47a)d(n) :=

∞∑

i,j=0

∣∣∣ q(n)ij

∣∣∣2 . (5.47b)

d and d(n) exist as ψ and ψ(n)2 are both L2. Dene the partial sums ak :=

∑km=0

∑i,j

i+j=m| qij |2and a(n)

k :=∑k

m=0

∑i,j

i+j=m

∣∣∣ q(n)ij

∣∣∣2. Equation (5.47b) an be rewritten to

∀ ε > 0 ∃k1 ∈ N : d ≥ ak > d− ε ∀ k ≥ k1 . (5.48a)We now have to nd a k2 su h that the following inequation holds∀ ε > 0 ∃k2 ∈ N : d(n) ≥ a

(n)k > d(n) − ε ∀n ∈ N ∀ k ≥ k2 . (5.48b)

114 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)If existing, k2 an be determined byk2 := max

n∈Nmink2 ∈ N | d(n) ≥ a

(n)k > d(n) − ε ∀ k ≥ k2

. (5.49)Then, we dene the oe ients q[k]ij byq[k]ij :=

qij if i+ j ≤ k,0 else.Sin e ψ ∈ H2 ⊆ H is L2, the fun tion dened by

ψ[k] :=∞∑

i,j=0

q[k]ij |ij〉is also L2 and thus, ψ[k] ∈ H. We ompute

∑

ij

∣∣∣ qij − q[k]ij

∣∣∣2

=

∞∑

m=k+1

∑

i,ji+j=m

| qij |2 = d− akand∑

ij

∣∣∣ q(k)ij − q[k]ij

∣∣∣2

=∞∑

m=k+1

∑

i,ji+j=m

∣∣∣ q(k)ij

∣∣∣2

= d(k) − a(k)k .Using equations (5.47a), (5.47b), (5.48a) and (5.48b), we on lude

∀ ε > 0 ∃k1 ∈ N : 0 ≤ d− ak < ε ∀ k ≥ k1 and∀ ε > 0 ∃k2 ∈ N : 0 ≤ d(k) − a

(k)k < ε ∀ k ≥ k2 .Thus, ∑ij

∣∣∣ qij − q[k]ij

∣∣∣2 and ∑ij

∣∣∣ q(k)ij − q[k]ij

∣∣∣2 both onverge towards zero and the limits of

∑ij

∣∣∣ q(n)ij

∣∣∣2 and ∑ij

∣∣∣ q[k]ij∣∣∣2 both equal d. Consequently, with the existen e of k2 we wouldknow ψ

(n)2 onverges towards ψ and spanQ∈QψQ2 is a dense subspa e of H2.(iii) Using the linearity in the iteration pro edure given by formula (5.42), we redu e theproblem (i) of ψ(n)

2 being a Cau hy sequen e in the following sense: For some (i1, j1) ∈ N2,i1 ≤ 1 or j1 ≤ 1, denote Q1 ∈ Q the oe ients' sequen e-four-tuple where the element atposition (i1, j1) equals 1 ∈ C and all other elements are zero. At rst, we have to show thatthere exists an estimate for a ψQ1

2 whi h is independent of these indi es i1 and j1 ( ontrary tothe estimates performed equation (5.44)). Let C denote that upper bound, i.e. ∣∣∣∣∣∣ψ(n)2

∣∣∣∣∣∣ ≤ Cfor all n, if existing. Subsequently, for n ≥ n′ and Q(n,n′) dened by q(n,n′)

ij := q(n)ij − q

(n′)ij , wehave ∞∑

i,j=0

∣∣∣ q(n,n′)

ij

∣∣∣2≤ C

∣∣∣∣∣∣Q(n,n′)

∣∣∣∣∣∣2 , (5.50)

5.4 THE HILBERT SPACE H2 ASSOCIATED WITH P2 115with ∣∣∣∣∣∣Q(n,n′)∣∣∣∣∣∣2 given by

∣∣∣∣∣∣Q(n,n′)

∣∣∣∣∣∣2

:=∑

i,jj≤1 or i≤1

∣∣∣ q(n,n′)

ij

∣∣∣2 .Thus, with the existen e of C, equation (5.50) proves that ψ(n)2 is a Cau hy sequen e sin e

ψ is L2.This C ould possibly be determined in the following way: Without loss of generality, let i1be an arbitrary integer and j1 = 0 (the onstru tions for the other three ases are equivalent).Considering formula (5.42) and the dependen e diagram on page 107, we observe that onlythe ir led elements of the following diagram are non-zero (the big point denotes the 1 atposition (i1, 0)). 1st diagonal2nd diagonal3rd diagonal4th diagonalAs we sear h for an estimate whi h is independent of the integer i1, we do not know thewidth of our triangle at the starting level. Therefore we have to sum up an innite numberof oe ients for ea h diagonal (the dashed lines in the diagram above). These diagonals ontain fa tors λ−3(k+l), where k is the element's number in some diagonal l (for ea h di-agonal start ounting with the element on the left). The main problem is to ompute allthe interdependen ies for ea h ir led element and then extra t these fa tors in some esti-mates.

116 5 THE COSTRATIFIED HILBERT SPACE FOR SU(3)

6 OutlookPau a sed matura. Few, but ripe. Carl Friedri h GauÿProposition 5.4.8 determines the Hilbert spa e H2 asso iated with P2 in an impli it way.Even though ψQ2 is uniquely determined by an element Q ∈ Q, the iterative pro edure leavessome aspe ts unanswered.An expli it formula for the oe ients of the fun tions spanning H2 would be very usefulfor a detailed study of the given toy model. It will provide a better handling of H2 and anadditionally yield a dire t approa h to orthogonality in H2.Without an expli it des ription for the oe ients, the question whi h hoi e of Q ∈ Q yieldsmutually orthogonal fun tions an be raised. I.e. how to hoose the nite set qi0, qi1, q0j andq1j to obtain mutually orthogonal fun tions. By answering this question, we will be able todetermine an orthogonal basis of H2.In addition, physi al questions on erning the strata have to be studied. E.g. an investigationof the probability ow under time evolution between the dierent strata will be useful foran in-depth understanding of the role of the singular strata. For that purpose, the timedependent S hrödinger equation has to be solved for the Hamiltonian dened in equation(3.17).Latti e gauge theory relies on the assumption that the limit of an innitely extended (thermo-dynami al limit) and innitely ne ( ontinuum limit) latti e re overs the ontinuum theory.The analysed toy model a single plaquette is still far from its onversion into ontinuumtheory. A heuristi dis ussion of the thermodynami al limit is performed in [KRT97.The rst step towards a ontinuum theory would be to expand the investigations of this thesisto larger latti es.

117

118 6 OUTLOOK

A SU(3) hara ter valuesThe appli ation of Weyl's hara ter formula for SUC(3) (proposition 4.4.2) to elementsparametrised by t = diag(z, z, z−2) with z ∈ C \ 0 hara terising the elements of P2obtained by polar de omposition, ompare se tion 5.4, leads toχCn1n2

(diag(z, z, z−2)) =(n1 + 1)z3n1+3n2+6 − (n1 + n2 + 2)z3n1+3 + n2 + 1

z2n1+n2 (z3 − 1)2.The rst 28 elements of χCn1n2

(diag(z, z, z−2)) are seen in table A.1.In order to get rid of the denominator, we sear h for hara ter ombinations whi h an elout all powers of z in the numerator whi h are smaller than those in the denominator. These ombinations are (3χCj+1,0 − χCj,2)(diag(z, z, z−2)) and (2χCj+1,1 − χCj,3)(diag(z, z, z−2)) (seese tion 5.4 for the proof). Compare tables A.2 and A.3 for the rst 12 values.

