Loop quantisation of supergravity theories

Schleifenquantisierung von Supergravitationstheorien

Der Naturwissenschaftlichen Fakult¨at der Friedrich-Alexander-Universit¨atErlangen-N¨urnberg zur Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von Norbert Bodendorfer

aus Waldbr¨ol Als Dissertation genehmigt von der Naturwissen- schaftlichen Fakult¨atder Friedrich-Alexander Universit¨at Erlangen-N¨urnberg

Tag der m¨undlichen Pr¨ufung: 30. April 2013

Vorsitzende/r der Promotionskommission: 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 15 3.1 The Ashtekar-Barbero variables ...... 15 3.2 Canonical analysis of the higher-dimensional Palatini action ...... 16 3.3 Gauge unfixing ...... 20 3.3.1 Toy model ...... 20 3.3.2 General theory ...... 21 3.3.3 Application to gravity ...... 23

4 Canonical transformation 24 4.1 Phase space extension and canonical transformation ...... 24 4.2 The Hamiltonian constraint in the new variables ...... 28

5 The Linear simplicity constraint 30 5.1 Introduction ...... 30 5.2 Introducing linear simplicity constraints ...... 31

II Loop quantum gravity in higher dimensions 34

6 LQG-techniques in higher dimensions 35 6.1 Kinematical quantisation techniques ...... 35 6.1.1 Holonomies, fluxes, and right invariant vector fields ...... 35 6.1.2 Solution of the Gauß and spatial diffeomorphism constraints ...... 37 6.2 Regularisation of the Hamiltonian constraint operator ...... 38

1 6.2.1 Volume operator ...... 38 6.2.2 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 45 8.1 Introduction ...... 45 8.2 The quadratic simplicity constraint operators ...... 47 8.2.1 A maximal closing subset of vertex constraints ...... 47 8.2.2 The solution space of the maximal closing subset ...... 49 8.2.3 Remarks ...... 50 8.3 The linear simplicity constraint operators ...... 53 8.3.1 Regularisation and anomaly freedom ...... 53 8.3.2 Solution on the vertices ...... 55 8.3.3 Edge constraints ...... 56 8.3.4 Remarks ...... 57 8.3.5 Mixed quantisation ...... 60 8.4 Comparison to existing approaches ...... 61 8.4.1 Continuum vs. discrete starting point ...... 62 8.4.2 Projected spin networks ...... 62 8.4.3 EPRL model ...... 64 8.5 Discussion and conclusions ...... 65

III Extensions to supergravity 67

9 Standard matter 68

10 Rarita-Schwinger field 70 10.1 Introduction ...... 70 10.2 Review of canonical supergravity ...... 72 10.2.1 Status of canonical supergravity ...... 72 10.2.2 Canonical supergravity in the time gauge ...... 73 10.3 Phase space extension ...... 75 10.3.1 Symplectic structure in the SO(D) theory ...... 75 10.3.2 SO(D + 1) gauge supergravity theory ...... 79 10.4 Background independent Hilbert space representations for Majorana fermions . . 86 10.5 Generalisations to different multiplets ...... 90 10.5.1 Majorana spin 1/2 fermions ...... 90 10.5.2 Mostly plus / mostly minus conventions ...... 91 10.5.3 Weyl fermions ...... 91 10.6 Conclusions ...... 92

11 p-form gauge fields 94 11.1 Introduction ...... 94 11.2 Classical Hamiltonian analysis of the 3-index-photon action ...... 95 11.3 Reduced phase space quantisation ...... 99 11.4 Conclusions ...... 104

2 IV Initial value quantisation of higher-dimensional general relativity 105

12 Loop quantum gravity without the Hamiltonian constraint 106 12.1 Introduction ...... 106 12.2 Classical analysis ...... 108 12.3 Quantisation ...... 111 12.4 Geometric operators ...... 111 12.5 Application to black hole entropy ...... 112 12.6 Concluding remarks ...... 112

V Isolated horizon boundaries in higher-dimensional LQG 114

13 Classical phase space description of isolated horizon boundaries 115 13.1 Introduction ...... 115 13.2 General strategy ...... 116 13.3 Higher-dimensional isolated horizons and Lagrangian framework ...... 118 13.4 SO(D + 1) as internal gauge group ...... 120 13.5 Inclusion of distortion ...... 122 13.5.1 Beetle-Engle method ...... 122 13.5.2 Perez-Pranzetti method ...... 123 13.6 Comments on quantisation ...... 124 13.7 Concluding remarks ...... 125

VI Conclusion 127

14 Concluding remarks and further research 128 14.1 Summary of the results ...... 128 14.2 Towards loop quantum supergravity: Where do we stand? ...... 129 14.3 Further research ...... 130

A Simple irreps of SO(D + 1) and square integrable functions on the sphere SD 134

Danksagung (german acknowledgements) 136

Bibliography 137

3 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 the official starting date is December 2009, when I obtained my physics diploma. The topic of this thesis 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 conducted at the Albert-Einstein-Institute in Potsdam, which I thank for hospitality. Also during my diploma thesis, I have been supported by the Friedrich-Naumann-Foundation, the Max-Weber- Programme of Bavaria, the Leonardo-Kolleg of the University of Erlangen-N¨urnberg, the Elite Network Bavaria, and e-Fellows. My undergraduate and doctoral studies have been conducted in the Physics Advanced programme of the Universities Erlangen-N¨urnberg and Regensburg.

The research which lead to this thesis has been performed at the Institute for Theoretical Physics III of the University of Erlangen-N¨urnberg. I strongly acknowledge financial and ideational support of the German National Merit Foundation. Also, I acknowledge support of the Elite Network Bavaria and e-Fellows.

4 Abstract

The aim of this thesis is to develop loop quantisation techniques for higher-dimensional supergravities in order to make progress towards comparing loop quantum gravity to superstring theory. The abstract idea for this comparison is to look at loop quantum gravity and string theory as two methods of quantising an underlying gravity theory. While loop quantum gravity circumvents the usual problems associated with standard Fock-type quantisations of gravity theories 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 distance divergences arising in perturbative quantum gravity. The most obvious problem with comparing these two approaches to quantum gravity is that while it was only known how to formulate loop quantum gravity in four (or three) dimensions, including all types of standard model matter couplings, string theory requires a ten-dimensional spacetime with supersymmetry. The possible reductions of string theory to four dimensions on the other hand are highly non-unique, thus it is very hard to compare these approaches at this level. A possible solution to this problem, which is investigated in this thesis, is to extend the techniques of loop quantum gravity to higher-dimensional supergravities, which arise as the low energy limits of string theories. These supergravities have the same low energy particle content and live in the same spacetime dimension as the corresponding string theories and are thus suited to circumvent the above problem. Since this solution does not seem to be overly sophisticated at a first glance, it is important to remark that, prior to recent work on which this thesis is based, it was 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 available in four dimensions. In this thesis, we will first develop a generalisation of this type of connection formulation for higher-dimensional general relativity. We will then show that the quantisation techniques from loop quantum gravity can be mainly carried over to this new formulation, while some new ingredients have to be introduced. Next, we are going to extend this formulation to higher-dimensional supergravities, thus providing a solution for the above problem. On a different route, we provide a partially reduced phase space quantisation of higher-dimensional general relativity conformally coupled to a scalar field based on the constant mean curvature gauge in the fourth part of this thesis. Finally, in the fifth part of this thesis, we will treat isolated horizons as boundaries of the spacetime manifold in order to take first steps to compare string theory and higher-dimensional loop quantum gravity in the presence of a black hole.

5 Zusammenfassung

Es ist das Ziel dieser Arbeit, Methoden zu entwickeln, die die Quantisierung von Super- gravitationstheorien mit Hilfe von Techniken aus der Schleifenquantengravitation erm¨oglichen, um Fortschritte beim Vergleichen der Schleifenquantengravitation mit der Superstringtheorie zu machen. Die abstrakte Idee dieses Vergleichs ist es, sowohl die Schleifenquantengravitation als auch die Superstringtheorie als Quantisierungen einer zugrundeliegenden Gravitationstheorie zu verstehen. W¨ahrenddie Schleifenquantengravitation die ¨ublichen Probleme bei Fock-Typ Quan- tisierungen von Gravitationstheorien vermeidet, indem sie von einer anderen Poisson-*-Algebra von Phasenraumfunktionen startet, postuliert die Superstringtheorie den String als neuen fun- damentalen Baustein um den UV-Divergenzen in der perturbativen Quantengravitation Herr zu werden, welche durch die Betrachtungen von beliebig kleinen Distanzen entstehen. Das offensichtlichste Problem, welches beim Vergleichen dieser beiden Theorien entsteht, ist dass w¨ahrendnur bekannt war, wie man Schleifenquantengravitation in vier (oder drei) Dimensionen formuliert und die Materiefelder des Standardmodells behandelt werden konnten, aber nicht notwendig waren, die Superstringtheorie eine zehndimensionale Raumzeit mit Supersymmetrie braucht. Andererseits sind m¨ogliche Reduktionen der Superstringtheorie auf vier Dimensionen hochgradig nicht-eindeutig und daher ist es sehr schwierig diese Reduktionen mit der Schleifen- quantengravitation zu vergleichen. Eine m¨ogliche L¨osung dieses Problems, welche in dieser Arbeit untersucht wird, ist es die Techniken der Schleifenquantengravitation auf h¨oherdimensionaleSupergravitationen 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 das oben beschriebene Problem zu umgehen. Da dieser L¨osungsvorschlag auf den ersten Blick nicht sehr tiefsinnig erscheint, muss angemerkt werden, dass es bis vor einigen k¨urzlich er- schienenen Publikationen, welche zu dieser Arbeit gef¨uhrthaben, nicht einmal bekannt war, wie und ob man h¨oherdimensionaleGravitation mit den Methoden der Schleifenquantengravi- tation behandeln k¨onnte. Das Problem an dieser Stelle war, dass der Hauptinput in der klas- sischen Theorie, die Ashtekar-Barbero-Variablen, nur in vier Dimensionen verf¨ugbarsind. In dieser Arbeit wird erst eine Verallgemeinerung dieser Art von Zusammenhangsformulierung auf h¨oherdimensionaleallgemeine Relativit¨atstheorieentwickelt werden. Danach wird gezeigt wer- den, dass die meisten Quantisierungstechniken der Schleifenquantengravitation auf diese neue Formulierung angewendet werden k¨onnen,aber auch ein paar neue Techniken gebraucht werden. Als n¨achstes werden diese Techniken noch auf Supergravitationen erweitert werden, und damit ein L¨osungsvorschlag f¨urdas obige Problem pr¨asentiert werden. Abseits dieses Hauptthemas wird eine partiell reduzierte Phasenraumquantisierung von h¨oherdimensionaler allgemeiner Rela- tivit¨atstheoriemit einem konform gekoppeltem Skalarfeld, basierend auf der Konstante-mittlere- Kr¨ummungs-Eichung, im vierten Teil dieser Arbeit vorgestellt werden. Abschließend werden im f¨unftenTeil dieser Arbeit Raumzeiten mit isolierten Horizonten als R¨andernbetrachtet, um erste Schritte hin zu einem Vergleich der Superstringtheorie mit der h¨oherdimensionalenSchleifen- quantengravitation in Raumzeiten mit schwarzen L¨ochern zu machen.

6 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) a ei SO(D) D-bein (0.11) a Ei densitised SO(D) D-bein (0.12) a eI SO(1,D) or SO(D + 1) (hybrid)-D-bein (0.13) a EI 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 )† = ηII γI (no summation here) (0.19) γIJ...K := γ[I γJ ...γK] (with total weight one) (0.20) ⊥ µ γ := nµγ (0.21) i ΣIJ := − γIJ (SO(D + 1) generators in spinor representation) (0.22) 2 i ∇ (A) χ := ∂ χ + A ΣIJ χ (0.23) a a 2 aIJ i ΣIJ , ΣKL = ηLJ ΣKI − ηLI ΣKJ − ηKJ ΣLI + ηKI ΣLJ (0.24) χ¯ := χ†γ0 (0.25) χ¯ = χT C (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.

7 Chapter 1

Introduction

The main topic of this thesis is to establish a connection between two different approaches to quantum gravity, namely the string theory1 / M-theory / supergravity side and loop quantum gravity (LQG). The motivation for this kind of research stems from the observation that although important 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 of experiments which could discriminate among these different theories, this situation would not be very problematic, however, since such experiments are missing today, the present quantum gravity research has to be conducted on purely theoretical grounds. Accordingly, if one expects to be successful in constructing quantum gravity by purely theoretical means, the theory should be reasonably unique. Focussing on string theory and loop quantum gravity, the aim of this thesis is to establish a connection between these two theories by applying LQG methods to supergravity 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 treated under the aspect of providing quantum theories with the supergravity theories in ten and eleven dimensions as their low energy limits. The arguments which lead one to the conclusion that supergravities are indeed the low energy limits of superstring and M-theory are mainly due to symmetry principles. As an example, Werner Nahm has classified all possible supergravity theories which do not lead to fields of spin greater than 2 upon dimensional reduction to four dimension [6]. Different superstring theories on the other hand yield the same massless particle spectrum as the corresponding supergravity theories in ten dimensions, from which one concludes by invoking Nahm’s results that dynamics of the theories also have to agree for the massless low energy limit of the superstring theories. Since the supergravity actions consist basically of the gravity action in higher dimension plus matter couplings, it seems straight forward to try to apply loop quantum gravity techniques to this system in order to compare this quantisation with the one of superstring and M-theory. While the quantisation of many higher-dimensional supergravity theories will be achieved in this thesis, the comparison with string theory will be left for further research. On the one hand, one could compare the results and explicit derivations of certain symmetry reduced situations like the black hole entropy calculations or cosmology. On the other hand, a more direct route would be to gain a better understanding of the quantum field theory on curved spacetime limit of loop quantum gravity, which would enable one to compare it to superstring theory also in the realm of scattering theory. Since, as said, the explicit form of superstring theory is not of importance for this thesis, we will not comment on it further. An introduction to supergravity theories and their canonical analysis is provided in the third part of 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-dimensional bosonic string theories in this thesis.

8 quantum gravity programme and comment on where the main obstacles on the way towards a loop quantisation of higher-dimensional supergravities lie. The starting point of the loop quantum gravity research programme was the seminal paper by Ashtekar [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 complex and results in complicated reality conditions which, up till now, could not be implemented on a 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, which promised to be an important step in making mathematical sense out of the Wheeler-deWitt equation [8, 9, 10], one of the drawbacks of which is that it is non-polynomial in the canonical variables and thus very hard to define as an operator equation. Since progress with the reality conditions could not be achieved, Barbero proposed a real version of Ashtekar’s connection variables [11], nowadays known as the Ashtekar-Barbero connection. This connection is currently favoured for constructing loop quantum gravity, since its reality conditions are automatically implemented on the Ashtekar-Lewandowski Hilbert space of loop quantum gravity. Although the Hamiltonian becomes non-polynomial in the Ashtekar-Barbero variables, Thiemann was able to 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 papers Ashtekar, Isham, Lewandowski, Marolf, Mour˜ao,and Thiemann [14, 15, 16, 17, 18, 19]. The key input of this construction was to make use of projective limits in order to be able to work with spin networks which are supported on finite graphs while maintaining invariance under arbitrary refinements of the graph. The compactness of the gauge group SU(2) used for the Asthekar-Barbero variables is the main input into this construction, necessary for constructing a measure on an infinitely refined graph. Moreover, the construction of the kinematical Hilbert space has been performed independently of the number of spatial dimensions and the compact gauge group used. Thus, the problem of kinematically quantising general relativity in higher dimensions had already been reduced to finding a connection formulation of general relativity in higher dimensions based on a compact gauge group. Matter coupling can be done rather naturally in the context of loop quantum gravity and several results are available. The technical framework for matter coupled LQG has been derived in [20, 21] and it has has been shown how to regularise the Hamiltonian constraint operator on the corresponding Hilbert space. However, due to the well known problem of time, it is difficult to interpret solutions to the Hamiltonian constraint operator and deparametrised models were developed [22, 23, 24, 25] in order to have a physical Hamiltonian encoding the “time” evolution with 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 in the 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 become physical coherent states [22]. Thus, a framework capable to complete the quantisation of general relativity coupled to interacting 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 are several open issues to deal with. Most prominently, the Hamiltonian constraint or the derived true Hamiltonians suffer from different quantisation ambiguities, the physical consequences of which are not very well understood. Since the action of a (true or constrained) Hamiltonian operator 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

9 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 removed is to construct a connection formulation of higher-dimensional general relativity with certain important properties, which will be the subject of the first part of the thesis. Following, in part two, we will extend the quantisation techniques developed for the four-dimensional case to higher dimensions. In a last step in part three, we extend the quantisation procedure for pure higher-dimensional general relativity to the matter fields appearing in higher-dimensional supergravities. In part four, we will further develop the quantisation of higher-dimensional general relativity (without supersymmetry) by constructing a partially reduced phase space quantisation with a geometric clock. In part five, we will further develop the classical phase space description in the presence of an isolated horizon boundary which is necessary for the derivation of black hole entropy within LQG. A more detailed overview of the research presented in this thesis is found in the next chapter.

10 Chapter 2

Overview of the results

In a series of papers [35, 36, 37, 38, 39, 40, 41], Thomas Thiemann, Andreas Thurn, and the author 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 is compact, 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-perturbative and 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 be briefly 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 discussed below. 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 was shown to obtain a boundary contribution of the higher-dimensional Chern-Simons type.

Part I: New canonical variables In [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 the canonical transformation which relates this new formulation to the ADM formulation [46] is the simplicity constraint familiar from spin foam models, which relates a “generalised” vielbein transforming 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. The resulting 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 a key input in the construction and it can be incorporated either in the quadratic or in the linear form. 1From the compact gauge group, construct the Ashtekar-Lewandowski measure, which is a positive linear functional on the holonomy-flux algebra based on the Yang-Mills connection and its conjugate momentum. The Ashtekar-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 Poisson bracket tricks.

11 Part II: Quantum theory A loop quantisation of higher-dimensional general relativity using the new canonical variables was spelled out in [37]. The existing quantisation techniques from the literature were either already 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 regularisation of the Hamiltonian constraint operator. Also, a quantisation of the simplicity constraint was performed 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 supergravity While 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 problem could be circumvented by acting with SO(d) on the fermionic representation space in such a way that the theory reduced to general relativity coupled to fermions in a certain gauge [38], the use of Majorana fermions in supergravity theories required more work since Majorana conditions are sensitive to the signature of spacetime. The solution proposed in [40] is the usage of a compound object of fermionic and gravitational degrees of freedom which obeys the Lorentzian Majorana condition 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 which appear in ten and eleven dimensional supergravities. Due to the presence of a Chern-Simons term for the 3-index photon, which is due to local supersymmetry, the theory is self-interacting and the application of a direct generalisation of the LQG Hilbert space to Abelian p-forms was not 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 quantisation In [43, 44], a reduced phase space quantisation of general relativity conformally coupled to a scalar 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 rescaling of 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 the second class partner D of the Hamiltonian constraint H, i.e. D is a gauge fixing for H. It turns out that, when coupling a conformally coupled scalar field (as opposed to a minimally coupled scalar field), D can be chosen to be the generator of local rescalings for both, the gravitational and the scalar field variables. Due to this fact, a new metric and conjugate momentum, which are invariant under the local rescaling, can be constructed as combined objects of the scalar field and the original metric and momentum. It follows that the Dirac bracket associated with gauge fixing the Hamiltonian constraint with the generator of local rescalings agrees, when using the 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 techniques without having to worry about the Hamiltonian constraint. The downside of this formulation are difficulties when trying to implement a time evolution, i.e. a true Hamiltonian with respect to 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 is globally 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 very different from the ones in the deparametrised models, since it consists of momenta instead of configuration space variables. Moreover, it is purely geometric.

Part V: Isolated horizon boundary degrees of freedom An immediate application of the results of the first three parts of this thesis is the calculation of black hole entropy in higher dimensions within the LQG framework. First steps towards this goal have been taken in [45], where the classical phase space description of the proposed connection 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 condition similar to the one known from the four-dimensional treatment can be derived. Furthermore, the symplectic structure obtains a boundary term which was shown to result in a higher-dimensional Chern-Simons symplectic structure. This boundary term was already familiar from [35], where it 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), as opposed to SO(1,D) in the treatment starting with the Palatini action.

13 Part I

Connection dynamics for classical higher-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 an understanding of its canonical structure. As in the four-dimensional case, the theory will turn out 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 or a 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 contrasts our formulation from the derivation of the Ashtekar-Barbero variables, which can be derived by supplementing the Palatini action with the Holst modification, which is however only available in 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]. Of course, one expects that a corresponding derivation at the Lagrangian level should exist and the corresponding derivation was given by Holst [51]. In order to arrive at the Ashtekar-Barbero variables, it is important to start with a modified version of the Palatini action, it has to be supplemented by the Holst modification as Z Z s 4 µI νJ 1 s 4 µI νJ KL SHolst = d X ee e FµνIJ (A) + d X ee e Fµν (A)IJKL , (3.1.1) 2 M 2γ 2 M | {z } | {z } Palatini action Holst modification 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 can be 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 a torsion term, which vanishes by the equations of motion. Thus, the classical equivalence to the Palatini 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 a canonical 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 canonical analysis of the higher-dimensional Palatini action. For now, we will just cite the main results in order to understand the main ideas. Also, we will only comment the canonical analysis with imposing the time gauge, as was originally done by Holst [51]. In Holst’s analysis, the second class constraints mentioned below do not appear and he simply obtains a first class Hamiltonian theory with Hamiltonian constraint, spatial diffeomorphism constraint, and Gauß constraint in terms of the Ashtekar-Barbero variables. The canonical analysis of the Holst action without the time gauge was first performed by Barros 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 the fundamental representation. Moreover, two second class constraints appear, the simplicity con- straint Sab and a second class partner Dab. The simplicity constraint essentially enforces that the conjugate momentum to the Palatini connection is constructed from a vielbein, thus killing superfluous degrees of freedom. Its second class Partner Dab is a consequence of the stability analysis and essentially enforces that a certain part of the torsion of the Palatini connection is zero. 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 quantum gravity quantisation methods. On the other hand, when solving the second class constraints, one obtains the Ashtekar-Barbero formulation of general relativity and thus the goal which we were aiming 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-formulation with vierbeins [47], a theory with SO(1, 3) gauge invariance, but without a connection, as noted by Peldan [47]. This problem also carries over to higher dimensions, as we will show later. It was already observed by Han, Ma, Ding, and Qin [56] that when imposing the time gauge prior to performing the canonical analysis of the Palatini action, one ends up with the ADM formulation with SO(D)-vielbeins also in higher dimensions. Moreover, adding a generalisation of the Holst term is difficult in higher dimensions. On the one hand, a naive extension of the Holst-modification does not seem to exist in higher dimensions, since the epsilon symbol used to construct it acquires additional indices for which a non-vanishing and gauge invariant contraction linear in the field strength could not be given so far. On the other hand, a generalisation of the Nieh-Yan topological density exists in 4n, n ∈ N , dimensions, but it contains higher powers of the field strength and would thus significantly complicate the canonical analysis and most likely not yield the 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 proceed by trying to gain a better understanding for the canonical structure of the Palatini action by performing a detailed canonical analysis. The starting point for this analysis is the higher- dimensional Palatini action Z s D+1 µI νJ SPalatini = d X ee e FµνIJ (A), (3.2.1) 2 M

16 µI µI where e is an SO(1,D) vielbein, e is the determinant of e , FµνIJ is the curvature of the SO(1,D) connection AaIJ , D is the number of spatial dimensions, and s = ±1 is the signature of the spacetime and we are using the mostly plus signature convention. We assume our spacetime M 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 a time and labels the Cauchy surfaces. Furthermore, we define the time evolution vector field T µ by LT t = 1, where L denotes the Lie derivative and split T according to

µ µ µ µ T = Nn + N , nµN = 0, (3.2.2)

µ where n is the unit normal on Σt. As in the usual treatments, we call N the lapse function and N µ the shift vector. Performing the D + 1 split, we obtain

Z Z Z 1 1  S = dt L = dt dDx π0aIJ L A − NH0 − N aH0 − λ G0IJ , (3.2.3) 2 T aIJ a 2 IJ σ e where all spacetime tensor indices are pulled back to σ (µ, ν, . . . → a, b, . . .) and

0aIJ [I k a|J] √ [I k a|J] I I µ √ π := 2n E := 2 qn e , n = eµn ,N := −N/ q , (3.2.4) e

0IJ 0aIJ 0aIJ 0a IJ G := Daπ := ∂aπ + [Aa, π ] , (3.2.5) 1 1 H0 := π0aIK π0bJ F and H0 := π0bIJ F . (3.2.6) 2 K abIJ a 2 abIJ 0 0 As we will see explicitly later on, H is the Hamiltonian constraint, Ha is the spatial diffeo- morphism constraint and G0IJ is the Gauß constraint. The primes appearing in the previous equations are to distinguish these constraints from constraints build purely in terms of AaIJ and its momentum, as we will do later on. Essentially, when performing the Legendre transform at this point, we will obtain the constraint

SaIJ = πaIJ − π0aIJ , (3.2.7)

aIJ where π is the momentum conjugate to AaIJ . Thus, our theory will have two sets of canon- aIJ a I ically conjugate pairs, (AaIJ , π ) and (EI ,Pa ), 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 canonical analysis including both pairs has been performed in [36] and we will not repeat it here, as it is not necessary for understanding the following. To prevent the second canonical pair from appearing, Peldan suggested [47], in close analogy to 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 simplicity constraint. The generalisation to higher dimensions of this constraint reads Z ab h M i D 1 M aIJ bKL SM cab = d x cab IJKLM π π ≈ 0, (3.2.8) σ 4 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 to constraints. In the context of spin foam models, this constraint has been considered in higher dimensions 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 proof 1 aIJ b ab over to the canonical setting and we obtain that if the constraint is zero and sπ πIJ ≈ 2qq is positive definite2, then a √ a πIJ = ±2 qn[I eJ]. (3.2.9) A complication arises in D = 3, where an additional topological sector appears. We will however neglect this sector in the following, since we are mainly interested in D > 3, where this sector does h i not exist. Adding Sab cM to (3.2.3), we can substitute all π0aIJ for πaIJ and thus completely M ab a free the action of any dependence on EI . Thus, the Legendre transform will yield only the aIJ canonical pair (AaIJ , π ), while by introducing the additional simplicity constraint we have ensured equivalence with general relativity. Beginning the Legendre transform, we read off the symplectic structure

bKL b K L {AaIJ , π } = 2δaδ[I δJ] (3.2.10) and the constraints 1 H := πaIK πbJ F (3.2.11) 2 K abIJ 1 H := πbIJ F (3.2.12) a 2 abIJ IJ aIJ aIJ a IJ G := Daπ := ∂aπ + [Aa, π ] (3.2.13) 1 Sab := cM  πaIJ πbKL. (3.2.14) M 4 ab IJKLM Applying the Dirac stability algorithm, we find one new constraint

ab cIJ  (a|KN b)L  DM = −IJKLM π π Dcπ N . (3.2.15) by calculating n ab M o ab h M i ab SM [cab ], H[N] = DM Ncab + SM [...]. (3.2.16) The relevant part of the remaining constraint algebra is given by

1 1  1 GIJ [f ], GKL[γ ] = GIJ λ γK − γ λK  (3.2.17) 2 IJ 2 KL 2 IK J IK J 1  GIJ [f ], H [N a] = 0, (3.2.18) 2 IJ a 1  GIJ [f ], H[N] = 0 (3.2.19) 2 IJ "D−3 # 1  0 IJ ab M ab X Mi M1...Mi−1Mi Mi+1...MD−3 G [λIJ ],S [c ] = S λ M 0 c (3.2.20) 2 M ab M i ab i=1 "D−3 # 1  0 IJ ab M ab X Mi M1...Mi−1Mi Mi+1...MD−3 G [λIJ ],D [d ] = D λ M 0 d (3.2.21) 2 M ab M i ab i=1

1At least at the classical level, the quantum constraint is more subtle as discussed in the second part of this thesis 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 n o 1 H [f a], H [N b] = H [(L N)a] − GIJ [f aN bF ] (3.2.22) a b a f 2 abIJ a IJ a b K {Ha[f ], H[N]} = H[Lf N] + G [Nf π I FabJK ] (3.2.23) n ˜ a ab M o ab h M i Ha[N ],SM [cab ] = −SM (LN c)ab (3.2.24) 1 h i {H[M], H[N]} = − H (M∂ N − N∂ M)πaIJ πb 2 a b b IJ 1 h i +s Sab (M∂ N − N∂ M) M πcIJ F KL . (3.2.25) 4 M a a IJKL cb It can furthermore be shown that Sab and Dab constitute a second class pair. However, many of M M these second class constraints are superfluous, as a count of the degrees of freedom they remove shows. In order to understand the structure of these constraints, we split ¯ ¯ AaIJ = ΓaIJ + KaIJ + 2n[I Ka|J], (3.2.26) where ΓaIJ is Peldan’s hybrid connection [47], see equation (4.1.11) for more details. The K¯ I notation indicates that all internal indices are orthogonal to nI , i.e. K¯aIJ n = 0. On the constraint surface of the simplicity constraint, we calculate

¯ IJKLM ab ¯ IJKLM cAB (a|C b)DE f(a|IJ π|b)KL DM = −f(a|IJ π|b)KL ABCDM π π EDcπ aIJ,bKL ¯ ≈ −(D − 3)!(D − 1)K¯aIJ F fbKL, (3.2.27) and ab D A cB (a|C b)E ¯ DM ≈ −2sABC M n E E E KcED, (3.2.28) where F aIJ,bKL = 4sEa[K η¯L][J Eb|I] (3.2.29) and its inverse s   F −1
= E E η¯ABη¯ η¯ − 2¯ηBη¯ η¯A . (3.2.30) aIJ,bKL 4 aA bB K[I J]L [I J][K L] are defined on tensors of the type f¯bKL which are antisymmetric in K,L, orthogonal on nI , and ¯bKL −1 aIJ tracefree in the sense f EbK = 0. πaIJ is defined as q qabπ , with qab being the inverse of qab, and thus purely as a function of πaIJ . We conclude that using the “reduced” multiplier dM = f¯ π IJKLM of the above tensor type is enough to fulfill the constraint Dab = 0. ab (a|IJ |b)KL M Using this ansatz, we further calculate Z Z D D ¯T IJKLM n ab cd o T MNOP N d x d y [f(a|IJ πb)KL ](x) SM (x),DN (y) [¯g(c|MN πd)OP  ](y) Z 2 2 D ¯T aIJ,bKL T ≈ 4(D − 1) ((D − 3)!) d x faIJ F g¯bKL. (3.2.31) and conclude that Sab and Dab constitute indeed a second class pair. The remaining Poisson M M brackets are not important, since we can ensure stability of Dab by adjusting the multiplier of M aI the simplicity constraint. It can furthermore be shown that the trace part K¯aIJ E , vanishes by the Gauß constraint [36], which shows that the all physical information about the extrinsic curvature has to be located in K¯aI . IJ Concluding, we found a constraint algebra with the first class subset H, Ha, and G as well as the second class constraints Sab and Dab . Since the quantisation of second class constraints M M is problematic, we will apply the algorithm of gauge unfixing in the next subsection in order to obtain 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 first and second class constrained systems to each other. Given a first class constraint, we can always perform a gauge fixing by introducing a new constraint which does not Poisson commute with the original constraint. This new constraint then fixes a gauge in the sense that the previously first class 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 introduced constraint. The idea of gauge unfixing is to relate the process of fixing a gauge by setting a certain phase space variable to zero, i.e. using second class constraints, with the process of cutting out all the dependence of the theory on this variable, i.e. using gauge invariant observables, leading to a first class system where the Hamiltonian and the observables are gauge invariant under the new first class constraints. As an elementary example, we consider the system with configuration variables x1, x2, i i momenta P1, P2, Poisson bracket {x ,Pj} = δj and the first class constraint S = P2 ≈ 0. A 1 2 phase space function f(x , x ,P1,P2) of this system is observable if and only if it is evaluated on the constraint surface S ≈ 0 and Poisson commutes with S. Poisson commutativity implies that 2 the phase space function does not depend on x . On the other hand, dependence on P2 = 0 is equivalent with non-dependence on P2 since we can Taylor-expand the phase space function into a series in P2 where only the constant term would survive. The argument also works when the 1 1 1 constraint says P2 = g(x ,P1), since every function f(x ,P1,P2 = g(x ,P1)) is already included 1 in the functions depending only on x and P1. The constraint surface of the second class theory, where a gauge fixing constraint is present, is 2-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 have therefore effectively the same size. It remains to be shown that the dynamics of the two theories are the same. Starting with the first class theory, the most general first class Hamiltonian reads

1 H = A(x ,P1) + λS (3.3.1) where A represents the gauge invariant part and λ is the Lagrange multiplier for the first class 1 2 constraint S. Observables are phase space functions f(x ,P1). Clearly, imposing D = x ≈ 1 h(x ,P1) does not alter the observables or the Hamiltonian except for setting λ = {h, A} to ensure stability of D. But the observables are independent of λ so that the gauge fixing does not alter the gauge invariant information of the system. The connection with the gauge variant system S ≈ 0 is made by considering the observable 2 part of the system. In the first class system we saw that S renders both x and P2 unphysical, 2 P2 by explicitly fixing P2 = 0 and x by modding out gauge orbits. In the second class system, 2 1 both x and P2 are explicitly set to functions of x and P1. Therefore, the effective phase space and the dynamics of observables are the same when going from the gauge variant to the gauge invariant 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 want to illustrate the idea in a toy model related to gravity. We start with the Hamiltonian system described by the previous phase space, the Hamiltonian 1 H = A(x1,P ) + (x2)2 + λS, (3.3.2) 1 2

20 2 and the constraint S = P2 ≈ 0. Stability of S yields the new constraint D = x ≈ 0. Stability of D sets λ = 0 and we have the consistent Hamiltonian system 1 H = A(x1,P ) + (x2)2, S ≈ D ≈ 0. (3.3.3) 1 2 We note that the Hamiltonian is only first class up to the second class constraint D when 1 2 Poisson commuted with S. We can change this by subtracting 2 D from the Hamiltonian since the square of a second class constraint is a first class constraint. We call this new Hamiltonian

1 H˜ = A(x ,P1) + λS (3.3.4) and note that it Poisson commutes with S independently of D. The idea is now that it shoud not matter to forget about the constraint D since observables of the first class theory S = 0 do not 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 on x2. 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˜ . On the other hand, we can gauge fix the first class system with D ≈ 0 and go back to the initial second class system.

