Loop quantisation of supergravity theories Schleifenquantisierung … · 2013-09-03 · and Andreas Thurn. The work started during my diploma thesis in February 2009, while the o

Loop quantisation of supergravity theories

Schleifenquantisierung von Supergravitationstheorien

Der Naturwissenschaftlichen Fakultatder Friedrich-Alexander-Universitat Erlangen-Nurnberg

zurErlangung des Doktorgrades Dr. rer. nat.

vorgelegt vonNorbert Bodendorfer

aus Waldbrol

Als Dissertation genehmigt von der Naturwissen-schaftlichen Fakultat der Friedrich-Alexander Universitat

Erlangen-Nurnberg

Tag der mundlichen Prufung: 30. April 2013

Vorsitzende/r derPromotionskommission: Prof. Dr. Johannes Barth

Erstberichterstatter/in: Prof. Dr. Thomas Thiemann

Zweitberichterstatter/in: Prof. Dr. Kristina Giesel

Contents

Table of contents 1

About this thesis 4

Abstract 5

Zusammenfassung (german abstract) 6

Notation 7

1 Introduction 8

2 Overview of the results 11

I Connection dynamics for classical higher-dimensional general relativity 14

3 Hints from the Palatini action 153.1 The Ashtekar-Barbero variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Canonical analysis of the higher-dimensional Palatini action . . . . . . . . . . . . 163.3 Gauge unfixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.1 Toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.2 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.3 Application to gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Canonical transformation 244.1 Phase space extension and canonical transformation . . . . . . . . . . . . . . . . 244.2 The Hamiltonian constraint in the new variables . . . . . . . . . . . . . . . . . . 28

5 The Linear simplicity constraint 305.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2 Introducing linear simplicity constraints . . . . . . . . . . . . . . . . . . . . . . . 31

II Loop quantum gravity in higher dimensions 34

6 LQG-techniques in higher dimensions 356.1 Kinematical quantisation techniques . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.1.1 Holonomies, fluxes, and right invariant vector fields . . . . . . . . . . . . . 356.1.2 Solution of the Gauß and spatial diffeomorphism constraints . . . . . . . 37

6.2 Regularisation of the Hamiltonian constraint operator . . . . . . . . . . . . . . . 38

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6.2.1 Volume operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.2.2 Poisson bracket identities and regularisation . . . . . . . . . . . . . . . . . 38

6.3 Regularisation of the simplicity constraint . . . . . . . . . . . . . . . . . . . . . . 39

7 Hilbert space techniques for the linear simplicity constraint 42

8 The simplicity constraint 458.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458.2 The quadratic simplicity constraint operators . . . . . . . . . . . . . . . . . . . . 47

8.2.1 A maximal closing subset of vertex constraints . . . . . . . . . . . . . . . 478.2.2 The solution space of the maximal closing subset . . . . . . . . . . . . . . 498.2.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8.3 The linear simplicity constraint operators . . . . . . . . . . . . . . . . . . . . . . 538.3.1 Regularisation and anomaly freedom . . . . . . . . . . . . . . . . . . . . . 538.3.2 Solution on the vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.3.3 Edge constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568.3.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578.3.5 Mixed quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

8.4 Comparison to existing approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 618.4.1 Continuum vs. discrete starting point . . . . . . . . . . . . . . . . . . . . 628.4.2 Projected spin networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628.4.3 EPRL model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8.5 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

III Extensions to supergravity 67

9 Standard matter 68

10 Rarita-Schwinger field 7010.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7010.2 Review of canonical supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

10.2.1 Status of canonical supergravity . . . . . . . . . . . . . . . . . . . . . . . 7210.2.2 Canonical supergravity in the time gauge . . . . . . . . . . . . . . . . . . 73

10.3 Phase space extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7510.3.1 Symplectic structure in the SO(D) theory . . . . . . . . . . . . . . . . . . 7510.3.2 SO(D + 1) gauge supergravity theory . . . . . . . . . . . . . . . . . . . . 79

10.4 Background independent Hilbert space representations for Majorana fermions . . 8610.5 Generalisations to different multiplets . . . . . . . . . . . . . . . . . . . . . . . . 90

10.5.1 Majorana spin 1/2 fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 9010.5.2 Mostly plus / mostly minus conventions . . . . . . . . . . . . . . . . . . . 9110.5.3 Weyl fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

10.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

11 p-form gauge fields 9411.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9411.2 Classical Hamiltonian analysis of the 3-index-photon action . . . . . . . . . . . . 9511.3 Reduced phase space quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . 9911.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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IV Initial value quantisation of higher-dimensional general relativity 105

12 Loop quantum gravity without the Hamiltonian constraint 10612.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10612.2 Classical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10812.3 Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11112.4 Geometric operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11112.5 Application to black hole entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 11212.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

V Isolated horizon boundaries in higher-dimensional LQG 114

13 Classical phase space description of isolated horizon boundaries 11513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11513.2 General strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11613.3 Higher-dimensional isolated horizons and Lagrangian framework . . . . . . . . . 11813.4 SO(D + 1) as internal gauge group . . . . . . . . . . . . . . . . . . . . . . . . . . 12013.5 Inclusion of distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

13.5.1 Beetle-Engle method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12213.5.2 Perez-Pranzetti method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

13.6 Comments on quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12413.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

VI Conclusion 127

14 Concluding remarks and further research 12814.1 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12814.2 Towards loop quantum supergravity: Where do we stand? . . . . . . . . . . . . . 12914.3 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

A Simple irreps of SO(D + 1) and square integrable functions on the sphere SD134

Danksagung (german acknowledgements) 136

Bibliography 137

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About this thesis

This thesis has grown out of collaborations with Alexander Stottmeister, Thomas Thiemann,and Andreas Thurn. The work started during my diploma thesis in February 2009, while theofficial starting date is December 2009, when I obtained my physics diploma. The topic of thisthesis has been proposed by my supervisor Prof. Dr. Thomas Thiemann.

The work for my diploma thesis, which contained parts of chapter 3, has been mainly conductedat the Albert-Einstein-Institute in Potsdam, which I thank for hospitality. Also during mydiploma thesis, I have been supported by the Friedrich-Naumann-Foundation, the Max-Weber-Programme of Bavaria, the Leonardo-Kolleg of the University of Erlangen-Nurnberg, the EliteNetwork Bavaria, and e-Fellows. My undergraduate and doctoral studies have been conductedin the Physics Advanced programme of the Universities Erlangen-Nurnberg and Regensburg.

The research which lead to this thesis has been performed at the Institute for Theoretical PhysicsIII of the University of Erlangen-Nurnberg. I strongly acknowledge financial and ideationalsupport of the German National Merit Foundation. Also, I acknowledge support of the EliteNetwork Bavaria and e-Fellows.

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Abstract

The aim of this thesis is to develop loop quantisation techniques for higher-dimensionalsupergravities in order to make progress towards comparing loop quantum gravity to superstringtheory. The abstract idea for this comparison is to look at loop quantum gravity and stringtheory as two methods of quantising an underlying gravity theory. While loop quantum gravitycircumvents the usual problems associated with standard Fock-type quantisations of gravitytheories by starting classically from a different Poisson-*-subalgebra of phase space functions,string theory introduces the string as a fundamental ingredient to cope with the short distancedivergences arising in perturbative quantum gravity. The most obvious problem with comparingthese two approaches to quantum gravity is that while it was only known how to formulate loopquantum gravity in four (or three) dimensions, including all types of standard model mattercouplings, string theory requires a ten-dimensional spacetime with supersymmetry. The possiblereductions of string theory to four dimensions on the other hand are highly non-unique, thus itis very hard to compare these approaches at this level.

A possible solution to this problem, which is investigated in this thesis, is to extend thetechniques of loop quantum gravity to higher-dimensional supergravities, which arise as the lowenergy limits of string theories. These supergravities have the same low energy particle contentand live in the same spacetime dimension as the corresponding string theories and are thus suitedto circumvent the above problem. Since this solution does not seem to be overly sophisticated ata first glance, it is important to remark that, prior to recent work on which this thesis is based, itwas not even known how to formulate loop quantum gravity for pure higher-dimensional gravity,since the main ingredient at the classical level, the Ashtekar-Barbero connection, is only availablein four dimensions. In this thesis, we will first develop a generalisation of this type of connectionformulation for higher-dimensional general relativity. We will then show that the quantisationtechniques from loop quantum gravity can be mainly carried over to this new formulation, whilesome new ingredients have to be introduced. Next, we are going to extend this formulationto higher-dimensional supergravities, thus providing a solution for the above problem. On adifferent route, we provide a partially reduced phase space quantisation of higher-dimensionalgeneral relativity conformally coupled to a scalar field based on the constant mean curvaturegauge in the fourth part of this thesis. Finally, in the fifth part of this thesis, we will treatisolated horizons as boundaries of the spacetime manifold in order to take first steps to comparestring theory and higher-dimensional loop quantum gravity in the presence of a black hole.

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Zusammenfassung

Es ist das Ziel dieser Arbeit, Methoden zu entwickeln, die die Quantisierung von Super-gravitationstheorien mit Hilfe von Techniken aus der Schleifenquantengravitation ermoglichen,um Fortschritte beim Vergleichen der Schleifenquantengravitation mit der Superstringtheorie zumachen. Die abstrakte Idee dieses Vergleichs ist es, sowohl die Schleifenquantengravitation alsauch die Superstringtheorie als Quantisierungen einer zugrundeliegenden Gravitationstheorie zuverstehen. Wahrend die Schleifenquantengravitation die ublichen Probleme bei Fock-Typ Quan-tisierungen von Gravitationstheorien vermeidet, indem sie von einer anderen Poisson-*-Algebravon Phasenraumfunktionen startet, postuliert die Superstringtheorie den String als neuen fun-damentalen Baustein um den UV-Divergenzen in der perturbativen Quantengravitation Herrzu werden, welche durch die Betrachtungen von beliebig kleinen Distanzen entstehen. Dasoffensichtlichste Problem, welches beim Vergleichen dieser beiden Theorien entsteht, ist dasswahrend nur bekannt war, wie man Schleifenquantengravitation in vier (oder drei) Dimensionenformuliert und die Materiefelder des Standardmodells behandelt werden konnten, aber nichtnotwendig waren, die Superstringtheorie eine zehndimensionale Raumzeit mit Supersymmetriebraucht. Andererseits sind mogliche Reduktionen der Superstringtheorie auf vier Dimensionenhochgradig nicht-eindeutig und daher ist es sehr schwierig diese Reduktionen mit der Schleifen-quantengravitation zu vergleichen.

Eine mogliche Losung dieses Problems, welche in dieser Arbeit untersucht wird, ist es dieTechniken der Schleifenquantengravitation auf hoherdimensionale Supergravitationen zu erweit-ern, welche als Niederenergielimites von Superstringtheorien auftreten. Diese Supergravita-tionen haben das selbe Niederenergieteilchenspektrum und leben in der selben Raumzeitdi-mension wie die entsprechenden Superstringtheorien und sind deshalb gut geeignet, um dasoben beschriebene Problem zu umgehen. Da dieser Losungsvorschlag auf den ersten Blicknicht sehr tiefsinnig erscheint, muss angemerkt werden, dass es bis vor einigen kurzlich er-schienenen Publikationen, welche zu dieser Arbeit gefuhrt haben, nicht einmal bekannt war,wie und ob man hoherdimensionale Gravitation mit den Methoden der Schleifenquantengravi-tation behandeln konnte. Das Problem an dieser Stelle war, dass der Hauptinput in der klas-sischen Theorie, die Ashtekar-Barbero-Variablen, nur in vier Dimensionen verfugbar sind. Indieser Arbeit wird erst eine Verallgemeinerung dieser Art von Zusammenhangsformulierung aufhoherdimensionale allgemeine Relativitatstheorie entwickelt werden. Danach wird gezeigt wer-den, dass die meisten Quantisierungstechniken der Schleifenquantengravitation auf diese neueFormulierung angewendet werden konnen, aber auch ein paar neue Techniken gebraucht werden.Als nachstes werden diese Techniken noch auf Supergravitationen erweitert werden, und damitein Losungsvorschlag fur das obige Problem prasentiert werden. Abseits dieses Hauptthemaswird eine partiell reduzierte Phasenraumquantisierung von hoherdimensionaler allgemeiner Rela-tivitatstheorie mit einem konform gekoppeltem Skalarfeld, basierend auf der Konstante-mittlere-Krummungs-Eichung, im vierten Teil dieser Arbeit vorgestellt werden. Abschließend werden imfunften Teil dieser Arbeit Raumzeiten mit isolierten Horizonten als Randern betrachtet, um ersteSchritte hin zu einem Vergleich der Superstringtheorie mit der hoherdimensionalen Schleifen-quantengravitation in Raumzeiten mit schwarzen Lochern zu machen.

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Notation

Here, we gather some important reoccurring notation as a reference.

≈ weakly equal (equal up to constraints) (0.1)

D spatial dimension (0.2)

d = D + 1 spacetime dimension (0.3)

s, ζ spacetime, internal signature (0.4)

ηIJ = diag(ζ, 1, 1, ...) (0.5)

I, J, ... SO(D, 1) or SO(D + 1) indices (short: SO(η)) (0.6)

i, j, ... SO(D) indices (0.7)

α, β, ... spinor indices (except for chapter 13) (0.8)

µ, ν, ... spacetime indices (0.9)

a, b, ... spatial indices (0.10)

eai SO(D) D-bein (0.11)

Eai densitised SO(D) D-bein (0.12)

eaI SO(1, D) or SO(D + 1) (hybrid)-D-bein (0.13)

EaI densitised SO(1, D) or SO(D + 1) (hybrid)-D-bein (0.14)

πaIJ densitised SO(1, D) or SO(D + 1) (hybrid)-D-bein (0.15)

in adjoint representation

AaIJ SO(1, D) or SO(D + 1) connection, (0.16)

canonically conjugate to πaIJ

nµ normal vector on spatial hypersurfaces (0.17)γI , γJ

= 2ηIJ (Clifford algebra) (0.18)

(γI)† = ηIIγI (no summation here) (0.19)

γIJ...K := γ[IγJ ...γK] (with total weight one) (0.20)

γ⊥ := nµγµ (0.21)

ΣIJ := − i2γIJ (SO(D + 1) generators in spinor representation) (0.22)

∇a(A) χ := ∂aχ+i

2AaIJΣIJχ (0.23)

i[ΣIJ ,ΣKL

]= ηLJΣKI − ηLIΣKJ − ηKJΣLI + ηKIΣLJ (0.24)

χ := χ†γ0 (0.25)

χ = χTC (Majorana condition) (0.26)

Note that the choice (0.19) is always possible [1]. Furthermore, note that in a Majorana repre-sentation the spinors will be real and then, from the Majorana condition (0.26) follows γ0 = C.

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Chapter 1

Introduction

The main topic of this thesis is to establish a connection between two different approaches toquantum gravity, namely the string theory1 / M-theory / supergravity side and loop quantumgravity (LQG). The motivation for this kind of research stems from the observation that althoughimportant progress has been made in many branches of quantum gravity over the past decades,there is a general lack of convergence between the different approaches. In the presence ofexperiments which could discriminate among these different theories, this situation would notbe very problematic, however, since such experiments are missing today, the present quantumgravity research has to be conducted on purely theoretical grounds. Accordingly, if one expectsto be successful in constructing quantum gravity by purely theoretical means, the theory shouldbe reasonably unique. Focussing on string theory and loop quantum gravity, the aim of thisthesis is to establish a connection between these two theories by applying LQG methods tosupergravity theories which arise as the low energy limits of superstring and M-theory.

In the context of this thesis, superstring and M-theory [2, 3, 4, 5] will be only treatedunder the aspect of providing quantum theories with the supergravity theories in ten and elevendimensions as their low energy limits. The arguments which lead one to the conclusion thatsupergravities are indeed the low energy limits of superstring and M-theory are mainly dueto symmetry principles. As an example, Werner Nahm has classified all possible supergravitytheories which do not lead to fields of spin greater than 2 upon dimensional reduction to fourdimension [6]. Different superstring theories on the other hand yield the same massless particlespectrum as the corresponding supergravity theories in ten dimensions, from which one concludesby invoking Nahm’s results that dynamics of the theories also have to agree for the masslesslow energy limit of the superstring theories. Since the supergravity actions consist basically ofthe gravity action in higher dimension plus matter couplings, it seems straight forward to tryto apply loop quantum gravity techniques to this system in order to compare this quantisationwith the one of superstring and M-theory. While the quantisation of many higher-dimensionalsupergravity theories will be achieved in this thesis, the comparison with string theory will beleft for further research. On the one hand, one could compare the results and explicit derivationsof certain symmetry reduced situations like the black hole entropy calculations or cosmology. Onthe other hand, a more direct route would be to gain a better understanding of the quantum fieldtheory on curved spacetime limit of loop quantum gravity, which would enable one to compareit to superstring theory also in the realm of scattering theory. Since, as said, the explicit formof superstring theory is not of importance for this thesis, we will not comment on it further. Anintroduction to supergravity theories and their canonical analysis is provided in the third partof this thesis. In the rest of this introduction, we will focus on the current state of the loop

1By string theory we will always mean superstring theory, i.e. we are not interested in the 26-dimensionalbosonic string theories in this thesis.

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quantum gravity programme and comment on where the main obstacles on the way towards aloop quantisation of higher-dimensional supergravities lie.

The starting point of the loop quantum gravity research programme was the seminal paper byAshtekar [7], in which he showed that general relativity can be written as a Yang-Mills theory.The connection proposed by him, nowadays known as the Ashtekar connection, is complexand results in complicated reality conditions which, up till now, could not be implemented ona Hilbert space. The strength of his proposal was however that the Hamiltonian constraint(with density weight two) becomes a fourth order polynomial in the canonical variables, whichpromised to be an important step in making mathematical sense out of the Wheeler-deWittequation [8, 9, 10], one of the drawbacks of which is that it is non-polynomial in the canonicalvariables and thus very hard to define as an operator equation. Since progress with the realityconditions could not be achieved, Barbero proposed a real version of Ashtekar’s connectionvariables [11], nowadays known as the Ashtekar-Barbero connection. This connection is currentlyfavoured for constructing loop quantum gravity, since its reality conditions are automaticallyimplemented on the Ashtekar-Lewandowski Hilbert space of loop quantum gravity. Althoughthe Hamiltonian becomes non-polynomial in the Ashtekar-Barbero variables, Thiemann was ableto construct a well defined operator from the Hamiltonian constraint with density weight one[12, 13], thus providing a well defined proposal for the quantum Einstein equations.

Meanwhile, a rigorous construction of the kinematical Hilbert space was provided in papersAshtekar, Isham, Lewandowski, Marolf, Mourao, and Thiemann [14, 15, 16, 17, 18, 19]. Thekey input of this construction was to make use of projective limits in order to be able to workwith spin networks which are supported on finite graphs while maintaining invariance underarbitrary refinements of the graph. The compactness of the gauge group SU(2) used for theAsthekar-Barbero variables is the main input into this construction, necessary for constructinga measure on an infinitely refined graph. Moreover, the construction of the kinematical Hilbertspace has been performed independently of the number of spatial dimensions and the compactgauge group used. Thus, the problem of kinematically quantising general relativity in higherdimensions had already been reduced to finding a connection formulation of general relativityin higher dimensions based on a compact gauge group.

Matter coupling can be done rather naturally in the context of loop quantum gravity andseveral results are available. The technical framework for matter coupled LQG has been derivedin [20, 21] and it has has been shown how to regularise the Hamiltonian constraint operator onthe corresponding Hilbert space. However, due to the well known problem of time, it is difficultto interpret solutions to the Hamiltonian constraint operator and deparametrised models weredeveloped [22, 23, 24, 25] in order to have a physical Hamiltonian encoding the “time” evolutionwith respect to a reference field. Also, the algebraic quantum gravity framework [26, 27, 28, 22]was developed in order to deal with the difficulties coming from the embedding of the graphs inthe standard treatment of LQG. Meanwhile, kinematical coherent states [29, 30, 31, 32, 33, 34]offered a possibility to test the classical limit of the Hamiltonian constraint (or Master constraint,true Hamiltonian, ...), and in the presence of a complete deparametrisation, they even becomephysical coherent states [22].

Thus, a framework capable to complete the quantisation of general relativity coupled tointeracting matter fields has been developed within LQG. Several explicit examples were con-structed and more can be given. Despite this success in treating matter coupled LQG, there areseveral open issues to deal with. Most prominently, the Hamiltonian constraint or the derivedtrue Hamiltonians suffer from different quantisation ambiguities, the physical consequences ofwhich are not very well understood. Since the action of a (true or constrained) Hamiltonianoperator on spin network functions strongly depends on the precise regularisation procedure,it is hard to judge the results of possible calculations involving this operator without having

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experimental results at one’s disposal.In this thesis, we will not focus on the issues concerning the quantisation ambiguities, but

show that a loop quantisation of many higher-dimensional supergravities, including the four,ten, and eleven dimensional ones, exists. The main obstacle that will have to be removedis to construct a connection formulation of higher-dimensional general relativity with certainimportant properties, which will be the subject of the first part of the thesis. Following, inpart two, we will extend the quantisation techniques developed for the four-dimensional caseto higher dimensions. In a last step in part three, we extend the quantisation procedure forpure higher-dimensional general relativity to the matter fields appearing in higher-dimensionalsupergravities. In part four, we will further develop the quantisation of higher-dimensionalgeneral relativity (without supersymmetry) by constructing a partially reduced phase spacequantisation with a geometric clock. In part five, we will further develop the classical phasespace description in the presence of an isolated horizon boundary which is necessary for thederivation of black hole entropy within LQG. A more detailed overview of the research presentedin this thesis is found in the next chapter.

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Chapter 2

Overview of the results

In a series of papers [35, 36, 37, 38, 39, 40, 41], Thomas Thiemann, Andreas Thurn, and theauthor of this thesis were able to provide Hamiltonian formulations of higher-dimensional super-gravities as gauge theories on a Yang-Mills phase space, such that the Yang-Mills gauge group iscompact, the canonical variables are real and obey simple commutation relations, and the con-straints of the theory form a closing Poisson algebra. It then follows1 that a non-perturbativeand mathematically well defined quantisation of these theories, including d = 10, 11 supergrav-ities, can be explicitly constructed. A summary is given in [42] and the main results will bebriefly sketched in the following.Also, together with Alexander Stottmeister and Andreas Thurn, a reduced phase space quanti-sation of general relativity conformally coupled to a scalar field has been proposed in [43, 44].The “time” function of this model is conceptually different from the deparametrised models[22, 23, 24, 25] and results in a different range of applicability, details of which are also discussedbelow.Finally, together with Thomas Thiemann and Andreas Thurn, higher-dimensional isolated hori-zons have been incorporated in the proposed connection formulation in [45]. A boundary con-dition similar to the four-dimensional case has been derived and the symplectic structure wasshown to obtain a boundary contribution of the higher-dimensional Chern-Simons type.

Part I: New canonical variablesIn [35, 36], it was shown that Hamiltonian Lorentzian general relativity in d = D + 1 space-time dimensions can be rewritten as an SO(d) Yang-Mills theory. The main ingredient of thecanonical transformation which relates this new formulation to the ADM formulation [46] isthe simplicity constraint familiar from spin foam models, which relates a “generalised” vielbeintransforming in the adjoint representation of SO(d) to a usual vielbein, transforming in the fun-damental representation. The usage of this generalised vielbein allowed it to construct an SO(d)connection by adding the extrinsic curvature contracted with the generalised vielbein to the“hybrid” [47] spin connection annihilating the vielbein in the fundamental representation. Theresulting connection formulation is, although classically equivalent, different from the Ashtekar-Barbero [7, 11] formulation. The simplicity constraint from spin foam models turns out to be akey input in the construction and it can be incorporated either in the quadratic or in the linearform.

1From the compact gauge group, construct the Ashtekar-Lewandowski measure, which is a positive linearfunctional on the holonomy-flux algebra based on the Yang-Mills connection and its conjugate momentum. TheAshtekar-Isham-Lewandowski representation of this algebra follows from the Gelfand-Naimark-Segal construction.The regularisation of the Hamiltonian and supersymmetry constraints can be done using Thiemann’s Poissonbracket tricks.

11

Part II: Quantum theoryA loop quantisation of higher-dimensional general relativity using the new canonical variableswas spelled out in [37]. The existing quantisation techniques from the literature were eitheralready formulated independently of the dimension, as in the case of the Hilbert space construc-tion, or straight forward generalisations existed, as for geometrical operators or the regularisationof the Hamiltonian constraint operator. Also, a quantisation of the simplicity constraint wasperformed and turned out to be anomalous. Different approaches for the non-anomalous imple-mentation of the simplicity constraint in the canonical theory were then discussed in [39].

Part III: Matter coupling and supergravityWhile bosonic matter couplings were already available for higher dimensions, the use of SO(d)as an internal gauge group initially posed a problem for coupling fermions. While this problemcould be circumvented by acting with SO(d) on the fermionic representation space in such a waythat the theory reduced to general relativity coupled to fermions in a certain gauge [38], the useof Majorana fermions in supergravity theories required more work since Majorana conditions aresensitive to the signature of spacetime. The solution proposed in [40] is the usage of a compoundobject of fermionic and gravitational degrees of freedom which obeys the Lorentzian Majoranacondition even when one acts on it with SO(d) [40]. Furthermore, the Abelian 3-form field(3-index photon) of d=11 supergravity [48] was discussed as an example for p-form fields whichappear in ten and eleven dimensional supergravities. Due to the presence of a Chern-Simonsterm for the 3-index photon, which is due to local supersymmetry, the theory is self-interactingand the application of a direct generalisation of the LQG Hilbert space to Abelian p-forms wasnot possible. Nevertheless, a reduced phase space quantisation with respect to the 3-index pho-ton Gauß constraint was shown to be possible by using a state of the Narnhofer-Thirring type[49].

Part IV: Reduced phase space quantisationIn [43, 44], a reduced phase space quantisation of general relativity conformally coupled to ascalar field has been achieved. While the quantisation can be seen as a variation of the model pro-posed in [23], it has some conceptual differences which result in a different range of applicability.The basic idea of the quantisation uses certain results from what has been called shape dynamics[50], i.e. that the Hamiltonian constraint can be traded for the generator D of a local rescalingof the canonical variables, which coincides with the CMC gauge condition. The physical mean-ing of this symmetry trade is to restrict oneself to a certain spatial slice which is given by thesecond class partner D of the Hamiltonian constraint H, i.e. D is a gauge fixing for H. It turnsout that, when coupling a conformally coupled scalar field (as opposed to a minimally coupledscalar field), D can be chosen to be the generator of local rescalings for both, the gravitationaland the scalar field variables. Due to this fact, a new metric and conjugate momentum, whichare invariant under the local rescaling, can be constructed as combined objects of the scalarfield and the original metric and momentum. It follows that the Dirac bracket associated withgauge fixing the Hamiltonian constraint with the generator of local rescalings agrees, when usingthe invariant variables, with the ADM Poisson bracket for the original metric and momentum.Thus, one can loop quantise the system using the well developed kinematical LQG techniqueswithout having to worry about the Hamiltonian constraint. The downside of this formulationare difficulties when trying to implement a time evolution, i.e. a true Hamiltonian with respectto interpreting the gauge fixing condition as a time function. Nevertheless, as outlined in [43],an interesting application to black hole physics is possible since the gauge fixing condition isglobally accessible in this case and it follows that state counting can be performed at the level

12

of the physical Hilbert space. In fact, the structure of the “time” function D employed is verydifferent from the ones in the deparametrised models, since it consists of momenta instead ofconfiguration space variables. Moreover, it is purely geometric.

Part V: Isolated horizon boundary degrees of freedomAn immediate application of the results of the first three parts of this thesis is the calculationof black hole entropy in higher dimensions within the LQG framework. First steps towardsthis goal have been taken in [45], where the classical phase space description of the proposedconnection formulation on a spacetime manifold with isolated horizon boundary was developed.It was shown that starting with the Palatini action on such a manifold, a boundary conditionsimilar to the one known from the four-dimensional treatment can be derived. Furthermore, thesymplectic structure obtains a boundary term which was shown to result in a higher-dimensionalChern-Simons symplectic structure. This boundary term was already familiar from [35], whereit arises as a boundary contribution for the canonical transformation to higher-dimensional con-nection variables. Using this, the internal gauge group could be switched to SO(D + 1), asopposed to SO(1, D) in the treatment starting with the Palatini action.

13

Part I

Connection dynamics for classicalhigher-dimensional general relativity

14

Chapter 3

Hints from the Palatini action

In this chapter, we will perform a canonical analysis of the Palatini action in order to gain anunderstanding of its canonical structure. As in the four-dimensional case, the theory will turnout to be subject to second class constraints which have to be solved prior to quantisation.Since a solution yields either the ADM formulation of general relativity in terms of vielbeins ora non-commuting connection when using the Dirac bracket, a rather non-standard procedure,called gauge unfixing, will be employed to arrive at a first class constraint system. This contrastsour formulation from the derivation of the Ashtekar-Barbero variables, which can be derived bysupplementing the Palatini action with the Holst modification, which is however only availablein four spacetime dimensions. The original work on which this chapter is based is [36].

3.1 The Ashtekar-Barbero variables

The Ashtekar-Barbero variables were originally derived via a canonical transformation [11]. Ofcourse, one expects that a corresponding derivation at the Lagrangian level should exist and thecorresponding derivation was given by Holst [51]. In order to arrive at the Ashtekar-Barberovariables, it is important to start with a modified version of the Palatini action, it has to besupplemented by the Holst modification as

SHolst =s

2

∫Md4X eeµIeνJFµνIJ(A)︸︷︷︸

Palatini action

+1

2γ

s

2

∫Md4X eeµIeνJFµν

KL(A)εIJKL︸︷︷︸Holst modification

, (3.1.1)

where γ ∈ R\0 is the Barbero-Immirzi parameter and s = ±1 denotes the spacetime signature.The Holst modification does not change the equations of motion at the classical level and it canbe shown to vanish by the Bianchi identity when the equations of motion are satisfied. Later,it was shown by Sengupta [52] that it differs from the Nieh-Yan topological density only by atorsion term, which vanishes by the equations of motion. Thus, the classical equivalence to thePalatini action and thus also to the Einstein-Hilbert action is manifest.

In order to derive the Asthekar-Barbero variables from this action, one first performs acanonical analysis. The steps involved are:

1. Split spacetime into space and time.

2. Perform the (singular) Legendre transform, which yields constraints.

3. Test the stability of the constraints using the Dirac algorithm [53].

15

We will go into more detail on this derivation later on, when we will perform the canonicalanalysis of the higher-dimensional Palatini action. For now, we will just cite the main resultsin order to understand the main ideas. Also, we will only comment the canonical analysis withimposing the time gauge, as was originally done by Holst [51]. In Holst’s analysis, the secondclass constraints mentioned below do not appear and he simply obtains a first class Hamiltoniantheory with Hamiltonian constraint, spatial diffeomorphism constraint, and Gauß constraint interms of the Ashtekar-Barbero variables.

The canonical analysis of the Holst action without the time gauge was first performed byBarros et Sa [54] and Alexandrov [55]. The analysis yields the usual set of first class constraints,the Hamiltonian constraint H, the spatial diffeomorphism constraint Ha, and the Gauß con-straint GIJ , where a is a spatial tensor index and I, J are SO(1, 3) Lie algebra indices in thefundamental representation. Moreover, two second class constraints appear, the simplicity con-straint Sab and a second class partner Dab. The simplicity constraint essentially enforces thatthe conjugate momentum to the Palatini connection is constructed from a vielbein, thus killingsuperfluous degrees of freedom. Its second class Partner Dab is a consequence of the stabilityanalysis and essentially enforces that a certain part of the torsion of the Palatini connection iszero.

Since the naive quantisation of second class constraints results in an empty Hilbert space[53], they have to be solved classically. When resorting to the Dirac bracket for this purpose,the connection becomes non-selfcommuting, which spoils the applicability of the loop quantumgravity quantisation methods. On the other hand, when solving the second class constraints, oneobtains the Ashtekar-Barbero formulation of general relativity and thus the goal which we wereaiming for. We remark that in order for the symplectic reduction with respect to Sab and Dab

to yield the Asthekar-Barbero formulation, it is mandatory to have a finite, non-zero Barbero-Immirzi parameter γ. Otherwise, the symplectic reduction would yield the ADM-formulationwith vierbeins [47], a theory with SO(1, 3) gauge invariance, but without a connection, as notedby Peldan [47].

This problem also carries over to higher dimensions, as we will show later. It was alreadyobserved by Han, Ma, Ding, and Qin [56] that when imposing the time gauge prior to performingthe canonical analysis of the Palatini action, one ends up with the ADM formulation withSO(D)-vielbeins also in higher dimensions. Moreover, adding a generalisation of the Holst termis difficult in higher dimensions. On the one hand, a naive extension of the Holst-modificationdoes not seem to exist in higher dimensions, since the epsilon symbol used to construct itacquires additional indices for which a non-vanishing and gauge invariant contraction linear inthe field strength could not be given so far. On the other hand, a generalisation of the Nieh-Yantopological density exists in 4n, n ∈ N , dimensions, but it contains higher powers of the fieldstrength and would thus significantly complicate the canonical analysis and most likely not yieldthe desired result.

3.2 Canonical analysis of the higher-dimensional Palatini action

Since no suitable extension of the Host term exists in higher dimensions exists, we will proceedby trying to gain a better understanding for the canonical structure of the Palatini action byperforming a detailed canonical analysis. The starting point for this analysis is the higher-dimensional Palatini action

SPalatini =s

2

∫MdD+1X eeµIeνJFµνIJ(A), (3.2.1)

16

where eµI is an SO(1, D) vielbein, e is the determinant of eµI , FµνIJ is the curvature of theSO(1, D) connection AaIJ , D is the number of spatial dimensions, and s = ±1 is the signature ofthe spacetime and we are using the mostly plus signature convention. We assume our spacetimeM to be globally hyperbolic and conclude that it is topologically of the form σ × R [57]. Thus,we can introduce a slicing of the spacetime manifold into Cauchy-hypersurfaces Σt := Xt(σ),where Xt : σ → M is an embedding defined by Xt(σ) = X(t, σ). t has the interpretation of atime and labels the Cauchy surfaces. Furthermore, we define the time evolution vector field Tµ

by LT t = 1, where L denotes the Lie derivative and split T according to

Tµ = Nnµ +Nµ, nµNµ = 0, (3.2.2)

where nµ is the unit normal on Σt. As in the usual treatments, we call N the lapse functionand Nµ the shift vector. Performing the D + 1 split, we obtain

S =

∫dtL =

∫dt

∫σdDx

(1

2π′aIJLTAaIJ −N˜H′ −NaH′a −

1

2λIJG

′IJ)

, (3.2.3)

where all spacetime tensor indices are pulled back to σ (µ, ν, . . .→ a, b, . . .) and

π′aIJ := 2n[I‖Ea|J ] := 2√qn[I‖ea|J ], nI = eIµn

µ, N˜ := −N/√q , (3.2.4)

G′IJ := Daπ′aIJ := ∂aπ

′aIJ + [Aa, π′a]IJ , (3.2.5)

H′ := 1

2π′aIKπ′bJKFabIJ and H′a :=

1

2π′bIJFabIJ . (3.2.6)

As we will see explicitly later on, H′ is the Hamiltonian constraint, H′a is the spatial diffeo-morphism constraint and G′IJ is the Gauß constraint. The primes appearing in the previousequations are to distinguish these constraints from constraints build purely in terms of AaIJ andits momentum, as we will do later on. Essentially, when performing the Legendre transform atthis point, we will obtain the constraint

SaIJ = πaIJ − π′aIJ , (3.2.7)

where πaIJ is the momentum conjugate to AaIJ . Thus, our theory will have two sets of canon-ically conjugate pairs, (AaIJ , π

aIJ) and (EaI , PIa ), and the canonical analysis will be more elab-

orate than when just dealing with the first pair, as we will do in the following. The canonicalanalysis including both pairs has been performed in [36] and we will not repeat it here, as it isnot necessary for understanding the following.

To prevent the second canonical pair from appearing, Peldan suggested [47], in close analogyto the Plebanski formulation of general relativity [58], to introduce a constraint which enforces(3.2.7), but only depends on πaIJ . This constraint is a canonical version of Plebanski’s constraint,which is heavily used in the spin foam literature [59, 60, 61] and referred to as the simplicityconstraint. The generalisation to higher dimensions of this constraint reads

SabM

[cMab

]=

∫σdDx

1

4cMab εIJKLMπ

aIJπbKL ≈ 0, (3.2.8)

the four dimensional version is obtained by removing the multi-index M := M1M2 . . .MD−3, i.e.setting D = 3. Here and in the following, we denote by ≈ a weak equality, i.e. equality up toconstraints. In the context of spin foam models, this constraint has been considered in higherdimensions by Freidel, Krasnov, and Puzio [62] and its kernel has been worked out both in the

17

classical context and for a spin foam quantisation. It is straight forward to carry their proofover to the canonical setting1 and we obtain that if the constraint is zero and sπaIJπbIJ ≈ 2qqab

is positive definite2, thenπaIJ = ±2

√qn[Ie

aJ ]. (3.2.9)

A complication arises in D = 3, where an additional topological sector appears. We will howeverneglect this sector in the following, since we are mainly interested inD > 3, where this sector does

not exist. Adding SabM

[cMab

]to (3.2.3), we can substitute all π′aIJ for πaIJ and thus completely

free the action of any dependence on EaI . Thus, the Legendre transform will yield only thecanonical pair (AaIJ , π

aIJ), while by introducing the additional simplicity constraint we haveensured equivalence with general relativity.

Beginning the Legendre transform, we read off the symplectic structure

AaIJ , πbKL = 2δbaδK[I δ

LJ ] (3.2.10)

and the constraints

H :=1

2πaIKπbJKFabIJ (3.2.11)

Ha :=1

2πbIJFabIJ (3.2.12)

GIJ := DaπaIJ := ∂aπ

aIJ + [Aa, πa]IJ (3.2.13)

SabM

:=1

4cMab εIJKLMπ

aIJπbKL. (3.2.14)

Applying the Dirac stability algorithm, we find one new constraint

DabM

= −εIJKLMπcIJ(π(a|KNDcπ

b)LN

). (3.2.15)

by calculating SabM

[cMab ],H[N ]

= DabM

[NcMab

]+ Sab

M[. . .]. (3.2.16)

The relevant part of the remaining constraint algebra is given by1

2GIJ [fIJ ],

1

2GKL[γKL]

=

1

2GIJ

[λIKγ

KJ − γIKλKJ

](3.2.17)

1

2GIJ [fIJ ],Ha[Na]

= 0, (3.2.18)

1

2GIJ [fIJ ],H[N ]

= 0 (3.2.19)

1

2GIJ [λIJ ], Sab

M[cMab ]

= Sab

M

[D−3∑i=1

λMiM ′icM1...Mi−1M

′iMi+1...MD−3

ab

](3.2.20)

1

2GIJ [λIJ ], Dab

M[dMab ]

= Dab

M

[D−3∑i=1

λMiM ′idM1...Mi−1M

′iMi+1...MD−3

ab

](3.2.21)

1At least at the classical level, the quantum constraint is more subtle as discussed in the second part of thisthesis

2This is a non-holonomic constraint and does not have to be taken into account in the constraint analysis.Essentially, it restricts the phase space to non-degenerate geometries.

18

Ha[fa],Hb[N b]

= Ha[(LfN)a]− 1

2GIJ [faN bFabIJ ] (3.2.22)

Ha[fa],H[N ] = H[LfN ] +GIJ [NfaπbIKFabJK ] (3.2.23)

Ha[Na], SabM

[cMab ]

= −SabM

[(LNc)Mab

](3.2.24)

H[M ],H[N ] = −1

2Ha[(M∂bN −N∂bM)πaIJπbIJ

]+s

1

4SabM

[(M∂aN −N∂aM)εIJKL

MπcIJFKLcb

]. (3.2.25)

It can furthermore be shown that SabM

and DabM

constitute a second class pair. However, many ofthese second class constraints are superfluous, as a count of the degrees of freedom they removeshows. In order to understand the structure of these constraints, we split

AaIJ = ΓaIJ + KaIJ + 2n[IKa|J ], (3.2.26)

where ΓaIJ is Peldan’s hybrid connection [47], see equation (4.1.11) for more details. The Knotation indicates that all internal indices are orthogonal to nI , i.e. KaIJn

I = 0. On theconstraint surface of the simplicity constraint, we calculate

f(a|IJπ|b)KLεIJKLMDab

M= −f(a|IJπ|b)KLε

IJKLM εABCDMπcABπ(a|C

EDcπb)DE

≈ −(D − 3)!(D − 1)KaIJFaIJ,bKLfbKL, (3.2.27)

andDabM≈ −2sεABC

DMn

AEcBE(a|CEb)EKcED, (3.2.28)

whereF aIJ,bKL = 4sEa[K ηL][JEb|I] (3.2.29)

and its inverse (F−1

)aIJ,bKL

=s

4EaAEbB

(ηAB ηK[I ηJ ]L − 2ηB[I ηJ ][K η

AL]

). (3.2.30)

are defined on tensors of the type f bKL which are antisymmetric in K,L, orthogonal on nI , andtracefree in the sense f bKLEbK = 0. πaIJ is defined as q−1qabπ

aIJ , with qab being the inverseof qab, and thus purely as a function of πaIJ . We conclude that using the “reduced” multiplierdMab = f(a|IJπ|b)KLε

IJKLM of the above tensor type is enough to fulfill the constraint DabM

= 0.Using this ansatz, we further calculate∫

dDx

∫dDy [fT(a|IJπb)KLε

IJKLM ](x)SabM

(x), DcdN

(y)

[gT(c|MNπd)OP εMNOPN ](y)

≈ 4(D − 1)2((D − 3)!)2

∫dDx fTaIJF

aIJ,bKLgTbKL. (3.2.31)

and conclude that SabM

and DabM

constitute indeed a second class pair. The remaining Poisson

brackets are not important, since we can ensure stability of DabM

by adjusting the multiplier of

the simplicity constraint. It can furthermore be shown that the trace part KaIJEaI , vanishes

by the Gauß constraint [36], which shows that the all physical information about the extrinsiccurvature has to be located in KaI .

Concluding, we found a constraint algebra with the first class subset H,Ha, and GIJ as wellas the second class constraints Sab

Mand Dab

M. Since the quantisation of second class constraints

is problematic, we will apply the algorithm of gauge unfixing in the next subsection in order toobtain a first class constraint algebra.

19

3.3 Gauge unfixing

3.3.1 Toy model

In this introduction, we want to illustrate the main idea of gauge unfixing which is to relate firstand second class constrained systems to each other. Given a first class constraint, we can alwaysperform a gauge fixing by introducing a new constraint which does not Poisson commute with theoriginal constraint. This new constraint then fixes a gauge in the sense that the previously firstclass constraint cannot generate gauge transformation any more (since it is now second class),but a certain representative of each gauge orbit is chosen which satisfies the newly introducedconstraint.

The idea of gauge unfixing is to relate the process of fixing a gauge by setting a certain phasespace variable to zero, i.e. using second class constraints, with the process of cutting out all thedependence of the theory on this variable, i.e. using gauge invariant observables, leading to afirst class system where the Hamiltonian and the observables are gauge invariant under the newfirst class constraints.

As an elementary example, we consider the system with configuration variables x1, x2,momenta P1, P2, Poisson bracket xi, Pj = δij and the first class constraint S = P2 ≈ 0. A

phase space function f(x1, x2, P1, P2) of this system is observable if and only if it is evaluated onthe constraint surface S ≈ 0 and Poisson commutes with S. Poisson commutativity implies thatthe phase space function does not depend on x2. On the other hand, dependence on P2 = 0 isequivalent with non-dependence on P2 since we can Taylor-expand the phase space function intoa series in P2 where only the constant term would survive. The argument also works when theconstraint says P2 = g(x1, P1), since every function f(x1, P1, P2 = g(x1, P1)) is already includedin the functions depending only on x1 and P1.

The constraint surface of the second class theory, where a gauge fixing constraint is present, is2-dimensional whereas the constraint surface of the first class theory is 3-dimensional. However,in the first class theory we have to mod out the gauge orbits of S. The different phase spaces havetherefore effectively the same size. It remains to be shown that the dynamics of the two theoriesare the same. Starting with the first class theory, the most general first class Hamiltonian reads

H = A(x1, P1) + λS (3.3.1)

where A represents the gauge invariant part and λ is the Lagrange multiplier for the first classconstraint S. Observables are phase space functions f(x1, P1). Clearly, imposing D = x2 ≈h(x1, P1) does not alter the observables or the Hamiltonian except for setting λ = h,A toensure stability of D. But the observables are independent of λ so that the gauge fixing doesnot alter the gauge invariant information of the system.