119

120 APPENDIX ATab. A.1: 28 elements of the omplex hara ters χCn1n2of SU(3) al ulated in the parametrisation

(diag(z, z, z−2)).(n1, n2) χCn1n2

(diag(z, z, z−2))

(0, 0) 1

(0, 1) z3+2z

(0, 2) z6+2z3+3z2

(0, 3) z9+2z6+3z3+4z3

(0, 4) z12+2z9+3z6+4z3+5z4

(0, 5) z15+2z12+3z9+4z6+5z3+6z5

(1, 0) 2z3+1z2

(1, 1) 2z6+4z3+2z3

(1, 2) 2z9+4z6+6z3+3z4

(1, 3) 2z12+4z9+6z6+8z3+4z5

(1, 4) 2z15+4z12+6z9+8z6+10z3+5z6

(1, 5) 2z18+4z15+6z12+8z9+10z6+12z3+6z7

(2, 0) 3z6+2z3+1z4

(2, 1) 3z9+6z6+4z3+2z5

(2, 2) 3z12+6z9+9z6+6z3+3z6

(2, 3) 3z15+6z12+9z9+12z6+8z3+4z7

(2, 4) 3z18+6z15+9z12+12z9+15z6+10z3+5z8

(2, 5) 3z21+6z18+9z15+12z12+15z9+18z6+12z3+6z9

(3, 0) 4z9+3z6+2z3+1z6

(3, 1) 4z12+8z9+6z6+4z3+2z7

(3, 2) 4z15+8z12+12z9+9z6+6z3+3z8

(3, 3) 4z18+8z15+12z12+16z9+12z6+8z3+4z9

(3, 4) 4z21+8z18+12z15+16z12+20z9+15z6+10z3+5z10

(3, 5) 4z24+8z21+12z18+16z15+20z12+24z9+18z6+12z3+6z11

(4, 0) 5z12+4z9+3z6+2z3+1z8

(4, 1) 5z15+10z12+8z9+6z6+4z3+2z9

(4, 2) 5z18+10z15+15z12+12z9+9z6+6z3+3z10

(4, 3) 5z21+10z18+15z15+20z12+16z9+12z6+8z3+4z11

SU(3) CHARACTER VALUES 121Tab. A.2: The rst 12 hara ter ombinations 3χCj+1,0 − χCj,2.j (3χCj+1,0 − χCj,2)(diag(z, z, z−2))

0 4z − z4

1 5z2 − 2z5

2 6z3 − 3z6

3 7z4 − 4z7

4 8z5 − 5z8

5 9z6 − 6z9

6 10z7 − 7z10

7 11z8 − 8z11

8 12z9 − 9z12

9 13z10 − 10z13

10 14z11 − 11z14

11 15z12 − 12z15Tab. A.3: The rst 12 hara ter ombinations 2χCj+1,1 − χCj,3.j (2χCj+1,1 − χCj,3)(diag(z, z, z−2))

0 −z6 + 2z3 + 5

1 −2z7 + 2z4 + 6z

2 −3z8 + 2z5 + 7z2

3 −4z9 + 2z6 + 8z3

4 −5z10 + 2z7 + 9z4

5 −6z11 + 2z8 + 10z5

6 −7z12 + 2z9 + 11z6

7 −8z13 + 2z10 + 12z7

8 −9z14 + 2z11 + 13z8

9 −10z15 + 2z12 + 14z9

10 −11z16 + 2z13 + 15z10

11 −12z17 + 2z14 + 16z11

122 APPENDIX A

B Proof of the spe ial ases of proposition 5.4.5In the proof of proposition 5.4.5, the assertion (j + 1) χCj,i · χCG = γCi,j+2 − γCi+1,j (equation(5.37)) was already shown for the most general ase i, j ≥ 4. Here, we ompute the remaining24 spe ial ases of produ ts

χC00·χCGχC10·χCG χC01·χCG

χC20·χCG χC11·χCG χC02·χCGχC30·χCG χC21·χCG χC12·χCG χC03·χCG

... χC31·χCG χC22·χCG χC13·χCG ...

χCj00·χCG ... χC32·χCG χC23·χCG ... χC0i0·χCG

χCj11·χCG ... χC33·χCG ... χC1i1·χCG

χCj22·χCG ... ... χC2i2·χCG

χCj33·χCG χC3i3·χCGwith j0, .., j3, i0, .., i3 ≥ 4.Using remark 5.4.6 (iii), we start by proving equation (5.37) for the ten types of produ ts

χC00 · χCG, χC01 · χCG, χC02 · χCG, χC03 · χCG, χC0i0 · χCG and χC10 · χCG, χC11 · χCG, χC12 · χCG, χC13 · χCG,χC1i1 · χCG. There holds

γCi,2 − γCi+1,0 = γCi,2 = 3χC0,i+3 − χC2,i+2 − 4χC1,i+1 + 3χC−2,i+1 + 2χC3,i + 11χC0,i − χC2,i−1

− χC4,i−2 − 4χC1,i−2 + 2χC3,i−3 + 3χC0,i−3 − χC2,i−4

= 3χC0,i+3 − χC2,i+2 − 4χC1,i+1 + 2χC3,i + 8χC0,i − χC2,i−1 − χC4,i−2

− 4χC1,i−2 + 2χC3,i−3 + 3χC0,i−3 − χC2,i−4 and123

124 APPENDIX BγCi,3 − γCi+1,1 = γCi,3 = 4χC1,i+3 − 2χC3,i+2 − 8χC0,i+2 − 4χC2,i+1 + 4χC4,i + 14χC1,i − 4χC3,i−1

− 8χC0,i−1 − 2χC5,i−2 − 4χC2,i−2 + 4χC4,i−3 + 4χC1,i−3 − 2χC3,i−4

= 2(2χC1,i+3 − χC3,i+2 − 4χC0,i+2 − 2χC2,i+1 + 2χC4,i + 7χC1,i − 2χC3,i−1

− 4χC0,i−1 − χC5,i−2 − 2χC2,i−2 + 2χC4,i−3 + 2χC1,i−3 − χC3,i−4

) .Hen e,γC02 − γC10 = 3χC03 − χC22 − 4χC11 + 2χC30 + 8χC00 − χC4,−2

− 4χC1,−2 + 2χC3,−3 + 3χC0,−3 − χC2,−4

= 3χC03 − χC22 − 6χC11 + 3χC30 + 15χC00= χC00 · χCG ≡ χCG ,

γC12 − γC20 = 3χC04 − χC23 − 4χC12 + 2χC31 + 8χC01 − χC20+ 2χC3,−2 + 3χC0,−2 − χC2,−3

= 3χC04 − χC23 − 4χC12 + 2χC31 + 9χC01 − 3χC20= χC01 · χCG ,

γC22 − γC30 = 3χC05 − χC24 − 4χC13 + 2χC32 + 8χC02 − χC21 − χC40− 4χC10 − χC2,−2

= 3χC05 − χC24 − 4χC13 + 2χC32 + 8χC02 − χC21 − χC40 − 3χC10= χC02 · χCG ,

γC32 − γC40 = 3χC06 − χC25 − 4χC14 + 2χC33 + 8χC03 − χC22 − χC41− 4χC11 + 2χC30 + 3χC00

= χC03 · χCG ,γCi0,2 − γCi0+1,0 = 3χC0,i0+3 − χC2,i0+2 − 4χC1,i0+1 + 2χC3,i0 + 8χC0,i0 − χC2,i0−1 − χC4,i0−2

− 4χC1,i0−2 + 2χC3,i0−3 + 3χC0,i0−3 − χC2,i0−4

= χC0,i0 · χCGandγC03 − γC11 = 2

(2χC13 − χC32 − 4χC02 − 2χC21 + 2χC40 + 7χC10

− χC5,−2 − 2χC2,−2 + 2χC4,−3 + 2χC1,−3 − χC3,−4

)

= 2(2χC13 − χC32 − 3χC02 − 4χC21 + 3χC40 + 9χC10)

= 2 χC10 · χCG ,

PROOF OF THE SPECIAL CASES OF PROPOSITION 5.4.5 125γC13 − γC21 = 2

(2χC14 − χC33 − 4χC03 − 2χC22 + 2χC41 + 7χC11 − 2χC30

− 4χC00 + 2χC4,−2 + 2χC1,−2 − χC3,−3

)

= 2(2χC14 − χC33 − 4χC03 − 2χC22 + 2χC41 + 8χC11 − 4χC30 − 6χC00)

= 2 χC11 · χCG ,γC23 − γC31 = 2

(2χC15 − χC34 − 4χC04 − 2χC23 + 2χC42 + 7χC12 − 2χC31

− 4χC01 − χC50 − 2χC20 − χC3,−2

)

= 2(2χC15 − χC34 − 4χC04 − 2χC23 + 2χC42 + 7χC12 − 2χC31 − 4χC01 − χC50 − χC20)

= 2 χC12 · χCG ,γC33 − γC41 = 2

(2χC16 − χC35 − 4χC05 − 2χC24 + 2χC43 + 7χC13

− 2χC32 − 4χC02 − χC51 − 2χC21 + 2χC40 + 2χC10)= 2 χC13 · χCG ,

γCi1,3 − γCi1+1,1 = 2(2χC1,i1+3 − χC3,i1+2 − 4χC0,i1+2 − 2χC2,i1+1 + 2χC4,i1 + 7χC1,i1 − 2χC3,i1−1

− 4χC0,i1−1 − χC5,i1−2 − 2χC2,i1−2 + 2χC4,i1−3 + 2χC1,i1−3 − χC3,i1−4

)

= 2 χC1,i1 · χCG .Using equation (5.38), we now determine the ve ombinations γC0,j+2 − γC1,j , γC1,j+2 − γC2,j,γC2,j+2 − γC3,j , γC3,j+2 − γC4,j and γC4,j+2 − γC5,j .