3.3.2 General theory After a seminal paper by Mitra and Rajaraman [63], the general theory of gauge unfixing was developed by Anishetty and Vytheeswaran [64, 65]. The main idea is to reverse the process of gauge fixing, which turns a first class system into a second class system by making a gauge choice. Neglecting complications coming from Gribov copies, gauge fixing consists of choosing a second class partner to a first class constraint which specifies a point in the gauge orbit of the first class constraint. We do not claim credit for the following, but try to give a pedagogical overview 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 with i i i generalised coordinates x , pi, Hamiltonian H(x , pi), subject to the two constraints S(x , pi) ≈ 0 i i and D(x , pi) ≈ 0 with Poisson bracket {S, D} = F (x , pi) 6= 0. We are now going to forget about D. What does this mean for the Hamiltonian system? Detection of Dirac observables: i In the second class system, a Dirac observable is a function f(x , pi) defined on the phase space surface satisfying S ≈ D ≈ 0. In the first class system, the function f is defined on the surface S ≈ 0 and has to satisfy {f, S} ≈ 0, where ≈ means weakly with respect to S only. At any given point P in phase space, we can, at least locally, use S and D as coordinates on the phase space so that D parametrises the gauge orbit of S. We can then expand f into a Taylor series in D as i 2 f(x , pi) = f0 + f1D + f2D + ... , (3.3.5) where the coefficients fn, n = 0,..., ∞ do not depend on D. We then have {fn,S} ≈ 0 and {f, S} ≈ 0 is equivalent to fn = 0 for all n ≥ 1. Thus, {f, S} ≈ 0 forces f not to depend on D. Accordingly, it does not matter if f is defined on the surface S ≈ 0 or S ≈ D ≈ 0. In other words, the set of functions defined on S ≈ D ≈ 0 is exactly the same set as the set of functions defined 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 i phase space functions f(x , pi) acting as 1 f˜ := Pf := f − D{F −1S, f} + D2{F −1S, {F −1S, f}} 2! 1 − D3{F −1S, {F −1S, {F −1S, f}}} + ... . (3.3.6) 3! 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, g}DB) 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 follows that 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 H might depend on D which would lead away from the surface S ≈ 0. We use the above projector to construct the gauge invariant Hamiltonian H˜ which ensures stability of S. Time evolution in the second class system using the Dirac bracket is equivalent to this treatment since the Dirac bracket also annihilates the second class constraints present in the Hamiltonian. A gauge invariant function has therefore the same time evolution using either H and the Dirac bracket or H˜ and the Poisson bracket. Global issues and uniqueness: Problems concerning uniqueness and global treatment might arise when D is not linear in i the coordinates it constrains, leading to ambiguities in the solution to D(x , pi) = 0. Since this will 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, maybe infinite, series, which does not seem to help. Surprisingly, this series contributes only one extra 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 of freedom is the same task as looking for a representation of the Dirac bracket. A rigorous quantisation 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 = 0 with Hamiltonian H˜ and the set of gauge variant functions is equipped with the original Poisson bracket, we can quantise this system and implement the constraint S at the level of the kinematical Hilbert space as an operator equation. Of course, the task of finding observables 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 set of 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 in the previous section bears many problems at first sight, including: Why should the power series involved 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 due to non-linearities in the gauge fixing condition? However, as it turns out, gauge unfixing works very well for general relativity due to the following special properties: First of all, the simplicity constraints constitute a natural choice for a first class subset, since they are Abelian among themselves. Furthermore, the gauge fixing condition Dab is then linear in the degrees of freedom it constrains, i.e. the trace free part of M K¯ vanishes. Also, the Dirac matrix will only depend on πaIJ , since Dab is at most linear in aIJ M AaIJ . On top of that, the Hamiltonian constraint is a second order polynomial in AaIJ , which means that the power series involved in calculating its gauge invariant extension terminates after the second term. A little subtlety at this point is that the Dirac matrix

ab cd ab cd {SM ,DN } = FM N (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 1 H˜ = H − Dab F −1
M N Dcd (3.3.11) 2 M ab cd N 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

−1
N M EF GHN −1
ABCDM F cd,ab = γ π(c|EF F d)GH,(a|AB πb)CD (3.3.12) for a free constant γ, where s   F −1
:= π π πcEC π D − sηCD
ηABηK[I ηJ]L − 2ηLAηB[I ηJ]K aIJ,bKL 4(D − 1) aAC bBD cE (3.3.13) is expressed purely in terms of πaIJ . The πs with a lower spatial tensorial index are understood as functions of πaIJ . This ansatz makes sense since the Dirac matrix is invertible on the subset of multipliers that we need to access the trace free part of K¯aIJ . This part on the other hand is exactly what we need to remove from the Hamiltonian constraint to make it Poisson commute with the simplicity constraint. After an explicit calculation, it turns out that 1 γ = (3.3.14) 4(D − 1)2((D − 3)!)2 solves the problem, i.e. H˜ does not depend on the trace free part of K¯aIJ on the constraint surface. The remaining constraints can be trivially gauge unfixed since they Poisson commute weakly with the simplicity constraint. Concluding, we have transformed the second class constrained system describing general relativity into an equivalent first class system, which is more suitable as a starting point for quantisation. However, regarding the loop quantum gravity quantisation techniques, the internal gauge group is still non-compact and the Hilbert space methods are therefore not available. In the next chapter, we will therefore generalise the canonical tranformation which leads from the ADM 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 directly by 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 very convenient since we can reuse results from earlier canonical treatments of general relativity and supergravity, on the other hand the canonical formulations which will turn out to be necessary to complete the quantisation of general relativity and supergravity in higher dimensions do not seem to have an (at least manifestly covariant) analogue on the Lagrangian side. We will choose physical signature s and internal signature ζ independently in this section to emphasise that they do not need to agree in the canonical descriptions. For quantisation purposes of Lorentzian general relativity and supergravity, we will choose s = −1 and ζ = 1. The original work on which 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 to obtain a gauge theory and then performs a canonical transformation to connection variables, we extend the ADM phase space using the variables

aIJ π and KbKL (4.1.1) subject to the constraints 1 GIJ := 2K [I πaK|J] ≈ 0 and Sab :=  πaIJ πbKL ≈ 0, (4.1.2) a K M 4 IJKLM as well as the Poisson bracket

n bKLo b K L L K
n aIJ bKLo KaIJ , π = δa δI δJ − δI δJ , π , π = {KaIJ ,KbKL} = 0. (4.1.3)

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

s    1 qqab := πaIJ πb , q = det qqab D−1 and q qbc = δc. (4.1.4) 2 IJ ab a The algebra of the constraints (4.1.2) equals the algebra of Gauß and simplicity constraints in the previous sections and thus is of first class. What remains to be checked is that the Poisson

24 cd brackets between the ADM variables qab, P , considered as functions on the extended phase space, remain unchanged, at least if GIJ ≈ Sab ≈ 0. We express the extrinsic curvature using M the new variables as s Kab := − √ πbKLK qac (4.1.5) 4 q cKL and P ab as √   1   P ab = −s q Kab − qabKc = qacπbKLK − qabπcKLK . (4.1.6) c 4 cKL cKL We then can verify that

n ab cdo n cdo c d {qab, qcd} = P ,P ≈ 0, qab,P ≈ δ(aδb), (4.1.7)

cd bKL where qab and P are understood as functions of KaIJ and π . The first Poisson bracket vanishes trivially, calculation of the third equality is straight forward. The bracket P ab,P cd yields on the surface Sab ≈ 0 terms proportional to the expression K[a πb]IJ ≈ 2ζK[a Eb]I , M IJ I which vanishes if GIJ ≈ 0 (The calculations are analogous to those given in [66] for extended  i b ADM with variables Ka,Ej ). Finally, the ADM constraints can be expressed in terms of the new variables as 1   H = −2q ∇ P bc = − ∇ K πbIJ − δbK πcIJ , (4.1.8) a ac b 2 b aIJ a cIJ  s  1  √  H = − √ q q − q q P abP cd + qR q ac bd D − 1 ab cd s   √ = − √ π[a|IJ πb]KLK K − qR, (4.1.9) 8 q bIJ aKL where ∇a is the covariant derivative annihilating the spatial metric, and the Levi-Civita con- c 1 cd nection Γab := 2 q (∂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 simplicity constraints. Moreover, the whole system of constraints is of the first class, since both, Ha and H, are Gauß and simplicity invariant. For the latter, note that both constraints only depend on bIJ the combination KaIJ π , which just gives another simplicity term when Poisson commuted with Scd. Thus we found a viable extension of the ADM phase space. We want to stress that M 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 will turn this extension into a connection formulation in the next steps. First we rescale the canonical variables as 1 (β)πaIJ := πaIJ and (β)K := βK (4.1.10) β bKL bKL 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 2 Γ := π nK n ∂ πb L + ζη¯M η¯ π ∂ πbLK + ζΓc πb π K . (4.1.11) aIJ D − 1 bKL [I a J] [I J]K bLM a ab K[I c|J]

An expression for the determinant of the spatial metric is given below, the πaIJ are defined as below (3.2.30) and can be explicitly written using the formula for the inversion of a matrix. Additionally, 1 nI n ≈ πaKI π − ζηI
andη ¯ = η − ζn n (4.1.12) J D − 1 aKJ J IJ IJ I J

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

ab aIJ [I a|J] SM ≈ 0 ⇔ π ≈ 2n E (4.1.13)

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

IJ (β) aIJ h (β) ai ∂a π + Γa, π ≈ 0 (4.1.14) and can rewrite the Gauß constraint as

IJ IJ IJ h(β) (β) ai (β) aIJ h(β) (β) ai G = 0 + Ka, π ≈ ∂a π + Aa, π , (4.1.15) with the new connection (β) (β) AaIJ := ΓaIJ + KaIJ . (4.1.16) We postulate the new brackets

n(β) (β) o b K L L K
n(β) aIJ (β) bKLo n(β) (β) o AaIJ , πbKL = δa δI δJ − δI δJ , π , π = AaIJ , AbKL = 0 (4.1.17) and express the extrinsic curvature as √ s   qK b := − (β)πbIJ (β)A − Γ . (4.1.18) a 4 aIJ aIJ We again have to check if the ADM Poisson brackets are unchanged when calculated on the extended phase space, i.e. repeat the above calculations with “K replaced by A − Γ” (we set β = 1, simplifying the following discussion; it can easily be generalised for any value of β). The only calculation affected by this replacement is the bracket P ab,P cd . Comparison with (4.1.6) ab cd bIJ shows that P consists of terms q (AaIJ − ΓaIJ ) π (up to index contraction). For these, we find

n cd bIJ gh fKLo q (AaIJ − ΓaIJ ) π , q (AeKL − ΓeKL) π (4.1.19) f cd gh bIJ b cd gh fIJ = 2δa q q (AeIJ − ΓeIJ ) π − 2δeq q (AaIJ − ΓaIJ ) π bIJ n cd o gh fKL + (AaIJ − ΓaIJ ) π q , (AeKL − ΓeKL) q π

cd bIJ n gho fKL +q π (AaIJ − ΓaIJ ) , q (AeKL − ΓeKL) π cd gh bIJ fKL −q q [{AaIJ , ΓeKL} + {ΓaIJ ,AeKL}] π π .

In all but the last line, we can now reverse the replacement “K → A − Γ” and obtain the result of our previous calculation, namely that these terms vanish if Sab ≈ 0 ≈ GIJ . For the remaining M

26 terms in the last line, we first show that

aIJ dKL 0 ≈ π (x) {AbIJ (x), ΓcKL(y)} π (y) aIJ n dKL o aIJ n dKL o ≈ π (x) AbIJ (x), ΓcKL(y)π (y) − π (x) AbIJ (x), π (y) ΓcKL(y)

aIJ n boost dKL o aIJ n dKL o ≈ π (x) AbIJ (x), ΓcKL (y)π (y) − π (x) AbIJ (x), π (y) ΓcKL(y)

aIJ n boost o dKL aIJ n dKL o boost ≈ π (x) AbIJ (x), ΓcKL (y) π (y) + π (x) AbIJ (x), π (y) ΓcKL (y)

aIJ n dKL o −π (x) AbIJ (x), π (y) ΓcKL(y)

aIJ n boost o dKL aIJ d boost ≈ π (x) AbIJ (x), ΓcKL (y) π (y) + 2π (x)δ(x − y)δb ΓcIJ (y) aIJ d boost −2π (x)δ(x − y)δb ΓcIJ (y) aIJ n boost o dKL ≈ π (x) AbIJ (x), ΓcKL (y) π (y). (4.1.20)

Here, we noted that on the constraint surface Sab ≈ 0, Γ πdKL contains only the boost part M cKL boost I ΓcKL of the spin connection (with respect to n ) given by the first summand in (4.1.11). For rot the other summands, which correspond to the rotational components ΓcKL, the combination rot dKL ΓcKLπ ≈ 0 thus has to be expressible in terms of the simplicity constraint. Since the left hand side of the above Poisson bracket is simplicity invariant, we see that the boost part is the only term which can possibly contribute. Thus, one only has to calculate the bracket in last line and show that it vanishes. We split the calculation in some intermediate steps as

aIJ cKL n boost o 0 ≈ π (x)π (y) AbIJ (x), ΓdKL (y) (4.1.21)     2   = πaIJ (x)πcKL(y) A (x), π nM n ∂ πe N  (y) , D − 1 bIJ  eMN [K d L]   | {z } | {z } | {z }   (I) (II) (III)  where we used (4.1.11) and (4.1.12). As a first step, the Poisson bracket between AaIJ and −1 c πbKL = q qbcπ KL yields D  −1  {AbIJ (x), πeMN (y)} ≈ δ (x − y) 2q qbeη¯[I|M η¯J]N − ζπbMN πeIJ . (4.1.22)

Note that in all terms where a πaIJ appears, it will be contracted such that the first, purely rotational summand in the above equation does not contribute on the surface Sab ≈ 0 ⇔ πaIJ ≈ M 2n[I Ea|J]. We find     2 aIJ cKL M e N   π (x)π (y)n (y)n[K (y)∂dπ L] (y) AbIJ (x), πeMN (y) (4.1.23) D − 1  | {z }  (I)  4 ≈ − δD(x − y) δaEcK ∂ n − qqacE ∂ nK
D − 1 b d K bK d and   2   πaIJ (x)πcKL(y)nM (y)n (y)π (y) A (x), ∂ πe N (y) (4.1.24) D − 1 [K eMN bIJ d L]  | {z }  (III)  4 ≈ δD(x − y) δaEcK ∂ n − qqacE ∂ nK
, D − 1 b d K bK d

27 and see that the terms (I) and (III) precisely cancel. What remains to be shown is that I (II) ≈ 0, which is straight forward using (4.1.22). We repeatedly used the identities n ∂bnI = 0 aI aI and nI ∂bE = −E ∂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 been derived [35] and the resulting symplectic potential reads Z 2 D a I d x∂a(EI δn ). (4.1.25) β σ (β) aIJ (β)
Note that this calculation does not directly prove that the transformation π , KbKL → (β) aIJ (β)
(β) aIJ (β)
π , AbKL is canonical. Instead, it proves that the system described by π , 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 ADM constraints can be treated as above and we again obtain a first class constraint algebra. Note c (β) aIJ that Ha and H will involve ΓaIJ ,Γab and R expressed in terms of π , and therefore become tremendously complicated. However, we can now choose for Ha and H preferably simple expres- (β) aIJ (β) sions constructed from π and AbKL which reduce to their corresponding ADM versions on the surface GIJ ≈ Sab ≈ 0, and the algebra will still close. In particular, we can exploit our M 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 internal signature, independently of the external (physical) signature. The key to obtain a connection formulation with a compact internal gauge group as well as a Lorentzian external signature is to add a correction term to the Hamiltonian constraint which changes the sign in front of the KKEE term in the ADM Hamiltonian (4.2.4). We observe that 1     (β)πaK (β)D (β)πcJL (β)π (β)πb (β)D (β)πcJL (β)π ≈ KI EaKJ Eb (D − 1)2 J b cKL KJ a cKL b I a J (4.2.1) and 2  1  2 (β)πaKI (β)π (β)D (β)πbIJ ≈ KI Ea
. (4.2.2) D − 1 bKJ a a I Now, in the case of Euclidean internal and Lorentzian external signature, the expression β2  1    H := √ −(β)H + (β)Dab (β)F −1 M N (β)Dcd − β2 + 1
K K Ea[I Eb|J] (4.2.3) q E 2 M ab cd N aI bJ reduces to the Lorentzian ADM Hamiltonian constraint 1 1 H ≈ − √ EaI EbJ R − √ Ea[I Eb|J]KI KJ . (4.2.4) 2 q abIJ q a b

aIJ The KKEE terms are understood as shown above as functions of π and AbKL and 1 (β)H := (β)πaIK (β)πbJK (β)F (4.2.5) E 2 abIJ denotes the Euclidean part of the Hamiltonian constraint. (A note for completeness: If internal and 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 gauge unfixed theory.) The diffeomorphism constraint 1 H = (β)πbIJ (β)F , (4.2.6) a 2 abIJ the Gauß constraint IJ (β) (β) aIJ G = Da π (4.2.7) and the simplicity constraint 1 Sab =  (β)πaIJ (β)πbKL (4.2.8) M 4 IJKLM 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 anomalies both in the covariant [67] as well as in the canonical approach [36, 37]. Therefore, we want to introduce a linear simplicity constraint in the canonical theory in the next chapter, inspired by the new spin foam models [68, 69, 60, 67, 61, 70]. The linear simplicity constraint will also become 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 constraint is 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 include a normal field N I as an independent variable, as opposed to the normal nI (π) used before. Due to the additional degrees of freedom in the normal field, the simplicity constraint is modified to be linear in both N I and πaIJ and it can be shown that it removes, together with a condition on the momentum of the normal, all the newly introduced degrees of freedom. The necessity for using this formulation will only be apparent in the third part of this thesis, were it will be a key ingredient to deal with Majorana fields necessary for supergravity theories. Also, the linear constraint brings the presented canonical quantisation closer to the new spin foam models. The original 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 time normal 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 is natural to ask whether we can also include such a time normal as an independent parameter in our canonical theory. With the power of hindsight, we also know that we have to introduce this kind of formulation in order to deal with supergravities in the third part of this thesis. To make contact to the covariant formulation, it is therefore of interest to ask whether, from the canonical point of view, (a) the theory of formulated in the previous chapters can be reformulated using a linear simplicity constraint, and (b) if so, whether the linear version of the constraint can be quantised without anomalies. Both of these questions will be answered affirmatively, the answer to (a) in this chapter and the answer to (b) in the second part of this thesis. The original references are [40] and [39]. As we will show in the third part of this thesis, the use of the linear simplicity constraints (already at the classical level) is probably the most convenient approach towards constructing a connection formulation for supergravity theories in D + 1 dimensions with compact gauge group. To answer (a) we will follow the approach of the previous chapter and construct the theory with linear simplicity constraint by an extension of the ADM phase space. 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 the Plebanski formulation of general relativity using constraints which involve an additional three form and which are linear in the bivectors, without giving a Hamiltonian formulation. This chapter 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. This will be exploited when quantising the theory in the second part of this thesis, where we fix ζ = 1 and therefore do not have to bother with the non-compact gauge group SO(1,D). There, we employ the Hilbert space representation for the normal field derived in chapter 7 and then we find quantum operators corresponding to the linear simplicity constraint and show that these operators (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 given by [62]1 Sab = 0 ⇔ (β)πaIJ = 2 n[I Ea|J] and that nI is no independent field but determined by M β the densitised hybrid vielbein EaI . We now postulate a new field N I , which will play the role of this normal, together with its conjugate momentum PI , subject to the linear simplicity and normalisation constraints

a J (β) aKL SIM := IJKLM N π , (5.2.1) I N := N NI − ζ. (5.2.2) The solution to the linear simplicity constraint in any dimension D ≥ 3 is given by2 (β)πaIJ = 2 [I a|J] aI (β) aIJ β N E , with NI E = 0. We see that on the solutions, the physical information of π 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 find N I = nI (E), i.e. the N I are no physical degrees of freedom. The same has to be assured for the 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 can be 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 we will show that when solving these constraints, the momenta PI are automatically removed from the theory. The theory we want to construct is very similar to the one in the previous chapter. It is defined by the Poisson brackets (4.1.17) and  I I  I J N ,PJ = δJ , N ,N = {PI ,PJ } = 0, (5.2.3) and, apart from the linear simplicity and normalisation constraints (5.2.1,5.2.2), is subject to 1 GIJ = (β)D (β)πaIJ + P [I N J], (5.2.4) 2 a 1 1   H = (β)πbIJ ∂ (β)A − ∂ (β)πbIJ (β)A + P ∂ N I , (5.2.5) a 2 a bIJ 2 b aIJ I a s h     i √ H = − √ (β)π[a|IJ (β)πb]KL (β)A − Γ (β)A − Γ − qR. (5.2.6) 8 q bIJ aKL 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 is excluded 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 2 √ √p ab scalar fields φ, which would obtain additional terms ∼ det q + det qq φ,aφ,b for the scalar field φ and its conjugate momentum p which are missing here. In fact, these terms would spoil the constraint algebra, since a {H,SIM } 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 Poisson ab (β) (β) aIJ I brackets of the ADM variables K , qab in terms of the new variables AaIJ , π , N , 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 and normalisation constraints, the solution for (β)πaIJ is the same as in the case of the quadratic (β) aIJ 2 [I a|J] simplicity (neglecting the topological sector), π = β n E , and terms which vanished due to the quadratic simplicity constraint still vanish in the case at hand. For the Gauß constraint, (β) [a (β) b]IJ note that the only terms appearing in the calculation are of the form ( A − Γ) IJ π , which already vanish on the surface defined by the vanishing of the rotational parts of the Gauß constraint [35]. Now, if the linear simplicity and normalisation constraints hold, we know that N I = nI (E), i.e. the modification of the Gauß constraint P [I N J] ≈ P¯[I nJ] on-shell just changes the boost part of the Gauß constraint. Thus, the ADM brackets are reproduced on the surface defined by the vanishing of GIJ , Sa and N . IM IJ Next, we will show that the algebra is of first class. Note that since G and Ha gen- erate gauge transformations and spatial diffeomorphisms by inspection, their algebra with all other constraints obviously closes. The algebra of the linear simplicity and the normalisation constraint is trivial. Moreover, the Hamilton constraint Poisson-commutes trivially with the normalisation constraint and, since it depends only on the combination (β)πaIJ (β)A , we find n o bIJ H[N],Sa [sIM ] = Sa [...]. We are left with the Poisson-bracket between two Hamilton IM a IM constraints. Since on-shell the ADM brackets are reproduced, the result is

0 ab {H[M], H[N]} ≈ Ha[q (MN,b − NM,b)], (5.2.7)

0 bc where Ha = −2qac∇bP now denotes the ADM diffeomorphism constraint. Furthermore, it can be shown that the vector constraint (5.2.5) correctly reduces to the ADM diffeomorphism 0 constraint if the Gauß and simplicity constraints hold, Ha ≈ Ha. Therefore, the algebra closes. What is left to show is that also the Hamilton constraint H on-shell reproduces the ADM constraint, which will be made explicit in the next paragraph. Note that because of the modified Gauß 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 and normalisation constraints leads to 2 (β)πaIJ = n[I Ea|J] and N I = nI (E). (5.2.8) β

(β) We make the Ansatz AaIJ = ΓaIJ + βKaIJ with ΓaIJ defined as in (4.1.11). Then, the symplectic potential reduces to [47] 1 (β)πaIJ (β)A˙ + P N˙ I 2 aIJ I aJ aJ I I ≈ K¯aJ E˙ − K¯aIJ E n˙ + P¯I n˙ h  I bK I i aJ ≈ K¯aJ − nJ EaI K¯bK E + P¯ E˙ 00 ˙ aJ := KaJ E , (5.2.9)

00 where we have dropped total time derivatives and divergences. The notation KaI is chosen to (β) make this section consistent with [36]. Upon this ansatz for AaIJ , the Hamiltonian constraint reduces 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 00 On the other hand, we note that the substitution KaI → KaI in the Hamiltonian constraint after (β) decomposing AaIJ and solving the normalisation and simplicity constraints does not change aI the constraint since the additional terms are proportional to nI E = 0. The Gauß and spatial 00 diffeomorphism constraints also reduce to the expressions given in [47] with KaI as the extrinsic curvature variable. In a last step, one could solve the SO(D + 1) Gauß constraint by going over ab to the spatial metric qab and its conjugate momentum P familiar from the ADM formulation as canonical variables. Thus, we have shown that we recovered the usual ADM formulation and thus the equivalence to general relativity. The counting of the number of physical degrees of freedom goes as follows: The full phase D2(D+1) space consists of |{A, π, N, P }| = 2 2 + 2(D + 1) degrees of freedom which are subject n o 2 to H , H,GIJ ,Sa , N = (D + 1) + D(D+1) + D (D−1) + 1 first class constraints. It is a IM 2 2 most convenient to compare this to Peldan’s [47] extended ADM formulation (transforming under SO(1,D)), with |{E,K}| = 2D(D + 1) phase space degrees of freedom and the first class  IJ D(D+1) constraints Ha, H,G = (D + 1) + 2 . In any dimension, the difference in phase space degrees of freedom is exactly removed by the simplicity and normalisation constraint, n o |{A, π, N, P }| − |{E,K}| = 2 Sa , N . IM We remark that related formulations of general relativity, where a time normal appears as an independent dynamical field, have already appeared in the literature [54, 72, 73]. The difference between these and our formulation is that while our formulation features both the simplicity constraint and the time normal at the same time, the time normal appears in the process of solving the simplicity constraint without solving the boost part of the Gauß constraint in the other approaches. In other words, the time normal is an integral part of the simplicity constraint in our approach, not a concept emerging after its solution.

33 Part II

Loop quantum gravity in higher dimensions

34 Chapter 6

LQG-techniques in higher dimensions

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 based on results from Thiemann [12, 13, 74]. While the kinematical quantisation had already been worked out independently of the dimension and compact gauge group, the regularisation in higher dimensions was worked out in [37] and the simplicity constraint was discussed originally in [37, 39]. In this chapter, we will deal with the kinematical Hilbert space, the regularisation of the Hamiltonian constraint, as well as the regularisation of the simplicity constraint. The next chapter will then be devoted to a closer investigation of the solution space of the simplicity constraint, which becomes necessary due to quantisation anomalies as we will see at the end of this 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 1 (τ )K = δK δ − δK δ
(6.1.1) IJ L 2 I JL J IL as a basis for the adjoint representation of the so(D + 1) Lie algebra. As usual, we define the holonomy hc(A) ∈ SO(D + 1) along c by the equation d h (A) = h (A)A(c(s)), h = 1 , h (A) = h (A), (6.1.2) ds cs cs c0 D+1 c c1

IJ a using the definitions cs(t) := c(st), s ∈ [0, 1], A(c(s)) := Aa (c(s))τIJ c˙ (s). The formal solution to this equation is given by

∞ Z  X Z 1 Z 1 Z 1 hc(A) = P exp A = 1D+1 + dt1 dt2 ... dtnA(c(t1)) ...A(c(tn)), (6.1.3) c n=1 0 t1 tn−1

35 with P being the usual path ordering symbol. In the following, the notion of a cylindrical function 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 |E| Fγ : SO(D + 1) → C. It is clear that we can define the same function also on a refined graph γ0 containing all the edges of γ as compositions of edges in γ0. This idea leads to the notion of cylindrical consistency which is discussed at length in [66]. Since we cannot add anything new to 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 gauge group SO(D + 1), and this fact is used in the quantisation procedure. Essentially, we will be using a Cauchy completion of the space of connections given by A := Hom(P, SO(D + 1)). Since the details of this procedure have been outlined in great detail in [66], we will refrain from going into detail here. The important observation for what follows is that one can define a measure on A := Hom(P, SO(D + 1)), as well as on A/G, G = SO(D + 1), a locally gauge invariant version thereof. This measure is the Asthekar-Lewandowski measure, defined by Z Z µ0[f(A)] = dµ0(A)f(A) = dµ0,γ(A)fγ (A) A/G A/G   Z Y
=  dµH (he) Fγ h1, ..., h|E| , (6.1.4) |E(γ)| SO(D+1) e∈E(γ) with µH being the Haar probability measure on SO(D + 1). This measure has the important property that it is cylindrically consistent, meaning that one can subdivide and reorient the edges 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 the straight forward generalisation of the usual fluxes used in LQG reads Z Z n IJ aIJ a1 aD−1 π (S) := nIJ (∗π) = nIJ π aa1...aD−1 dx ∧ ... ∧ dx . (6.1.5) S S Following the standard regularisation procedure outlined in [37], we obtain for the action of the Hamiltonian vector field associated to πn(S)

X ∂Fγ   Y n (S)[f ] = (e, S)[n(b(e)) h (A)] S h (A), ..., h γS γS e AB e1 e|E(γ )|(A) ∂he(A)AB S e∈E(γS ) X IJ e = (e, S) n (e ∩ S) RIJ fγS , (6.1.6)

e∈E(γS ) where γS is a graph adapted to the surface S, (e, S) the usual type indicator function, b(e) the e beginning of the edge e, and A, B are SO(D + 1) indices in the representation of he. RIJ is the right invariant vector field associated to the edge e, defined by

 d  tτIJ (RIJ f)(he) := f(e he). (6.1.7) dt t=0 They generate the so(D + 1) Lie algebra if acting on the same edge as

h e e0 i 1 e e e e R ,R = δ 0 (η R + η R − η R − η R ). (6.1.8) IJ KL 2 e,e JK IL IL JK IK JL JL IK

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 of solving the constraint result in the requirement that one should use only gauge invariant spin networks as a basis in the gauge invariant Hilbert space. The spin networks associated to our gauge group SO(D + 1) are constructed in the very same way as in the usual SU(2) case, so we 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 the original literature [19] and quantise it as a finite spatial diffeomorphism. Essentially, the spatial diffeomorphisms 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 deeper reason behind this is that the scalar product induced by the Ashtekar-Lewandowski measure is not strongly continuous. As an example, we consider the scalar product between two equal spin 0 networks hfγ, fγi = 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 0 D 0 E loop, resulting in fγ0 , we have fγ, fγ0 = 0, independently of how small the additional edge is. Thus, in the limit of shrinking the additional edge e0 of γ0 to zero, we have, morally speaking, D 0 E lime0→0 fγ, fγ0 = 0 6= hfγ, fγi. It follows that we cannot recover an operator corresponding to the connection by shrinking a holonomy to zero length. Consequently, we do not have an operator for the field strength of the connection and thus also not an operator corresponding to the generator of spatial diffeomorphisms, which is expressed in terms of the field strength. This problem is bypassed in the definition of the Hamiltonian constraint by defining the Hamiltonian constraint on a finite triangulation which is then shrunk to zero. However, in the process of taking the limit, the action of the constraint on spin network functions does not change any more after reaching a suitable refinement of the triangulation and when evaluated on diffeomorphism invariant distributions. On the other hand, the finite spatial diffeomorphisms φ have a natural representation Uˆ(φ) on ˆ spin network functions given by U(φ)fγ = fφ(γ), where φ(γ) is the action of the diffeomorphism on 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 spatial dimension. We just briefly summarise the main ideas here. The intuitive idea of the solution space 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 cylindrical function any more. Nevertheless, a precise sense can given to such a construction by going to the ∗ dual space Hkin of the kinematical Hilbert space Hkin, consisting of all functionals η : Hkin → C. ∗ On Hkin, an average over the spatial diffeomorphism group is possible and the result of taking ∗ a spatial diffeomorphism average of a typical element of Hkin, e.g. hfγ, ·i, gives, up to some 0 subtleties discussed below, a finite expression when evaluated on a spin network function fγ0 since the scalar product will only give a non-zero value when the graphs γ and γ0 exactly match. The subtleties mentioned above come from factoring out certain diffeomorphisms correctly which leave the graph invariant. These are on the one hand diffeomorphisms with trivial action on the graph. On the other hand, these diffeomorphisms could be graph symmetries. We refer to the original 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. The operator 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 modified algebraic structure, since the vielbeins from which the operator is constructed are in the adjoint representation 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 detailed regularisation procedure. Starting from the classical expression for the volume of a region R, Z √ V (R) := dDx q, (6.2.1) R and expressing it in terms of fluxes, it turns out that the higher-dimensional generalisation of the Ashtekar-Lewandowski volume operator is given for D + 1 even by Z Z D D Vˆ (R) = d p |det(\q)(p)| γ = d pVˆ (p)γ, (6.2.2) R R D   D−1 X Vˆ (p) = ~ δD(p, v)Vˆ , (6.2.3) γ 2 v,γ v∈V (γ) 1 D−1 D ˆ i X Vv,γ = s(e1, . . . , eD)qe ,...,e , (6.2.4) D! 1 D e1,...,eD∈E(γ), e1∩...∩eD=v 1 q =  RIJ RI1K1 RJ1 ...RInKn RJn (6.2.5) e1,...,eD IJI1J1I2J2...InJn e e1 e0 K1 en e0 Kn 2 1 n and for D + 1 odd by Z Z D D Vˆ (R) = d p |det(\q)(p)| γ = d pVˆ (p)γ, (6.2.6) R R D   D−1 X Vˆ (p) = ~ δD(p, v)Vˆ , (6.2.7) γ 2 v,γ v∈V (γ) D I i X I Vˆ = s(e1, . . . , e )q , (6.2.8) v,γ D! D e1,...,eD e1,...,eD∈E(γ), e1∩...∩eD=v 1 ˆ ˆ I ˆ 2D−2 Vv,γ = Vv,γVI v,γ , (6.2.9)

I I1K1 J1 InKn Jn qe ,...,e = II1J1I2J2...InJn Re R 0 K1 ...Re R 0 Kn . (6.2.10) 1 D 1 e1 n en

6.2.2 Poisson bracket identities and regularisation The regularisation procedure for the Hamiltonian constraint relies on several Poisson bracket identities 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 be found in [37]. Essentially, when looking at the expression for the Hamiltonian constraint with density weight one β2  1    H˜ := √ −(β)H + (β)Dab (β)F −1 M N (β)Dcd − β2 + 1
K K Ea[I Eb|J] , (6.2.11) q E 2 M ab cd N aI bJ

38 with (β) [a|IK (β) b]J π π K (β)H = √ (β)F , (6.2.12) E q abIJ 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 , I which could be obtained by quantising an expression of type KaI = (A−Γ)aIJ n , which however would result in a very complicated operation of the constraint operator. Both problems can be resolved by using the Poisson bracket tricks √ qπaIJ (x) = −(D − 1){AaIJ ,V (x, )}, (6.2.13) D − 1 K(x) := EaI (x)K (x) ≈ {H (x),V (x, )} , (6.2.14) aI D E (D − 1) EbI (x)K (x) ≈ πbKL(x) {A (x), {H [1](x, ),V (x, )}} , (6.2.15) aI 2D aKL E (β) first introduced by Thiemann for D = 3 [12], which first allow to quantise HE for D + 1 even by using

[a|IK b]J π π K 1 √ (x) ≈ abca1b1...an−1bn−1 IJKLI1J1...In−1Jn−1 sgn(det e)(x) (6.2.16) q 4(D − 2)!