The connection with the gauge variant system S ≈ 0 is made by considering the observablepart of the system. In the first class system we saw that S renders both x2 and P2 unphysical,P2 by explicitly fixing P2 = 0 and x2 by modding out gauge orbits. In the second class system,both x2 and P2 are explicitly set to functions of x1 and P1. Therefore, the effective phase spaceand the dynamics of observables are the same when going from the gauge variant to the gaugeinvariant system. Note however that this statement is not true for non-observable quantities,i.e. functions depending on x2.

Before proceeding to the general theory and its application to general relativity, we wantto illustrate the idea in a toy model related to gravity. We start with the Hamiltonian systemdescribed by the previous phase space, the Hamiltonian

H = A(x1, P1) +1

2(x2)2 + λS, (3.3.2)

20

and the constraint S = P2 ≈ 0. Stability of S yields the new constraint D = x2 ≈ 0. Stabilityof D sets λ = 0 and we have the consistent Hamiltonian system

H = A(x1, P1) +1

2(x2)2, S ≈ D ≈ 0. (3.3.3)

We note that the Hamiltonian is only first class up to the second class constraint D whenPoisson commuted with S. We can change this by subtracting 1

2D2 from the Hamiltonian since

the square of a second class constraint is a first class constraint. We call this new Hamiltonian

H = A(x1, P1) + λS (3.3.4)

and note that it Poisson commutes with S independently of D. The idea is now that it shoudnot matter to forget about the constraint D since observables of the first class theory S = 0 donot depend on the value of x2 and the Hamiltonian is also first class in this theory.

The reason why this procedure works so well in this example is the choice of starting Hamil-tonian. As seen before, the constraint S ≈ 0 demands that the physics does not depend onx2. Therefore, the Hamiltonian should also not depend on x2, but it does. In the toy model,stability of S yields D = x2 ≈ 0 which can then be used to go to the new Hamiltonian H. Onthe other hand, we can gauge fix the first class system with D ≈ 0 and go back to the initialsecond class system.

3.3.2 General theory

After a seminal paper by Mitra and Rajaraman [63], the general theory of gauge unfixing wasdeveloped by Anishetty and Vytheeswaran [64, 65]. The main idea is to reverse the processof gauge fixing, which turns a first class system into a second class system by making a gaugechoice. Neglecting complications coming from Gribov copies, gauge fixing consists of choosinga second class partner to a first class constraint which specifies a point in the gauge orbit of thefirst class constraint. We do not claim credit for the following, but try to give a pedagogicaloverview in the case of a single second class pair. The general case can be found in [65].

In analogy to gravity, we will illustrate gauge unfixing using a Hamiltonian system withgeneralised coordinates xi, pi, Hamiltonian H(xi, pi), subject to the two constraints S(xi, pi) ≈ 0and D(xi, pi) ≈ 0 with Poisson bracket S,D = F (xi, pi) 6= 0. We are now going to forgetabout D. What does this mean for the Hamiltonian system?

Detection of Dirac observables:In the second class system, a Dirac observable is a function f(xi, pi) defined on the phase

space surface satisfying S ≈ D ≈ 0. In the first class system, the function f is defined on thesurface S ≈ 0 and has to satisfy f, S ≈ 0, where ≈ means weakly with respect to S only. Atany given point P in phase space, we can, at least locally, use S and D as coordinates on thephase space so that D parametrises the gauge orbit of S. We can then expand f into a Taylorseries in D as

f(xi, pi) = f0 + f1D + f2D2 + . . . , (3.3.5)

where the coefficients fn, n = 0, . . . ,∞ do not depend on D. We then have fn, S ≈ 0 andf, S ≈ 0 is equivalent to fn = 0 for all n ≥ 1. Thus, f, S ≈ 0 forces f not to depend onD. Accordingly, it does not matter if f is defined on the surface S ≈ 0 or S ≈ D ≈ 0. In otherwords, the set of functions defined on S ≈ D ≈ 0 is exactly the same set as the set of functionsdefined on S ≈ 0 with f, S ≈ 0.

21

Construction of Dirac observables:We introduce a linear projector P (modulo the first class constraint S ≈ 0) on the set of

phase space functions f(xi, pi) acting as

f := Pf := f − DF−1S, f+1

2!D2F−1S, F−1S, f

− 1

3!D3F−1S, F−1S, F−1S, f+ . . . . (3.3.6)

Given convergence, it is easy to verify that f , S ≈ 0. It has the properties

P2 ≈ P, P(αf + βg) = αP(f) + βP(g) and P(fg) = P(f)P(g), (3.3.7)

where α, β are constants and f, g phase space functions. Furthermore, we have

P(f),P(g) = P (f, gDB) and (3.3.8)

P(f), P(g),P(h)+ P(g), P(h),P(f)+ P(h), P(f),P(g) ≈ 0, (3.3.9)

where , DB denotes the Dirac bracket constructed from the second class pair (S,D). It followsthat P is a Dirac bracket homomorphism and therefore a canonical transformation.

Hamiltonian evolution:The Hamiltonian evolution has to preserve the constraint surface S ≈ 0. In the Hamiltonian

system we started with, this was the case on the surface S ≈ D ≈ 0. The problem is that Hmight depend on D which would lead away from the surface S ≈ 0. We use the above projectorto construct the gauge invariant Hamiltonian H which ensures stability of S. Time evolutionin the second class system using the Dirac bracket is equivalent to this treatment since theDirac bracket also annihilates the second class constraints present in the Hamiltonian. A gaugeinvariant function has therefore the same time evolution using either H and the Dirac bracketor H and the Poisson bracket.

Global issues and uniqueness:Problems concerning uniqueness and global treatment might arise when D is not linear in

the coordinates it constrains, leading to ambiguities in the solution to D(xi, pi) = 0. Since thiswill not be the case in general relativity, we will not deal with these problems here.

Remarks:

• In general, the result of applying the projection operator is a very complicated, maybeinfinite, series, which does not seem to help. Surprisingly, this series contributes only oneextra term to the Hamiltonian of (higher-dimensional) general relativity.

• Concerning quantisation, it seems that nothing is gained because the projector P is clas-sically a Dirac bracket homomorphism. Quantisation of the gauge invariant degrees offreedom is the same task as looking for a representation of the Dirac bracket. A rigorousquantisation of the projector in the case of general relativity is hopeless.

• However, a new option emerges in this framework: Since the first class system S = 0with Hamiltonian H and the set of gauge variant functions is equipped with the originalPoisson bracket, we can quantise this system and implement the constraint S at the levelof the kinematical Hilbert space as an operator equation. Of course, the task of findingobservables at the quantum level remains open on this route. Nevertheless, a (kinematical)quantisation can still be achieved, as opposed to starting with the Dirac bracket or the setof gauge invariant functions at the classical level.

22

3.3.3 Application to gravity

The application of gauge unfixing to the canonical formulation of general relativity derived inthe previous section bears many problems at first sight, including: Why should the power seriesinvolved in calculating the gauge invariant extension of the Hamiltonian constraint terminate?If not, does it converge? Is there a natural first class subset? What about problems arising dueto non-linearities in the gauge fixing condition?

However, as it turns out, gauge unfixing works very well for general relativity due to thefollowing special properties: First of all, the simplicity constraints constitute a natural choicefor a first class subset, since they are Abelian among themselves. Furthermore, the gauge fixingcondition Dab

Mis then linear in the degrees of freedom it constrains, i.e. the trace free part of

KaIJ vanishes. Also, the Dirac matrix will only depend on πaIJ , since DabM

is at most linear inAaIJ . On top of that, the Hamiltonian constraint is a second order polynomial in AaIJ , whichmeans that the power series involved in calculating its gauge invariant extension terminates afterthe second term. A little subtlety at this point is that the Dirac matrix

SabM, Dcd

N = F ab

McdN

(3.3.10)

is only invertible on a subset of its multipliers as described above. However, motivated by(3.2.31), we can still define an inverse as described in equation (3.3.12) in the next paragraph.Using this definition for the inverse Dirac Matrix, we calculate

H = H − 1

2DabM

(F−1

)Mab

Ncd D

cdN

(3.3.11)

for the gauge invariant extension of the Hamiltonian constraint.In order to work out the remaining problem of the non-invertibility, we make the ansatz(

F−1)Ncd,Mab = γεEFGHNπ(c|EF

(F−1

)d)GH,(a|AB πb)CDε

ABCDM (3.3.12)

for a free constant γ, where(F−1

)aIJ,bKL

:=s

4(D − 1)πaACπbBD

(πcECπcE

D − sηCD) (ηABηK[IηJ ]L − 2ηLAηB[IηJ ]K

)(3.3.13)

is expressed purely in terms of πaIJ . The πs with a lower spatial tensorial index are understoodas functions of πaIJ . This ansatz makes sense since the Dirac matrix is invertible on the subsetof multipliers that we need to access the trace free part of KaIJ . This part on the other hand isexactly what we need to remove from the Hamiltonian constraint to make it Poisson commutewith the simplicity constraint. After an explicit calculation, it turns out that

γ =1

4(D − 1)2((D − 3)!)2(3.3.14)

solves the problem, i.e. H does not depend on the trace free part of KaIJ on the constraintsurface. The remaining constraints can be trivially gauge unfixed since they Poisson commuteweakly with the simplicity constraint.

Concluding, we have transformed the second class constrained system describing generalrelativity into an equivalent first class system, which is more suitable as a starting point forquantisation. However, regarding the loop quantum gravity quantisation techniques, the internalgauge group is still non-compact and the Hilbert space methods are therefore not available. Inthe next chapter, we will therefore generalise the canonical tranformation which leads from theADM phase space to this formulation to be based on the compact gauge group SO(D + 1).

23

Chapter 4

Canonical transformation

The theory obtained via the canonical analysis and gauge unfixing can be constructed directlyby a canonical transformation starting from the ADM phase space. The use of canonical trans-formations will prove very useful in the following parts of this thesis as it is on the one hand veryconvenient since we can reuse results from earlier canonical treatments of general relativity andsupergravity, on the other hand the canonical formulations which will turn out to be necessaryto complete the quantisation of general relativity and supergravity in higher dimensions do notseem to have an (at least manifestly covariant) analogue on the Lagrangian side. We will choosephysical signature s and internal signature ζ independently in this section to emphasise thatthey do not need to agree in the canonical descriptions. For quantisation purposes of Lorentziangeneral relativity and supergravity, we will choose s = −1 and ζ = 1. The original work onwhich this chapter is based is [35].

4.1 Phase space extension and canonical transformation

In analogy to the usual 3 + 1-dimensional treatment where one first introduces the dreibein toobtain a gauge theory and then performs a canonical transformation to connection variables, weextend the ADM phase space using the variables

πaIJ and KbKL (4.1.1)

subject to the constraints

GIJ := 2Ka[IKπ

aK|J ] ≈ 0 and SabM

:=1

4εIJKLMπ

aIJπbKL ≈ 0, (4.1.2)

as well as the Poisson bracketKaIJ , π

bKL

= δba(δKI δ

LJ − δLI δKJ

),πaIJ , πbKL

= KaIJ ,KbKL = 0. (4.1.3)

The lowering of the spatial indices is performed using the spatial metric qab defined by

qqab :=s

2πaIJπbIJ , q =

(det(qqab

)) 1D−1

and qabqbc = δca. (4.1.4)

The algebra of the constraints (4.1.2) equals the algebra of Gauß and simplicity constraints inthe previous sections and thus is of first class. What remains to be checked is that the Poisson

24

brackets between the ADM variables qab, Pcd, considered as functions on the extended phase

space, remain unchanged, at least if GIJ ≈ SabM≈ 0. We express the extrinsic curvature using

the new variables asKab := − s

4√qπbKLKcKLq

ac (4.1.5)

and P ab as

P ab = −s√q(Kab − qabKc

c

)=

1

4

(qacπbKLKcKL − qabπcKLKcKL

). (4.1.6)

We then can verify that

qab, qcd =P ab, P cd

≈ 0,

qab, P

cd≈ δc(aδ

db), (4.1.7)

where qab and P cd are understood as functions of KaIJ and πbKL. The first Poisson bracketvanishes trivially, calculation of the third equality is straight forward. The bracket

P ab, P cd

yields on the surface Sab

M≈ 0 terms proportional to the expression K [a

IJπb]IJ ≈ 2ζK [a

IEb]I ,

which vanishes if GIJ ≈ 0 (The calculations are analogous to those given in [66] for extended

ADM with variables(Kia, E

bj

)). Finally, the ADM constraints can be expressed in terms of the

new variables as

Ha = −2qac∇bP bc = −1

2∇b(KaIJπ

bIJ − δbaKcIJπcIJ)

, (4.1.8)

H = −[s√q

(qacqbd −

1

D − 1qabqcd

)P abP cd +

√qR

]= − s

8√q

(π[a|IJπb]KLKbIJKaKL

)−√qR, (4.1.9)

where ∇a is the covariant derivative annihilating the spatial metric, and the Levi-Civita con-nection Γcab := 1

2qcd (∂aqbd + ∂bqad − ∂dqab) as well as the Ricci scalar R are expressed in terms

of πaIJ in the rather complicated but obvious way. By construction, their algebra on the ex-tended phase space remains unchanged up to terms proportional to the Gauß and simplicityconstraints. Moreover, the whole system of constraints is of the first class, since both, Ha andH, are Gauß and simplicity invariant. For the latter, note that both constraints only depend onthe combination KaIJπ

bIJ , which just gives another simplicity term when Poisson commutedwith Scd

M. Thus we found a viable extension of the ADM phase space. We want to stress that

the signature of the internal metric ηIJ can be chosen independently of the spacetime signature.In particular, we can formulate Lorentzian gravity with internal Euclidean signature. We willturn this extension into a connection formulation in the next steps.

First we rescale the canonical variables as

(β)πaIJ :=1

βπaIJ and (β)KbKL := βKbKL (4.1.10)

with a constant β ∈ R \ 0. Clearly, the Poisson brackets remain the same.In order to obtain a connection formulation, we use Peldan’s hybrid connection [47] expressed

as a function of πaIJ as

ΓaIJ :=2

D − 1πbKLn

Kn[I∂aπbJ ]L + ζηM[I ηJ ]KπbLM∂aπ

bLK + ζΓcabπbK[Iπc|J ]

K . (4.1.11)

An expression for the determinant of the spatial metric is given below, the πaIJ are defined asbelow (3.2.30) and can be explicitly written using the formula for the inversion of a matrix.Additionally,

nInJ ≈1

D − 1

(πaKIπaKJ − ζηIJ

)and ηIJ = ηIJ − ζnInJ (4.1.12)

25

complete the set of the needed expressions.This hybrid spin connection coincides on the constraint surface

SabM≈ 0 ⇔ πaIJ ≈ 2n[IEa|J ] (4.1.13)

with Peldan’s original hybrid connection [47] annihilating EaJ . Since ΓaIJ is a homogeneousfunction of degree zero in πaIJ , it does not matter whether or not we use the rescaled variablesto define it. Since nI is a function of EaJ , we have

∂a(β)πaIJ +

[Γa,

(β)πa]IJ≈ 0 (4.1.14)

and can rewrite the Gauß constraint as

GIJ = 0 +[

(β)Ka,(β)πa

]IJ≈ ∂a(β)πaIJ +

[(β)Aa,

(β)πa]IJ

, (4.1.15)

with the new connection(β)AaIJ := ΓaIJ + (β)KaIJ . (4.1.16)

We postulate the new brackets(β)AaIJ ,

(β)πbKL

= δba

(δKI δ

LJ − δLI δKJ

),

(β)πaIJ , (β)πbKL

=

(β)AaIJ ,(β)AbKL

= 0

(4.1.17)and express the extrinsic curvature as

√qKa

b := −s4

(β)πbIJ(

(β)AaIJ − ΓaIJ

). (4.1.18)

We again have to check if the ADM Poisson brackets are unchanged when calculated on theextended phase space, i.e. repeat the above calculations with “K replaced by A − Γ” (we setβ = 1, simplifying the following discussion; it can easily be generalised for any value of β). Theonly calculation affected by this replacement is the bracket

P ab, P cd

. Comparison with (4.1.6)

shows that P ab consists of terms qcd (AaIJ − ΓaIJ)πbIJ (up to index contraction). For these, wefind

qcd (AaIJ − ΓaIJ)πbIJ , qgh (AeKL − ΓeKL)πfKL

(4.1.19)

= 2δfaqcdqgh (AeIJ − ΓeIJ)πbIJ − 2δbeq

cdqgh (AaIJ − ΓaIJ)πfIJ

+ (AaIJ − ΓaIJ)πbIJqcd, (AeKL − ΓeKL)

qghπfKL

+qcdπbIJ

(AaIJ − ΓaIJ) , qgh

(AeKL − ΓeKL)πfKL

−qcdqgh [AaIJ ,ΓeKL+ ΓaIJ , AeKL]πbIJπfKL.

In all but the last line, we can now reverse the replacement “K → A−Γ” and obtain the resultof our previous calculation, namely that these terms vanish if Sab

M≈ 0 ≈ GIJ . For the remaining

26

terms in the last line, we first show that

0 ≈ πaIJ(x) AbIJ(x),ΓcKL(y)πdKL(y)

≈ πaIJ(x)AbIJ(x),ΓcKL(y)πdKL(y)

− πaIJ(x)

AbIJ(x), πdKL(y)

ΓcKL(y)

≈ πaIJ(x)AbIJ(x),Γboost

cKL (y)πdKL(y)− πaIJ(x)

AbIJ(x), πdKL(y)

ΓcKL(y)


cKL (y)πdKL(y) + πaIJ(x)

AbIJ(x), πdKL(y)

ΓboostcKL (y)

−πaIJ(x)AbIJ(x), πdKL(y)

ΓcKL(y)


cKL (y)πdKL(y) + 2πaIJ(x)δ(x− y)δdbΓboost

cIJ (y)

−2πaIJ(x)δ(x− y)δdbΓboostcIJ (y)


cKL (y)πdKL(y). (4.1.20)

Here, we noted that on the constraint surface SabM≈ 0, ΓcKLπ

dKL contains only the boost part

ΓboostcKL of the spin connection (with respect to nI) given by the first summand in (4.1.11). For

the other summands, which correspond to the rotational components ΓrotcKL, the combination

ΓrotcKLπ

dKL ≈ 0 thus has to be expressible in terms of the simplicity constraint. Since the lefthand side of the above Poisson bracket is simplicity invariant, we see that the boost part is theonly term which can possibly contribute. Thus, one only has to calculate the bracket in last lineand show that it vanishes. We split the calculation in some intermediate steps as

0 ≈ πaIJ(x)πcKL(y)AbIJ(x),Γboost

dKL (y)

(4.1.21)

=2

D − 1πaIJ(x)πcKL(y)

AbIJ(x),

πeMN︸︷︷︸(I)

nMn[K︸︷︷︸(II)

∂dπeL]N︸︷︷︸

(III)

(y)

,

where we used (4.1.11) and (4.1.12). As a first step, the Poisson bracket between AaIJ andπbKL = q−1qbcπ

cKL yields

AbIJ(x), πeMN (y) ≈ δD(x− y)[2q−1qbeη[I|M ηJ ]N − ζπbMNπeIJ

]. (4.1.22)

Note that in all terms where a πaIJ appears, it will be contracted such that the first, purelyrotational summand in the above equation does not contribute on the surface Sab

M≈ 0⇔ πaIJ ≈

2n[IEa|J ]. We find

2

D − 1πaIJ(x)πcKL(y)nM (y)n[K(y)∂dπ

eL]N (y)

AbIJ(x), πeMN (y)︸︷︷︸(I)

(4.1.23)

≈ − 4

D − 1δD(x− y)

(δabE

cK∂dnK − qqacEbK∂dnK)

and

2

D − 1πaIJ(x)πcKL(y)nM (y)n[K(y)πeMN (y)

AbIJ(x), ∂dπeL]N (y)︸︷︷︸

(III)

(4.1.24)

≈ 4

D − 1δD(x− y)

(δabE

cK∂dnK − qqacEbK∂dnK)

,

27

and see that the terms (I) and (III) precisely cancel. What remains to be shown is that(II) ≈ 0, which is straight forward using (4.1.22). We repeatedly used the identities nI∂bnI = 0and nI∂bE

aI = −EaI∂bnI .As a side remark which will become important in the fifth part of this thesis, the canonical

transformation presented here results in a boundary term which we assumed to vanish here,e.g. we are working on compact spatial slices without boundary. The boundary term has beenderived [35] and the resulting symplectic potential reads

2

β

∫σdDx∂a(E

aI δn

I). (4.1.25)

Note that this calculation does not directly prove that the transformation(

(β)πaIJ , (β)KbKL

)→(

(β)πaIJ , (β)AbKL)

is canonical. Instead, it proves that the system described by(

(β)πaIJ , (β)AbKL)

constitutes another viable extension of the ADM phase space, which is sufficient for our pur-poses. A full proof of the canonicity is given in [35] using a generating functional. The ADMconstraints can be treated as above and we again obtain a first class constraint algebra. Notethat Ha and H will involve ΓaIJ , Γcab and R expressed in terms of (β)πaIJ , and therefore becometremendously complicated. However, we can now choose for Ha and H preferably simple expres-sions constructed from (β)πaIJ and (β)AbKL which reduce to their corresponding ADM versionson the surface GIJ ≈ Sab

M≈ 0, and the algebra will still close. In particular, we can exploit our

knowledge from the previous chapter and modify the “simpler” constraints obtained there.

4.2 The Hamiltonian constraint in the new variables

Like in the previous case, the extension works equally well for Euclidean or Lorentzian internalsignature, independently of the external (physical) signature. The key to obtain a connectionformulation with a compact internal gauge group as well as a Lorentzian external signature isto add a correction term to the Hamiltonian constraint which changes the sign in front of theKKEE term in the ADM Hamiltonian (4.2.4). We observe that

1

(D − 1)2(β)πaKJ

((β)Db

(β)πcJL)

(β)πcKL(β)πbKJ

((β)Da

(β)πcJL)

(β)πcKL ≈ KIbE

aIK

JaE

bJ

(4.2.1)and (

1

D − 1(β)πaKI (β)πbKJ

(β)Da(β)πbIJ

)2

≈(KIaE

aI

)2. (4.2.2)

Now, in the case of Euclidean internal and Lorentzian external signature, the expression

H :=β2

√q

(−(β)HE +

1

2(β)Dab

M

((β)F−1

)Mab

Ncd

(β)DcdN−(β2 + 1

)KaIKbJE

a[IEb|J ]

)(4.2.3)

reduces to the Lorentzian ADM Hamiltonian constraint

H ≈ − 1

2√qEaIEbJRabIJ −

1√qEa[IEb|J ]KI

aKJb . (4.2.4)

The KKEE terms are understood as shown above as functions of πaIJ and AbKL and

(β)HE :=1

2(β)πaIK (β)πbJK (β)FabIJ (4.2.5)

denotes the Euclidean part of the Hamiltonian constraint. (A note for completeness: If internaland physical signature match, all terms in the Hamiltonian constraint H except β2KKEE

28

change sign, producing a (β2 − 1)KKEE term. For the choice β = ±1, we recover the gaugeunfixed theory.)

The diffeomorphism constraint

Ha =1

2(β)πbIJ (β)FabIJ , (4.2.6)

the Gauß constraintGIJ = (β)Da

(β)πaIJ (4.2.7)

and the simplicity constraint

SabM

=1

4εIJKLM

(β)πaIJ (β)πbKL (4.2.8)

can easily be written in the new variables and are unaffected up to a constant rescaling.From the classical point of view, this formulation is a genuine connection formulation of

general relativity. In the quantum theory, the quadratic simplicity constraint leads to anomaliesboth in the covariant [67] as well as in the canonical approach [36, 37]. Therefore, we wantto introduce a linear simplicity constraint in the canonical theory in the next chapter, inspiredby the new spin foam models [68, 69, 60, 67, 61, 70]. The linear simplicity constraint will alsobecome important when dealing with supergravity in the third part of this thesis. Nevertheless,a mechanism for resolving the anomaly resulting from using the quadratic simplicity constraintis discussed in chapter 8.

29

Chapter 5

The Linear simplicity constraint

In this chapter, we will generalise the canonical formulation of the previous chapters to includea normal field N I as an independent variable, as opposed to the normal nI(π) used before. Dueto the additional degrees of freedom in the normal field, the simplicity constraint is modified tobe linear in both N I and πaIJ and it can be shown that it removes, together with a conditionon the momentum of the normal, all the newly introduced degrees of freedom. The necessityfor using this formulation will only be apparent in the third part of this thesis, were it will be akey ingredient to deal with Majorana fields necessary for supergravity theories. Also, the linearconstraint brings the presented canonical quantisation closer to the new spin foam models. Theoriginal work from which this chapter is taken is [39].

5.1 Introduction

The new spin foam models [68, 69, 60, 67, 61, 70] of loop quantum gravity heavily use a timenormal as a key input to the construction of their theories, in the EPRL model, one of them,the simplicity constraint also turns out to be linear in the fluxes. From this perspective, it isnatural to ask whether we can also include such a time normal as an independent parameter inour canonical theory. With the power of hindsight, we also know that we have to introduce thiskind of formulation in order to deal with supergravities in the third part of this thesis. To makecontact to the covariant formulation, it is therefore of interest to ask whether, from the canonicalpoint of view, (a) the theory of formulated in the previous chapters can be reformulated usinga linear simplicity constraint, and (b) if so, whether the linear version of the constraint can bequantised without anomalies. Both of these questions will be answered affirmatively, the answerto (a) in this chapter and the answer to (b) in the second part of this thesis. The originalreferences are [40] and [39]. As we will show in the third part of this thesis, the use of the linearsimplicity constraints (already at the classical level) is probably the most convenient approachtowards constructing a connection formulation for supergravity theories in D + 1 dimensionswith compact gauge group. To answer (a) we will follow the approach of the previous chapterand construct the theory with linear simplicity constraint by an extension of the ADM phasespace.

Note that the linear constraints already have been introduced in a continuum theory in [71],yet the considerations there are rather different. The authors reformulate the action of thePlebanski formulation of general relativity using constraints which involve an additional threeform and which are linear in the bivectors, without giving a Hamiltonian formulation. Thischapter on the other hand will deal exclusively with the Hamiltonian framework.

Notice that we denote by s the spacetime signature and by ζ the internal signature, which

30

can be chosen independently as in the previous chapter. In particular, the gauge group SO(η)(with η = diag(ζ, 1, 1, ...)) can be chosen compact, irrespective of the spacetime signature. Thiswill be exploited when quantising the theory in the second part of this thesis, where we fix ζ = 1and therefore do not have to bother with the non-compact gauge group SO(1, D). There, weemploy the Hilbert space representation for the normal field derived in chapter 7 and then wefind quantum operators corresponding to the linear simplicity constraint and show that theseoperators (b) actually are of the first class and therefore can be implemented strongly.

5.2 Introducing linear simplicity constraints

Recall that the solution to the (quadratic) simplicity constraint in dimensions D ≥ 3 is givenby [62]1 Sab

M= 0 ⇔ (β)πaIJ = 2

βn[IEa|J ] and that nI is no independent field but determined by

the densitised hybrid vielbein EaI . We now postulate a new field N I , which will play the roleof this normal, together with its conjugate momentum PI , subject to the linear simplicity andnormalisation constraints

SaIM

:= εIJKLM NJ (β)πaKL, (5.2.1)

N := N INI − ζ. (5.2.2)

The solution to the linear simplicity constraint in any dimension D ≥ 3 is given by2 (β)πaIJ =2βN

[IEa|J ], with NIEaI = 0. We see that on the solutions, the physical information of (β)πaIJ

is encoded in the vielbein EaI , which in turn fixes the direction of N I completely. The re-maining freedom in choosing its length is fixed by the normalisation constraint N and we findN I = nI(E), i.e. the N I are no physical degrees of freedom. The same has to be assured forthe momenta P I , i.e. we should add additional constraints P I = 0. However, these extra con-ditions can be interpreted as (partial) gauge fixing conditions for (5.2.1,5.2.2), which then canbe removed by applying the procedure of gauge unfixing. We will take a short-cut and directly“guess” the theory such that the constraints (5.2.1,5.2.2) are implemented as first class, and wewill show that when solving these constraints, the momenta PI are automatically removed fromthe theory.

The theory we want to construct is very similar to the one in the previous chapter. It isdefined by the Poisson brackets (4.1.17) and

N I , PJ

= δIJ ,N I , NJ

= PI , PJ = 0, (5.2.3)

and, apart from the linear simplicity and normalisation constraints (5.2.1,5.2.2), is subject to

GIJ =1

2(β)Da

(β)πaIJ + P [INJ ], (5.2.4)

Ha =1

2(β)πbIJ∂a

(β)AbIJ −1

2∂b

((β)πbIJ (β)AaIJ

)+ PI∂aN

I , (5.2.5)

H = − s

8√q

[(β)π[a|IJ (β)πb]KL

((β)A− Γ

)bIJ

((β)A− Γ

)aKL

]−√qR. (5.2.6)

Note that the Hamilton constraint is the same3 as in equation (4.2.3), whereas the Gauß and

1In D = 3, an additional topological sector exists [62]. The above results hold in D = 3 only if this sector isexcluded by hand.

2Using the linear simplicity constraints, we automatically exclude the topological sector in D = 3.3In particular, we want to point out that it is not the Hamilton constraint for gravity coupled to standard

scalar fields φ, which would obtain additional terms ∼ p2√det q

+√

det qqabφ,aφ,b for the scalar field φ and itsconjugate momentum p which are missing here. In fact, these terms would spoil the constraint algebra, sinceH, Sa

IM and H,N would not vanish weakly.

31

vector constraint differ and are chosen such that they obviously generate SO(η) gauge transfor-mations and spatial diffeomorphisms respectively on all phase space variables. In the following,we prove its equivalence to the ADM formulation. First of all, we will show that the Poissonbrackets of the ADM variables Kab, qab in terms of the new variables (β)AaIJ , (β)πaIJ , N I ,and PI are still reproduced on the extended phase space up to constraints. This is non-trivial,since we changed both the simplicity and the Gauß constraint. For the linear simplicity andnormalisation constraints, the solution for (β)πaIJ is the same as in the case of the quadraticsimplicity (neglecting the topological sector), (β)πaIJ = 2

βn[IEa|J ], and terms which vanished due

to the quadratic simplicity constraint still vanish in the case at hand. For the Gauß constraint,note that the only terms appearing in the calculation are of the form ((β)A − Γ)[a

IJ(β)πb]IJ ,

which already vanish on the surface defined by the vanishing of the rotational parts of the Gaußconstraint [35]. Now, if the linear simplicity and normalisation constraints hold, we know thatN I = nI(E), i.e. the modification of the Gauß constraint P [INJ ] ≈ P [InJ ] on-shell just changesthe boost part of the Gauß constraint. Thus, the ADM brackets are reproduced on the surfacedefined by the vanishing of GIJ , Sa

IMand N .

Next, we will show that the algebra is of first class. Note that since GIJ and Ha gen-erate gauge transformations and spatial diffeomorphisms by inspection, their algebra with allother constraints obviously closes. The algebra of the linear simplicity and the normalisationconstraint is trivial. Moreover, the Hamilton constraint Poisson-commutes trivially with thenormalisation constraint and, since it depends only on the combination (β)πaIJ (β)AbIJ , we findH[N ], Sa

IM[sIMa ]

= Sa

IM[...]. We are left with the Poisson-bracket between two Hamilton

constraints. Since on-shell the ADM brackets are reproduced, the result is

H[M ],H[N ] ≈ H′a[qab (MN,b −NM,b)], (5.2.7)

where H′a = −2qac∇bP bc now denotes the ADM diffeomorphism constraint. Furthermore, itcan be shown that the vector constraint (5.2.5) correctly reduces to the ADM diffeomorphismconstraint if the Gauß and simplicity constraints hold, Ha ≈ H′a. Therefore, the algebra closes.What is left to show is that also the Hamilton constraint H on-shell reproduces the ADMconstraint, which will be made explicit in the next paragraph. Note that because of the modifiedGauß and simplicity constraints, again this is non-trivial since H is identical with (4.2.3).

As already pointed out at the beginning of this section, solving the linear simplicity andnormalisation constraints leads to

(β)πaIJ =2

βn[IEa|J ] and N I = nI(E). (5.2.8)

We make the Ansatz (β)AaIJ = ΓaIJ + βKaIJ with ΓaIJ defined as in (4.1.11). Then, thesymplectic potential reduces to [47]

1

2(β)πaIJ (β)AaIJ + PIN

I

≈ KaJ EaJ − KaIJE

aJ nI + PI nI

≈[KaJ − nJEaI

(KbK

IEbK + P I)]EaJ

:= K ′′aJ EaJ , (5.2.9)

where we have dropped total time derivatives and divergences. The notation K ′′aI is chosen tomake this section consistent with [36]. Upon this ansatz for (β)AaIJ , the Hamiltonian constraintreduces correctly to the ADM Hamiltonian constraint with internal SO(D + 1) (or SO(1, D))gauge symmetry as given in [47], however with KaI as the variable for the extrinsic curvature.

32

On the other hand, we note that the substitution KaI → K ′′aI in the Hamiltonian constraint afterdecomposing (β)AaIJ and solving the normalisation and simplicity constraints does not changethe constraint since the additional terms are proportional to nIE

aI = 0. The Gauß and spatialdiffeomorphism constraints also reduce to the expressions given in [47] with K ′′aI as the extrinsiccurvature variable. In a last step, one could solve the SO(D+ 1) Gauß constraint by going overto the spatial metric qab and its conjugate momentum P ab familiar from the ADM formulationas canonical variables. Thus, we have shown that we recovered the usual ADM formulation andthus the equivalence to general relativity.

The counting of the number of physical degrees of freedom goes as follows: The full phase

space consists of |A, π,N, P| = 2D2(D+1)

2 + 2(D + 1) degrees of freedom which are subject

to∣∣∣Ha,H, GIJ , SaIM ,N∣∣∣ = (D + 1) + D(D+1)

2 + D2(D−1)2 + 1 first class constraints. It is

most convenient to compare this to Peldan’s [47] extended ADM formulation (transformingunder SO(1, D)), with |E,K| = 2D(D+ 1) phase space degrees of freedom and the first class

constraints∣∣Ha,H, GIJ∣∣ = (D + 1) + D(D+1)

2 . In any dimension, the difference in phasespace degrees of freedom is exactly removed by the simplicity and normalisation constraint,

|A, π,N, P| − |E,K| = 2∣∣∣Sa

IM,N∣∣∣.

We remark that related formulations of general relativity, where a time normal appears as anindependent dynamical field, have already appeared in the literature [54, 72, 73]. The differencebetween these and our formulation is that while our formulation features both the simplicityconstraint and the time normal at the same time, the time normal appears in the process ofsolving the simplicity constraint without solving the boost part of the Gauß constraint in theother approaches. In other words, the time normal is an integral part of the simplicity constraintin our approach, not a concept emerging after its solution.

33

Part II

Loop quantum gravity in higherdimensions

34

Chapter 6

LQG-techniques in higherdimensions

The quantisation techniques for loop quantum gravity can be divided into two main sectors:First, kinematical techniques which deal with the construction of the kinematical Ashtekar-Lewandowski Hilbert space as well as the solution of the kinematical constraints, the Gauß,spatial diffeomorphism, and the simplicity constraints. In a second step, the Hamiltonian con-straint operator, or the Master constraint operator respectively, have to be regularised thereon.The original literature for the first part includes [14, 15, 16, 17, 18, 19], the second part is basedon results from Thiemann [12, 13, 74]. While the kinematical quantisation had already beenworked out independently of the dimension and compact gauge group, the regularisation inhigher dimensions was worked out in [37] and the simplicity constraint was discussed originallyin [37, 39]. In this chapter, we will deal with the kinematical Hilbert space, the regularisationof the Hamiltonian constraint, as well as the regularisation of the simplicity constraint. Thenext chapter will then be devoted to a closer investigation of the solution space of the simplicityconstraint, which becomes necessary due to quantisation anomalies as we will see at the end ofthis chapter. The original work on which this chapter is based is [37].

6.1 Kinematical quantisation techniques

6.1.1 Holonomies, fluxes, and right invariant vector fields

Since AaIJ is a Lie-algebra valued one form, it is natural to smear it over curves c : [0, 1] → σ.We choose the generators

(τIJ)K L =1

2

(δKI δJL − δKJ δIL

)(6.1.1)

as a basis for the adjoint representation of the so(D + 1) Lie algebra. As usual, we define theholonomy hc(A) ∈ SO(D + 1) along c by the equation

d

dshcs(A) = hcs(A)A(c(s)), hc0 = 1D+1, hc(A) = hc1(A), (6.1.2)

using the definitions cs(t) := c(st), s ∈ [0, 1], A(c(s)) := AIJa (c(s))τIJ ca(s). The formal solution

to this equation is given by

hc(A) = P exp

(∫cA

)= 1D+1 +

∞∑n=1

∫ 1

0dt1

∫ 1

t1

dt2 . . .

∫ 1

tn−1

dtnA(c(t1)) . . . A(c(tn)), (6.1.3)

35

with P being the usual path ordering symbol. In the following, the notion of a cylindricalfunction will be central. Essentially, this is a function which only depends on a finite amount of

holonomies defined along the edges of a graph γ, i.e. fγ(A) = Fγ

(he1(A), ..., he|E|(A)

), where

Fγ : SO(D + 1)|E| → C. It is clear that we can define the same function also on a refined graphγ′ containing all the edges of γ as compositions of edges in γ′. This idea leads to the notion ofcylindrical consistency which is discussed at length in [66]. Since we cannot add anything newto this topic, we will refer the reader to this reference and the short exposition given in [37].

Abstractly, the holonomies constitute a homomorphism of the path groupoid into the gaugegroup SO(D + 1), and this fact is used in the quantisation procedure. Essentially, we will beusing a Cauchy completion of the space of connections given by A := Hom(P,SO(D + 1)). Sincethe details of this procedure have been outlined in great detail in [66], we will refrain from goinginto detail here. The important observation for what follows is that one can define a measure onA := Hom(P,SO(D + 1)), as well as on A/G, G = SO(D + 1), a locally gauge invariant versionthereof. This measure is the Asthekar-Lewandowski measure, defined by

µ0[f(A)] =

∫A/G

dµ0(A)f(A) =

∫A/G

dµ0,γ(A)fγ (A)

=

∫SO(D+1)|E(γ)|

∏e∈E(γ)

dµH(he)

Fγ(h1, ..., h|E|

), (6.1.4)

with µH being the Haar probability measure on SO(D + 1). This measure has the importantproperty that it is cylindrically consistent, meaning that one can subdivide and reorient theedges along which the cylindrical functions f(A) are calculated without changing µ0[f(A)].

It is furthermore natural to smear πaIJ over (D − 1)-surfaces S since it is dual to a (D − 1)form. The Lie algebra indices are contracted with a smearing function nIJ(x), so that thestraight forward generalisation of the usual fluxes used in LQG reads

πn(S) :=

∫SnIJ(∗π)IJ =

∫SnIJπ

aIJεaa1...aD−1dxa1 ∧ . . . ∧ dxaD−1 . (6.1.5)

Following the standard regularisation procedure outlined in [37], we obtain for the action of theHamiltonian vector field associated to πn(S)

Y nγS

(S) [fγS ] =∑

e∈E(γS)

ε(e, S) [n(b(e)) he(A)]AB∂FγS

∂he(A)AB

(he1(A), ..., he|E(γS)|(A)

)=

∑e∈E(γS)

ε(e, S) nIJ(e ∩ S) ReIJfγS , (6.1.6)

where γS is a graph adapted to the surface S, ε(e, S) the usual type indicator function, b(e) thebeginning of the edge e, and A,B are SO(D + 1) indices in the representation of he. R

eIJ is the

right invariant vector field associated to the edge e, defined by

(RIJf) (he) :=

(d

dt

)t=0

f(etτIJhe). (6.1.7)

They generate the so(D + 1) Lie algebra if acting on the same edge as[ReIJ , R

e′KL

]=

1

2δe,e′ (ηJKR

eIL + ηILR

eJK − ηIKReJL − ηJLReIK) . (6.1.8)

36

6.1.2 Solution of the Gauß and spatial diffeomorphism constraints

The Gauß constraint can be solved either classically or quantum mechanically. Both ways ofsolving the constraint result in the requirement that one should use only gauge invariant spinnetworks as a basis in the gauge invariant Hilbert space. The spin networks associated to ourgauge group SO(D + 1) are constructed in the very same way as in the usual SU(2) case, sowe will refrain from repeating the exact definition here. A detailed description can be found in[66], for a short definition especially for the gauge group SO(D + 1) see [37]. In the following,we denote the gauge invariant Hilbert space by Hkin.

As for the spatial diffeomorphism constraint, we employ the same point of view as in theoriginal literature [19] and quantise it as a finite spatial diffeomorphism. Essentially, the spatialdiffeomorphisms have to be quantised as finite diffeomorphisms since the generator of infinitesi-mal diffeomorphisms does not exist as an operator on the kinematical Hilbert space. The deeperreason behind this is that the scalar product induced by the Ashtekar-Lewandowski measure isnot strongly continuous. As an example, we consider the scalar product between two equal spinnetworks 〈fγ , fγ〉 = 1. Now, if we add an additional edge e′ carrying a non-trivial represen-tation of the gauge group to one of the spin networks in a gauge invariant way, e.g. a wilson

loop, resulting in f ′γ′ , we have⟨fγ , f

′γ′

⟩= 0, independently of how small the additional edge is.

Thus, in the limit of shrinking the additional edge e′ of γ′ to zero, we have, morally speaking,

lime′→0

⟨fγ , f

′γ′

⟩= 0 6= 〈fγ , fγ〉. It follows that we cannot recover an operator corresponding

to the connection by shrinking a holonomy to zero length. Consequently, we do not have anoperator for the field strength of the connection and thus also not an operator corresponding tothe generator of spatial diffeomorphisms, which is expressed in terms of the field strength. Thisproblem is bypassed in the definition of the Hamiltonian constraint by defining the Hamiltonianconstraint on a finite triangulation which is then shrunk to zero. However, in the process oftaking the limit, the action of the constraint on spin network functions does not change any moreafter reaching a suitable refinement of the triangulation and when evaluated on diffeomorphisminvariant distributions.

On the other hand, the finite spatial diffeomorphisms φ have a natural representation U(φ) onspin network functions given by U(φ)fγ = fφ(γ), where φ(γ) is the action of the diffeomorphismon the graph γ. It turns out that this action can be extended to A and is free of anomalies[66]. The corresponding solution space has already been formulated independent of the spatialdimension. We just briefly summarise the main ideas here. The intuitive idea of the solutionspace would be a space consisting of spatial diffeomorphism averages of spin networks functions.Since the spatial diffeomorphisms are uncountable, such an expression would not be a cylindricalfunction any more. Nevertheless, a precise sense can given to such a construction by going to thedual space H∗kin of the kinematical Hilbert space Hkin, consisting of all functionals η : Hkin → C.On H∗kin, an average over the spatial diffeomorphism group is possible and the result of takinga spatial diffeomorphism average of a typical element of H∗kin, e.g. 〈fγ , ·〉, gives, up to somesubtleties discussed below, a finite expression when evaluated on a spin network function f ′γ′since the scalar product will only give a non-zero value when the graphs γ and γ′ exactly match.The subtleties mentioned above come from factoring out certain diffeomorphisms correctly whichleave the graph invariant. These are on the one hand diffeomorphisms with trivial action on thegraph. On the other hand, these diffeomorphisms could be graph symmetries. We refer to theoriginal literature [19] as well as the textbook treatment [66] for details.

37

6.2 Regularisation of the Hamiltonian constraint operator

6.2.1 Volume operator

In order to regularise the Hamiltonian constraint operator, we will have to resort to Pois-son bracket techniques for which the volume operator in higher dimensions is necessary. Theoperator has been constructed in [37] and its main features are the same as for the Ashtekar-Lewandowski volume operator in four dimensions. The main difference is is a slightly modifiedalgebraic structure, since the vielbeins from which the operator is constructed are in the adjointrepresentation in higher dimensions, while they are in the fundamental one in the usual four-dimensional formulation. Here, we just cite the main result and refer to [37] for the detailedregularisation procedure. Starting from the classical expression for the volume of a region R,

V (R) :=

∫RdDx√q, (6.2.1)

and expressing it in terms of fluxes, it turns out that the higher-dimensional generalisation ofthe Ashtekar-Lewandowski volume operator is given for D + 1 even by

V (R) =

∫RdDp |det(q)(p)| γ =

∫RdDpV (p)γ , (6.2.2)

V (p)γ =

(~2

) DD−1 ∑

v∈V (γ)

δD(p, v)Vv,γ , (6.2.3)

Vv,γ =

∣∣∣∣∣∣ iD

D!