γC0,j+2 − γC1,j = (j + 1)(−χCj−2,4 + 2χCj,3 + 2χCj−3,3 − χCj+2,2 − 2χCj−1,2 − χCj−4,2

− 2χCj+1,1 − 2χCj−2,1 + 2χCj+3,0 + 6χCj,0 + 2χCj−3,0

− χCj+4,−2 − 2χCj+1,−2 − χCj−2,−2 + 2χCj+3,−3 + 2χCj,−3 − χCj+2,−4

)

= (j + 1)(−χCj−2,4 + 2χCj,3 + 2χCj−3,3 − χCj+2,2 − χCj−1,2 − χCj−4,2

− 4χCj+1,1 − 4χCj−2,1 + 3χCj+3,0 + 8χCj,0 + 3χCj−3,0

) ,γC1,j+2 − γC2,j = (j + 1)

(−χCj−2,5 + 2χCj,4 + 2χCj−3,4 − χCj+2,3 − 2χCj−1,3 − χCj−4,3

− 2χCj+1,2 − 2χCj−2,2 + 2χCj+3,1 + 6χCj,1 + 2χCj−3,1 − 2χCj+2,0 − 2χCj−1,0

+ 2χCj+3,−2 + 2χCj,−2 − χCj+2,−3

)

= (j + 1)(−χCj−2,5 + 2χCj,4 + 2χCj−3,4 − χCj+2,3 − 2χCj−1,3 − χCj−4,3

− 2χCj+1,2 − 2χCj−2,2 + 2χCj+3,1 + 7χCj,1 + 2χCj−3,1 − 4χCj+2,0 − 4χCj−1,0

) ,

126 APPENDIX BγC2,j+2 − γC3,j = (j + 1)

(−χCj−2,6 + 2χCj,5 + 2χCj−3,5 − χCj+2,4 − 2χCj−1,4 − χCj−4,4

− 2χCj+1,3 − 2χCj−2,3 + 2χCj+3,2 + 6χCj,2 + 2χCj−3,2 − 2χCj+2,1 − 2χCj−1,1

− χCj+4,0 − 2χCj+1,0 − χCj−2,0 − χCj+2,−2

)

= (j + 1)(−χCj−2,6 + 2χCj,5 + 2χCj−3,5 − χCj+2,4 − 2χCj−1,4 − χCj−4,4

− 2χCj+1,3 − 2χCj−2,3 + 2χCj+3,2 + 6χCj,2 + 2χCj−3,2 − 2χCj+2,1 − 2χCj−1,1

− χCj+4,0 − χCj+1,0 − χCj−2,0

) ,γC3,j+2 − γC4,j = (j + 1)

(−χCj−2,7 + 2χCj,6 + 2χCj−3,6 − χCj+2,5 − 2χCj−1,5 − χCj−4,5

− 2χCj+1,4 − 2χCj−2,4 + 2χCj+3,3 + 6χCj,3 + 2χCj−3,3 − 2χCj+2,2 − 2χCj−1,2

− χCj+4,1 − 2χCj+1,1 − χCj−2,1 + 2χCj+3,0 + 2χCj,0) .Therefore, with j0, .., j3 ≥ 4, we ompute the twelve asesγC04 − γC12 = 3

(−χC04 + 2χC23 − χC42 − χC12 − χC−2,2 − 4χC31 − 4χC01 + 3χC50 + 8χC20)

= 3(−χC04 + 2χC23 − χC42 − χC12 − 4χC31 − 3χC01 + 3χC50 + 8χC20)

= 3 χC20 · χCG ,γC05 − γC13 = 4

(−χC14 + 2χC33 + 2χC03 − χC52 − χC22 − 4χC41 − 4χC11 + 3χC60 + 8χC30 + 3χC00)

= 4 χC30 · χCG ,γC0,j0+2 − γC1,j0 = (j0 + 1)

(−χCj0−2,4 + 2χCj0,3 + 2χCj0−3,3 − χCj0+2,2 − χCj0−1,2 − χCj0−4,2

− 4χCj0+1,1 − 4χCj0−2,1 + 3χCj0+3,0 + 8χCj0,0 + 3χCj0−3,0

)

= (j0 + 1) χCj0,0 · χCG ,γC14 − γC22 = 3

(−χC05 + 2χC24 − χC43 − 2χC13 − χC−2,3

− 2χC32 − 2χC02 + 2χC51 + 7χC21 − 4χC40 − 4χC10)= 3

(−χC05 + 2χC24 − χC43 − 2χC13 − 2χC32 − χC02 + 2χC51 + 7χC21 − 4χC40 − 4χC10)

= 3 χC21 · χCG ,γC15 − γC23 = 4

(−χC15 + 2χC34 + 2χC04 − χC53 − 2χC23

− 2χC42 − 2χC12 + 2χC61 + 7χC31 + 2χC01 − 4χC50 − 4χC20)= 4 χC31 · χCG ,

PROOF OF THE SPECIAL CASES OF PROPOSITION 5.4.5 127γC1,j1+2 − γC2,j1 = (j1 + 1)

(−χCj1−2,5 + 2χCj1,4 + 2χCj1−3,4 − χCj1+2,3 − 2χCj1−1,3 − χCj1−4,3

− 2χCj1+1,2 − 2χCj1−2,2 + 2χCj1+3,1 + 7χCj1,1 + 2χCj1−3,1 − 4χCj1+2,0 − 4χCj1−1,0

)

= (j1 + 1) χCj1,1 · χCG ,γC24 − γC32 = 3

(−χC06 + 2χC25 − χC44 − 2χC14 − χC−2,4 − 2χC33 − 2χC03 + 2χC52

+ 6χC22 − 2χC41 − 2χC11 − χC60 − χC30 − χC00)= 3

(−χC06 + 2χC25 − χC44 − 2χC14 − 2χC33 − χC03 + 2χC52

+ 6χC22 − 2χC41 − 2χC11 − χC60 − χC30 − χC00)= 3 χC22 · χCG ,

γC25 − γC33 = 4(−χC16 + 2χC35 + 2χC05 − χC54 − 2χC24 − 2χC43 − 2χC13 + 2χC62

+ 6χC32 + 2χC02 − 2χC51 − 2χC21 − χC70 − χC40 − χC10)= 4 χC32 · χCG ,

γC2,j2+2 − γC3,j2 = (j2 + 1)(−χCj2−2,6 + 2χCj2,5 + 2χCj2−3,5 − χCj2+2,4 − 2χCj2−1,4 − χCj2−4,4

− 2χCj2+1,3 − 2χCj2−2,3 + 2χCj2+3,2 + 6χCj2,2 + 2χCj2−3,2 − 2χCj2+2,1 − 2χCj2−1,1

− χCj2+4,0 − χCj2+1,0 − χCj2−2,0

)

= (j2 + 1) χCj2,2 · χCGandγC34 − γC42 = 3

(−χC07 + 2χC26 − χC45 − 2χC15 − χC−2,5 − 2χC34 − 2χC04 + 2χC53 + 6χC23

− 2χC42 − 2χC12 − χC61 − 2χC31 − χC01 + 2χC50 + 2χC20)= 3

(−χC07 + 2χC26 − χC45 − 2χC15 − 2χC34 − χC04 + 2χC53 + 6χC23

− 2χC42 − 2χC12 − χC61 − 2χC31 − χC01 + 2χC50 + 2χC20)= 3 χC23 · χCG ,

γC35 − γC43 = 4(−χC17 + 2χC36 + 2χC06 − χC55 − 2χC25 − 2χC44 − 2χC14 + 2χC63 + 6χC33

+ 2χC03 − 2χC52 − 2χC22 − χC71 − 2χC41 − χC11 + 2χC60 + 2χC30)= 4 χC33 · χCG ,

128 APPENDIX BγC3,j3+2 − γC4,j3 = (j3 + 1)

(−χCj3−2,7 + 2χCj3,6 + 2χCj3−3,6 − χCj3+2,5 − 2χCj3−1,5 − χCj3−4,5

− 2χCj3+1,4 − 2χCj3−2,4 + 2χCj3+3,3 + 6χCj3,3 + 2χCj3−3,3 − 2χCj3+2,2 − 2χCj3−1,2

− χCj3+4,1 − 2χCj3+1,1 − χCj3−2,1 + 2χCj3+3,0 + 2χCj3,0)= (j3 + 1) χCj3,3 · χCG .We now he k the remaining two types of produ ts χC2,i2 · χCG and χC3,i3 · χCG with i2, i3 ≥ 4.Using equation (5.38), there holds

γCi2,4 − γCi2+1,2 = 3(−χC0,i2+4 + 2χC2,i2+3 − χC4,i2+2 − 2χC1,i2+2 − χC−2,i2+2

− 2χC3,i2+1 − 2χC0,i2+1 + 2χC5,i2 + 6χC2,i2 − 2χC4,i2−1 − 2χC1,i2−1

− χC6,i2−2 − 2χC3,i2−2 − χC0,i2−2 + 2χC5,i2−3 + 2χC2,i2−3 − χC4,i2−4

)

= 3(−χC0,i2+4 + 2χC2,i2+3 − χC4,i2+2 − 2χC1,i2+2

− 2χC3,i2+1 − χC0,i2+1 + 2χC5,i2 + 6χC2,i2 − 2χC4,i2−1 − 2χC1,i2−1

− χC6,i2−2 − 2χC3,i2−2 − χC0,i2−2 + 2χC5,i2−3 + 2χC2,i2−3 − χC4,i2−4

)

= 3 χC2,i2 · χCG andγCi3,5 − γCi3+1,3 = 4

(−χC1,i3+4 + 2χC3,i3+3 + 2χC0,i3+3 − χC5,i3+2 − 2χC2,i3+2

− 2χC4,i3+1 − 2χC1,i3+1 + 2χC6,i3 + 6χC3,i3 + 2χC0,i3 − 2χC5,i3−1 − 2χC2,i3−1

− χC7,i3−2 − 2χC4,i3−2 − χC1,i3−2 + 2χC6,i3−3 + 2χC3,i3−3 − χC5,i3−4

)

= 4 χC3,i3 · χCG .Hen e, all 24 spe ial ases hold and proposition 5.4.5 is shown.