K1 Kn−1 √ D−2 πcKL(x)πa1I1K1 (x)πb1J1 (x) . . . πan−1In−1Kn−1 (x)πbn−1Jn−1 (x) q (x). and for D + 1 odd by

[a|IK b]J π π K 1 √ ≈ aba1b1...anbn IJKI1J1...InJn sgn(det e) q 2(D − 2)!

K1 Kn √ D−2 nK πa1I1K1 πb1J1 . . . πanInKn πbnJn q (6.2.17) with 1 √ nI (x) ≈ a1b1...an+1bn+1 II1J1...In+1Jn+1 sgn(det e)(x) qD−1(x) D! K1 Kn+1 πa1I1K1 (x)πb1J1 (x) . . . πan+1In+1Kn+1 (x)πbn+1Jn+1 (x). (6.2.18) The basic idea is that since the volume operator and the holonomy are well defined operators on (β) the kinematical Hilbert space, we can now express HE purely in terms of volume operators (β) and holonomies, which results in a well defined quantisation for HE. Using this operator, we can further use the remaining Poisson bracket identities to construct the rest of the Hamiltonian constraint. The explicit form of these operators and a toolbox to construct them has been spelled 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 operator which has been performed in [37]. The regularisation of the linear simplicity constraint operator follows easily from these considerations and is sketched in section 8.3.1. In order to obtain a well defined quantum operator associated to the quadratic simplicity constraint Sab(x) = M 1 aIJ bKL aIJ 4 IJKLM π (x)π (x), we have to smear the momenta π (x) over (D − 1)-hypersurfaces Sx, S0x as x 0x 1 IJ x KL 0x C (S ,S ) := lim 0  π (S )π (S0 ), (6.3.1) M ,0→0 (D−1) (D−1) IJKLM

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

ˆ x 0x 1 X X C (S ,S )γ 0 fγ 0 = lim 0  M SS SS ,0→0 (D−1) (D−1) IJKLM   0 0 e∈E(γSS0 );b(e)=x e ∈E(γSS0 );b(e )=x x 0 0x IJ KL 0 (e, S )(e ,S )Re Re fγSS0 , (6.3.2) 0 where b(e) is the beginning of e and γSS0 is a graph adapted to both S and S such that all edges are outgoing from x. At first sight, one might be worried that this operator is divergent since the lower line does not depend on the regulators , 0, whereas the first line is divergent for  → 0 or 0 → 0. However, since the quantum operator is defined as the limit of the regularised operator, it vanishes if and only if

X X x 0 0x IJ KL 0 (e, S )(e ,S )Re Re fγSS0 = 0 (6.3.3) 0 0 e∈E(γSS0 );b(e)=x e ∈E(γSS0 );b(e )=x and diverges otherwise. It therefore makes sense to ask for its kernel, which is well defined by the above equation. In the following, we will drop the superscript x from the hypersurfaces to simplify notation. The kernel of the simplicity constraint acting on interior points of an edge can be easily hP e i calculated, since, due to gauge invariance, e∈E(γ); v=b(e) RIJ fγSS0 = 0, the above expression reduces to CˆM (S, S0, x)p∗ f = ±p∗ 4IJKLM Re1 Re1 p∗ f (6.3.4) γ γ γSS0 IJ KL γSS0 γ γ when splitting the edge e at x into two outgoing edges e1 and e2. This implies that the operator vanishes if and only if τ[IJ τKL] = 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 on the 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 the quantum level, but the commutator between two simplicity constraints acting on an edge gives another 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, the situation can be simplified a lot by showing that at a vertex v, the simplicity constraint operator already implies that

e e0 0 00 00 R[IJ RKL]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 fields lie in the linear span of the flux vector fields. The full proof has been given in [37] and we only remark that its key idea is to use small deformations of the hypersurfaces used to construct the fluxes in order to single out specific edges at the vertex. An appropriate summing over these fluxes then yields the desired result. The vertex simplicity constraint thus reduces to

IJKLM e e0  RIJ RKLfγ = 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 of vertex simplicity constraint leads to the unique Barret-Crane intertwiner, which is however in conflict with intertwiner degrees of freedom present in loop quantum gravity when solving the simplicity constraint at the classical level in four dimensions. One therefore expects an anomaly to be present for the vertex simplicity constraint operator which results in too many degrees of freedom to be constrained. In order to explicitly show this anomaly, we calculate

h IJKLM e e0 ABCDE e0 e00 i ABCE IJ K 00 0  RIJ RKL,  RABRCD ∼ δIJKM (Re )AB(Re) (Re ) C . (6.3.8)

As discussed in [37], the projection of the right hand side onto the vertex simplicity constraint operators is vanishing, thus it cannot be expressed as a sum of simplicity constraint operators and 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 thesis and try to understand this anomaly in more detail in section 8.2. It will be shown there that the anomaly can be avoided by restricting to a certain subclass of vertex simplicity constraints which form a closed algebra.

41 Chapter 7

Hilbert space techniques for the linear simplicity constraint

In this relatively short chapter, we will derive new Hilbert space techniques necessary for the implementation of the theory based on the linear simplicity constraint. The construction from the previous chapter will have to be supplemented by a Hilbert space which carries operators I associated to the, now independent, normal N and its conjugate momentum PI . The basic idea of 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 construction of the Ashtekar-Lewandowski Hilbert space, it is possible to define a projective limit of Hilbert I spaces, each of which carries the operators corresponding to N and PI at a finite collection of points. Taking the limit, which exists due to the normalisability of the Haar measure on the D-sphere, we arrive at a the desired Hilbert space. The original work from which this chapter is taken is [40].

We restrict to the case ζ = 1 in the following, because the kinematical Hilbert space for canonical loop quantum gravity has been defined rigorously only for compact gauge groups like SO(D + 1). For scalar fields like the Higgs field, two different constructions to obtain a kinematical Hilbert space have been given. In the first one [21], a crucial role is played by point IJ
IJ holonomies Ux(Φ) := exp Φ (x)τIJ . The field Φ , which is assumed to transform according to the adjoint representation of G, is contracted with the basis elements τIJ of the Lie algebra of G and then exponentiated. Point holonomies are better suited for background independent quantisation than the field variables ΦIJ themselves, since the latter are real valued rather than valued in a compact set. Therefore, a Gaußian measure would be a natural choice for the inner product 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 the I I case at hand there is a more natural route. On the constraint surface N = N NI − 1 = 0, N is valued in the compact set SD. In this case the measure problems can be circumvented by solving N classically. The obvious choice of Hilbert space is then of course the space of square integrable functions on the D-sphere.

42 I To solve N , we choose a second class partner N˜ := N PI ,

 I J I D D N (x)NI (x) − 1,N (y)PJ (y) = N (x)NI (x)δ (x − y) ≈ δ (x − y), (7.1) where terms ∝ N have been dropped. N˜ weakly Poisson commutes with the constraints: It is Gauß invariant and transforms diffeomorphism covariantly, it trivially Poisson commutes with I H (which neither depends on N nor on PI ), and a short calculations yields

n a IM ˜ o a IM SIM [sa ], N [˜n] = SIM [˜nsa ]. (7.2)

Therefore, the algebra of the remaining constraints is unchanged when we solve N , N˜ using the 2 J Dirac bracket. Letη ¯IJ = ηIJ − NI NJ /||N|| whenceη ¯IJ N = 0 also when ||N|| 6= 1. Then J P¯I =η ¯IJ P Poisson commutes with the normalisation constraint and thus is an observable just I I as N . Since the Dirac matrix is diagonal, the Dirac brackets of P¯I ,N coincide with their Poisson brackets. We find

 I N (x),NJ (y) DB = 0,  I ¯ I D N (x), PJ (y) DB =η ¯ J (x)δ (x − y),  ¯I ¯J ¯[I J] D P (x), P (y) DB = 2P (x)N (x)δ (x − y), (7.3) while the remaining brackets are unaffected. We see that unfortunately the Poisson alge- I bra of the N and P¯I does not close, it automatically generates also the rotation generator ¯ LIJ = 2N[I PJ] = 2N[I PJ]. We therefore have to include it into our algebra. On the other hand obviously {LIJ , N} = 0 so that LIJ is also an observable and moreover the LIJ generate the Lie algebra so(D + 1) while {LIJ ,NK } = −2N[I δJ]K so that the algebra of the NI ,LIJ already J 2 I closes. Finally we have the identity LIJ N = −||N|| P¯I so that the N ,LIJ already determine P¯I . We conclude that nothing is gained by considering the P¯I and that it is better to consider I the overcomplete set of observables N ,LIJ instead.

Consider, similar as in LQG, cylindrical functions F [N] of the form F [N] = Fp1,..,pn (N(p1), .., N(pn)) I where Fp1..pn is a polynomial with complex coefficients of the N (pk), k = 1, .., n; I = 0, .., D+1. IJ We define the operator NˆI (x) to be multiplication by NI (x) on this space. Let also Λ be a smooth antisymmetric matrix valued function of compact support and define the operator Z ˆ D IJ ˆ ˆ L[Λ] := 2 d x Λ (x) N[I (x) PJ](x), (7.4) where PˆJ (x) = iδ/δNJ (x). Notice that no factor ordering problems arise. The operator Lˆ[Λ] has a well defined action on cylindrical functions, specifically

n ˆ X IJ ˆ L[Λ] F [N] = 2i Λ (pk)N[I (pk)∂/∂NJ](pk) F [N] (7.5) k=1 and annihilates constant functions. D+1 2 D Let dν(N) := cDd Nδ(||N|| − 1) the SO(D + 1) invariant measure on S where the constant cD makes it a probability measure. For a function cylindrical over the finite point set {p1, .., pn} we define the following positive linear functional Z

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 thus has a unique σ−additive extension to the projective limit of the finite Cartesian products of D Q D copies of S which in this case is just the infinite Cartesian product N := x S of copies of SD [76], one for each spatial point. This space can be considered as a space of distributional normals because a generic point in it is a collection of vectors (N(x))x without any continuity properties. The operator NˆI (x) is bounded and trivially self-adjoint because NI (x) is real valued D IJ and S is compact. To see that Lˆ[Λ] is self adjoint we let gΛ(p) = exp(Λ (p)τIJ ) where τIJ are the generators of SO(D + 1). We define the operator

 ˆ  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 explicitly that 1 d Lˆ[Λ] = [ ] Uˆ[tΛ], (7.8) i dt t=0 whence Lˆ[Λ] is self-adjoint by Stone’s theorem [77]. Finally it is straightforward to check that besides the ∗-relations also the commutator relations hold, i.e. they reproduce i times the classical Poisson bracket. We conclude that we have found a suitable background independent representation of the normal field sector. D At each point p ∈ Σ, an orthonormal basis in the Hilbert space Hp = L2(S , dν) is given K~ by the generalisations of spherical harmonics to higher dimensions Ξl (N), which are shortly introduced in A (see [78] for a comprehensive treatment). An orthonormal basis for HN is given ~ Q Kv by spherical harmonic vertex functions F ~ (N) := Ξ (N). Any cylindrical function F~v ~v,~l,K~ v∈~v lv can be written as a mean-convergent series of the form X F~v(N) = a ~ F ~ (N) (7.9) ~v,~l,K~ ~v,~l,K~ ~ ~l,K~ for complex coefficients a ~ . The sum here runs for each v ∈ ~v over all values l ∈ N0 and ~v,~l,K~ for each l over all K~ compatible with l. Following the construction in [21] we obtain the combined Hilbert space for the scalar field and the connection simply by the tensor product, SO(D+1) SO(D+1) HT = Hgrav ⊗HN = L2(A , dµAL )⊗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 coming from the normal field. This of course leads to an obvious modification of the definition of the intertwiners 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 linear simplicity constraints in the context of the canonical formulations of higher-dimensional general relativity and supergravity described in this thesis. Since the canonical quadratic simplicity constraint 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 are non-anomalous among themselves and allow for a natural unitary map of the spin networks in the kernel of these simplicity constraint operators to the SU(2)-based Ashtekar-Lewandowski Hilbert space in D = 3. The linear constraint operators on the other hand are non-anomalous by themselves, however their solution space will be shown to differ in D = 3 from the expected Ashtekar-Lewandowski Hilbert space. We comment on possible strategies to make a connection to the quadratic theory. Also, we comment on the relation of our proposals to existing work in the spin foam literature and how these works could be used in the canonical theory. We emphasise that many ideas developed in this chapter are certainly incomplete and should be considered as suggestions for possible starting points for more satisfactory treatments in the future. The original work from which this chapter is taken is [39]. Parts of the ideas are based on [37].

8.1 Introduction

As shown in the first part of this thesis, gravity in any dimension D + 1 ≥ 3 can be formulated as a gauge theory of SO(1,D) or of the compact group SO(D + 1), irrespective of the spacetime signature. The resulting theory has been obtained on two different routes, a Hamiltonian analysis of the Palatini action making use of the procedure of gauge unfixing, and on the canonical side by an 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. the antisymmetrised product of two vectors. Originally introduced in Plebanski’s [58] formulation of general relativity as a constrained BF theory in 3 + 1 dimensions, they have been generalised to arbitrary dimension in [62]. Moreover, discrete versions of the simplicity constraints are a standard 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 of simplicity constraints are considered in the literature, which are either quadratic or linear in the bivector fields. The quantum operators corresponding to the quadratic simplicity constraints have 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 imply erroneous elimination of physical degrees of freedom [53]. This triggered the development of the new spin foam models [68, 69, 60, 67, 61, 70], in which the quadratic simplicity constraints are replaced by linear simplicity constraints. The linear version of the constraints is slightly stronger than the quadratic constraints, since in 3 + 1 dimensions the topological solution is absent. The corresponding quantum√ operators are still anomalous (unless the Barbero-Immirzi parameter takes 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 the anomaly. Also, the newly developed U(N) tools [83, 84, 85] have been recently applied to solve the simplicity constraints [86, 87]. In this chapter, we are, for the most part, not going to import techniques for solving the simplicity 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 how to 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 that will not arrive at the conclusion that a certain kind of imposition will be the correct one and thus further research, centered around consistency considerations and the classical limit, has to be performed to find a satisfactory treatment for the simplicity constraints. Of course, in the end 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 mathematical consistency of the approach, as well as comparison with the classical implementation of the simplicity constraints in D = 3, where the usual SU(2) Ashtekar-Barbero variables exist. If a satisfactory implementation in D = 3 can be constructed, the hope would then be that this procedure has a natural generalisation to higher dimensions. Since parts of the very promising results developed from the spin foam literature are restricted to four dimensions, we will restrict ourselves 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 operators which have been shown to be anomalous in section 6.3. It will be illustrated that choosing a recoupling scheme for the intertwiner naturally leads to a maximal closing subset of simplicity constraint operators. Next, the solution to this subset will be shown to allow for a natural unitary map to the SU(2) based Ashtekar-Lewandowski Hilbert space in D = 3 and we will finish the first part with several remarks on this quantisation procedure. In section 8.3, we will analyse the strong implementation of the linear simplicity constraint operators since they are non-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 discuss several 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 a constraint ensuring the equality of the normals N I and nI (π). In section 8.4, we will compare our results to existing approaches from the spin foam literature. Finally, we will give a critical evaluation of our results and conclude in section 8.5.

1 Strongly here means that the constraint operator annihilates physical states, Cˆ |ψi = 0 ∀ |ψi ∈ 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 quadratic simplicity constraint operator acting on a vertex v are given by

e e0 0 00 00 R[IJ RKL]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 foam models, see e.g. [62, 67]. Since not all of these building blocks commute with each other, i.e. the ones sharing exactly one edge, we will have to resort to a non-standard procedure in order to avoid an anomaly in the quantum theory. The strong imposition of the above constraints, leading to the Barrett-Crane intertwiner [59], was discussed in [62]. A master constraint formulation of the vertex simplicity constraint operator was proposed in [37], however apart from providing a precise 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 quadratic vertex 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 kinematical level, lead to the same Hilbert space as the quantisation of the classical theory without a simplicity constraint. More precisely, there should exist a natural unitary map from the solution 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 a classical connection formulation of general relativity in the first part of this thesis was originally developed in the context of anomalous gauge theory, where it was observed that first class constraints can turn into second class constraints after quantisation [63, 89, 90, 91, 92]. This is however precisely what is happening in our case: The classically Abelian simplicity constraints become 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 simplicity constraint operators look like? Theorem 1. Given a N-valent vertex v ∈ γ, the set

 RIJ RKL = ... =  RIJ RKL = 0 (8.2.2) IJKLM e1 e1 IJKLM eN eN

IJ IJ
KL KL
IJKLM Re1 + Re2 Re1 + Re2 = 0 IJ IJ IJ
KL KL KL
IJKLM Re1 + Re2 + Re3 Re1 + Re2 + Re3 = 0 ...      RIJ + ... + RIJ RKL + ... + RKL = 0 (8.2.3) IJKLM e1 eN−2 e1 eN−2 generates a closed algebra of vertex simplicity constraint operators. Under the assumption that no linear combinations with different multi-indices are allowed 2, the set is maximal in the sense that adding new vertex constraint operators spoils closure.

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

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

h e e0 i 1 e e e e R ,R = δ 0 (η R + η R − η R − η R ) (8.2.4) IJ KL 2 e,e JK IL IL JK IK JL JL IK 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. The same is also true for the elements in (8.2.3): Taking the ordering from above, every constraint operates as an infinitesimal rotation on all constraints prior in the list. Since the commutator is antisymmetric 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 simplicity constraints as given in (8.2.2) and (8.2.3), adding any other linear combination of the building blocks (8.2.1) spoils the closure of the algebra. To this end, we make the most general Ansatz

X IJ KL αij IJKLM Ri Rj (8.2.5) 1≤i

 IJ KL AB CD AB CD
 IJKLM R12 R12 , ABCDE α13R1 R3 + α23R2 R3 + ... N−1 X IJ KL AB MN CD ≈ 2α1j IJKLM R2 f MN R1 ABCDERj j=3 N−1 X IJ KL AB MN CD + 2α2j IJKLM R1 f MN R2 ABCDERj j=3 N−1 X IJ KL AB MN CD ≈ 2(α1j − α2j) IJKLM R2 f MN R1 ABCDERj , (8.2.6) j=3 where we dropped terms proportional to (8.2.2) in the first and in the second step. For a closing algebra, the right hand side of (8.2.6) necessarily has to be proportional to (a linear combination of) 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 the contributions 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) are proportional to ABC IJ K δIJK (Rj)AB(R2) (R1) C , (8.2.7) where we used the notation δI1...In := n! δI1 δI2 ...δIn . To show that this expression can not be J1...Jn [J1 J2 Jn] 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

ABCE IJ K δIJKM (Rj)AB(R2) (R1) C . (8.2.8)

48 Whatever multi-index E we might have chosen in the Ansatz (8.2.5), we can always restrict attention to those simplicity constraints in the maximal set which have the same multi-index M = E. Then, the same calculation as in the case of D = 3 shows that the antisymmetrisations of 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 of IJ KL IJKLM R1 R2 . The second option (b) is, for j = 3, excluded by our choice of αij and we must 0 have α13 = α23 = 0. Next, consider j = 4 and suppose we have α14 = α24 := α 6= 0. Then, 0 0 0 IJ KL 0 IJ KL we can define α34 := α34 − α and find the terms α IJKLM R123R4 + α34IJKLM R3 R4 in (8.2.5). The first term again is already in the chosen set, which implies we can set α14 = α24 = 0 0 w.l.o.g. by changing α34 → α34 (We will drop the prime in the following). This immediately generalises to j > 4, and we have w.l.o.g. α1j = α2j = 0 (3 ≤ j < N). IJ KL Suppose we have calculated the commutators of IJKLM R1...iR1...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,

  N−1  IJ KL X AB CD IJKLM R1...(n+1)R1...(n+1), ABCDE  α(n+1)jR(n+1)Rj + ... j=n+2 N−1 X IJ KL AB MN CD ≈ 2α(n+1)j IJKLM R1...nf MN R(n+1)ABCDERj , (8.2.9) j=(n+2) which, by the reasoning above, again is not a linear combination of any simplicity building blocks for 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 the same as the diagonal simplicity constraints acting on edges of γ and can be solved by demanding the 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 the intertwiner ι at v: Couple the representations on e1 and e2, then couple this representation to e3, and so forth, see fig. 8.1. We call the intermediate virtual edges e12, e123, ... and denote the highest weights of the representations thereon by Λ~ 12, Λ~ 123,... Since we can use gauge invariance at all the intermediate intertwiners in the recoupling scheme, e.g., Re1 + Re2 = Re12 , we have IJ IJ
KL KL
IJ KL IJKLM Re1 + Re2 Re1 + Re2 = IJKLM Re12 Re12 = 0 (8.2.10) and thus that the representation on e12 has to be simple, i.e.

Λ~ 12 = (λ12, 0, ..., 0) λ12 = 0, 1, 2, ... (8.2.11) Using the same procedure, all intermediate representations are required to be simple and the intertwiner is labeled by N − 3 “spins” λi ∈ N0. We call an intertwiner where all internal lines are labeled with simple representations simple. SU(2) Spin(D+1) Denote by IN the set of N-valent SU(2) intertwiners and by Is,N the set of N-valent simple Spin(D + 1) intertwiners. Recalling that an N-valent SU(2) intertwiner can be expressed in the same recoupling basis and calling the intermediate spins ji, we see that the map Spin(D+1) SU(2) F : Is,N → IN 1 λ 7→ j (8.2.12) 2 i i

49  3  4  2  5  123   12 1234

  1 N

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-Lewandowski measures, see [37]). The motivation for the factor 1/2 comes from the fact that Λ~ = (1, 0) in 3 D = 3 corresponds to the familiar j+ = j− = 1/2 and the area spacings of the SO(4) and the 4 SU(2) based theories agree using this identification, cf [37].  2 8.2.3 Remarks  5  l  l 123 l12 1. Since the choice of the maximall closing123 subset of the simplicity constraint operators is 1234  arbitrary, no recoupling basis is preferred a priori. Onl12 the SU(2) level, a change in the  recoupling scheme amounts to al change1234 of basis in the intertwiner space and therefore poses no problems. On the level of simple Spin(D + 1) representations however, a choice in the recoupling scheme affects the property “simple”, since the non-commutativity of constraint  1 operators belongingN to different recoupling schemes means that kinematical states cannot have the property simple in both schemes.

2. There exist recoupling schemes which are not included in the above procedure, e.g., take N = 6 and the constraints R12R12 = R34R34 = R56R56 = 0 and couple the three re- sulting simple representations. The theorem should however generalise to those additional recoupling 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 use a projector on the space of simple intertwiners in a certain recoupling scheme to restrict the 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 a procedure 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 solved in the weak operator topology (matrix elements with respect to solutions to the maximal subset).

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 r X 1 Ar [S] :=  πIJ (S )πKL(S )|. (8.2.13) M 4 IJKLM U U U∈U Notice, however, that for D > 3 these generalised area operators, just as the simplicity constraints, 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 disposal whose respective symplectic reduction reproduces the ADM phase space. One of them is the Ashtekar-Barbero-Immirzi connection formulation in terms of the gauge group SU(2) with additional SU(2) Gauß constraint next to spatial diffeomorphism and Hamiltonian constraint, and the other is our connection formulation in terms of SO(4) with additional SO(4) Gauß constraint and simplicity constraint. Both formulations are classically com- pletely equivalent and thus one should expect that also the quantum theories are equivalent in 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) theory is unitarily equivalent to the kernel of the SU(2) Gauß constraint of the SU(2) theory. To investigate this first from the classical perspective, we split the SO(4) connection and its IJ j ± conjugate momentum (A , πIJ ) into self-dual and anti-selfdual parts A±, πj ) which then turn out to be conjugate pairs again. It is easy to see that the SO(4) Gauß constraint GIJ ± splits into two SU(2) Gauß constraints Gj , one involving only self-dual variables and the other only anti-selfdual ones which therefore mutually commute as one would expect. The SO(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 yield + − a kinematical Hilbert space with an orthonormal basis Ts+ ⊗Ts− where S± are usual SU(2) invariant spin networks. The simplicity constraint, which in D = 3 is Gauß invariant and can be imposed after solving the Gauß constraint, from classical perspective asks that the ab a b jk double density inverse metrics q± = πj±πk±δ are identical. This is classically equivalent to the statement that corresponding area functions Ar±(S) are identical for every S. The corresponding statement in the quantum theory is, however, again anomalous because it is 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 only + − be satisfied on vectors of the form Ts+ ⊗ Ts− 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 the area operator is diagonal with an eigenvalue ∝ pj(j + 1) and we can always arrange such an intersection situation by choosing suitable S. By a similar argument one can show that the intertwiners at the edges have to be the same. But this is only a sufficient condition because 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 Ts ⊗ Ts for the same spin network s = s+ = s−. The desired unitary map between the Hilbert spaces would therefore simply + − be Ts 7→ Ts ⊗ Ts . This can be justified abstractly as follows: From all possible area operators pick a maximal ± commuting subset Arα using the axiom of choice (i.e. pick a corresponding maximal set ± of surfaces Sα). We may construct an adapted orthonormal basis Tλ diagonalising all of

51 3 ± ± ± them such that Arα Tλ = λαTλ . Now the constraint + − Arα ⊗ 1 = 1 ⊗ Arα can be solved on vectors T + ⊗ T − by demanding λ = λ . The desired unitary map λ+ λ− + − + − would then be Tλ 7→ Tλ ⊗Tλ . 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. We leave this to future research. 7. In D 6= 3 the afore mentioned split into selfdual and anti-selfdual sector is meaningless and we 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 the graphs. Of course, before imposing the spatial diffeomorphism constraint these graphs are embedded into spatial slices of different dimension and thus provide different amounts of degrees of freedom. However, after implementation of the diffeomorphism constraint, most of the embedding information will be lost and the graphs can be treated almost as abstract combinatorial objects. Let us neglect here, for the sake of the argument, the possibility of certain remaining moduli, depending on the amount of diffeomorphism invariance that one imposes, which could a priori be different in different dimensions. In the case that the proposed 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 concept dictated by the choice of semiclassical states which provide the necessary embedding infor- mation. A possible caveat to this argument is the remaining Hamiltonian constraint and the algebra of Dirac observables which critically depend on the dimension (for instance through the volume operator or dimension dependent coefficients, see [35, 36]) and which could 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 be forced on us by demanding anomaly freedom of the simplicity constraint operators. Of course, 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 therefore the speculation of dimension emergence could fail exactly when we try to construct the semiclassical sector with the solutions to the simplicity constraints advertised above. This would mean that our scheme is wrong. On the other hand, there are indications [93] that the semiclassical sector of the LQG Hilbert space already in D = 3 is entirely described in terms of 6-valent vertices. Therefore, the higher valent graphs which in D = 3 could correspond 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 limit to the valence of a graph, this would mean that already the D = 3 theory contains all higher-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 that the 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 room for degeneracies, that is the λα completely specify the eigenvector. We will assume this to be the case for the following argument.

52 because there is no well defined scheme that attributes quantum to classical degrees of freedom unless the Hilbert space takes the form of a tensor product, where each factor corresponds 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 from the constraint surface per point on the spatial slice. In order to count the quantum degrees of freedom that are removed by the quantum simplicity constraint when acting on a spin network 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 graph γ which is attributed to the spatial slice. This point is equipped with degrees of freedom labelled by edge representations and the intertwiner. Every edge incident at v is shared by 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. D+1 We take as edge degrees of freedom the b 2 c Casimir eigenvalues of SO(D + 1) labelling the irreducible representation. The edge simplicity constraint removes all but one of these D−1 Casimir eigenvalues, thus per edge b 2 c edge degrees of freedom are removed. Further, a gauge invariant intertwiner is labelled by a recoupling scheme involving N − 3 irreducible representations not fixed by the irreducible representations carried by the edges adjacent to the vertex in question, which are fully attributed to the vertex (there are N − 2 virtual edges coming from coupling 1,2 then 3 etc. until N but the last one is fixed due to gauge invariance). We take as vertex degrees of freedom these N − 3 irreducible representa- D+1 tions each of which is labelled again by b 2 c Casimir eigenvalues. The vertex simplicity D−1 constraint again deletes all but one of these eigenvalues, thus it removes (N − 3)b 2 c quantum degrees of freedom. We conclude that the quantum simplicity constraint removes N D − 1 (N − 3 + )b c 2 2 quantum degrees of freedom per point (N valent vertex) where N −3 accounts for the vertex and N/2 for the N edges counted with half weight as argued above. This is to be compared with the classical simplicity constraint which removes D2(D −1)/2−D degrees of freedom per point. Requiring equality we see that vertices of a definitive valence ND are preferred in D spatial dimensions which for large D grows quadratically with D. Specifically for D = 3 we find N3 = 6. Thus, our naive counting astonishingly yields the same preference for 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 and one thus would rather expect ND = 2D (hypercubulations) and this may indicate that our counting 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 weight one, it is most naturally smeared over (D − 1)-dimensional surfaces S. The regularisation of the objects Z b IJKLM I aJK b1 bD−1 S (S) := bLM (x) N (x)π (x)ab1...bD−1 dx ∧ ... ∧ dx , (8.3.1) S

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

ˆ ˆ X Sˆb(S)f = Yˆ bN (S)f = p∗ Yˆ bN (S)f = p∗ (e, S)IJKLM b (b(e))Nˆ I (b(e))RJK f γS γS γS γS LM e γS e∈γS (8.3.2) has to annihilate physical states for all surfaces S ⊂ σ and all semianalytic functions bIM of compact support, where pγ denotes the cylindrical projection and γS is a graph adapted to the surface 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 of the flux vector fields. Therefore, it is necessary and sufficient to demand that

IJKLM ˆ I JK  bLM (b(e)) N (b(e)) Re · fγ = 0 (8.3.3) for all points of γ (which can be be seen as the beginning point of edges by suitably subdividing and inverting edges). Since Nˆ I acts by multiplication and commutes with the right invariant vector fields, see [40] for details, the condition is equivalent to4

¯IJ Re · fγ = 0, (8.3.4) i.e. the generators of rotations stabilising N I have to annihilate physical states. Before imposing these 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 linear and the quadratic simplicity constraints are Poisson self-commuting. The quadratic constraint is 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 anomalous when using a non-zero Immirzi parameter (at least if γ 6= 1 in the Euclidean theory. But γ = 1 is ill-defined for SO(4), see e.g., [94]). Surprisingly, in the case at hand and without an Immirzi parameter in four dimensions, we do not find an anomaly. But that is just because the generators of rotations stabilising N I form a closed subalgebra! Direct calculation yields, choosing (without 0 loss of generality) γSS0 to be a graph adapted to both surfaces S, S ,   h b b0 0 i X IJ X AB Sˆ (S), Sˆ (S ) f = ... R¯ , ... R¯ 0 f γSS0 γSS0 γSS0  e e  γSS0 0 e∈γSS0 e ∈γSS0 X  ¯IJ ¯AB = ... Re , Re fγSS0 e∈γSS0 X I J A B KLCD MN = ... η¯ K η¯ Lη¯ C η¯ D f MN Re fγSS0 e∈γSS0 X I J A B  L][C D] [K  MN = ... η¯ K η¯ Lη¯ C η¯ D η δ [M δN] Re fγSS0 e∈γSS0 X ¯MN = ... Re fγSS0 , (8.3.5) e∈γSS0 where the operator in the last line is in the linear span of the vector fields Sˆb(S). The classical constraint algebra is not reproduced exactly (the commutator does not vanish identically), but

4 ¯ K L ¯ J Use the decomposition of XIJ into its rotational (XIJ :=η ¯I η¯J XKL) and “boost” parts (XI := −ζN XIJ ) with respect to N I in (8.3.3).