∑e1,...,eD∈E(γ), e1∩...∩eD=v

s(e1, . . . , eD)qe1,...,eD

∣∣∣∣∣∣1

D−1

, (6.2.4)

qe1,...,eD =1

2εIJI1J1I2J2...InJnR

IJe R

I1K1e1 RJ1

e′1K1 . . . R

InKnen RJne′nKn

(6.2.5)

and for D + 1 odd by

V (R) =


∫RdDpV (p)γ , (6.2.6)

V (p)γ =

(~2

) DD−1 ∑

v∈V (γ)

δD(p, v)Vv,γ , (6.2.7)

V Iv,γ =

iD

D!

∑e1,...,eD∈E(γ), e1∩...∩eD=v

s(e1, . . . , eD)qIe1,...,eD , (6.2.8)

Vv,γ =∣∣∣V Iv,γ VI v,γ

∣∣∣ 12D−2

, (6.2.9)

qIe1,...,eD = εI I1J1I2J2...InJnRI1K1e1 RJ1

e′1K1 . . . R

InKnen RJne′nKn

. (6.2.10)

6.2.2 Poisson bracket identities and regularisation

The regularisation procedure for the Hamiltonian constraint relies on several Poisson bracketidentities which can be seen as generalisations of the identities used in the standard treatment[13]. We just briefly cite those identities here and comment on the regularisation. Details can befound in [37]. Essentially, when looking at the expression for the Hamiltonian constraint withdensity weight one

H :=β2

√q

(−(β)HE +

1

2(β)Dab

M

((β)F−1

)Mab

Ncd

(β)DcdN−(β2 + 1

)KaIKbJE

a[IEb|J ]

), (6.2.11)

38

with

(β)HE =(β)π[a|IK (β)πb]JK√

q(β)FabIJ , (6.2.12)

the problematic terms contain the inverse of the square root of the determinant of the metric,which is ill defined at points where the metric is degenerate, and the extrinsic curvature KaI ,which could be obtained by quantising an expression of type KaI = (A−Γ)aIJn

I , which howeverwould result in a very complicated operation of the constraint operator. Both problems can beresolved by using the Poisson bracket tricks

√qπaIJ(x) = −(D − 1)AaIJ , V (x, ε), (6.2.13)

K(x) := EaI(x)KaI(x) ≈ D − 1

DHE(x), V (x, ε) , (6.2.14)

EbI(x)KaI(x) ≈ (D − 1)

2DπbKL(x) AaKL(x), HE [1](x, ε), V (x, ε) , (6.2.15)

first introduced by Thiemann for D = 3 [12], which first allow to quantise (β)HE for D+ 1 evenby using

π[a|IKπb]JK√q

(x) ≈ 1

4(D − 2)!εabca1b1...an−1bn−1εIJKLI1J1...In−1Jn−1sgn(det e)(x) (6.2.16)

πcKL(x)πa1I1K1(x)πb1J1K1(x) . . . πan−1In−1Kn−1(x)πbn−1Jn−1

Kn−1(x)√qD−2(x).

and for D + 1 odd by

π[a|IKπb]JK√q

≈ 1

2(D − 2)!εaba1b1...anbnεIJKI1J1...InJnsgn(det e)

nKπa1I1K1πb1J1K1 . . . πanInKnπbnJn

Kn√qD−2 (6.2.17)

with

nI(x) ≈ 1

D!εa1b1...an+1bn+1εII1J1...In+1Jn+1sgn(det e)(x)

√qD−1(x)

πa1I1K1(x)πb1J1K1(x) . . . πan+1In+1Kn+1(x)πbn+1Jn+1

Kn+1(x). (6.2.18)

The basic idea is that since the volume operator and the holonomy are well defined operators onthe kinematical Hilbert space, we can now express (β)HE purely in terms of volume operatorsand holonomies, which results in a well defined quantisation for (β)HE . Using this operator, wecan further use the remaining Poisson bracket identities to construct the rest of the Hamiltonianconstraint. The explicit form of these operators and a toolbox to construct them has beenspelled out in great detail in [37] and we refrain from repeating the discussion here.

6.3 Regularisation of the simplicity constraint

In this section, we sketch the regularisation of the quadratic simplicity constraint operatorwhich has been performed in [37]. The regularisation of the linear simplicity constraint operatorfollows easily from these considerations and is sketched in section 8.3.1. In order to obtaina well defined quantum operator associated to the quadratic simplicity constraint Sab

M(x) =

14εIJKLMπ

aIJ(x)πbKL(x), we have to smear the momenta πaIJ(x) over (D − 1)-hypersurfacesSx, S′x as

CM (Sx, S′x) := limε,ε′→0

1

ε(D−1)ε′(D−1)εIJKLMπ

IJ(Sxε )πKL(S′xε′ ), (6.3.1)

39

where epsilon denotes a regulator which vanishes in the limit of the hypersurfaces shrinking tothe point x. Inserting the definition of the fluxes and quantising, we obtain

CM (Sx, S′x)γSS′fγSS′ = limε,ε′→0

1

ε(D−1)ε′(D−1)εIJKLM

∑e∈E(γSS′ );b(e)=x

∑e′∈E(γSS′ );b(e

′)=x

ε(e, Sx)ε(e′, S′x)RIJe RKLe′ fγSS′ , (6.3.2)

where b(e) is the beginning of e and γSS′ is a graph adapted to both S and S′ such that alledges are outgoing from x. At first sight, one might be worried that this operator is divergentsince the lower line does not depend on the regulators ε, ε′, whereas the first line is divergent forε→ 0 or ε′ → 0. However, since the quantum operator is defined as the limit of the regularisedoperator, it vanishes if and only if∑

e∈E(γSS′ );b(e)=x

∑e′∈E(γSS′ );b(e

′)=x

ε(e, Sx)ε(e′, S′x)RIJe RKLe′ fγSS′ = 0 (6.3.3)

and diverges otherwise. It therefore makes sense to ask for its kernel, which is well defined bythe above equation. In the following, we will drop the superscript x from the hypersurfaces tosimplify notation.

The kernel of the simplicity constraint acting on interior points of an edge can be easily

calculated, since, due to gauge invariance,[∑

e∈E(γ); v=b(e)ReIJ

]fγSS′ = 0, the above expression

reduces toCM (S, S′, x)p∗γfγ = ±p∗γSS′4ε

IJKLMRe1IJRe1KLp

∗γSS′γ

fγ (6.3.4)

when splitting the edge e at x into two outgoing edges e1 and e2. This implies that the operatorvanishes if and only if

τ[IJτKL] = 0, (6.3.5)

i.e. the constraint restricts the allowed representations of the edge through this equation con-taining the generators of SO(D + 1). The solution to this equation has been found by Freidel,Krasnov and Puzio [62], it restricts the representation to be “simple” (or in a different termi-nology, to be of class 1). The quantum algebra of Gauß and simplicity constraints acting onthe interior point of an edge can be shown to be non-anomalous by an explicit calculation. In-terestingly, the simplicity constraint exhibits a “soft” anomaly, in that it is not Abelian on thequantum level, but the commutator between two simplicity constraints acting on an edge givesanother simplicity constraint [37].

The action of the simplicity constraint operator on a vertex is significantly more complicated,since gauge invariance does not allow us to get rid of the sums over the edges. Nevertheless, thesituation can be simplified a lot by showing that at a vertex v, the simplicity constraint operatoralready implies that

Re[IJRe′

KL]fγ = 0 ∀e, e′ ∈ e′′ ∈ E(γ); v = b(e′′). (6.3.6)

In simple terms, in order to show this, we need to prove that the right invariant vector fieldslie in the linear span of the flux vector fields. The full proof has been given in [37] and we onlyremark that its key idea is to use small deformations of the hypersurfaces used to construct thefluxes in order to single out specific edges at the vertex. An appropriate summing over thesefluxes then yields the desired result. The vertex simplicity constraint thus reduces to

εIJKLMReIJRe′KLfγ = 0, (6.3.7)

since this equation is clearly sufficient for the vanishing of the full simplicity constraint operator.

40

It has been shown in [62] in the context of spin foam models that the strong solution ofvertex simplicity constraint leads to the unique Barret-Crane intertwiner, which is however inconflict with intertwiner degrees of freedom present in loop quantum gravity when solving thesimplicity constraint at the classical level in four dimensions. One therefore expects an anomalyto be present for the vertex simplicity constraint operator which results in too many degrees offreedom to be constrained. In order to explicitly show this anomaly, we calculate[

εIJKLMReIJRe′KL, ε

ABCDERe′ABR

e′′CD

]∼ δABCE

IJKM(Re′′)AB(Re)

IJ(Re′)KC . (6.3.8)

As discussed in [37], the projection of the right hand side onto the vertex simplicity constraintoperators is vanishing, thus it cannot be expressed as a sum of simplicity constraint operatorsand the anomaly is manifest. At this point, one could resort to a master constraint treatment[75] as proposed in [37]. However, we are going to take a different point of view in this thesisand try to understand this anomaly in more detail in section 8.2. It will be shown there thatthe anomaly can be avoided by restricting to a certain subclass of vertex simplicity constraintswhich form a closed algebra.

41

Chapter 7

Hilbert space techniques for thelinear simplicity constraint

In this relatively short chapter, we will derive new Hilbert space techniques necessary for theimplementation of the theory based on the linear simplicity constraint. The construction fromthe previous chapter will have to be supplemented by a Hilbert space which carries operatorsassociated to the, now independent, normal N I and its conjugate momentum PI . The basic ideaof the construction rests on the fact that by restricting to the case of Euclidean internal signa-ture, N I lives on the D-sphere, which is a compact space. Thus, analogous to the constructionof the Ashtekar-Lewandowski Hilbert space, it is possible to define a projective limit of Hilbertspaces, each of which carries the operators corresponding to N I and PI at a finite collection ofpoints. Taking the limit, which exists due to the normalisability of the Haar measure on theD-sphere, we arrive at a the desired Hilbert space. The original work from which this chapteris taken is [40].

We restrict to the case ζ = 1 in the following, because the kinematical Hilbert space forcanonical loop quantum gravity has been defined rigorously only for compact gauge groupslike SO(D + 1). For scalar fields like the Higgs field, two different constructions to obtain akinematical Hilbert space have been given. In the first one [21], a crucial role is played by pointholonomies Ux(Φ) := exp

(ΦIJ(x)τIJ

). The field ΦIJ , which is assumed to transform according

to the adjoint representation of G, is contracted with the basis elements τIJ of the Lie algebraof G and then exponentiated. Point holonomies are better suited for background independentquantisation than the field variables ΦIJ themselves, since the latter are real valued rather thanvalued in a compact set. Therefore, a Gaußian measure would be a natural choice for the innerproduct for ΦIJ , but this is in conflict with diffeomorphism invariance [21]. In the case at hand,this framework is not applicable, since N I transforms in the defining representation of SO(D+1)and therefore, there is no (or, at least no obvious) way to construct point holonomies from N I

in such a way that the exponentiated objects transform “nicely” under gauge transformations.The second avenue [66] for background independent quantisation of scalar fields leads to a dif-feomorphism invariant Fock representation and can be applied in principle. However, in thecase at hand there is a more natural route. On the constraint surface N = N INI − 1 = 0, N I

is valued in the compact set SD. In this case the measure problems can be circumvented bysolving N classically. The obvious choice of Hilbert space is then of course the space of squareintegrable functions on the D-sphere.

42

To solve N , we choose a second class partner N := N IPI ,N I(x)NI(x)− 1, NJ(y)PJ(y)

= N I(x)NI(x)δD(x− y) ≈ δD(x− y), (7.1)

where terms ∝ N have been dropped. N weakly Poisson commutes with the constraints: It isGauß invariant and transforms diffeomorphism covariantly, it trivially Poisson commutes withH (which neither depends on N I nor on PI), and a short calculations yields

SaIM

[sIMa ], N [n]

= SaIM

[nsIMa ]. (7.2)

Therefore, the algebra of the remaining constraints is unchanged when we solve N , N using theDirac bracket. Let ηIJ = ηIJ − NINJ/||N ||2 whence ηIJN

J = 0 also when ||N || 6= 1. ThenPI = ηIJP

J Poisson commutes with the normalisation constraint and thus is an observable justas N I . Since the Dirac matrix is diagonal, the Dirac brackets of PI , N

I coincide with theirPoisson brackets. We find

N I(x), NJ(y)DB

= 0,N I(x), PJ(y)

DB

= ηIJ(x)δD(x− y),P I(x), P J(y)

DB

= 2P [I(x)NJ ](x)δD(x− y), (7.3)

while the remaining brackets are unaffected. We see that unfortunately the Poisson alge-bra of the N I and PI does not close, it automatically generates also the rotation generatorLIJ = 2N[IPJ ] = 2N[I PJ ]. We therefore have to include it into our algebra. On the other handobviously LIJ ,N = 0 so that LIJ is also an observable and moreover the LIJ generate theLie algebra so(D + 1) while LIJ , NK = −2N[IδJ ]K so that the algebra of the NI , LIJ already

closes. Finally we have the identity LIJNJ = −||N ||2PI so that the N I , LIJ already determine

PI . We conclude that nothing is gained by considering the PI and that it is better to considerthe overcomplete set of observables N I , LIJ instead.

Consider, similar as in LQG, cylindrical functions F [N ] of the form F [N ] = Fp1,..,pn(N(p1), .., N(pn))where Fp1..pn is a polynomial with complex coefficients of the N I(pk), k = 1, .., n; I = 0, .., D+1.We define the operator NI(x) to be multiplication by NI(x) on this space. Let also ΛIJ be asmooth antisymmetric matrix valued function of compact support and define the operator

L[Λ] := 2

∫dDx ΛIJ(x) N[I(x) PJ ](x), (7.4)

where PJ(x) = iδ/δNJ(x). Notice that no factor ordering problems arise. The operator L[Λ]has a well defined action on cylindrical functions, specifically

L[Λ] F [N ] = 2in∑k=1

ΛIJ(pk)N[I(pk)∂/∂NJ ](pk) F [N ] (7.5)

and annihilates constant functions.Let dν(N) := cDd

D+1Nδ(||N ||2 − 1) the SO(D + 1) invariant measure on SD where theconstant cD makes it a probability measure. For a function cylindrical over the finite point setp1, .., pn we define the following positive linear functional

µ[F ] :=

∫dν(N1) .. dν(Nn) Fp1..pn(N1, .., Nn). (7.6)

43

Just as in LQG it is straightforward to show that the measure is consistently defined and thushas a unique σ−additive extension to the projective limit of the finite Cartesian products ofcopies of SD which in this case is just the infinite Cartesian product N :=

∏x S

D of copies ofSD [76], one for each spatial point. This space can be considered as a space of distributionalnormals because a generic point in it is a collection of vectors (N(x))x without any continuityproperties. The operator NI(x) is bounded and trivially self-adjoint because NI(x) is real valuedand SD is compact. To see that L[Λ] is self adjoint we let gΛ(p) = exp(ΛIJ(p)τIJ) where τIJare the generators of SO(D + 1). We define the operator(

U(Λ)F)

[N ] := Fp1..pn (gΛ(p1)N(p1), .., gΛ(pn)N(pn)) , (7.7)

which can be verified to be unitary and strongly continuous in Λ. It may be verified explicitlythat

L[Λ] =1

i[d

dt]t=0U [tΛ], (7.8)

whence L[Λ] is self-adjoint by Stone’s theorem [77]. Finally it is straightforward to check thatbesides the ∗-relations also the commutator relations hold, i.e. they reproduce i times theclassical Poisson bracket. We conclude that we have found a suitable background independentrepresentation of the normal field sector.

At each point p ∈ Σ, an orthonormal basis in the Hilbert space Hp = L2(SD, dν) is given

by the generalisations of spherical harmonics to higher dimensions Ξ~Kl (N), which are shortly

introduced in A (see [78] for a comprehensive treatment). An orthonormal basis for HN is given

by spherical harmonic vertex functions F~v,~l,

~~K(N) :=

∏v∈~v Ξ

~Kvlv

(N). Any cylindrical function F~v

can be written as a mean-convergent series of the form

F~v(N) =∑~l,~~K

a~v,~l,

~~KF~v,~l,

~~K(N) (7.9)

for complex coefficients a~v,~l,

~~K. The sum here runs for each v ∈ ~v over all values l ∈ N0 and

for each l over all ~K compatible with l. Following the construction in [21] we obtain thecombined Hilbert space for the scalar field and the connection simply by the tensor product,

HT = Hgrav⊗HN = L2(ASO(D+1), dµ

SO(D+1)AL )⊗L2(N , dµN ). An orthonormal basis in this space

is given by a slight generalisation of the usual gauge-variant spin network states (cf., e.g., [21]),where each vertex is labelled by an additional simple SO(D+1) irreducible representation comingfrom the normal field. This of course leads to an obvious modification of the definition of theintertwiners which also have to contract the indices coming from this additional representation.

44

Chapter 8

The simplicity constraint

In this chapter, we are going to discuss several approaches to solve the quadratic and linearsimplicity constraints in the context of the canonical formulations of higher-dimensional generalrelativity and supergravity described in this thesis. Since the canonical quadratic simplicityconstraint operators have been shown to be anomalous in any dimension D ≥ 3 in section 6.3,non-standard methods have to be employed to avoid inconsistencies in the quantum theory.We show that one can choose a subset of quadratic simplicity constraint operators which arenon-anomalous among themselves and allow for a natural unitary map of the spin networks inthe kernel of these simplicity constraint operators to the SU(2)-based Ashtekar-LewandowskiHilbert space in D = 3. The linear constraint operators on the other hand are non-anomalousby themselves, however their solution space will be shown to differ in D = 3 from the expectedAshtekar-Lewandowski Hilbert space. We comment on possible strategies to make a connectionto the quadratic theory. Also, we comment on the relation of our proposals to existing workin the spin foam literature and how these works could be used in the canonical theory. Weemphasise that many ideas developed in this chapter are certainly incomplete and should beconsidered as suggestions for possible starting points for more satisfactory treatments in thefuture. The original work from which this chapter is taken is [39]. Parts of the ideas are basedon [37].

8.1 Introduction

As shown in the first part of this thesis, gravity in any dimension D + 1 ≥ 3 can be formulatedas a gauge theory of SO(1, D) or of the compact group SO(D+ 1), irrespective of the spacetimesignature. The resulting theory has been obtained on two different routes, a Hamiltonian analysisof the Palatini action making use of the procedure of gauge unfixing, and on the canonical side byan extension of the ADM phase space. The additional constraints appearing in this formulation,the simplicity constraints, are well known. They constrain bivectors to be simple, i.e. theantisymmetrised product of two vectors. Originally introduced in Plebanski’s [58] formulationof general relativity as a constrained BF theory in 3 + 1 dimensions, they have been generalisedto arbitrary dimension in [62]. Moreover, discrete versions of the simplicity constraints are astandard ingredient of the spin foam approaches to quantum gravity [59, 60, 61], see [79, 80]for reviews, and recently were also used in group field theory [81]. Two different versions ofsimplicity constraints are considered in the literature, which are either quadratic or linear in thebivector fields. The quantum operators corresponding to the quadratic simplicity constraintshave been found to be anomalous both in the covariant [67] as well as in the canonical picture[82, 37]. On the covariant side, this lead to one of the major points of critique about the

45

Barrett-Crane model [59]: The anomalous constraints are imposed strongly1, which may implyerroneous elimination of physical degrees of freedom [53]. This triggered the development of thenew spin foam models [68, 69, 60, 67, 61, 70], in which the quadratic simplicity constraints arereplaced by linear simplicity constraints. The linear version of the constraints is slightly strongerthan the quadratic constraints, since in 3 + 1 dimensions the topological solution is absent. Thecorresponding quantum operators are still anomalous (unless the Barbero-Immirzi parametertakes the values γ = ±

√ζ, where ζ denotes the internal signature, or γ = ∞). Therefore, in

the new models (parts of) the simplicity constraints are implemented weakly to account for theanomaly. Also, the newly developed U(N) tools [83, 84, 85] have been recently applied to solvethe simplicity constraints [86, 87].

In this chapter, we are, for the most part, not going to import techniques for solving thesimplicity constraints which were developed in other contexts, but we are going to take an un-biased look at them from the canonical perspective in the hope of finding new clues for howto implement the constraints correctly. Afterwards, we will compare our results to existing ap-proaches from the spin foam literature and outline similarities and differences. We stress thatwill not arrive at the conclusion that a certain kind of imposition will be the correct one andthus further research, centered around consistency considerations and the classical limit, has tobe performed to find a satisfactory treatment for the simplicity constraints. Of course, in theend an experiment will have to decide which implementation, if any, will be the correct one.Since such experiments are missing up to now, the general guidelines are of course mathematicalconsistency of the approach, as well as comparison with the classical implementation of thesimplicity constraints in D = 3, where the usual SU(2) Ashtekar-Barbero variables exist. If asatisfactory implementation in D = 3 can be constructed, the hope would then be that thisprocedure has a natural generalisation to higher dimensions. Since parts of the very promisingresults developed from the spin foam literature are restricted to four dimensions, we will restrictourselves to dimension independent treatments in the main part of this chapter.

This chapter will be divided into three sections.In section 8.2, we will begin with investigating the quadratic simplicity constraint operatorswhich have been shown to be anomalous in section 6.3. It will be illustrated that choosing arecoupling scheme for the intertwiner naturally leads to a maximal closing subset of simplicityconstraint operators. Next, the solution to this subset will be shown to allow for a naturalunitary map to the SU(2) based Ashtekar-Lewandowski Hilbert space in D = 3 and we willfinish the first part with several remarks on this quantisation procedure. In section 8.3, we willanalyse the strong implementation of the linear simplicity constraint operators since they arenon-anomalous from start. The resulting intertwiner space will be shown to be one-dimensional,which is problematic because this forbids the construction of a natural 1-1 map to the SU(2)based Ashtekar-Lewandowski Hilbert space. In contrast to the quadratic case, the linear simplic-ity constraint operators will be shown to be problematic when acting on edges. We will discussseveral possibilities of how to resolve these problems and finally introduce a mixed quantisation,in which the linear simplicity constraints will be substituted by the quadratic constraints plus aconstraint ensuring the equality of the normals N I and nI(π). In section 8.4, we will compareour results to existing approaches from the spin foam literature. Finally, we will give a criticalevaluation of our results and conclude in section 8.5.

1Strongly here means that the constraint operator annihilates physical states, C |ψ〉 = 0 ∀ |ψ〉 ∈ Hphys.

46

8.2 The quadratic simplicity constraint operators

8.2.1 A maximal closing subset of vertex constraints

It has been shown in section 6.3 that the necessary and sufficient building blocks of the quadraticsimplicity constraint operator acting on a vertex v are given by

Re[IJRe′

KL]fγ = 0 ∀e, e′ ∈ e′′ ∈ E(γ); v = b(e′′). (8.2.1)

We note that these are exactly the off-diagonal simplicity constraints familiar from spin foammodels, see e.g. [62, 67]. Since not all of these building blocks commute with each other, i.e. theones sharing exactly one edge, we will have to resort to a non-standard procedure in order toavoid an anomaly in the quantum theory. The strong imposition of the above constraints, leadingto the Barrett-Crane intertwiner [59], was discussed in [62]. A master constraint formulation ofthe vertex simplicity constraint operator was proposed in [37], however apart from providing aprecise definition of the problem, this approach has not lead to a concrete solution up to now.

In this section, we are going to explore a different strategy for implementing the quadraticvertex simplicity constraint operators which is guided by two natural requirements:

1. The imposition of the constraints should be non-anomalous.

2. The imposition of the simplicity constraint operator should, at least on the kinematicallevel, lead to the same Hilbert space as the quantisation of the classical theory without asimplicity constraint. More precisely, there should exist a natural unitary map from thesolution space of the quadratic simplicity constraint operators Hsimple to the Ashtekar-Lewandowski Hilbert space HAL in D = 3.

The concept of gauge unfixing [88, 64, 65] which was successfully used in order to derive aclassical connection formulation of general relativity in the first part of this thesis was originallydeveloped in the context of anomalous gauge theory, where it was observed that first classconstraints can turn into second class constraints after quantisation [63, 89, 90, 91, 92]. This ishowever precisely what is happening in our case: The classically Abelian simplicity constraintsbecome a set of non-commuting operators due to the regularisation procedure used for the fluxes.The natural question arising is thus: How does a set of maximally commuting vertex simplicityconstraint operators look like?

Theorem 1. Given a N -valent vertex v ∈ γ, the set

εIJKLMRIJe1 R

KLe1 = . . . = εIJKLMR

IJeNRKLeN = 0 (8.2.2)

εIJKLM(RIJe1 +RIJe2

) (RKLe1 +RKLe2

)= 0

εIJKLM(RIJe1 +RIJe2 +RIJe3

) (RKLe1 +RKLe2 +RKLe3

)= 0

. . .

εIJKLM

(RIJe1 + . . .+RIJeN−2

)(RKLe1 + . . .+RKLeN−2

)= 0 (8.2.3)

generates a closed algebra of vertex simplicity constraint operators. Under the assumption thatno linear combinations with different multi-indices are allowed 2, the set is maximal in the sensethat adding new vertex constraint operators spoils closure.

2A superposition of different multi-indices seems to be highly unnatural since an anomaly with the Gaußconstraint has to be expected. We are however currently not aware of a proof which excludes this possibility fromthe viewpoint of a maximal closing set.

47

Proof. Closure can be checked by explicit calculation. In order to understand why the calculationworks, recall that right invariant vector fields generate the Lie algebra so(D + 1) as [37][

ReIJ , Re′KL

]=

1

2δe,e′ (ηJKR

eIL + ηILR

eJK − ηIKReJL − ηJLReIK) (8.2.4)

and thus infinitesimal rotations. The commutativity of (8.2.2) has been discussed in [37]. Fur-ther, we see that every element of (8.2.3) operates on (8.2.2) as an infinitesimal rotation. Thesame is also true for the elements in (8.2.3): Taking the ordering from above, every constraintoperates as an infinitesimal rotation on all constraints prior in the list. Since the commutator isantisymmetric in the exchange of its arguments, closure, i.e. commutativity up to constraints,of (8.2.3) follows.

To prove maximality of the set, we will show that, having chosen a subset of simplicityconstraints as given in (8.2.2) and (8.2.3), adding any other linear combination of the buildingblocks (8.2.1) spoils the closure of the algebra. To this end, we make the most general Ansatz∑

1≤i<j<Nαij εIJKLMR

IJi R

KLj (8.2.5)

for an N -valent vertex. Note that the diagonal terms (i = j) are proportional to (8.2.2) andtherefore do not have to be taken into account in the above sum, and that RN =

∑N−1i=1 Ri can

be dropped due to gauge invariance. Moreover, αij can be chosen such that for fixed j′ not allαij′ (i < j′) are equal. Otherwise, with αij′ := αj′ we find the term αj′εIJKLMR

IJ1...(j′−1)R

KLj′

in the sum, which can be expressed as a linear combination of (8.2.2) and (8.2.3) and thereforecan be dropped. Consider[

εIJKLMRIJ12R

KL12 , εABCDE

(α13R

AB1 RCD3 + α23R

AB2 RCD3 + ...

)]≈

N−1∑j=3

2α1j εIJKLMRIJ2 fKL AB

MNRMN1 εABCDER

CDj

+N−1∑j=3

2α2j εIJKLMRIJ1 fKL AB

MNRMN2 εABCDER

CDj

≈N−1∑j=3

2(α1j − α2j) εIJKLMRIJ2 fKL AB

MNRMN1 εABCDER

CDj , (8.2.6)

where we dropped terms proportional to (8.2.2) in the first and in the second step. For a closingalgebra, the right hand side of (8.2.6) necessarily has to be proportional to (a linear combinationof) simplicity building blocks (8.2.1). Terms containing Rj (j ≥ 3) have to vanish separately(In general, one could make use of gauge invariance to “mix” the contributions of different Rj .However, in the case at hand this will produce terms containing RN , which do not vanish if thecontributions of different Rjs did not already vanish separately).

We start with the case D = 3. The summands on the right hand sides of (8.2.6) areproportional to

δABCIJK (Rj)AB(R2)IJ(R1)KC , (8.2.7)

where we used the notation δI1...InJ1...Jn:= n! δI1[J1

δI2J2...δInJn]. To show that this expression can not be

rewritten as a linear combination of the of building blocks (8.2.1) we antisymmetrise the indices[ABIJ ], [ABKC] and [IJKC] and find in each case that the result is zero.

For D > 3, the summands are proportional to

δABCEIJKM

(Rj)AB(R2)IJ(R1)KC . (8.2.8)

48

Whatever multi-index E we might have chosen in the Ansatz (8.2.5), we can always restrictattention to those simplicity constraints in the maximal set which have the same multi-indexM = E. Then, the same calculation as in the case of D = 3 shows that the antisymmetrisationsof the indices [ABIJ ], [ABKC] and [IJKC] vanish.

Therefore, the only possibilities are (a) the trivial solution α1j = α2j = 0 or (b) α1j =α2j(6= 0), which implies that the terms on the right hand side of (8.2.6) are a rotated version ofεIJKLMR

IJ1 RKL2 . The second option (b) is, for j = 3, excluded by our choice of αij and we must

have α13 = α23 = 0. Next, consider j = 4 and suppose we have α14 = α24 := α′ 6= 0. Then,we can define α′34 := α34 − α′ and find the terms α′εIJKLMR

IJ123R

KL4 + α′34εIJKLMR

IJ3 RKL4 in

(8.2.5). The first term again is already in the chosen set, which implies we can set α14 = α24 = 0w.l.o.g. by changing α34 → α′34 (We will drop the prime in the following). This immediatelygeneralises to j > 4, and we have w.l.o.g. α1j = α2j = 0 (3 ≤ j < N).

Suppose we have calculated the commutators of εIJKLMRIJ1...iR

KL1...i (i = 2, ..., n) with (8.2.5)

and found that for closure, we need αij = 0 for 1 ≤ i ≤ n and i < j < N . Then,εIJKLMRIJ1...(n+1)RKL1...(n+1), εABCDE

N−1∑j=n+2

α(n+1)jRAB(n+1)R

CDj + ...

≈

N−1∑j=(n+2)

2α(n+1)j εIJKLMRIJ1...nf

KL ABMNR

MN(n+1)εABCDER

CDj , (8.2.9)

which, by the reasoning above, again is not a linear combination of any simplicity building blocksfor any choice of α(n+1)j , and therefore only the trivial solution α(n+1)j = 0 (n + 1 < j < N)leads to closure of the algebra.

8.2.2 The solution space of the maximal closing subset

In order to interpret this set of constraints, recall from [37] that the constraints in (8.2.2) are thesame as the diagonal simplicity constraints acting on edges of γ and can be solved by demandingthe edge representations to be simple, i.e. the highest weight ~Λ has the form (λ, 0, . . . , 0), λ ∈ N0.The remaining constraints (8.2.3) can be interpreted as specifying a recoupling scheme for theintertwiner ι at v: Couple the representations on e1 and e2, then couple this representation toe3, and so forth, see fig. 8.1. We call the intermediate virtual edges e12, e123, . . . and denote thehighest weights of the representations thereon by ~Λ12, ~Λ123, . . . Since we can use gauge invarianceat all the intermediate intertwiners in the recoupling scheme, e.g., Re1 +Re2 = Re12 , we have

εIJKLM(RIJe1 +RIJe2

) (RKLe1 +RKLe2

)= εIJKLMR

IJe12RKLe12

= 0 (8.2.10)

and thus that the representation on e12 has to be simple, i.e.

~Λ12 = (λ12, 0, ..., 0) λ12 = 0, 1, 2, ... (8.2.11)

Using the same procedure, all intermediate representations are required to be simple and theintertwiner is labeled by N − 3 “spins” λi ∈ N0. We call an intertwiner where all internal linesare labeled with simple representations simple.

Denote by ISU(2)N the set of N -valent SU(2) intertwiners and by ISpin(D+1)

s,N the set of N -valentsimple Spin(D + 1) intertwiners. Recalling that an N -valent SU(2) intertwiner can be expressedin the same recoupling basis and calling the intermediate spins ji, we see that the map

F : ISpin(D+1)s,N → ISU(2)

N

1

2λi 7→ ji (8.2.12)

49

12

1231234

4

2

1N

5

3

4

2

1N

5

3

l12

l12l123

l123

l1234

l1234

Figure 8.1: Recoupling scheme corresponding to the subset of quadratic vertex simplicity con-straint operators (8.2.3).

is unitary (with respect the scalar products induced by the respective Ashtekar-Lewandowskimeasures, see [37]). The motivation for the factor 1/2 comes from the fact that ~Λ = (1, 0) inD = 3 corresponds to the familiar j+ = j− = 1/2 and the area spacings of the SO(4) and theSU(2) based theories agree using this identification, cf [37].

8.2.3 Remarks

1. Since the choice of the maximal closing subset of the simplicity constraint operators isarbitrary, no recoupling basis is preferred a priori. On the SU(2) level, a change in therecoupling scheme amounts to a change of basis in the intertwiner space and therefore posesno problems. On the level of simple Spin(D + 1) representations however, a choice in therecoupling scheme affects the property “simple”, since the non-commutativity of constraintoperators belonging to different recoupling schemes means that kinematical states cannothave the property simple in both schemes.

2. There exist recoupling schemes which are not included in the above procedure, e.g., takeN = 6 and the constraints εR12R12 = εR34R34 = εR56R56 = 0 and couple the three re-sulting simple representations. The theorem should however generalise to those additionalrecoupling schemes.

3. It is doubtful if the action of the Hamiltonian constraint leaves the space of simple inter-twiners in a certain recoupling scheme invariant. To avoid this problem, one could usea projector on the space of simple intertwiners in a certain recoupling scheme to restrictthe Hamiltonian constraint on this subspace and average later on over the different recou-pling schemes if they turn out to yield different results. The possible drawbacks of such aprocedure are however presently unclear to the author and we refer to further research.

4. It would be interesting to check whether the dropped constraints are automatically solvedin the weak operator topology (matrix elements with respect to solutions to the maximalsubset).

5. The imposition of the constraints can be stated as the search for the joint kernel of a

50

maximal set of commuting generalised area operators

ArM [S] :=∑U∈U

√1

4εIJKLMπ

IJ(SU )πKL(SU )|. (8.2.13)

Notice, however, that for D > 3 these generalised area operators, just as the simplicityconstraints, are not gauge invariant while in D = 3 they are.

6. In D = 3 we have the following special situation:We have two classically equivalent extensions of the ADM phase space at our disposalwhose respective symplectic reduction reproduces the ADM phase space. One of them isthe Ashtekar-Barbero-Immirzi connection formulation in terms of the gauge group SU(2)with additional SU(2) Gauß constraint next to spatial diffeomorphism and Hamiltonianconstraint, and the other is our connection formulation in terms of SO(4) with additionalSO(4) Gauß constraint and simplicity constraint. Both formulations are classically com-pletely equivalent and thus one should expect that also the quantum theories are equivalentin the sense that they have the same semiclassical limit. Let us ask a stronger condition,namely that the joint kernel of SO(4) Gauß and simplicity constraint of the SO(4) theoryis unitarily equivalent to the kernel of the SU(2) Gauß constraint of the SU(2) theory. Toinvestigate this first from the classical perspective, we split the SO(4) connection and itsconjugate momentum (AIJ , πIJ) into self-dual and anti-selfdual parts Aj±, π

±j ) which then

turn out to be conjugate pairs again. It is easy to see that the SO(4) Gauß constraint GIJsplits into two SU(2) Gauß constraints G±j , one involving only self-dual variables and theother only anti-selfdual ones which therefore mutually commute as one would expect. TheSO(4) Gauß constraint now asks for separate SU(2) gauge invariance for these two sec-tors. Thus a quantisation in the Ashtekar-Isham-Lewandowski representation would yielda kinematical Hilbert space with an orthonormal basis T+

s+⊗T−s− where S± are usual SU(2)

invariant spin networks. The simplicity constraint, which in D = 3 is Gauß invariant andcan be imposed after solving the Gauß constraint, from classical perspective asks that thedouble density inverse metrics qab± = πaj±π

bk±δ

jk are identical. This is classically equivalentto the statement that corresponding area functions Ar±(S) are identical for every S. Thecorresponding statement in the quantum theory is, however, again anomalous because itis well known that area operators do not commute with each other. On the other hand,neglecting this complication for a moment, it is clear that the quantum constraint can onlybe satisfied on vectors of the form T+

s+ ⊗ T−s− for all S if s+, s− share the same graph and

SU(2) representations on the edges because if S cuts a single edge transversally then thearea operator is diagonal with an eigenvalue ∝

√j(j + 1) and we can always arrange such

an intersection situation by choosing suitable S. By a similar argument one can show thatthe intertwiners at the edges have to be the same. But this is only a sufficient conditionbecause in a sense there are too many quantum simplicity constraints due to the anomaly.However, the discussion suggests that the joint kernel of both SO(4) and simplicity con-straint is the closed linear span of vectors of the form T+

s ⊗ T−s for the same spin networks = s+ = s−. The desired unitary map between the Hilbert spaces would therefore simplybe Ts 7→ T+

s ⊗ T−s .

This can be justified abstractly as follows: From all possible area operators pick a maximalcommuting subset Ar±α using the axiom of choice (i.e. pick a corresponding maximal setof surfaces Sα). We may construct an adapted orthonormal basis T±λ diagonalising all of

51

them3 such that Ar±αT±λ = λαT

±λ . Now the constraint

Ar+α ⊗ 1 = 1⊗Ar−α

can be solved on vectors T+λ+⊗ T−λ− by demanding λ+ = λ−. The desired unitary map

would then be Tλ 7→ T+λ ⊗T

−λ . Thus the question boils down to asking whether a maximal

closing subset can be chosen such that the eigenvalues λ are just the spin networks s. Weleave this to future research.

7. In D 6= 3 the afore mentioned split into selfdual and anti-selfdual sector is meaningless andwe must stick with the dimension independent scheme outlined above. An astonishing fea-ture of this scheme is that after the proposed implementation of the simplicity constraints,the size of the kinematical Hilbert space is the same for all dimensions D ≥ 3! By “size”,we mean that the spin networks are labelled by the same sets of quantum numbers on thegraphs. Of course, before imposing the spatial diffeomorphism constraint these graphs areembedded into spatial slices of different dimension and thus provide different amounts ofdegrees of freedom. However, after implementation of the diffeomorphism constraint, mostof the embedding information will be lost and the graphs can be treated almost as abstractcombinatorial objects. Let us neglect here, for the sake of the argument, the possibilityof certain remaining moduli, depending on the amount of diffeomorphism invariance thatone imposes, which could a priori be different in different dimensions. In the case that theproposed quantisation would turn out to be correct, that is, allow for the correct semiclas-sical limit, this would mean that the dimensionality of space would be an emergent conceptdictated by the choice of semiclassical states which provide the necessary embedding infor-mation. A possible caveat to this argument is the remaining Hamiltonian constraint andthe algebra of Dirac observables which critically depend on the dimension (for instancethrough the volume operator or dimension dependent coefficients, see [35, 36]) and whichcould require to delete different amounts of degrees of freedom depending on the dimension.

This idea of dimension emergence is not new in the field of quantum gravity, however,it is interesting to possibly see here a concrete technical realisation which appears to beforced on us by demanding anomaly freedom of the simplicity constraint operators. Ofcourse, these speculations should be taken with great care: The number of degrees of free-dom of the classical theory certainly does strongly depend on the dimension and thereforethe speculation of dimension emergence could fail exactly when we try to construct thesemiclassical sector with the solutions to the simplicity constraints advertised above. Thiswould mean that our scheme is wrong. On the other hand, there are indications [93] thatthe semiclassical sector of the LQG Hilbert space already in D = 3 is entirely describedin terms of 6-valent vertices. Therefore, the higher valent graphs which in D = 3 couldcorrespond to pure quantum degrees of freedom, could account for the semiclassical de-grees of freedom of higher-dimensional general relativity. Since there is no upper limitto the valence of a graph, this would mean that already the D = 3 theory contains allhigher-dimensional theories!

Obviously, this puzzle asks for thorough investigation in future research.

8. The discussion reveals that we should compare the amount of degrees of freedom thatthe classical and the quantum simplicity constraint removes. This is a difficult subject,

3If the maximal set still separates the points of the classical configurations space, this should leave no roomfor degeneracies, that is the λα completely specify the eigenvector. We will assume this to be the case for thefollowing argument.

52

because there is no well defined scheme that attributes quantum to classical degrees offreedom unless the Hilbert space takes the form of a tensor product, where each factorcorresponds to precisely one of the classical configuration degrees of freedom. The follow-ing “counting” therefore is highly heuristic and speculative:

In the case D = 3, the classical simplicity constraints remove 6 degrees of freedom fromthe constraint surface per point on the spatial slice. In order to count the quantum degreesof freedom that are removed by the quantum simplicity constraint when acting on a spinnetwork function, we make the following, admittedly naive analogy:We attribute to a point on the spatial slice an N -valent vertex v of the underlying graphγ which is attributed to the spatial slice. This point is equipped with degrees of freedomlabelled by edge representations and the intertwiner. Every edge incident at v is sharedby exactly one other vertex (or returns to v which however does not change the result).Therefore, only half of the degrees of freedom of an edge can be attributed to one vertex.We take as edge degrees of freedom the bD+1

2 c Casimir eigenvalues of SO(D+ 1) labellingthe irreducible representation. The edge simplicity constraint removes all but one of theseCasimir eigenvalues, thus per edge bD−1

2 c edge degrees of freedom are removed. Further, agauge invariant intertwiner is labelled by a recoupling scheme involving N − 3 irreduciblerepresentations not fixed by the irreducible representations carried by the edges adjacentto the vertex in question, which are fully attributed to the vertex (there are N − 2 virtualedges coming from coupling 1,2 then 3 etc. until N but the last one is fixed due to gaugeinvariance). We take as vertex degrees of freedom these N − 3 irreducible representa-tions each of which is labelled again by bD+1

2 c Casimir eigenvalues. The vertex simplicityconstraint again deletes all but one of these eigenvalues, thus it removes (N − 3)bD−1

2 cquantum degrees of freedom. We conclude that the quantum simplicity constraint removes

(N − 3 +N

2)bD − 1

2c

quantum degrees of freedom per point (N valent vertex) where N−3 accounts for the vertexand N/2 for the N edges counted with half weight as argued above. This is to be comparedwith the classical simplicity constraint which removes D2(D−1)/2−D degrees of freedomper point. Requiring equality we see that vertices of a definitive valence ND are preferredin D spatial dimensions which for large D grows quadratically with D. Specifically forD = 3 we find N3 = 6. Thus, our naive counting astonishingly yields the same preferencefor 6-valent graphs in D = 3 as has been obtained in [93] by completely different methods.From the analysis of [93], it transpires that N3 = 6 has an entirely geometric origin andone thus would rather expect ND = 2D (hypercubulations) and this may indicate that ourcounting is incorrect.

8.3 The linear simplicity constraint operators

8.3.1 Regularisation and anomaly freedom

Since the linear simplicity constraint as defined in equation (5.2.1) is a vector of density weightone, it is most naturally smeared over (D− 1)-dimensional surfaces S. The regularisation of theobjects

Sb(S) :=

∫SbLM (x)εIJKLMN I(x)πaJK(x)εab1...bD−1

dxb1 ∧ ... ∧ dxbD−1 , (8.3.1)

53

where bLM denotes an arbitrary semianalytic smearing function of compact support, is thereforecompletely analogous to the case of flux vector fields. The corresponding quantum operator

Sb(S)f = Y εbN (S)f = p∗γS YεbNγS

(S)fγS = p∗γS

∑e∈γS

ε(e, S)εIJKLMbLM (b(e))N I(b(e))RJKe fγS

(8.3.2)

has to annihilate physical states for all surfaces S ⊂ σ and all semianalytic functions bIM ofcompact support, where pγ denotes the cylindrical projection and γS is a graph adapted to thesurface S. Since we can always choose surfaces which intersect a given graph only in one point,this implies that the constraint has to vanish when acting on single points of a given graph. In[37], it has been shown that the right invariant vector fields actually are in the linear span ofthe flux vector fields. Therefore, it is necessary and sufficient to demand that

εIJKLM bLM (b(e)) N I(b(e)) RJKe · fγ = 0 (8.3.3)

for all points of γ (which can be be seen as the beginning point of edges by suitably subdividingand inverting edges). Since N I acts by multiplication and commutes with the right invariantvector fields, see [40] for details, the condition is equivalent to4

RIJe · fγ = 0, (8.3.4)

i.e. the generators of rotations stabilising N I have to annihilate physical states. Before imposingthese conditions on the quantum states, we have to consider the possibility of an anomaly.Classically and before using the singular smearing of holonomies and fluxes, both, the linearand the quadratic simplicity constraints are Poisson self-commuting. The quadratic constraintis known to be anomalous both in the spin foam [67] as well as in the canonical picture [82, 37]and thus should not be imposed strongly. Also the linear simplicity constraint is anomalouswhen using a non-zero Immirzi parameter (at least if γ 6= 1 in the Euclidean theory. But γ = 1is ill-defined for SO(4), see e.g., [94]). Surprisingly, in the case at hand and without an Immirziparameter in four dimensions, we do not find an anomaly. But that is just because the generatorsof rotations stabilising N I form a closed subalgebra! Direct calculation yields, choosing (withoutloss of generality) γSS′ to be a graph adapted to both surfaces S, S′,

[SbγSS′ (S), Sb

′γSS′

(S′)]fγSS′ =

∑e∈γSS′

. . . RIJe ,∑

e′∈γSS′

. . . RABe′

fγSS′=

∑e∈γSS′

. . .[RIJe , R

ABe

]fγSS′

=∑e∈γSS′

. . . ηIK ηJLη

AC η

BD fKL CD

MNRMNe fγSS′

=∑e∈γSS′

. . . ηIK ηJLη

AC η

BD

(ηL][CδD]

[MδN ][K)RMNe fγSS′

=∑e∈γSS′

. . . RMNe fγSS′ , (8.3.5)

where the operator in the last line is in the linear span of the vector fields Sb(S). The classicalconstraint algebra is not reproduced exactly (the commutator does not vanish identically), but

4Use the decomposition of XIJ into its rotational (XIJ := ηKI ηLJXKL) and “boost” parts (XI := −ζNJXIJ)

with respect to NI in (8.3.3).