C Expli it al ulation of the 15 types of fun tionsspanning V2Let k, l, k0, .., k3, l2, l3 ≥ 4. The 15 types of non-zero fun tions spanning V2 in the S hrödingerrepresentation areµ20

µ30 µ21

... µ31 µ22

µk00 ... µ32 µ23

µk11 ... µ33 ...

µk22 ... ... µ2l2

µk33 ... µ3l3

... ...

µkl .For the subsequent al ulations, we use equation (5.39). The elements of the rst diagonalrow expli itly read(~π)−2 |µ20 〉 = − λ ˜22 |22〉 + 2 λ ˜30 |30〉 + 3 λ ˜03 |03〉 − λ ˜4,−2 |4,−2〉 − 4 λ ˜11 |11〉

+ 2 λ ˜3,−3 |3,−3〉 + 8 λ ˜00 |00〉 − 4 λ ˜1,−2 |1,−2〉 − λ ˜2,−4 |2,−4〉+ 3 λ ˜0,−3 |0,−3〉

= − λ ˜22 |22〉 + 3 λ ˜30 |30〉 + 3 λ ˜03 |03〉 − 6 λ ˜11 |11〉 + 15 λ ˜00 |00〉 ,(~π)−2 |µ21 〉 = − λ ˜23 |23〉 + 2 λ ˜31 |31〉 + 3 λ ˜04 |04〉 − 4 λ ˜12 |12〉 − λ ˜20 |20〉

+ 2 λ ˜3,−2 |3,−2〉 + 8 λ ˜01 |01〉 − λ ˜2,−3 |2,−3〉 + 3 λ ˜0,−2 |0,−2〉129

130 APPENDIX C= − λ ˜23 |23〉 + 2 λ ˜31 |31〉 + 3 λ ˜04 |04〉 − 4 λ ˜12 |12〉 − 3 λ ˜20 |20〉

+ λ ˜10 |10〉 + 8 λ ˜01 |01〉 ,(~π)−2 |µ22〉 = − λ ˜24 |24〉 + 2 λ ˜32 |32〉 + 3 λ ˜05 |05〉 − λ ˜40 |40〉 − 4 λ ˜13 |13〉

− λ ˜21 |21〉 + 8 λ ˜02 |02〉 − 4 λ ˜10 |10〉 − λ ˜2,−2 |2,−2〉= − λ ˜24 |24〉 + 2 λ ˜32 |32〉 + 3 λ ˜05 |05〉 − λ ˜40 |40〉 − 4 λ ˜13 |13〉

− λ ˜21 |21〉 + 8 λ ˜02 |02〉 − 3 λ ˜10 |10〉 ,(~π)−2 |µ23〉 = − λ ˜25 |25〉 + 2 λ ˜33 |33〉 + 3 λ ˜06 |06〉 − λ ˜41 |41〉 − 4 λ ˜14 |14〉

− λ ˜22 |22〉 + 2 λ ˜30 |30〉 + 8 λ ˜03 |03〉 − 4 λ ˜11 |11〉 + 3 λ ˜00 |00〉 ,(~π)−2 |µ2,l2 〉 = − λ ˜2,l2+2 |2, l2 + 2〉 + 2 λ ˜3,l2 |3, l2 〉 + 3 λ ˜0,l2+3 |0, l2 + 3〉

− λ ˜4,l2−2 |4, l2 − 2〉 − 4 λ ˜1,l2+1 |1, l2 + 1〉 − λ ˜2,l2−1 |2, l2 − 1〉+ 2 λ ˜3,l2−3 |3, l2 − 3〉 + 8 λ ˜0,l2 |0, l2 〉 − 4 λ ˜1,l2−2 |1, l2 − 2〉− λ ˜2,l2−4 |2, l2 − 4〉 + 3 λ ˜0,l2−3 |0, l2 − 3〉The elements of the se ond diagonal row are given by

(~π)−2 |µ30〉 = − 2 λ ˜32 |32〉 + 4 λ ˜40 |40〉 + 4 λ ˜13 |13〉 − 2 λ ˜5,−2 |5,−2〉− 4 λ ˜21 |21〉 − 8 λ ˜02 |02〉 + 4 λ ˜4,−3 |4,−3〉 + 14 λ ˜10 |10〉− 4 λ ˜2,−2 |2,−2〉 − 2 λ ˜3,−4 |3,−4〉 + 4 λ ˜1,−3 |1,−3〉

= − 2 λ ˜32 |32〉 + 6 λ ˜40 |40〉 + 4 λ ˜13 |13〉− 8 λ ˜21 |21〉 − 6 λ ˜02 |02〉 + 18 λ ˜10 |10〉 ,

(~π)−2 |µ31〉 = − 2 λ ˜33 |33〉 + 4 λ ˜41 |41〉 + 4 λ ˜14 |14〉 − 4 λ ˜22 |22〉− 4 λ ˜30 |30〉 − 8 λ ˜03 |03〉 + 4 λ ˜4,−2 |4,−2〉 + 14 λ ˜11 |11〉− 2 λ ˜3,−3 |3,−3〉 − 8 λ ˜00 |00〉 + 4 λ ˜1,−2 |1,−2〉

= − 2 λ ˜33 |33〉 + 4 λ ˜41 |41〉 + 4 λ ˜14 |14〉 − 4 λ ˜22 |22〉− 8 λ ˜30 |30〉 − 8 λ ˜03 |03〉 + 16 λ ˜11 |11〉 − 12 λ ˜00 |00〉 ,

(~π)−2 |µ32〉 = − 2 λ ˜34 |34〉 + 4 λ ˜42 |42〉 + 4 λ ˜15 |15〉 − 2 λ ˜50 |50〉− 4 λ ˜23 |23〉 − 4 λ ˜31 |31〉 − 8 λ ˜04 |04〉 + 14 λ ˜12 |12〉− 4 λ ˜20 |20〉 − 2 λ ˜3,−2 |3,−2〉 − 8 λ ˜01 |01〉

= − 2 λ ˜34 |34〉 + 4 λ ˜42 |42〉 + 4 λ ˜15 |15〉 − 2 λ ˜50 |50〉− 4 λ ˜23 |23〉 − 4 λ ˜31 |31〉 − 8 λ ˜04 |04〉 + 14 λ ˜12 |12〉− 2 λ ˜20 |20〉 − 8 λ ˜01 |01〉 ,

EXPLICIT CALCULATION OF THE 15 TYPES OF FUNCTIONS SPANNING V2 131(~π)−2 |µ33 〉 = − 2 λ ˜35 |35〉 + 4 λ ˜43 |43〉 + 4 λ ˜16 |16〉 − 2 λ ˜51 |51〉

− 4 λ ˜24 |24〉 − 4 λ ˜32 |32〉 − 8 λ ˜05 |05〉 + 4 λ ˜40 |40〉+ 14 λ ˜13 |13〉 − 4 λ ˜21 |21〉 − 8 λ ˜02 |02〉 + 4 λ ˜10 |10〉 ,

(~π)−2 |µ3,l3 〉 = − 2 λ ˜3,l3+2 |3, l3 + 2〉 + 4 λ ˜4,l3 |4, l3 〉 + 4 λ ˜1,l3+3 |1, l3 + 3〉− 2 λ ˜5,l3−2 |5, l3 − 2〉 − 4 λ ˜2,l3+1 |2, l3 + 1〉 − 4 λ ˜3,l3−1 |3, l3 − 1〉− 8 λ ˜0,l3+2 |0, l3 + 2〉 + 4 λ ˜4,l3−3 |4, l3 − 3〉 + 14 λ ˜1,l3 |1, l3 〉− 4 λ ˜2,l3−2 |2, l3 − 2〉 − 2 λ ˜3,l3−4 |3, l3 − 4〉 − 8 λ ˜0,l3−1 |0, l3 − 1〉+ 4 λ ˜1,l3−3 |1, l3 − 3〉 .The elements of the remaining diagonal row read