54 the algebra of quantum simplicity constraints closes, they are of the first class. Therefore, strong imposition of the quantum constraints does make mathematical sense. Note that up to now, we did not solve the Gauß constraint. The quantum constraint algebra of the simplicity and the Gauß constraint can easily be calculated and reproduces the classical result h i Sˆb(S), GˆAB[Λ ] p∗ f AB γS γS    ∗ X 0 IJKLM I JK X AB AB =p (e ,S) b (v)Nˆ (v)R 0 , Λ (v) R + R f γS  LM e AB  e N  γS 0 0 e ∈E(γS ),v=b(e ) e∈E(γS ),v=b(e) X   =p∗ Λ (v)IJKLM b (v) (e, S) Nˆ I (v) RJK ,RAB + ηI[AN B](v)RJK f γS AB LM e e e γS e∈E(γS ),v=b(e) X   =p∗ Λ (v)IJKLM b (v) (e, S) Nˆ I (v)2ηKARJB + ηI[AN B](v)RJK f γS AB LM e e γS e∈E(γS ),v=b(e) =Sˆ(−Λ·b)(S)p∗ f . (8.3.6) γS γS   where we used RAB := 1 N A ∂ − N B ∂ . It follows that the simplicity constraint operator N 2 ∂NB ∂NA does not preserve the Gauß invariant subspace (in other words, as in the classical theory, the Gauß constraint does not generate an ideal in the constraint algebra). This implies that the joint kernel of both Gauß and simplicity constraint must be a proper subspace of the Gauß invariant subspace. It is therefore most convenient to look for the joint kernel in the kinematical (non Gauß invariant) Hilbert space.

8.3.2 Solution on the vertices Consider a slight modification of the usual gauge-variant spin network functions, where the I intertwiners iv = iv(N) are square integrable functions of N . Let v be a vertex of γ and e1, . . . , en the edges of γ incident at v, where all orientations are chosen such that the edges are all outgoing at v. Then we can write the modified spin network functions

n Y   T ~~(A, N) := (iv(N)) ~ ~ πle (hei (A)) (Mv) ~ 0 ~ 0 γ,l,i K1...Kn i K~ K~ 0 K1...Kn i=1 i i   = tr i (N) · ⊗n π (h (A)) · M , (8.3.7) v i=1 lei ei v where Mv contracts the indices corresponding to the endpoints of the edges ei and represents the rest of the graph γ. These states span the combined Hilbert space for the normal field and the connection HT = Hgrav ⊗ HN (cf. [40]) and they will prove convenient for solving the simplicity constraints. Choose the surface S0 such that it intersects a given graph γ0 only in the vertex 0 0 ˆb 0 0 v ∈ γ . The action of S (S ) on the vertex v of a spin network Tγ0,~l,~i(A, N) implies with (8.3.4) that

ˆb 0 S (S )γ0 Tγ0,~l,~i(A, N) = 0  IJ  n  0 ⇐⇒ tr iv(N)¯τ · ⊗ π (he (A)) · Mv = 0 ∀e at v , (8.3.8) πle i=1 lei i where τ IJ here denote the generators of SO(D + 1) in the representation π of the edge e and πle le the bar again denotes the restriction to rotational components (w.r.t. N I ). The above equation implies that the intertwiner iv, seen as a vector transforming in the representationπ ¯le dual to

55 I πle of the edge e, has to be invariant under the SO(D)N subgroup which stabilises the N . By definition [78], the only representations of SO(D + 1) which have in their space nonzero vectors 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 spin foams [62]. It is easy to see that the dual representations of simple representations are simple representations. 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 vector

ξle (N) in the representationπ ¯le is unique, which implies that the allowed intertwiners iv(N) are given by the tensor product of the invariant vectors of all n edges and potentially an additional square integrable function F (N), i (N) = ξ (N) ⊗ ... ⊗ ξ (N) ⊗ F (N). Going over to v v le1 len v normalised gauge invariant spin network functions implies that Fv(N) = 1, and the resulting intertwiner space solving the simplicity and Gauß constraint becomes one-dimensional, spanned by I (N) := ξ (N)⊗...⊗ξ (N). We will call these intertwiners and vertices coloured by them v le1 len linear-simple. For an instructive example of the linear-simple intertwiners, consider the defining representation (which is simple since the highest weight vector is Λ = (1, 0,..., 0), cf. appendix A). The unit vector invariant under rotations (w.r.t. N I ) is given by N I and for edges in the IJ defining representation incoming at v we simply contract he NJ . If the constraint is acting on an interior point of an analytic edge, this point can be considered as a trivial two-valent vertex and the above result applies. Since this has to be true for all surfaces, a spin network function solving the constraint would need to have linear-simple intertwiners at every point of its graph γ, 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 is problematic, 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. The problem with this type of solution is that all intertwiners, even trivial intertwiners at all interior points of edges, have to be linear-simple, which is however in conflict with the definition of a cylindrical function, in other words, there would be no holonomies left in a spin network because every point would be a N-dependent vertex. It could be possible to resolve this issue using a rigging map construction [95, 96, 97] of the type

D pγ E η(Tγ,~l,~l ,~i)[Tγ0,~l0,~l0 ,~i0 ] := lim C pγ,Tγ,Tγ0 T ~~ ~ ,Tγ0,~l0,~l0 ,~i0 , (8.3.9) N N Pγ 3pγ →∞ γ,l,lN ,i N kin

N where Pγ is the set of finite point sets p of a graph γ, p = {{xi}i=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 the pγ sense of net convergence with respect to Pγ. By the prescription T we mean the projection γ,~l,~l ,~i N
of T ~~ onto linear-simple intertwiners at every point in p and C pγ,Tγ,Tγ0 is a numerical γ,l,lN ,~i factor. Assuming this to work, consider any surface S intersecting γ0. We (heuristically) find

ˆb
D p ˆb E η(Tγ)[S (S)Tγ0 ] = lim C pγ,Tγ,Tγ0 Tγ , S (S)Tγ0 Pγ 3pγ →∞ kin
D ˆb † p E = lim C pγ,Tγ,Tγ0 [S (S)] Tγ ,Tγ0 Pγ 3pγ →∞ kin
D ˆb p E = lim C pγ,Tγ,Tγ0 S (S)Tγ ,Tγ0 = 0, (8.3.10) Pγ 3pγ →∞ kin

56 b since the intersection points of S with γ will eventually be in pγ and Sˆ (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 to implement the requirement, that the rigging map has to commute with observables. It therefore seems plausible to look for non-standard quantisation schemes for the linear simplicity constraint operators, at least when acting on edges. Comparison with the quadratic simplicity constraint suggests that also the linear constraint should enforce simple representations on the edges, see the 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 imposed simplicity 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 taking into account holonomies along edges e0 originating at v and ending in a 1-valent vertex v0. Since e0 and v0 are assigned in a unique fashion to v if the valence of v is at least 2, we can consider the set {v, e0, v0} as a new “non-local” intertwiner. Since we can label e0 with an arbitrary simple representation, we get an infinite set of intertwiners which are orthogonal in the above scalar product. This interpretation however does not mimic the classical imposition of the simplicity constraints or the above imposition of the quadratic simplicity constraint operators.

2. The main difference between the formulation of the theory with quadratic and linear simplicity constraint respectively is the appearance of the additional normal field sector in the linear case. Thus one could expect that one would recover the quadratic simplicity constraint formulation by ad hoc averaging the solutions of the linear constraint over the normal field dependence with the probability measure νN defined in [40]. Indeed, if one does so, then one recovers the solutions to the quadratic simplicity constraints in terms of the Barrett-Crane intertwiners in D = 3 and higher-dimensional analogs thereof as has been shown long ago by Freidel, Krasnov, and Puzio [62]. Such an average also deletes the solutions with “open ends” of the previous item by an appeal to Schur’s lemma. Since after such an average the N dependence of all solutions disappears, we can drop the µN integral in the kinematical inner product since µN is a probability measure. The resulting effective physical scalar product would then be the Ashtekar-Lewandowski scalar product of the theory between the solutions to the quadratic simplicity constraints. Such an averaging would also help with the solution of the edge constraints, since a 2-valent linear-simple intertwiner is averaged as Z ¯α β 1 αβ dν(N) ξl (N)ξl (N) = δ , (8.3.11) SD dπl thus yielding a projector on simple representations.

3. It can be easily checked that the volume operator as defined in [37], and therefore also more general operators like the Hamiltonian constraint, do not leave the solution space to the linear (vertex) simplicity constraints invariant. A possible cure would be to introduce a projector PS on the solution space and redefine the volume operator as Vˆ := PSVˆ PS. Such procedures are however questionable on the general ground that anomalies can always be removed by projectors.

57 4. If one accepts the usage of the projector PS, calculations involving the volume operator simplify tremendously since the intertwiner space is one-dimensional. We will give a few examples which can be calculated by hand in a few lines, restricting ourselves to the defining representation of SO(D + 1), where the SO(D)N invariant unit vector is given by N I . Having direct access to N I , one can base the quantisation of the volume operator on the classical expression

1 1 D−1 I a1K1J1
aDKDJD
det q = IJ ...J N π NK ... π NK a ...a . (8.3.12) D! 1 D 1 D 1 D 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 Z Z D D Vˆ (R) = d p |det(\q)(p)| γ = d p Vˆ (p)γ, (8.3.13) R R D   D−1 X Vˆ (p) = ~ δD(p, v)Vˆ , (8.3.14) γ 2 v,γ v∈V (γ) 1 (D−1) D ˆ i X Vv,γ = s(e1, . . . , eD)ˆqeq,...,e , (8.3.15) D! D e1,...,eD∈E(γ), e1∩...∩eD=v     qˆ =  Nˆ I RK1J1 Nˆ ... RKDJD Nˆ . (8.3.16) e1,...,eD IJ1...JD e1 K1 eD KD

Note that the operatorq ˆe1,...,eD is built from D right invariant vector fields. Since these are antisymmetric,q ˆT = (−1)Dqˆ . In the case at hand, we have to use the e1,...,eD e1,...,eD projectors PS to project on the allowed one-dimensional intertwiner space, the operator PSqˆPS therefore has to vanish for the case D + 1 even (an antisymmetric matrix on a one- 2 2 dimensional space is equal to 0). However, the volume operator depends onq ˆ , and PSqˆ PS actually is a non-zero operator in any dimension, though trivially diagonal. Therefore, also Vˆ is diagonal. The simplest non-trivial calculation involves a D-valent non-degenerate (i.e. no three tangents to edges at v lie in the same plane) vertex v where all edges are labelled by the defining representation of SO(D +1) and thus the unique intertwiner which we will denote A A by N 1 ...N D . We find

 1D A1 AD IA1...AD qˆe ,...,e N ...N = s(e1, ..., eD) − NI  , 1 D 2 1D IA1...AD A1 AD qˆe ,...,e NI  = s(e1, ..., eD) D! N ...N , 1 D 2 ! 1 1D 2(D−1) ˆ A1 AD A1 AD Vv N ...N = D! N ...N , (8.3.17) 4

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

58 i.e. for those special vertices, the volume operator preserves the simple vertices. For vertices of higher valence and/or other representations, we need to use the projectors. Of special interest are the vertices of valence D + 1 (triangulation) and 2D, where every edge has exactly one partner which is its analytic continuation through v (cubulation). We find ! 1 1D 2(D−1) ˆ A1 AD+1 A1 AD+1 Vv N ...N = (D + 1)! N ...N , 4 ! 1 1D 2(D−1) ˆ A1 A2D A1 A2D Vv N ...N , cubic = (D)! N ...N , cubic .(8.3.18) 2

The dimensionality of the spatial slice now appears as a quantum number like the spins labelling the representations on the edges and it could be interesting to consider a large dimension 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 linear J aKL (γ) bKL b K L (D) constraint IJKLN π ≈ 0 while having {AaIJ (x), π (y)} = 2δaδ[I δJ]δ (x − (γ) aIJ aIJ IJKL a y) with π = π + 1/(2γ)√ π KL, the linear simplicity constraint operators become 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 that I KL KLMN MN these anomalous constraints require IJKLN (Re − 1/(2γ) Re ) · fγ = 0. Since I KL KLMN MN IJKLN (Re − 1/(2γ) Re ) do not generate a subgroup, the constraint can not be 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 for the quadratic simplicity constraints, to weaken the imposition of the constraints at the quantum level. The basic difference between the linear and the quadratic simplicity constraints is that the time normal N I is left arbitrary in the quadratic case and fixed in the linear case. In order to 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 the quadratic constraints, we could choose the subset J KL KL
IJKLM N Re1 + Re2 = 0 J KL KL KL
IJKLM N Re1 + Re2 + Re3 = 0 ...    N J RKL + ... + RKL = 0 (8.3.19) IJKLM e1 eN−2 for each N-valent vertex plus the edge constraints. As above, the choice of the subset specifies a recoupling scheme and the imposition of the constraints leads to the contraction of the virtual edges and virtual intertwiners of the recoupling scheme with the SO(D)N -invariant vectors ξ (N) and their complex conjugates ξ¯ (N), see fig. 8.2. Gauge invariance can still be used lei lei P ¯ at each (virtual) vertex in this calculation in the form i Rei = 0, which is sufficient since ¯ only Rei appears in the linear simplicity constraints. If we now integrate over each pair of ξ (N) “generated” by the elements of the proposed subset of the simplicity constraint operators lei separately, we obtain projectors on simple representations for each of the virtual edges in the recoupling scheme. The integration over N I for the edge constraints yields projectors on simple representations in the same manner. Finally, we obtain the simple intertwiners of the quadratic operators in addition to solutions where incoming edges are contracted with SO(D)N -invariant vectors ξ (N). A few remarks are appropriate: lei

59  3  4  2  5  123   12 1234

  1 N

 3  4  2  5  l  l 123 l12 l 123 1234  l12  l1234

  1 N

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

6. Although this procedure yields a promising result, it contains several non-standard and ah-hoc steps which have to be justified. One could argue that the “correct” quantisation of 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-equivalent quantum 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 vacuum theory, integration over every point in σ gives the Barrett-Crane intertwiner for the edges contracted with SO(D)N -invariant vectors. This type of integration would also get rid of the 1-valent vertices and thus allow for a natural unitary map to the quadratic solutions as already mentioned above.

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

8.3.5 Mixed quantisation Since the implementation of the quadratic simplicity constraints described above yields a more promising result than the implementation of the linear constraints, we can try to perform a mixed quantisation by noting that we can classically express the linear constraints for even D in the form 1  πaIJ πbKL ≈ 0,N I − nI (π) ≈ 0. (8.3.20) 4 IJKLM The phase space extension derived in [40] remains valid when interchanging the linear simplicity constraint for the above constraints. The reason for restricting D to be even is that we have an explicit expression for nI (π), see [35, 36]. Since a quantisation of nI (π) will most likely not commute with the Hamiltonian constraint operator, we resort to a master constraint. Note that

60 the expression I I √ D−1
J J √ D−1
0 (N − n (π)) q δIJ (N − n (π)) q M := √ , (8.3.21) N q2D−3 which is the densitised square of N I − nI (π), can be quantised as

√ q ˆ 0 \3−2D I I M N = 2 q ( |Vˆ VˆI | − NI Vˆ ), (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 √ A1 A2 AD BA1...AD s(e1, ..., eD) D!|N N ...N > +|NB > (8.3.23) is annihilated by Mˆ N , which can be easily checked when using the results of the volume operator acting on the solution space of the full set of linear simplicity constraint operators. In order to turn the expression into a well defined master constraint operator, we have to square it again and to adjust the density weight, leading to

 q † q √\5/2−2D I I √\5/2−2D I I Mˆ N = 4 q ( |Vˆ VˆI | − NI Vˆ ) q ( |Vˆ VˆI | − NI Vˆ ), (8.3.24) which is by construction a self-adjoint operator with non-negative spectrum. We remark that it was necessary to use the fourth power of the classical constraint for quantisation, because the second power, having the desired property that its solution space is not empty, does not qualify as a well defined master constraint operator in the ordering we have chosen. There exists however no a priori reason why one should not take into account master constraint operators constructed from higher powers of classical constraints [75]. Curiously, the quadratic simplicity constraint operators as given above do not annihilate the solution displayed. Clearly, the calculations will become much harder as soon as vertices with a valence higher than D are used, since the building blocks 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 application to using EPRL intertwiners is outlined. In contrast to the earlier assumption of D being even in order to have an explicit expression nI (π), we can also perform the mixed quantisation using I J I J I J n n (π) as given in (4.1.12) and the constraint NJ (n n (π) − N N ) ≈ 0. For the application proposed, we will only need that the corresponding master constraint can be regularised such that 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 foam literature to the proposals in this chapter. In short, the main conclusion will be that in the case of four spacetime dimensions, many results from the spin foam literature can be used also in the canonical framework. However, they fail to work in higher dimensions due to special properties of the four dimensional rotation group which are heavily used in spin foams. We will not comment on results based on coherent state techniques [61, 86, 87, 98] since we do not see a resemblance 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 the reformulation of general relativity as a BF theory subject to the simplicity constraints, and thus similar to the point of view taken in this thesis. The crucial difference however is that while spin foam models start classically from discretised general relativity, the canonical approach discussed here starts from its continuum formulation. When looking at the simplicity constraints, this difference 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 of spacetime, the set of (D − 1)-surfaces is fixed by foliating the discretised spacetime. Restricting to 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 can IJ IJ R IJ take the bivectors B integrated over the individual faces of a tetrahedron, Bf (t) := f B , as the basic variables and the quadratic (off)-diagonal simplicity constraints read [67]

IJ KL Cff := IJKLBf (t) Bf (t) diagonal simplicity, ∀f ∈ t (8.4.1) IJ KL 0 Cff 0 := IJKLBf (t) Bf 0 (t) off-diagonal simplicity, ∀f, f ∈ t. (8.4.2)

In the continuum formulation however, we have to consider all possible (D − 1)-surfaces, and thus also hypersurfaces containing the vertex v dual to the tetrahedron t. The resulting flux IJ operators a priori contain a sum of right invariant vector fields Re acting on all the edges e connected to v. While this poses no problem for the diagonal simplicity constraints which act on edges of the spin networks as shown in section 6.3, the off-diagonal simplicity constraints arising when both surfaces contain v are not given by (8.4.2), but by sums over different Cff 0 , see [37] for details. It can however be shown by suitable superpositions of simplicity constraints associated to different surfaces that (8.4.2) is actually implied also by the quadratic simplicity constraints 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 also in 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 properly regularised 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 the canonical treatment since the generalised vielbeins do not have timelike tensorial indices after being pulled back to the spatial hypersurfaces.

8.4.2 Projected spin networks Projected spin networks were originally introduced in [99, 100] to describe Lorentz covariant formulations of loop quantum gravity, meaning that the internal gauge group is SO(1, 3) (or SL(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 quantum gravity formulated in terms of the usual SU(2) connection and its conjugate momentum are orthogonal to the time normal field, one performs projections in the spin networks from the full gauge group SO(1, 3) to a subgroup stabilising the time normal. Since the projector transforms covariantly under SO(1, 3), a (gauge invariant) projected spin network is already defined by its evaluation for a specific choice of the time normals and the resulting effective gauge invariance is only SU(2), which exemplifies the relation to the usual SU(2) formulation in the time gauge xI (= 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 the canonical framework discussed in this thesis which we will explain now. While the extension of projected spin networks to different gauge groups has already been discussed in [100], there is a subtle problem associated with the part of the connection which is projected out by the projections, that could not have been anticipated by looking at loop quantum gravity in terms of the Ashtekar-Barbero variables. There, the physical information in the connection, the extrinsic curvature, is located in the rotational components of the connection. To see this, consider in four dimensions the 2-parameter family of connections discussed in [36]6, 1 A = Γhyb + βK + γ  KLK , (8.4.3) aIJ aIJ aIJ 2 IJ aKL where γ corresponds to the Barbero-Immirzi parameter restricted to four dimensions and β is the new free parameter appearing in any dimension. As in (3.2.26), we decompose KaIJ as

¯ ¯ trace ¯ trace free KaIJ = 2N[I Ka|J] + KaIJ + KaIJ , (8.4.4)

I J where K¯aIJ means that N K¯aIJ = N K¯aIJ = 0 and the trace / traceless split is performed with respect to the hybrid vielbein. The extrinsic curvature which we need to recover from AaIJ ¯ ¯ trace ¯ trace free is located in KaJ , whereas KaIJ vanishes by the Gauß constraint and KaIJ 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 dimensions or 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 if one would just project the projected spin network at the intertwiners, but when one tries to go to fully projected spin networks as proposed in [99]. Then, since one would take a limit of inserting projectors at every point of the spin network, the physical information in the connection would be completely lost. Next to this problem, there are other problems associated to taking an infinite refinement limit for projected spin networks as discussed by Alexandrov [99] and Livine [100], e.g. that fully projected spin networks are not spin networks any more (since they only contain vertices and no edges) and, connected with this problem, that the trivial bivalent vertex, the Kronecker delta, 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 to projected spin networks where one projects the incoming and outgoing edges at the intertwiner in 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 I N J at every point of the spin network, whereas one would need to insert the the Kronecker delta δIJ to achieve cylindrical consistency 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 fact that we do not know of an analogue of the Barbero-Immirzi parameter in higher dimensions which would allow us to put the extrinsic curvature also in the rotational components of the connection. In four dimensions on the other hand, this problem would be absent and one would be 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 prefer this parametrisation to make our point clear.

63 section 8.3. Therefore, using projected spin networks in four dimensions with non-vanishing Barbero-Immirzi parameter is an option for the canonical framework developed in this thesis and 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 are implemented with a master constraint construction [67] or weakly [101]. Furthermore, the Barbero-Immirzi parameter is a necessary ingredient. We restrict here to the Euclidean model since its group theory is much closer to the connection formulation with compact gauge group SO(D + 1). While the diagonal simplicity constraints give the well known relation

γ + 12 (j+)2 = (j−)2, (8.4.5) γ − 1

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

 2j− 2  2j+ 2 k2 = = (8.4.6) 1 − γ 1 + γ where k is the quantum number associated to the Casimir operator of the SU(2) subgroup stabilising 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 networks with arbitrary valency [70] is then constructed by first coupling the two SU(2) subgroups of SO(4) holonomies in the representations (j+, j−), calculated along incoming and outgoing edges to the intertwiner, to the k representation. Then, the k representations associated to each edge are coupled via an SU(2) intertwiner and the complete construction is then projected into the set 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 D E ˆ ext φ C ψ = 0 ∀ |φi , |ψi ∈ H . (8.4.7)

In this approach, one can also show that the volume operator restricted to Hext has the same spectrum as in the canonical theory, which is an important test to establish a relation between the canonical theory and the EPRL model. Closely related to what we already observed in the previous subsection on projected spin networks, 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 would have to restrict to four dimensions with non-vanishing γ if we would want to use EPRL solution to the simplicity constraints. One upside of this solution when comparing to our proposition for solving the quadratic constraints is that no choice problem occurs, i.e. if we map the quantum numbers of the EPRL intertwiners to SU(2) spin networks, a change of recoupling basis in the SU(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

7 Note that these ~ corrections are necessary since the master constraint, by construction, has the same solution space as the original constraint [75], i.e. Cˆ†Cˆ |ψi = 0 implies Cˆ |ψi = 0. In the master constraint language, one subtracts an operator from the master constraint which vanishes in the classical limit to obtain a sufficiently large solution space.

64 of the Hamiltonian constraint is however, to the best of our knowledge, not circumvented when using EPRL intertwiners. Also, in order to use the EPRL solution in the canonical framework, one would have to discuss exactly 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 guaranteed at the classical level if one changes from the quadratic constraints to linear constraints from one point on the spatial hypersurface to another. The mixed quantisation proposed in section 8.3.5 can be seen as an attempt to using both the time normal as an independent variable as well as quadratic simplicity constraints. In this case, the main difference is the presence of an additional constraint relating the time normal constructed from the generalised vielbeins to the independent time normal (which could be used in the linear simplicity constraints). In section 8.3.5, this additional constraint was regularised as a master constraint which acts only on vertices. Taking the point of view that one can freely change between using the quadratic constraints plus this additional constraint or the linear constraints, one could choose the linear constraints for vertices and the quadratic constraints for edges. Since we can use a factor ordering for the master constraint where a commutator between a holonomy and a volume operator is ordered to the right, the master constraint would vanish on edges and only the quadratic simplicity constraints would have to be implemented, which are however not problematic. At vertices, 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 considers it natural or not to use both linear and quadratic constraints in the same formulation is a matter of personal taste. Nevertheless, it would be desirable to have only one kind of simplicity constraints. As a last remark, we point out that the non-commutativity of the linear simplicity constraints in the EPRL model results from using γ 6= 0 and thus we are not faced with this problem in higher dimensions. Essentially, as discussed in more detailed in remark 5 of section 8.3.4, while the rotations stabilising N I form an SO(D) subgroup of SO(D +1), the linear simplicity constraints in 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 any indication that the solution space is identical to previous results (up to the fact that it has the same “size” as SU(2) spin networks). In the spin foam literature, the linear simplicity constraints are cornerstones of the new spin foam models and have been introduced since the quadratic simplicity constraints acting on vertices do not commute. While the methods for treating supergravity introduced in the third part of this thesis necessarily need an independent time 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 in four dimensions). Therefore, one should not dismiss the quadratic constraints, especially since the linear constraints come with their own problems in the canonical approach. The solution presented in section 8.2 is certainly not free of problems, most prominently the choice of the maximal 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) intertwiners make it look very promising. The linear simplicity constraints come with their own set of problems, many of which were already 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-dimensional supergravity which will be developed in the third part of this thesis makes it necessary to use an independent time normal as an additional phase space variable. This time normal would naturally point towards using linear simplicity constraints, although the mixed quantisation of section 8.3.5 could avoid this. Since there is no anomaly appearing when using the linear simplicity 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 this point, it seems to be best to let oneself be guided by physical intuition and the results from the 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 however stress that these methods are, as said, ad hoc and they don’t follow from standard quantisation procedures. The mixed quantisation discussed at the end of section 8.3 also does not seem completely 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, in section 8.4.3, an application to EPRL intertwiners is outlined which could avoid this problem by using linear simplicity constraints for the vertices. The strength of the mixed quantisation is thus that it provides a mechanism to incorporate both the quadratic simplicity constraints as well as an independent time normal in the same canonical framework, which is what is done on the path integral side in the EPRL model. A comparison to results from the spin foam literature, especially projected spin networks and the EPRL model, shows that many of the problems connected with using the linear simplicity constraints have already been known, partly in different guises. While using these known results in our framework seems to be a viable option in four dimensions, we are unaware of possible ways to 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 constraints which appear in the connection formulation of general relativity derived in this thesis in any dimension D ≥ 3 and found that none of the presented ideas are entirely satisfactory at this point and further research on the open questions needs to be conducted. We hope that the discussion 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 freedom in loop quantum gravity type quantisations of higher-dimensional general relativity discussed in this thesis. While the techniques developed for scalar fields and gauge fields directly carry over to higher dimensions, some care has to be taken when dealing with fermions, since despite their Lorentzian nature, they will have to be incorporated in the canonical description of the first part of this thesis, thus transforming under Spin(D + 1) as opposed to Spin(1,D). The original work on which this chapter is based is [38].

As for gauge fields with compact gauge group, the treatment is identical to the one for the standard (3+1)-dimensional case as discussed in [20, 21]. Essentially, one constructs holonomies from the gauge fields in direct analogy to the construction of the Ashtekar-Lewandowski Hilbert space. Due to the compactness of the gauge groups, the projective limit associated to an infinite refinement of the graph can be taken and we can construct generalised spin networks which 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 edges and intertwiners for the matter gauge groups at the vertices. The emerging picture is very similar to lattice gauge theory, with the main difference that the graphs on which the holonomies are defined are dynamical objects of the theory. Also for scalars, the techniques of [20, 21] directly carry over to higher dimensions. Here, it is important to either restrict to scalars which transform in the trivial or the adjoint representation of 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 these representations. The standard model however falls into this class of theories, so that we can easily live with this restriction. For spin 1/2 fields, the situation is more complicated since they transform in the spinor representation of SO(1,D) or, in the (time) gauge fixed version, of SO(D). Thus, a naive extension to SO(D + 1) as the internal gauge group for the gravitational degrees of freedom seems to be problematic since the fermions living at the vertices of the spin networks cannot be contracted any more with the incoming holonomies to form gauge invariant objects, since gauge invariance would be spoiled due to the different gauge groups. However, it turns out that, at least 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 reality conditions need to be satisfied for Dirac spinors, we do not get any problems due to different factors of i in which the generators of SO(1,D) and SO(D + 1) differ. In order to establish equivalence between the theory where SO(1,D) and SO(D +1) act, we will require that they are identical 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 an extended ADM phase space which is invariant under a SO(D) gauge symmetry. For this, we a start at a phase space coordinatised by a densitised D-bein Ei and its conjugate momentum i Ka. The relation to the ADM phase space is given by

ab a b ij i √ −1 c qq = Ei Ej δ ,Kab = K(aqb)c q Ei (9.1) and the ADM constraints are expressed accordingly. It is a well known result that using the new Poisson bracket i b (D) b i {Ka(x),Ej (y)} = δ (x − y)δaδj (9.2) a amounts to a canonical transformation when introducing the Gauß constraint E[iKa|j] = 0 and thus 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 dimension of the internal space is just one dimension larger in the SO(D + 1) case, it is enough to use a normal field nI which is orthogonal to to all objects in the SO(D) theory, but transforms under the 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 normal nI . The precise canonical transformation associated to this train of thought is given by

aI I aKJ J K E¯ = ζη¯ J π nK , K¯aI = ζη¯I (A − Γ)aKJ n , (9.3)

aIJ I where π and AaIJ are the new canonical variables, ΓaIJ is the hybrid connection, and n is the normal which can be constructed from πaIJ as described in equation (4.1.12). We remark that it is only necessary to construct nI nJ (π) because nI always appears in such a combination in all the constraints for even D + 1, and we can use (4.1.12) for D + 1 odd. The explicit calculation of the canonicity of the transformation however can rely on abstract properties of nI , as demonstrated in [38]. It is also possible to introduce an independent normal N I as described in section 5.2. Es- bj
bKL I
sentially, the extension of Kai,E with SO(D) Gauß constraint to AaIJ , π ,N ,PJ with SO(η) Gauß, linear simplicity and normalisation constraint works exactly the same way and the calculations can be copied nearly verbatim. We can even choose to simplify the replacement of the vielbein and extrinsic curvature using the normal N I ,

aI I aKJ J K E¯ = ζη¯ J π NK , K¯aI = ζη¯I (A − Γ)aKJ N , (9.4)

I whereη ¯IJ now is understood as a function of N . 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 second part of this chapter and amended by the corresponding techniques for fermions as introduced in [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 problem arising in the construction of such an expression is that many counterterms are needed when one 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 from a non-vanishing trace free part of K¯aIJ in the presence of fermions. Furthermore, the rather unnatural SO(D + 1) gauge invariance also leads to counterterms when one wants to establish equality 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 general relativity in D + 1 dimensions with compact gauge group SO(D + 1) such that the connection is Poisson commuting, which implies that loop quantum gravity quantisation methods apply. We also provided the coupling to standard matter. In this and the next chapter, we extend our methods to derive a connection formulation of a 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 from a Hamiltonian formulation in the time gauge which yields a Spin(D) theory, a major challenge is 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 Majorana fermion for the Lorentzian Clifford algebra and Majorana representations of the Clifford algebra are 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 non trivial Dirac antibracket ∗-algebra for the Majorana field which significantly differs from the analogous construction for Dirac fields already available in the literature. The original work from 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 formed field of supergravity, driven by the hope to construct a theory of quantum gravity without the shortcoming of perturbative non-renormalisability. Werner Nahm classified in 1977 all possible supergravities, arriving at the result that, under certain assumptions, d = 11 was the maximal number of Minkowski signature spacetime dimensions in which supergravities could exist [6]. In the following year, d = 11 supergravity was constructed by Cremmer, Julia and Scherk [48] in order to obtain d = 4, N = 8 maximal supergravity by dimensional reduction. Various forms of supergravity were derived in dimensions d ≤ 11 and relations among them were discovered in the subsequent years [103]. While the initial hope linked with perturbative supergravity was vanishing due to results suggesting its non-renormalisability [104] and the community turned to superstring theory, loop quantum gravity (LQG) emerged as a new candidate theory for quantum gravity after Ashtekar discovered his new variables in 1986 [7]. As described before, LQG is formulated in an entirely non-perturbative and background-independent way and suggests the appearance of a quantum geometry 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 would be very interesting to compare and merge the results coming from these two different approaches to 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 are only 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 respective supergravities. It is therefore interesting to study quantum supergravity as a means of probing the 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], where the closed bosonic string has been quantised using rigorous background-independent techniques, resulting in a new solution of the representation problem which differs from standard string theory. Also, the Hamiltonian formulation of the algebraic first order bosonic string and its relation 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, contrary to 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-dimensional general relativity, whose action is an integral part of all supergravity theories, has been given in the 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 much linked 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 have simple and metric independent Poisson brackets for the Rarita-Schwinger field ψα, one should √ a α 4 a α use half-densitised internally projected fields φi := qei ψa . This field redefinition has to be changed in order to work in the new internal space, more specifically we have to ensure that the number 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 to ensure that no inconsistencies arise when using SO(D + 1) instead of SO(1,D) as the internal gauge group. Concretely, this will be achieved in dimensions where Majorana representation for the 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 SUGRA theories does not pose any problems for this classical canonical transformation. However, we must provide background independent representations for these fields in the quantum theory which, 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-index photon present in 11d SUGRA with Chern-Simons term. Scalars, fermions and connections of compact, 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 also mention 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 canonical supergravity 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 structure of the Rarita-Schwinger field in the time gauge in convenient variables which will be crucial for a 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 Dirac anti bracket geared to Majorana spinor fields rather than Dirac spinor fields. In section 10.5 we show 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 supergravity theories. 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 will settle the notation.