54

the algebra of quantum simplicity constraints closes, they are of the first class. Therefore, strongimposition of the quantum constraints does make mathematical sense.

Note that up to now, we did not solve the Gauß constraint. The quantum constraint algebraof the simplicity and the Gauß constraint can easily be calculated and reproduces the classicalresult[

Sb(S), GAB[ΛAB]]p∗γSfγS

=p∗γS

∑e′∈E(γS),v=b(e′)

ε(e′, S)εIJKLMbLM (v)N I(v)RJKe′ ,ΛAB(v)

∑e∈E(γS),v=b(e)

RABe +RABN

fγS=p∗γSΛAB(v)εIJKLMbLM (v)

∑e∈E(γS),v=b(e)

ε(e, S)(N I(v)

[RJKe , RABe

]+ ηI[ANB](v)RJKe

)fγS

=p∗γSΛAB(v)εIJKLMbLM (v)∑

e∈E(γS),v=b(e)

ε(e, S)(N I(v)2ηKARJBe + ηI[ANB](v)RJKe

)fγS

=S(−Λ·b)(S)p∗γSfγS . (8.3.6)

where we used RABN := 12

(NA ∂

∂NB−NB ∂

∂NA

). It follows that the simplicity constraint operator

does not preserve the Gauß invariant subspace (in other words, as in the classical theory, theGauß constraint does not generate an ideal in the constraint algebra). This implies that the jointkernel of both Gauß and simplicity constraint must be a proper subspace of the Gauß invariantsubspace. It is therefore most convenient to look for the joint kernel in the kinematical (nonGauß invariant) Hilbert space.

8.3.2 Solution on the vertices

Consider a slight modification of the usual gauge-variant spin network functions, where theintertwiners iv = iv(N) are square integrable functions of N I . Let v be a vertex of γ ande1, . . . , en the edges of γ incident at v, where all orientations are chosen such that the edges areall outgoing at v. Then we can write the modified spin network functions

Tγ,~l,~i

(A,N) := (iv(N)) ~K1... ~Kn

n∏i=1

(πlei (hei(A))

)~Ki ~K′i

(Mv) ~K′1... ~K′n

= tr(iv(N) · ⊗ni=1πlei (hei(A)) ·Mv

), (8.3.7)

where Mv contracts the indices corresponding to the endpoints of the edges ei and represents therest of the graph γ. These states span the combined Hilbert space for the normal field and theconnection HT = Hgrav⊗HN (cf. [40]) and they will prove convenient for solving the simplicityconstraints. Choose the surface S′ such that it intersects a given graph γ′ only in the vertexv′ ∈ γ′. The action of Sb(S′) on the vertex v′ of a spin network T

γ′,~l,~i(A,N) implies with (8.3.4)that

Sb(S′)γ′Tγ′,~l,~i(A,N) = 0

⇐⇒ tr((iv(N)τ IJπle

)· ⊗ni=1πlei (hei(A)) ·Mv

)= 0 ∀e at v′, (8.3.8)

where τ IJπle here denote the generators of SO(D + 1) in the representation πle of the edge e and

the bar again denotes the restriction to rotational components (w.r.t. N I). The above equationimplies that the intertwiner iv, seen as a vector transforming in the representation πle dual to

55

πle of the edge e, has to be invariant under the SO(D)N subgroup which stabilises the N I .By definition [78], the only representations of SO(D + 1) which have in their space nonzerovectors which are invariant under a SO(D) subgroup are of the representations of class one(cf. also appendix A), and they exactly coincide with the simple representations used in spinfoams [62]. It is easy to see that the dual representations of simple representations are simplerepresentations. Therefore, all edges must be labelled by simple representations of SO(D + 1).Moreover, SO(D) is a massive subgroup of SO(D + 1) [78], so that the (unit) invariant vectorξle(N) in the representation πle is unique, which implies that the allowed intertwiners iv(N) aregiven by the tensor product of the invariant vectors of all n edges and potentially an additionalsquare integrable function Fv(N), iv(N) = ξle1 (N) ⊗ ... ⊗ ξlen (N) ⊗ Fv(N). Going over tonormalised gauge invariant spin network functions implies that Fv(N) = 1, and the resultingintertwiner space solving the simplicity and Gauß constraint becomes one-dimensional, spannedby Iv(N) := ξle1 (N)⊗ ...⊗ξlen (N). We will call these intertwiners and vertices coloured by themlinear-simple. For an instructive example of the linear-simple intertwiners, consider the definingrepresentation (which is simple since the highest weight vector is Λ = (1, 0, . . . , 0), cf. appendixA). The unit vector invariant under rotations (w.r.t. N I) is given by N I and for edges in thedefining representation incoming at v we simply contract hIJe NJ . If the constraint is acting onan interior point of an analytic edge, this point can be considered as a trivial two-valent vertexand the above result applies. Since this has to be true for all surfaces, a spin network functionsolving the constraint would need to have linear-simple intertwiners at every point of its graphγ, i.e. at infinitely many points, which is in conflict with the definition of cylindrical functions(cf. [21]). In the next section, we comment on a possibility of how to implement this idea.

8.3.3 Edge constraints

As noted above, the imposition of the linear simplicity constraint operators acting on edges isproblematic, because it does not, as one might have expected, single out simple representations,but demand that at every point where it acts, there should be a linear-simple intertwiner. Theproblem with this type of solution is that all intertwiners, even trivial intertwiners at all interiorpoints of edges, have to be linear-simple, which is however in conflict with the definition of acylindrical function, in other words, there would be no holonomies left in a spin network becauseevery point would be a N -dependent vertex.

It could be possible to resolve this issue using a rigging map construction [95, 96, 97] of thetype

η(Tγ,~l,~lN ,~i

)[Tγ′,~l′,~l′N ,

~i′ ] := limPγ3pγ→∞

C(pγ , Tγ , Tγ′

) ⟨Tpγ

γ,~l,~lN ,~i, T

γ′,~l′,~l′N ,~i′

⟩kin

, (8.3.9)

where Pγ is the set of finite point sets p of a graph γ, p = xiNi=1|xi ∈ γ ∀ i,N < ∞. Pγ

is partially ordered by inclusion, q p if p is a subset of q, so that the limit is meant in thesense of net convergence with respect to Pγ . By the prescription T

pγ

γ,~l,~lN ,~iwe mean the projection

of Tγ,~l,~lN ,~i

onto linear-simple intertwiners at every point in p and C(pγ , Tγ , Tγ′

)is a numerical

factor. Assuming this to work, consider any surface S intersecting γ′. We (heuristically) find

η(Tγ)[Sb(S)Tγ′ ] = limPγ3pγ→∞


) ⟨T pγ , S

b(S)Tγ′⟩

kin

= limPγ3pγ→∞


) ⟨[Sb(S)]†T pγ , Tγ′

⟩kin

= limPγ3pγ→∞


) ⟨Sb(S)T pγ , Tγ′

⟩kin

= 0, (8.3.10)

56

since the intersection points of S with γ will eventually be in pγ and Sb(S) is self-adjoint.We were however not able to find such a rigging map with satisfactory properties. It is

especially difficult to handle observables with respect to the linear simplicity constraint and toimplement the requirement, that the rigging map has to commute with observables. It thereforeseems plausible to look for non-standard quantisation schemes for the linear simplicity constraintoperators, at least when acting on edges. Comparison with the quadratic simplicity constraintsuggests that also the linear constraint should enforce simple representations on the edges, seethe following remarks as well as section 8.3.5 for ideas on how to reach this goal.

8.3.4 Remarks

The intertwiner space at each vertex is one-dimensional and thus the strong solution of the un-altered linear simplicity constraint operator contrasts the quantisation of the classically imposedsimplicity constraint at first sight. A few remarks are appropriate:

1. One could argue that the intertwiner space at a vertex v is infinite-dimensional by takinginto account holonomies along edges e′ originating at v and ending in a 1-valent vertexv′. Since e′ and v′ are assigned in a unique fashion to v if the valence of v is at least 2,we can consider the set v, e′, v′ as a new “non-local” intertwiner. Since we can label e′

with an arbitrary simple representation, we get an infinite set of intertwiners which areorthogonal in the above scalar product. This interpretation however does not mimic theclassical imposition of the simplicity constraints or the above imposition of the quadraticsimplicity constraint operators.

2. The main difference between the formulation of the theory with quadratic and linearsimplicity constraint respectively is the appearance of the additional normal field sectorin the linear case. Thus one could expect that one would recover the quadratic simplicityconstraint formulation by ad hoc averaging the solutions of the linear constraint over thenormal field dependence with the probability measure νN defined in [40]. Indeed, if onedoes so, then one recovers the solutions to the quadratic simplicity constraints in termsof the Barrett-Crane intertwiners in D = 3 and higher-dimensional analogs thereof as hasbeen shown long ago by Freidel, Krasnov, and Puzio [62]. Such an average also deletes thesolutions with “open ends” of the previous item by an appeal to Schur’s lemma. Since aftersuch an average the N dependence of all solutions disappears, we can drop the µN integralin the kinematical inner product since µN is a probability measure. The resulting effectivephysical scalar product would then be the Ashtekar-Lewandowski scalar product of thetheory between the solutions to the quadratic simplicity constraints. Such an averagingwould also help with the solution of the edge constraints, since a 2-valent linear-simpleintertwiner is averaged as ∫

SDdν(N) ξαl (N)ξβl (N) =

1

dπlδαβ, (8.3.11)

thus yielding a projector on simple representations.

3. It can be easily checked that the volume operator as defined in [37], and therefore alsomore general operators like the Hamiltonian constraint, do not leave the solution space tothe linear (vertex) simplicity constraints invariant. A possible cure would be to introduce aprojector PS on the solution space and redefine the volume operator as V := PSV PS . Suchprocedures are however questionable on the general ground that anomalies can always beremoved by projectors.

57

4. If one accepts the usage of the projector PS , calculations involving the volume operatorsimplify tremendously since the intertwiner space is one-dimensional. We will give a fewexamples which can be calculated by hand in a few lines, restricting ourselves to thedefining representation of SO(D+ 1), where the SO(D)N invariant unit vector is given byN I .

Having direct access to N I , one can base the quantisation of the volume operator on theclassical expression

det q =

∣∣∣∣ 1

D!εIJ1...JDN

I(πa1K1J1NK1

). . .(πaDKDJDNKD

)εa1...aD

∣∣∣∣ 1D−1

. (8.3.12)

In the case D + 1 uneven, this choice is much easier than the expression quantised in[37]. In the case D + 1 even, the above choice is of the same complexity5 as the one in[37], but leads to a formula applicable in any dimension and therefore, for us, is favoured.Proceeding as in [37], we obtain for the volume operator

V (R) =


∫RdDp V (p)γ , (8.3.13)

V (p)γ =

(~2

) DD−1 ∑

v∈V (γ)

δD(p, v)Vv,γ , (8.3.14)

Vv,γ =

∣∣∣∣∣∣ iD

D!

∑e1,...,eD∈E(γ), e1∩...∩eD=v

s(e1, . . . , eD)qeq ,...,eD

∣∣∣∣∣∣1

(D−1)

, (8.3.15)

qe1,...,eD = εIJ1...JDNI(RK1J1e1 NK1

). . .(RKDJDeD

NKD

). (8.3.16)

Note that the operator qe1,...,eD is built from D right invariant vector fields. Since theseare antisymmetric, qTe1,...,eD = (−1)D qe1,...,eD . In the case at hand, we have to use theprojectors PS to project on the allowed one-dimensional intertwiner space, the operatorPS qPS therefore has to vanish for the case D+ 1 even (an antisymmetric matrix on a one-dimensional space is equal to 0). However, the volume operator depends on q2, and PS q2PSactually is a non-zero operator in any dimension, though trivially diagonal. Therefore, alsoV is diagonal.

The simplest non-trivial calculation involves a D-valent non-degenerate (i.e. no threetangents to edges at v lie in the same plane) vertex v where all edges are labelled by thedefining representation of SO(D+1) and thus the unique intertwiner which we will denoteby∣∣NA1 . . . NAD

⟩. We find

qe1,...,eD∣∣NA1 . . . NAD

⟩= s(e1, ..., eD)

(−1

2

)D ∣∣NIεIA1...AD

⟩,

qe1,...,eD∣∣NIε

IA1...AD⟩

= s(e1, ..., eD)

(1

2

)DD!∣∣NA1 . . . NAD

⟩,

Vv∣∣NA1 . . . NAD

⟩=

((1

4

)DD!

) 12(D−1) ∣∣NA1 . . . NAD

⟩, (8.3.17)

5Up to (N)D+1, but in the chosen representation N acts by multiplication and therefore is less problematicthan additional powers of right invariant vector fields.

58

i.e. for those special vertices, the volume operator preserves the simple vertices. Forvertices of higher valence and/or other representations, we need to use the projectors. Ofspecial interest are the vertices of valence D+ 1 (triangulation) and 2D, where every edgehas exactly one partner which is its analytic continuation through v (cubulation). We find

Vv∣∣NA1 . . . NAD+1

⟩=

((1

4

)D(D + 1)!

) 12(D−1) ∣∣NA1 . . . NAD+1

⟩,

Vv∣∣NA1 . . . NA2D , cubic

⟩=

((1

2

)D(D)!

) 12(D−1) ∣∣NA1 . . . NA2D , cubic

⟩.(8.3.18)

The dimensionality of the spatial slice now appears as a quantum number like the spinslabelling the representations on the edges and it could be interesting to consider a largedimension limit in the spirit of the large N limit in QCD.

5. When introducing an Barbero-Immirzi parameter in D = 3 [36], i.e. using the linearconstraint εIJKLN

JπaKL ≈ 0 while having AaIJ(x), (γ)πbKL(y) = 2δbaδK[I δ

LJ ]δ

(D)(x −y) with (γ)πaIJ = πaIJ + 1/(2γ)εIJKLπaKL, the linear simplicity constraint operatorsbecome anomalous unless γ = ±

√ζ, the (anti)self-dual case, which however results in

non-invertibility of the prescription (γ). Repeating the steps in section 8.3.1, we find thatthese anomalous constraints require εIJKLN

I(RKLe − 1/(2γ)εKLMNRMNe ) · fγ = 0. Since

εIJKLNI(RKLe − 1/(2γ)εKLMNRMN

e ) do not generate a subgroup, the constraint can notbe satisfied strongly if the edge e transforms in an irreducible representation of SO(D+ 1)(by definition, the representation space does not contain an invariant vector).

In order to figure out the “correct” quantisation, one can try, in analogy to the strategy forthe quadratic simplicity constraints, to weaken the imposition of the constraints at the quantumlevel. The basic difference between the linear and the quadratic simplicity constraints is thatthe time normal N I is left arbitrary in the quadratic case and fixed in the linear case. In orderto loose this dependence in the linear case, one could average over all N I at each point in σ,which however leads to the Barrett-Crane intertwiners as described above. In analogy to thequadratic constraints, we could choose the subset

εIJKLMNJ(RKLe1 +RKLe2

)= 0

εIJKLMNJ(RKLe1 +RKLe2 +RKLe3

)= 0

. . .

εIJKLMNJ(RKLe1 + . . .+RKLeN−2

)= 0 (8.3.19)

for each N -valent vertex plus the edge constraints. As above, the choice of the subset specifiesa recoupling scheme and the imposition of the constraints leads to the contraction of the virtualedges and virtual intertwiners of the recoupling scheme with the SO(D)N -invariant vectorsξlei (N) and their complex conjugates ξlei (N), see fig. 8.2. Gauge invariance can still be used

at each (virtual) vertex in this calculation in the form∑

i Rei = 0, which is sufficient sinceonly Rei appears in the linear simplicity constraints. If we now integrate over each pair ofξlei (N) “generated” by the elements of the proposed subset of the simplicity constraint operatorsseparately, we obtain projectors on simple representations for each of the virtual edges in therecoupling scheme. The integration over N I for the edge constraints yields projectors on simplerepresentations in the same manner. Finally, we obtain the simple intertwiners of the quadraticoperators in addition to solutions where incoming edges are contracted with SO(D)N -invariantvectors ξlei (N). A few remarks are appropriate:

59

12

1231234

4

2

1N

5

3

4

2

1N

5

3

l12

l12l123

l123

l1234

l1234

Figure 8.2: Recoupling scheme corresponding to the subset of linear vertex simplicity constraintoperators (8.3.19).

6. Although this procedure yields a promising result, it contains several non-standard andah-hoc steps which have to be justified. One could argue that the “correct” quantisationof the linear and quadratic simplicity constraints should give the same quantum theory,however, as is well known, classically equivalent theories result in general in non-equivalentquantum theories, which nevertheless can have the same classical limit.

7. It is unclear how to proceed with “integrating out” N I in the general case. For the vacuumtheory, integration over every point in σ gives the Barrett-Crane intertwiner for the edgescontracted with SO(D)N -invariant vectors. This type of integration would also get rid ofthe 1-valent vertices and thus allow for a natural unitary map to the quadratic solutionsas already mentioned above.

8. When introducing fermions, there is the possibility for non-trivial gauge-invariant functionsof N I at the vertices which immediately results in the question of how to integrate outthis N I -dependence. Next to including those N I in the above integration or to integrateout the remaining N I separately, one could transfer this integration back into the scalarproduct. Since the author is presently not aware of an obvious way to decide about theseissues, we will leave them for further research.

8.3.5 Mixed quantisation

Since the implementation of the quadratic simplicity constraints described above yields a morepromising result than the implementation of the linear constraints, we can try to perform amixed quantisation by noting that we can classically express the linear constraints for even Din the form

1

4εIJKLMπ

aIJπbKL ≈ 0, N I − nI(π) ≈ 0. (8.3.20)

The phase space extension derived in [40] remains valid when interchanging the linear simplicityconstraint for the above constraints. The reason for restricting D to be even is that we havean explicit expression for nI(π), see [35, 36]. Since a quantisation of nI(π) will most likely notcommute with the Hamiltonian constraint operator, we resort to a master constraint. Note that

60

the expression

M ′N :=

((N I − nI(π))

√qD−1

)δIJ((NJ − nJ(π))

√qD−1

)√q2D−3

, (8.3.21)

which is the densitised square of N I − nI(π), can be quantised as

M ′N = 2√q3−2D(

√|V I VI | −NI V

I), (8.3.22)

when using a suitable factor ordering, where a quantisation of√q3−2D is described in [37]. The

solution space is not empty since the intertwiner

s(e1, ..., eD)√D!|NA1NA2 . . . NAD > +|NBε

BA1...AD > (8.3.23)

is annihilated by MN , which can be easily checked when using the results of the volume operatoracting on the solution space of the full set of linear simplicity constraint operators. In order toturn the expression into a well defined master constraint operator, we have to square it againand to adjust the density weight, leading to

MN = 4

(√q5/2−2D(


I)

)† √q5/2−2D(


I), (8.3.24)

which is by construction a self-adjoint operator with non-negative spectrum. We remark that itwas necessary to use the fourth power of the classical constraint for quantisation, because thesecond power, having the desired property that its solution space is not empty, does not qualifyas a well defined master constraint operator in the ordering we have chosen. There exists howeverno a priori reason why one should not take into account master constraint operators constructedfrom higher powers of classical constraints [75]. Curiously, the quadratic simplicity constraintoperators as given above do not annihilate the solution displayed. Clearly, the calculations willbecome much harder as soon as vertices with a valence higher than D are used, since the buildingblocks of the volume operator will not be diagonal on the intertwiner space.

This type of quantisation is further discussed in section 8.4.3, where a possible applicationto using EPRL intertwiners is outlined. In contrast to the earlier assumption of D being evenin order to have an explicit expression nI(π), we can also perform the mixed quantisation usingnInJ(π) as given in (4.1.12) and the constraint NJ(nInJ(π)−N INJ) ≈ 0. For the applicationproposed, we will only need that the corresponding master constraint can be regularised suchthat it vanishes when not acting on non-trivial vertices, which can be achieved as before.

8.4 Comparison to existing approaches

In this section, we are going to comment on the relation of existing results from the spin foamliterature to the proposals in this chapter. In short, the main conclusion will be that in thecase of four spacetime dimensions, many results from the spin foam literature can be used alsoin the canonical framework. However, they fail to work in higher dimensions due to specialproperties of the four dimensional rotation group which are heavily used in spin foams. We willnot comment on results based on coherent state techniques [61, 86, 87, 98] since we do not see aresemblance to our results which do not make use of coherent states in any way. Nevertheless,similarities could be present as the relation between the EPRL [67] and FK [61] models show.

61

8.4.1 Continuum vs. discrete starting point

The starting point for introducing the simplicity constraints in the spin foam models is thereformulation of general relativity as a BF theory subject to the simplicity constraints, and thussimilar to the point of view taken in this thesis. The crucial difference however is that while spinfoam models start classically from discretised general relativity, the canonical approach discussedhere starts from its continuum formulation. When looking at the simplicity constraints, thisdifference manifests itself in the choice of (D−1)-surfaces over which the the generalised vielbeins(i.e. the bivectors in spin foam models) have to be smeared. Starting from a discretisation ofspacetime, the set of (D− 1)-surfaces is fixed by foliating the discretised spacetime. Restrictingto a simplicial decomposition of a four-dimensional spacetime as an example, these would e.g.be the faces f of a tetrahedron t in the boundary of the discretisation. It follows that one cantake the bivectors BIJ integrated over the individual faces of a tetrahedron, BIJ

f (t) :=∫f B

IJ ,

as the basic variables and the quadratic (off)-diagonal simplicity constraints read [67]

Cff := εIJKLBf (t)IJBf (t)KL diagonal simplicity, ∀f ∈ t (8.4.1)

Cff ′ := εIJKLBf (t)IJBf ′(t)KL off-diagonal simplicity, ∀f, f ′ ∈ t. (8.4.2)

In the continuum formulation however, we have to consider all possible (D − 1)-surfaces, andthus also hypersurfaces containing the vertex v dual to the tetrahedron t. The resulting fluxoperators a priori contain a sum of right invariant vector fields RIJe acting on all the edges econnected to v. While this poses no problem for the diagonal simplicity constraints which acton edges of the spin networks as shown in section 6.3, the off-diagonal simplicity constraintsarising when both surfaces contain v are not given by (8.4.2), but by sums over different Cff ′ ,see [37] for details. It can however be shown by suitable superpositions of simplicity constraintsassociated to different surfaces that (8.4.2) is actually implied also by the quadratic simplicityconstraints arising from a proper regularisation in the canonical framework. This statement,briefly mentioned in section 6.3, is non-trivial and had to be proved in [37]. Thus, we can alsoin the canonical theory consider the individual building blocks (8.2.1) as done in section 8.2.Furthermore, the same is also true when using linear simplicity constraints, i.e. the properlyregularised linear simplicity constraints in the canonical theory imply that all building blocks(8.3.3) vanish.

We also note that there is no analogue of the normalisation simplicity constraints [62] in thecanonical treatment since the generalised vielbeins do not have timelike tensorial indices afterbeing pulled back to the spatial hypersurfaces.

8.4.2 Projected spin networks

Projected spin networks were originally introduced in [99, 100] to describe Lorentz covariantformulations of loop quantum gravity, meaning that the internal gauge group is SO(1, 3) (orSL(2,C)) instead of SU(2). The basic idea is that next to the connection, the time normal field,often called x or χ in the spin foam literature, becomes a multiplication operator since it Poisson-commutes classically with the connection. Since the physical degrees of freedom of loop quantumgravity formulated in terms of the usual SU(2) connection and its conjugate momentum areorthogonal to the time normal field, one performs projections in the spin networks from the fullgauge group SO(1, 3) to a subgroup stabilising the time normal. Since the projector transformscovariantly under SO(1, 3), a (gauge invariant) projected spin network is already defined by itsevaluation for a specific choice of the time normals and the resulting effective gauge invarianceis only SU(2), which exemplifies the relation to the usual SU(2) formulation in the time gaugexI(= N I) = (1, 0, 0, 0).

62

Despite its close relation to the techniques used in this chapter and its merits for the four-dimensional treatment, there are several problems connected with using this approach in thecanonical framework discussed in this thesis which we will explain now. While the extensionof projected spin networks to different gauge groups has already been discussed in [100], thereis a subtle problem associated with the part of the connection which is projected out by theprojections, that could not have been anticipated by looking at loop quantum gravity in terms ofthe Ashtekar-Barbero variables. There, the physical information in the connection, the extrinsiccurvature, is located in the rotational components of the connection. To see this, consider infour dimensions the 2-parameter family of connections discussed in [36]6,

AaIJ = ΓhybaIJ + βKaIJ + γ

1

2εIJ

KLKaKL, (8.4.3)

where γ corresponds to the Barbero-Immirzi parameter restricted to four dimensions and β isthe new free parameter appearing in any dimension. As in (3.2.26), we decompose KaIJ as

KaIJ = 2N[IKa|J ] + KtraceaIJ + Ktrace free

aIJ , (8.4.4)

where KaIJ means that N IKaIJ = NJKaIJ = 0 and the trace / traceless split is performedwith respect to the hybrid vielbein. The extrinsic curvature which we need to recover from AaIJis located in KaJ , whereas Ktrace

aIJ vanishes by the Gauß constraint and Ktrace freeaIJ is pure gauge

from the simplicity gauge transformations.Now setting β = 0 and N I = (1, 0, 0, 0) in four dimensions, we recover the Ashtekar-Barbero

connection and see that the physical information is located in the rotational components of AaIJ .It thus makes sense to project onto this subspace in the projected spin network construction, i.e.we are not loosing physical information. On the other hand, setting γ = 0 in four dimensionsor going to higher dimensions, we see that a projection onto the subspace orthogonal to N I

annihilates the physical components of the connection. This would not be necessarily an issue ifone would just project the projected spin network at the intertwiners, but when one tries to go tofully projected spin networks as proposed in [99]. Then, since one would take a limit of insertingprojectors at every point of the spin network, the physical information in the connection wouldbe completely lost.

Next to this problem, there are other problems associated to taking an infinite refinementlimit for projected spin networks as discussed by Alexandrov [99] and Livine [100], e.g. thatfully projected spin networks are not spin networks any more (since they only contain verticesand no edges) and, connected with this problem, that the trivial bivalent vertex, the Kroneckerdelta, is not an allowed intertwiner. Similar problems have been encountered in section 8.3,i.e. while the vertex simplicity constraints could be solved by a construction very similar toprojected spin networks where one projects the incoming and outgoing edges at the intertwinerin the direction of the time normal N I , imposing the linear simplicity constraint on the edges,one would have to insert “trivial” bivalent vertices of the form N INJ at every point of the spinnetwork, whereas one would need to insert the the Kronecker delta δIJ to achieve cylindricalconsistency while maintaining a spin network containing edges and not only vertices.

Thus, the main problem with using (fully) projected spin networks is connected to the factthat we do not know of an analogue of the Barbero-Immirzi parameter in higher dimensionswhich would allow us to put the extrinsic curvature also in the rotational components of theconnection. In four dimensions on the other hand, this problem would be absent and one wouldbe left with the issue of refining the projected spin networks, which is however also present in

6Note that the definitions of the parameters are different in [36] for calculational simplicity, but here we preferthis parametrisation to make our point clear.

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section 8.3. Therefore, using projected spin networks in four dimensions with non-vanishingBarbero-Immirzi parameter is an option for the canonical framework developed in this thesisand the known issues discussed above should be addressed in further research.

8.4.3 EPRL model

The basic idea of the EPRL model is to implement the diagonal simplicity constraints as usual,but to replace the off-diagonal simplicity constraints by linear simplicity constraints which areimplemented with a master constraint construction [67] or weakly [101]. Furthermore, theBarbero-Immirzi parameter is a necessary ingredient. We restrict here to the Euclidean modelsince its group theory is much closer to the connection formulation with compact gauge groupSO(D + 1). While the diagonal simplicity constraints give the well known relation

(j+)2 =

(γ + 1

γ − 1

)2

(j−)2, (8.4.5)

the master constraint for the linear constraints gives [67], up to ~ corrections7,

k2 =

(2j−

1− γ

)2

=

(2j+

1 + γ

)2

(8.4.6)

where k is the quantum number associated to the Casimir operator of the SU(2) subgroupstabilising N I . Depending on the value of the Barbero-Immirzi parameter, either k = j+ + j−

or k = |j+ − j−| is selected by this constraint. The EPRL intertwiner for SO(4) spin networkswith arbitrary valency [70] is then constructed by first coupling the two SU(2) subgroups ofSO(4) holonomies in the representations (j+, j−), calculated along incoming and outgoing edgesto the intertwiner, to the k representation. Then, the k representations associated to each edgeare coupled via an SU(2) intertwiner and the complete construction is then projected into theset of SO(4) intertwiners.

An alternative derivation proposed by Ding and Rovelli [101] makes use of weakly imple-menting the linear simplicity constraints, i.e. restricting to a subspace Hext such that⟨

φ∣∣∣ C ∣∣∣ ψ⟩ = 0 ∀ |φ〉 , |ψ〉 ∈ Hext. (8.4.7)

In this approach, one can also show that the volume operator restricted to Hext has the samespectrum as in the canonical theory, which is an important test to establish a relation betweenthe canonical theory and the EPRL model.

Closely related to what we already observed in the previous subsection on projected spinnetworks, the EPRL model makes heavy use of the fact that SO(4) splits into two SU(2) sub-groups and that the Barbero-Immirzi parameter is available in four dimensions. Thus, we wouldhave to restrict to four dimensions with non-vanishing γ if we would want to use EPRL solutionto the simplicity constraints. One upside of this solution when comparing to our proposition forsolving the quadratic constraints is that no choice problem occurs, i.e. if we map the quantumnumbers of the EPRL intertwiners to SU(2) spin networks, a change of recoupling basis in theSU(2) spin networks results again in EPRL intertwiners solving the same simplicity constraints.The problem of stability of the solution space Hext of the simplicity constraint under the action

7Note that these ~ corrections are necessary since the master constraint, by construction, has the same solutionspace as the original constraint [75], i.e. C†C |ψ〉 = 0 implies C |ψ〉 = 0. In the master constraint language, onesubtracts an operator from the master constraint which vanishes in the classical limit to obtain a sufficiently largesolution space.

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of the Hamiltonian constraint is however, to the best of our knowledge, not circumvented whenusing EPRL intertwiners.

Also, in order to use the EPRL solution in the canonical framework, one would have to discussexactly what it means to use linear and quadratic simplicity constraints in the same formulation,i.e. if one can freely interchange them and how continuity of the time normal field is guaranteedat the classical level if one changes from the quadratic constraints to linear constraints fromone point on the spatial hypersurface to another. The mixed quantisation proposed in section8.3.5 can be seen as an attempt to using both the time normal as an independent variable aswell as quadratic simplicity constraints. In this case, the main difference is the presence ofan additional constraint relating the time normal constructed from the generalised vielbeinsto the independent time normal (which could be used in the linear simplicity constraints). Insection 8.3.5, this additional constraint was regularised as a master constraint which acts onlyon vertices. Taking the point of view that one can freely change between using the quadraticconstraints plus this additional constraint or the linear constraints, one could choose the linearconstraints for vertices and the quadratic constraints for edges. Since we can use a factor orderingfor the master constraint where a commutator between a holonomy and a volume operatoris ordered to the right, the master constraint would vanish on edges and only the quadraticsimplicity constraints would have to be implemented, which are however not problematic. Atvertices, we would be left with the linear constraints and could use the EPRL intertwiners.Thus, the EPRL solution seems to be a viable option in four dimensions. Whether one considersit natural or not to use both linear and quadratic constraints in the same formulation is amatter of personal taste. Nevertheless, it would be desirable to have only one kind of simplicityconstraints.

As a last remark, we point out that the non-commutativity of the linear simplicity constraintsin the EPRL model results from using γ 6= 0 and thus we are not faced with this problem in higherdimensions. Essentially, as discussed in more detailed in remark 5 of section 8.3.4, while therotations stabilising N I form an SO(D) subgroup of SO(D+1), the linear simplicity constraintsin four dimensions with γ 6= 0 and β 6= 0 do not generate such a subgroup.

8.5 Discussion and conclusions

Let us briefly discuss the results of this chapter and judge the different approaches.First, the mechanism for avoiding the non-commutativity in the quadratic simplicity con-

straints discussed in section 8.2 is new to the best of our knowledge and we do not see anyindication that the solution space is identical to previous results (up to the fact that it hasthe same “size” as SU(2) spin networks). In the spin foam literature, the linear simplicityconstraints are cornerstones of the new spin foam models and have been introduced since thequadratic simplicity constraints acting on vertices do not commute. While the methods fortreating supergravity introduced in the third part of this thesis necessarily need an independenttime normal and thus suggest using linear simplicity constraints, there is no need for the lin-ear constraints in pure gravity (except for the fact that they exclude the topological sector infour dimensions). Therefore, one should not dismiss the quadratic constraints, especially sincethe linear constraints come with their own problems in the canonical approach. The solutionpresented in section 8.2 is certainly not free of problems, most prominently the choice of themaximal commuting subset, but its close relation the SU(2) based theory and the (natural)unitarity (at the level of Hilbert space elements) of the intertwiner map to SU(2) intertwinersmake it look very promising.

The linear simplicity constraints come with their own set of problems, many of which werealready known in the spin foam literature. While the results of section 8.2 would naturally

65

lead us to consider the quadratic constraints, the connection formulation of higher-dimensionalsupergravity which will be developed in the third part of this thesis makes it necessary to usean independent time normal as an additional phase space variable. This time normal wouldnaturally point towards using linear simplicity constraints, although the mixed quantisationof section 8.3.5 could avoid this. Since there is no anomaly appearing when using the linearsimplicity constraints (with γ = 0 in four dimensions), we should implement them strongly.However, this leads to a solution space very different from the SU(2) spin networks. At thispoint, it seems to be best to let oneself be guided by physical intuition and the results fromthe quadratic simplicity constraints as well as the desired resemblance to SU(2) spin networks.Ad hoc methods for getting close to this goal have been discussed in section 8.3.4. We howeverstress that these methods are, as said, ad hoc and they don’t follow from standard quantisationprocedures. The mixed quantisation discussed at the end of section 8.3 also does not seemcompletely satisfactory, especially since the master constraint ensuring the equality of the inde-pendent normal N I and the derived normal nI(π) is very complicated to solve. Nevertheless, insection 8.4.3, an application to EPRL intertwiners is outlined which could avoid this problemby using linear simplicity constraints for the vertices. The strength of the mixed quantisationis thus that it provides a mechanism to incorporate both the quadratic simplicity constraints aswell as an independent time normal in the same canonical framework, which is what is done onthe path integral side in the EPRL model.

A comparison to results from the spin foam literature, especially projected spin networks andthe EPRL model, shows that many of the problems connected with using the linear simplicityconstraints have already been known, partly in different guises. While using these known resultsin our framework seems to be a viable option in four dimensions, we are unaware of possible waysto extend them also to higher dimensions since main ingredients are a non-vanishing Barbero-Immirzi parameter as well as special properties of SO(4).

In conclusion, we reported on several new ideas of how to treat the simplicity constraintswhich appear in the connection formulation of general relativity derived in this thesis in anydimension D ≥ 3 and found that none of the presented ideas are entirely satisfactory at thispoint and further research on the open questions needs to be conducted. We hope that thediscussion presented will be useful for an eventually consistent formulation.

66

Part III

Extensions to supergravity

67

Chapter 9

Standard matter

In this chapter, we will shortly discuss the inclusion of standard model matter degrees of freedomin loop quantum gravity type quantisations of higher-dimensional general relativity discussed inthis thesis. While the techniques developed for scalar fields and gauge fields directly carry overto higher dimensions, some care has to be taken when dealing with fermions, since despite theirLorentzian nature, they will have to be incorporated in the canonical description of the firstpart of this thesis, thus transforming under Spin(D+ 1) as opposed to Spin(1, D). The originalwork on which this chapter is based is [38].

As for gauge fields with compact gauge group, the treatment is identical to the one for thestandard (3+1)-dimensional case as discussed in [20, 21]. Essentially, one constructs holonomiesfrom the gauge fields in direct analogy to the construction of the Ashtekar-Lewandowski Hilbertspace. Due to the compactness of the gauge groups, the projective limit associated to aninfinite refinement of the graph can be taken and we can construct generalised spin networkswhich carry, next to the irreducible representations of SO(D + 1) on the edges and SO(D + 1)intertwiners on the vertices, irreducible representations of the matter gauge groups on the edgesand intertwiners for the matter gauge groups at the vertices. The emerging picture is very similarto lattice gauge theory, with the main difference that the graphs on which the holonomies aredefined are dynamical objects of the theory.

Also for scalars, the techniques of [20, 21] directly carry over to higher dimensions. Here, it isimportant to either restrict to scalars which transform in the trivial or the adjoint representationof the matter gauge groups, since the construction of point holonomies is essential in [20, 21]and gauge covariance requires the field to be exponentiated to transform in either of theserepresentations. The standard model however falls into this class of theories, so that we caneasily live with this restriction.

For spin 1/2 fields, the situation is more complicated since they transform in the spinorrepresentation of SO(1, D) or, in the (time) gauge fixed version, of SO(D). Thus, a naiveextension to SO(D + 1) as the internal gauge group for the gravitational degrees of freedomseems to be problematic since the fermions living at the vertices of the spin networks cannot becontracted any more with the incoming holonomies to form gauge invariant objects, since gaugeinvariance would be spoiled due to the different gauge groups. However, it turns out that, atleast for Dirac fermions, this problem can be evaded rather easily by realising that SO(1, D)and SO(D + 1) act on the same spinorial representation spaces. Furthermore, since no realityconditions need to be satisfied for Dirac spinors, we do not get any problems due to differentfactors of i in which the generators of SO(1, D) and SO(D + 1) differ. In order to establishequivalence between the theory where SO(1, D) and SO(D+1) act, we will require that they areidentical when restricting to the time gauge N I = (1, 0, . . . , 0). The original work summarised

68

here has been published in [38] and contains the missing details.As a first step in constructing the SO(D+ 1)-invariant theory including spinors, we need to

extend the canonical transformation introduced in the first part of this thesis to start from anextended ADM phase space which is invariant under a SO(D) gauge symmetry. For this, westart at a phase space coordinatised by a densitised D-bein Eai and its conjugate momentumKia. The relation to the ADM phase space is given by

qqab = Eai Ebjδij , Kab = Ki

(aqb)c√q−1Eci (9.1)

and the ADM constraints are expressed accordingly. It is a well known result that using thenew Poisson bracket

Kia(x), Ebj (y) = δ(D)(x− y)δbaδ

ij (9.2)

amounts to a canonical transformation when introducing the Gauß constraint Ea[iKa|j] = 0 andthus results in the same physics.

In order to establish a map between these variables and an SO(D + 1) invariant extension,we first have to account for the increased dimension of the internal space. Since the dimensionof the internal space is just one dimension larger in the SO(D + 1) case, it is enough to use anormal field nI which is orthogonal to to all objects in the SO(D) theory, but transforms underthe full SO(D+ 1). Then, it is possible to act with SO(D+ 1) also on the objects of the SO(D)theory, since a rotation taking one out of the SO(D) invariant subspace also changes the normalnI . The precise canonical transformation associated to this train of thought is given by

EaI = ζηIJπaKJnK , KaI = ζηI

J(A− Γ)aKJnK , (9.3)

where πaIJ and AaIJ are the new canonical variables, ΓaIJ is the hybrid connection, and nI isthe normal which can be constructed from πaIJ as described in equation (4.1.12). We remarkthat it is only necessary to construct nInJ(π) because nI always appears in such a combinationin all the constraints for even D + 1, and we can use (4.1.12) for D + 1 odd. The explicitcalculation of the canonicity of the transformation however can rely on abstract properties ofnI , as demonstrated in [38].

It is also possible to introduce an independent normal N I as described in section 5.2. Es-sentially, the extension of

(Kai, E

bj)

with SO(D) Gauß constraint to(AaIJ , π

bKL, N I , PJ)

withSO(η) Gauß, linear simplicity and normalisation constraint works exactly the same way and thecalculations can be copied nearly verbatim. We can even choose to simplify the replacement ofthe vielbein and extrinsic curvature using the normal N I ,

EaI = ζηIJπaKJNK , KaI = ζηI

J(A− Γ)aKJNK , (9.4)

where ηIJ now is understood as a function of N I .After having performed this canonical transformation, it remains to find an expression for the

Hamiltonian constraint which can easily be quantised with the methods introduced in the secondpart of this chapter and amended by the corresponding techniques for fermions as introducedin [20, 21]. A proposal for an expression of the Hamiltonian constraint has been provided in[38], however it is rather lengthy and we refrain from displaying it here explicitly. A problemarising in the construction of such an expression is that many counterterms are needed whenone wants to use a field strength tensor in the gravitational part of the Hamiltonian constraint,since the gauge unfixing part of the Hamiltonian constraint obtains new terms which come froma non-vanishing trace free part of KaIJ in the presence of fermions. Furthermore, the ratherunnatural SO(D + 1) gauge invariance also leads to counterterms when one wants to establishequality of the constraints upon performing the reduction to the SO(D) invariant theory.

69

Chapter 10

Rarita-Schwinger field

In the previous chapters, we managed to derive a connection formulation of Lorentzian generalrelativity in D+ 1 dimensions with compact gauge group SO(D+ 1) such that the connection isPoisson commuting, which implies that loop quantum gravity quantisation methods apply. Wealso provided the coupling to standard matter.In this and the next chapter, we extend our methods to derive a connection formulation ofa large class of Lorentzian signature supergravity theories, in particular 11d SUGRA and 4d,N = 8 SUGRA, which was in fact the motivation to consider higher dimensions. Starting froma Hamiltonian formulation in the time gauge which yields a Spin(D) theory, a major challengeis to extend the internal gauge group to Spin(D + 1) in presence of the Rarita-Schwinger field.This is non-trivial because SUSY typically requires the Rarita-Schwinger field to be a Majoranafermion for the Lorentzian Clifford algebra and Majorana representations of the Clifford algebraare not available in the same spacetime dimension for both Lorentzian and Euclidean signature.We resolve the arising tension and provide a background independent representation of the nontrivial Dirac antibracket ∗-algebra for the Majorana field which significantly differs from theanalogous construction for Dirac fields already available in the literature. The original workfrom which this chapter is taken is [40].

10.1 Introduction

During the years after the discovery of D + 1 = 3 + 1 supergravity by Freedman, Ferrara,and van Nieuwenhuizen in 1976 [102], there has been a lot of activity in the newly formedfield of supergravity, driven by the hope to construct a theory of quantum gravity without theshortcoming of perturbative non-renormalisability. Werner Nahm classified in 1977 all possiblesupergravities, arriving at the result that, under certain assumptions, d = 11 was the maximalnumber of Minkowski signature spacetime dimensions in which supergravities could exist [6]. Inthe following year, d = 11 supergravity was constructed by Cremmer, Julia and Scherk [48] inorder to obtain d = 4, N = 8 maximal supergravity by dimensional reduction. Various formsof supergravity were derived in dimensions d ≤ 11 and relations among them were discovered inthe subsequent years [103].