(~π)−2 |µk0,0〉 = − (k0 − 1) λ ˜k0,2 |k0, 2〉 + (2k0 − 2) λ ˜k0+1,0 |k0 + 1, 0〉+ (k0 + 1) λ ˜k0−2,l+3 |k0 − 2, 3〉 − (k0 − 1) λ ˜k0+2,−2 |k0 + 2,−2〉− 4 λ ˜k0−1,1 |k0 − 1, 1〉− (2k0 + 2) λ ˜k0−3,2 |k0 − 3, 2〉 + (2k0 − 2) λ ˜k0+1,−3 |k0 + 1,−3〉+ (3k0 + 5) λ ˜k0−2,0 |k0 − 2, 0〉 − 4 λ ˜k0−1,−2 |k0 − 1,−2〉+ (k0 + 1) λ ˜k0−4,1 |k0 − 4, 1〉 − (k0 − 1) λ ˜k0,−4 |k0,−4〉+ (k0 + 1) λ ˜k0−2,−3 |k0 − 2,−3〉

= − (k0 − 1) λ ˜k0,2 |k0, 2〉 + (3k0 − 3) λ ˜k0+1,0 |k0 + 1, 0〉+ (k0 + 1) λ ˜k0−2,3 |k0 − 2, 3〉 − (2k0 + 2) λ ˜k0−1,1 |k0 − 1, 1〉− (k0 + 3) λ ˜k0−3,2 |k0 − 3, 2〉 + (3k0 + 9) λ ˜k0−2,0 |k0 − 2, 0〉 ,

(~π)−2 |µk1,1〉 = − (k1 − 1) λ ˜k1,3 |k1, 3〉 + (2k1 − 2) λ ˜k1+1,1 |k1 + 1, 1〉+ (k1 + 1) λ ˜k1−2,4 |k1 − 2, 4〉− 4 λ ˜k1−1,2 |k1 − 1, 2〉 − (3k1 − 5) λ ˜k1,0 |k1, 0〉− (2k1 + 2) λ ˜k1−3,3 |k1 − 3, 3〉 + (2k1 − 2) λ ˜k1+1,−2 |k1 + 1,−2〉+ (3k1 + 5) λ ˜k1−2,1 |k1 − 2, 1〉+ (k1 + 1) λ ˜k1−4,2 |k1 − 4, 2〉 − (k1 − 1) λ ˜k1,−3 |k1,−3〉− (2k1 + 2) λ ˜k1−3,0 |k1 − 3, 0〉 + (k1 + 1) λ ˜k1−2,−2 |k1 − 2,−2〉

= − (k1 − 1) λ ˜k1,3 |k1, 3〉 + (2k1 − 2) λ ˜k1+1,1 |k1 + 1, 1〉+ (k1 + 1) λ ˜k1−2,4 |k1 − 2, 4〉 − 4 λ ˜k1−1,2 |k1 − 1, 2〉− (5k1 − 7) λ ˜k1,0 |k1, 0〉 − (2k1 + 2) λ ˜k1−3,3 |k1 − 3, 3〉+ (4k1 + 4) λ ˜k1−2,1 |k1 − 2, 1〉 + (k1 + 1) λ ˜k1−4,2 |k1 − 4, 2〉− (3k1 + 3) λ ˜k1−3,0 |k1 − 3, 0〉 ,

132 APPENDIX C(~π)−2 |µk2,2〉 = − (k2 − 1) λ ˜k2,4 |k2, 4〉 + (2k2 − 2) λ ˜k2+1,2 |k2 + 1, 2〉

+ (k2 + 1) λ ˜k2−2,5 |k2 − 2, 5〉 − (k2 − 1) λ ˜k2+2,0 |k2 + 2, 0〉− 4 λ ˜k2−1,3 |k2 − 1, 3〉 − (3k2 − 5) λ ˜k2,1 |k2, 1〉− (2k2 + 2) λ ˜k2−3,4 |k2 − 3, 4〉+ (3k2 + 5) λ ˜k2−2,2 |k2 − 2, 2〉 − 4 λ ˜k2−1,0 |k2 − 1, 0〉+ (k2 + 1) λ ˜k2−4,3 |k2 − 4, 3〉 − (k2 − 1) λ ˜k2,−2 |k2,−2〉− (2k2 + 2) λ ˜k2−3,1 |k2 − 3, 1〉

= − (k2 − 1) λ ˜k2,4 |k2, 4〉 + (2k2 − 2) λ ˜k2+1,2 |k2 + 1, 2〉+ (k2 + 1) λ ˜k2−2,5 |k2 − 2, 5〉 − (k2 − 1) λ ˜k2+2,0 |k2 + 2, 0〉− 4 λ ˜k2−1,3 |k2 − 1, 3〉 − (3k2 − 5) λ ˜k2,1 |k2, 1〉− (2k2 + 2) λ ˜k2−3,4 |k2 − 3, 4〉 + (3k2 + 5) λ ˜k2−2,2 |k2 − 2, 2〉+ (k2 − 5) λ ˜k2−1,0 |k2 − 1, 0〉 + (k2 + 1) λ ˜k2−4,3 |k2 − 4, 3〉− (2k2 + 2) λ ˜k2−3,1 |k2 − 3, 1〉 and

(~π)−2 |µk3,3〉 = − (k3 − 1) λ ˜k3,5 |k3, 5〉 + (2k3 − 2) λ ˜k3+1,3 |k3 + 1, 3〉+ (k3 + 1) λ ˜k3−2,6 |k3 − 2, 6〉 − (k3 − 1) λ ˜k3+2,1 |k3 + 2, 1〉− 4 λ ˜k3−1,4 |k3 − 1, 4〉 − (3k3 − 5) λ ˜k3,2 |k3, 2〉− (2k3 + 2) λ ˜k3−3,5 |k3 − 3, 5〉 + (2k3 − 2) λ ˜k3+1,0 |k3 + 1, 0〉+ (3k3 + 5) λ ˜k3−2,3 |k3 − 2, 3〉 − 4 λ ˜k3−1,1 |k3 − 1, 1〉+ (k3 + 1) λ ˜k3−4,4 |k3 − 4, 4〉− (2k3 + 2) λ ˜k3−3,2 |k3 − 3, 2〉 + (k3 + 1) λ ˜k3−2,0 |k3 − 2, 0〉 .Finally, the general ase is given by

(~π)−2 |µk,l〉 = − (k − 1) λ ˜k,l+2 |k, l + 2〉 + (2k − 2) λ ˜k+1,l |k + 1, l〉+ (k + 1) λ ˜k−2,l+3 |k − 2, l + 3〉 − (k − 1) λ ˜k+2,l−2 |k + 2, l − 2〉− 4 λ ˜k−1,l+1 |k − 1, l + 1〉 − (3k − 5) λ ˜k,l−1 |k, l − 1〉− (2k + 2) λ ˜k−3,l+2 |k − 3, l + 2〉 + (2k − 2) λ ˜k+1,l−3 |k + 1, l − 3〉+ (3k + 5) λ ˜k−2,l |k − 2, l〉 − 4 λ ˜k−1,l−2 |k − 1, l − 2〉+ (k + 1) λ ˜k−4,l+1 |k − 4, l + 1〉 − (k − 1) λ ˜k,l−4 |k, l − 4〉− (2k + 2) λ ˜k−3,l−1 |k − 3, l − 1〉 + (k + 1) λ ˜k−2,l−3 |k − 2, l − 3〉 .

D Mathemati a 6.0 ode(**********************************************************************************)(* tunnelling probability between H10, H11 and H12 *********************************)(**********************************************************************************)TO = 60; (* summation limit, TO=60 re ommended *)g[n1_, n2_ := 2/3*(n1^2 + n2^2 + n1*n2 + 3 (n1 + n2 + 1))d[n1_, n2_ := (n1 + 1) (n2 + 1) (1/2*(n1 + n2) + 1)S[y_, n1_, n2_, m_ :=Exp[-y*g[n1, n2 - I*2 m/3 Pi (n1 + 2 n2) d[n1, n2 (*the parameter y represents \[HBar\[Beta *)SC[y_, n1_, n2_, m_ :=Exp[-y*g[n1, n2 + I*2 m/3 Pi (n1 + 2 n2) d[n1, n2Nm[y_ :=Sum[Exp[-2 y*g[n1, n2*d[n1, n2^2, n1, 0, TO, n2, 0, TOS12[y_ :=1/Nm[y*Sum[SC[y, n1, n2, 1*S[y, n1, n2, 2, n1, 0, TO, n2, 0, TOPlot[Abs[S12[x^2, 1, x, 0, 2.5, PlotRange -> -0.04, 1.04,Aspe tRatio -> 1/GoldenRatio, Axes -> True,AxesLabel -> "G1x", "G1y", AxesStyle -> Dire tive[Bla k, 12,PlotStyle -> Dire tive[Bla k, Thi kness[0.003,Dire tive[Bla k, Thin, Dashed,Ti ks -> 0, 0.5, 1, 1.5, 2, 2.5, 0, 0.2, 0.4, 0.6,0.8, 1, "1.0"S13[y_ :=1/Nm[y*Sum[SC[y, n1, n2, 1*S[y, n1, n2, 3, n1, 0, TO, n2, 0, TOPlot[Abs[S13[x^2, 1, x, 0, 2.5, PlotRange -> -0.04, 1.04,Aspe tRatio -> 1/GoldenRatio, Axes -> True,AxesLabel -> "G1x", "G1y", AxesStyle -> Dire tive[Bla k, 12,PlotStyle -> Dire tive[Bla k, Thi kness[0.003,Dire tive[Bla k, Thin, Dashed,Ti ks -> 0, 0.5, 1, 1.5, 2, 2.5, 0, 0.2, 0.4, 0.6,0.8, 1, "1.0"S23[y_ := 133