10.2.1 Status of canonical supergravity Hamiltonian formulations of supergravity are a tedious business due to the complexity of the Lagrangians and the appearance of constraints. Nevertheless, the canonical structure emerging is very similar for the explicitly known Hamiltonian formulations. To the best of our knowledge, the D +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 of D + 1 = 3 + 1 supergravity was first computed by Henneaux [117] up to terms quadratic in the constraints [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 calculation of the constraint algebra of D + 1 = 3 + 1 supergravity in [120] where a term quadratic in the Gauß constraint appears in the Poisson bracket of two supersymmetry constraints. The constraint algebra for D + 1 = 9 + 1, N = 1 supergravity coupled to supersymmetric Yang-Mills theory was calculated by de Azeredo Campos and Fisch in [121]. Shortly after the introduction of the complex Ashtekar variables, Jacobson generalised the construction to d = 4, N = 1 supergravity [122]. In the following, different authors including F¨ul¨op[123], Gorobey and Lukyanenko [124], as well as Matschull [125], explored the subject further. Armand-Ugon, Gambini, Obr´egonand Pullin [126] formulated the theory in terms of 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 the GSU(2) connection were studied in detail. In the above works, complex Ashtekar variables are employed for which the methods developed in [14, 15, 16, 17, 18, 19] are not available. Also, the Ashtekar 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 explored non-perturbative formulations of certain parts of eleven dimensional supergravity [130, 131]. The generalisation of the loop quantum gravity methods to antisymmetric tensors was considered by 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 the Nieh-Yan topological density. An attempt to construct Ashtekar-type variables for d = 11 Supergravity has already been made by Melosch and Nicolai using an SO(1, 2) × SO(16) invariant reformulation of the original CJS theory [134]. In this formulation, the connection is not Poisson commuting, thus forbidding LQG techniques. In a paper on canonical supergravity in 2 + 1 dimensions [135], Matschull and Nicolai 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 was observed in [21], this problem can be avoided by using half-densitised fermions as canonical variables. The general picture emerging is that the canonical decomposition S = R dt (pq˙ − H) in the time gauge leads to Z Z  D ˙ a i √ ¯ a⊥b ˙ S = d x dt Ei Ka + i qψaγ ψb + tensors + vectors + spin 1/2 + scalars σ  a ij −NH − N Ha − λijG − ψ¯tS − tensor constraints , (10.2.1) where the Hamiltonian constraint H, the spatial diffeomorphism constraint Ha, the Spin(D) Gauß constraint Gij, the supersymmetry constraint S and the tensor constraints form a first class a i algebra. Ei is the densitised vielbein and Ka its canonical momentum. ψa denotes the Rarita- a Schwinger field with suppressed spinor indices. N, N , λij and ψ¯t are Lagrange multipliers for the respective constraints. With tensor constraints we mean constraints acting only on additional tensor fields such as the three-index photon of D + 1 = 10 + 1 supergravity. The remaining 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, we have 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 thesis and extended to the case of spin 1/2 fermions in chapter 9. The Rarita-Schwinger field turns out to be more difficult to deal with than the spin 1/2 fermions. On the one hand, it leads to second class constraints [113], which encode the reality conditions, with a structure which is different from the case of Dirac spinors1. On the other hand, as the other fermions, it has to be i treated as a half-density in order to commute with Ka[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 = 5 supergravity and higher-dimensional pure gravity [139, 140] and d = 10, N = 1 supergravity coupled to supersymmetric Yang-Mills theory [141, 142]. The relation of the covariant canonical formalism and the conventional canonical analysis is discussed in [143] using the example of four dimensional 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 by Sawaguchi [120] in order to give the reader a feeling for what is happening during the canonical decomposition. The resulting picture generalises to all dimensions. The symplectic potential derived in this context is exemplary for the supergravity theories of our interest and we will continue with the general treatment in the next section. We remark that in 3 + 1 spacetime dimensions, the relations CT = −C and CγI C−1 = −(γI )T hold, where C denotes the charge conjugation matrix. The action for 3 + 1, N = 1 first order supergravity is given by Z 4 s µI νJ ¯ µρσ  S = d X ee e FµνIJ (A) + is eψµγ ∇ρ(A)ψσ . (10.2.2) M 2 1 ¯ While for Dirac spinors, the second class constraints are of the form πψ¯ ∝ ψ, πψ ∝ ψ, in the Majorana case we obtain an equation of the form πψ ∝ ψ, where πx denotes the momenta conjugate to x.

73 µρσ IJK µ ρ σ Using the conventions introduced above and γ = γ eI eJ eK , one can explicitly check that the action is real. The 3+1 decomposition is done like in the previous chapters, and the notation used can be found there. We obtain Z 4 s µI νJ ¯ µρσ  S = d X ee e FµνIJ (A) + s eψµγ ∇ρ(A)ψσ M 2 Z Z 1 √   = dt d3x πaIJ L A − i q ψ¯ γ⊥abL ψ − N Hgrav − iq ψ¯ γabc∇ (A)ψ 2 T aIJ a T b a b c R σ e  √  1  √  −N a Hgrav + 3i qψ¯ γ⊥bc∇ (A)ψ + A GIJ − i qψ¯ γ⊥ab[iΣIJ ]ψ a [a b c] 2 tIJ grav a b ¯ √ ⊥ab √ ⊥ab i −iψt qγ ∇b(A)ψa + q∇b(A)(γ ψa) . (10.2.3)

IJ From there, one can read off the constraints H, Ha, G and S. We will choose time gauge I I n = δ0 at this point to simplify the further discussion. For the symplectic potential, we find Z Z   3 1 aIJ √ ¯ ⊥ab dt d x π LT AaIJ − i qψaγ LT ψb R σ 2 Z Z 3  ai ˙ † ab ˙  → dt d x E Kai − iφaγ φb R σ Z Z  h i 3 ai ˙ † ij ˙ ˙b k = dt d x E Kai − iφi γ φj − (Ej )Eb φk R σ Z Z  h i 3 ai ˙ j ˙ ˙b k = dt d x E Kai − π φj − (Ej )Eb φk R σ Z Z 3  ai ˙ j j  = dt d x E (Kai − πiEaφj) − π φ˙j R σ Z Z 3  ai ˙ 0 j ˙  = dt d x E Kai − π φj , (10.2.4) R σ where we successively defined √ 1 φ := 4 qψ , φ := √ Eaφ , πi := iφ†γji and K0 := K − π Ejφ . (10.2.5) a a i q i a j ai ai i a j

In the second line, we chose time gauge and half-densities as fermionic variables [21]. Then, we transformed 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 extrinsic 0 curvature and we have to define a new variable Kai. The Gauß constraint becomes under these changes of variables

ij [i a|j] k  ij G = 2Ka E + π iΣ φk  0 [i [i k  a|j] k  ij = 2 Ka + π Ea φk E + π iΣ φk 0 [i a|j] [i j] k  ij = Ka E + 2π φ + π iΣ φk. (10.2.6)

The generator of spatial diffeomorphisms H˜a is given by the following linear combination of constraints 1 H˜ := H + A GIJ + ψ¯ S. (10.2.7) a a 2 aIJ a

74 It becomes bj  bj  b  b  H˜a = E ∂aKbj − ∂b E Kaj − π ∂aφb + ∂b π φa

bj  0 k   bj  0 k  = E ∂a Kbj + πjEb φk − ∂b E Kaj + πjEa φk 1 √   bi 4 j
bi j − √ πiE ∂a qφ Ebj + ∂b πiE φ Eaj 4 q   √  1  bj 0 bj 0 4 i = E ∂aK − ∂b E Kaj + q∂a √ πi φ bj 4 q   1 1 = Ebj∂ K0 − ∂ EbjK0 + ∂ (π ) φi − π ∂ φi. (10.2.8) a bj b aj 2 a i 2 i a i For the last step, note that π φi = 0. Thus, these constraints exactly change as one would expect under the performed change of variables. The other constraints also can be rewritten in terms of the new variables, but this is less instructive and their explicit form is not important for what follows. We only want to remark that they depend on the contorsion Kaij, which is not 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 time gauge this is a SO(D) theory which is the subject of the first part. In the second part we will perform a phase space extension to a SO(D + 1) theory where special attention must be paid to the 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 impose I I the time gauge n = δ0 prior to the D + 1 split and restrict to dimensions where a Majorana representation of the γ-matrices exists, which we will use. This allows us to set C = γ0 which simplifies the following analysis. The generic terms important for this chapter appearing in supergravity theories are Z D+1 s µI νJ ¯ µρσ  Sgrav.+RS = d X ee e FµνIJ (A) + is eψµγ ∇ρ(A)ψσ (10.3.1) M 2 in case of a first order formulation and analogous terms for a second order formulation. This difference in defining the theory will not be important in what follows, since as demonstrated above for the 3 + 1 dimensional case, the symplectic potential of these actions in the time gauge turns out to be Z Z D  ai √ ¯ ⊥ab  dt d x E LT Kai − i qψaγ LT ψb R σ Z Z D  ai ˙ 0 j ˙  = dt d x E Kai − π φj , (10.3.2) R σ where we used the same definitions as in (10.2.5). From (10.3.2) we can read off the non-vanishing Poisson brackets2 n 0 bjo b j n α j o α j Kai,E = δaδi and φi , πβ = −δβ δi . (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 for notational simplicity in what follows with the usual rules for the interplay between the Poisson brackets for integral and half-integral spin respectively. See e.g. [144, 145] for an account.

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

i i T 0 ji † T 0 Ω := π + iφj Cγ γ = 0 and φi = −φi Cγ . (10.3.4) In order to be able to introduce a connection variable along the lines of [35], we need to enlarge the internal space, i.e. replacing the gauge group SO(D) by either SO(1,D) or SO(D + 1). In view of subsequent quantisation, SO(D + 1) is favoured because of its compactness and will be our choice in the following. This enlargement can be done consistently if also additional spinorial degrees of freedom are added as well as additional constraints which remove the newly introduced fermions. Finally, the extension has to be consistent with the reality conditions. All this turns out to be rather hard to achieve, and the final version of the theory looks rather different from what a “first guess” might have been. To motivate it, we will review the whole process of finding the theory, showing where the straight-forward ideas lead to dead ends, and how they can be modified to arrive at a consistent theory. We will only discuss the fermionic variables 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 this end, we calculate the Dirac matrix i Cij = Ωi, Ωj = −2iCγ0γij ,(C−1) = −γ0 ((2 − D) η + γ ) C−1, (10.3.5) ij 2(D − 1) ij ij and thus find for the Dirac bracket

n ko −1 n l o −1 {φi, φj}DB = − φi, Ω (C )kl Ω , φj = −(C )ij. (10.3.6)

To simplify the subsequent discussion, in the following we will consider real representations of the Dirac matrices only, which implies C = γ0. Then the above equations read i Cij = 2iγij ,(C−1) = − ((2 − D) η + γ ), {φ , φ } = −(C−1) . (10.3.7) ij 2(D − 1) ij ij i j DB ij

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 the internal space. (a) immediately leads to problems. The symmetry of the Poisson brackets n o α β ˜−1 αβ ˜−1 φI , φJ ∝ (C )IJ implies that matrix C is symmetric under the exchange of (I, α) ↔ (J, β). −1 i The naive extension (C )IJ = − 2(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 in T different, more “unnatural” ways, e.g. containing terms like γJ γI etc. and “cure” this problem for a moment, but also the Gauß constraint will be problematic. The SO(D) constraint contains ij i 1 T ji ˆIJ C (since we used π = − 2 φj C ) and this matrix should also be replaced by some C , such IJ ij that φI transforms covariantly and G reduces correctly to G if we choose time gauge and solve its boost part. This implies restrictions on Cˆ and further restrictions on C˜−1. We did not succeed in finding matrices which fulfil all these requirements. In the following, we therefore will follow the second route (b) and simplify the Dirac brackets before doing the enlargement of the 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 and i symmetric, hence there always exists a real, orthogonal matrix O j such that under the i 0i i j change of variables φ → φ := O jφ the brackets becomes i times a real diagonal matrix.

76 However, now the new fundamental degrees of freedom φ0i in general do not transform −1 i 0j nicely under SO(D) gauge transformations, only (O ) jφ do. More severely, it is unclear i I how the extension O j → O J should be done. 2. To assure that the fundamental degrees of freedom still transform nicely under SO(D) 0i ij ij ij ij transformations, we can use the Ansatz φ := M φj with M := (αδ 1+βΣ ). Matrices of this form are in general invertible (cf. point 3. below for two exceptions) and, since they are constructed from intertwining matrices, φ0i will transform nicely under gauge transformations. Moreover, now there is a chance to generalise the matrix to one dimension higher. For the Dirac brackets to become diagonal, α and β have to be determined by solving MC−1M T = i1. The problem is that there is no solution for both parameters being real, at least one is necessarily complex. More general Ans¨atzefor M ij (e.g. involving γfive in even dimensions) share the same problem. Thus we exchanged the problem of complicated 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 internal space. Given the difficulties just mentioned, the optimal approach in the desire to simplify the Poisson brackets is to find orthogonal projections onto subspaces of the real Graßmann vector space which are built from δij1 and Σij such that the symplectic structure becomes block diagonal on those subspaces. One can then define simple Poisson brackets and add the projection constraints as secondary constraints which leads to corresponding Dirac brackets which will be proportional to those projectors. As we will see, the fact that these are projectors makes it possible to find a Hilbert space representation of the corresponding Dirac bracket.

We define in any dimension D 1 D − 1 2i Pij := ηijδ − (γiγj) = ηijδ − Σij , (10.3.8) αβ αβ D αβ D αβ D αβ 1 1 2i Qij := (γiγj) = ηijδ + Σij . (10.3.9) αβ D αβ D αβ D αβ Those matrices are both real (we are using Majorana representations) and built from intertwiners, but they are not invertible. It is easy to check that

ij βγ ij βγ iγ ij βγ iγ PαβQjk = 0, PαβPjk = Pαk, QαβQjk = Qαk, and P + Q = 1η, (10.3.10) i.e. the above equations define projectors. By construction, P projects on “trace-free” ij β α ij components w.r.t. γi, i.e. Pαβγj = 0 = γi Pαβ. Using these projectors, we can decompose the Rarita-Schwinger field as follows 1 φ = P φj + Q φj =: ρ + γ σ, (10.3.11) i ij ij i D i j i 3 with ρi := Pijφ and σ := γ φi . Using the reality conditions (10.3.4) for φi, we find

T T ρ¯i = ρi C andσ ¯ = σ C. (10.3.12) Moreover, using γij = −Pij + (D − 1)Qij, (10.3.13)

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

77 the symplectic potential becomes i ˙ † ji ˙ −π φi = −iφjγ φi † ji ji
˙ = −iφj −P + (D − 1)Q φi †  j ki j ki ˙ = −iφj −P kP + (D − 1)Q kQ φi

j
† ki˙ j
† ki˙ = i Pk φj (P φi) − i(D − 1) Qk φj (Q φi) D − 1 = iρ†ρ˙i − i σ†σ˙ i D D − 1 = −iρT Cγ0ρ˙i + i σT Cγ0σ˙ i D D − 1 = iρT ρ˙i − i σT σ˙ , (10.3.14) i D where in the second to last line we used the reality conditions (10.3.12) and in the last line we restricted to a real representation, C = γ0. This motivates the definition of the brackets i D ρ , ρi = − 1δi and {σ, σ} = i 1, (10.3.15) j 2 j 2(D − 1) ∗ ∗ together with the reality conditions ρi = ρi, σ = σ (cf. (10.3.12)) and additionally introduced constraints to account for the superfluous fermionic degrees of freedom, i β Λα := γαβρi ≈ 0. (10.3.16)

We need to check that the extension is valid, i.e. that the Poisson brackets of the φi, considered as functions on the extended phase space, are equal to the Dirac brackets (10.3.6) of the system j 1 before we did the extension. Using φi = Pijρ + D γiσ (cf. 10.3.11) and the Poisson brackets 1 (10.3.15), this can be checked explicitly (this calculation shows why the factors of 2 in (10.3.15) are needed). Using this, we can express the constraints H and S in terms of the new variables in the obvious way and know that their algebra is unchanged. In particular, since the projectors are built from intertwiners, we find for the fermionic part of the Gauß constraint   h i  D − 1  Gij = ... + −iρkT 2η[iηj] + iΣijη ρl + i σT iΣij σ, (10.3.17) k l kl D which allows for an easy generalisation to SO(D+1) or SO(1,D) as a gauge group. Furthermore, i j since ρ in the other constraints only appears in the combination Pijρ , they automatically Poisson commute with Λα. i Note that if we now would calculate the Dirac bracket, we would get {ρi, ρj}DB = − 2 Pij, which again is non-trivial. Instead, we directly enlarge the phase space from {ρi, σ} to {ρI , σ}, i D with, as a first guess, the brackets {ρI , ρJ } = − 2 ηIJ 1, {σ, σ} = i 2(D−1) 1, the reality conditions ∗ ∗ ρI = ρI , σ = σ and the constraints I I N ρI ≈ 0 and γ ρI ≈ 0. (10.3.18) Unfortunately, this immediately leads to an inconsistency in the case of the compact gauge group SO(D + 1), since for our choice of Dirac matrices, γ0 necessarily is complex in the Euclidean case. Therefore, the reality conditions again are not SO(D + 1) covariant and the constraints I I 4 (10.3.18) only are consistent in the time gauge N = δ0 . With a more elaborate choice of reality condition it is possible to define a consistent theory, which will be the subject of the next section. 4 I γ ρI ≈ 0 is a complex constraint and thus equal to two real constraints. Only in time gauge, its imaginary I part is already solved by demanding N ρI ≈ 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 group from SO(D) to SO(D + 1) is that the real vector space V of real SO(1,D) Majorana spinors is not preserved under SO(D + 1) whose spinor representations are necessarily on complex vector spaces. Let VC be the complexification of V . Now SO(D + 1) acts on VC but the theory we started 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 structure on VC that is VC = VR ⊕ iVR. The mathematical problem left is therefore to add the reality condition 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 where B is a “Euclidean boost” in the 0j planes and R a rotation that preserves the internal vector I I n0 := δ0. 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 , n that is, we need a kind of polar decomposition. If VC would be just a vector subspace of some C we could do this by standard methods. But this involves squaring of and dividing by complex numbers 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 then I J the D boost parameters can be extracted from the D rotation angles in the normal N = BIJ n0 to which we have access because N is part of the extended phase space. To be explicit, let e(A) D+1 (A) A (A) be the standard base of R , that is, eI = δI . We construct another orthonormal basis b of RD+1 as follows: Let b(0) := N and

(0) (0) b0 = sin(φ1).. sin(φD), bj = 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 polar coordinates. Define ∂b(0)/∂φ b(j) = I j , (10.3.22) I (0) ||∂b /∂φj|| where the denominator denotes the Euclidean norm of the numerator. Then it maybe checked by straightforward computation that

IJ (A) (B) AB δ bI bJ = δ . (10.3.23) We consider now the SO(D + 1) matrix

D −1 X (A) (A) (A(N) )IJ := bI eJ , (10.3.24) A=0 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) −1 (0) with ||N|| = 1 and θ ∈ VR as A(N) · (e , ρ) where ρ ∈ V . We need to investigate how SO(D + 1) acts on this parametrisation. On the one hand we have

−1 X (A) (A) [g A(N) ]IJ = (gb )I eJ . (10.3.25) A

On the other hand we can construct A(g · N)−1 by following the above procedure, that is, (A) computing the polar coordinates θgj of g · N and defining the bj (g · N) via the derivatives with (0) respect to the θgj. The common element of both bases is g · N = g · b . Therefore, there exists an element R(g, N) ∈ SO(D) such that

j (k) g · b (N) = Rkj(g, N)b (g · N), (10.3.26) or with R00 = 1,R0i = Ri0 = 0

A (B) g · b (N) = RBA(g, N)b (g · N) (10.3.27) defines a rotation in SO(D + 1) preserving e(0). Putting these findings together we obtain

−1 X (B) (A) X (A) [g · A(N) ]IJ = RBA(g, N) bI (g · N) eJ = RAJ (g, N) bI (g · N) A,B A X (A) (A) −1 = RKJ (g, N) bI (g · N) δK = [A(g · N) R(g, N)]IJ . (10.3.28) A

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 rotation −1 that preserves V and thus altogether the decomposition of VR = {A(N) · V ; ||N|| = 1} is preserved with the expected covariant action of SO(D + 1) on N. It therefore makes sense to impose the reality condition that A(N) θ is a real spinor. In the subsequent construction, this idea will be implemented together with an extension of the phase space ρj → ρI subject to the I constraint N ρI = 0. All these constraints and the reality conditions are second class and we will show explicitly that the symplectic structure reduces to the time gauge theory. Despite the fact that we end up with a non trivial Dirac (anti-) bracket, it can nevertheless be quantised and non 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

I J I A J N = δ0. (10.3.29)

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

¯ J I ¯ J A0I = NI and AIJ X = δi AIJ X (10.3.30) for XJ arbitrary. The corresponding rotation on spinors will be denoted by A. This matrix ro- I I tates the normal N into its time gauge value δ0 without imposing time gauge explicitly, which we will 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 better suited for certain problems. In D = 3, we can, e.g., construct A(N) as a linear function of the components of I I N by using A0I = NI and subsequently interchanging the components of N with appropriate signs for the remaining columns of A(N).

80 bKL I ∗ ∗ we do not choose time gauge. We introduce the set of variables (AaIJ , π ,N ,PJ , ρI , ρJ , σ, σ ) together with the following non-vanishing Poisson brackets

n bKL o b [K L] D  I I D AaIJ (x), π (y) = 2δaδI δJ δ (x − y), N (x),PJ (y) = δJ δ (x − y), D {ρ (x), ρ∗ (y)} = −iη 1δD(x − y), {σ(x), σ∗(y)} = i 1δD(x − y), I J IJ D − 1 (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 into time gauge. Notice that before imposing the constraints, ρ, θ are complex Graßmann variables and 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 when replacing the above Poisson bracket by the corresponding Dirac bracket. Additionally, we want that the variables transform nicely under spatial diffeomorphisms and gauge transformations, thus we add D − 1  GIJ := D πaIJ + 2P [I N J] − 2iρ†[I ρJ] − iρ† [iΣIJ ]ρK + i σ† [iΣIJ ]σ + ... (10.3.33) a K D 1 1   H˜ := πbIJ ∂ A − ∂ πbIJ A + P I ∂ N a 2 a bIJ 2 b aIJ a I i i D − 1 D − 1 − ∂ (ρ†I )ρ + ρ†I ∂ ρ + i ∂ (σ†)σ − i σ†∂ σ + ... . (10.3.34) 2 a I 2 a I 2D a 2D a The old variables are expressed in terms of the new ones by

ai iJ aIK I J E := ζA η¯JK π NI , Kai = ζAi η¯IK (AaKJ − ΓaKJ (π))N , 1 1 ρ = A η¯JK (Aρ + A∗ρ∗ ), σ = (Aσ + A∗σ∗), (10.3.35) i 2 iJ K K 2 I where the bar here means rotational components w.r.t N ,η ¯IJ := ηIJ − ζNI NJ . To remove unnecessary degrees of freedom, we need the constraints

a J aKL SIM := IJKLM N π , I N := N NI − ζ, I JK ∗ ∗ JK −1 ∗ ∗ Λ := γ AIJ η¯ (AρK + A ρK ) = AγJ η¯ (ρK + A A ρK ), I ∗ ∗ Θ := N (AρI + A ρI ), (10.3.36) together with the Hamilton and supersymmetry constraints, where we replace the old by the new variables as shown above. To prove that this theory is equivalent to supergravity and can possibly be quantised, we have to answer the following questions: • Are the reality conditions (10.3.32) consistent? I. e., do they transform under gauge transformations in a sensible way and do they (weakly) Poisson commute with the other constraints? • 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. do the newly introduced constraints (10.3.36) fit “nicely” in the set of the old constraints? If not, do at least the constraints which were of the first class before the enlargement of the gauge group retain this property?

81 • Do the constraints, especially the Gauß and spatial diffeomorphism constraint, reduce correctly? • 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. I • The orthogonal matrix AIJ is a function of N only as we have seen above. We have A0K = NK , but the remaining components of the matrix are complicated functions of the I components of the vector N . Thus, the whole matrix AIJ will have a rather awkward transformation behaviour under the action of GIJ . The reality conditions (10.3.32) as a whole, however, transform in a “nice” way under SO(D + 1) gauge transformations (we will discuss ρI in the following, σ can be treated analogously). For g ∈ SO(D + 1), the reality condition transforms as follows: J ∗ J∗ JK ∗JK ∗ ∗ ∗ A(N)ρ = A(N) ρ −→ g A(g · N)gρK = g A(g · N) g ρK . (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 can be split up in a part which leaves N I invariant and a “Euclidean boost” changing N I . For I −1 I −1 K ij∗ ij the rotations, A is invariant and we find using Aγ¯ A =η ¯ J AJK γ and Σ = −Σ ¯ ¯ JK ¯ ¯ JK −1 ¯ JK −1 −1 LM δΛ¯ AρI = iΛJK AΣ ρI = iΛJK AΣ A AρI = iΛ A[J|LAK]M Σ AρI = ¯ JK LM ¯ JK lm = iAL[J AM|K]Λ Σ AρI = iAl[J Am|K]Λ Σ AρI , (10.3.38) ∗ ¯ ¯ JK ∗ ¯ JK lm ∗ δΛ¯ (AρI ) = (iΛJK AΣ ρI ) = (iAl[J Am|K]Λ Σ AρI ) = ¯ JK lm∗ ∗ ∗ ¯ JK lm ∗ ∗ = −iAl[J Am|K]Λ Σ A ρI = iAl[J Am|K]Λ Σ A ρI . (10.3.39) I For finite transformationsg ¯ ∈ SO(D)N stabilising N , we thus have AρI → Agρ¯ I = g0AρI , where g0 ∈ SO(D)0 stabilises the zeroth component and thus is, with our choice of representation, a real matrix. Hence, reality conditions transform again into reality conditions under rotations. For a boost b the situation is a bit more complicated. Under a boost AIJ will transform intricately, but we know that a) the matrix remains orthogonal L L by construction, and b) that A0K = NK → ΛK NL = −A0LΛ K . The most general KL −1 −1 N b −1 M transformation compatible with the above is AIJ → (g0)IK A (¯g )LN (b ) M ( g¯ ) J where g0 ∈ SO(D)0 is some group element which does not change the zeroth component, I b g¯ ∈ SO(D)N is in the stabiliser of N and g¯ ∈ SO(D)b·N . Since we have SO(D)N = −1 b b SO(D)b·N b, we can eliminate g¯ by a redefinition ofg ¯. By definition of a representation, −1 −1 we then also have A → g0Ag¯ b and thus −1 −1 −1 I AρI → g0Ag¯ b bρI = g0Ag¯ ρI =g ˜0Aρ , (10.3.40)

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

 ∗ ∗ ∗ ∗ h † ∗ T i (AρI − A ρI ) , (AρJ + A ρJ ) = −iηIJ +AA − A A = 0. (10.3.41)

82 ai • The brackets between E 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 and the 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 problems 6 I J because of its orthogonality. For the fermionic variables, we find using Ai Aj η¯IJ = ηij and A†A = 1 i {ρ (x), ρ (y)} = − A I A J η¯K η¯L [{Aρ (x),A∗ρ∗ (y)} + {A∗ρ∗ (x), Aρ (y)}] i j 4 i j I J K L K L i h i = − A I A J η¯ AA† + A∗AT δD(x − y) 4 i j IJ i = − δ 1 + 1T  δD(x − y) 4 ij i = − δ 1δD(x − y), (10.3.42) 2 ij D {σ(x), σ(y)} = i 1δD(x − y). (10.3.43) 2(D − 1)

This automatically implies that the algebra of H and S remains unchanged if we replace the old variables by (10.3.35). From (10.3.35), it is also clear that H and S Poisson commute with Sa and N . By inspection, all constraints transform covariant under IM spatial diffeomorphisms. More surprisingly, all constraints Poisson commute with GIJ . This can be seen quite easily for GIJ , H˜ , Sa , N and also for Λ and Θ (note that a IM −1 ∗ ∗
A, AIJ are invertible and that ρI + A A ρI transforms like ρI which can be shown using the methods above). But for H and S this is, at first sight, a small miracle, since the replacement rules (10.3.35) of all old variables depend on A(N), which is known to transform oddly. But the matrices A are placed such that they, in fact, either a) appear −1 ∗ ∗ † T T in the combinations (ρI + A A ρI ) or (ρI + ρI A A), which can easily be shown to I † transform like ρ and ρI respectively with the methods above, or b) all cancel out! The J I −1 ∗ ∗ T general situation is the following: ρi is replaced by ρi = Ai η¯J A(ρI + A A ρI ), ρi by T J I † T T −1 ρi = Ai η¯J (ρI + ρ A A)A , where the expression in brackets transform sensible (cf. ai k above). The free internal indices of E , Kbj and ρ are either contracted with each other, i then in the replacement the AiJ s will cancel because of orthogonality, or with γ , which will 7 −1 −1 J be contracted from both sides with A(N) and all As cancel due to (A )IJ A γ A = γI . Cancelling the As makes H gauge invariant and S gauge covariant by inspection, if we replace all γ0 by iN/ . Thus we are left with Θ and Λ, which are their own second class partners but Poisson commute with everything else, which can be seen as follows. For Θ, JK ∗ ∗ note that H, S and Λ only depend onη ¯ (AρK + A ρK ), which Poisson commutes with Θ due to the projectorη ¯. For Λ, the situation again is more complicated. Remember i j i that H and S in the time gauge only depended on X Pijρ for some X . Whatever i IJ X may be, under (10.3.35) it will be replaced by something of the form A X¯J and the ¯ JI KL M ∗ ∗ 1 whole expression will become XI A PJK A η¯L (AρM + A ρM ) with PIJ = ηIJ − D γI γJ . Crucial for the following calculation is the property (10.3.30), which will be used several

6 J J Because of orthogonality, we trivially have AIJ AK = ηIK . Additionally, AiJ η¯K = AiK , which can be seen K I J I from AiK N = 0. Therefore, Ai Aj η¯IJ = Ai AjI = 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 ψ¯t → ψ¯tA to make it hold.

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

 ¯ JI KL M ∗ ∗ N OP ∗ ∗ XI A PJK A η¯L (AρM + A ρM ) , γ ANOη¯ (AρP + A ρP ) JI KL M N
T = −iX¯I A PJK A η¯L γ ANM jI kL M n T = −iX¯I A PjkA η¯L (γ ) AnM jI kL n jI k = −iX¯I A PjkA γ AnL = −iX¯I A Pjkγ = 0. (10.3.44)

The constraint algebra is summarised in table 10.1.

First class constraints Second class constraints

GIJ , H˜ , H, S, Sa and N Λ, Θ, χ and χ a IM I

Table 10.1: List of first and second class constraints.

I I • By construction, H and S reduce correctly if we choose time gauge N = δ0, which automatically implies AIJ → (g0)IJ ∈ SO(D)0. Since the theory is SO(D)0 invariant, a I I gauge transformation g0 → 1 can be performed, which implies ρ = ρr. From this one IJ easily deduces that G and H˜a 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 ) } h † ∗ T i = −iηIJ −AA − A A = 2i1ηIJ (10.3.45) 1 (C−1)IJ = − 1ηIJ (10.3.46) i ∗ −1 KL ∗ {ρI , ρJ }DB = − {ρI , AρK − (AρK ) } (C ) {AρL − (AρL) , ρJ } = i = − η A†A∗, (10.3.47) 2 IJ and for σ analogously. We now can choose new variables which have simpler brackets. Motivated from the original replacement (10.3.35), we define

I IJ ρr := A AρJ , σr := Aσ, (10.3.48) I
∗ IJ ∗ ∗ IJ ∗ ∗ −1 I ∗ ρr = A A ρJ = A A ((A ) AρJ ) = ρr, σr = σr, (10.3.49) with the Dirac brackets i i ρI , ρJ = AIK Aρ ,AJLAρ = − ηIJ AA†A∗AT = − ηIJ 1,(10.3.50) r r DB K L DB 2 2 D {σ , σ } = i 1. (10.3.51) r r 2(D − 1)

I Thus, the Dirac brackets of the ρr, σr are simple as are the reality conditions. Only the transformation 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 at once.