While the initial hope linked with perturbative supergravity was vanishing due to resultssuggesting its non-renormalisability [104] and the community turned to superstring theory, loopquantum gravity (LQG) emerged as a new candidate theory for quantum gravity after Ashtekardiscovered his new variables in 1986 [7]. As described before, LQG is formulated in an entirelynon-perturbative and background-independent way and suggests the appearance of a quantumgeometry at the Planck scale. It is therefore in a sense dual to the perturbative descriptions

70

coming from conventional quantum (super)gravities and superstring- / M-theory and it wouldbe very interesting to compare and merge the results coming from these two different approachesto quantum gravity. The main conceptual obstacle in comparing these two methods of quanti-sation has been the spacetime they are formulated in. While the Ashtekar-Barbero variables areonly defined in 3 + 1 dimensions, where also an extension to supersymmetry exists, superstring-/ M-theory favours 9 + 1 / 10 + 1 dimensions and is regarded as a quantisation of the respectivesupergravities. It is therefore interesting to study quantum supergravity as a means of probingthe low-energy limit of superstring- / M-theory with different quantisation techniques, both per-turbative and non-perturbative. A somewhat different approach has been taken in [105], wherethe closed bosonic string has been quantised using rigorous background-independent techniques,resulting in a new solution of the representation problem which differs from standard stringtheory. Also, the Hamiltonian formulation of the algebraic first order bosonic string and itsrelation to self-dual gravity have been recently investigated in [106, 107].

Apart from contact with superstring- / M-theory, new results from perturbative d = 4,N = 8 supergravity [108, 109, 110] suggest that the theory might be renormalisable, contraryto prior believes. It is therefore interesting in its own right to study the loop quantised d = 4,N = 8 theory and compare the results with the perturbative expansion.

A possible solution for the problem of quantising standard matter coupled higher-dimensionalgeneral relativity, whose action is an integral part of all supergravity theories, has been given inthe previous chapters of this thesis. The purpose of this chapter is to generalise this transfor-mation to supergravity theories. The problem arising in these generalisations are not so muchlinked to the appearance of additional tensor fields and spin 1/2 fermions, but to the Rarita-Schwinger field which obeys a Majorana condition. It is well known [111] that in order to havesimple and metric independent Poisson brackets for the Rarita-Schwinger field ψαa , one shoulduse half-densitised internally projected fields φαi := 4

√qeaiψ

αa . This field redefinition has to be

changed in order to work in the new internal space, more specifically we have to ensure that thenumber of degrees of freedom still matches by imposing suitable constraints. Also, the Majo-rana conditions are sensitive to the dimensionality and signature of spacetime. We thus have toensure that no inconsistencies arise when using SO(D + 1) instead of SO(1, D) as the internalgauge group. Concretely, this will be achieved in dimensions where Majorana representation forthe γ-matrices exists, which covers many interesting supergravity theories (d = 4, 8, 9, 10, 11).

The presence of additional tensors, vectors, scalars and spin 1/2 fermions in various SUGRAtheories does not pose any problems for this classical canonical transformation. However, wemust provide background independent representations for these fields in the quantum theorywhich, to the best of our knowledge, has not been done yet for all of them. As an example,in the next chapter we consider the quantisation of Abelian p-form fields such as the 3-indexphoton present in 11d SUGRA with Chern-Simons term. Scalars, fermions and connections ofcompact, possibly non-Abelian, gauge groups have already been treated in [21].

This chapter is organised as follows:Section 10.2 is subdivided into two parts. In the first, we review prior work on canonical super-gravity theories in various dimensions and identify their common structural elements. We alsomention the basic difficulties in our goal to match these canonical formulations to the reformula-tions of the gravitational sector in the first part of this thesis. In the second we display canonicalsupergravity explicitly in the time gauge paying special attention to the Rarita-Schwinger sector.Section 10.3 is also subdivided into two parts. In the first we display the symplectic structureof the Rarita-Schwinger field in the time gauge in convenient variables which will be crucial fora later quantisation of the theory. In the second, following the strategy of chapters 4 and 9,we will perform an extension of the phase space subject to additional second class constraints

71

ensuring that we are dealing with the same theory while the internal gauge group can be ex-tended from SO(D) to SO(D + 1). In section 10.4 we construct a representation of the Diracanti bracket geared to Majorana spinor fields rather than Dirac spinor fields. In section 10.5 weshow that our formalism easily extends without additional complications to chiral supergravities(Majorana-Weyl spinors) and to spin 1/2 Majorana fields which are present in some supergravitytheories. In section 10.6 we summarise and conclude.

10.2 Review of canonical supergravity

In the first part of this section we summarise the status of canonical supergravity and its quan-tisation. In the second we display the details of the theory to the extent we need it which willsettle the notation.

10.2.1 Status of canonical supergravity

Hamiltonian formulations of supergravity are a tedious business due to the complexity of theLagrangians and the appearance of constraints. Nevertheless, the canonical structure emerging isvery similar for the explicitly known Hamiltonian formulations. To the best of our knowledge, theD+1 split for D ≥ 3 has been explicitly performed for D+1 = 3+1, N = 1 [111, 112, 113, 114],D + 1 = 9 + 1, N = 1 [115], and D + 1 = 10 + 1, N = 1 [116]. The algebra of constraints ofD + 1 = 3 + 1 supergravity was first computed by Henneaux [117] up to terms quadratic in theconstraints [118]. The same method was applied by Diaz [119] to D + 1 = 10 + 1 supergravity,also neglecting terms quadratic in the constraints. Sawaguchi performed an explicit calculationof the constraint algebra of D + 1 = 3 + 1 supergravity in [120] where a term quadratic inthe Gauß constraint appears in the Poisson bracket of two supersymmetry constraints. Theconstraint algebra for D+ 1 = 9 + 1, N = 1 supergravity coupled to supersymmetric Yang-Millstheory was calculated by de Azeredo Campos and Fisch in [121].

Shortly after the introduction of the complex Ashtekar variables, Jacobson generalised theconstruction to d = 4, N = 1 supergravity [122]. In the following, different authors includingFulop [123], Gorobey and Lukyanenko [124], as well as Matschull [125], explored the subjectfurther. Armand-Ugon, Gambini, Obregon and Pullin [126] formulated the theory in termsof a GSU(2) connection and thus unified bosonic and fermionic variables in a single connec-tion. Building on these works, Ling and Smolin published a series of papers on the subject[127, 128, 129], where, among other topics, supersymmetric spin networks coming from theGSU(2) connection were studied in detail. In the above works, complex Ashtekar variables areemployed for which the methods developed in [14, 15, 16, 17, 18, 19] are not available. Also, theAshtekar variables are restricted to four spacetime dimensions and thus not applicable to higher-dimensional supergravities. Aiming at a unification of string theory and LQG, Smolin explorednon-perturbative formulations of certain parts of eleven dimensional supergravity [130, 131]. Thegeneralisation of the loop quantum gravity methods to antisymmetric tensors was consideredby Arias, di Bartolo, Fustero, Gambini, and Trias [132]. The full canonical analysis of d = 4,N = 1 supergravity using real Ashtekar-Barbero variables was first performed by Sawaguchi[120]. Kaul and Sengupta [133] considered a Lagrangian derivation of this formulation using theNieh-Yan topological density.

An attempt to construct Ashtekar-type variables for d = 11 Supergravity has already beenmade by Melosch and Nicolai using an SO(1, 2)× SO(16) invariant reformulation of the originalCJS theory [134]. In this formulation, the connection is not Poisson commuting, thus forbiddingLQG techniques. In a paper on canonical supergravity in 2 + 1 dimensions [135], Matschull andNicolai discovered a similar noncommutativity property which they avoided by adding a purely

72

imaginary fermionic bilinear to the connection, leading to a complexified gauge group. As wasobserved in [21], this problem can be avoided by using half-densitised fermions as canonicalvariables.

The general picture emerging is that the canonical decomposition S =∫dt (pq −H) in the

time gauge leads to

S =

∫σdDx

∫dt

(EaiK

ia + i

√qψaγ

a⊥bψb + tensors + vectors + spin 1/2 + scalars

−NH−NaHa − λijGij − ψtS − tensor constraints

), (10.2.1)

where the Hamiltonian constraint H, the spatial diffeomorphism constraint Ha, the Spin(D)Gauß constraintGij , the supersymmetry constraint S and the tensor constraints form a first classalgebra. Eai is the densitised vielbein and Ki

a its canonical momentum. ψa denotes the Rarita-Schwinger field with suppressed spinor indices. N , Na, λij and ψt are Lagrange multipliersfor the respective constraints. With tensor constraints we mean constraints acting only onadditional tensor fields such as the three-index photon of D + 1 = 10 + 1 supergravity. Theremaining terms in the first line are kinetic terms appearing in the decomposition of the action.Since we will not deal with them explicitly in this thesis, we refer to [115, 116] for details.

In order to apply the techniques developed for loop quantum gravity to this system, wehave to turn it into a connection formulation in the spirit of the Ashtekar-Barbero variables.Concerning the purely gravitational part, this has been achieved in the first part of this thesisand extended to the case of spin 1/2 fermions in chapter 9. The Rarita-Schwinger field turnsout to be more difficult to deal with than the spin 1/2 fermions. On the one hand, it leadsto second class constraints [113], which encode the reality conditions, with a structure which isdifferent from the case of Dirac spinors1. On the other hand, as the other fermions, it has to betreated as a half-density in order to commute with Ki

a[111].Apart from the conventional canonical analysis, where time and space are treated differ-

ently, there exists a covariant canonical formalism treating space and time on an equal footing[136]. It has been applied to vielbein gravity [137], d = 4, N = 1 supergravity [138], d = 5supergravity and higher-dimensional pure gravity [139, 140] and d = 10, N = 1 supergravitycoupled to supersymmetric Yang-Mills theory [141, 142]. The relation of the covariant canonicalformalism and the conventional canonical analysis is discussed in [143] using the example of fourdimensional supergravity coupled to supersymmetric Yang-Mills theory.

10.2.2 Canonical supergravity in the time gauge

We will illustrate the 3+1 split of N = 1 supergravity in first order formulation as performed bySawaguchi [120] in order to give the reader a feeling for what is happening during the canonicaldecomposition. The resulting picture generalises to all dimensions. The symplectic potentialderived in this context is exemplary for the supergravity theories of our interest and we willcontinue with the general treatment in the next section. We remark that in 3 + 1 spacetimedimensions, the relations CT = −C and CγIC−1 = −(γI)T hold, where C denotes the chargeconjugation matrix.

The action for 3 + 1, N = 1 first order supergravity is given by

S =

∫Md4X

(s2eeµIeνJFµνIJ(A) + is eψµγ

µρσ∇ρ(A)ψσ

). (10.2.2)

1While for Dirac spinors, the second class constraints are of the form πψ ∝ ψ, πψ ∝ ψ, in the Majorana casewe obtain an equation of the form πψ ∝ ψ, where πx denotes the momenta conjugate to x.

73

Using the conventions introduced above and γµρσ = γIJKeµI eρJeσK , one can explicitly check that

the action is real. The 3+1 decomposition is done like in the previous chapters, and the notationused can be found there. We obtain

S =

∫Md4X

(s2eeµIeνJFµνIJ(A) + s eψµγ

µρσ∇ρ(A)ψσ

)=

∫Rdt

∫σd3x

[1

2πaIJLTAaIJ − i

√q ψaγ

⊥abLTψb −N˜(Hgrav − iq ψaγabc∇b(A)ψc

)−Na

(Hgrava + 3i

√qψ[aγ

⊥bc∇b(A)ψc]

)+

1

2AtIJ

(GIJgrav − i

√qψaγ

⊥ab[iΣIJ ]ψb

)−iψt

(√qγ⊥ab∇b(A)ψa +

√q∇b(A)(γ⊥abψa)

)]. (10.2.3)

From there, one can read off the constraints H, Ha, GIJ and S. We will choose time gaugenI = δI0 at this point to simplify the further discussion. For the symplectic potential, we find∫

Rdt

∫σd3x

(1

2πaIJLTAaIJ − i

√qψaγ

⊥abLTψb)

→∫

Rdt

∫σd3x

(EaiKai − iφ†aγabφb

)=

∫Rdt

∫σd3x

(EaiKai − iφ†iγ

ij[φj − ˙(Ebj )E

kb φk

])=

∫Rdt

∫σd3x

(EaiKai − πj

[φj − ˙(Ebj )E

kb φk

])=

∫Rdt

∫σd3x

(Eai

˙(Kai − πiEjaφj)− πjφj

)=

∫Rdt

∫σd3x

(EaiK ′ai − πjφj

), (10.2.4)

where we successively defined

φa := 4√qψa , φi :=

1√qEai φa , πi := iφ†jγ

ji and K ′ai := Kai − πiEjaφj . (10.2.5)

In the second line, we chose time gauge and half-densities as fermionic variables [21]. Then, wetransformed the spatial index of the fermions into an internal one using the vielbein, but pre-serving the fermionic density weight [111]. This second transformation also affects the extrinsiccurvature and we have to define a new variable K ′ai. The Gauß constraint becomes under thesechanges of variables

Gij = 2K [iaE

a|j] + πk[iΣij

]φk

= 2(K ′a

[i + π[iEkaφk

)Ea|j] + πk

[iΣij

]φk

= K ′a[iEa|j] + 2π[iφj] + πk

[iΣij

]φk. (10.2.6)

The generator of spatial diffeomorphisms Ha is given by the following linear combination ofconstraints

Ha := Ha +1

2AaIJG

IJ + ψaS. (10.2.7)

74

It becomes

Ha = Ebj∂aKbj − ∂b(EbjKaj

)− πb∂aφb + ∂b

(πbφa

)= Ebj∂a

(K ′bj + πjE

kb φk

)− ∂b

(Ebj

(K ′aj + πjE

kaφk

))− 1

4√qπiE

bi∂a(

4√qφjEbj

)+ ∂b

(πiE

biφjEaj

)= Ebj∂aK

′bj − ∂b

(EbjK ′aj

)+ 4√q∂a

(14√qπi

)φi

= Ebj∂aK′bj − ∂b

(EbjK ′aj

)+

1

2∂a (πi)φ

i − 1

2πi∂aφ

i. (10.2.8)

For the last step, note that πiφi = 0. Thus, these constraints exactly change as one wouldexpect under the performed change of variables. The other constraints also can be rewritten interms of the new variables, but this is less instructive and their explicit form is not importantfor what follows. We only want to remark that they depend on the contorsion Kaij , which isnot dynamical and has to be solved for in terms of φi. This can be done explicitly.

10.3 Phase space extension

In this section we focus on the symplectic structure of the Rarita-Schwinger sector. In the timegauge this is a SO(D) theory which is the subject of the first part. In the second part we willperform a phase space extension to a SO(D+ 1) theory where special attention must be paid tothe reality conditions.

10.3.1 Symplectic structure in the SO(D) theory

The 3+1 split described above generalises directly to higher dimensions. We will always imposethe time gauge nI = δI0 prior to the D + 1 split and restrict to dimensions where a Majoranarepresentation of the γ-matrices exists, which we will use. This allows us to set C = γ0 whichsimplifies the following analysis. The generic terms important for this chapter appearing insupergravity theories are

Sgrav.+RS =

∫MdD+1X

(s2eeµIeνJFµνIJ(A) + is eψµγ

µρσ∇ρ(A)ψσ

)(10.3.1)

in case of a first order formulation and analogous terms for a second order formulation. Thisdifference in defining the theory will not be important in what follows, since as demonstratedabove for the 3 + 1 dimensional case, the symplectic potential of these actions in the time gaugeturns out to be ∫

Rdt

∫σdDx

(EaiLTKai − i

√qψaγ

⊥abLTψb)

=

∫Rdt

∫σdDx

(EaiK ′ai − πjφj

), (10.3.2)

where we used the same definitions as in (10.2.5). From (10.3.2) we can read off the non-vanishingPoisson brackets2

K ′ai, Ebj

= δbaδji and

φαi , π

jβ

= −δαβ δ

ji . (10.3.3)

2More precisely we should call them Poisson anti-brackets which are symmetric under exchange of the ar-guments and which are to be quantised by anti commutators. We will call them Poisson brackets anyway fornotational simplicity in what follows with the usual rules for the interplay between the Poisson brackets for integraland half-integral spin respectively. See e.g. [144, 145] for an account.

75

Additionally, we have the following second class constraints and reality conditions

Ωi := πi + iφTj Cγ0γji = 0 and φ†i = −φTi Cγ0. (10.3.4)

In order to be able to introduce a connection variable along the lines of [35], we need to enlargethe internal space, i.e. replacing the gauge group SO(D) by either SO(1, D) or SO(D + 1). Inview of subsequent quantisation, SO(D + 1) is favoured because of its compactness and willbe our choice in the following. This enlargement can be done consistently if also additionalspinorial degrees of freedom are added as well as additional constraints which remove the newlyintroduced fermions. Finally, the extension has to be consistent with the reality conditions. Allthis turns out to be rather hard to achieve, and the final version of the theory looks ratherdifferent from what a “first guess” might have been. To motivate it, we will review the wholeprocess of finding the theory, showing where the straight-forward ideas lead to dead ends, andhow they can be modified to arrive at a consistent theory. We will only discuss the fermionicvariables in this chapter, the gravitational part is treated in the first two parts of this thesis.

Before we enlarge the internal space, we will get rid of the second class constraints. To thisend, we calculate the Dirac matrix

Cij =

Ωi,Ωj

= −2iCγ0γij , (C−1)ij = −γ0 i

2(D − 1)((2−D) ηij + γij)C

−1, (10.3.5)

and thus find for the Dirac bracket

φi, φjDB = −φi,Ω

k

(C−1)kl

Ωl, φj

= −(C−1)ij . (10.3.6)

To simplify the subsequent discussion, in the following we will consider real representations ofthe Dirac matrices only, which implies C = γ0. Then the above equations read

Cij = 2iγij , (C−1)ij = − i

2(D − 1)((2−D) ηij + γij) , φi, φjDB = −(C−1)ij . (10.3.7)

Now we can either (a) try to enlarge the internal space and afterwards choose new vari-ables which have simpler brackets, or (b) we simplify the Dirac bracket before enlarging theinternal space. (a) immediately leads to problems. The symmetry of the Poisson bracketsφαI , φ

βJ

∝ (C−1)αβIJ implies that matrix C−1 is symmetric under the exchange of (I, α)↔ (J, β).

The naive extension (C−1)IJ = − i2(D−1) ((2−D) ηIJ + γIJ) however does not have this sym-

metry. Its symmetric part C−1 + (C−1)T is not invertible. Of course, one can extend C−1 indifferent, more “unnatural” ways, e.g. containing terms like γTJ γI etc. and “cure” this problemfor a moment, but also the Gauß constraint will be problematic. The SO(D) constraint containsCij (since we used πi = −1

2φTj C

ji) and this matrix should also be replaced by some CIJ , such

that φI transforms covariantly and GIJ reduces correctly to Gij if we choose time gauge andsolve its boost part. This implies restrictions on C and further restrictions on C−1. We did notsucceed in finding matrices which fulfil all these requirements. In the following, we thereforewill follow the second route (b) and simplify the Dirac brackets before doing the enlargement ofthe internal space.

There are several possible ways how to simplify the Dirac brackets:

1. Note that the matrix C−1 on the right hand side of the Dirac brackets is imaginary andsymmetric, hence there always exists a real, orthogonal matrix Oij such that under thechange of variables φi → φ′i := Oijφ

j the brackets becomes i times a real diagonal matrix.

76

However, now the new fundamental degrees of freedom φ′i in general do not transformnicely under SO(D) gauge transformations, only (O−1)ijφ

′j do. More severely, it is unclearhow the extension Oij → OIJ should be done.

2. To assure that the fundamental degrees of freedom still transform nicely under SO(D)transformations, we can use the Ansatz φ′i := M ijφj with M ij := (αδij1+βΣij). Matricesof this form are in general invertible (cf. point 3. below for two exceptions) and, sincethey are constructed from intertwining matrices, φ′i will transform nicely under gaugetransformations. Moreover, now there is a chance to generalise the matrix to one dimensionhigher. For the Dirac brackets to become diagonal, α and β have to be determined bysolving MC−1MT = i1. The problem is that there is no solution for both parameters beingreal, at least one is necessarily complex. More general Ansatze for M ij (e.g. involvingγfive in even dimensions) share the same problem. Thus we exchanged the problem ofcomplicated brackets with complicated reality conditions, which again are hard to quantise.

3. The third route, which will lead to the consistent theory, in the end implies the introduc-tion of additional fermionic degrees of freedom already before enlargement of the internalspace. Given the difficulties just mentioned, the optimal approach in the desire to simplifythe Poisson brackets is to find orthogonal projections onto subspaces of the real Graßmannvector space which are built from δij1 and Σij such that the symplectic structure becomesblock diagonal on those subspaces. One can then define simple Poisson brackets and addthe projection constraints as secondary constraints which leads to corresponding Diracbrackets which will be proportional to those projectors. As we will see, the fact that theseare projectors makes it possible to find a Hilbert space representation of the correspondingDirac bracket.

We define in any dimension D

Pijαβ := ηijδαβ −1

D(γiγj)αβ =

D − 1

Dηijδαβ −

2i

DΣijαβ, (10.3.8)

Qijαβ :=1

D(γiγj)αβ =

1

Dηijδαβ +

2i

DΣijαβ. (10.3.9)

Those matrices are both real (we are using Majorana representations) and built fromintertwiners, but they are not invertible. It is easy to check that

PijαβQβγjk = 0, PijαβPβγjk = Piγαk, QijαβQβγjk = Qiγαk, and P + Q = 1η, (10.3.10)

i.e. the above equations define projectors. By construction, P projects on “trace-free”components w.r.t. γi, i.e. Pijαβγ

βj = 0 = γαi Pijαβ. Using these projectors, we can decompose

the Rarita-Schwinger field as follows

φi = Pijφj + Qijφ

j =: ρi +1

Dγiσ, (10.3.11)

with ρi := Pijφj and σ := γiφi3. Using the reality conditions (10.3.4) for φi, we find

ρi = ρTi C and σ = σTC. (10.3.12)

Moreover, usingγij = −Pij + (D − 1)Qij , (10.3.13)

3When considering the free Rarita-Schwinger action, this decomposition also appears to isolate the physicaldegrees of freedom, cf. e.g. [111]. The “trace part” σ is unphysical for the free field.

77

the symplectic potential becomes

−πiφi = −iφ†jγjiφi

= −iφ†j(−Pji + (D − 1)Qji

)φi

= −iφ†j(−PjkP

ki + (D − 1)QjkQki)φi

= i(Pk

jφj)† ˙(Pkiφi)− i(D − 1)

(Qk

jφj)† ˙(Qkiφi)

= iρ†i ρi − iD − 1

Dσ†σ

= −iρTi Cγ0ρi + iD − 1

DσTCγ0σ

= iρTi ρi − iD − 1

DσT σ, (10.3.14)

where in the second to last line we used the reality conditions (10.3.12) and in the last linewe restricted to a real representation, C = γ0.

This motivates the definition of the bracketsρj , ρ

i

= − i2

1δij and σ, σ = iD

2(D − 1)1, (10.3.15)

together with the reality conditions ρ∗i = ρi, σ∗ = σ (cf. (10.3.12)) and additionally introduced

constraints to account for the superfluous fermionic degrees of freedom,

Λα := γiαβρβi ≈ 0. (10.3.16)

We need to check that the extension is valid, i.e. that the Poisson brackets of the φi, consideredas functions on the extended phase space, are equal to the Dirac brackets (10.3.6) of the systembefore we did the extension. Using φi = Pijρj + 1

Dγiσ (cf. 10.3.11) and the Poisson brackets(10.3.15), this can be checked explicitly (this calculation shows why the factors of 1

2 in (10.3.15)are needed). Using this, we can express the constraints H and S in terms of the new variablesin the obvious way and know that their algebra is unchanged. In particular, since the projectorsare built from intertwiners, we find for the fermionic part of the Gauß constraint

Gij = ...+(−iρkT

) [2η

[ik η

j]l + iΣijηkl

]ρl +

(iD − 1

DσT)[

iΣij]σ, (10.3.17)

which allows for an easy generalisation to SO(D+1) or SO(1, D) as a gauge group. Furthermore,since ρi in the other constraints only appears in the combination Pijρj , they automaticallyPoisson commute with Λα.

Note that if we now would calculate the Dirac bracket, we would get ρi, ρjDB = − i2Pij ,

which again is non-trivial. Instead, we directly enlarge the phase space from ρi, σ to ρI , σ,with, as a first guess, the brackets ρI , ρJ = − i

2ηIJ1, σ, σ = i D2(D−1)1, the reality conditions

ρ∗I = ρI , σ∗ = σ and the constraints

N IρI ≈ 0 and γIρI ≈ 0. (10.3.18)

Unfortunately, this immediately leads to an inconsistency in the case of the compact gauge groupSO(D + 1), since for our choice of Dirac matrices, γ0 necessarily is complex in the Euclideancase. Therefore, the reality conditions again are not SO(D + 1) covariant and the constraints(10.3.18) only are consistent in the time gauge N I = δI0

4. With a more elaborate choice ofreality condition it is possible to define a consistent theory, which will be the subject of the nextsection.

4γIρI ≈ 0 is a complex constraint and thus equal to two real constraints. Only in time gauge, its imaginarypart is already solved by demanding NIρI ≈ 0.

78

10.3.2 SO(D + 1) gauge supergravity theory

As we just have seen, the remaining obstacle on our road of extending the internal gauge groupfrom SO(D) to SO(D + 1) is that the real vector space V of real SO(1, D) Majorana spinors isnot preserved under SO(D + 1) whose spinor representations are necessarily on complex vectorspaces. Let VC be the complexification of V . Now SO(D + 1) acts on VC but the theory westarted from is not VC but rather the SO(D + 1) orbit of V . This is the real vector subspace

VR = θ ∈ VC; ∃ ρ ∈ V, g ∈ SO(D + 1) 3 θ = g · ρ, (10.3.19)

where g· denotes the respective representation of SO(D + 1). This defines a reality structureon VC that is VC = VR ⊕ iVR. The mathematical problem left is therefore to add the realitycondition that we are dealing with VR rather than VC.

In order to implement this, recall that any g ∈ SO(D + 1) can be written as g = BR whereB is a “Euclidean boost” in the 0j planes and R a rotation that preserves the internal vectornI0 := δI0 . The spinor representation of R just needs γj which is real valued. It follows that(10.3.19) can be replaced by

VR = θ ∈ VC; ∃ ρ ∈ V, B ∈ SO(D + 1) 3 θ = B · ρ. (10.3.20)

The problem boils down to extracting from a given θ ∈ VR the boost B and the element ρ ∈ V ,that is, we need a kind of polar decomposition. If VC would be just a vector subspace of some Cn

we could do this by standard methods. But this involves squaring of and dividing by complexnumbers and these operations are ill defined for our VC since Graßmann numbers are nilpotent.Thus, we need to achieve this by different methods.

The natural solution lies in the observation that if we use the linear simplicity constraint thenthe D boost parameters can be extracted from the D rotation angles in the normal N I = BIJn

J0

to which we have access because N is part of the extended phase space. To be explicit, let e(A)

be the standard base of RD+1, that is, e(A)I = δAI . We construct another orthonormal basis b(A)

of RD+1 as follows:Let b(0) := N and

b(0)0 = sin(φ1).. sin(φD), b

(0)j = sin(φ1).. sin(φD−j) cos(φD+1−j); j = 1..D, (10.3.21)

with φ1, ..φD−1 ∈ [0, π] and φD ∈ [0, 2π] modulo usual identifications and singularities of polarcoordinates. Define

b(j)I =

∂b(0)I /∂φj

||∂b(0)/∂φj ||, (10.3.22)

where the denominator denotes the Euclidean norm of the numerator. Then it maybe checkedby straightforward computation that

δIJ b(A)I b

(B)J = δAB. (10.3.23)

We consider now the SO(D + 1) matrix

(A(N)−1)IJ :=

D∑A=0

b(A)I e

(A)J , (10.3.24)

which has the property that A(N)−1 · e(0) = N .Now starting from the time gauge, g ∈ SO(D + 1) acts on V and produces N = g · e(0)

and θ = g · ρ. We decompose g = A(N)−1R(N) where A(N)−1 is the boost defined above and

79

R · e(0) = e(0) is a rotation preserving e(0). It follows that we may parametrise any pair (N, θ)with ||N || = 1 and θ ∈ VR as A(N)−1 · (e(0), ρ) where ρ ∈ V . We need to investigate howSO(D + 1) acts on this parametrisation. On the one hand we have

[g A(N)−1]IJ =∑A

(gb(A))I e(A)J . (10.3.25)

On the other hand we can construct A(g · N)−1 by following the above procedure, that is,

computing the polar coordinates θgj of g ·N and defining the b(A)j (g ·N) via the derivatives with

respect to the θgj . The common element of both bases is g ·N = g · b(0). Therefore, there existsan element R(g,N) ∈ SO(D) such that

g · bj(N) = Rkj(g,N)b(k)(g ·N), (10.3.26)

or with R00 = 1, R0i = Ri0 = 0

g · bA(N) = RBA(g,N)b(B)(g ·N) (10.3.27)

defines a rotation in SO(D + 1) preserving e(0). Putting these findings together we obtain

[g ·A(N)−1]IJ =∑A,B

RBA(g,N) b(B)I (g ·N) e

(A)J =

∑A

RAJ(g,N) b(A)I (g ·N)

=∑A

RKJ(g,N) b(A)I (g ·N) δ

(A)K = [A(g ·N)−1R(g,N)]IJ . (10.3.28)

Hence the matrix A(N)−1 plays the role of a filter in the sense that the action of SO(D + 1)on A(N)−1 · ρ can be absorbed into the matrix A−1 parametrised by g · N modulo a rotationthat preserves V and thus altogether the decomposition of VR = A(N)−1 · V ; ||N || = 1 ispreserved with the expected covariant action of SO(D + 1) on N . It therefore makes sense toimpose the reality condition that A(N) θ is a real spinor. In the subsequent construction, thisidea will be implemented together with an extension of the phase space ρj → ρI subject to theconstraint N IρI = 0. All these constraints and the reality conditions are second class and wewill show explicitly that the symplectic structure reduces to the time gauge theory. Despite thefact that we end up with a non trivial Dirac (anti-) bracket, it can nevertheless be quantised andnon trivial Hilbert space representations can be found as we will demonstrate in the next section.

We define A(N) ∈ SO(D + 1) quite generally5 in the spin 1 representation by the equation

AIJNJ = δI0 . (10.3.29)

It is determined up to SO(D) rotations. From the above equation, it follows that

A0I = NI and AIJXJ = δIiAIJX

J (10.3.30)

for XJ arbitrary. The corresponding rotation on spinors will be denoted by A. This matrix ro-tates the normalN I into its time gauge value δI0 without imposing time gauge explicitly, which wewill use to circumvent the reality problems of the SO(D+1) theory mentioned above appearing if

5There exist other possible choices apart from the construction using polar coordinates which might be bettersuited for certain problems. In D = 3, we can, e.g., construct A(N) as a linear function of the components ofNI by using A0I = NI and subsequently interchanging the components of NI with appropriate signs for theremaining columns of A(N).

80

we do not choose time gauge. We introduce the set of variables (AaIJ , πbKL, N I , PJ , ρI , ρ

∗J , σ, σ

∗)together with the following non-vanishing Poisson brackets

AaIJ(x), πbKL(y)

= 2δbaδ[KI δ

L]J δ

D(x− y),N I(x), PJ(y)

= δIJδ

D(x− y),

ρI(x), ρ∗J(y) = −iηIJ1δD(x− y), σ(x), σ∗(y) = iD

D − 11δD(x− y),

(10.3.31)

and the reality conditions

χI := AρI − (AρI)∗ = 0, χ := Aσ − (Aσ)∗ = 0 , (10.3.32)

which just say that the fermionic variables are real as soon as the normal N I gets rotated intotime gauge. Notice that before imposing the constraints, ρ, θ are complex Graßmann variablesand only the Poisson brackets between these and their complex conjugates are non-vanishing.The non-vanishing brackets between themselves of the previous section will be recovered whenreplacing the above Poisson bracket by the corresponding Dirac bracket.

Additionally, we want that the variables transform nicely under spatial diffeomorphisms andgauge transformations, thus we add

GIJ := DaπaIJ + 2P [INJ ] − 2iρ†[IρJ ] − iρ†K [iΣIJ ]ρK + i

(D − 1

Dσ†)

[iΣIJ ]σ + . . . (10.3.33)

Ha :=1

2πbIJ∂aAbIJ −

1

2∂b

(πbIJAaIJ

)+ P I∂aNI

− i2∂a(ρ

†I)ρI +i

2ρ†I∂aρI + i

D − 1

2D∂a(σ

†)σ − iD − 1

2Dσ†∂aσ + . . . . (10.3.34)

The old variables are expressed in terms of the new ones by

Eai := ζAiJ ηJKπaIKNI , Kai = ζAi

I ηIK(AaKJ − ΓaKJ(π))NJ ,

ρi =1

2AiJ η

JK (AρK +A∗ρ∗K) , σ =1

2(Aσ +A∗σ∗) , (10.3.35)

where the bar here means rotational components w.r.t N I , ηIJ := ηIJ − ζNINJ . To removeunnecessary degrees of freedom, we need the constraints

SaIM

:= εIJKLMNJπaKL,

N := N INI − ζ,

Λ := γIAIJ ηJK(AρK +A∗ρ∗K) = AγJ η

JK(ρK +A−1A∗ρ∗K),

Θ := N I(AρI +A∗ρ∗I), (10.3.36)

together with the Hamilton and supersymmetry constraints, where we replace the old by thenew variables as shown above. To prove that this theory is equivalent to supergravity and canpossibly be quantised, we have to answer the following questions:

• Are the reality conditions (10.3.32) consistent? I. e., do they transform under gaugetransformations in a sensible way and do they (weakly) Poisson commute with the otherconstraints?

• Are the Poisson brackets of the old variables when expressed in terms of the new ones(10.3.35) equal to those on the old phase space? Does the constraint algebra close, i.e. dothe newly introduced constraints (10.3.36) fit “nicely” in the set of the old constraints? Ifnot, do at least the constraints which were of the first class before the enlargement of thegauge group retain this property?

81

• Do the constraints, especially the Gauß and spatial diffeomorphism constraint, reducecorrectly?

• Which Dirac brackets arise from the reality conditions? In view of a later quantisation,can we find variables such that the Dirac brackets become simple?

We will answer these questions in the order they were posed above.

• The orthogonal matrix AIJ is a function of N I only as we have seen above. We haveA0K = NK , but the remaining components of the matrix are complicated functions of thecomponents of the vector N I . Thus, the whole matrix AIJ will have a rather awkwardtransformation behaviour under the action of GIJ . The reality conditions (10.3.32) as awhole, however, transform in a “nice” way under SO(D + 1) gauge transformations (wewill discuss ρI in the following, σ can be treated analogously). For g ∈ SO(D + 1), thereality condition transforms as follows:

A(N)ρJ = A(N)∗ρJ∗ −→ gJKA(g ·N)gρK = g∗JKA(g ·N)∗g∗ρ∗K . (10.3.37)

Since gIJ is real, it is sufficient to consider the transformation behaviour of the spinor AρI ,so we will skip the action on internal indices in the following. Note that every rotation canbe split up in a part which leaves N I invariant and a “Euclidean boost” changing N I . Forthe rotations, A is invariant and we find using AγIA−1 = ηIJA

−1JKγ

K and Σij∗ = −Σij

δΛAρI = iΛJKAΣJKρI = iΛJKAΣJKA−1AρI = iΛJKA−1[J |LA

−1K]MΣLMAρI =

= iAL[JAM |K]ΛJKΣLMAρI = iAl[JAm|K]Λ

JKΣlmAρI , (10.3.38)

δΛ(AρI)∗ = (iΛJKAΣJKρI)

∗ = (iAl[JAm|K]ΛJKΣlmAρI)

∗ =

= −iAl[JAm|K]ΛJKΣlm∗A∗ρ∗I = iAl[JAm|K]Λ

JKΣlmA∗ρ∗I . (10.3.39)

For finite transformations g ∈ SO(D)N stabilising N I , we thus have AρI → AgρI =g0AρI , where g0 ∈ SO(D)0 stabilises the zeroth component and thus is, with our choiceof representation, a real matrix. Hence, reality conditions transform again into realityconditions under rotations. For a boost b the situation is a bit more complicated. Under aboost AIJ will transform intricately, but we know that a) the matrix remains orthogonalby construction, and b) that A0K = NK → ΛK

LNL = −A0LΛLK . The most generaltransformation compatible with the above is AIJ → (g0)IKA

KL(g−1)LN (b−1)NM (bg−1)MJ

where g0 ∈ SO(D)0 is some group element which does not change the zeroth component,g ∈ SO(D)N is in the stabiliser of N I and bg ∈ SO(D)b·N . Since we have SO(D)N =b−1SO(D)b·Nb, we can eliminate bg by a redefinition of g. By definition of a representation,we then also have A→ g0Ag

−1b−1 and thus

AρI → g0Ag−1b−1bρI = g0Ag

−1ρI = g0AρI , (10.3.40)

where in the last step we used the result we obtained for rotations above. Since g0 ∈SO(D)0 is real, we see that under a “Euclidean boost” the reality condition can onlyget rotated. What remains to be checked is that the reality condition Poisson commuteswith all other constraints. It transforms covariantly under spatial diffeomorphisms byinspection and, as we have just proven, it forms a closed algebra with SO(D + 1) gaugetransformations. Concerning all other constraints, note that they, by construction, dependonly on <(AρJ) (cf. the replacement (10.3.35) and the new constraints (10.3.36)), whilethe reality condition demands that =(AρJ) vanishes. But real and imaginary parts Poissoncommute, which can be checked explicitly,

(AρI −A∗ρ∗I) , (AρJ +A∗ρ∗J)

= −iηIJ[+AA† −A∗AT

]= 0. (10.3.41)

82

• The brackets between Eai and Kbj have already been shown to yield the right results in[38]. The only modifications in the case at hand are a) the replacement of nI(π) by N I andthe corresponding replacement of the quadratic by the linear simplicity constraint, which,in fact, simplifies the calculations, and b) the matrix AIJ , which does not lead to problemsbecause of its orthogonality. For the fermionic variables, we find using6 Ai

IAjJ ηIJ = ηij

and A†A = 1

ρi(x), ρj(y) = − i4Ai

IAjJ ηKI η

LJ [AρK(x), A∗ρ∗L(y)+ A∗ρ∗K(x), AρL(y)]

= − i4Ai

IAjJ ηIJ

[AA† +A∗AT

]δD(x− y)

= − i4δij[1 + 1T

]δD(x− y)

= − i2δij1δ

D(x− y), (10.3.42)

σ(x), σ(y) = iD

2(D − 1)1δD(x− y). (10.3.43)

This automatically implies that the algebra of H and S remains unchanged if we replacethe old variables by (10.3.35). From (10.3.35), it is also clear that H and S Poissoncommute with Sa

IMand N . By inspection, all constraints transform covariant under

spatial diffeomorphisms. More surprisingly, all constraints Poisson commute with GIJ .This can be seen quite easily for GIJ , Ha, SaIM , N and also for Λ and Θ (note that

A, AIJ are invertible and that(ρI +A−1A∗ρ∗I

)transforms like ρI which can be shown

using the methods above). But for H and S this is, at first sight, a small miracle, sincethe replacement rules (10.3.35) of all old variables depend on A(N), which is known totransform oddly. But the matrices A are placed such that they, in fact, either a) appear

in the combinations (ρI + A−1A∗ρ∗I) or (ρ†I + ρTI ATA), which can easily be shown to

transform like ρI and ρ†I respectively with the methods above, or b) all cancel out! Thegeneral situation is the following: ρi is replaced by ρi = Ai

J ηJIA(ρI + A−1A∗ρ∗I), ρ

Ti by

ρTi = AiJ ηJ

I(ρ†I + ρTATA)A−1, where the expression in brackets transform sensible (cf.above). The free internal indices of Eai, Kbj and ρk are either contracted with each other,then in the replacement the AiJs will cancel because of orthogonality, or with γi, which willbe contracted from both sides7 with A(N) and all As cancel due to (A−1)IJA

−1γJA = γI .Cancelling the As makes H gauge invariant and S gauge covariant by inspection, if wereplace all γ0 by i /N . Thus we are left with Θ and Λ, which are their own second classpartners but Poisson commute with everything else, which can be seen as follows. For Θ,note that H, S and Λ only depend on ηJK(AρK + A∗ρ∗K), which Poisson commutes withΘ due to the projector η. For Λ, the situation again is more complicated. Rememberthat H and S in the time gauge only depended on XiPijρj for some Xi. WhateverXi may be, under (10.3.35) it will be replaced by something of the form AIJXJ and thewhole expression will become XIA

JIPJKAKLηLM (AρM +A∗ρ∗M ) with PIJ = ηIJ− 1DγIγJ .

Crucial for the following calculation is the property (10.3.30), which will be used several

6Because of orthogonality, we trivially have AIJAKJ = ηIK . Additionally, AiJ η

JK = AiK , which can be seen

from AiKNK = 0. Therefore, Ai

IAjJ ηIJ = Ai

IAjI = ηij .7Strictly speaking, this is true only for H, since it has no free indices. For S we may change the definition of

the Lagrange multiplier ψt → ψtA to make it hold.

83

times. Then we find that the generic term is Poisson commuting with Λ,XIA

JIPJKAKLηL

M (AρM +A∗ρ∗M ) , γNANOηOP (AρP +A∗ρ∗P )

= −iXIA

JIPJKAKLηL

M(γN)TANM

= −iXIAjIPjkA

kLηLM (γn)T AnM

= −iXIAjIPjkA

kLγnAnL = −iXIAjIPjkγ

k = 0. (10.3.44)

The constraint algebra is summarised in table 10.1.

First class constraints Second class constraints

GIJ , Ha, H, S, SaIM

and N Λ, Θ, χI and χ

Table 10.1: List of first and second class constraints.

• By construction, H and S reduce correctly if we choose time gauge N I = δI0 , whichautomatically implies AIJ → (g0)IJ ∈ SO(D)0. Since the theory is SO(D)0 invariant, agauge transformation g0 → 1 can be performed, which implies ρI = ρIr . From this oneeasily deduces that GIJ and Ha also reduce correctly. Since the theory was SO(D + 1)invariant in the beginning, these results do not depend on the gauge choice.

• For the Dirac matrix, we find8

CIJ = AρI − (AρI)∗, AρJ − (AρJ)∗

= −iηIJ[−AA† −A∗AT

]= 2i1ηIJ (10.3.45)

(C−1)IJ = −1

i1ηIJ (10.3.46)

ρI , ρJDB = −ρI , AρK − (AρK)∗ (C−1)KL AρL − (AρL)∗, ρJ =

= − i2ηIJA

†A∗, (10.3.47)

and for σ analogously. We now can choose new variables which have simpler brackets.Motivated from the original replacement (10.3.35), we define

ρIr := AIJAρJ , σr := Aσ, (10.3.48)(ρIr)∗

= AIJA∗ρ∗J = AIJA∗((A∗)−1AρJ) = ρIr , σ∗r = σr, (10.3.49)

with the Dirac bracketsρIr , ρ

Jr

DB

=AIKAρK , A

JLAρLDB

= − i2ηIJAA†A∗AT = − i

2ηIJ1,(10.3.50)

σr, σr = iD

2(D − 1)1. (10.3.51)

Thus, the Dirac brackets of the ρIr , σr are simple as are the reality conditions. Only thetransformation behaviour of the new variables under SO(D + 1) rotations is complicated

8Note that the Dirac matrix is block diagonal. Therefore, we do not need to consider the full Dirac matrix atonce.