134 APPENDIX D1/Nm[y*Sum[SC[y, n1, n2, 2*S[y, n1, n2, 3, n1, 0, TO, n2, 0, TOPlot[Abs[S23[x^2, 1, x, 0, 2.5, PlotRange -> -0.04, 1.04,Aspe tRatio -> 1/GoldenRatio, Axes -> True,AxesLabel -> "G1x", "G1y", AxesStyle -> Dire tive[Bla k, 12,PlotStyle -> Dire tive[Bla k, Thi kness[0.003,Dire tive[Bla k, Thin, Dashed,Ti ks -> 0, 0.5, 1, 1.5, 2, 2.5, 0, 0.2, 0.4, 0.6,0.8, 1, "1.0"(**********************************************************************************)(* wave fun tions spanning the 3 strata H10, H11 and H12 **************************)(**********************************************************************************)TO = 100; (* summation limit *) hi[z1_, z2_, n1_,n2_ := ((z1 - z1^(-1))*(z2^3 + z2^(-3) - z1 -z1^(-1)))^(-1)*(z2^(n1 + 2 n2 + 3) (z1^(n1 + 1) -z1^(-n1 - 1)) +z2^(-2 n1 - n2 - 3) (z1^(n2 + 1) - z1^(-n2 - 1)) -z2^(n1 - n2) (z1^(n1 + n2 + 2) - z1^(-n1 - n2 - 2)))S [x1_, x2_, y_, m_, n1_, n2_ :=Exp[-y*(2/3*(n1^2 + n2^2 + n1*n2 + 3 (n1 + n2 + 1))) -I*2/3*m*Pi*(n1 + 2 n2)*(n1 + 1) (n2 + 1) (1/2*(n1 + n2) + 1)* hi[Exp[-I/2 (x1 - x2), Exp[-I/2 (x1 + x2), n1, n2 (*the parameter y represents \[HBar\[Beta *)Nn [y_, to_ :=Sqrt[Sum[Exp[-2*y*(2/3*(n1^2 + n2^2 + n1*n2 + 3 (n1 + n2 + 1)))*((n1 +1) (n2 + 1) (1/2*(n1 + n2) + 1))^2, n1, 0, to, n2, 0,toN1 = N[Nn [1, 800,1000; (* square root of the normalisation fa tor for \[HBar\[Beta=1 *)N4 = N[Nn [1/4, 800,1000; (* square root of the normalisation fa tor for \[HBar\[Beta=1/4 *)N16 = N[Nn [1/16, 800,1000; (* square root of the normalisation fa tor for \[HBar\[Beta=1/16 *)(* to=300 results in 200 signifi ant digits for Nn , when to=500 there are 600 signifi ant digits *)(* the fun tions spanning the 3 strata; these fun tions are a omposition of \parameters / fun tions defined above *)S P0[x1_, x2_, y_, N_, to_ :=1/N*Sum[S [x1, x2, y, 0, n1, n2, n1, 0, to, n2, 0, toS P1[x1_, x2_, y_, N_, to_ :=

MATHEMATICA 6.0 CODE 1351/N*Sum[S [x1, x2, y, 1, n1, n2, n1, 0, to, n2, 0, toS P2[x1_, x2_, y_, N_, to_ :=1/N*Sum[S [x1, x2, y, 2, n1, n2, n1, 0, to, n2, 0, to(* sour e ode of the singel 3D plots, TO=140 re ommended *)Plot3D[Abs[S P0[x1, x2, 1, N1, TO*Sin[(x1 - x2)/2*Sin[(x1 + 2 x2)/2*Sin[(2 x1 + x2)/2, x1, -Pi, Pi, x2, -Pi, Pi,PlotRange -> 0, 2.8, Axes -> True, True, False,AxesLabel -> "G2x", "G2y",Ti ks -> -Pi, -Pi/2, 0, Pi/2, Pi, -Pi, -Pi/2, 0, Pi/2, Pi,Boxed -> False, MaxRe ursion -> 4,Mesh -> -Pi, -3 Pi/4, -Pi/2, -Pi/4, 0, Pi/4, Pi/2, 3 Pi/4,Pi, -Pi, -3 Pi/4, -Pi/2, -Pi/4, 0, Pi/4, Pi/2, 3 Pi/4, Pi,MeshStyle -> Thi kness[0.0001Plot3D[Abs[S P0[x1, x2, 1/4, N4, TO*Sin[(x1 - x2)/2*Sin[(x1 + 2 x2)/2*Sin[(2 x1 + x2)/2, x1, -Pi, Pi, x2, -Pi, Pi,PlotRange -> 0, 2.8, Axes -> True, True, False,AxesLabel -> "G2x", "G2y",Ti ks -> -Pi, -Pi/2, 0, Pi/2, Pi, -Pi, -Pi/2, 0, Pi/2, Pi,Boxed -> False, MaxRe ursion -> 4,Mesh -> -Pi, -3 Pi/4, -Pi/2, -Pi/4, 0, Pi/4, Pi/2, 3 Pi/4,Pi, -Pi, -3 Pi/4, -Pi/2, -Pi/4, 0, Pi/4, Pi/2, 3 Pi/4, Pi,MeshStyle -> Thi kness[0.0001Plot3D[Abs[S P0[x1, x2, 1/16, N16, TO*Sin[(x1 - x2)/2*Sin[(x1 + 2 x2)/2*Sin[(2 x1 + x2)/2, x1, -Pi, Pi, x2, -Pi, Pi,PlotRange -> 0, 2.8, Axes -> True, True, False,AxesLabel -> "G2x", "G2y",Ti ks -> -Pi, -Pi/2, 0, Pi/2, Pi, -Pi, -Pi/2, 0, Pi/2, Pi,Boxed -> False, MaxRe ursion -> 4,Mesh -> -Pi, -3 Pi/4, -Pi/2, -Pi/4, 0, Pi/4, Pi/2, 3 Pi/4,Pi, -Pi, -3 Pi/4, -Pi/2, -Pi/4, 0, Pi/4, Pi/2, 3 Pi/4, Pi,MeshStyle -> Thi kness[0.0001(* sour e ode of the 3-in-1 3D plot, TO=100 re ommended *)Plot3D[Abs[S P0[x1, x2, 1/16, N16, TO*Sin[(x1 - x2)/2*Sin[(x1 + 2 x2)/2*Sin[(2 x1 + x2)/2,Abs[S P1[x1, x2, 1/16, N16, 100*Sin[(x1 - x2)/2*Sin[(x1 + 2 x2)/2*Sin[(2 x1 + x2)/2,Abs[S P2[x1, x2, 1/16, N16, 100*Sin[(x1 - x2)/2*Sin[(x1 + 2 x2)/2*Sin[(2 x1 + x2)/2, x1, -Pi, Pi, x2, -Pi,Pi, PlotRange -> 0, 2.8, Axes -> True, True, False,AxesLabel -> "G2x", "G2y",Ti ks -> -Pi, -Pi/2, 0, Pi/2, Pi, -Pi, -Pi/2, 0, Pi/2, Pi,Boxed -> False, MaxRe ursion -> 4,Mesh -> -Pi, -3 Pi/4, -Pi/2, -Pi/4, 0, Pi/4, Pi/2, 3 Pi/4,Pi, -Pi, -3 Pi/4, -Pi/2, -Pi/4, 0, Pi/4, Pi/2, 3 Pi/4, Pi,