84  I J because of the appearance of the rotation A in their definition. Note that also P ,P DB,  I J  I ˜I P , ρr DB and P , σr DB will be non-zero. Therefore, we also choose a new variable P with simple Dirac brackets, which can most easily be found by performing the symplectic reduction. After that, we can simply read it off the symplectic potential. We find using −1 J † J T ρI = AJI A ρr and ρI = AJI (ρr ) A D − 1 +iρ†ρ˙I − i σ†σ˙ + P I N˙ I D I D − 1 = iA I (ρJ )T A(A A˙−1ρK ) − i σT A(A−˙1σ ) + P I N˙ J r KI r D r r I D − 1 = i(ρJ )T ρ˙ − i σT σ˙ + P I N˙ + r Jr D r r I  −1 −1  L J T ∂AKL K J T ∂A D − 1 T ∂A ˙ +i AJ (ρr ) ρr + (ρr ) A ρJr − σr A σr NI ∂NI ∂NI D ∂NI D − 1 = i(ρJ )T ρ˙ − i σT σ˙ + P˜I N˙ , (10.3.52) r Jr D r r I

−1 −1 I I L J T ∂AKL K J T ∂A D−1 T ∂A with P˜ := P + iAJ (ρ ) ρ + i(ρ ) A ρJr − i σ A σr. It can be r ∂NI r r ∂NI D r ∂NI ˜I I † J † checked explicitly that P , expressed in the old variables (P ,NJ , ρI , ρ , σ , σ), Poisson commutes with the reality conditions and with itself, and therefore has nice Dirac brackets. For the spatial diffeomorphism constraint, a short calculation yields i i D − 1 D − 1 H˜ = P I ∂ N − ∂ (ρ†I )ρ + ρ†I ∂ ρ + i ∂ (σ†)σ − i σ†∂ σ + ... = a a I 2 a I 2 a I 2D a 2D a D − 1 = P˜I ∂ N + i(ρI )T ∂ ρ − i σT ∂ σ + ... , (10.3.53) a I r a Ir D r a r which by inspection generates spatial diffeomorphisms on the new variables. The con- straints Λ and Θ become

i 0 Λ = γiρr ≈ 0 and Θ = ρr ≈ 0, (10.3.54) which look utterly non-covariant, but which by construction still Poisson commute with the SO(D + 1) Gauß constraint. It therefore has to have a complicated form. We find D − 1  GIJ = 2P [I N J] − 2iρ†[I ρJ] − iρ† [iΣIJ ]ρK + i σ† [iΣIJ ]σ + ... = K D ˜[I J] TK [I |J] L T IJ −1 K = 2P N − 2iρr AK AL ρr − iρKrA[iΣ ]A ρr  D − 1 T IJ −1 L M T ∂AKL K +i σr A[iΣ ]A σr − 2i AM (ρr ) ρr + D ∂N[I −1 −1  N T ∂A D − 1 T ∂A J] + (ρr ) A ρNr − σr A σr N + ... (10.3.55) ∂N[I D ∂N[I

Finally, we solve the remaining second class constraints Λ and Θ which after a short calcu- lations results in the final Dirac brackets i ρi , ρj = − Pij, ρ0, ρj = 0, ρ0, ρ0 = 0. r r DB 2 r r DB r r DB As a consistency check, we can consider    i j i 1 i j 1 j −1 ij φ , φ DB = ρr + γ σr, ρr + γ σr = −(C ) , (10.3.56) D D DB

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 in D = 3, N = 1 supergravity from [120] adapted to our notation,      i abc k ˆ 1 l ˆ 1 k l S = −  γ5 γkeaDb √ ecφl + Db √ γkeaecφl 2 4 q 4 q 1   + abc ei K j − iφ¯ γ0γljEmφ γkenφ , (10.3.57) 2 ijk a b l b m c n

ˆ j i kl √i k ¯ ¯
where Daφi = ∂aφi +ω ˆaijφ + 2 ωˆaklΣ φi,ω ˆaij = Γaij + 4 q ea φiγkφj + 2φ[iγj]φk , and Γaij is the spin-connection annihilating the triad. An explicit expression for S in terms of the extended variables (A, π, N, P, ρ, ρ∗, σ, σ∗) can be found using (10.3.11), (10.3.35). The corresponding constraint operator is obtained using the methods in section 10.4, 7 and [20, 37].

10.4 Background independent Hilbert space representations for Majorana fermions

Background independent Hilbert space representations for Dirac spinor fields were constructed in [21]. One may think that for the Rarita-Schwinger field or more generally for Majorana fermion fields one can reduce to this construction as follows: Consider the following variables 1 −i ξIα = √ ρ2α+1 + iρ2α+2
, πIα = √ ρ2α+1 − iρ2α+2
, α = 1,..., 2b(D+1)/2c, (10.4.1) 2 r r 2 r r which have the non-vanishing Dirac brackets n o ξIα(x), πJβ(y) = −iηIJ δαβδ(D)(x − y) (10.4.2) and the simple reality condition π¯ = −iξ. (10.4.3) The elements of the Hilbert space are field theoretic extensions of holomorphic (i.e. they only depend on θα) functions on the Graßmann space spanned by the Graßmann numbers θα and their adjoints θ¯α, the operators corresponding to the phase space variables act as d ξfˆ := θf, πfˆ := i f, (10.4.4) dθ and the scalar product Z ¯ < 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- rα rβ αβ ond class constraints, that is, {ρi , ρj }DB = −i/2 Pij . Recall that one must solve the second class constraints before quantisation, hence it is not sufficient to consider the quantisation of the 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; θA = θA. We consider the finite dimensional, complex vector space V of polynomials in the θA with 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 the complex dimension of V . We may write a polynomial f ∈ V in several equivalent ways which are useful in different contexts. Let f (n) , 0 ≤ n ≤ N be a completely skew complex valued A1..An tensor (n-form) then f can be written as

N N X 1 (n) X X (n) f = f θA1 ..θAn = f θA1 ..θAn . (10.4.6) n! A1..An A1..An n=0 n=0 1≤A1<..

An equivalent way of writing f is by considering for σk ∈ {0, 1} and A1 < .. < An the relabelled coefficients  (n) 1 k ∈ {A1, .., An} fσ ..σ := f , σk := (10.4.7) 1 N A1..An 0 else. It follows X σ1 σN f = fσ1..σN θ1 ..θN (10.4.8) σ1,..,σN ∈{0,1} 0 with the convention θA := 1. On V we define the obvious positive definite sesqui-linear form

N X X (n) (n)0 X < f, f 0 >:= f f = f f 0 (10.4.9) A1..AN A1..AN σ1..σn σ1..σN n=0 A1<..

l [θA · f](θ) := θA f(θ), [∂A · f](θ) := ∂ f(θ)/∂A, (10.4.10) where the latter denotes the left derivative on Graßmann space (see, e.g., [144] for precise definitions). Notice the relations ∂(A∂B) = 0, 2∂(AθB) = δAB which can be verified by applying them to arbitrary polynomials f. We claim that the operators (10.4.10) satisfy the adjointness relation † θA = ∂A. (10.4.11) The easiest way to verify this is to use the presentation (10.4.8). We find explicitly

X σ1+..+σA−1 σ1 σN θA · f = fσ1..σN (−1) δσA,0 θ1 ..θA..θN σ1,..,σN

X σ1+..+σA−1 σ1 σN = [fσ1..σA−1..σN (−1) δσA,1] θ1 ..θN σ1,..,σN X =: f˜A θσ1 ..θσN , σ1..σN 1 N σ1,..,σN

X σ1+..+σA−1 σ1 σN ∂A · f = fσ1..σN (−1) δσA,1 θ1 ..θcA..θN σ1,..,σN

X σ1+..+σA−1 σ1 σN = [fσ1..σA+1..σN (−1) δσA,0] θ1 ..θN σ1,..,σN X =: fˆA θσ1 ..θσN , (10.4.12) σ1..σN 1 N σ1,..,σN

87 where the wide hat in the fourth line denotes omission of the variable. We conclude X < f, θ f 0 > = f f˜A0 A σ1..σN σ1..σN σ1,..,σN X 0 = f (−1)σ1+..+σA−1 f˜ δ σ1..σN σ1..σA−1..σN σA,1 σ1,..,σN X 0 = f (−1)σ1+..+σA−1 δ f˜ σ1..σA+1..σN σA,0 σ1..σN σ1,..,σN X = fˆA f˜0 =< ∂ f, f 0 > . (10.4.13) σ1..σN σ1..σN A σ1,..,σN

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 general Ansatz for a “measure” is dµ = dθ1..dθN , µ(θ) with µ ∈ V fails to reproduce (10.4.9) if we 9 R σ apply the usual rule for the Berezin integral dθ θ = δσ,1. Notice that from this we induce R R dθA dθB = − dθB dθA as one quickly verifies when applying to V . However, there exists a non-trivial differential kernel

N−1 N−1 K := (θ1 + (−1) ∂1)..(θN + (−1) ∂N ) (10.4.14) such that Z 0 ∗ 0 < f, f >= dθN ..dθ1 f K f , (10.4.15) where we emphasise that f ∗ is the Graßmann involution

X σ X PN−1 PN σ ∗ N σ1 k=1 σk l=k+1 σl σ1 N f = fσ1..σN θN ..θ1 = fσ1..σN (−1) θ1 ..θN (10.4.16) σ1..σN σ1..σN 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

(−1)N(N−1)/2 Z < f, f 0 >= dθ ..dθ f ∗D ..D f 0 (10.4.17) N! A1 AN A1 AN where N−1 DA = θA + (−1) ∂A. (10.4.18) The presentation (10.4.19) establishes that the linear functional is invariant under U(N) acting on V by f 7→ U · f;[U · f](n) = f (n) U ..U , (10.4.19) A1..AN B1..BN B1A1 BN AN which is of course also clear from (10.4.9). Notice that (10.4.19) formally corresponds to θA 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. In this case it is sufficient to restrict to real valued coefficients in f and now the real dimension of V 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 for k = 1, .., N X σk+1 σN Fσ1..σk := fσ1..σN θk+1 ..θN , (10.4.20) σk+1..σN whence Fσ1..σN = fσ1..σN . Notice that Fσ1..σk no longer depends on θ1, .., θk. Using this we N compute with d θ := dθN ..dθ1 and using anticommutativity at various places Z dN θ f ∗ K f 0 Z N ∗ ∗ N−1 0 0 = d θ [F0 + F1 θ1](θ1 + (−1) ∂1) D2 .. DN [F0 + θ1 F1] Z N n ∗ N−1 N−1 0 0 = d θ F0 (−1) D2..DN (θ1 + (−1) ∂1)[F0 + θ1F1]

∗ N−1 0 0 o + F1 (−1) D2 .. DN θ1∂1[F0 + θ1 F1] Z N n ∗ N−1 0 N−1 0 ∗ N−1 0o = d θ F0 (−1) D2..DN [θ1F0 + (−1) F1] + F1 (−1) D2 .. DN θ1 F1 Z N n ∗ 0 ∗ 0o = d θ F0 θ1 D2..DN F0 + F1 θ1 D2 .. DN F1 , (10.4.21) where we used that the second term no longer is linear in θ1 and therefore drops out from the Berezin 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 Z N ∗ 0 d θ F0 θ1 D2..DN F0 Z N ∗ ∗ N−1 0 0 = d θ [F00 + F01θ2] θ1 (θ2 + (−1) ∂2) D3..DN [F00 + θ2F01] Z N n ∗ N−2 N−1 0 0 = d θ F00 (−1) θ1 D3..DN (θ2 + (−1) ∂2)[F00 + θ2F01]

∗ N−2 0 0 o + F01 (−1) θ1 D3..DN θ2∂2 [F00 + θ2F01] Z N n ∗ N−2 0 N−1 0 ∗ N−2 0 o = d θ F00 (−1) θ1 D3..DN [θ2F00 + (−1) F01] + F01 (−1) θ1 D3..DN θ2 F01] Z N n ∗ 0 ∗ 0 o = d θ F00 θ1θ2 D3..DN F00 + F01 θ1θ2 D3..DN F01 . (10.4.22)

Similarly for the second term in (10.4.21) Z Z N ∗ 0 N n ∗ 0 ∗ 0 o d θ F1 θ1 D2..DN F1 = d θ F10 θ1θ2 D3..DN F10 + F11 θ1θ2 D3..DN F11 . (10.4.23)

It is transparent how the computation continues: We continue expanding Fσ1..σk = Fσ1..σk0 + 10 θk+1Fσ1..σk1 and see by exactly the same computation as above that the signs match up to the effect that Z X Z dN θ F ∗ θ ..θ D ..D F 0 = dN θ F ∗ θ ..θ D ..D F 0 , σ1..σk 1 k k+1 N σ1..σk σ1..σk+1 1 k+1 k+2 N σ1..σk+1 σk+1 (10.4.24)

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

89 R N from which the claim follows using d θ 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 index A = (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 the auxiliary operator √ ρˆα := ~[θα + ∂α], (10.4.25) j 2 j j which by virtue of (10.4.11) is self adjoint and satisfies the anticommutator relation

[ˆρα, ρˆβ] = ~δ δαβ. (10.4.26) j k + 2 jk α rα rα rβ However,ρ ˆj is not yet a representation of ρj which satisfies the Dirac antibracket {ρj , ρk }DB = i αβ rα ∗ rα D ∗ − 2 Pjk and the reality condition (ρj ) = ρj . Similarly, {σα, σβ}DB = i 2(D−1) δαβ, σα = σα. α ∗ Correspondingly, what we need is a representation π(ρj ), π(σα) of the abstract CAR -algebra defined by canonical quantisation, that is, D [ρrα, ρrβ] = ~Pαβ, (ρrα)∗ = ρrα, [σr , σr ] = ~ δ , (σr )∗ = σr (10.4.27) j k + 2 jk j j α β + 2(D − 1) αβ α α

11 αβ all other anticommutators vanishing . Fortunately, using that Pjk is a real valued projector (in particular symmetric and positive semidefinite) we can now write the following faithful ∗ representation of our abstract -algebra (10.4.27) on the Hilbert H = VDM ⊗ VM defined above: 1r D π(ρrα) := Pαβρˆβ, π(σr ) := ~ [θ + ∂ ]. (10.4.28) j jk k α 2 D − 1 α α

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 point x and taking as representation space either the inductive limit of the finite tensor products n Hx1,..,xn = ⊗k=1 Hxk [66] or the infinite tensor product [146] H = ⊗xHx of which the former is just a tiny subspace. The ∗-algebra (10.4.27) is then simply extended by adding labels x to the operators and to ask that anticommutators between operators at points x, y be proportional to δx,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 Hilbert space12. 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 also present in supergravity theories, e.g. D + 1 = 9 + 1, N = 2a non-chiral supergravity [147]. They

11 This corresponds to the quantisation rule that the anticommutator is +i~ times the Dirac bracket in the ρ sector and −i~ times the Dirac bracket in the σ sector. This is the only possible choice of signs because the anticommutator of the same operator which in our case is self adjoint is a positive operator. The other choice of signs would yield a mathematical contradiction. 12 In the case of the inductive limit, a vector in v ∈ Hx1,..,xn is embedded in any larger Hx1,..,xn,y1,..,ym by v 7→ v ⊗ ⊗m1 where 1 is the constant polynomial equal to one. This way any operator at x acts in a well defined way on any vector in the Hilbert space.

90 are described by actions of the type Z D+1 ¯ µ SMajorana, 1/2 = d X iλγ ,Dµλ (10.5.1) which, using time gauge and a real representation for the γ-matrices, lead to the canonical ∗ brackets {λα, λβ} ∼ iδαβ with the reality conditions λ = λ. They can thus also be treated with the above techniques by substituting ρi with λ and removing the AIJ matrices as well as the η¯IJ 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 with the 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 necessary when 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, we define D(D+1) +1 L γfive := i 2 γ0 γ1 . . . γD (10.5.2) 2 † with the properties γfive = 1, γfive = γfive and [γI , γfive]+ = 0 (which follows from our conventions L 2 2 † for the gamma matrices (γ0 ) = −1, γi = 1, γI = ηII γI ). We introduce the chiral projectors 1 P± = (1 ± γ ) , (10.5.3) 2 five which fulfil the relations P±P± = P±, P±P∓ = 0, P+ +P− = 1 and (P±)† = P±. These follow directly 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 Z   D+1 i µ I + i µ I + SF = − d X ΨeI γ DµP Ψ − DµΨeI γ P Ψ . (10.5.4) M 2 2 The 3+1 split is performed analogous to [37]. Choosing time gauge, we obtain the non-vanishing Poisson brackets n ± ±o ± Ψα , Πβ = −Pαβ, (10.5.5)

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

− χα := Πα , (10.5.6) where we used the notation Ψ± := P±Ψ. The first class property of this constraint follows from the fact that the action (10.5.4) and therefore all resulting constraints do not depend on Ψ− at all. In the quantum theory, the Hilbert space for the chiral fermions can be constructed similar to the case of non-chiral ones [21]. We obtain a faithful representation of the Poisson

91 algebra by replacing the operators θˆ (acting by multiplication) and θ¯ˆ = d defined in [21] by α α dθα ˆ+ + ˆ ¯ˆ+ ¯ˆ + θα := Pαβθβ and θα := θβPβα, as can be seen by

hˆ+ ¯ˆ+i + θα , θβ = Pαβ. (10.5.7) + The reality conditions are implemented if we use the unique measure constructed in [21]. We then have to impose the condition ¯ˆ− θα f({θβ}) = 0, (10.5.8) + which restricts the Hilbert space to functions f such that f({θα}) = f({Pαβθβ}). Classically, − observables do not depend on Ψα . In the quantum theory, they become operators which do not ˆ− ¯ˆ− contain θα and therefore commute with θα .

Spin 3/2 Majorana-Weyl fermions Majorana-Weyl spin 3/2 fermions appear in chiral supergravity theories, e.g., D + 1 = 9 + 1, T N = 1 [148]. In general, in a real representation (γI = ηII γI ) or in a completely imaginary D(D+1) T T 2 +1 D+1 representation (γI = −ηII γI ) we have γfive = (−1) γfive. Therefore, if 2 is odd, we ± T ± D+1 ± T ∓ have (P ) = P , and if 2 is even, (P ) = P . In the case at hand (D = 9), there exists a real representation and the chiral projectors will be symmetric, (P±)T = P±. Again, we will just consider the kinetic term for a chiral Rarita-Schwinger field, Z D+1 µρσ +
S = d X is eψµγ DρP ψσ . (10.5.9) M

+ + T ji + The 3 + 1 split is performed like above. We find the second class constraint πi = i(φj ) γ P − − and the first class constraint πi = 0. We introduce a second class partner φi = 0 for the first class constraint. Then we can solve all the constraints using the Dirac bracket

n + +o + −1 + φi , φj = −P (C )ijP , (10.5.10) DB and all other brackets are vanishing. From here, we can copy the enlargement of the internal space from above, which results in the same theory with all variables projected with P+. (Note + 1 JK + ∗ ∗ 1 JK + ∗ + ∗
that equations like e.g. ρi = 2 AiJ η¯ P (AρK + A ρK ) = 2 AiJ η¯ AρK + A (ρ )K are consistent. This can be seen by the fact that the matrix A(N) can be written as an infinite IJ sum of even powers of gamma matrices, A(N) ∝ exp(iΛIJ (N)Σ ) and therefore it commutes ± + + with the projectors P .) The quantisation of the resulting theory with variables ρr I and σ is similar to the non-chiral case, with chiral projectors P+ added in observables, and modifications of 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 to extend 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 graviton sector 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 where this 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

92 space requires an additional normal field N and that the constraints depend non-trivially on a matrix A(N) which transforms in a complicated fashion under SO(D + 1) but which in the present formulation is crucial in order to formulate the reality conditions for the Majorana fermions in the SO(D + 1) theory. One would expect that the field N is superfluous and that the matrix A(N) would simply drop out when performing an extension to SO(1,D) because then no non-trivial reality conditions need to be imposed. One would expect that one only needs the quadratic and not the linear simplicity constraint and that, just as it happened in the gravitational sector, the Hamiltonian phase space extension method simply coincides with the direct Hamiltonian formulation obtained by an D+1 split of the SO(1,D) action followed by a gauge unfixing step in order to obtain a first class formulation. Surprisingly, this is not the case. The basic difficulty is that when performing the 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 of similar complexity. It therefore appears that there is no advantage of the SO(1,D) extension as compared to the SO(D + 1) even as far as the classical theory is concerned. Of course, the quantum 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 spinors coming from SO(1,D) on the one hand and an SO(D +1) extension of the theory which actually needs complex valued spinors on the other, is most probably far from unique nor the most elegant one. Several other solutions have suggested themselves in the course of our analysis but the corresponding reformulation is not yet complete at this point. Hence, we may revisit this issue in the future and simplify the presentation. Furthermore, it would be interesting to investigate if the extensions of the gauge group SO(D) → SO(D + 1) also is possible in the case of symplectic Majorana fermions, which would permit access to even more supergravity theories. To the best of our knowledge, the background independent Hilbert space representation of the Rarita-Schwinger field presented in section 10.4 is also new. Apart from the fact that this has to be done for half-density valued Majorana spinors whose tensor index is transformed into an 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 deal with the non-trivial Dirac bracket.

93 Chapter 11 p-form gauge fields

In the previous chapter, we focussed on the quantisation of the Rarita-Schwinger sector of supergravity theories in various dimensions by using an extension of loop quantum gravity to all spacetime dimensions. In this chapter, we extend this analysis by considering the quantisation of additional bosonic fields necessary to obtain a complete SUSY multiplet next to graviton and gravitino in various dimensions. As a generic example, we study concretely the quantisation of the 3-index photon of maximal 11d SUGRA, but our methods easily extend to more general p-form fields. Due to the presence of a Chern-Simons term for the 3-index photon, which is due to local SUSY, 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 Gauß constraint is possible. Specifically, the Weyl algebra of observables, which deviates from the usual CCR Weyl algebras by an interesting twist contribution proportional to the level of the Chern-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 relativity coupled to standard matter in terms of connection variables for a compact gauge group without second class constraints in order that loop quantum gravity (LQG) quantisation methods, so far formulated only in three and four spacetime dimensions [149, 66], apply. The new field content of supergravity theories as compared to standard matter Lagrangians are 1. Majorana (or Majorana-Weyl) spinor fields of spin 1/2, 3/2 including the Rarita-Schwinger field (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 under consideration. 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 supergravity theories by considering the quantisation of the additional bosonic fields, in particular, p-form fields. 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 photon form 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 use methods from higher gauge theory [150, 151, 152, 153, 154] such as p-groups, p-holonomies etc., 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 of this additional Gauß constraint, the quantisation of the theory is not straightforward and cannot be 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 presence is dictated by supersymmetry and which makes the theory in fact self-interacting, that is, the Hamiltonian is a fourth order polynomial in the 3-connection and its conjugate momentum just like in Yang-Mills theory. In particular, while one can define a holonomy-flux algebra as for Abelian Maxwell-theory, the Ashtekar-Isham-Lewandowski representation [14, 15] is inadequate because 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 terms of a twisted holonomy-flux algebra, which is in fact Gauß invariant. We were not able to find a background independent representation of the corresponding Heisenberg algebra, which also differs by a twist from the usual one, however, one succeeds when formulating the quantum theory 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 that it admits a state of the Narnhofer-Thirring type [49], whence the Hilbert space representation follows 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-contained fashion for the benefit of the reader and in order to settle our notation. We also describe in detail why one cannot straightforwardly apply methods from LQG as mentioned above. In section 11.3, we display the reduced phase space quantisation solution in terms of the twisted holonomy-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]. We will review the analysis of the contribution of the 3-index-photon 3-form Aµνρ = A[µνρ] to the 11d SUGRA Lagrangian with Chern-Simons term. This part of the Lagrangian is given up to a numerical constant by 1 c L = − |g|1/2F F µ1..µ4 −α|g|1/2F J µ1..µ4 − |g|1/2F F A µ1..µ4ν1..ν4ρ1..ρ3. C 2 µ1..µ4 µ1..µ4 2 µ1..µ4 ν1..ν4 ρ1..ρ3 (11.2.1)

Here, F = dA, Fµ1..µ4 = ∂[µ1 Aµ2..µ4] is the curvature of the 3-index-photon and indices are moved with the spacetime metric gµν. Furthermore, J is a totally skew tensor current bilinear in the graviton field not containing derivatives, whose explicit form does not need to concern us here, except that it does not depend on any other fields. Finally, c, α are positive numerical constants whose value is fixed by the requirement of local supersymmetry [156]. The number c could 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 coordinates t, xa; a = 1, .., D adapted to a foliation of the spacetime manifold. The result of a tedious

95 calculation is given by

µ1..µ4 ta1..a3 a1..a4 Fµ1..µ4 F = 4Fta1..a3 F + Fa1..a4 F , ta1..a3 a1..a3,b1..b3 Fta1..a3 F = G Fta1..a3 Ftb1..b3 a1..a3,b1..b4 −M Fta1..a3 Fb1..b4 , Ga1..a3,b1..b3 = gttga1b1 ga2b2 ga3b3 − 3gta1 gtb1 ga2b2 ga3b3 , M a1..a3,b1..b4 = ga1b2 ga2b2 ga3b3 gtb4 ,

a1..a4 a1..a3,b1..b4 Fa1..a4 F = V1 − 4M Fta1..a3 Fb1..b4 , a1b1 a4b4 V1 = g ..g Fa1..a4 Fb1..b4 , µ1..µ4ν1..ν4ρ1..ρ3 a1..a3b1..b4c1..c3 Fµ1..µ4 Fν1..ν4 Aρ1..ρ3  = 8 Fta1..a3 Fb1..b4 Ac1..c3 a1..a4b1..b4c1c2 +3 Fa1..a4 Fb1..b4 Atc1c2 , µ1..µ4 a1..a3 J Fµ1..µ4 = 4j Fta1..a3 + V2, a1..a4 V2 = J Fa1..a4 , (11.2.2) a ..a ta ..a a ..a ta ..a where we used  1 D =  1 D and defined j 1 3 := J 1 3 . The potential terms V1,V2 only depend on the spatial components of the curvature and do not contain time derivatives. Using 1 F = [A˙ − 3∂ A ], (11.2.3) ta1..a3 4 a1..a3 [a1 a2a3]t we may perform the Legendre transform. The momentum conjugate to A reads ∂L πa1..a3 = ˙ ∂Aa1..a3 1/2 a1..a3,b1..b3 a1..a3,b1..b4 a1..a3 = −|g| [G Ftb1..b3 − M Fb1..b4 + αj ] a1..a3b1..b4c1..c3 −c  Fb1..b4 Ac1..c3 . (11.2.4)

We may solve (11.2.4) for Fta1a2a3

1/2 b1..b3 b1..b3 1/2 b1..b3 Fta1..a3 = −|g| Ga1..a3,b1..b3 [π + B + α|g| j ], a1..a3 a1..a3b1..b4c1..c3 1/2 a1..a3,b1..b4 B = c Fb1..b4 Ac1..c3 − |g| M Fb1..b4 , (11.2.5) where c1..c3,b1..b3 b1 b2 b3 Ga ..a ,c ..c G = δ δ δ (11.2.6) 1 3 1 3 [a1 a2 a3] 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 a longer calculation Z n o 10 ˙ a1..a3 H = d x Aa1..a3 π − L Z n 10 a1a2 −1/2 a1..a3 b1..b3 = − d x 3Ata1a2 GC + 2|g| Ga1..a3,b1..b3 [π + B + αj] [π + B + αj]

1/2 o + |g| [V1/2 + αV2] , c Ga1a2 := ∂ πa1..a3 − a1a2b1..b4c1..c4 F F , (11.2.7) C a3 2 b1..b4 c1..c4 where an integration by parts has been performed in order to isolate the Lagrange multiplier

Ata1a2 . Using the ADM frame metric components gtt = −1/N 2, gta = N a/N 2, gab = qab − N aN b/N 2; 2 a b b gtt = −N + qabN N , gta = qabN , gab = qab, (11.2.8)

96 a with qab the induced metric on the spatial slices and lapse respectively shift functions N,N a1a2 we can easily decompose the piece of H independent of the 3-index Gauß constraint GC into a the contributions N HCa + NHC to the spatial diffeomorphism constraint and Hamiltonian constraint, 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 the p-form sector. We smear the Gauß constraint with a 2-form Λ, that is Z 10 ab G[Λ] := d x Λab G (11.2.9)

abc and study the gauge transformation behaviour of the canonical pair (Aabc, π ) with non- vanishing Poisson brackets

{πa1..a3 (x),A (y)} = δ(10)(x, y) δa1 δa2 δa3 . (11.2.10) b1..b3 [b1 b2 b3] We find

{G[Λ],Aa1..a3 } = −∂[a1 Λa2a3], a1..a3 a1..a3b1..b3c1..c4 {G[Λ], π } = c  ∂[b1 Λb2b3]Fc1..c4 . (11.2.11) These equations can be written more compactly in differential form language, in terms of which they are easier to memorise. Introducing the dual 7-pseudo-form1 1 (∗π) :=  πb1..b3 (11.2.12) a1..a7 3! 7! b1..b3a1..a7 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 of the theory can be coordinatised by integrals of A and ∗π respectively over closed 3-submanifolds or 7-submanifolds respectively. The G(Λ) generate an Abelian ideal in the constraint algebra since

G[Λ],G[Λ0] = 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 contributions to 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 behaviour of ∗π differs from the transformation behaviour with respect to the higher-dimensional analog a1..a3 of the usual Maxwell type of Gauß law, which would be just the divergence term ∂a1 π . In particular, πabc itself is not gauge invariant. This “twisted” Gauß constraint (11.2.7) can be written in the form c Ga1a2 := ∂ [πa1..a3 − a1..a3b1..b3c1..c4 A F ] =: ∂ π0a1..a3 , (11.2.15) a3 2 b1..b3 c1..c4 a3 which suggests to introduce a new momentum π0. Unfortunately, this does not work because ∗(π0 − π) = A ∧ F does not have a generating functional K with δK/δA = A ∧ F , since the only possible candidate K = R A ∧ A ∧ F ≡ 0 identically vanishes in the dimensions considered here. Since this is not the case, the Poisson brackets of π0 with itself do not vanish and neither

1 abc 3 Notice that π is a tensor density of type T0 and density weight one.