84

because of the appearance of the rotation A in their definition. Note that alsoP I , P J

DB

,P I , ρJr

DB

andP I , σr

DB

will be non-zero. Therefore, we also choose a new variable P I

with simple Dirac brackets, which can most easily be found by performing the symplecticreduction. After that, we can simply read it off the symplectic potential. We find usingρI = AJIA

−1ρJr and ρ†I = AJI(ρJr )TA

+iρ†I ρI − iD − 1

Dσ†σ + P INI

= iAJI(ρJr )TA ˙(AKIA−1ρKr )− iD − 1

DσTr A

˙(A−1σr) + P INI

= i(ρJr )T ρJr − iD − 1

DσTr σr + P INI +

+i

(AJ

L(ρJr )T∂AKL∂NI

ρKr + (ρJr )TA∂A−1

∂NIρJr −

D − 1

DσTr A

∂A−1

∂NIσr

)NI

= i(ρJr )T ρJr − iD − 1

DσTr σr + P INI , (10.3.52)

with P I := P I + iAJL(ρJr )T ∂AKL∂NI

ρKr + i(ρJr )TA∂A−1

∂NIρJr − iD−1

D σTr A∂A−1

∂NIσr. It can be

checked explicitly that P I , expressed in the old variables (P I , NJ , ρ†I , ρ

J , σ†, σ), Poissoncommutes with the reality conditions and with itself, and therefore has nice Dirac brackets.For the spatial diffeomorphism constraint, a short calculation yields

Ha = P I∂aNI −i

2∂a(ρ

†I)ρI +i

2ρ†I∂aρI + i

D − 1

2D∂a(σ

†)σ − iD − 1

2Dσ†∂aσ + . . . =

= P I∂aNI + i(ρIr)T∂aρIr − i

D − 1

DσTr ∂aσr + . . . , (10.3.53)

which by inspection generates spatial diffeomorphisms on the new variables. The con-straints Λ and Θ become

Λ = γiρir ≈ 0 and Θ = ρ0

r ≈ 0, (10.3.54)

which look utterly non-covariant, but which by construction still Poisson commute withthe SO(D + 1) Gauß constraint. It therefore has to have a complicated form. We find

GIJ = 2P [INJ ] − 2iρ†[IρJ ] − iρ†K [iΣIJ ]ρK + i

(D − 1

Dσ†)

[iΣIJ ]σ + . . . =

= 2P [INJ ] − 2iρTKr AK[IAL

|J ]ρLr − iρTKrA[iΣIJ ]A−1ρKr

+iD − 1

DσTr A[iΣIJ ]A−1σr − 2i

(AM

L(ρMr )T∂AKL∂N[I

ρKr +

+ (ρNr )TA∂A−1

∂N[IρNr −

D − 1

DσTr A

∂A−1

∂N[Iσr

)NJ ] + . . . (10.3.55)

Finally, we solve the remaining second class constraints Λ and Θ which after a short calcu-lations results in the final Dirac brackets

ρir, ρjr

DB

= − i2

Pij ,ρ0r , ρ

jr

DB

= 0,ρ0r , ρ

0r

DB

= 0.

As a consistency check, we can considerφi, φj

DB

=

ρir +

1

Dγiσr, ρ

jr +

1

Dγjσr

DB

= −(C−1)ij , (10.3.56)

85

which coincides with the Dirac brackets obtained in (10.3.6).The form of the Hamiltonian and supersymmetry constraints H, S strongly depends on the

supergravity theory under consideration. Exemplarily, we cite the supersymmetry constraint inD = 3, N = 1 supergravity from [120] adapted to our notation,

S = − i2εabcγ5

[γke

kaDb

(14√qelcφl

)+ Db

(14√qγke

kaelcφl

)]+

1

2εabcεijke

ia

(Kb

j − iφlγ0γljEmb φm

)γkenc φn, (10.3.57)

where Daφi = ∂aφi + ωaijφj + i

2 ωaklΣklφi, ωaij = Γaij + i

4√qeka

(φiγkφj + 2φ[iγj]φk

), and Γaij is

the spin-connection annihilating the triad. An explicit expression for S in terms of the extendedvariables (A, π,N, P, ρ, ρ∗, σ, σ∗) can be found using (10.3.11), (10.3.35). The correspondingconstraint operator is obtained using the methods in section 10.4, 7 and [20, 37].

10.4 Background independent Hilbert space representations forMajorana fermions

Background independent Hilbert space representations for Dirac spinor fields were constructedin [21]. One may think that for the Rarita-Schwinger field or more generally for Majoranafermion fields one can reduce to this construction as follows: Consider the following variables

ξIα =1√2

(ρ2α+1r + iρ2α+2

r

), πIα =

−i√2

(ρ2α+1r − iρ2α+2

r

), α = 1, . . . , 2b(D+1)/2c, (10.4.1)

which have the non-vanishing Dirac bracketsξIα(x), πJβ(y)

= −iηIJδαβδ(D)(x− y) (10.4.2)

and the simple reality conditionπ = −iξ. (10.4.3)

The elements of the Hilbert space are field theoretic extensions of holomorphic (i.e. they onlydepend on θα) functions on the Graßmann space spanned by the Graßmann numbers θα andtheir adjoints θα, the operators corresponding to the phase space variables act as

ξf := θf, πf := id

dθf, (10.4.4)

and the scalar product

< f, g >:=

∫eθθfg dθ dθ (10.4.5)

faithfully implements the reality conditions. There are, however, two drawbacks to this:1. Due to the arbitrary split of the variables into two halves, the scalar product is not SO(D)invariant which makes it difficult to solve the Gauß constraint.2. The scalar product given above fails to implement the Dirac bracket resulting from the sec-ond class constraints, that is, ρrαi , ρ

rβj DB = −i/2 Pαβij . Recall that one must solve the second

class constraints before quantisation, hence it is not sufficient to consider the quantisation ofthe Poisson bracket as was done above.

In what follows we develop a background independent Hilbert space representation that is SO(D)

86

invariant, implements the Dirac bracket and is geared to real valued (Majorana) spinor fields.We begin quite generally with N real valued Graßmann variables θA, A = 1, .., N ; θ(AθB) =0; θ∗A = θA. We consider the finite dimensional, complex vector space V of polynomials in the θAwith complex valued coefficients. Notice that f ∈ V depends on all real Graßmann coordinates,it is not holomorphic as in the case of the Dirac spinor field [21]. Thus dim(V ) = 2N is thecomplex dimension of V . We may write a polynomial f ∈ V in several equivalent ways which

are useful in different contexts. Let f(n)A1..An

, 0 ≤ n ≤ N be a completely skew complex valuedtensor (n-form) then f can be written as

f =

N∑n=0

1

n!f

(n)A1..An

θA1 ..θAn =N∑n=0

∑1≤A1<..<An≤N

f(n)A1..An

θA1 ..θAn . (10.4.6)

An equivalent way of writing f is by considering for σk ∈ 0, 1 and A1 < .. < An the relabelledcoefficients

fσ1..σN := f(n)A1..An

, σk :=

1 k ∈ A1, .., An0 else.

(10.4.7)

It followsf =

∑σ1,..,σN∈0,1

fσ1..σN θσ11 ..θσNN (10.4.8)

with the convention θ0A := 1.

On V we define the obvious positive definite sesqui-linear form

< f, f ′ >:=N∑n=0

∑A1<..<An

f(n)A1..AN

f(n)′A1..AN

=∑σ1..σN

fσ1..σn f′σ1..σN

(10.4.9)

as well as the operators

[θA · f ](θ) := θA f(θ), [∂A · f ](θ) := ∂lf(θ)/∂A, (10.4.10)

where the latter denotes the left derivative on Graßmann space (see, e.g., [144] for precisedefinitions). Notice the relations ∂(A∂B) = 0, 2∂(AθB) = δAB which can be verified by applyingthem to arbitrary polynomials f . We claim that the operators (10.4.10) satisfy the adjointnessrelation

θ†A = ∂A. (10.4.11)

The easiest way to verify this is to use the presentation (10.4.8). We find explicitly

θA · f =∑

σ1,..,σN

fσ1..σN (−1)σ1+..+σA−1 δσA,0 θσ11 ..θA..θ

σNN

=∑

σ1,..,σN

[fσ1..σA−1..σN (−1)σ1+..+σA−1 δσA,1] θσ11 ..θσNN

=:∑

σ1,..,σN

fAσ1..σNθσ1

1 ..θσNN ,

∂A · f =∑

σ1,..,σN

fσ1..σN (−1)σ1+..+σA−1 δσA,1 θσ11 ..θA..θ

σNN

=∑

σ1,..,σN

[fσ1..σA+1..σN (−1)σ1+..+σA−1 δσA,0] θσ11 ..θσNN

=:∑

σ1,..,σN

fAσ1..σNθσ1

1 ..θσNN , (10.4.12)

87

where the wide hat in the fourth line denotes omission of the variable. We conclude

< f, θAf′ > =

∑σ1,..,σN

fσ1..σN fA′σ1..σN

=∑

σ1,..,σN

fσ1..σN (−1)σ1+..+σA−1 f ′σ1..σA−1..σNδσA,1

=∑

σ1,..,σN

fσ1..σA+1..σN (−1)σ1+..+σA−1δσA,0 f′σ1..σN

=∑

σ1,..,σN

fAσ1..σNf ′σ1..σN

=< ∂Af, f′ > . (10.4.13)

Although not strictly necessary, it is interesting to see whether the scalar product (10.4.9)can be expressed in terms of a Berezin integral, perhaps with a non-trivial measure as in [21]for complex Graßmann variables. The answer turns out to be negative: The most generalAnsatz for a “measure” is dµ = dθ1..dθN , µ(θ) with µ ∈ V fails to reproduce (10.4.9) if weapply the usual rule for the Berezin integral9

∫dθ θσ = δσ,1. Notice that from this we induce∫

dθA dθB = −∫dθB dθA as one quickly verifies when applying to V . However, there exists a

non-trivial differential kernel

K := (θ1 + (−1)N−1∂1)..(θN + (−1)N−1∂N ) (10.4.14)

such that

< f, f ′ >=

∫dθN ..dθ1 f

∗ K f ′, (10.4.15)

where we emphasise that f∗ is the Graßmann involution

f∗ =∑σ1..σN

fσ1..σN θσNN ..θσ1

1 =∑σ1..σN

fσ1..σN (−1)∑N−1k=1 σk

∑Nl=k+1 σl θσ1

1 ..θσNN (10.4.16)

and not just complex conjugation of the coefficients of f . Notice also that due to total antisym-metry we may rewrite (10.4.15) in the form

< f, f ′ >=(−1)N(N−1)/2

N !

∫dθA1 ..dθAN f∗DA1 ..DAN f ′ (10.4.17)

whereDA = θA + (−1)N−1∂A. (10.4.18)

The presentation (10.4.19) establishes that the linear functional is invariant under U(N) actingon V by

f 7→ U · f ; [U · f ](n)A1..AN

= f(n)B1..BN

UB1A1 ..UBNAN , (10.4.19)

which is of course also clear from (10.4.9). Notice that (10.4.19) formally corresponds toθA 7→ UABθB but this is not an action on real Graßmann variables unless U is real valued.If we want to have an action on the linear polynomials with real coefficients then we must re-strict U(N) to O(N) or a subgroup thereof which will precisely the case in our application. Inthis case it is sufficient to restrict to real valued coefficients in f and now the real dimension ofV is 2N .

9Rather a linear functional on V which is of course also a non-Abelian Graßmann algebra.

88

We sketch the proof that (10.4.14) accomplishes (10.4.15). We introduce the notation fork = 1, .., N

Fσ1..σk :=∑

σk+1..σN

fσ1..σN θσk+1

k+1 ..θσNN , (10.4.20)

whence Fσ1..σN = fσ1..σN . Notice that Fσ1..σk no longer depends on θ1, .., θk. Using this wecompute with dNθ := dθN ..dθ1 and using anticommutativity at various places∫

dNθ f∗ K f ′

=

∫dNθ [F ∗0 + F ∗1 θ1] (θ1 + (−1)N−1∂1) D2 .. DN [F ′0 + θ1 F

′1]

=

∫dNθ

F ∗0 (−1)N−1 D2..DN (θ1 + (−1)N−1∂1)[F ′0 + θ1F

′1]

+ F ∗1 (−1)N−1 D2 .. DN θ1∂1[F ′0 + θ1 F′1]

=

∫dNθ

F ∗0 (−1)N−1 D2..DN [θ1F

′0 + (−1)N−1F ′1] + F ∗1 (−1)N−1 D2 .. DN θ1 F

′1

=

∫dNθ

F ∗0 θ1 D2..DN F ′0 + F ∗1 θ1 D2 .. DN F ′1

, (10.4.21)

where we used that the second term no longer is linear in θ1 and therefore drops out from theBerezin integral. The calculation explains why the factor (−1)N−1 in (10.4.18) is necessary.

Next consider the first term in the last line of (10.4.21). We have∫dNθ F ∗0 θ1 D2..DN F ′0

=

∫dNθ [F ∗00 + F ∗01θ2] θ1 (θ2 + (−1)N−1∂2) D3..DN [F ′00 + θ2F

′01]

=

∫dNθ

F ∗00 (−1)N−2 θ1 D3..DN (θ2 + (−1)N−1∂2)[F ′00 + θ2F

′01]

+ F ∗01 (−1)N−2 θ1 D3..DN θ2∂2 [F ′00 + θ2F′01]

=

∫dNθ

F ∗00 (−1)N−2 θ1 D3..DN [θ2F

′00 + (−1)N−1F ′01] + F ∗01 (−1)N−2 θ1 D3..DN θ2 F

′01]

=

∫dNθ

F ∗00 θ1θ2 D3..DN F ′00 + F ∗01 θ1θ2 D3..DN F ′01

. (10.4.22)

Similarly for the second term in (10.4.21)∫dNθ F ∗1 θ1 D2..DN F ′1 =

∫dNθ

F ∗10 θ1θ2 D3..DN F ′10 + F ∗11 θ1θ2 D3..DN F ′11

. (10.4.23)

It is transparent how the computation continues: We continue expanding Fσ1..σk = Fσ1..σk0 +θk+1Fσ1..σk1 and see by exactly the same computation as above that10 the signs match up to theeffect that∫

dNθ F ∗σ1..σkθ1..θk Dk+1..DN F ′σ1..σk

=∑σk+1

∫dNθ F ∗σ1..σk+1

θ1..θk+1 Dk+2..DN F ′σ1..σk+1,

(10.4.24)

10A strict proof would proceed by induction which we leave as an easy exercise for the interested reader.

89

from which the claim follows using∫dNθ θ1..θN = 1. In our applications N will be even so

that DA = θA − ∂A.

For our application to the Rarita-Schwinger field we consider the compound indexA = (j, α), j =1, .., D; α = 1, .., 2[(D+1)/2c or just A = α whence N = DM or N = M is even. We consider theauxiliary operator

ραj :=

√~

2[θαj + ∂αj ], (10.4.25)

which by virtue of (10.4.11) is self adjoint and satisfies the anticommutator relation

[ραj , ρβk ]+ =

~2δjk δ

αβ. (10.4.26)

However, ραj is not yet a representation of ρrαj which satisfies the Dirac antibracket ρrαj , ρrβk DB =

− i2Pαβjk and the reality condition (ρrαj )∗ = ρrαj . Similarly, σα, σβDB = i D

2(D−1)δαβ, σ∗α = σα.

Correspondingly, what we need is a representation π(ραj ), π(σα) of the abstract CAR ∗-algebradefined by canonical quantisation, that is,

[ρrαj , ρrβk ]+ =

~2

Pαβjk , (ρrαj )∗ = ρrαj , [σrα, σrβ]+ =

D~2(D − 1)

δαβ, (σrα)∗ = σrα (10.4.27)

all other anticommutators vanishing11. Fortunately, using that Pαβjk is a real valued projector(in particular symmetric and positive semidefinite) we can now write the following faithfulrepresentation of our abstract ∗-algebra (10.4.27) on the Hilbert H = VDM ⊗ VM defined above:

π(ρrαj ) := Pαβjk ρβk , π(σrα) :=

1

2

√D~D − 1

[θα + ∂α]. (10.4.28)

So far we have considered just one point on the spatial slice corresponding to a quantum me-chanical system. The field theoretical generalisation now proceeds exactly as in [21] and con-sists in considering copies Hx of the Hilbert space just constructed, one for every spatial pointx and taking as representation space either the inductive limit of the finite tensor productsHx1,..,xn = ⊗nk=1 Hxk [66] or the infinite tensor product [146] H = ⊗xHx of which the former isjust a tiny subspace. The ∗-algebra (10.4.27) is then simply extended by adding labels x to theoperators and to ask that anticommutators between operators at points x, y be proportional toδx,y in agreement with the classical bracket. It is easy to see that adding the label x to (10.4.28)correctly reproduces this Kronecker symbol and that they satisfy all relations on the Hilbertspace12. Finally notice that the corresponding scalar product is locally SO(D) invariant.

10.5 Generalisations to different multiplets

10.5.1 Majorana spin 1/2 fermions

The above construction generalises immediately to Majorana spin 1/2 fermions which are alsopresent in supergravity theories, e.g. D+ 1 = 9 + 1, N = 2a non-chiral supergravity [147]. They

11This corresponds to the quantisation rule that the anticommutator is +i~ times the Dirac bracket in the ρsector and −i~ times the Dirac bracket in the σ sector. This is the only possible choice of signs because theanticommutator of the same operator which in our case is self adjoint is a positive operator. The other choice ofsigns would yield a mathematical contradiction.

12In the case of the inductive limit, a vector in v ∈ Hx1,..,xn is embedded in any larger Hx1,..,xn,y1,..,ym byv 7→ v⊗ ⊗m1 where 1 is the constant polynomial equal to one. This way any operator at x acts in a well definedway on any vector in the Hilbert space.

90

are described by actions of the type

SMajorana, 1/2 =

∫dD+1X iλγµ,Dµλ (10.5.1)

which, using time gauge and a real representation for the γ-matrices, lead to the canonicalbrackets λα, λβ ∼ iδαβ with the reality conditions λ∗ = λ. They can thus also be treated withthe above techniques by substituting ρi with λ and removing the AIJ matrices as well as theηIJ projectors.

10.5.2 Mostly plus / mostly minus conventions

The convention used for the internal signature, i.e. mostly plus or mostly minus and the asso-ciated purely real or purely imaginary representations of the γ-matrices, does not interfere withthe above construction. The important property we are using is the reality of iΣIJ for SO(1, D),i.e. that the Gauß constraint is consistent with real spinors. The substitution γI → iγI necessarywhen changing the signature convention does not influence these considerations.

10.5.3 Weyl fermions

In dimensions D + 1 even, we also need to consider the case of Weyl fermions. To this end, wedefine

γfive := iD(D+1)

2+1γL0 γ1 . . . γD (10.5.2)

with the properties γ2five = 1, γ†five = γfive and [γI , γfive]+ = 0 (which follows from our conventions

for the gamma matrices (γL0 )2 = −1, γ2i = 1, γ†I = ηIIγI). We introduce the chiral projectors

P± =1

2(1± γfive) , (10.5.3)

which fulfil the relations P±P± = P±, P±P∓ = 0, P+ +P− = 1 and (P±)† = P±. These followdirectly from the properties of γfive.

Spin 1/2 Dirac-Weyl fermions

The kinetic term of the action for a chiral Dirac spinor is given by

SF = −∫MdD+1X

(i

2ΨeµI γ

IDµP+Ψ− i

2DµΨeµI γ

IP+Ψ

). (10.5.4)

The 3+1 split is performed analogous to [37]. Choosing time gauge, we obtain the non-vanishingPoisson brackets

Ψ±α ,Π±β

= −P±αβ, (10.5.5)

where Π±β = −i(Ψ±)†β, and the first class constraint

χα := Π−α , (10.5.6)

where we used the notation Ψ± := P±Ψ. The first class property of this constraint followsfrom the fact that the action (10.5.4) and therefore all resulting constraints do not depend onΨ− at all. In the quantum theory, the Hilbert space for the chiral fermions can be constructedsimilar to the case of non-chiral ones [21]. We obtain a faithful representation of the Poisson

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algebra by replacing the operators θα (acting by multiplication) and ˆθα = ddθα

defined in [21] by

θ+α := P+

αβ θβ and ˆθ+α := ˆθβP+

βα, as can be seen by[θ+α ,

ˆθ+β

]+

= P+αβ. (10.5.7)

The reality conditions are implemented if we use the unique measure constructed in [21]. Wethen have to impose the condition

ˆθ−α f(θβ) = 0, (10.5.8)

which restricts the Hilbert space to functions f such that f(θα) = f(P+αβθβ). Classically,

observables do not depend on Ψ−α . In the quantum theory, they become operators which do not

contain θ−α and therefore commute with ˆθ−α .

Spin 3/2 Majorana-Weyl fermions

Majorana-Weyl spin 3/2 fermions appear in chiral supergravity theories, e.g., D + 1 = 9 + 1,N = 1 [148]. In general, in a real representation (γTI = ηIIγI) or in a completely imaginary

representation (γTI = −ηIIγI) we have γTfive = (−1)D(D+1)

2+1γfive. Therefore, if D+1

2 is odd, wehave (P±)T = P±, and if D+1

2 is even, (P±)T = P∓. In the case at hand (D = 9), there existsa real representation and the chiral projectors will be symmetric, (P±)T = P±. Again, we willjust consider the kinetic term for a chiral Rarita-Schwinger field,

S =

∫MdD+1X

(is eψµγ

µρσDρP+ψσ)

. (10.5.9)

The 3 + 1 split is performed like above. We find the second class constraint π+i = i(φ+

j )TγjiP+

and the first class constraint π−i = 0. We introduce a second class partner φ−i = 0 for the firstclass constraint. Then we can solve all the constraints using the Dirac bracket

φ+i , φ

+j

DB

= −P+(C−1)ijP+, (10.5.10)

and all other brackets are vanishing. From here, we can copy the enlargement of the internalspace from above, which results in the same theory with all variables projected with P+. (Notethat equations like e.g. ρ+

i = 12AiJ η

JKP+ (AρK +A∗ρ∗K) = 12AiJ η

JK(Aρ+

K +A∗(ρ+)∗K)

areconsistent. This can be seen by the fact that the matrix A(N) can be written as an infinitesum of even powers of gamma matrices, A(N) ∝ exp(iΛIJ(N)ΣIJ) and therefore it commuteswith the projectors P±.) The quantisation of the resulting theory with variables ρ+

r I and σ+ issimilar to the non-chiral case, with chiral projectors P+ added in observables, and modificationsof the Hilbert space similar to the ones given above for Dirac-Weyl fermions.

10.6 Conclusions

In the present chapter, we have demonstrated that the complications arising when trying toextend canonical supergravity in the time gauge from the gauge group SO(D) to SO(D + 1)in order to achieve a seamless match to the canonical connection formulation of the gravitonsector outlined in the first part of this thesis can be resolved. Since we worked with a Ma-jorana representation of the γ-matrices, our analysis is restricted to those dimensions wherethis representation is available, which, however, covers many interesting supergravity theories(d = 4, 8, 9, 10, 11). The price to pay for the enlargement of the gauge group is that the phase

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space requires an additional normal field N and that the constraints depend non-trivially ona matrix A(N) which transforms in a complicated fashion under SO(D + 1) but which in thepresent formulation is crucial in order to formulate the reality conditions for the Majoranafermions in the SO(D + 1) theory.

One would expect that the field N is superfluous and that the matrix A(N) would simplydrop out when performing an extension to SO(1, D) because then no non-trivial reality conditionsneed to be imposed. One would expect that one only needs the quadratic and not the linearsimplicity constraint and that, just as it happened in the gravitational sector, the Hamiltonianphase space extension method simply coincides with the direct Hamiltonian formulation obtainedby an D+1 split of the SO(1, D) action followed by a gauge unfixing step in order to obtain a firstclass formulation. Surprisingly, this is not the case. The basic difficulty is that when performingthe D + 1 split without time gauge, the symplectic structure turns out to be unmanageable.A treatment similar to the one carried out in this chapter is possible but turns out to be ofsimilar complexity. It therefore appears that there is no advantage of the SO(1, D) extensionas compared to the SO(D + 1) even as far as the classical theory is concerned. Of course, thequantum theory of the SO(1, D) extension is beyond any control at this point.

The solution to the tension presented in this chapter, between having real Majorana spinorscoming from SO(1, D) on the one hand and an SO(D+1) extension of the theory which actuallyneeds complex valued spinors on the other, is most probably far from unique nor the mostelegant one. Several other solutions have suggested themselves in the course of our analysisbut the corresponding reformulation is not yet complete at this point. Hence, we may revisitthis issue in the future and simplify the presentation. Furthermore, it would be interesting toinvestigate if the extensions of the gauge group SO(D)→ SO(D+ 1) also is possible in the caseof symplectic Majorana fermions, which would permit access to even more supergravity theories.

To the best of our knowledge, the background independent Hilbert space representation ofthe Rarita-Schwinger field presented in section 10.4 is also new. Apart from the fact that thishas to be done for half-density valued Majorana spinors whose tensor index is transformed intoan external one by contracting with a vielbein, as compared to Dirac spinors there is no repre-sentation in terms of holomorphic functions [21] of the Graßmann variables and one had to dealwith the non-trivial Dirac bracket.

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Chapter 11

p-form gauge fields

In the previous chapter, we focussed on the quantisation of the Rarita-Schwinger sector ofsupergravity theories in various dimensions by using an extension of loop quantum gravity to allspacetime dimensions. In this chapter, we extend this analysis by considering the quantisationof additional bosonic fields necessary to obtain a complete SUSY multiplet next to graviton andgravitino in various dimensions. As a generic example, we study concretely the quantisation ofthe 3-index photon of maximal 11d SUGRA, but our methods easily extend to more generalp-form fields.Due to the presence of a Chern-Simons term for the 3-index photon, which is due to localSUSY, the theory is self-interacting and its quantisation far from straightforward. Nevertheless,we show that a reduced phase space quantisation with respect to the 3-index photon Gaußconstraint is possible. Specifically, the Weyl algebra of observables, which deviates from theusual CCR Weyl algebras by an interesting twist contribution proportional to the level of theChern-Simons theory, admits a background independent state of the Narnhofer-Thirring type.The original work from which this chapter is taken is [41].

11.1 Introduction

In the previous chapters of this thesis, we studied the canonical formulation of general relativitycoupled to standard matter in terms of connection variables for a compact gauge group withoutsecond class constraints in order that loop quantum gravity (LQG) quantisation methods, so farformulated only in three and four spacetime dimensions [149, 66], apply.

The new field content of supergravity theories as compared to standard matter Lagrangiansare 1. Majorana (or Majorana-Weyl) spinor fields of spin 1/2, 3/2 including the Rarita-Schwingerfield (gravitino) and 2. additional bosonic fields that appear in order to obtain a complete su-persymmetry multiplet in the dimension and the amount N of supersymmetry charges underconsideration. The treatment of the Rarita-Schwinger sector was subject to the previous chapter.In this chapter, we complete the quantisation of the extra matter content of many supergravitytheories by considering the quantisation of the additional bosonic fields, in particular, p-formfields. Specifically, for reasons of concreteness, we quantise the 3-index photon of 11d Super-gravity but it will transpire that the methods employed generalise to arbitrary p.

What makes the quantisation possible is that the Gauß constraints of the 3-index photonform an Abelian ideal in the constraint algebra. If this ideal (or subalgebra) would be non-Abelian, then our methods would be insufficient and we most probably would have to usemethods from higher gauge theory [150, 151, 152, 153, 154] such as p-groups, p-holonomiesetc., a subject which at the moment is not yet sufficiently developed from the mathematical

94

perspective (see [155] for the state of the art of the subject). Despite the Abelian character ofthis additional Gauß constraint, the quantisation of the theory is not straightforward and cannotbe performed in complete analogy to the treatment of the Abelian Gauß constraint of standard 1-form matter [20]. This is due to a Chern-Simons term in the supergravity action, whose presenceis dictated by supersymmetry and which makes the theory in fact self-interacting, that is, theHamiltonian is a fourth order polynomial in the 3-connection and its conjugate momentum justlike in Yang-Mills theory. In particular, while one can define a holonomy-flux algebra as forAbelian Maxwell-theory, the Ashtekar-Isham-Lewandowski representation [14, 15] is inadequatebecause the Abelian gauge group does not preserve the holonomy-flux algebra.

A solution to the problem lies in performing a reduced phase space quantisation in termsof a twisted holonomy-flux algebra, which is in fact Gauß invariant. We were not able to finda background independent representation of the corresponding Heisenberg algebra, which alsodiffers by a twist from the usual one, however, one succeeds when formulating the quantumtheory in terms of the corresponding Weyl elements. The resulting Weyl algebra is not of stan-dard form and to the best of our knowledge it has not been quantised before. We show thatit admits a state of the Narnhofer-Thirring type [49], whence the Hilbert space representationfollows by the GNS construction. The Hamiltonian (constraint) can be straightforwardly ex-pressed in terms of the Weyl elements, in fact it is quadratic in terms of the classical observables.

This chapter’s architecture is as follows:In section 11.2, we sketch the Hamiltonian analysis of the 3-index photon in a self-containedfashion for the benefit of the reader and in order to settle our notation. We also describe indetail why one cannot straightforwardly apply methods from LQG as mentioned above. Insection 11.3, we display the reduced phase space quantisation solution in terms of the twistedholonomy-flux algebra. Finally, in section 11.4, we summarise and conclude.

11.2 Classical Hamiltonian analysis of the 3-index-photon ac-tion

The Hamiltonian analysis of the full 11d SUGRA Lagrangian has been performed in [116]. Wewill review the analysis of the contribution of the 3-index-photon 3-form Aµνρ = A[µνρ] to the11d SUGRA Lagrangian with Chern-Simons term. This part of the Lagrangian is given up to anumerical constant by

LC = −1

2|g|1/2Fµ1..µ4 F

µ1..µ4−α|g|1/2Fµ1..µ4 Jµ1..µ4− c

2|g|1/2Fµ1..µ4 Fν1..ν4 Aρ1..ρ3ε

µ1..µ4ν1..ν4ρ1..ρ3.

(11.2.1)Here, F = dA, Fµ1..µ4 = ∂[µ1

Aµ2..µ4] is the curvature of the 3-index-photon and indices aremoved with the spacetime metric gµν . Furthermore, J is a totally skew tensor current bilinearin the graviton field not containing derivatives, whose explicit form does not need to concernus here, except that it does not depend on any other fields. Finally, c, α are positive numericalconstants whose value is fixed by the requirement of local supersymmetry [156]. The number ccould be called the level of the Chern-Simons theory in analogy to d = D + 1 = 3.

We proceed to the D + 1 split of this Lagrangian in a coordinate system with coordinatest, xa; a = 1, .., D adapted to a foliation of the spacetime manifold. The result of a tedious

95

calculation is given by

Fµ1..µ4 Fµ1..µ4 = 4Fta1..a3 F

ta1..a3 + Fa1..a4Fa1..a4 ,

Fta1..a3 Fta1..a3 = Ga1..a3,b1..b3 Fta1..a3 Ftb1..b3

−Ma1..a3,b1..b4 Fta1..a3 Fb1..b4 ,

Ga1..a3,b1..b3 = gttga1b1ga2b2ga3b3 − 3gta1gtb1ga2b2ga3b3 ,

Ma1..a3,b1..b4 = ga1b2ga2b2ga3b3gtb4 ,

Fa1..a4 Fa1..a4 = V1 − 4Ma1..a3,b1..b4Fta1..a3Fb1..b4 ,

V1 = ga1b1 ..ga4b4 Fa1..a4 Fb1..b4 ,

Fµ1..µ4 Fν1..ν4 Aρ1..ρ3εµ1..µ4ν1..ν4ρ1..ρ3 = 8εa1..a3b1..b4c1..c3Fta1..a3Fb1..b4Ac1..c3

+3εa1..a4b1..b4c1c2Fa1..a4Fb1..b4Atc1c2 ,

Jµ1..µ4Fµ1..µ4 = 4ja1..a3Fta1..a3 + V2,

V2 = Ja1..a4Fa1..a4 , (11.2.2)

where we used εa1..aD = εta1..aD and defined ja1..a3 := J ta1..a3 . The potential terms V1, V2 onlydepend on the spatial components of the curvature and do not contain time derivatives.

Using

Fta1..a3 =1

4[Aa1..a3 − 3∂[a1

Aa2a3]t], (11.2.3)

we may perform the Legendre transform. The momentum conjugate to A reads

πa1..a3 =∂L

∂Aa1..a3

= −|g|1/2[Ga1..a3,b1..b3Ftb1..b3 −Ma1..a3,b1..b4Fb1..b4 + αja1..a3 ]

−c εa1..a3b1..b4c1..c3Fb1..b4Ac1..c3 . (11.2.4)

We may solve (11.2.4) for Fta1a2a3

Fta1..a3 = −|g|1/2Ga1..a3,b1..b3 [πb1..b3 +Bb1..b3 + α|g|1/2jb1..b3 ],

Ba1..a3 = cεa1..a3b1..b4c1..c3Fb1..b4Ac1..c3 − |g|1/2 Ma1..a3,b1..b4Fb1..b4 , (11.2.5)

whereGa1..a3,c1..c3G

c1..c3,b1..b3 = δb1[a1δb2a2δb3a3] (11.2.6)

defines the inverse of G.Inverting (11.2.3) for A and using (11.2.4) and (11.2.2) we obtain for the Hamiltonian after alonger calculation

H =

∫d10x

Aa1..a3π

a1..a3 − L

= −∫

d10x

3Ata1a2 Ga1a2C + 2|g|−1/2Ga1..a3,b1..b3 [π +B + αj]a1..a3 [π +B + αj]b1..b3

+ |g|1/2[V1/2 + αV2]

,

Ga1a2C := ∂a3π

a1..a3 − c

2εa1a2b1..b4c1..c4Fb1..b4Fc1..c4 , (11.2.7)

where an integration by parts has been performed in order to isolate the Lagrange multiplierAta1a2 . Using the ADM frame metric components

gtt = −1/N2, gta = Na/N2, gab = qab −NaN b/N2;

gtt = −N2 + qabNaN b, gta = qabN

b, gab = qab, (11.2.8)

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with qab the induced metric on the spatial slices and lapse respectively shift functions N,Na

we can easily decompose the piece of H independent of the 3-index Gauß constraint Ga1a2C into

the contributions NaHCa + NHC to the spatial diffeomorphism constraint and Hamiltonianconstraint, however, we will not need this at this point.We will drop the subscript C in what follows, since in this chapter we are only interested in thep-form sector. We smear the Gauß constraint with a 2-form Λ, that is

G[Λ] :=

∫d10x Λab G

ab (11.2.9)

and study the gauge transformation behaviour of the canonical pair (Aabc, πabc) with non-

vanishing Poisson brackets

πa1..a3(x), Ab1..b3(y) = δ(10)(x, y) δa1

[b1δa2b2δa3

b3]. (11.2.10)

We find

G[Λ], Aa1..a3 = −∂[a1Λa2a3],

G[Λ], πa1..a3 = c εa1..a3b1..b3c1..c4∂[b1Λb2b3]Fc1..c4 . (11.2.11)

These equations can be written more compactly in differential form language, in terms of whichthey are easier to memorise. Introducing the dual 7-pseudo-form1

(∗π)a1..a7 :=1

3! 7!εb1..b3a1..a7π

b1..b3 (11.2.12)

we may write (11.2.11) as

δΛA = −dΛ, δΛ ∗ π = c (dΛ) ∧ F . (11.2.13)

Since the right hand side of (11.2.13) is closed, in fact exact, it would seem that the observables ofthe theory can be coordinatised by integrals of A and ∗π respectively over closed 3-submanifoldsor 7-submanifolds respectively.

The G(Λ) generate an Abelian ideal in the constraint algebra sinceG[Λ], G[Λ′]

= 0, G[Λ], H(x) = 0, (11.2.14)

where H(x) is the integrand of H in (11.2.7) and since the only π or A dependent contributionsto the Hamiltonian and spatial diffeomorphism constraints are contained in H(x).

We see that due to the non-vanishing Chern-Simons constant c, the transformation behaviourof ∗π differs from the transformation behaviour with respect to the higher-dimensional analogof the usual Maxwell type of Gauß law, which would be just the divergence term ∂a1π

a1..a3 . Inparticular, πabc itself is not gauge invariant. This “twisted” Gauß constraint (11.2.7) can bewritten in the form

Ga1a2 := ∂a3 [πa1..a3 − c

2εa1..a3b1..b3c1..c4Ab1..b3Fc1..c4 ] =: ∂a3π

′a1..a3 , (11.2.15)

which suggests to introduce a new momentum π′. Unfortunately, this does not work because∗(π′ − π) = A ∧ F does not have a generating functional K with δK/δA = A ∧ F , since theonly possible candidate K =

∫A ∧A ∧ F ≡ 0 identically vanishes in the dimensions considered

here. Since this is not the case, the Poisson brackets of π′ with itself do not vanish and neither

1Notice that πabc is a tensor density of type T 30 and density weight one.

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is π′ gauge invariant as we will see below, so that there is no advantage of working with π′ ascompared to π.

The presence of the twist term in the Gauß constraint leads to the following difficulty whentrying to quantise the theory on the usual LQG type kinematical Hilbert space:Such a Hilbert space would roughly be generated by a holonomy-flux algebra constructed fromholonomies

A(e) = exp(i

∫eA), π(S) =

∫S∗π, (11.2.16)

where e and S are oriented 3-dimensional and 7-dimensional submanifolds respectively, whichwe call “edges” and surfaces in what follows. One could then study the GNS Hilbert spacerepresentation generated by the LQG type of positive linear functional

ω(fπ(S1)..π(Sn)) = 0, ω(f) = µ[f ], (11.2.17)

where µ is an LQG type measure on a space of generalised connections A. One can define itabstractly by requiring that the charge network functions

Tγ,n =∏e∈γ

A(e)ne , ne ∈ Z (11.2.18)

form an orthonormal basis in the corresponding H = L2(A, µ), see [66] for details. Here, agraph γ is a collection of edges which are disjoint up to intersections in “vertices”, which areoriented 2-manifolds. The possible intersection structure of these cobordisms should be tamedby requiring that all submanifolds are semi-analytic.Up to here everything is in full analogy with LQG. The problem is now to isolate the Gaußinvariant subspace of the Hilbert space: While the connection transforms as in a theory withuntwisted Gauß constraint, it appears that we can solve it by requiring that charges add up tozero at vertices. However, this does not work because while such a vector is annihilated by thedivergence term in Gab, it is not by the second term ∝ A ∧ F . Even more disastrous, the termA∧F does not exist in this representation which is strongly discontinuous in the holonomies sothat operators A,F do not exist. Finally, although π is not Gauß invariant, it leaves this wouldbe gauge invariant subspace invariant, which reveals that this subspace is not the kernel of thetwisted Gauß constraint.

We therefore must be more sophisticated. Since the A dependent terms in G cannot bequantised on the kinematical Hilbert space, we must exponentiate it:Consider the Hamiltonian flow of G[Λ]

exp(G[Λ], ·)A = A− dΛ, exp(G[Λ], ·) ∗ π = ∗π + c(dΛ) ∧ F , (11.2.19)

which is a Poisson automorphism αΛ (canonical transformation) and one would like to securethat an implementation of the corresponding automorphism group αΛ αΛ′ = αΛ+Λ′ by unitaryoperators U(Λ) exists. The U(Λ) would correspond to the desired exponentiation of the Gaußconstraint. One way of securing this is by looking for an invariant state ω = ω αΛ on theholonomy - flux algebra (see [157]) for the details for this construction). This would thenopen the possibility that the Gauß constraint can be solved by group averaging methods. Thefirst problem is that the automorphisms do not preserve the holonomy-flux algebra becausethere appears an F on the right hand side of (11.2.19) which should appear exponentiated inorder that the algebra closes. This forces us to pass to exponentiated fluxes, that is, to thecorresponding Weyl algebra defined by exponentials of π,A. This algebra is now preserved bythe automorphisms, as one can see by an appeal to the Baker-Campbell-Hausdorff formula.However, we now see that the state (11.2.17) is not invariant, because

ω(eiπ(S)) = 1, ω(αΛ(eiπ(S))) = ω(ei[π(S)+c∫S dΛ∧F ]) = 0 (11.2.20)

98

for suitable choices of Λ. In the GNS Hilbert space we would like to have unitary operatorsU(Λ) such that for any element W in the Weyl algebra we have U(Λ)π(W )U(Λ)∗ = π(αΛ(W )).Then (11.2.20) is compatible with unitarity only if the LQG vacuum Ω is not invariant underU(Λ). Now the operator U(Λ) should correspond to exp(iG[Λ]) and using a calculation similarto (11.2.14) and the BCH formula one shows that on the LQG vacuum Ω = 1 it reduces formallyto U(Λ)Ω = exp(ic/2

∫Λ ∧ F ∧ F )Ω which is ill defined as it stands. We must therefore define

U(Λ)Ω to be some state in the GNS Hilbert space which has a component orthogonal to thevacuum and such that the representation property U(Λ)U(Λ′) = U(Λ + Λ′), U(Λ)∗ = U(−Λ)(possibly up to a projective twist) holds. We did not succeed to find a solution to this problemindicating that a unitary implementation of the Gauss constraint is impossible in the LQGrepresentation and even it were possible, the strategy outlined in the next section is certainlymore natural. We also remark that solving the constraint by group averaging methods becomesnon-trivial if not impossible in case of the non-existence of U(Λ). Even if we could somehowconstruct the Gauß invariant Hilbert space, the observables A(e), exp(iπ(S)) with ∂e = ∂S = ∅,which leave the physical Hilbert space invariant, are insufficient to approximate (for small e, S)the π dependent terms appearing in the Hamiltonian (11.2.7), as one can check explicitly.

11.3 Reduced phase space quantisation

In the previous section, we established that a quantisation in strict analogy to the procedurefollowed in LQG does not work. While a rigorous kinematical Hilbert space can be constructed,the Dirac operator constraint method of looking for the kernel of the Gauß constraint is prob-lematic. As an alternative, a reduced phase space quantisation suggests itself. This has a chanceto work due to the observation (11.2.14) which demonstrates that H(x) only depends on observ-ables. Indeed, H(x) depends, except for Gab which is a trivial observable since it is constrainedto vanish, only on the combination π + B + αj. Obviously j trivially Poisson commutes withG. Unpacking B from (11.2.5), we see that π + B is a linear combination (with only metricdependent coefficients) of F and

P abc := πabc + cεabcd1..d4e1..e3Fd1..d4Ae1..e3 ⇔ ∗P = ∗π + c A ∧ F , (11.3.1)

which suggests that G(Λ), P abc(x) = 0 because F is already invariant. This indeed can beverified using (11.2.13)

δΛ ∗ P = δΛ ∗ π + cδΛA ∧ F = 0. (11.3.2)

Our classical observables therefore are coordinatised by the 4-form and 7-form F = dA and∗P = ∗π + cA ∧ F respectively. Since F is exact, it is determined entirely by a 3-form moduloan exact form, which in turn is parametrised by a 2-form. This 2-form worth of gauge freedommatches the number of Gauß constraints which can be read as a condition on π. Thus, on theconstraint surface, the number of degrees of freedom contained in F and P match.

We compute the observable algebra. Let f be a 3-form and h a 6-form with dual ∗h (atotally skew 4-times contravariant tensor pseudo density) and smear the observables with these

P [f ] :=

∫d10x fa1..a3 P

a1..a3 =

∫f ∧ ∗P, F [h] :=

∫d10x (∗h)a1..a4Fa1..a4 =

∫h ∧ F .

(11.3.3)Then, we find after a short computation

F [h], F [h′] = 0, P [f ], F [h] =

∫h ∧ df, P [f ], P [f ′] = −3c F [f ∧ f ′]. (11.3.4)

Thus, the observable algebra closes but P is not conjugate to F .

99

The form of the observable algebra (11.3.4) reveals the following:Typically, background independent representations tend to be discontinuous in at least one ofthe configuration or the momentum variable. For instance, in LQG electric fluxes exist in non-exponentiated form, but connections do not. Let us assume that we find such a representationin which F [h] does not exist so that we have to consider instead its exponential (Weyl element).Then (11.3.4) tells us that in such a representation automatically also P [f ] cannot be defined,because if it could, then its commutator would exist, which however is proportional to some Fwhich is a contradiction. Hence, either both F, P exist or only both of their corresponding Weylelements.

We did not manage to find a representation in which the Weyl elements

W [h, f ] := exp (i(F [h] + P [f ])) (11.3.5)

are strongly continuous operators in both f, h. However, we did find one in which they arediscontinuous in both h, f . This representation was studied in the context of QED in [49] andwas applied to an LQG type of quantisation of the closed bosonic string in [105]. Before wedefine it, we must first define the Weyl algebra generated by the Weyl elements (11.3.5). The∗-relations are obvious,

W [h, f ]∗ = W [−h,−f ]. (11.3.6)

However, the product relations are very interesting and non-trivial, because they require thegeneralisation of the Baker-Campbell-Hausdorff formula [158, 159, 160, 161, 162, 163] to highercommutators [164]. Suppose that X,Y are operators on some Hilbert space such that the triplecommutators [X, [X,Y ]] and [Y, [Y,X]] commute with both X and Y . This formally applies toour case with X = F [h] + P [f ], Y = F [h′] + P [f ′], which obey the canonical commutationrelations (we set ~ = 1 for simplicity)

[X,Y ] := iX,Y = i

[

∫(h′ ∧ df − h ∧ df ′)] 1− 3cF [f ∧ f ′]

. (11.3.7)

From this follows for the triple commutators

[X, [X,Y ]] = −3c(i)2 P [f ], F [f ∧ f ′] = 3c

∫f ∧ f ′ ∧ df 1,

[Y, [Y,X]] = 3c(i)2 P [f ′], F [f ∧ f ′] = −3c

∫f ∧ f ′ ∧ df ′ 1, (11.3.8)

which thus are in the centre of the algebra.The BCH formula for the case of all triple commutators commuting with X,Y reads

eX eY = eX+Y+ 12

[X,Y ]+ 112

([X,[X,Y ]]+[Y,[Y,X]]), (11.3.9)

which can also be proved using elementary methods. From this it is easy to derive the alsouseful Zassenhaus formula [164]

eX+Y = eX eY e−12

[X,Y ] e−16

([X,[X,Y ]]+2[Y,[X,Y ]]). (11.3.10)

Putting all these together, we obtain the Weyl relations

W [h, f ] W [h′, f ′] = W [h+ h′ +3c

2f ∧ f ′, f + f ′]

× exp

(i

4

∫ [2(h ∧ df ′ − h′ ∧ df)− cf ∧ f ′ ∧ d(f − f ′)

]). (11.3.11)

100

Hence also the Weyl relations get twisted as compared to the situation with c = 0. Notice thatthe first term in the phase is antisymmetric under the exchange (h, f) ↔ (h′, f ′), while thesecond is symmetric.