136 APPENDIX DMeshStyle -> Thi kness[0.0001,PlotStyle -> Blend[LightBlue, Blue, 0.1, White,Blend[LightBlue, Blue, 0.24(* sour e ode of the ontour plots, TO=50 re ommended *)ContourPlot[-Abs[S P0[x1, x2, 1, N1, TO*Sin[(x1 - x2)/2*Sin[(x1 + 2 x2)/2*Sin[(2 x1 + x2)/2, x1, -Pi, Pi, x2, -Pi, Pi,PlotRange -> -2.8, 0, Frame -> True, FrameLabel -> "G2x", "G2y",RotateLabel -> False,FrameTi ks -> -Pi, -Pi/2, 0, Pi/2, Pi, -Pi, -Pi/2, 0, Pi/2, Pi,MaxRe ursion -> 4, Contours -> 11, ContourStyle -> NoneContourPlot[-Abs[S P0[x1, x2, 1/4, N4, TO*Sin[(x1 - x2)/2*Sin[(x1 + 2 x2)/2*Sin[(2 x1 + x2)/2, x1, -Pi, Pi, x2, -Pi, Pi,PlotRange -> -2.8, 0, Frame -> True, FrameLabel -> "G2x", "G2y",RotateLabel -> False,FrameTi ks -> -Pi, -Pi/2, 0, Pi/2, Pi, -Pi, -Pi/2, 0, Pi/2, Pi,MaxRe ursion -> 4, Contours -> 9, ContourStyle -> NoneContourPlot[-Abs[S P0[x1, x2, 1/16, N16, TO*Sin[(x1 - x2)/2*Sin[(x1 + 2 x2)/2*Sin[(2 x1 + x2)/2, x1, -Pi, Pi, x2, -Pi, Pi,PlotRange -> -2.8, 0, Frame -> True, FrameLabel -> "G2x", "G2y",RotateLabel -> False,FrameTi ks -> -Pi, -Pi/2, 0, Pi/2, Pi, -Pi, -Pi/2, 0, Pi/2, Pi,MaxRe ursion -> 4, Contours -> 8, ContourStyle -> None(**********************************************************************************)(* al ulations for H2 ************************************************************)(**********************************************************************************)TO = 6; hiFS[n1_, n2_, z_ := (z^(-2 n1 - n2) (1 + n2 - (2 + n1 + n2) z^(3 + 3 n1) + (1 + n1) z^(3 (2 + n1 + n2))))/(-1 + z^3)^2Table[Together[Expand[ hiFS[i1, i2, z, i1, 0, TO, i2, 0, TOSimplify[-1.4 (-5 z^2 + 2 z^5) -0.6 (-8 z^5 + 5 z^8) + (-7 z^2 - 2 z^5 + 3 z^8)Table[Expand[Together[ hiFS[j, 2 + i, z - (3 - i) hiFS[1 + j, i, z, j,2 + i, 1 + j, i, i, 0, 1, j, 0, 10(*i1,0,TO,i2,0,TO,j1,0,TO,j2,0,TO*)Table[Expand[Together[ hiFS[3 i, 2, z - 3 hiFS[3 i + 1, 0, z,3 i, 2, i + 1, 0, i, 0, 20

MATHEMATICA 6.0 CODE 137Sum[Expand[Together[1/((3 i + 1) (3 i + 4))*(3 hiFS[3 i + 1, 0, z - hiFS[3 i, 2, z), i, 0, Infinity(* general formula, equation 5.36 *)Simplify[(j + 1) hiFS[j - 2, i + 3, z - (j - 1) hiFS[j, i + 2, z - (2 j + 2) hiFS[j - 3, i + 2, z -4 hiFS[j - 1, i + 1, z + (j + 1) hiFS[j - 4, i + 1, z + (2 j - 2) hiFS[j + 1, i, z + (3 j + 5) hiFS[j - 2, i, z - (3 j - 5) hiFS[j, i - 1, z - (2 j + 2) hiFS[j - 3, i - 1, z - (j - 1) hiFS[j + 2, i - 2, z -4 hiFS[j - 1, i - 2, z + (2 j - 2) hiFS[j + 1, i - 3, z + (j + 1) hiFS[j - 2, i - 3, z - (j - 1) hiFS[j, i - 4, z(**********************************************************************************)(* LaTeX generating routines ******************************************************)(**********************************************************************************)TO = 10; hiFS[n1_, n2_, z_ := (z^(-2 n1 - n2) (1 + n2 - (2 + n1 + n2) z^(3 + 3 n1) + (1 + n1) z^(3 (2 + n1 + n2))))/(-1 + z^3)^2s = ""; n1 = 0; n2 = 0; (* table A.1 *)While[n1 <= TO, n2 = 0;While[n2 <= TO,s = s <> "$\" <> ToString[n1 <> "," <> ToString[n2 <>"\$ & $" <>ToString[TeXForm[ExpandNumerator[Together[ hiFS[n1, n2, z <>"$\\\\\n";n2++;n1++;Export["tex1.txt", s;s = ""; n1 = 0; n2 = 0; (* table A.1, alternative sequen e *)While[n1 <= TO, n2 = 0;While[n2 <= n1,s = s <> "$\" <> ToString[n1 <> "," <> ToString[n2 <>"\$ & $" <>ToString[TeXForm[ExpandNumerator[Together[ hiFS[n1, n2, z <>"$\\\\\n";If[n1 > n2,s = s <> "$\" <> ToString[n2 <> "," <> ToString[n1 <>"\$ & $" <> ToString[TeXForm[Together[ hiFS[n2, n1, z <>"$\\\\\n", y; n2++;n1++;Export["tex1.txt", s;

138 APPENDIX Ds = ""; j = 0; (* table A.3, table A.2 is analogue *)While[j <= TO,s = s <> "$" <> ToString[j <> "$ & $" <>ToString[TeXForm[ExpandNumerator[Together[2 hiFS[j + 1, 1, z - hiFS[j, 3, z <>"$\\\\\n";j++;Export["tex3.txt", s;

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140 REFERENCES[DV80 Daniel, M. and Viallet, C.M.: The geometri al setting of gauge theories of theYang-Mills type. Reviews of Modern Physi s 52 (1980), no. 1, p. 175-197[EGH80 Egu hi, T. ; Gilkey, P.B. and Hanson, A.J.: Gravitation, gauge theories anddierential geometry. Physi s Reports 66 (1980), no. 6, p. 213-393[Fis04 Fis her, E.: Singular Marsden-Weinstein Redu tion in Latti e Gauge Theory.Diploma thesis, University of Leipzig, 2004[Hal94 Hall, B.C.: The Segal-Bargmann oherent state transform for ompa t Liegroups. Journal of Fun tional Analysis 122 (1994), no. 1, p. 103-151[Hal96 Hall, B.C.: Phase spa e bounds for quantum me hani s on a ompa t Lie group.Communi ations in Mathemati al Physi s 184 (1996), no. 1, p. 233-250[Hal02 Hall, B.C.: Geometri quantization and the generalized Segal-Bargmann trans-form for Lie groups of ompa t type. Communi ations in Mathemati al Physi s226 (2002), no. 2, p. 233-268 arXiv:quant-ph/0012105v3[Hel84 Helgason, S.: Groups and Geometri Analysis Integral Geometry, InvariantDierential Operators, and Spheri al Fun tions. A ademi Press, In ., Orlando,FL, 1984 ISBN 0-12-338301-3[HN91 Hilgert, J. and Neeb, K.-H.: Lie-Gruppen und Lie-Algebren. Verlag Vieweg,Brauns hweig, 1991 ISBN 3-528-06432-3[HRS07 Hübs hmann, J. ; Rudolph, G. and S hmidt, M.: A gauge model for quantumme hani s on a stratied spa e. (2007) arXiv:hep-th/0702017v2[Hsi00 Hsiang, W.Y.: Le tures on Lie Groups. World S ienti , Singapore, 2000 (Serieson University Mathemati s, 2) ISBN 981-02-3529-1[Hüb01 Hübs hmann, J.: Kähler spa es, nilpotent orbits, and singular redu tion. Mem-oirs of the Ameri an Mathemati al So iety 172 (2001), no. 814, p. 75-109 arXiv:math.DG/0104213v4[Hüb04 Hübs hmann, J.: Stratied Kähler stru tures on adjoint quotients. (2004) arXiv:math.DG/0404141v3[Hüb05 Hübs hmann, J.: Singular Poisson-Kähler geometry of ertain adjoint quo-tients. Dierential Geometry and its Appli ations 23 (2005), no. 1, p. 79-93 arXiv:math.SG/0610614v1

REFERENCES 141[Hüb06 Hübs hmann, J.: Kähler quantization and redu tion. Journal fürdie reine und angewandte Mathematik 2006 (2006), no. 591, p. 75-109 arXiv:math.SG/0207166v5[Hüb07 Hübs hmann, J.: The holomorphi Peter-Weyl theorem and the Blattner-Kostant-Sternberg pairing. (2007) arXiv:math.DG/0610613v4[JKR05 Jarvis, P.D. ; Kijowski, J. andRudolph, G.: On the stru ture of the observablealgebra of QCD on the latti e. Journal of Physi s A: Mathemati al and General38 (2005), no. 23, p. 5359-5377 arXiv:hep-th/0412143v1[KN96a Kobayashi, S. and Nomizu, K.: Foundations of Dierential Geometry. Volume 1.Wiley Classi s Library Edition, John Wiley & Sons, In ., New York, NY, 1996 ISBN 0-471-15733-3[KN96b Kobayashi, S. and Nomizu, K.: Foundations of Dierential Geometry. Volume 2.Wiley Classi s Library Edition, John Wiley & Sons, In ., New York, NY, 1996 ISBN 0-471-15732-5[KRS98 Kijowski, J. ; Rudolph, G. and liwa, C.: On the stru ture of the observablealgebra for QED on the latti e. Letters in Mathemati al Physi s 43 (1998), no. 4,p. 299-308[KRT97 Kijowski, J. ; Rudolph, G. and Thielmann, A.: Algebra of observables and harge supersele tion se tors for QED on the latti e. Communi ations in Mathe-mati al Physi s 188 (1997), no. 3, p. 535-564[KS75 Kogut, J. and Susskind, L.: Hamiltonian formulation of Wilson's latti e gaugetheories. Physi al Review D 11 (1975), no. 2, p. 395-408[P01 Pflaum, M.J.: Analyti and Geometri Study of Stratied Spa es. Springer-Verlag, Berlin, 2001 ISBN 3-540-42626-4[RSV02 Rudolph, G. ; S hmidt, M. and Volobuev, I.P.: Classi ation of gauge orbittypes for SU(n)-gauge theories. Mathemati al Physi s, Analysis and Geometry 5(2002), no. 3, p. 201-241 arXiv:math-ph/0003044v1[Rud05 Rudolph, G.: Mathematis he Physik 1 Mannigfaltigkeiten, Tensorfelder undHamiltons he Systeme. Le ture, University of Leipzig, 2005[Rud06 Rudolph, G.: Mathematis he Physik 2 Faserbündel, Zusammenhänge undEi htheorie. Le ture, University of Leipzig, 2006