97 is π0 gauge invariant as we will see below, so that there is no advantage of working with π0 as compared to π. The presence of the twist term in the Gauß constraint leads to the following difficulty when trying 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 from holonomies Z Z A(e) = exp(i A), π(S) = ∗π, (11.2.16) e S where e and S are oriented 3-dimensional and 7-dimensional submanifolds respectively, which we call “edges” and surfaces in what follows. One could then study the GNS Hilbert space representation 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 it abstractly by requiring that the charge network functions

Y ne Tγ,n = A(e) , ne ∈ Z (11.2.18) e∈γ form an orthonormal basis in the corresponding H = L2(A, µ), see [66] for details. Here, a graph γ is a collection of edges which are disjoint up to intersections in “vertices”, which are oriented 2-manifolds. The possible intersection structure of these cobordisms should be tamed by requiring that all submanifolds are semi-analytic. Up to here everything is in full analogy with LQG. The problem is now to isolate the Gauß invariant subspace of the Hilbert space: While the connection transforms as in a theory with untwisted Gauß constraint, it appears that we can solve it by requiring that charges add up to zero at vertices. However, this does not work because while such a vector is annihilated by the divergence term in Gab, it is not by the second term ∝ A ∧ F . Even more disastrous, the term A ∧ F does not exist in this representation which is strongly discontinuous in the holonomies so that operators A, F do not exist. Finally, although π is not Gauß invariant, it leaves this would be gauge invariant subspace invariant, which reveals that this subspace is not the kernel of the twisted Gauß constraint. We therefore must be more sophisticated. Since the A dependent terms in G cannot be quantised 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 secure that an implementation of the corresponding automorphism group αΛ ◦ αΛ0 = αΛ+Λ0 by unitary operators U(Λ) exists. The U(Λ) would correspond to the desired exponentiation of the Gauß constraint. One way of securing this is by looking for an invariant state ω = ω ◦ αΛ on the holonomy - flux algebra (see [157]) for the details for this construction). This would then open the possibility that the Gauß constraint can be solved by group averaging methods. The first problem is that the automorphisms do not preserve the holonomy-flux algebra because there appears an F on the right hand side of (11.2.19) which should appear exponentiated in order that the algebra closes. This forces us to pass to exponentiated fluxes, that is, to the corresponding Weyl algebra defined by exponentials of π, A. This algebra is now preserved by the 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

iπ(S) iπ(S) i[π(S)+c R dΛ∧F ] ω(e ) = 1, ω(αΛ(e )) = ω(e S ) = 0 (11.2.20)

98 for suitable choices of Λ. In the GNS Hilbert space we would like to have unitary operators ∗ U(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 under U(Λ). Now the operator U(Λ) should correspond to exp(iG[Λ]) and using a calculation similar to (11.2.14) and the BCH formula one shows that on the LQG vacuum Ω = 1 it reduces formally to U(Λ)Ω = exp(ic/2 R 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 the vacuum and such that the representation property U(Λ)U(Λ0) = U(Λ + Λ0),U(Λ)∗ = U(−Λ) (possibly up to a projective twist) holds. We did not succeed to find a solution to this problem indicating that a unitary implementation of the Gauss constraint is impossible in the LQG representation and even it were possible, the strategy outlined in the next section is certainly more natural. We also remark that solving the constraint by group averaging methods becomes non-trivial if not impossible in case of the non-existence of U(Λ). Even if we could somehow construct 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 procedure followed 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 chance to 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 constrained to vanish, only on the combination π + B + αj. Obviously j trivially Poisson commutes with G. Unpacking B from (11.2.5), we see that π + B is a linear combination (with only metric dependent coefficients) of F and

abc abc abcd1..d4e1..e3 P := π + c Fd1..d4 Ae1..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 be verified 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 modulo an exact form, which in turn is parametrised by a 2-form. This 2-form worth of gauge freedom matches the number of Gauß constraints which can be read as a condition on π. Thus, on the constraint 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 (a totally skew 4-times contravariant tensor pseudo density) and smear the observables with these Z Z Z Z 10 a1..a3 10 a1..a4 P [f] := d x fa1..a3 P = f ∧ ∗P,F [h] := d x (∗h) Fa1..a4 = h ∧ F . (11.3.3) Then, we find after a short computation Z {F [h],F [h0]} = 0, {P [f],F [h]} = h ∧ df, {P [f],P [f 0]} = −3c F [f ∧ f 0]. (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 of the 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 representation in 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 F which is a contradiction. Hence, either both F,P exist or only both of their corresponding Weyl elements. 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 are discontinuous in both h, f. This representation was studied in the context of QED in [49] and was applied to an LQG type of quantisation of the closed bosonic string in [105]. Before we define 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 the generalisation of the Baker-Campbell-Hausdorff formula [158, 159, 160, 161, 162, 163] to higher commutators [164]. Suppose that X,Y are operators on some Hilbert space such that the triple commutators [X, [X,Y ]] and [Y, [Y,X]] commute with both X and Y . This formally applies to our case with X = F [h] + P [f],Y = F [h0] + P [f 0], which obey the canonical commutation relations (we set ~ = 1 for simplicity)  Z  [X,Y ] := i{X,Y } = i [ (h0 ∧ df − h ∧ df 0)] 1 − 3cF [f ∧ f 0] . (11.3.7)

From this follows for the triple commutators Z [X, [X,Y ]] = −3c(i)2 {P [f],F [f ∧ f 0]} = 3c f ∧ f 0 ∧ df 1, Z [Y, [Y,X]] = 3c(i)2 {P [f 0],F [f ∧ f 0]} = −3c f ∧ f 0 ∧ df 0 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

X Y X+Y + 1 [X,Y ]+ 1 ([X,[X,Y ]]+[Y,[Y,X]]) e e = e 2 12 , (11.3.9) which can also be proved using elementary methods. From this it is easy to derive the also useful Zassenhaus formula [164]

X+Y X Y − 1 [X,Y ] − 1 ([X,[X,Y ]]+2[Y,[X,Y ]]) e = e e e 2 e 6 . (11.3.10)

Putting all these together, we obtain the Weyl relations 3c W [h, f] W [h0, f 0] = W [h + h0 + f ∧ f 0, f + f 0] 2  i Z  × exp 2(h ∧ df 0 − h0 ∧ df) − cf ∧ f 0 ∧ d(f − f 0) . (11.3.11) 4

100 Hence also the Weyl relations get twisted as compared to the situation with c = 0. Notice that the first term in the phase is antisymmetric under the exchange (h, f) ↔ (h0, f 0), while the second is symmetric. In order to obtain a representation of this ∗-algebra A generated by the Weyl elements, it is sufficient to find a positive linear functional. We consider the Narnhofer-Thirring type of functional  1 h = f = 0 ω(W (h, f)) = (11.3.12) 0 else and show that it is positive definite on A. Let

N X a := ck W [zk] (11.3.13) k=1 be a general element in A, where N ∈ N, ck ∈ C and the zk = (hk, fk) are arbitrary, where without loss of generality zk 6= zl for k 6= l. We have

N ∗ X ω(a a) = c¯k cl ω(W [−zk] W [zl]) k,l=1 N X = c¯k cl ω(W [zkl]) exp(iαkl), k,l=1 3 z = (−h + h − cf ∧ f , −f + f ), kl k l 2 k l k l 1 Z α = [2(−h ∧ df + h ∧ df ) − cf ∧ f ∧ d(f + f )]. (11.3.14) kl 4 k l l k k l k l

For k = l, we have zkl = αkl = 0 because fk, fl are 3-forms. For k 6= l, we must have either fk 6= fl or hk 6= hl or both. If fk 6= fl, then obviously zkl 6= 0. If fk = fl, then necessarily hk 6= hl and zkl = (−hk + hl, 0) 6= 0. By definition (11.3.12) then

N ∗ X 2 ∗ ω(a a) = |ck| ≥ 0; ω(a a) = 0 ⇔ a = 0 (11.3.15) k=1 is positive definite. Thus, the left ideal I = {a ∈ A; ω(a∗a) = 0} = {0} is trivial and the Hilbert 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 scalar product < a, b >:= ω(a∗b) and the representation is simply π(a)b := ab on the common dense domain D = A. The representation is evidently strongly discontinuous in both h, f and while cyclic, it is not irreducible. 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 defined when restricting the smearing functions (h, f) to the form factors of 4-surfaces and 7-surfaces respectively. The bearing of this question is that in the Hamiltonian constraint the functions F and ∗P appear in such a way, that in a discretisation of it, which results from replacing the integral by Riemann sums in the spirit of [13], these functions are naturally smeared over 4-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”) Z S4 b1 b4 ha1..a6 (x) := a1..a6b1..b4 dy ∧ dy δ(x, y), S4 Z S7 b1 b7 fa1..a3 (x) := a1..a3b1..b7 dy ∧ dy δ(x, y). (11.3.16) S7

Then Z Z F [hS4 ] = F,P [f S7 ] = ∗P . (11.3.17) S4 S7 Thus, the natural integrals of F,P over surfaces can be reexpressed in terms of distributional 6 forms and 4-forms respectively. It remains to check whether the exterior derivative and product combinations 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 product expressions appear. The first is, using formally Stokes theorem Z Z Z hS4 ∧ df S4 = df S7 = f S7 S4 ∂S4 Z Z a1 a3 b1 b7 = dx ∧ .. ∧ dx a1..a3b1..b7 dy ∧ .. ∧ dy δ(x, y) =: σ(∂S4,S7). (11.3.18) ∂S4 S7

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 the surfaces 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 takes the values ±1. This can be justified by the same regularisation as in LQG for the holonomy-flux algebra [66]. S S0 The second type of integral is given by F [f 7 ∧ f 7 ]. The support of the integral will be S S0 on S 7 ∩ S 7 and in D = 10 dimensions this will decompose into components that are at least 4-dimensional. By the same regularisation as in [66], one can remove the higher-dimensional components and thus keep only the 4-dimensional ones. In what follows, we thus assume that 0 S4 := S7 ∩ S7 is a single 4-dimensional component, otherwise the non vanishing contributions are over a sum of those. We have Z S S0 S0 F [f 7 ∧ f 7 ] = F ∧ f 7 (11.3.19) S7 Z Z a1 a4 b1 b3 c1 c7 = dx ∧ .. ∧ dx ∧ dx ∧ .. ∧ dx b1..b3c1..c7 dy ∧ .. ∧ dy δ(x, y) Fa1..a4 (x). 0 S7 S7 By assumption, we have embeddings

0 X : U → S ; Y 0 : V → S ; Z : W → S , (11.3.20) S7 7 S7 7 S4 4 with open subsets U, V of R7 and an open subset W of R4 respectively, whose coordinates will be denoted by u, v, w respectively. The condition X (u) = Y 0 (v) = Z (w) is solved by solving u, v S7 S7 S4 for w, which leads to u = u(w), v = v(w). Since the integrals are reparametrisation invariant, 0 in the neighbourhood of S4 on both S7 and S7 therefore we may use adapted coordinates so that I I I I I w = u = v ,I = 1, .., 4 on S4 and u , v ,I = 5, .., 7 denote the transversal coordinates, which 0 take the value 0 on S4. In this parametrisation both U, V are of the form U = W × U ,V = W × V 0 for some 3-dimensional subsets U 0,V 0 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 and I I u = v = 0,I = 5, .., 7 and we have in the neighbourhood of S4

4 7 a a X a  I I  X  a I a I  X (u) − Y (v) = − YI (u, 0) u − v + XI (u, 0)u − YI (u, 0)v . (11.3.21) I=1 I=5 We can now solve the δ distribution in (11.3.19) by performing the integral over u5, .., u7, v1, .., v7 a a I a a I and find with the notation X = ∂X (u)/∂u and Y = ∂Y 0 (v)/∂v etc. I S7 I S7 Z Z S S0 7 I ..I h a a b b i 7 J ..J h b b i F [f 7 ∧ f 7 ] = d u  1 7 X 1 ..X 4 X 1 ..X 3 (u) d v  1 7 Y 4 ..Y 10 (v) × I1 I4 I5 I7 b1..b10 J1 J7 U V

δ (X(u),Y (v)) Fa1..a4 (X(u)) Z h i = − d4w I1..I7 Za1 ..Za4 (w) J1..J7 F (Z(w))  × I1 I4 a1..a4 I5..I7J1..J7 W " !#  ∂(X(u) − Y (v))  sgn det 5 7 1 7 ∂(u , .., u , v , .., v ) vI =uI =wI ;I=1,..,4;vI =uI =0;I=5,..,7

0 S4 =: − 3! 7!˜σ(S7,S7)F [h ], (11.3.22) where the 10d antisymmetric symbol is in terms of the coordinates u5, .., u7, v1, .., v7 and in the last step we noticed that the range of I1..I4 is restricted to 1..4. Also, we assumed that the 0 sign function under the integral is constant and equal toσ ˜(S7,S7) on S4 (which defines this function), otherwise we must decompose S4 further. Under this assumption, we conclude the form factor identity 0 S ∩S0 f ∧ f 0 = −3! 7!σ ˜(S ,S )h 7 7 . (11.3.23) S7 S7 7 7 Finally, we consider the integral of the third type, which now combining (11.3.18) and (11.3.24) is easily calculated Z Z 0 S ∩S0 S 0 0 f ∧ f 0 ∧ df = −3! 7!σ ˜(S ,S ) h 7 7 ∧ df 7 = −3! 7!σ ˜(S ,S )σ(∂(S ∩ S ),S ) = 0, S7 S7 S7 7 7 7 7 7 7 7 (11.3.24) 0 because ∂(S7 ∩ S7) ⊂ S7 for which σ vanishes by definition. In order to make this restricted Weyl algebra close, we now have to decide whether the form 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 algebra already closes. In the former case we must replace the form factors f S7 by √ 1 f S7 , such 3! 7! 3c/2 that in the simplest situation we have  i  W [S ,S ] W [S0 ,S0 ] = W [S +S0 −σ˜(S ,S0 )S ∩S0 ,S +S0 ] exp σ(∂S ,S0 ) − σ(∂S0 ,S ) , 4 7 4 7 4 4 7 7 7 7 7 7 2 4 7 4 7 (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, wherever P or F appear in the Hamiltonian constraint, we follow the general regularisation procedure outlined 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 of P [S7] and F [S4] = A[∂S4], which we approximate for instance by sin(P [S7]), sin(F [S4]) similar as in LQG. The details are obvious and are left to the interested reader.

103 11.4 Conclusions

Supergravity theories typically need additional bosonic fields next to the graviton, in order to obtain a SUSY multiplet (representation) containing the gravitino. In this chapter, we focussed on 11d, N = 1 SUGRA for reasons of concreteness (and its relevance for lower dimensional SUGRA theories), which contains the 3-index photon in the bosonic sector. However, our analysis is easily generalised to arbitrary p-form fields. Without the Chern-Simons term in the action (i.e. c = 0) the analysis would be straightforward and in complete analogy to the background independent treatment of Maxwell theory in D + 1 = 4 dimensions [20]. In particular, the Hamiltonian constraint would be quadratic in the 3-form field and its conjugate momentum, 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 in the connection and thus becomes self-interacting even when switching off gravity, just like in non-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 is self-interacting. In fact, in terms of the observables, the Hamiltonian constraint is a quadratic polynomial, however, the price to pay is that the observable algebra is non-standard. Yet, the resulting Weyl algebra can be computed in closed form and we found at least one non-trivial and background independent representation thereof, which nicely fits into the background indepen- dent quantisation of the gravitational degrees of freedom in the contribution to the Hamiltonian constraint depending on the 3-index-photon. There are many open questions arising from the present study. One of them concerns the reducibility of the GNS representation found, which involves a mixed state. It would be nice to have control over the superselection sectors of the theory and, in particular, to analyse whether the 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 representations for 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 theory of relevance when restricting the Weyl algebra to distributional 4-form and 7-form factors as smearing functions which is only sketched in this chapter. We plan to revisit these questions in future publications.

104 Part IV

Initial value quantisation of higher-dimensional general relativity

105 Chapter 12

Loop quantum gravity without the Hamiltonian constraint

In this chapter, we show that under certain technical assumptions, including the existence of a CMC slice and strict positivity of the scalar field, general relativity conformally coupled to a scalar field can be quantised on a partially reduced phase space, meaning reduced only with respect 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 acting on both, the metric and the scalar field, which coincides with the CMC gauge condition. A new metric, which is invariant under this transformation, is constructed and used to define connection variables which can be quantised by standard loop quantum gravity methods. Since this connection is invariant under the local conformal transformation, the generator of which is shown to be a good gauge fixing for the Hamiltonian constraint, the Dirac bracket associated with 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. The physical 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 to address dynamical problems in this framework (due to the complicated “time” function), it seems, due to good accessibility properties of the CMC gauge, to be well suited for problems such as the computation of black hole entropy, where actual physical states can be counted and the dynamics is only of indirect importance. Also, the interpretation of the geometric operators gets an interesting twist, which exemplifies the deep relationship between observables and the choice of a time function. The original work from which this chapter is taken is [43]. Parts of the ideas are based on [44].

12.1 Introduction

In the second part of this thesis, we have shown that the quantisation techniques developed for 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 well defined 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 well understood. Even more so, we do not have access to a set of Dirac observables commuting with the Hamiltonian constraint operator (except for the ADM-charges). The master constraint approach [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, the issue of quantum observables remains open also in this approach. This is even more true in the case of higher dimensions considered in this thesis, since the Hamiltonian constraint operator is even more complicated than when using Ashtekar-Barbero variables and the issue of anomalies arises 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 of observables. However, a reduced phase space quantisation of a given theory is generally very problematic due to the complexity of the representation problem resulting from a non-trivial observable algebra, as is e.g. the case for pure general relativity. In order to make progress with the above mentioned issues, relational models were introduced in LQG, pioneered by Giesel and Thiemann [22] using the Brown-Kuchar dust model [170]. Due to the additional matter fields which are used as clocks and rods, it is possible to construct an observable algebra with the same algebraic structure as the holonomy-flux algebra, so that the methods developed for the constrained quantisation can be directly applied also to the reduced phase space. Further models 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 (and rod) variables. Consequently, this choice of clock is only valid as long as the clock matter fields remain classical in the experiments which we are describing. This situation might hold if the clock matter fields couple sufficiently weak, or not at all, to e.g. other particles in a scattering experiment. Nevertheless, it would be desirable to have a clock which is purely geometrical and thus 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 on the particular choice of clock. While such true Hamiltonians have been derived in the available reduced phase space quantisations, there are situations, e.g. state counting in the derivation of the black hole entropy, where the quantum dynamics are not relevant, but only access to the physical 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 the specific 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 slice defined by the gauge fixing condition. By restricting to such a quantisation on a fixed spatial slice, we neglect the problem of quantum dynamics, but nevertheless have a quantisation of the observable algebra (up to the problem of implementing the spatial diffeomorphisms). Such observables 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 be changed, since the Dirac bracket associated to implementing the gauge condition D = 0 will be non-trivial. A resulting change of spectrum of the geometric operators in loop quantum gravity at 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 other available reduced phase space quantisations do not change spectrum due to the matter field nature of the clock variables. In this chapter, we show how such a reduced phase space quantisation can be constructed for general 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 local rescaling 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 Hamiltonian constraint 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 the Gauß and spatial diffeomorphism constraints. The spatial diffeomorphism constraint poses the same difficulties as in standard LQG, e.g. the construction of spatially diffeomorphism invariant operators 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 can be treated by the proposed method, thus allowing black hole state counting at the level of the physical 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 the Hamiltonian constraint for the type of static black hole solutions under consideration in this chapter. E.g. since {H, D} would be linear in the momenta for scalar field or dust clocks D, we have {H, D} = 0 in static situations at least for the type of foliations (vanishing momentum of the 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 the most 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. The action of the scalar field conformally coupled to general relativity is given by 1 Z √ S = dD+1X gR(D+1)a(Φ) κ M Z 1 D+1 √ µν + d X gg (∇µΦ)(∇νΦ), (12.2.1) 2λ M where we defined 1 − D κ(D − 1) a(Φ) := 1 − αΦ2, ∆Φ := ∆g, α := − . 4 8λD

The scalar field part of the action, i.e. S − SEinstein-Hilbert, is invariant under the conformal transformation ∆g ∆Φ gµν → Ω gµν, Φ → Ω Φ. (12.2.2) The D + 1 split of this action gives Z Z D h ab a i S = dt d x P q˙ab + πΦΦ˙ − N Ha − NH , (12.2.3) R σ

108 where we have defined 1 √ 4α√ π := − q(L Φ) + qΦK, Φ λ n κ 1 √   2α√ P ab := a(Φ) q Kab − qabK + qqabΦ(L Φ), κ κ n 1 P ab := P ab − qabP cdq , tf D cd Z a D h ab i Ha[N ] := d x P (LN q)ab + πΦ(LN Φ) , (12.2.4) σ " # Z κ D2 H[N] := dDxN H + H − , Grav Φ g2 √ σ ∆ D(D − 1) q κ2 √ κH := √ P tf P ab − qR(D), Grav qa(Φ) ab tf

κ √  λ2 1 κH := q − π2 − qab(D Φ)(D Φ) Φ 2λ q Φ D a b D − 1 1 ∆Φ  + ΦD DaΦ + R(D)Φ2 , D a D ∆g √ ∆g(1 − D) q D := ∆gP + ∆Φπ Φ = K. (12.2.5) Φ κ n denotes the normal vector on the spatial slices, L the Lie derivative, Kab the extrinsic curvature, ab ab Da the covariant derivative compatible with the spatial metric qab, P = P qab, and K = K qab. 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 constructed from specific initial data (a spatial metric, a transversal trace free second rank tensor field and a constant value for the mean curvature) by performing a conformal rescaling of the fields with a scaling factor satisfying the Lichnerowicz-York equation. On the other hand, if, morally speaking, only conformal equivalence classes of initial data would be specified, one could perform a conformal transformation to initial data satisfying the Hamiltonian constraint without leaving the equivalence class, i.e. without changing the initial data. It therefore transpires that one should try to exchange the equation H = 0 for invariance under a local conformal rescaling. Parts of this idea have been implemented in shape dynamics [50], however, it was not possible so far to find a general solution to the conformal invariance condition which could also be quantised in a satisfactory way. In this chapter, we take this last step by realising that a conformally coupled scalar field, as opposed to obstructions arising from other matter fields [175], allows for a non-trivial conformal weight in the generator of the local conformal transformation, and thus for the construction of a conformally invariant metric. We remark that an earlier account of introducing 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, which is 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 the Hamiltonian constraint at least locally and restricting to spatial slices which allow for the D = 0 gauge. 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[...] Z  2  D √ g a κ tf ab 1 (D) + d x (D − 1) qρ∆ DaD − 2 PabPtf − R N (12.2.6) σ 2qa(Φ) 2 and conclude that D = 0 is locally a good gauge fixing if the elliptic partial differential operator κ2 1 D Da − P tf P ab − R(D) (12.2.7) a 2qa(Φ)2 ab tf 2 is invertible. By a standard argument from the theory of partial differential equations [178] and using the assumption of a compact spatial slice without boundary (boundaries and non-compact spatial slices are treated in [44]), it is sufficient to show that κ2 1 P tf P ab + R(D) > 0. (12.2.8) 2qa(Φ)2 ab tf 2 ν µ Demanding the dominant energy condition (−Tµ ζ is a future causal vector for all future timelike vectors ζ) and using the field equations as well as the vanishing of the constraints, it follows that 1 1 κ2 R(D) > [K Kab − K2] ≈ P tf P ab, (12.2.9) 2 2 ab 2qa(Φ)2 ab tf and thus κ2 1 κ2 P tf P ab + R(D) > P tf P ab ≥ 0. (12.2.10) 2q a(Φ)2 ab tf 2 qa(Φ)2 ab tf We remark that the dominant energy condition does not generally hold for the conformally coupled scalar field [179] and we have to restrict to spacetimes where it is satisfied, e.g. the MST 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

4 φ˜ ab − 4 φ˜ ab q˜ab := e D−1 qab, P˜ := e D−1 P , 1 φ˜ := ln Φ, π˜ := D. (12.2.11) φ˜ ∆Φ Indeed, the new Poisson brackets read ˜cd c d ˜ {q˜ab, P } = δ(aδb), {φ, π˜φ˜} = 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 the spacetimes which we want to describe follows. Next, we pass to the Dirac bracket {·, ·}DB associated with implementing H = D = 0 ab simultaneously, which solves these constraints classically. We note thatq ˜ab and P˜ are enough ab functions to parametrise the reduced phase space, and since {q˜ab, D} = {P˜ , D} = 0, the non-vanishing Dirac brackets among them are ˜cd ˜cd c d {q˜ab, P }DB = {q˜ab, P } = δ(aδb). (12.2.13) The remaining constraint algebra simply reads a b a {Ha[N ], Hb[M ]}DB = Ha[(LN M) ]. (12.2.14) Up to the missing Hamiltonian constraint and the unphysical remaining scalar field, this system is identical to the ADM formulation [46] of general relativity and we can thus use standard techniques from loop quantum gravity in order to quantise it.

110 12.3 Quantisation

From the above ADM-type phase space, we can perform a canonical transformation to real connection 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 quantum gravity methods [149, 66] and the uniqueness result on the representation [180] when demanding a 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 complications associated 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 have nothing new to add to the usual treatment, see [66] for an exposition. Further research for a better 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, can be constructed in the usual manner from the invariant connection. However, their interpretation now changes since their spectrum has to be related with the geometry based on the non-invariant metric. It follows that, morally speaking,

Aˆinv = Φ2AˆLQG, Vˆ inv = Φ2D/(D−1)Vˆ LQG, (12.4.1) where the usual LQG operators measure the actual geometry while the invariant operators have the familiar discrete spectrum. A similar, although conceptually different, observation has been made by Ashtekar and Corichi in [181]. We remark that the possible occurrence of such a phenomenon has been emphasised by Dittrich and Thiemann [171]: The geometric operators of LQG might change their spectrum when taking into account the Hamiltonian constraint. This has to be seen in contrast to the result of [22, 23], where the spectra remain unchanged. The change 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 invariant geometric operators, which have to Poisson commute with the time function at the classical level. Further discussion on this issue is given in [44]. It is interesting to note that when using a Higgs-type potential for Φ which leads to a non-vanishing vacuum expectation value hΦi, one could approximate the invariant geometric operators by the LQG geometric operators times a constant which changes the fundamental geometric scale by a factor of 1/ hΦi in Planck units. While this might present a mechanism to increase the fundamental scale of LQG and make it thus more accessible to experiments, we caution the reader that such an interpretation is strongly tied to the type of foliation we are using and that the associated dynamics have to be investigated to check for consistency with current experiments, thus making further research necessary before jumping to conclusions. Also, the proposed quantisation of Φ would be very different than in the standard model, since Φ would be quantised as a part of the invariant metric 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 integration to physical values of other matter fields. We leave this issue for further research.

111 12.5 Application to black hole entropy

One of the major open problems in the calculation of the black hole entropy in the loop quantum gravity framework is the treatment of the Hamiltonian constraint. While the constraint vanishes on the black hole horizon [182] and therefore does not have to be taken into account there, it still acts on the bulk. In the entropy calculations, it is assumed that every horizon state has at least 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 this chapter, the problem of implementing the Hamiltonian constraint in the bulk does not even arise, since it is solved classically. We briefly sketch some aspects of the black hole entropy calculation in our framework, details will be reported elsewhere. First, we remark that several black hole solutions for general relativity conformally coupled to a scalar field exist, which avoid the no-hair theorems in 3 + 1 dimensions and allow for non-trivial horizon topologies, see [183] for an overview. In order to treat them in our framework, we first have to check if the gauge D = 0 is accessible. For simplicity, let us restrict to static, i.e. the metric 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 follows since all the momenta, and thus D, vanish in this case. Next, we have to check if the gauge is well behaved, i.e. (12.2.7) has trivial kernel (an extension to non-compact spatial slices is treated in [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 shown in [44] that D can be supplemented with an additional term to imply that (12.2.7) has trivial kernel. 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 kernel still follows. The corresponding black hole solution has been found by Mart´ınez,Staforelli and Troncoso, it describes an asymptotically locally AdS black hole and admits non-trivial horizon topologies 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 thermodynamics of this class of black holes. Building on the results of [185, 181], the entropy can be calculated by counting the horizon states which are in agreement with the macroscopic properties of the black 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, because D = 0 can select any t = const hypersurface. However, the transformation between different t = const hypersurfaces is not a gauge transformation but an asymptotic symmetry and thus not a constraint which we have to fix, see also [44]. This can be seen by the fact that the corresponding lapse function would not vanish sufficiently fast at asymptotic infinity, thus leading to 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 Hamiltonian constraint. While a very important question, the quantum dynamics do not seem to be easily addressable in this framework. It is nevertheless possible to ask questions which only depend on the physical Hilbert space and operator spectra. The examples which have been explicitly provided are on the one hand the calculation of black hole entropy, where physical states can be counted as opposed to kinematical states in the standard treatment, and the calculation of the 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

112 of the geometric operators has been conjectured by Dittrich and Thiemann in [171], but it has not been explicitly shown to be feature of loop quantum gravity before. This result raises the question if the fundamental geometric scale of loop quantum gravity could be significantly raised by choosing specific time functions, i.e. performing experiments which measure the geometric properties of areas and volumes which are suitably embedded into spacetime. The example of a black hole to which the gauge fixing discussed in this chapter can be applied should have generalisations in higher dimensions. In the next chapter, we are going to extend the connection formulation derived in the first part of this thesis to the presence of higher-dimensional isolated horizons 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 formalism is that a conformally coupled scalar field allows for a non-trivial conformal scaling and thus for the construction of an invariant metric from which quantisation can start. Also, we do not have a global Hamiltonian as in [50] since we are not restricting to volume preserving conformal transformations.

• An extension to standard model matter, a cosmological constant, and non-compact spatial slices is discussed in [44].

• Since dilaton fields are naturally appearing in supergravity, we plan to investigate an extension of our framework to this setting. Here, it will be interesting to check what extent of supersymmetry is compatible with the D = 0 gauge fixing or how one could also gauge fix the supersymmetry constraint.

113 Part V

Isolated horizon boundaries in higher-dimensional LQG

114 Chapter 13

Classical phase space description of isolated 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-distorted isolated 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 the horizon is a higher-dimensional SO(2(n + 1))-Chern-Simons theory, which has local degrees of freedom. We comment briefly on a possible quantisation of the horizon theory and emphasize that the horizon degrees of freedom which are “visible” from the outside of a black hole form a Poisson-subalgebra, which could significantly simplify the quantisation problem for black hole entropy calculations in higher-dimensional LQG if suitable arguments supporting this treatment can 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 and supergravity. Most prominently, the derivation of black hole entropy within string theory was first performed for a five dimensional black hole [186]. Also, no-hair theorems familiar from d = 4 spacetime dimensions generally fail in higher dimensions, resulting in a large variety of black hole solutions with new (exotic) properties, see [187] for a review. Accordingly, higher dimensions are especially interesting for the area of black hole thermodynamics, since, e.g. the entropy of the black hole depends on its topology, which is considerably richer for d > 4 than in the usual 4 dimensions. While this fact has been appreciated in, e.g. string theory, it was not possible so far to perform these calculations in the context of loop quantum gravity, since the 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-dimensional supergravity derived in this thesis opens the window to treat higher-dimensional black holes also with the methods of loop quantum gravity. The treatment of horizons and black hole entropy within loop quantum gravity can be dated back 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, more precisely SU(2) Chern-Simons theory. While heuristic at certain points, this seminal work already contained many of the ideas which were later necessary to give a rigorous derivation of the black hole entropy within LQG. An entropy associated to a surface which is proportional to the

115 area was first calculated in a paper by Krasnov [189], where the important conceptual ingredient was that the punctures of horizon were distinguishable. Further work which strengthened the relation 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 quantum gravity was derived by Ashtekar and collaborators [182, 191, 192, 193], where the notion of isolated horizon turned out to be crucial in order to have a local description of a black hole horizon. While a classical gauge fixing from SU(2) to U(1) was performed in order to derive the results of [182, 191, 192, 193], it was later shown by Perez and collaborators [194] that the derivation could be extended to an SU(2) invariant framework. The precise state counting for the derivation 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-dimensional black hole entropy using loop quantum gravity methods by deriving a generalisation of isolated horizon boundary condition F ∝ Σ first proposed in [188] and derived rigorously in [182]. We further show that the canonical transformation to higher-dimensional connection variables in- duces a higher-dimensional Chern-Simons symplectic structure on the intersection of the spatial slice with the black hole horizon. Also, we shortly comment on the quantisation of the resulting theory on the black hole horizon, a higher-dimensional Chern-Simons theory. The derivations in 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 spacetime dimensions, the horizon then is odd-dimensional and a Chern-Simons theory can arise. The corresponding classical higher-dimensional black hole solution (with spherical symmetry) was found by Tangherlini [197] and generalises the Scharzschild solution to higher dimensions, see also [187] for an overview. Since, in the loop quantum gravity treatment, the notion of isolated horizon is more central than that of a classical black hole solution, we will not go into details about 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 horizon boundary 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 finding an analogue of the isolated horizon boundary condition in higher dimensions. In section 13.3, we will derive the boundary condition as well as the horizon symplectic structure from the Palatini action 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 the isolated horizon boundary condition and the Chern-Simons symplectic structure independently of 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 of freedom 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 horizon boundary condition. It will turn out that there is merely a single reasonable possibility for the general structure of the boundary condition for which a numerical prefactor and an expression for the connection on the horizon have to be fixed by an actual calculation. However, the connection

116 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 black hole horizon. Let us start with some hints for the boundary condition based on the new connection variables derived in [35, 36]

• Tensorial structure: i ci The 3 + 1 dimensional SU(2) based boundary condition Fab ∝ E abc does not generalise trivially to higher dimensions due to the tensorial structure, i.e. a vector density is dual to a (D − 1)-form in D spatial dimensions, which is a two-form only for D = 3. Since, in analogy to the 3 + 1 dimensional case, we expect to get a theory which is purely defined in terms of a connection on the horizon, the easiest expression with the correct tensorial structure to write down is

aIJ ab1c1...bncn IJK1L1...KnLn π ∝   Fb1c1K1L1 ...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 new variables 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 momentum π and the field strengths F , but the other obvious choice δJ][K1 δL1][K2 . . . δLn−1][Kn δLn][I results in a vanishing right hand side for even n and does not allow for the construction performed in this chapter in the other cases. The open question at this point is mostly on which connection the field strengths should be based.

• Topological invariants: Up to a constant prefactor, the derivation of the boundary condition in three spatial dimensions and spherical symmetry can be easily accomplished by appealing to special properties of curvature tensors in two dimensions. More precisely, the Riemann tensor Rµνρσ on a two-dimensional manifold, e.g. a spatial slice of a black hole horizon, is, due (2) (2) (4) (4) to its symmetries, given by Rµνρσ ∝ R gµ[ρgσ]ν. Thus, after obtaining FµνIJ = RµνIJ = (4) ρσ (4) ρσ (2) ρσ RµνρσΣIJ from the field equations and since Rµν ρσΣIJ = RµνρσΣIJ when using the IH ⇐= ⇐ boundary conditions, it directly follows that F (4) ∝ R(2)Σ , where denotes the ⇐µνIJ ⇐µνIJ ⇐ pullback from the spacetime manifold to a spatial slice of the horizon. In the further discussion of IHs in 4 dimensional LQG, it is of importance that in 2 dimensions, the integral over the Ricci curvature actually is a topological invariant by the Gauß-Bonnet theorem. The question thus is, by which topological invariant that role will be played in higher dimensions. (2) (2) From the above calculation, we expect that only the step using Rµνρσ ∝ R gµ[ρgσ]ν does not straight forwardly generalise to higher dimensions. However, this formula is equivalent (2) αβ IJ (2) to R ∝   RαβIJ , and in this form can be generalised to even dimensions and one is lead to consider the Euler topological density [201]

(D+1) µ1ν1...µnνn I1J1...InJn E :=   Rµ1ν1I1J1 ...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 black hole horizon while the internal gauge group is inherited from the bulk, thus having a

117 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 the boundary, the field strength of which will be inherently “orthogonal” on πaIJ and thus allowing for a precise implementation of the above idea for a boundary condition based on the Euler topological density. We remark at this point that our normalisation of the Euler topological density does not coincide with the standard definition leading to the Euler 1 R (2n) characteristic, but χ = (8π)nn! E 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 defined I1J1 InJn by dLCS = gI1J1...InJn F ∧ ... ∧ F , where d is the exterior derivative and g inter- twines n = (D + 1)/2 adjoint representations of so(D + 1), see [202]. The right hand side of the previous equation can easily be seen to be the Euler topological density for

gI1J1...InJn = I1J1...InJn . The equations of motion derived from this Lagrangian are given I2J2 InJn by gI1J1...InJn F ∧ ... ∧ F = 0, thus fitting nicely in the LQG quantisation scheme for black holes, i.e. the straight forward generalisation of F IJ = 0 at points of the horizon I2J2 InJn which are not punctured by spin networks is given by I1J1...InJn F ∧ ... ∧ F = 0. As the canonical analysis of higher-dimensional Chern-Simons theory reveals [202], the I2J2 InJn theory has local degrees of freedom, e.g. gI1J1...InJn F ∧ ... ∧ F = 0 does not imply F 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 of hyb Peldan’s hybrid connection ΓaIJ [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 that there might be other connections, e.g. a one-parameter family depending on a free parameter unrelated 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 derived from 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 content us in this chapter with providing the definition of a non-distorted isolated horizon and refer to [45] for a detailed discussion of the definition and its consequences. Also, we will be rather sketchy 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 ∆ but normal to the two-sphere cross-sections). We introduce a coordinate v on ∆ such that k = −dv is the other future oriented null vector field normal to the preferred foliation. We extend k uniquely to M at points of ∆ by requiring that it is null. Furthermore, we fix l, k up to mutually inverse and constant rescaling and require that ilk = −1.