In order to obtain a representation of this ∗-algebra A generated by the Weyl elements, itis sufficient to find a positive linear functional. We consider the Narnhofer-Thirring type offunctional

ω(W (h, f)) =

1 h = f = 00 else

(11.3.12)

and show that it is positive definite on A. Let

a :=N∑k=1

ck W [zk] (11.3.13)

be a general element in A, where N ∈ N, ck ∈ C and the zk = (hk, fk) are arbitrary, wherewithout loss of generality zk 6= zl for k 6= l. We have

ω(a∗a) =

N∑k,l=1

ck cl ω(W [−zk] W [zl])

=N∑

k,l=1

ck cl ω(W [zkl]) exp(iαkl),

zkl = (−hk + hl −3

2cfk ∧ fl, −fk + fl),

αkl =1

4

∫[2(−hk ∧ dfl + hl ∧ dfk)− cfk ∧ fl ∧ d(fk + fl)]. (11.3.14)

For k = l, we have zkl = αkl = 0 because fk, fl are 3-forms. For k 6= l, we must have eitherfk 6= fl or hk 6= hl or both. If fk 6= fl, then obviously zkl 6= 0. If fk = fl, then necessarilyhk 6= hl and zkl = (−hk + hl, 0) 6= 0. By definition (11.3.12) then

ω(a∗a) =N∑k=1

|ck|2 ≥ 0; ω(a∗a) = 0 ⇔ a = 0 (11.3.15)

is positive definite. Thus, the left ideal I = a ∈ A; ω(a∗a) = 0 = 0 is trivial and theHilbert space representation is given by the GNS data [157]:The cyclic vector is Ω = 1, the Hilbert space H is the Cauchy completion of A in the scalarproduct < a, b >:= ω(a∗b) and the representation is simply π(a)b := ab on the common densedomain D = A.The representation is evidently strongly discontinuous in both h, f and while cyclic, it is notirreducible. Equivalently, ω is not a pure state [165, 166].

The question left open to answer is whether the algebra and the state ω are still well definedwhen restricting the smearing functions (h, f) to the form factors of 4-surfaces and 7-surfacesrespectively. The bearing of this question is that in the Hamiltonian constraint the functionsF and ∗P appear in such a way, that in a discretisation of it, which results from replacingthe integral by Riemann sums in the spirit of [13], these functions are naturally smeared over4-surfaces and 7-surfaces respectively. They could thus be approximated by Weyl elements.

101

To answer this question, let S4, S7 be general 4 and 7 surfaces respectively. Consider the distri-butional forms (“form factors”)

hS4a1..a6

(x) :=

∫S4

εa1..a6b1..b4dyb1 ∧ dyb4 δ(x, y),

fS7a1..a3

(x) :=

∫S7

εa1..a3b1..b7dyb1 ∧ dyb7 δ(x, y). (11.3.16)

Then

F [hS4 ] =

∫S4

F, P [fS7 ] =

∫S7

∗P . (11.3.17)

Thus, the natural integrals of F, P over surfaces can be reexpressed in terms of distributional6 forms and 4-forms respectively. It remains to check whether the exterior derivative and productcombinations of these distributional forms appearing in the multiple Poisson brackets of (11.3.17)and in the Weyl relations remain meaningful. Three types of exterior derivative and productexpressions appear. The first is, using formally Stokes theorem∫

hS4 ∧ dfS4 =

∫S4

dfS7 =

∫∂S4

fS7

=

∫∂S4

dxa1 ∧ .. ∧ dxa3εa1..a3b1..b7

∫S7

dyb1 ∧ .. ∧ dyb7 δ(x, y) =: σ(∂S4, S7). (11.3.18)

The integral is supported on ∂S4 ∩ S7 and we can decompose this set into components (sub-manifolds) which are 0,1,2,3-dimensional. The number of these components will be finite if thesurfaces are semianalytic. We define the intersection number σ(∂S4, S7) to be zero for the 1,2,3-dimensional components and by (11.3.18) for the isolated intersection points, which then takesthe values ±1. This can be justified by the same regularisation as in LQG for the holonomy-fluxalgebra [66].

The second type of integral is given by F [fS7 ∧ fS′7 ]. The support of the integral will beon SS7 ∩ SS′7 and in D = 10 dimensions this will decompose into components that are at least4-dimensional. By the same regularisation as in [66], one can remove the higher-dimensionalcomponents and thus keep only the 4-dimensional ones. In what follows, we thus assume thatS4 := S7 ∩ S′7 is a single 4-dimensional component, otherwise the non vanishing contributionsare over a sum of those. We have

F [fS7 ∧ fS′7 ] =

∫S7

F ∧ fS′7 (11.3.19)

=

∫S7

dxa1 ∧ .. ∧ dxa4 ∧ dxb1 ∧ .. ∧ dxb3εb1..b3c1..c7∫S′7

dyc1 ∧ .. ∧ dyc7 δ(x, y) Fa1..a4(x).

By assumption, we have embeddings

XS7 : U → S7; YS′7 : V → S′7; ZS4 : W → S4, (11.3.20)

with open subsets U, V of R7 and an open subset W of R4 respectively, whose coordinates will bedenoted by u, v, w respectively. The conditionXS7(u) = YS′7(v) = ZS4(w) is solved by solving u, vfor w, which leads to u = u(w), v = v(w). Since the integrals are reparametrisation invariant,in the neighbourhood of S4 on both S7 and S′7 therefore we may use adapted coordinates so thatwI = uI = vI , I = 1, .., 4 on S4 and uI , vI , I = 5, .., 7 denote the transversal coordinates, whichtake the value 0 on S4. In this parametrisation both U, V are of the form U = W × U ′, V =W × V ′ for some 3-dimensional subsets U ′, V ′ of R3. It follows Z(w) = X(w, 0) = Y (w, 0)

102

in this parametrisation. The δ distribution is then supported on uI = vI , I = 1, .., 4 anduI = vI = 0, I = 5, .., 7 and we have in the neighbourhood of S4

Xa(u)− Y a(v) = −4∑I=1

Y aI (u, 0)

[uI − vI

]+

7∑I=5

[XaI (u, 0)uI − Y a

I (u, 0)vI]

. (11.3.21)

We can now solve the δ distribution in (11.3.19) by performing the integral over u5, .., u7, v1, .., v7

and find with the notation XaI = ∂Xa

S7(u)/∂uI and Y a

I = ∂Y aS′7

(v)/∂vI etc.

F [fS7 ∧ fS′7 ] =

∫Ud7u εI1..I7

[Xa1I1..Xa4

I4Xb1I5..Xb3

I7

](u)εb1..b10

∫Vd7v εJ1..J7

[Y b4J1..Y b10

J7

](v) ×

δ (X(u), Y (v)) Fa1..a4(X(u))

= −∫W

d4w εI1..I7[Za1I1..Za4

I4

](w) εJ1..J7 Fa1..a4(Z(w)) εI5..I7J1..J7 ×[

sgn

(det

(∂(X(u)− Y (v))

∂(u5, .., u7, v1, .., v7)

)vI=uI=wI ;I=1,..,4;vI=uI=0;I=5,..,7

)]

=:− 3! 7!σ(S7, S′7)F [hS4 ], (11.3.22)

where the 10d antisymmetric symbol is in terms of the coordinates u5, .., u7, v1, .., v7 and in thelast step we noticed that the range of I1..I4 is restricted to 1..4. Also, we assumed that thesign function under the integral is constant and equal to σ(S7, S

′7) on S4 (which defines this

function), otherwise we must decompose S4 further. Under this assumption, we conclude theform factor identity

fS7 ∧ fS′7 = −3! 7! σ(S7, S′7)hS7∩S′7 . (11.3.23)

Finally, we consider the integral of the third type, which now combining (11.3.18) and(11.3.24) is easily calculated∫

fS7 ∧ fS′7 ∧ dfS7 = −3! 7! σ(S7, S′7)

∫hS7∩S′7 ∧ dfS7 = −3! 7! σ(S7, S

′7)σ(∂(S7 ∩ S′7), S7) = 0,

(11.3.24)because ∂(S7 ∩ S′7) ⊂ S7 for which σ vanishes by definition.

In order to make this restricted Weyl algebra close, we now have to decide whether theform factors should only be added with integer valued coefficients [20] or with real valued ones[167, 168, 169]. In the latter case we do not need to do anything and the restricted Weyl algebraalready closes. In the former case we must replace the form factors fS7 by 1√

3! 7! 3c/2fS7 , such

that in the simplest situation we have

W [S4, S7] W [S′4, S′7] = W [S4+S′4−σ(S7, S

′7)S7∩S′7, S7+S′7] exp

(i

2

[σ(∂S4, S

′7)− σ(∂S′4, S7)

]),

(11.3.25)from which the general case can be easily deduced.

We conclude that the restricted Weyl algebra is well defined in either case. Thus, whereverP or F appear in the Hamiltonian constraint, we follow the general regularisation procedureoutlined in [13], which employs a combination of spatial diffeomorphism invariance and an infi-nite refinement limit of a Riemann sum approximation of the Hamiltonian constraint in terms ofP [S7] and F [S4] = A[∂S4], which we approximate for instance by sin(P [S7]), sin(F [S4]) similaras in LQG. The details are obvious and are left to the interested reader.

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11.4 Conclusions

Supergravity theories typically need additional bosonic fields next to the graviton, in order toobtain a SUSY multiplet (representation) containing the gravitino. In this chapter, we focussedon 11d, N = 1 SUGRA for reasons of concreteness (and its relevance for lower dimensionalSUGRA theories), which contains the 3-index photon in the bosonic sector. However, ouranalysis is easily generalised to arbitrary p-form fields. Without the Chern-Simons term inthe action (i.e. c = 0) the analysis would be straightforward and in complete analogy tothe background independent treatment of Maxwell theory in D + 1 = 4 dimensions [20]. Inparticular, the Hamiltonian constraint would be quadratic in the 3-form field and its conjugatemomentum, which thus would reduce to a free field theory when switching off gravity. However,with the Chern-Simons term (c 6= 0) the Hamiltonian constraint becomes in fact quartic inthe connection and thus becomes self-interacting even when switching off gravity, just like innon-Abelian Yang-Mills theories.

It is therefore the more astonishing that we can quantise the resulting ∗-algebra of ob-servables (with respect to the 3-index-Gauß constraint) rigorously, even though the theory isself-interacting. In fact, in terms of the observables, the Hamiltonian constraint is a quadraticpolynomial, however, the price to pay is that the observable algebra is non-standard. Yet, theresulting Weyl algebra can be computed in closed form and we found at least one non-trivial andbackground independent representation thereof, which nicely fits into the background indepen-dent quantisation of the gravitational degrees of freedom in the contribution to the Hamiltonianconstraint depending on the 3-index-photon.

There are many open questions arising from the present study. One of them concerns thereducibility of the GNS representation found, which involves a mixed state. It would be nice tohave control over the superselection sectors of the theory and, in particular, to analyse whetherthe cyclic GNS vector is not already cyclic for the Abelian subalgebra generated by the W [h, 0].Next, it is worthwhile to study the question whether this algebra admits regular representationsfor both P and F , because then the GNS Hilbert space would admit a measure theoretic inter-pretation as an L2 space. Finally, it is certainly necessary to work out the cobordism theoryof relevance when restricting the Weyl algebra to distributional 4-form and 7-form factors assmearing functions which is only sketched in this chapter. We plan to revisit these questions infuture publications.

104

Part IV

Initial value quantisation ofhigher-dimensional general relativity

105

Chapter 12

Loop quantum gravity without theHamiltonian constraint

In this chapter, we show that under certain technical assumptions, including the existence ofa CMC slice and strict positivity of the scalar field, general relativity conformally coupled toa scalar field can be quantised on a partially reduced phase space, meaning reduced only withrespect to the Hamiltonian constraint and a proper gauge fixing. More precisely, we introduce,in close analogy to shape dynamics, the generator of a local conformal transformation actingon both, the metric and the scalar field, which coincides with the CMC gauge condition. Anew metric, which is invariant under this transformation, is constructed and used to defineconnection variables which can be quantised by standard loop quantum gravity methods. Sincethis connection is invariant under the local conformal transformation, the generator of which isshown to be a good gauge fixing for the Hamiltonian constraint, the Dirac bracket associatedwith implementing these constraints coincides with the Poisson bracket for the connection. Thus,the well developed kinematical quantisation techniques for loop quantum gravity are available,while the Hamiltonian constraint has been solved (more precisely, gauge fixed) classically. Thephysical interpretation of this system is that of general relativity on a fixed spatial CMC slice,the associated “time” of which is given by the constant mean curvature. While it is hard toaddress dynamical problems in this framework (due to the complicated “time” function), itseems, due to good accessibility properties of the CMC gauge, to be well suited for problemssuch as the computation of black hole entropy, where actual physical states can be counted andthe dynamics is only of indirect importance. Also, the interpretation of the geometric operatorsgets an interesting twist, which exemplifies the deep relationship between observables and thechoice of a time function. The original work from which this chapter is taken is [43]. Parts ofthe ideas are based on [44].

12.1 Introduction

In the second part of this thesis, we have shown that the quantisation techniques developedfor loop quantum gravity formulated in terms of the Ashtekar-Barbero variables in 3 + 1 di-mensions can be generalised to higher dimensions. Despite this success in writing down a welldefined quantum theory including the closure of the constraint algebra in a suitable sense [12],the solution space of the Hamiltonian constraint and its interpretation remain not very wellunderstood. Even more so, we do not have access to a set of Dirac observables commutingwith the Hamiltonian constraint operator (except for the ADM-charges). The master constraintapproach [75] was introduced to improve on this situation by providing an explicit method for

106

constructing the physical Hilbert space including a scalar product thereon. Nevertheless, theissue of quantum observables remains open also in this approach. This is even more true in thecase of higher dimensions considered in this thesis, since the Hamiltonian constraint operator iseven more complicated than when using Ashtekar-Barbero variables and the issue of anomaliesarises when considering the quantum simplicity constraints.

A possible way around these problems is to solve the Hamiltonian constraint and possi-bly the spatial diffeomorphism constraints classically and to quantise the resulting algebra ofobservables. However, a reduced phase space quantisation of a given theory is generally veryproblematic due to the complexity of the representation problem resulting from a non-trivialobservable algebra, as is e.g. the case for pure general relativity. In order to make progress withthe above mentioned issues, relational models were introduced in LQG, pioneered by Gieseland Thiemann [22] using the Brown-Kuchar dust model [170]. Due to the additional matterfields which are used as clocks and rods, it is possible to construct an observable algebra withthe same algebraic structure as the holonomy-flux algebra, so that the methods developed forthe constrained quantisation can be directly applied also to the reduced phase space. Furthermodels based on different choices of matter fields [22, 23, 24, 25] have been given.

Although these models are very promising for calculating reliable predictions from the theory,a possible issue which might be raised is that of all these models use matter fields as clock (androd) variables. Consequently, this choice of clock is only valid as long as the clock matter fieldsremain classical in the experiments which we are describing. This situation might hold if theclock matter fields couple sufficiently weak, or not at all, to e.g. other particles in a scatteringexperiment. Nevertheless, it would be desirable to have a clock which is purely geometrical andthus could serve as a good clock in particle scattering experiments.

On the other hand, applications to scattering theory would require us to derive a true Hamil-tonian corresponding to the chosen geometrical clock, which could be very difficult depending onthe particular choice of clock. While such true Hamiltonians have been derived in the availablereduced phase space quantisations, there are situations, e.g. state counting in the derivation ofthe black hole entropy, where the quantum dynamics are not relevant, but only access to thephysical Hilbert space is needed. For this, one would need a gauge fixing D (a time function)of the Hamiltonian constraint H, i.e. H,D = invertible, which is accessible (at least for thespecific situations under consideration) and leads to a manageable Dirac bracket. The inter-pretation of such a formulation would be to consider general relativity on a fixed spatial slicedefined by the gauge fixing condition. By restricting to such a quantisation on a fixed spatialslice, we neglect the problem of quantum dynamics, but nevertheless have a quantisation ofthe observable algebra (up to the problem of implementing the spatial diffeomorphisms). Suchobservables include e.g. the geometric operators evaluated on this fixed spatial slice. However,since the clock is of geometric origin, we expect the (algebra of) geometric operators to bechanged, since the Dirac bracket associated to implementing the gauge condition D = 0 will benon-trivial. A resulting change of spectrum of the geometric operators in loop quantum gravityat the level of the physical Hilbert space has already been conjectured by Dittrich and Thiemann[171], however an explicit example has been missing since the geometric operators in the otheravailable reduced phase space quantisations do not change spectrum due to the matter fieldnature of the clock variables.

In this chapter, we show how such a reduced phase space quantisation can be constructed forgeneral relativity conformally coupled to a scalar field by using ideas from shape dynamics [50].First, we show that the generator of a local conformal symmetry (i.e., in what follows, a localrescaling of the canonical variables) is a good gauge fixing for the Hamiltonian constraint. Inter-estingly, the generator coincides with the constant mean curvature (CMC) gauge condition [172],thus being a purely geometric clock. A new metric, which is invariant under the local conformal

107

transformation, is constructed as a compound object of the original metric and the scalar field.Due to this invariance, the Dirac bracket with respect to implementing both the Hamiltonianconstraint and the generator of the conformal symmetry coincides with the Poisson bracket.Passing to Ashtekar-Barbero type connection variables, the Ashtekar-Isham-Lewandowski rep-resentation of loop quantum gravity can be employed. At the quantum level, we are left with theGauß and spatial diffeomorphism constraints. The spatial diffeomorphism constraint poses thesame difficulties as in standard LQG, e.g. the construction of spatially diffeomorphism invariantoperators and the associated question of graph preservation need further research.

As an application, we show explicitly that there exists a family of black holes which canbe treated by the proposed method, thus allowing black hole state counting at the level of thephysical Hilbert space. Although access to the physical Hilbert space is also given in the models[22, 23, 24, 25], it is unclear if their associated time functions can be good gauge fixings for theHamiltonian constraint for the type of static black hole solutions under consideration in thischapter. E.g. since H,D would be linear in the momenta for scalar field or dust clocks D, wehave H,D = 0 in static situations at least for the type of foliations (vanishing momentum ofthe scalar field, see the section on black holes) considered in this chapter. Using other foliations,this objection would not hold, however, the CMC foliations considered here seem to be themost natural ones for static situations. Of course, in other situations, the time functions of[22, 23, 24, 25] might be better suited than the one presented here.

12.2 Classical analysis

The presentation of the calculations in this chapter is concise, see [44] for more details. Theaction of the scalar field conformally coupled to general relativity is given by

S =1

κ

∫MdD+1X

√gR(D+1)a(Φ)

+1

2λ

∫MdD+1X

√ggµν(∇µΦ)(∇νΦ), (12.2.1)

where we defined

a(Φ) := 1− αΦ2, ∆Φ :=1−D

4∆g, α := −κ(D − 1)

8λD.

The scalar field part of the action, i.e. S − SEinstein-Hilbert, is invariant under the conformaltransformation

gµν → Ω∆ggµν , Φ→ Ω∆Φ

Φ. (12.2.2)

The D + 1 split of this action gives

S =

∫Rdt

∫σdDx

[P abqab + πΦΦ−NaHa −NH

], (12.2.3)

108

where we have defined

πΦ := − 1

λ

√q(LnΦ) +

4α

κ

√qΦK,

P ab :=1

κa(Φ)

√q(Kab − qabK

)+

2α

κ

√qqabΦ(LnΦ),

P abtf := P ab − 1

DqabP cdqcd,

Ha[Na] :=

∫σdDx

[P ab(LNq)ab + πΦ(LNΦ)

], (12.2.4)

H[N ] :=

∫σdDxN

[HGrav +HΦ −

κ D2

∆g2D(D − 1)√q

],

κHGrav :=κ2

√qa(Φ)

P tfabP

abtf −

√qR(D),

κHΦ :=κ

2λ

√q

[−λ

2

qπ2

Φ −1

Dqab(DaΦ)(DbΦ)

+D − 1

DΦDaD

aΦ +1

D

∆Φ

∆gR(D)Φ2

],

D := ∆gP + ∆ΦπΦΦ =∆g(1−D)

√q

κK. (12.2.5)

n denotes the normal vector on the spatial slices, L the Lie derivative, Kab the extrinsic curvature,Da the covariant derivative compatible with the spatial metric qab, P = P abqab, and K = Kabqab.It is easy to see that D is the generator of local conformal transformations.

The underlying idea of what follows originates in the work of Lichnerowicz [173] and York[174]: Good initial data (satisfying H = 0 = Ha) for general relativity can be constructedfrom specific initial data (a spatial metric, a transversal trace free second rank tensor fieldand a constant value for the mean curvature) by performing a conformal rescaling of the fieldswith a scaling factor satisfying the Lichnerowicz-York equation. On the other hand, if, morallyspeaking, only conformal equivalence classes of initial data would be specified, one could performa conformal transformation to initial data satisfying the Hamiltonian constraint without leavingthe equivalence class, i.e. without changing the initial data. It therefore transpires that oneshould try to exchange the equationH = 0 for invariance under a local conformal rescaling. Partsof this idea have been implemented in shape dynamics [50], however, it was not possible so farto find a general solution to the conformal invariance condition which could also be quantisedin a satisfactory way. In this chapter, we take this last step by realising that a conformallycoupled scalar field, as opposed to obstructions arising from other matter fields [175], allows fora non-trivial conformal weight in the generator of the local conformal transformation, and thusfor the construction of a conformally invariant metric. We remark that an earlier account ofintroducing a conformal symmetry in canonical quantum gravity has been given in [176, 177],however, to the best of our knowledge, the kernel of the quantised conformal constraint, whichis a part of the constraint algebra, has not been studied so far.

The main result of this section is that the CMC gauge D = 0 is a good gauge fixing for theHamiltonian constraint at least locally and restricting to spatial slices which allow for the D = 0gauge. While we restrict to zero constant mean curvature in this chapter, the general case is

109

developed in [44]. More precisely, we calculate

κH[N ],D[ρ] = H[. . .] +D[. . .]

+

∫σdDx (D − 1)

√qρ∆g

[DaD

a − κ2

2qa(Φ)2P tfabP

abtf −

1

2R(D)

]N (12.2.6)

and conclude that D = 0 is locally a good gauge fixing if the elliptic partial differential operator

DaDa − κ2

2qa(Φ)2P tfabP

abtf −

1

2R(D) (12.2.7)

is invertible. By a standard argument from the theory of partial differential equations [178] andusing the assumption of a compact spatial slice without boundary (boundaries and non-compactspatial slices are treated in [44]), it is sufficient to show that

κ2

2qa(Φ)2P tfabP

abtf +

1

2R(D) > 0. (12.2.8)

Demanding the dominant energy condition (−Tµνζµ is a future causal vector for all futuretimelike vectors ζ) and using the field equations as well as the vanishing of the constraints, itfollows that

1

2R(D) >

1

2[KabK

ab −K2] ≈ κ2

2qa(Φ)2P tfabP

abtf , (12.2.9)

and thus

κ2

2q a(Φ)2P tfabP

abtf +

1

2R(D) >

κ2

qa(Φ)2P tfabP

abtf ≥ 0. (12.2.10)

We remark that the dominant energy condition does not generally hold for the conformallycoupled scalar field [179] and we have to restrict to spacetimes where it is satisfied, e.g. theMST black hole discussed later on.

The next step is to construct a new metric invariant under local conformal transformations,which is achieved by the canonical transformation

qab := e4

D−1φqab, P ab := e−

4D−1

φP ab,

φ := ln Φ, πφ :=1

∆ΦD. (12.2.11)

Indeed, the new Poisson brackets read

qab, P cd = δc(aδdb), φ, πφ = 1. (12.2.12)

Here, we restricted ourselves to Φ > 0, which can be interpreted as a dilaton-type field Φ = exp φ.This restriction is necessary in order not to divide by zero and an according restriction on thespacetimes which we want to describe follows.

Next, we pass to the Dirac bracket ·, ·DB associated with implementing H = D = 0simultaneously, which solves these constraints classically. We note that qab and P ab are enoughfunctions to parametrise the reduced phase space, and since qab,D = P ab,D = 0, thenon-vanishing Dirac brackets among them are

qab, P cdDB = qab, P cd = δc(aδdb). (12.2.13)

The remaining constraint algebra simply reads

Ha[Na],Hb[M b]DB = Ha[(LNM)a]. (12.2.14)

Up to the missing Hamiltonian constraint and the unphysical remaining scalar field, this systemis identical to the ADM formulation [46] of general relativity and we can thus use standardtechniques from loop quantum gravity in order to quantise it.

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12.3 Quantisation

From the above ADM-type phase space, we can perform a canonical transformation to realconnection variables as in [7, 11], or in all dimensions D ≥ 2 along the lines of [35]. A math-ematically rigorous quantisation of this classical system can be accomplished by loop quantumgravity methods [149, 66] and the uniqueness result on the representation [180] when demandinga unitary representation of the spatial diffeomorphisms remains valid since the spatial diffeomor-phism constraint still has to be quantised. The difference to loop quantum gravity is, however,that the Hamiltonian constraint has been solved already classically and the usual complicationsassociated with its quantisation do not arise.

The Gauß and spatial diffeomorphism constraint can be solved by standard methods [66].As for spatially diffeomorphism invariant operators, in our case physical observables, we havenothing new to add to the usual treatment, see [66] for an exposition. Further research for abetter understanding of these operators, especially graph-changing ones, is nevertheless needed.

12.4 Geometric operators

The geometric operators of loop quantum gravity, such as the area and volume operators, canbe constructed in the usual manner from the invariant connection. However, their interpretationnow changes since their spectrum has to be related with the geometry based on the non-invariantmetric. It follows that, morally speaking,

Ainv = Φ2ALQG, V inv = Φ2D/(D−1)V LQG, (12.4.1)

where the usual LQG operators measure the actual geometry while the invariant operators havethe familiar discrete spectrum. A similar, although conceptually different, observation has beenmade by Ashtekar and Corichi in [181]. We remark that the possible occurrence of such aphenomenon has been emphasised by Dittrich and Thiemann [171]: The geometric operators ofLQG might change their spectrum when taking into account the Hamiltonian constraint. Thishas to be seen in contrast to the result of [22, 23], where the spectra remain unchanged. Thechange in spectrum has to be attributed to the different choice of equal time hypersurfaces,i.e. D = 0 in our case and, e.g. Φ − const = 0 in [23], and the different resulting invariantgeometric operators, which have to Poisson commute with the time function at the classicallevel. Further discussion on this issue is given in [44]. It is interesting to note that when usinga Higgs-type potential for Φ which leads to a non-vanishing vacuum expectation value 〈Φ〉, onecould approximate the invariant geometric operators by the LQG geometric operators times aconstant which changes the fundamental geometric scale by a factor of 1/ 〈Φ〉 in Planck units.While this might present a mechanism to increase the fundamental scale of LQG and makeit thus more accessible to experiments, we caution the reader that such an interpretation isstrongly tied to the type of foliation we are using and that the associated dynamics have tobe investigated to check for consistency with current experiments, thus making further researchnecessary before jumping to conclusions. Also, the proposed quantisation of Φ would be verydifferent than in the standard model, since Φ would be quantised as a part of the invariantmetric instead of a usual scalar field.

Of course, at this point, these invariant operators still do not commute with the spatial diffeo-morphism constraint, which could for example be achieved by tying their domain of integrationto physical values of other matter fields. We leave this issue for further research.

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12.5 Application to black hole entropy

One of the major open problems in the calculation of the black hole entropy in the loop quantumgravity framework is the treatment of the Hamiltonian constraint. While the constraint vanisheson the black hole horizon [182] and therefore does not have to be taken into account there, itstill acts on the bulk. In the entropy calculations, it is assumed that every horizon state has atleast one extension into the bulk which is annihilated by the Hamiltonian constraint, a proof,however, has not been given so far. On the other hand, using the techniques developed in thischapter, the problem of implementing the Hamiltonian constraint in the bulk does not evenarise, since it is solved classically. We briefly sketch some aspects of the black hole entropycalculation in our framework, details will be reported elsewhere.First, we remark that several black hole solutions for general relativity conformally coupled to ascalar field exist, which avoid the no-hair theorems in 3 + 1 dimensions and allow for non-trivialhorizon topologies, see [183] for an overview. In order to treat them in our framework, we firsthave to check if the gauge D = 0 is accessible. For simplicity, let us restrict to static, i.e. themetric and the scalar field do not depend on the time coordinate t, 3 + 1 dimensional solutions.Choosing the t = const. hypersurfaces as the leaves of our foliation, accessibility directly followssince all the momenta, and thus D, vanish in this case. Next, we have to check if the gauge iswell behaved, i.e. (12.2.7) has trivial kernel (an extension to non-compact spatial slices is treatedin [44], where we also discuss global aspects). In the case of vanishing cosmological constant Λ,the scalar field is diverging at the horizon and we neglect this case. For Λ > 0, it was shownin [44] that D can be supplemented with an additional term to imply that (12.2.7) has trivialkernel. However, this additional term would spoil the accessibility of the gauge for the t = const.foliation. On the other hand, for Λ < 0, D may remain unaltered and triviality of the kernelstill follows. The corresponding black hole solution has been found by Martınez, Staforelli andTroncoso, it describes an asymptotically locally AdS black hole and admits non-trivial horizontopologies of the form H2/Γ, where Γ is a freely acting discrete subgroup of O(2, 1) [184].Due to the non-trivial horizon topology, it will be very interesting to study the thermodynamicsof this class of black holes. Building on the results of [185, 181], the entropy can be calculatedby counting the horizon states which are in agreement with the macroscopic properties of theblack hole prescribed by the invariant area operator instead of the usual LQG area operator.One might object that the gauge is not fixed completely in the above static spacetime, becauseD = 0 can select any t = const hypersurface. However, the transformation between differentt = const hypersurfaces is not a gauge transformation but an asymptotic symmetry and thusnot a constraint which we have to fix, see also [44]. This can be seen by the fact that thecorresponding lapse function would not vanish sufficiently fast at asymptotic infinity, thus leadingto boundary terms which spoil the invertibility of (12.2.7).

12.6 Concluding remarks

In conclusion, we have constructed a quantisation of the phase space of general relativity con-formally coupled to a scalar field which has been reduced with respect to the Hamiltonianconstraint. While a very important question, the quantum dynamics do not seem to be easilyaddressable in this framework. It is nevertheless possible to ask questions which only dependon the physical Hilbert space and operator spectra. The examples which have been explicitlyprovided are on the one hand the calculation of black hole entropy, where physical states canbe counted as opposed to kinematical states in the standard treatment, and the calculation ofthe spectra of geometric operators, whose spectrum is multiplied by a power of the scalar field.While the result for the state counting does not come as a real surprise, the change in spectrum

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of the geometric operators has been conjectured by Dittrich and Thiemann in [171], but it hasnot been explicitly shown to be feature of loop quantum gravity before. This result raises thequestion if the fundamental geometric scale of loop quantum gravity could be significantly raisedby choosing specific time functions, i.e. performing experiments which measure the geometricproperties of areas and volumes which are suitably embedded into spacetime. The example ofa black hole to which the gauge fixing discussed in this chapter can be applied should havegeneralisations in higher dimensions. In the next chapter, we are going to extend the connectionformulation derived in the first part of this thesis to the presence of higher-dimensional isolatedhorizons which will then enable us to look at entropy calculations also in higher dimensions.

We close with some final remarks.

• We underline that the original idea of trading the Hamiltonian constraint for a local con-formal invariance originated in shape dynamics [50]. The main new input in our formalismis that a conformally coupled scalar field allows for a non-trivial conformal scaling and thusfor the construction of an invariant metric from which quantisation can start. Also, we donot have a global Hamiltonian as in [50] since we are not restricting to volume preservingconformal transformations.

• An extension to standard model matter, a cosmological constant, and non-compact spatialslices is discussed in [44].

• Since dilaton fields are naturally appearing in supergravity, we plan to investigate anextension of our framework to this setting. Here, it will be interesting to check whatextent of supersymmetry is compatible with the D = 0 gauge fixing or how one could alsogauge fix the supersymmetry constraint.

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Part V

Isolated horizon boundaries inhigher-dimensional LQG

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Chapter 13

Classical phase space description ofisolated horizon boundaries

In this chapter, we generalise the treatment of isolated horizons in loop quantum gravity, result-ing in a Chern-Simons theory on the boundary in the four-dimensional case, to non-distortedisolated horizons in 2(n + 1)-dimensional spacetimes. The key idea is to generalise the four-dimensional horizon boundary condition by using the Euler topological density E(2n) of a spa-tial slice of the black hole horizon as a measure of the distortion. The resulting theory on thehorizon is a higher-dimensional SO(2(n + 1))-Chern-Simons theory, which has local degrees offreedom. We comment briefly on a possible quantisation of the horizon theory and emphasizethat the horizon degrees of freedom which are “visible” from the outside of a black hole form aPoisson-subalgebra, which could significantly simplify the quantisation problem for black holeentropy calculations in higher-dimensional LQG if suitable arguments supporting this treatmentcan be given. The original work from which parts of this chapter are taken is [45].

13.1 Introduction

Black holes in higher dimensions are a subject of great interest in both general relativity andsupergravity. Most prominently, the derivation of black hole entropy within string theory wasfirst performed for a five dimensional black hole [186]. Also, no-hair theorems familiar fromd = 4 spacetime dimensions generally fail in higher dimensions, resulting in a large variety ofblack hole solutions with new (exotic) properties, see [187] for a review. Accordingly, higherdimensions are especially interesting for the area of black hole thermodynamics, since, e.g. theentropy of the black hole depends on its topology, which is considerably richer for d > 4 thanin the usual 4 dimensions. While this fact has been appreciated in, e.g. string theory, it wasnot possible so far to perform these calculations in the context of loop quantum gravity, sincethe Ashtekar-Barbero variables [7, 11] necessary for loop quantisation are restricted to d = 3, 4.On the other hand, the extension of this type of connection formulation to higher-dimensionalsupergravity derived in this thesis opens the window to treat higher-dimensional black holes alsowith the methods of loop quantum gravity.

The treatment of horizons and black hole entropy within loop quantum gravity can be datedback to a remarkable paper by Smolin [188], in which it was shown that under some (natural)assumptions, boundaries of spacetime are described by a topological quantum field theory, moreprecisely SU(2) Chern-Simons theory. While heuristic at certain points, this seminal workalready contained many of the ideas which were later necessary to give a rigorous derivation of theblack hole entropy within LQG. An entropy associated to a surface which is proportional to the

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area was first calculated in a paper by Krasnov [189], where the important conceptual ingredientwas that the punctures of horizon were distinguishable. Further work which strengthened therelation of this type of calculation to black hole entropy and improved the arguments of [189]on several points was done by Rovelli [190].

The rigorous technical framework for calculating the black hole entropy within loop quantumgravity was derived by Ashtekar and collaborators [182, 191, 192, 193], where the notion ofisolated horizon turned out to be crucial in order to have a local description of a black holehorizon. While a classical gauge fixing from SU(2) to U(1) was performed in order to derivethe results of [182, 191, 192, 193], it was later shown by Perez and collaborators [194] that thederivation could be extended to an SU(2) invariant framework. The precise state counting for thederivation of the black hole entropy has been extensively studied by Barbero and collaborators,see [195] and references therein. Also, [196] provides a recent extensive review of the subject,including a comparison of the U(1) and SU(2) treatments.

In this chapter, we are going to take first steps towards the derivation of higher-dimensionalblack hole entropy using loop quantum gravity methods by deriving a generalisation of isolatedhorizon boundary condition F ∝ Σ first proposed in [188] and derived rigorously in [182]. Wefurther show that the canonical transformation to higher-dimensional connection variables in-duces a higher-dimensional Chern-Simons symplectic structure on the intersection of the spatialslice with the black hole horizon. Also, we shortly comment on the quantisation of the resultingtheory on the black hole horizon, a higher-dimensional Chern-Simons theory. The derivationsin this chapter will be restricted to even spacetime dimensions, since the Euler topological den-sity, which will play a key role in the construction, does not exist otherwise. In even spacetimedimensions, the horizon then is odd-dimensional and a Chern-Simons theory can arise. Thecorresponding classical higher-dimensional black hole solution (with spherical symmetry) wasfound by Tangherlini [197] and generalises the Scharzschild solution to higher dimensions, seealso [187] for an overview. Since, in the loop quantum gravity treatment, the notion of isolatedhorizon is more central than that of a classical black hole solution, we will not go into detailsabout the classical black hole solutions. As the notion of isolated horizon has already been gen-eralised to higher dimensions in [198, 199], we can concentrate on deriving the isolated horizonboundary condition and the symplectic structure in this chapter.

This chapter is organised as follows:We start in section 13.2 with an outline of the general strategy used in this chapter for findingan analogue of the isolated horizon boundary condition in higher dimensions. In section 13.3, wewill derive the boundary condition as well as the horizon symplectic structure from the Palatiniaction by making use of the definition of a non-distorted isolated horizon in higher dimensions.In order to make the connection to SO(D + 1) as the internal gauge group, we rederive theisolated horizon boundary condition and the Chern-Simons symplectic structure independentlyof the internal signature in section 13.4, this time purely within the Hamiltonian framework.We shortly comment about the generalisation of the proposed framework to non-distorted hori-zons in section 13.5. Finally, we will discuss a possible quantisation of the boundary degrees offreedom in section 13.6 and conclude in section 13.7.

13.2 General strategy

In this section, we will briefly comment on the general strategy of deriving the isolated horizonboundary condition. It will turn out that there is merely a single reasonable possibility for thegeneral structure of the boundary condition for which a numerical prefactor and an expression forthe connection on the horizon have to be fixed by an actual calculation. However, the connection

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used on the boundary is not necessarily unique as already observed in the four dimensional case[200], where one is free to choose an independent Barbero-Immirzi-type parameter on the blackhole horizon.

Let us start with some hints for the boundary condition based on the new connection variablesderived in [35, 36]

• Tensorial structure:The 3 + 1 dimensional SU(2) based boundary condition F iab ∝ Eciεabc does not generalisetrivially to higher dimensions due to the tensorial structure, i.e. a vector density is dualto a (D − 1)-form in D spatial dimensions, which is a two-form only for D = 3. Since, inanalogy to the 3 + 1 dimensional case, we expect to get a theory which is purely definedin terms of a connection on the horizon, the easiest expression with the correct tensorialstructure to write down is

πaIJ ∝ εab1c1...bncnεIJK1L1...KnLnFb1c1K1L1 . . . FbncnKnLn , (13.2.1)

where a, b, c are spatial tensorial indices and I, J,K,L are fundamental so(D+ 1) indices,n = (D− 1)/2, and πaIJ is the momentum conjugate to the connection on which the newvariables in higher dimensions [35, 36] are based. More generally, one could also use a differ-ent invariant tensor to intertwine the adjoint so(D+ 1) representations on the momentumπ and the field strengths F , but the other obvious choice δJ ][K1δL1][K2 . . . δLn−1][KnδLn][I

results in a vanishing right hand side for even n and does not allow for the constructionperformed in this chapter in the other cases. The open question at this point is mostly onwhich connection the field strengths should be based.

• Topological invariants:Up to a constant prefactor, the derivation of the boundary condition in three spatialdimensions and spherical symmetry can be easily accomplished by appealing to specialproperties of curvature tensors in two dimensions. More precisely, the Riemann tensorRµνρσ on a two-dimensional manifold, e.g. a spatial slice of a black hole horizon, is, due

to its symmetries, given by R(2)µνρσ ∝ R(2) gµ[ρgσ]ν . Thus, after obtaining F

(4)µνIJ = R

(4)µνIJ =

R(4)µνρσΣρσ

IJ from the field equations and since R(4)µν⇐=

ρσΣρσIJ = R

⇐(2)µνρσΣρσ

IJ when using the IH

boundary conditions, it directly follows that F⇐

(4)

µνIJ∝ R(2)Σ

⇐µνIJ, where

⇐denotes the

pullback from the spacetime manifold to a spatial slice of the horizon. In the furtherdiscussion of IHs in 4 dimensional LQG, it is of importance that in 2 dimensions, theintegral over the Ricci curvature actually is a topological invariant by the Gauß-Bonnettheorem. The question thus is, by which topological invariant that role will be played inhigher dimensions.

From the above calculation, we expect that only the step using R(2)µνρσ ∝ R(2) gµ[ρgσ]ν does

not straight forwardly generalise to higher dimensions. However, this formula is equivalent

to R(2) ∝ εαβεIJR(2)αβIJ , and in this form can be generalised to even dimensions and one is

lead to consider the Euler topological density [201]

E(D+1) := εµ1ν1...µnνnεI1J1...InJnRµ1ν1I1J1 . . . RµnνnInJn (13.2.2)

as a generalisation. Although this looks already very similar to the above boundary con-dition (13.2.1), the Euler density would have to be defined on the spatial slices of blackhole horizon while the internal gauge group is inherited from the bulk, thus having a

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representation space which is two dimensions larger than the tangent space of the spa-tial slice of the horizon. Later in this chapter, we will chose a special connection on theboundary, the field strength of which will be inherently “orthogonal” on πaIJ and thusallowing for a precise implementation of the above idea for a boundary condition based onthe Euler topological density. We remark at this point that our normalisation of the Eulertopological density does not coincide with the standard definition leading to the Eulercharacteristic, but χ = 1

(8π)nn!

∫E(2n) holds, leading to an Euler characteristic of χ = 2

for spheres.

• Higher-dimensional Chern-Simons theory:The notion of 2 + 1 dimensional Chern-Simons theory has a straight forward generalisa-tion to higher dimensions, i.e. a higher-dimensional Chern-Simons Lagrangian is definedby dLCS = gI1J1...InJnF

I1J1 ∧ . . . ∧ F InJn , where d is the exterior derivative and g inter-twines n = (D + 1)/2 adjoint representations of so(D + 1), see [202]. The right handside of the previous equation can easily be seen to be the Euler topological density forgI1J1...InJn = εI1J1...InJn . The equations of motion derived from this Lagrangian are givenby gI1J1...InJnF

I2J2 ∧ . . . ∧ F InJn = 0, thus fitting nicely in the LQG quantisation schemefor black holes, i.e. the straight forward generalisation of F IJ = 0 at points of the horizonwhich are not punctured by spin networks is given by εI1J1...InJnF

I2J2 ∧ . . . ∧ F InJn = 0.

As the canonical analysis of higher-dimensional Chern-Simons theory reveals [202], thetheory has local degrees of freedom, e.g. gI1J1...InJnF

I2J2 ∧ . . .∧ F InJn = 0 does not implyF IJ = 0. The implications for a potential quantisation are discussed in section 13.6.

Based on this outline, we will now give a precise derivation of the above proposed generalisa-tion of the isolated horizon boundary condition. The connection used will be a generalisation ofPeldan’s hybrid connection Γhyb

aIJ [47], which was already used in the construction of the connec-tion variables in higher dimensions in the first part of this thesis. We want to stress again thatthere might be other connections, e.g. a one-parameter family depending on a free parameterunrelated to the Barbero-Immirzi parameter, which satisfy an analogous boundary condition,as observed in [200] in the four-dimensional case.

13.3 Higher-dimensional isolated horizons and Lagrangian frame-work

In this section, we will sketch how the isolated horizon boundary condition can be derivedfrom the Palatini action. The definition of an isolated horizon stems from the seminal works[182, 191, 192, 203] and was generalised to higher dimensions in [198, 199, 204]. We will contentus in this chapter with providing the definition of a non-distorted isolated horizon and referto [45] for a detailed discussion of the definition and its consequences. Also, we will be rathersketchy in the derivation and also refer to [45] for the details of the calculation.

Definition 1. An undistorted non-rotating isolated horizon (UDNRIH) is a submanifold ∆ of(M, g) subject to the following conditions:

1. ∆ is foliated by a (preferred) family of (D−1) – spheres, that is, it is topologically R×SD−1.