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IndexAsso iated bundle, see Prin ipal bre bundleCartan integers, see RootsCartan matrix, see RootsCartan subalgebra, see Lie algebraChara ter of a representation omplex, 46denition, 46irredu ible, 46of a torus, 47SU(3), see SU(3)Weyl's hara ter formula, 67Weyl's integral formula, 51Chara terisation of a representation, 4148Conne tion, see Prin ipal bre bundleCurvature, see Prin ipal bre bundle

D-de omposition, 21Darboux oordinates, 14theorem, 13Field strength, see Prin ipal bre bundleG-a tion

G-manifold, see G-manifolddenition, 11faithful / ee tive, 12free, 12Hamiltonian G-a tion, 16momentum map, see Momentum mapisotropy group, 11lift, 18orbit spa e, 11orbit type, 13orbits, 11

proper, 12transitive, 12G-manifold, 1113denition, 11equivariant maps, 12Hamiltonian G-manifold, 17symple ti G-manifold, 16Gauge potential, see Prin ipal bre bundleGauge theory lassi al, 3233 lassi al redu tion, 3739on ompa t onne ted Lie groups, 3839

SU(3), see SU(3)denition, 33gauge transformation, 32Hamiltonian formulation, 3334latti e gauge model, 3436tree gauge, 3637verti al automorphism, 32Haar integral, see Lie group, integrationHaar measure, see Lie group, integrationHamiltonian derivation, 15Hamiltonian system, 15Hamiltonian ve tor eld, 14Hilbert dire t sum, 48Kähler stru ture, see ManifoldKilling form, 10su(2), see su(2)Killing ve tor eld, 12Krone ker's theorem, 42Lie algebra, 5Cartan subalgebra, 53143

144 INDEXrank, 53 omplex, 5 omplexi ation, 53exponential map, 7Lie bra ket, 5 ommon, 6of a Lie group, 6real, 5representation, see Representationroots, see Rootssemi-simple, 9simple, 9Lie group, 5automorphism, 5homomorphism, 5integration, 42Haar integral, 43Haar measure, 43isomorphism, 5maximal torus, see Maximal torusrepresentation, see Representationsemi-simple, 10simple, 10torus, see TorusManifoldG-manifold, see G-manifoldKähler stru ture, 76Kähler manifold, 75Kähler potential, 76stratied Kähler stru ture of our model,see ModelSymple ti G-manifold, see G-manifoldSymple ti manifold, see Symple ti man-ifoldMarsden-Weinstein redu tion, 1924regular, 20redu ed Hamiltonian, 20redu ed spa e, 20redu tion map, 20theorem, 20shifting tri k, 19singular, 2124

onstru tion of the Poisson algebra, 2324 onstru tion of the strata, 2223de omposed spa e, see D-de ompositionstratied symple ti spa e, 21theorem, 21Maximal torus, 4952Cartan's theorem, 49denition, 49of U(n) and SU(n), 52torus, see TorusWeyl group W , 49, 57Model ostratied Hilbert spa easso iated with P , 82denition, 81Hilbert spa e and its measure, 7980Hilbert spa e H2 asso iated with P2, 88115basis of V2, 98fun tional des ription of z, 91generating fun tion of V2, 93generating fun tion of V2, de omposi-tion, 9497impli it des ription, 105orthogonal fun tions on V2, 103106orthogonal fun tions on V2, oe ientiteration, 104105orthogonal fun tions on V2, derivation,102104Hilbert spa es H1m asso iated with P1m,8388expli it des ription, 85representation in L2([−π, π]2, dx1dx2),87tunnelling between the H1m, 86Kähler quantisation on GC, 79orbit type strati ation, 76orthonormal basis of H, 81redu ed onguration spa e, 76S hrödinger quantisation of T∗G, 79stratied Kähler stru ture on P , 78Momentum map, 17, 19

INDEX 145One-parameter group of dieomorphisms, 6Parti le elds, see Prin ipal bre bundlePeter-Weyl theorem, see RepresentationPoisson algebradenition, 15non-degenerate, 15Poisson bra ket, 15Poisson ideal, 15Poisson map, 16Polar de omposition of GC, 75Prin ipal bre bundle, 2527asso iated bre bundle, 31base manifold, 26 onne tion, 2729TpP = Hp ⊕ Vp denition, 28 onne tion one-form denition, 28eld strength, 30gauge potential, 29ve tor potential, 29 urvature, 30denition, 30denition, 26global se tion, 27lo al se tion, 27lo al trivialisation, 26parti le elds, 31proje tion map, 26stru ture group, 26total spa e, 26transition fun tion, 26trivial, 27RepresentationAdjoint representation, 9adjoint representation, 9 hara ter, see Chara ter of a representa-tion oAdjoint representation, 9 oadjoint representation, 9denition, 8equivalen e, 8equivalen e, G, 45intertwining operator / morphism, 43

irredu ible, 43Ja obian of the onjugation map, 50Lie algebra, 8Lie group, 8 ompa t abelian, 41of abelian stru tures, 44Peter-Weyl theorem, 48holomorphi , 80redu ible, 43 ompletely, 45representation spa e, 8S hur's lemma, 44SU(n), see SU(n)

su(n), see su(n)unitary, 43G-invariant inner produ t, 43Weyl theorem, 44Roots, 5361Cartan integers, 55Cartan matrix, 56denition, 53ordering, 56positive roots, 55root subspa e, 54simple roots, 55

su(3), see su(3)Weyl group W , see Maximal torusWeyl ree tions, 57S hur's lemma, see RepresentationSU(3) lassi al redu tion in gauge theory, 39 omplex hara ters, 6674rst 12 hara ter ombinations 2χCj+1,1−

χCj,3, 119rst 12 hara ter ombinations 3χCj+1,0−χCj,2, 119rst 28 elements of χCn1n2

(diag(z, z, z−2)),119formula, rst parametrisation, 69formula, se ond parametrisation, 70redu tion of dire t produ ts of irredu iblerepresentations (S. Coleman), 93

146 INDEXSU(n)denition, 7su(2)Killing form, 10su(3)Cartan matrix, 60highest weight, 66irredu ible representations, 6566Killing form, 58roots, 5861Weyl group, 61su(n)denition, 7Lie algebra of SU(n), 7Submodule, 43Symple ti manifolddenition, 13stratied symple ti spa e, see Marsden-Weinstein redu tion, singularsymple ti omplement, 14symple ti form, 13 anoni al, 14Torusdenition, 41generator, 41Ve tor eldleft-invariant, 6pull-ba k, 16push-forward, 6Ve tor potential, see Prin ipal bre bundleWeights, 6166denitionLie algebra, 62Lie group, 62fundamental weights, 64highest weight, 64of the adjoint representation, 63

su(3), see su(3)Weyl group, see Maximal torusWeyl ree tions, see RootsWeyl theorem, see Representation

Weyl's hara ter formula, see Chara ter of arepresentationWeyl's integral formula, see Chara ter of a rep-resentation

SelbstständigkeitserklärungHiermit erkläre i h, dass i h die vorliegende Arbeit selbstständig verfasst habe und keine an-deren als die angegebenen Quellen und Hilfsmittel benutzt habe. Alle Stellen der Arbeit, diewörtli h oder sinngemäÿ aus Veröentli hungen oder aus anderweitigen fremden Äuÿerungenentnommen wurden, sind als sol he kenntli h gema ht.Ferner erkläre i h, dass die Arbeit no h ni ht in einem anderen Studiengang als Prüfungsleis-tung verwendet wurde.Leipzig, den 03.10.2008EinverständniserklärungHiermit erkläre i h mi h einverstanden, dass meine Diplomarbeit na h positiver Beguta htungder Zweigstelle Physik der Universitätsbibliothek Leipzig zur Verfügung gestellt wird.Leipzig, den 03.10.2008