2. ∆ is a null surface and l its future oriented null normal (which is tangential to ∆ butnormal to the two-sphere cross-sections). We introduce a coordinate v on ∆ such thatk = −dv is the other future oriented null vector field normal to the preferred foliation. Weextend k uniquely to M at points of ∆ by requiring that it is null. Furthermore, we fix l, kup to mutually inverse and constant rescaling and require that ilk = −1.

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3. (a) l is expansion-free.

(b) k is shear-free with nowhere vanishing spherically symmetric expansion and vanishingNewman - Penrose coefficients πJ = lµmν

J∇µkν on ∆.

4. All field equations hold at ∆.

5. −Tµν lν is a causal vector.

6. The ratio E(D−1)/√h, where E(D−1) is the Euler density of the (D − 1) – sphere cross

sections and the h the determinant of the induced metric thereon, is constant on the (D−1)– sphere cross sections.

Let us introduce and review some notation first. We will denote by S the sphere crosssections of ∆. As in the first part of this thesis, nI denotes the internal normal on the Cauchyslice σ. Local coordinates on S will be denoted by lower case Greek letters from the beginningof the alphabet, α, β, . . .. Let sa be the normal on the boundary of σ, pointing outwards of σ,i.e. inwards of S, and sI = saeIa its internal version. The pullback of the spatial co-(D+ 1)-beineIα←

to S is denoted by mIα. With

⇐we mean a pullback from M to S. The induced metric on S

is denoted by hαβ = mIαmβI . Also, we use

⟨E(D−1)

⟩:=∫S E

(D−1)dD−1x to denote the mean of

the Euler topological density. It is related to the Euler characteristic via χ = 1(8π)nn!

⟨E(D−1)

⟩.

From this definition, several properties of l, k, and the Riemann tensor can be deducted, asshown in [45]. The important observation is that, after a long calculation, it follows that

F⇐µνIJ

= R⇐

(D+1)µνIJ = hµ

⇐=

µ′ hν⇐=

ν′R(D+1)µ′ν′ρσe

ρIeσJ = hµ⇐=

µ′ hν⇐=

ν′R(D−1)µ′ν′ρσm

ρImσJ , (13.3.1)

i.e. the spacetime curvature can be expressed in terms of the induced Riemann tensor of thesphere cross section of the isolated horizon. Thus, we are already very close to the envisaged

boundary condition in (13.2.1). However, we still need to relate R(D−1)µ′ν′ρσm

ρImσJ to a fieldstrength of an SO(1, D) connection defined on S. This can be accomplished by noting thatPeldan’s hybrid connection from the previous parts of this thesis can be generalised to higher-dimensional internal spaces, as shown in [45]. Essentially, on a (D − 1)-dimensional manifoldwith a given (D + 1)-bein mI

α, one can construct a unique connection Γ0αIJ by requiring that

the resulting covariant derivative annihilates, next to the (D + 1)-bein, also two additionallychosen normals. We will use nI , the internal normal on the Cauchy slices of our spacetime, andsI = saeIa, the internal normal on S. It follows for the resulting field strength, denoted by RαβIJ ,

that RαβIJnI = RαβIJs

I = 0 and RαβIJ = R(D−1)αβρσ m

ρImσJ . Concluding, we immediately find

εK1L1...KnLnIJ ε⇐

µ1ν1...µnνnF⇐µ1ν1K1L1 ...F⇐µnνnKnLn

=1√hε⇐ρ1σ1...ρnσn ε

⇐µ1ν1...µnνnR(D−1)

µ1ν1ρ1σ1...R(D−1)

µnνnρnσn2n[IsJ ]

=1√hεα1β1...αnβnεIJK1L1...KnLnRα1β1K1L1 ...RαnβnKnLn ≈

E(2n)

√hπaIJsa , (13.3.2)

which is the boundary condition which we were looking for.Next, we want to derive the boundary symplectic structure induced on S. For this, we

employ the covariant phase space techniques [205, 206, 207] and start by the Palatini action

s

2

∫Md4X eeµIeνJFµνIJ(A) =

∫M

ΣIJ ∧ F IJ (13.3.3)

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with the definitions F = 1/2Fµνdxµ ∧ dxν , Fµν

IJ = 2∂[µAν]IJ + [Aµ, Aν ]IJ , Σ := −∗ (e∧ e), and

− ∗ (e ∧ e)µ1...µD−1IJ = 1(D−1)!e

K1µ1. . . e

KD−1µD−1 εIJK1...KD−1

. It is well known that when calculatingthe first variation of this action, the boundary term∫

∆Σ←IJ∧ δA←IJ , (13.3.4)

arises. However, as shown in [45], this term vanishes by the isolated horizon boundary conditions.The second variation of the Palatini action yields the well known symplectic current δ[1ΣIJδ2]AIJ .

By the covariant phase methods, it is closed, and it follows that

(

∫σ2

−∫σ1

+

∫∆

)δ[1ΣIJδ2]AIJ = 0. (13.3.5)

Since the symplectic structure should be independent of the choice of spatial hypersurfaces, wehave to show that the integral over ∆ results in two boundary terms, integrals over the spherecross sections S2 and S1 bounding σ2 and σ1. Indeed, in [45], it is shown that∫

∆δ[1Σ←IJδ2]A←IJ

= ΩS2CS(δ1, δ2)− ΩS1

CS(δ1, δ2), (13.3.6)

where

ΩSCS(δ1, δ2) =

nAS⟨E(2n)

⟩ ∫SεIJKLM2N2...MnNn

(δ[1A⇐IJ

)∧(δ2]A⇐KL

)∧ F⇐M2N2 ∧ ... ∧ F⇐MnNn

(13.3.7)

with 2n = D − 1. The details involve a rather lengthy calculation with the intermediate resultthat

ΩSCS(δ1, δ2) =

∫S

2(δ[1sI)(δ2]nI), (13.3.8)

where sI =√hsI . This equation is also the key in order to connect the derivation in this section

with the canonical transformation to SO(D + 1) connection variables proposed in the first partof this thesis: The above symplectic structure arises from the boundary term (4.1.25) whenperforming the canonical transformation. With this observation, we can proceed to the nextsection, where we will use SO(D+1) as a gauge group instead of SO(1, D) in this section, whichis enforced upon us by the Lorentzian Palatini action.

13.4 SO(D + 1) as internal gauge group

In the previous section, we have derived the isolated horizon boundary condition relating theconnection on the horizon with the bulk degrees of freedom, as well as the symplectic structureon the horizon, which coincides with the one of higher-dimensional Chern-Simons theory. Sincewe started from the spacetime covariant Palatini action, the internal gauge group was fixed toSO(1, D). In the light of quantising the bulk degrees of freedom however, it was pointed outin the first part of this thesis that one can change the internal gauge group to SO(D + 1) by acanonical transformation from the ADM phase space. After this reformulation, the quantisationof the bulk degrees of freedom can be performed with standard loop quantum gravity methodsas spelled out in chapter 6. Thus, we are interested in reformulating the horizon boundarycondition and the horizon symplectic structure so that it fits in the SO(D + 1) scheme.

The general idea of this change in gauge group is the same as for the bulk degrees of freedom,i.e. we construct a canonical transformation from the ADM phase space. We start by realising

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that the construction of the boundary condition (13.3.2) can be taken over from the Lorentziancase by simply changing the internal group to SO(D + 1) and using the generalised Peldanhybrid connection Γ0

αIJ from the previous section as a connection on S. As was noted at theend of section 4.1, the canonical transformation to connection variables in the bulk leads to aboundary term which we have to take into account on the isolated horizon. In a first step, onenotes that this boundary term leads to the Euclidean version of the right hand side of (13.3.8).What remains to be done then is to show that (13.3.8) is also valid in the case of SO(D + 1)as internal group and using Γ0 instead of A as a connection in (13.3.7). It then follows thatthe canonical transformation induces a higher-dimensional SO(D+ 1) Chern-Simons symplectic

structure on S. The non-distortion condition δE(2n)√h

= 0 has to be kept also in the SO(D + 1)

framework and is independent of the chosen internal group since it can solely be expressed interms of hαβ. The link to the spacetime formulation is of course missing in this picture, i.e.(13.3.1) does not make sense any more due to the different internal groups.

As for the boundary condition, the generalisation to the Euclidean internal group is straightforward, since the construction of the connection Γ0 provided in [45] works independently ofthe internal signature. Thus, constructing Γ0 such that it annihilates both nK and sK = sae

aK

additionally to mKα = e

←Kα in the SO(D + 1) framework, the horizon boundary condition

εK1L1...KnLnIJεα1β1...αnβn

R0α1β1K1L1 ...R

0αnβnKnLn =

E(2n)

√qπaIJsa (13.4.1)

follows immediately from the fact that R0αβKLn

K = R0αβKLs

K = 0.In order to derive the new symplectic structure, we first perform a symplectic reduction of the

theory derived in the previous chapters by solving the Gauß and simplicity constraint. This leadsus to the ADM phase space, from which we can perform further canonical transformations. Thisstep is important since it tells us that using an isolated horizon as a boundary of our manifold, wewill have a vanishing horizon symplectic structure when using ADM variables. We remark thatthis does not follow trivially for any boundary if one starts with the Einstein-Hilbert action andperforms the Legendre transform, since one is picking up boundary terms in the Gauß-Codazziequation which are neglected in order to arrive at the standard ADM symplectic structure.

As remarked earlier, it was shown in [35] that the canonical transformation to SO(D + 1)connection variables leads to the boundary term (4.1.25) which upon pulling back to S andperforming a second variation yields the symplectic structure

ΩS(δ1, δ2) =2

β

∫SdD−1x δ[1sIδ2]n

I . (13.4.2)

Furthermore, under the non-distortion condition δE(2n)√h

= 0, i.e. restricting to the part of phase

space where E(2n)√h

=〈E(2n)〉AS

is constant, a lengthy calculation detailed in [45] shows that

2E(2n)

√h

(δ[1sI)(δ2]nI) = nεIJKLM2N2...MnNnεαβα2β2...αnβn

×(δ[1Γ0

αIJ

) (δ2]Γ

0βKL

)R0α2β2M2N2

...R0αnβnMnNn , (13.4.3)

which results in the Chern-Simons type boundary symplectic structure

ΩSCS(δ1, δ2) =

nAS

β⟨E(2n)

⟩ ∫SεIJKLM2N2...MnNnεαβα2β2...αnβn

×(δ[1Γ0

αIJ

) (δ2]Γ

0βKL

)R0α2β2M2N2

...R0αnβnMnNn . (13.4.4)

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Concluding, we have shown that also for the case of SO(D + 1) as an internal gauge group,one arrives at a higher-dimensional Chern-Simons symplectic structure at the isolated horizonboundary of σ.

13.5 Inclusion of distortion

In this section, we are going to comment on the generalisation of the isolated horizon boundarycondition derived in the non-distorted case to general isolated horizons with spherical topology.The seminal work on this subject has been a paper by Asthekar, Engle, and Van Den Broeck[208], where treatment was generalised to axi-symmetric horizons. For the generalisation toarbitrary spherical horizons, two methods by Perez and Pranzetti [209] and Beetle and Engle[210] exist in four dimensions. We will discuss them in the following and show that an extensionof them to higher dimensions is not straight forward.

13.5.1 Beetle-Engle method

In order to derive the symplectic structure on a spatial slice S of the horizon, it is key tothe derivation that E(2n)/

√h is a constant on S. Otherwise, unwanted terms appear due to the

variation of E(2n)/√h. Of course, this observation has already been made in the four-dimensional

case and a solution of this problem for generic horizons in case of U(1) as the gauge group on Shas been proposed by Beetle and Engle [210]. Essentially, they construct a new U(1) connectionon S as

V α:=

1

2θα − εαβhβγDγΨ, (13.5.1)

where 12θα is the U(1) connection used for spherically symmetric isolated horizons and Ψ is a

curvature potential defined by the equation

∆Ψ = R− 〈R〉 , (13.5.2)

where R is the intrinsic scalar curvature which is proportional to E(2)/√h. Calculating the

curvature ofV α, the terms proportional to R drop out and one gets

dV= −〈R〉

4ε = − 2π

ASΣir

i. (13.5.3)

Thus,V α mimics the spin connection of a spherically symmetric horizon, although being defined

for any horizon of spherical topology.The trick of Beetle and Engle can be generalised to this framework for the case of four

dimensions by using the connection

AαIJ = Γ0αIJ + 2mα[Imβ|J ]h

βγ(Dγψ). (13.5.4)

Insertion into the boundary condition

εαβεIJKLRαβKL(A) = 2〈E(2)〉n[I sJ ] (13.5.5)

yields

∆ψ =1

4

(E(2)

√h− 〈E(2)〉

). (13.5.6)

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As shown in [45], it follows that

2〈E(2)〉(δ[1sI)(δ2]nI) = εIJKLεαβ

(δ[1AαIJ

) (δ2]AβKL

). (13.5.7)

The problem with generalising this trick to higher dimensions is that it leads to a non-linearpartial differential equation for Ψ, for which, as opposed to the Laplace operator ∆, a well devel-oped theory ensuring the existence of a solution does not exist. Thus, although a generalisationto higher dimensions seems straight forward, we cannot proceed due to the resulting non-linearpartial differential equation.

13.5.2 Perez-Pranzetti method

The basic idea of Perez and Pranzetti [209] in order to solve the problem of a varying scalarcurvature on S is to use two Chern-Simons connections on S, defined by

Aiγ = Γi + γei, Aiσ = Γi + σei. (13.5.8)

For the boundary conditions, it follows that

F i(Aγ) = Ψ2Σi +1

2(γ2 + c)Σi, F i(Aσ) = Ψ2Σi +

1

2(σ2 + c)Σi, (13.5.9)

where the Newman-Penrose coefficient Ψ2 is proportional to the scalar curvature. Subtractingthese two equations, Perez and Pranzetti find

F i(Aγ)− F i(Aσ) =1

2(γ2 − σ2)Σi, (13.5.10)

which can be used to derive the symplectic structure of two SU(2) Chern-Simons connectionson S, since the scalar curvature disappeared from this new boundary condition. Furthermore,they take the additional constraint into account which follows from adding the above two fieldstrengths, which requires to first find a suitable quantisation of the scalar curvature.

The first steps of this treatment generalise to higher dimensions in a straight forward way:Introduce (D − 1) Chern-Simons connections as

A(a1)αIJ = ΓαIJ + 2

√a1s[Imα|J ], . . . , A

(aD−1)αIJ = ΓαIJ + 2

√aD−1s[Imα|J ]. (13.5.11)

For their field strengths, it follows that

F(ai)αβIJ = RαβIJ − 2mα[Imβ|J ]ai. (13.5.12)

With the abbreviation

EIJ(ai)(A(ai)) := εβ1γ1...βnγnεIJK1L1...KnLnF

(ai)β1γ1K1L1

. . . F(ai)βnγnKnLn

, (13.5.13)

the straight forward generalisation of the boundary condition of Perez and Pranzetti reads

D−1∑i=1

(−1)i+1EIJ(ai)(A(ai))

!= 2n[I sJ ] (13.5.14)

yields (D−1)/2 equations only involving the coefficients ai. However, even if we can solve thoseequations, we still have to take into account further constraints resulting from different linearcombinations of EIJ(ai)

(A(ai)). As opposed to the four-dimensional case, i.e. two-dimensional

isolated horizons, these equations will generically contain terms of the type R ∧ . . . ∧ R ∧ (m ∧m) ∧ . . . ∧ (m ∧m), which seem to be very hard to quantise.

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13.6 Comments on quantisation

The quantisation of the three-dimensional Chern-Simons theories in the calculation of the blackhole entropy for D = 3 are based on Witten’s quantisation [211] of Chern-Simons theory. Thefact that the corresponding quantisation can be performed rests strongly on the fact that three-dimensional Chern-Simons theory is a topological field theory, i.e. local degrees of freedom areabsent. Also, it is key to the derivation that the lapse function vanishes on the horizon and theHamiltonian constraint does not have to be taken into account. This point continues to be truein higher dimensions. However, when taking into account higher dimensions, the Chern-Simonstheory generally admits local degrees of freedom since the equations of motion

εI1J1...InJnFI2J2 ∧ . . . ∧ F InJn = 0 (13.6.1)

do not constrain the connection to be flat [202]. Thus, a quantisation of the higher-dimensionalChern-Simons theories appearing on the horizons does not seem straight forward, at least notfor the complete theory.

Nevertheless, a quantisation of a subsector of the Chern-Simons theory which is similarto the three-dimensional case might still be possible, as we will argue below. Whether theentropy calculation can be performed solely by considering this subsector is presently unclearto the author since it, at best, captures only a subsector of the full Hilbert space of quantisedSO(D + 1) Chern-Simons theory. The following analogy could therefore be on a purely formallevel, however in case satisfying arguments could be given for its validity, the entropy calculationcould be performed in direct analogy to the four-dimensional case.

We will now outline the above mentioned formal analogy. Using the language of Engle,Noui, Perez, and Pranzetti [200], the Chern-Simons equations of motion (13.6.1) are modifiedby “particle degrees of freedom” which are induced by the spin networks puncturing the horizonas

EI1J1(x) := εI1J1...InJnFI2J2(x) ∧ . . . ∧ FInJn(x) ∝ saπaI1J1(x), (13.6.2)

where the operator on the right hand side symbolises to the flux operator which acts non-triviallyonly at points where a spin network punctures the horizon. Using the Dirac brackets obtainedfrom solving the second class constraints of the higher-dimensional Chern-Simons theory1, wecan explicitly calculate the algebra of these “particle excitations” as

EIJ(x), EKL(y)∝ δ(D−1)(x− y)f IJ,KL,MNE

MN (x), , (13.6.3)

where f are the structure constants of SO(D + 1).Since a representation of this algebra is just a representation of the Lie algebra so(D + 1)

for each puncture, the problems which have to be discussed for the quantisation are mainlyconnected with finding the right subspace of the tensor product of the individual so(D + 1)representation spaces which is selected by the criterion of compatibility with the bulk spinnetworks and the horizon topology. As opposed to the U(1) or SU(2) based constructions infour dimensions, the restrictions imposed by the simplicity constraint will also have to be takeninto account properly. While the simplicity constraint is solved on the horizon at the classicallevel by using the variables nI , sI , and eIα

←to construct Γ0

αIJ , there might still be non-trivial

restrictions coming from imposing the quantum simplicity constraint in the bulk. One of them

1Actually, in order to construct the Dirac brackets of the Chern-Simons theory, we would have to identify theset of second class constraints. This is non-trivial and depends on the choice of invariant tensor, as emphasised in[202]. There, it is also stated that for the epsilon tensor used in this chapter, our choice of second class constraintsis correct at least in six spacetime dimensions (εIJKLMN is “generic” in the language of [202]). On the other hand,we can use the horizon boundary condition and the symplectic structure (13.4.2) to calculate the same algebra.

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is to restrict the representations carried by the punctures to be the same as in the bulk, i.e.simple (spherical / class 1) SO(D + 1) representations. However, the more interesting questionwill be if there is a restriction resulting from implementing the vertex simplicity constraints. Weleave this question for further research.

As mentioned before, this analogy to the four-dimensional entropy calculation is only formalat the current level. A well-motivated calculation from first principles would amount to findinga complete quantisation of the higher-dimensional Chern-Simons theory and then to performthe entropy calculation taking into account all horizon degrees of freedom. From a differentpoint of view, one could also argue that we are only interested in an effective four-dimensionalentropy where the additional spatial dimensions are compactified. Then, one could also performa Kaluza-Klein reduction before quantisation which is legitimate since the whole LQG entropycalculation is only semiclassical due to the classical notion of the isolated horizon. We leavethese issues for further research.

13.7 Concluding remarks

In this chapter, we derived a generalisation of the isolated horizon boundary condition to non-distorted isolated horizons in even dimensional spacetimes and showed that the canonical trans-formation to SO(D + 1) connection variables leads to a higher-dimensional Chern-Simons sym-plectic structure on the boundary of the spatial slice. While the classical treatment from fourspacetime dimensions generalises rather directly, the quantisation of the resulting system is lessobvious, since, to the best of the author’s knowlege, there are no known generalisations of thequantisation of 2 + 1-dimensional Chern-Simons theory to higher dimensions. On the otherhand, it could be the case that it is not necessary to quantise “all of the Chern-Simons theory”,but just a subalgebra of phase space functions which result as topological defects induced bypuncturing the horizon with a spin network, as discussed in the previous section.

One of the most important questions which should be answered by a suitable quantisation ofthe theory on the black hole horizon is the treatment of the simplicity constraint. A preliminaryanalysis shows that the classical simplicity constraint fits nicely into the picture of a Chern-Simons theory with particles as proposed in [200]. While a quantisation of the edge simplicityconstraints would just restrict the group representations on the particle defects in the same wayas it restricts the edge representations, it might be that a proper quantisation of the horizondegrees of freedom gives us a hint on what the correct implementation of the simplicity constrainton a vertex is. The reason for this comes from the seemingly very effective treatment of a blackhole as a single intertwiner, see [212] and more recently also [213]. One would now expect thatthe correct treatment of the higher-dimensional vertex simplicity constraints should somehowbe reflected in this treatment, i.e. that the same restrictions on the intertwiner should also bereflected by the quantisation of the horizon degrees of freedom.

Additionally, it will be interesting to check to what extend the connection on the horizoncan be generalised, e.g. as in [200], where a new free parameter can be associated to thehorizon connection which can rescale the entropy to A/4 without fixing the Barbero-Immirziparameter. The consequences of introducing a two parameter family of connections in the bulkin four dimensions as proposed in [36] should also be investigated and a generalisation to generalnon-symmetric black holes has to be developed.

As a last remark, we want to emphasise that the study of different horizon topologies promisesto yield a deeper insight into the subject at hand. The reason is that there are results thatthe black hole entropy should depend on the horizon topology, more precisely on the Eulercharacteristic [214]. However, we caution the reader that before jumping to comparisons, theblack hole entropy calculations based on the presented framework will have to be worked out in

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detail. One of the problems arising is that the classical treatment assumes the non-distortioncondition, which cannot be satisfied for general topologies. Examples different from sphericaltopology where the classical derivation should however be applicable are black holes of constantnegative curvature, which we plan to investigate in the future.

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Part VI

Conclusion

127

Chapter 14

Concluding remarks and furtherresearch

In this chapter, we shortly comment on what has been achieved in this thesis and what directionsfurther research on this subject should explore.

14.1 Summary of the results

In this thesis, a canonical formulation of higher dimensional general relativity and many higherdimensional supergravity theories has been constructed which allows for an application of thequantisation techniques developed in the context of loop quantum gravity. The existence of thistype of connection formulation however came as a surprise, since the results from four dimensionsindicated that in order to remove the superfluous degrees of freedom from the connection, onewould need additional constraints (the simplicity constraints), which produced secondary secondclass constraints when stabilised in the Dirac procedure. The key idea on the gravitational sidewas to use the procedure of gauge unfixing in order to get rid of the second class constraints.Although this procedure generally produces an infinite series for the gauge invariant extensionsof the remaining constraints, in the case of general relativity it terminates after the second termand leaves us with a Hamiltonian constraint to which the quantisation techniques familiar fromfour dimensions could be extended rather straight forwardly.

In order to construct a well defined quantum theory, it was necessary to switch to the com-pact internal gauge group SO(D+ 1) instead of the Lorentz group. In retrospect, it seems clearthat general relativity would allow for such a description at the Hamiltonian level, however alsothis fact came as a surprise. When including fermions in the treatment, one has to generalisethe internal gauge group to Spin(D + 1) and act with this group also on the fermionic repre-sentation space. While this works without major obstacles for Dirac fermions, more work isrequired for Majorana fermions due to their reality conditions. In fact, since the existence ofMajorana fermions in Lorentzian signature excludes their existence in Euclidean signature in thesame number of dimensions, it was unclear if the Majorana fermions necessary for completingthe supergravity multiplets could be incorporated into the Spin(D + 1) framework. The keyinput for them to work was a modified reality condition which matches with the Lorentzianreality condition when choosing the time gauge N I = (1, 0, . . . , 0). Due to Spin(D + 1) gaugecovariance of this reality condition, the full Spin(D + 1)-invariant theory therefore agrees withthe corresponding Lorentzian supergravity. The inclusion of the three-index photon of eleven

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dimensional supergravity also posed an additional obstacle due to the twisted holonomy-fluxalgebra, which could however be circumvented using a reduced phase space quantisation withrespect to the three-index photon Gauß law. The interesting part of this quantisation is thatit seems to require a background independent representation which is discontinuous in boththe connection and its momentum, thus contrasting itself to the Ashtekar-Isham-Lewandowskirepresentation which is only discontinuous in the connection.

In the fourth part of this thesis, we have provided a partially reduced phase space quantisationof general relativity conformally coupled to a scalar field. The focus of this quantisation hasbeen on non-dynamical issues since the time function used, the CMC gauge, results in a verycomplicated generator of time evolution, but allows for good control over the existence of CMCslices. It has been explicitly shown that there exist interesting black hole solutions which canbe incorporated in the reduced phase space quantisation and thus their entropy calculations canbe based on the counting of physical states. Furthermore, the geometric operators which arephysical with respect to the classically reduced Hamiltonian constraint do not agree with thekinematical geometric operators or their counterparts in the deparametrised models. Thus, anexplicit example has been provided that the spectrum of the geometric operators depends on thechosen spatial hypersurface determined by the time function. While similar effects have alreadybeen conjectured in [171], an explicit example in full loop quantum gravity was not given so far.

Finally, we took first steps toward the derivation of higher dimensional black hole entropy inpart five. There, an isolated horizon boundary condition was derived which closely resembles theresult from the four-dimensional treatment. Moreover, it was shown that using the SO(D + 1)connection variables, a higher dimensional Chern-Simons symplectic structure arises on the iso-lated horizon. Similarly to four-dimensional treatment, a quantisation of the boundary conditiontogether with this symplectic structure results in a quantised higher-dimensional Chern-Simonstheory describing the isolated horizon. Since higher dimensional Chern-Simons theories containlocal degrees of freedom, the quantisation procedures familiar from four dimensions cannot beused directly.

14.2 Towards loop quantum supergravity: Where do we stand?

The main result of this thesis has been an existence proof that many higher dimensional su-pergravities can be quantised with loop quantum gravity methods. While explicit ways ofconstructing the Hamiltonian constraint and supersymmetry constraint have been given, theresulting operators are very complicated and the physical effects of the choices involved in theregularisations are poorly understood in general. In order to make progress on this issue, abetter understanding of dynamics resulting from loop quantum gravity type quantisations isneeded, i.e. one has to study the physical consequences of the regularisation ambiguities. How-ever, already in 3 + 1 dimensions, this is a very hard problem which has not gotten the requiredattention up till now. Thus, the main point of criticism which has to be raised at this momentis that more control over the dynamics of the theory is necessary, especially to judge its physicalsignificance. The more complicated form of the Hamiltonian constraint in higher dimensions in-creases the difficulty of this question even more, since the new gauge unfixing part is significantlymore complicated than the other terms.

On the other hand, the situation in 3 + 1 dimensions puts us in the position to compare theformulation developed in this thesis with the usual construction based on the Ashtekar-Barberovariables, thus having two quantisations of general relativity based on classically different vari-ables. This is especially interesting for the simplicity constraint, because we can discriminatebetween different impositions of the constraint by requiring dynamical equivalence of the theoriesbased on Ashtekar-Barbero variables and the ones introduced in this thesis.

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Another point which deserves attention is the question of the signature of the internal gaugegroup. In this thesis, we have derived a formulation of Lorentzian (D + 1)-dimensional generalrelativity with SO(D + 1) as an internal gauge group. Although this has been achieved by acanonical transformation, one might argue that it would be more natural to use SO(1, D) as theinternal gauge group. While this is also appealing from the point of view of the dynamics of thetheory since the Hamiltonian constraint will be simpler, the Hilbert space techniques developedfor loop quantum gravity are only available for compact gauge groups. However, if one couldextend these techniques also to non-compact gauge groups, SO(1, D) might become the gaugegroup of our choice.

As for the 3-index photon of eleven dimensional supergravity studied in this thesis, it ismandatory to gain a better understanding for the Narnhofer-Thirring type state used in theconstruction of the quantum theory. Its physics is rather unclear and, due to its discontinuityin both the 3-index photon and its momentum, it might lead to more quantisation ambiguitiesthan the Asthekar-Lewandowski measure on which loop quantum gravity is based.

Concluding, it has been shown that the methods of loop quantum gravity can be applied tomany higher dimensional supergravities, thus providing a rigorous quantisation of these theories.However, the dynamics of the resulting theories are poorly understood and progress on this issueis mandatory before continuing with the initial aim of this thesis, which was to establish a contactwith superstring theory.

14.3 Further research

In the following, several approaches for further research which suggest themselves at the currentstage of the loop quantum supergravity programme will be outlined. While the first three pro-posed projects deal with specific symmetry reduced situations and dualities, and are thus directapplications of the framework, the other proposed projects aim at gaining a better understand-ing of the loop quantum gravity dynamics and their quantum field theory on curved spacetimelimit in general.

Black holesInvestigating the quantisation of higher dimensional and supersymmetric black holes usingLQG methods is an important application of the results developed in this thesis. The cor-responding isolated horizon frameworks for higher dimensions [199, 198] and supersymmetry[215, 216, 217] have already been developed, and we were able to derive the classical phase spacedescription in higher dimensions in part five of this thesis under the non-distortion assumptionδ(E(D−1)/

√h) = 0. While Chern-Simons theory in 2 + 1 dimensions is topological, it ceases to

be so in higher dimensions [202] and new quantisation methods have to be developed for thehigher dimensional Chern-Simons theory on the horizon. Next, one needs to consider supersym-metry, however it seems unclear what role it will play on the horizon. While supersymmetricChern-Simons theories are available [218], it is also conceivable that the supersymmetry con-straint will not need to be taken into account on the horizon as in the case of the Hamiltonianconstraint [182]. Furthermore, the Rarita-Schwinger field does not produce a boundary term inthe derivation of the new variables for supergravity.Interestingly, the horizon topology of a black hole is not confined to be spherical in higher di-mensions. Accordingly, one should study what implications this has for the derivation of theboundary term and the induced Chern-Simons degrees of freedom. However, this is problematicsince the non-distortion condition cannot hold for an arbitrary topology.

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CosmologyThe study of higher dimensional and supersymmetric cosmological models is interesting since itmight provide a window on observable effects of higher dimensions and supersymmetry in cos-mological observables which are sensitive to quantum effects, see [219] for calculations in 3 + 1dimensional loop quantum cosmology.In order to extend the framework of loop quantum cosmology to supergravity, it seems reasonableto start with higher dimensional gravity first. While the kinematical framework should gener-alise easily to higher dimensions, a new term appearing in the Hamiltonian constraint in the newhigher dimensional formulation will have to be investigated for its dynamical consequences. In anext step, one has to think about including the remaining matter fields from the correspondingsupergravity multiplet and about how to implement the supersymmetry constraint operator, see[220] for a treatment in the Wheeler-DeWitt framework. This is conceptually different from thenon-supersymmetric case since the supersymmetry constraint is situated above the Hamiltonianconstraint in the “hierarchy” of constraints, in the sense that a solution to the supersymmetryconstraint operator is automatically a solution to the Hamiltonian constraint operator (up toanomalies). Thus, the difference-type equation [221, 222] from standard loop quantum cosmol-ogy has to be generalised in the case of supergravity.Since one is finally interested in theories which appear 3+1 dimensional at least at not too smallscales and low enough energies, a dimensional reduction has to be performed, see for example[223]. It is especially here where the spectral properties of the higher dimensional geometricoperators will enter and the results should become sensitive to extra dimensions.

Gauge / gravity correspondenceFurther down the road, an application of the new quantisation techniques for supergravity de-veloped in this thesis is the gauge / gravity correspondence, see [224] for a review. The basicidea is that since a non-perturbative sector of quantum supergravity can be constructed byusing LQG techniques, one should try to relate it to a certain sector of a dual gauge theory.The results of the fifth part of this thesis seem especially appealing in this context since thecanonical transformation used to derive the new variables yields a boundary term which inducesa (Chern-Simons) gauge theory on the boundary and relates it to the quantum gravity theory inthe bulk. To establish a precise dictionary is certainly a long shot, see however [225] for resultsfrom 2 + 1 dimensional quantum gravity. Along this road, several open technical problems, likenon-compact boundary conditions for spin-networks, which are important in their own right,have to be attacked. Despite the (at the moment) rather vague starting point, the potentialbenefits of such a line of research, including the application of the gauge / gravity correspondenceto real world physics like quark-gluon plasmas or condensed matter physics, greatly outweighits risks, especially since it is connected to black hole physics at least on a technical level.

Use existing work from loop quantum cosmology and the quantum constraint alge-braIn [226], it was shown that one can regularise the spatial diffeomorphsim constraint operatorin such a way that the Dirac algebra for spatial diffeomorphisms is reproduced at the quantumlevel. In order to achieve this result, it was necessary to consider in the regularisation for thefield strength, next to small closed loop holonomies, also fluxes and an explicit dependence onthe labels of the edges which the constraint acts upon. Further work to generalise these regu-larisation techniques also to the Hamiltonian constraint is currently performed by the authorsof [226]. It should be the goal to incorporate these findings into the regularisations of the trueHamiltonian operators for the deparametrised models, which will hopefully lead to a better un-derstanding of their action on spin-network functions.

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A conceptually related result has been obtained in the framework of the improved dynamicsof loop quantum cosmology (µ-scheme [222]), where the regularised operator corresponding tothe curvature has to depend on both, the connection and the geometry. Also here, it will beinteresting to investigate possible consequences for improved regularisations of the (true or con-strained) Hamiltonian operator in the full theory.

Supergravity as simplified matter coupled general relativitySupergravity is essentially general relativity coupled to specific matter in such a way that a newfermionic symmetry generator is present in the theory which maps fermions into bosons and viceversa. The simplification in supergravity, as opposed to matter coupled general relativity, stemsfrom the presence of this new symmetry, however not form a reduction to a finite amount ofdegrees of freedom as in many cosmological models. For flat space, the new symmetry can be in-tegrated into the Poincare group in a non-trivial way1, leading to the super-Poincare group. Thecounterpart in a Hamiltonian formulation of supergravity is a non-trivial extension of the Diracalgebra, where a new constraint, the supersymmetry constraint S, is present, which squares toall the other constraints as S,S = S +H +Ha [117]. At the quantum level, one would expectthat this relation is not implemented for a general regularisation of the constraint operators,but one might hope that a (unique) regularisation can be fixed by demanding the correspondingquantum operators to reproduce the classical relation. As opposed to matter coupled generalrelativity without supersymmetry, the above relation should yield new insights into how theHamiltonian constraint should be regularised. In order to attack this problem, it would be bestto start with supersymmetric quantum cosmology, where the calculations, as in loop quantumcosmology, will be easier to handle. Here, despite the symmetry reduction, the non-trivial super-Dirac algebra is still present [227] and, analogous to results from loop quantum cosmology, thegoal of this exercise will be to extract information on how exactly the Hamiltonian constraintincluding matter fields should be regularised. In the easiest case of minimal D + 1 = 4, N = 1supergravity, this will give only information about fermion couplings, but when increasing N ,the number of supersymmetry generators, up to a value of 8, all matter fields up to spin 2 areincorporated in the supersymmetry multiplet and appearing in the regularisation.

Coherent statesCoherent states are of great importance for the study of matter coupled loop quantum gravity,not only for checking the classical limit of the Hamiltonian constraint or the true Hamiltonians,but also for practical calculations in the deparametrised models. Since, in general, the generatorof time translations respective to a chosen clock is a complicated operator, e.g. a square root asin [22, 23], coherent states can be employed to approximate this operator, e.g. the square root,around its classical expectation value and to calculate quantum corrections.An important open problem concerning coherent states is recovering the usual background de-pendent Fock representation of quantum field theory. Here, seminal work for the Abelian caseis available [228, 229, 230], however, in the case of non-Abelian gauge groups the literature israther scarce [231]. Thus, it is very important to make progress on these questions, building onthe research started in [29, 30, 31, 32, 33, 34, 231].

Choice of time functionOne of the crucial ingredients of a reduced phase space quantisation is the choice of a timefunction. This time function dictates the choice of foliation in the spacetime manifold and all

1This works despite the Coleman-Mandula theorem (basically, in QFT all extensions of the Poincare groupare trivial.), which is circumvented by the fact that the supersymmetry generators have odd Graßmann parity.

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experimental questions that can be asked2 in the reduced phase space quantisation are restrictedto experiments happening on the leaves of this specific foliation. Thus, a given result from a re-duced phase space quantisation based on a certain time function, e.g. the kinematical geometricoperators of LQG becoming physical observables [22], might not carry over to a quantisationbased on a different time function. A specific example has been given in the fourth part of thisthesis, where, due to the choice of a time function which is a sum of momenta, the usual LQGoperators are not physical observables, but only a product of them with a certain power of ascalar field.This observation leads one to think about which time functions correspond to certain physicalsituations which one wants to describe, e.g. particle scatterings in the quantum field theory oncurved spacetime limit, which is one of the important limits that LQG has to reproduce. Sincethe time function dictates the form of the true Hamiltonian, choosing a time function is not amere choice of convenience, but it should be adapted to the physical question one is asking.

2At least in a practical way, meaning that one does not extrapolate from results being calculated on the naturaltime slices of a given deparametrised model to other time slices. Especially the result of [43], that the physicalgeometric operators can change due to a different choice of time function, exemplifies that it is very unlikely thatsuch an extrapolation would give a result which would agree with directly using the different time function.

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Appendix A

Simple irreps of SO(D + 1) andsquare integrable functions on thesphere SD

In this appendix, we cite several important results concerning simple irreducible representationsof SO(D + 1). This appendix is taken from the original work [39].

There is a natural action of SO(D + 1) on F ∈ H := L2(SD, dµ) given by π(g)F (N) :=F (g−1N). The π(g) are called quasi-regular representations of SO(D+1). The generators in thisrepresentation are of the form τIJ = 1

2( ∂∂NINJ − ∂

∂NJNI) and are known to satisfy the quadraticsimplicity constraint τ[IJτKL] = 0 [62]. These representations are reducible. The representation

space can be decomposed into spaces of harmonic homogeneous polynomials HD+1,l of degreel in D + 1 variables, L2(SD) =

∑∞l=0 H

D+1,l. The restriction of π(g) to these subspaces gives

irreducible representations of SO(D + 1) with highest weight ~Λ = (l, 0, ..., 0), l ∈ N. These are(up to equivalence) the only irreducible representations of SO(D + 1) satisfying the quadraticsimplicity constraint [62] and therefore are mostly called simple representations in the spinfoam community. Note that these representations have been studied quite extensively in themathematical literature, where they are called most degenerate representations [232, 233, 234],(completely) symmetric representations [233, 235, 236, 237] or representations of class one (withrespect to a SO(D) subgroup) [78]. The latter is due to the fact that these representations ofSO(D+1) are the only ones which have in their representations space a non-zero vector invariantunder a SO(D) subgroup, which is exactly the definition of being of class one w.r.t. a subgroupgiven in [78]. An orthonormal basis in HD+1,l is given by generalisations of spherical harmonics

to higher dimensions [78] which we denote Ξ~Kl (N),∫

SDΞ~Kl (N) Ξ

~Ml′ (N) dN = δll′δ

~K~M

, (A.1)

where ~K denotes an integer sequence ~K := (K1, . . . ,KD−2,±KD−1) satisfying l ≥ K1 ≥ . . . ≥KD−1 ≥ 0 and analogously defined ~M . Fl(N) ∈ HD+1,l can be decomposed as Fl(N) =∑

~K a ~KΞ~Kl (N) where the sum runs over those integer sequences ~K allowed by the above in-

equality. Since L2(SD) =∑∞

l=0 HD+1,l, any square integrable function F (N) on the sphere can

be expanded in a mean-convergent series of the form [78]

F (N) =

∞∑l=0

∑~Kl

al~KlΞ~Kll (N). (A.2)

134

Consider a recoupling basis [238] for the ONB of the tensor product of N irreps: Choose alabelling of the irreps ~Λ1, ..., ~ΛN . Then, consider the ONB∣∣∣~Λ1, ..., ~ΛN ; ~Λ12, ~Λ123, ..., ~Λ1...N−1; ~Λ, ~M

⟩, (A.3)

(with certain restrictions on the values of the intermediate and final highest weights). A basisin the intertwiner space is given by∣∣∣~Λ1, ..., ~ΛN ; ~Λ12, ~Λ123, ..., ~Λ1...N−1; 0, 0

⟩, (A.4)

(with certain restrictions). A change of recoupling scheme corresponds to a change of basis inthe intertwiner space. A basis in the intertwiner space of N simple irreps is given by∣∣∣Λ1, ...,ΛN ; ~Λ12, ~Λ123, ..., ~Λ1...N−1; 0, 0

⟩, (A.5)

(with certain restrictions), since in the tensor product of two simple irreps, non-simple irrepsappear in general [237, 236],

(λ1, 0, ..., 0)⊗ (λ2, 0, ..., 0) =

λ2∑k=0

λ2−k∑l=0

(λ1 + λ2 − 2k − l, l, 0, ..., 0) (λ2 ≤ λ1). (A.6)

135

Danksagung

An dieser Stelle mochte ich mich bei allen bedanken, die mich wahrend meiner Promotionsphaseunterstutzt haben und mir diesen Lebensweg ermoglicht haben.

Zu allererst gilt mein Dank meinen Eltern, die mich von Beginn meines Studiums an unterstutzthaben, ideell, finanziell, und auch tatkraftig bei den vielen Umzugen. Der Wert ihrer Un-terstutzung fur mich wurde sich nicht angemessen in Worte fassen lassen.

Desweiteren gilt mein Dank meinem Studienkollegen, Mitdoktoranden und nicht zuletzt FreundAndreas Thurn. Der Wert einer guten Kollaboration zusammen mit einem guten Freund hatsich in den letzten Jahren fur mich als unschatzbar groß erwiesen. Das gleiche gilt fur die Zusam-menarbeit mit Alexander Stottmeister bei den im vierten Teil dieser Doktorarbeit behandeltenErgebnissen.

Meinem Betreuer Thomas Thiemann mochte ich fur die vier schonen Jahre wahrend meinerDiplomarbeit und Promotionsphase danken, wahrend derer ich viel von ihm gelernt habe unddas besonders gute Arbeitsklima in der Gruppe, zuerst am Albert-Einstein-Institut in Potsdam,dann in der Theoretischen Physik III in Erlangen, genossen habe. Sein Vorschlag des Arbeits-themas als langeres Projekt fur Andreas Thurn und mich, seine intensive Betreuung sowie diegute Zusammenarbeit mit ihm haben maßgebend zum Erfolg dieser Promotion beigetragen.

Fur die Unterstutzung bei meinen Bewerbungen fur eine Postdoktorandenstelle, in Form vonmuhevoll geschriebenen Gutachten und wichtigen Tips, mochte ich mich bei Kristina Giesel,Jerzy Lewandowski und Hanno Sahlmann bedanken. An dieser Stelle mochte ich auch nocheinmal meinen Betreuer Thomas Thiemann hervorheben, dessen viele Hilfe und Tips bei denBewerbungen unerlasslich waren.

An dieser Stelle darf auch der “beschleunigte” Physikstudiengang “Physics Advanced” im Rah-men des Elitenetzwerks Bayern nicht unerwahnt bleiben, welcher der eigentliche Grund furmeine Entscheidung war, in Erlangen Physik zu studieren. Der fruher von Klaus Rith undheute von Klaus Mecke geleitete Studiengang hat mir ein zugiges und forschungsorientiertesStudium ermoglicht und ist in dieser Doktorarbeit gemundet. Mein Dank gilt allen fur diesenStudiengang engagierten Professoren und Studenten, welchen ich als wichtige Bereicherung furdie deutsche Hochschullandschaft sehe. Auch bei Frieder Lenz, der wahrend meiner Studienzeitals Mentor fungierte und mich an Thomas Thiemann empfohlen hat, mochte ich mich hier nocheinmal bedanken.

Ich bedanke mich weiterhin fur finanzielle und ideelle Unterstutzung wahrend meines Studiumsund meiner Doktorarbeit bei der Friedrich-Naumann-Stiftung, dem Max-Weber-Programm, demLeonardo-Kolleg der Universitat Erlangen-Nurnberg, e-Fellows, dem Elitenetzwerk Bayern, undder Studienstiftung des deutschen Volkes.

Abschließend mochte ich mich bei Emanuele Alesci, Enrique Fernandez Borja, Yuriy Davy-gora, Jonathan Engle, Christian Fitzner, Inaki Garay, Kristina Giesel, Muxin Han, SuzanneLanery, Karl-Hermann Neeb, Hanno Sahlmann, Alexander Stottmeister, Eckhard Strobel, Jo-hannes Tambornino, Derek Wise, Antonia Zipfel, und vielen anderen fur viele interessante undhilfreiche Diskussionen uber Physik bedanken.

136

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