118 3. (a) l is expansion-free. (b) k is shear-free with nowhere vanishing spherically symmetric expansion and vanishing µ ν Newman - Penrose coefficients πJ = l mJ ∇µkν on ∆. 4. All field equations hold at ∆.

µ ν 5. −Tν l 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 cross sections of ∆. As in the first part of this thesis, nI denotes the internal normal on the Cauchy slice σ. Local coordinates on S will be denoted by lower case Greek letters from the beginning of the alphabet, α, β, . . .. Let sa be the normal on the boundary of σ, pointing outwards of σ, I a I i.e. inwards of S, and s = s ea its internal version. The pullback of the spatial co-(D + 1)-bein I I eα to S is denoted by mα. With we mean a pullback from M to S. The induced metric on S ← ⇐ I (D−1) R (D−1) D−1 is denoted by hαβ = mαmβI . Also, we use E := S E d x to denote the mean of 1 (D−1) the Euler topological density. It is related to the Euler characteristic via χ = (8π)nn! E . From this definition, several properties of l, k, and the Riemann tensor can be deducted, as shown in [45]. The important observation is that, after a long calculation, it follows that

(D+1) µ0 ν0 (D+1) ρI σJ µ0 ν0 (D−1) ρI σJ F µνIJ = RµνIJ = hµ hν Rµ0ν0ρσe e = hµ hν Rµ0ν0ρσm m , (13.3.1) ⇐ ⇐ ⇐= ⇐= ⇐= ⇐= i.e. the spacetime curvature can be expressed in terms of the induced Riemann tensor of the sphere cross section of the isolated horizon. Thus, we are already very close to the envisaged (D−1) ρI σJ boundary condition in (13.2.1). However, we still need to relate Rµ0ν0ρσm m to a field strength of an SO(1,D) connection defined on S. This can be accomplished by noting that Peldan’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 manifold I 0 with a given (D + 1)-bein mα, one can construct a unique connection ΓαIJ by requiring that the resulting covariant derivative annihilates, next to the (D + 1)-bein, also two additionally chosen normals. We will use nI , the internal normal on the Cauchy slices of our spacetime, and I a I s = s ea, the internal normal on S. It follows for the resulting field strength, denoted by RαβIJ , I I (D−1) ρI σJ that RαβIJ n = RαβIJ s = 0 and RαβIJ = Rαβρσ m m . Concluding, we immediately find

µ ν ...µ ν K1L1...KnLnIJ 1 1 n n   F µ ν K L ...F µ ν K L ⇐ ⇐ 1 1 1 1 ⇐ n n n n 1 = √  ρ1σ1...ρnσn  µ1ν1...µnνn R(D−1) ...R(D−1) 2n[I sJ] h ⇐ ⇐ µ1ν1ρ1σ1 µnνnρnσn 1 E(2n) α1β1...αnβn IJK1L1...KnLn aIJ = √   Rα β K L ...Rα β K L ≈ √ π sa , (13.3.2) h 1 1 1 1 n n n n h 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 Z Z s 4 µI νJ IJ d X ee e FµνIJ (A) = ΣIJ ∧ F (13.3.3) 2 M M

119 µ ν IJ IJ IJ with the definitions F = 1/2Fµνdx ∧ dx , Fµν = 2∂[µAν] + [Aµ,Aν] , Σ := − ∗ (e ∧ e), and K − ∗ (e ∧ e) = 1 eK1 . . . e D−1  . It is well known that when calculating µ1...µD−1IJ (D−1)! µ1 µD−1 IJK1...KD−1 the first variation of this action, the boundary term Z IJ ΣIJ ∧ δA , (13.3.4) ∆ ← ← arises. However, as shown in [45], this term vanishes by the isolated horizon boundary conditions. IJ The second variation of the Palatini action yields the well known symplectic current δ[1Σ δ2]AIJ . By the covariant phase methods, it is closed, and it follows that Z Z Z IJ ( − + )δ[1Σ δ2]AIJ = 0. (13.3.5) σ2 σ1 ∆ Since the symplectic structure should be independent of the choice of spatial hypersurfaces, we have to show that the integral over ∆ results in two boundary terms, integrals over the sphere cross sections S2 and S1 bounding σ2 and σ1. Indeed, in [45], it is shown that Z IJ S2 S1 δ[1Σ δ2]AIJ = ΩCS(δ1, δ2) − ΩCS(δ1, δ2), (13.3.6) ∆ ← ← where nA Z     ΩS (δ , δ ) = S IJKLM2N2...MnNn δ A ∧ δ A ∧ F ∧ ... ∧ F CS 1 2 (2n) [1 IJ 2] KL M2N2 MnNn E S ⇐ ⇐ ⇐ ⇐ (13.3.7) with 2n = D − 1. The details involve a rather lengthy calculation with the intermediate result that Z S I ΩCS(δ1, δ2) = 2(δ[1s˜ )(δ2]nI ), (13.3.8) S √ wheres ˜I = 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 part of this thesis: The above symplectic structure arises from the boundary term (4.1.25) when performing the canonical transformation. With this observation, we can proceed to the next section, where we will use SO(D + 1) as a gauge group instead of SO(1,D) in this section, which is 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 the connection on the horizon with the bulk degrees of freedom, as well as the symplectic structure on the horizon, which coincides with the one of higher-dimensional Chern-Simons theory. Since we started from the spacetime covariant Palatini action, the internal gauge group was fixed to SO(1,D). In the light of quantising the bulk degrees of freedom however, it was pointed out in the first part of this thesis that one can change the internal gauge group to SO(D + 1) by a canonical transformation from the ADM phase space. After this reformulation, the quantisation of the bulk degrees of freedom can be performed with standard loop quantum gravity methods as spelled out in chapter 6. Thus, we are interested in reformulating the horizon boundary condition 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

120 that the construction of the boundary condition (13.3.2) can be taken over from the Lorentzian case by simply changing the internal group to SO(D + 1) and using the generalised Peldan 0 hybrid connection ΓαIJ from the previous section as a connection on S. As was noted at the end of section 4.1, the canonical transformation to connection variables in the bulk leads to a boundary term which we have to take into account on the isolated horizon. In a first step, one notes 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 that the canonical transformation induces a higher-dimensional SO(D + 1) Chern-Simons symplectic (2n) structure on S. The non-distortion condition δ E√ = 0 has to be kept also in the SO(D + 1) h framework and is independent of the chosen internal group since it can solely be expressed in terms 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 straight forward, since the construction of the connection Γ0 provided in [45] works independently of 0 K K aK the internal signature. Thus, constructing Γ such that it annihilates both n and s = sae additionally to mK = e K in the SO(D + 1) framework, the horizon boundary condition α ←α

(2n) α β ...αnβn E K1L1...KnLnIJ  1 1 R0 ...R0 = √ πaIJ s (13.4.1) α1β1K1L1 αnβnKnLn q a

0 K 0 K follows immediately from the fact that R αβKLn = R αβKLs = 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 leads us to the ADM phase space, from which we can perform further canonical transformations. This step is important since it tells us that using an isolated horizon as a boundary of our manifold, we will have a vanishing horizon symplectic structure when using ADM variables. We remark that this does not follow trivially for any boundary if one starts with the Einstein-Hilbert action and performs the Legendre transform, since one is picking up boundary terms in the Gauß-Codazzi equation 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 and performing a second variation yields the symplectic structure Z S 2 D−1 I Ω (δ1, δ2) = d x δ[1s˜I δ2]n . (13.4.2) β S

(2n) Furthermore, under the non-distortion condition δ E√ = 0, i.e. restricting to the part of phase h (2n) hE(2n)i space where E√ = is constant, a lengthy calculation detailed in [45] shows that h AS

E(2n) I IJKLM2N2...MnNn αβα2β2...αnβn 2 √ (δ[1s˜ )(δ2]nI ) = n  h 0
0
0 0 × δ[1ΓαIJ δ2]ΓβKL Rα2β2M2N2 ...RαnβnMnNn , (13.4.3) which results in the Chern-Simons type boundary symplectic structure nA Z ΩS (δ , δ ) = S IJKLM2N2...MnNn αβα2β2...αnβn CS 1 2 (2n) β E S 0
0
0 0 × δ[1ΓαIJ δ2]ΓβKL Rα2β2M2N2 ...RαnβnMnNn . (13.4.4)

121 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 horizon boundary of σ.

13.5 Inclusion of distortion

In this section, we are going to comment on the generalisation of the isolated horizon boundary condition 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 to arbitrary 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 extension of 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 to (2n) the derivation that√E / 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 S has been proposed by Beetle and Engle [210]. Essentially, they construct a new U(1) connection on S as ◦ 1 V := θ −  hβγD Ψ, (13.5.1) α 2 α αβ γ 1 where 2 θα is the U(1) connection used for spherically symmetric isolated horizons and Ψ is a curvature potential defined by the equation

∆Ψ = R − hRi , (13.5.2) √ where R is the intrinsic scalar curvature which is proportional to E(2)/ h. Calculating the ◦ curvature of V α, the terms proportional to R drop out and one gets

◦ hRi 2π i d V = −  = − Σir . (13.5.3) 4 AS

◦ 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

0 βγ AαIJ = ΓαIJ + 2mα[I mβ|J]h (Dγψ). (13.5.4)

Insertion into the boundary condition

αβ IJKL (2) [I J]   RαβKL(A) = 2hE in s˜ (13.5.5) yields ! 1 E(2) ∆ψ = √ − hE(2)i . (13.5.6) 4 h

122 As shown in [45], it follows that

(2) I IJKL αβ

2hE i(δ[1s˜ )(δ2]nI ) =   δ[1AαIJ δ2]AβKL . (13.5.7) The problem with generalising this trick to higher dimensions is that it leads to a non-linear partial 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 generalisation to higher dimensions seems straight forward, we cannot proceed due to the resulting non-linear partial 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 scalar curvature on S is to use two Chern-Simons connections on S, defined by

i i i i i i Aγ = Γ + γe ,Aσ = Γ + σe . (13.5.8) For the boundary conditions, it follows that 1 1 F i(A ) = Ψ Σi + (γ2 + c)Σi,F i(A ) = Ψ Σi + (σ2 + c)Σi, (13.5.9) γ 2 2 σ 2 2 where the Newman-Penrose coefficient Ψ2 is proportional to the scalar curvature. Subtracting these two equations, Perez and Pranzetti find 1 F i(A ) − F i(A ) = (γ2 − σ2)Σi, (13.5.10) γ σ 2 which can be used to derive the symplectic structure of two SU(2) Chern-Simons connections on 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 field strengths, 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

(a1) √ (aD−1) √ AαIJ = ΓαIJ + 2 a1s[I mα|J],...,AαIJ = ΓαIJ + 2 aD−1s[I mα|J]. (13.5.11) For their field strengths, it follows that

(ai) FαβIJ = RαβIJ − 2mα[I mβ|J]ai. (13.5.12) With the abbreviation

(a ) (a ) EIJ (A(ai)) := β1γ1...βnγn IJK1L1...KnLn F i ...F i , (13.5.13) (ai) β1γ1K1L1 βnγnKnLn the straight forward generalisation of the boundary condition of Perez and Pranzetti reads

D−1 X ! (−1)i+1EIJ (A(ai)) = 2n[I s˜J] (13.5.14) (ai) i=1 yields (D − 1)/2 equations only involving the coefficients ai. However, even if we can solve those equations, we still have to take into account further constraints resulting from different linear combinations of EIJ (A(ai)). As opposed to the four-dimensional case, i.e. two-dimensional (ai) 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.

123 13.6 Comments on quantisation

The quantisation of the three-dimensional Chern-Simons theories in the calculation of the black hole entropy for D = 3 are based on Witten’s quantisation [211] of Chern-Simons theory. The fact 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 are absent. Also, it is key to the derivation that the lapse function vanishes on the horizon and the Hamiltonian constraint does not have to be taken into account. This point continues to be true in higher dimensions. However, when taking into account higher dimensions, the Chern-Simons theory generally admits local degrees of freedom since the equations of motion

I2J2 InJn I1J1...InJn F ∧ ... ∧ F = 0 (13.6.1) do not constrain the connection to be flat [202]. Thus, a quantisation of the higher-dimensional Chern-Simons theories appearing on the horizons does not seem straight forward, at least not for the complete theory. Nevertheless, a quantisation of a subsector of the Chern-Simons theory which is similar to the three-dimensional case might still be possible, as we will argue below. Whether the entropy calculation can be performed solely by considering this subsector is presently unclear to the author since it, at best, captures only a subsector of the full Hilbert space of quantised SO(D + 1) Chern-Simons theory. The following analogy could therefore be on a purely formal level, however in case satisfying arguments could be given for its validity, the entropy calculation could 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 modified by “particle degrees of freedom” which are induced by the spin networks puncturing the horizon as EI1J1 (x) := I1J1...InJn F (x) ∧ ... ∧ F (x) ∝ s πaI1J1 (x), (13.6.2) I2J2 InJn ab where the operator on the right hand side symbolises to the flux operator which acts non-trivially only at points where a spin network punctures the horizon. Using the Dirac brackets obtained from solving the second class constraints of the higher-dimensional Chern-Simons theory1, we can explicitly calculate the algebra of these “particle excitations” as

 IJ KL (D−1) IJ,KL, MN E (x),E (y) ∝ δ (x − y)f MN E (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 mainly connected 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 spin networks and the horizon topology. As opposed to the U(1) or SU(2) based constructions in four dimensions, the restrictions imposed by the simplicity constraint will also have to be taken into account properly. While the simplicity constraint is solved on the horizon at the classical I I I 0 level by using the variables n , s , and eα to construct ΓαIJ , 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 the set 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 constraints is correct at least in six spacetime dimensions (IJKLMN 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.

124 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 question will be if there is a restriction resulting from implementing the vertex simplicity constraints. We leave this question for further research. As mentioned before, this analogy to the four-dimensional entropy calculation is only formal at the current level. A well-motivated calculation from first principles would amount to finding a complete quantisation of the higher-dimensional Chern-Simons theory and then to perform the entropy calculation taking into account all horizon degrees of freedom. From a different point of view, one could also argue that we are only interested in an effective four-dimensional entropy where the additional spatial dimensions are compactified. Then, one could also perform a Kaluza-Klein reduction before quantisation which is legitimate since the whole LQG entropy calculation is only semiclassical due to the classical notion of the isolated horizon. We leave these 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 four spacetime dimensions generalises rather directly, the quantisation of the resulting system is less obvious, since, to the best of the author’s knowlege, there are no known generalisations of the quantisation of 2 + 1-dimensional Chern-Simons theory to higher dimensions. On the other hand, 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 by puncturing 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 of the theory on the black hole horizon is the treatment of the simplicity constraint. A preliminary analysis 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 simplicity constraints would just restrict the group representations on the particle defects in the same way as it restricts the edge representations, it might be that a proper quantisation of the horizon degrees of freedom gives us a hint on what the correct implementation of the simplicity constraint on a vertex is. The reason for this comes from the seemingly very effective treatment of a black hole as a single intertwiner, see [212] and more recently also [213]. One would now expect that the correct treatment of the higher-dimensional vertex simplicity constraints should somehow be reflected in this treatment, i.e. that the same restrictions on the intertwiner should also be reflected by the quantisation of the horizon degrees of freedom. Additionally, it will be interesting to check to what extend the connection on the horizon can be generalised, e.g. as in [200], where a new free parameter can be associated to the horizon connection which can rescale the entropy to A/4 without fixing the Barbero-Immirzi parameter. The consequences of introducing a two parameter family of connections in the bulk in four dimensions as proposed in [36] should also be investigated and a generalisation to general non-symmetric black holes has to be developed. As a last remark, we want to emphasise that the study of different horizon topologies promises to yield a deeper insight into the subject at hand. The reason is that there are results that the black hole entropy should depend on the horizon topology, more precisely on the Euler characteristic [214]. However, we caution the reader that before jumping to comparisons, the black hole entropy calculations based on the presented framework will have to be worked out in

125 detail. One of the problems arising is that the classical treatment assumes the non-distortion condition, which cannot be satisfied for general topologies. Examples different from spherical topology where the classical derivation should however be applicable are black holes of constant negative curvature, which we plan to investigate in the future.

126 Part VI

Conclusion

127 Chapter 14

Concluding remarks and further research

In this chapter, we shortly comment on what has been achieved in this thesis and what directions further 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 higher dimensional supergravity theories has been constructed which allows for an application of the quantisation techniques developed in the context of loop quantum gravity. The existence of this type of connection formulation however came as a surprise, since the results from four dimensions indicated that in order to remove the superfluous degrees of freedom from the connection, one would need additional constraints (the simplicity constraints), which produced secondary second class constraints when stabilised in the Dirac procedure. The key idea on the gravitational side was 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 extensions of the remaining constraints, in the case of general relativity it terminates after the second term and leaves us with a Hamiltonian constraint to which the quantisation techniques familiar from four 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 clear that general relativity would allow for such a description at the Hamiltonian level, however also this fact came as a surprise. When including fermions in the treatment, one has to generalise the 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 is required for Majorana fermions due to their reality conditions. In fact, since the existence of Majorana fermions in Lorentzian signature excludes their existence in Euclidean signature in the same number of dimensions, it was unclear if the Majorana fermions necessary for completing the supergravity multiplets could be incorporated into the Spin(D + 1) framework. The key input for them to work was a modified reality condition which matches with the Lorentzian reality condition when choosing the time gauge N I = (1, 0,..., 0). Due to Spin(D + 1) gauge covariance of this reality condition, the full Spin(D + 1)-invariant theory therefore agrees with the corresponding Lorentzian supergravity. The inclusion of the three-index photon of eleven

128 dimensional supergravity also posed an additional obstacle due to the twisted holonomy-flux algebra, which could however be circumvented using a reduced phase space quantisation with respect to the three-index photon Gauß law. The interesting part of this quantisation is that it seems to require a background independent representation which is discontinuous in both the connection and its momentum, thus contrasting itself to the Ashtekar-Isham-Lewandowski representation which is only discontinuous in the connection. In the fourth part of this thesis, we have provided a partially reduced phase space quantisation of general relativity conformally coupled to a scalar field. The focus of this quantisation has been on non-dynamical issues since the time function used, the CMC gauge, results in a very complicated generator of time evolution, but allows for good control over the existence of CMC slices. It has been explicitly shown that there exist interesting black hole solutions which can be incorporated in the reduced phase space quantisation and thus their entropy calculations can be based on the counting of physical states. Furthermore, the geometric operators which are physical with respect to the classically reduced Hamiltonian constraint do not agree with the kinematical geometric operators or their counterparts in the deparametrised models. Thus, an explicit example has been provided that the spectrum of the geometric operators depends on the chosen spatial hypersurface determined by the time function. While similar effects have already been 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 in part five. There, an isolated horizon boundary condition was derived which closely resembles the result 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 condition together with this symplectic structure results in a quantised higher-dimensional Chern-Simons theory describing the isolated horizon. Since higher dimensional Chern-Simons theories contain local degrees of freedom, the quantisation procedures familiar from four dimensions cannot be used 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 of constructing the Hamiltonian constraint and supersymmetry constraint have been given, the resulting operators are very complicated and the physical effects of the choices involved in the regularisations are poorly understood in general. In order to make progress on this issue, a better understanding of dynamics resulting from loop quantum gravity type quantisations is needed, 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 required attention up till now. Thus, the main point of criticism which has to be raised at this moment is that more control over the dynamics of the theory is necessary, especially to judge its physical significance. 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 significantly more complicated than the other terms. On the other hand, the situation in 3 + 1 dimensions puts us in the position to compare the formulation developed in this thesis with the usual construction based on the Ashtekar-Barbero variables, thus having two quantisations of general relativity based on classically different vari- ables. This is especially interesting for the simplicity constraint, because we can discriminate between different impositions of the constraint by requiring dynamical equivalence of the theories based on Ashtekar-Barbero variables and the ones introduced in this thesis.

129 Another point which deserves attention is the question of the signature of the internal gauge group. In this thesis, we have derived a formulation of Lorentzian (D + 1)-dimensional general relativity with SO(D + 1) as an internal gauge group. Although this has been achieved by a canonical transformation, one might argue that it would be more natural to use SO(1,D) as the internal gauge group. While this is also appealing from the point of view of the dynamics of the theory since the Hamiltonian constraint will be simpler, the Hilbert space techniques developed for loop quantum gravity are only available for compact gauge groups. However, if one could extend these techniques also to non-compact gauge groups, SO(1,D) might become the gauge group of our choice. As for the 3-index photon of eleven dimensional supergravity studied in this thesis, it is mandatory to gain a better understanding for the Narnhofer-Thirring type state used in the construction of the quantum theory. Its physics is rather unclear and, due to its discontinuity in both the 3-index photon and its momentum, it might lead to more quantisation ambiguities than 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 to many 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 issue is mandatory before continuing with the initial aim of this thesis, which was to establish a contact with superstring theory.

14.3 Further research

In the following, several approaches for further research which suggest themselves at the current stage 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 direct applications 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 spacetime limit in general.

Black holes Investigating the quantisation of higher dimensional and supersymmetric black holes using LQG 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 space description√ in higher dimensions in part five of this thesis under the non-distortion assumption δ(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 the higher 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 supersymmetric Chern-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 Hamiltonian constraint [182]. Furthermore, the Rarita-Schwinger field does not produce a boundary term in the 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 the boundary term and the induced Chern-Simons degrees of freedom. However, this is problematic since the non-distortion condition cannot hold for an arbitrary topology.

130 Cosmology The study of higher dimensional and supersymmetric cosmological models is interesting since it might 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 + 1 dimensional loop quantum cosmology. In order to extend the framework of loop quantum cosmology to supergravity, it seems reasonable to 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 new higher dimensional formulation will have to be investigated for its dynamical consequences. In a next step, one has to think about including the remaining matter fields from the corresponding supergravity 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 the non-supersymmetric case since the supersymmetry constraint is situated above the Hamiltonian constraint in the “hierarchy” of constraints, in the sense that a solution to the supersymmetry constraint operator is automatically a solution to the Hamiltonian constraint operator (up to anomalies). 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 small scales 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 geometric operators will enter and the results should become sensitive to extra dimensions.

Gauge / gravity correspondence Further 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 basic idea is that since a non-perturbative sector of quantum supergravity can be constructed by using 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 the canonical transformation used to derive the new variables yields a boundary term which induces a (Chern-Simons) gauge theory on the boundary and relates it to the quantum gravity theory in the bulk. To establish a precise dictionary is certainly a long shot, see however [225] for results from 2 + 1 dimensional quantum gravity. Along this road, several open technical problems, like non-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 potential benefits of such a line of research, including the application of the gauge / gravity correspondence to real world physics like quark-gluon plasmas or condensed matter physics, greatly outweigh its 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- bra In [226], it was shown that one can regularise the spatial diffeomorphsim constraint operator in such a way that the Dirac algebra for spatial diffeomorphisms is reproduced at the quantum level. In order to achieve this result, it was necessary to consider in the regularisation for the field strength, next to small closed loop holonomies, also fluxes and an explicit dependence on the 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 authors of [226]. It should be the goal to incorporate these findings into the regularisations of the true Hamiltonian operators for the deparametrised models, which will hopefully lead to a better un- derstanding of their action on spin-network functions.

131 A conceptually related result has been obtained in the framework of the improved dynamics of loop quantum cosmology (¯µ-scheme [222]), where the regularised operator corresponding to the curvature has to depend on both, the connection and the geometry. Also here, it will be interesting 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 relativity Supergravity is essentially general relativity coupled to specific matter in such a way that a new fermionic symmetry generator is present in the theory which maps fermions into bosons and vice versa. The simplification in supergravity, as opposed to matter coupled general relativity, stems from the presence of this new symmetry, however not form a reduction to a finite amount of degrees of freedom as in many cosmological models. For flat space, the new symmetry can be in- tegrated into the Poincar´egroup in a non-trivial way1, leading to the super-Poincar´egroup. The counterpart in a Hamiltonian formulation of supergravity is a non-trivial extension of the Dirac algebra, where a new constraint, the supersymmetry constraint S, is present, which squares to all the other constraints as {S, S} = S + H + Ha [117]. At the quantum level, one would expect that 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 corresponding quantum operators to reproduce the classical relation. As opposed to matter coupled general relativity without supersymmetry, the above relation should yield new insights into how the Hamiltonian constraint should be regularised. In order to attack this problem, it would be best to start with supersymmetric quantum cosmology, where the calculations, as in loop quantum cosmology, 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, the goal of this exercise will be to extract information on how exactly the Hamiltonian constraint including matter fields should be regularised. In the easiest case of minimal D + 1 = 4, N = 1 supergravity, 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 are incorporated in the supersymmetry multiplet and appearing in the regularisation.

Coherent states Coherent 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 generator of time translations respective to a chosen clock is a complicated operator, e.g. a square root as in [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 case is available [228, 229, 230], however, in the case of non-Abelian gauge groups the literature is rather scarce [231]. Thus, it is very important to make progress on these questions, building on the research started in [29, 30, 31, 32, 33, 34, 231].

Choice of time function One of the crucial ingredients of a reduced phase space quantisation is the choice of a time function. 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 Poincar´egroup are trivial.), which is circumvented by the fact that the supersymmetry generators have odd Graßmann parity.

132 experimental questions that can be asked2 in the reduced phase space quantisation are restricted to 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 geometric operators of LQG becoming physical observables [22], might not carry over to a quantisation based on a different time function. A specific example has been given in the fourth part of this thesis, where, due to the choice of a time function which is a sum of momenta, the usual LQG operators are not physical observables, but only a product of them with a certain power of a scalar field. This observation leads one to think about which time functions correspond to certain physical situations which one wants to describe, e.g. particle scatterings in the quantum field theory on curved spacetime limit, which is one of the important limits that LQG has to reproduce. Since the time function dictates the form of the true Hamiltonian, choosing a time function is not a mere 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 natural time slices of a given deparametrised model to other time slices. Especially the result of [43], that the physical geometric operators can change due to a different choice of time function, exemplifies that it is very unlikely that such an extrapolation would give a result which would agree with directly using the different time function.

133 Appendix A

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

In this appendix, we cite several important results concerning simple irreducible representations of SO(D + 1). This appendix is taken from the original work [39].

D There is a natural action of SO(D + 1) on F ∈ H := L2(S , 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 this 1 ∂ ∂ representation are of the form τIJ = 2 ( ∂N I NJ − ∂N J NI ) and are known to satisfy the quadratic simplicity constraint τ[IJ τKL] = 0 [62]. These representations are reducible. The representation space can be decomposed into spaces of harmonic homogeneous polynomials HD+1,l of degree D P∞ D+1,l l in D + 1 variables, L2(S ) = l=0 H . 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 quadratic simplicity constraint [62] and therefore are mostly called simple representations in the spin foam community. Note that these representations have been studied quite extensively in the mathematical literature, where they are called most degenerate representations [232, 233, 234], (completely) symmetric representations [233, 235, 236, 237] or representations of class one (with respect to a SO(D) subgroup) [78]. The latter is due to the fact that these representations of SO(D+1) are the only ones which have in their representations space a non-zero vector invariant under a SO(D) subgroup, which is exactly the definition of being of class one w.r.t. a subgroup given in [78]. An orthonormal basis in HD+1,l is given by generalisations of spherical harmonics K~ to higher dimensions [78] which we denote Ξl (N), Z K~ M~ l K~ Ξ (N) Ξ (N) dN = δ 0 δ , (A.1) l l0 l M~ SD where K~ denotes an integer sequence K~ := (K1,...,KD−2, ±KD−1) satisfying l ≥ K1 ≥ ... ≥ D+1,l KD−1 ≥ 0 and analogously defined M~ . Fl(N) ∈ H can be decomposed as Fl(N) = P K~ ~ K~ aK~ Ξl (N) where the sum runs over those integer sequences K allowed by the above in- D P∞ D+1,l equality. Since L2(S ) = l=0 H , any square integrable function F (N) on the sphere can be expanded in a mean-convergent series of the form [78] ∞ ~ X X l Kl F (N) = a ~ Ξ (N). (A.2) Kl l l=0 ~ Kl

134 Consider a recoupling basis [238] for the ONB of the tensor product of N irreps: Choose a labelling of the irreps Λ~ 1, ..., Λ~ N . Then, consider the ONB E ~ ~ ~ ~ ~ ~ ~ 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 basis in the intertwiner space is given by E ~ ~ ~ ~ ~ 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 in the intertwiner space. A basis in the intertwiner space of N simple irreps is given by E ~ ~ ~ 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 irreps appear in general [237, 236],

λ λ −k X2 X2 (λ1, 0, ..., 0) ⊗ (λ2, 0, ..., 0) = (λ1 + λ2 − 2k − l, l, 0, ..., 0) (λ2 ≤ λ1). (A.6) k=0 l=0

135 Danksagung

An dieser Stelle m¨ochte ich mich bei allen bedanken, die mich w¨ahrendmeiner Promotionsphase unterst¨utzthaben und mir diesen Lebensweg erm¨oglicht haben.

Zu allererst gilt mein Dank meinen Eltern, die mich von Beginn meines Studiums an unterst¨utzt haben, ideell, finanziell, und auch tatkr¨aftigbei den vielen Umz¨ugen. Der Wert ihrer Un- terst¨utzungf¨urmich w¨urdesich nicht angemessen in Worte fassen lassen.

Desweiteren gilt mein Dank meinem Studienkollegen, Mitdoktoranden und nicht zuletzt Freund Andreas Thurn. Der Wert einer guten Kollaboration zusammen mit einem guten Freund hat sich in den letzten Jahren f¨urmich als unsch¨atzbargroß erwiesen. Das gleiche gilt f¨urdie Zusam- menarbeit mit Alexander Stottmeister bei den im vierten Teil dieser Doktorarbeit behandelten Ergebnissen.

Meinem Betreuer Thomas Thiemann m¨ochte ich f¨urdie vier sch¨onenJahre w¨ahrendmeiner Diplomarbeit und Promotionsphase danken, w¨ahrendderer ich viel von ihm gelernt habe und das 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 l¨angeresProjekt f¨urAndreas Thurn und mich, seine intensive Betreuung sowie die gute Zusammenarbeit mit ihm haben maßgebend zum Erfolg dieser Promotion beigetragen.

F¨urdie Unterst¨utzungbei meinen Bewerbungen f¨ureine Postdoktorandenstelle, in Form von m¨uhevoll geschriebenen Gutachten und wichtigen Tips, m¨ochte ich mich bei Kristina Giesel, Jerzy Lewandowski und Hanno Sahlmann bedanken. An dieser Stelle m¨ochte ich auch noch einmal meinen Betreuer Thomas Thiemann hervorheben, dessen viele Hilfe und Tips bei den Bewerbungen unerl¨asslich waren.

An dieser Stelle darf auch der “beschleunigte” Physikstudiengang “Physics Advanced” im Rah- men des Elitenetzwerks Bayern nicht unerw¨ahnt bleiben, welcher der eigentliche Grund f¨ur meine Entscheidung war, in Erlangen Physik zu studieren. Der fr¨uhervon Klaus Rith und heute von Klaus Mecke geleitete Studiengang hat mir ein z¨ugigesund forschungsorientiertes Studium erm¨oglicht und ist in dieser Doktorarbeit gem¨undet.Mein Dank gilt allen f¨urdiesen Studiengang engagierten Professoren und Studenten, welchen ich als wichtige Bereicherung f¨ur die deutsche Hochschullandschaft sehe. Auch bei Frieder Lenz, der w¨ahrendmeiner Studienzeit als Mentor fungierte und mich an Thomas Thiemann empfohlen hat, m¨ochte ich mich hier noch einmal bedanken.

Ich bedanke mich weiterhin f¨urfinanzielle und ideelle Unterst¨utzungw¨ahrendmeines Studiums und meiner Doktorarbeit bei der Friedrich-Naumann-Stiftung, dem Max-Weber-Programm, dem Leonardo-Kolleg der Universit¨atErlangen-N¨urnberg, e-Fellows, dem Elitenetzwerk Bayern, und der Studienstiftung des deutschen Volkes.

Abschließend m¨ochte ich mich bei Emanuele Alesci, Enrique Fernandez Borja, Yuriy Davy- gora, Jonathan Engle, Christian Fitzner, I˜nakiGaray, Kristina Giesel, Muxin Han, Suzanne Lan´ery, Karl-Hermann Neeb, Hanno Sahlmann, Alexander Stottmeister, Eckhard Strobel, Jo- hannes Tambornino, Derek Wise, Antonia Zipfel, und vielen anderen f¨urviele interessante und hilfreiche Diskussionen ¨uber Physik bedanken.

136 Bibliography

[1] P. van Nieuwenhuizen, “An Introduction to Simple Supergravity and the Kaluza-Klein Program,” in Relativity, Groups & Topology, II: Proceedings of the Les Houches Summer School, Session XL (B. S. DeWitt and R. Stora, eds.), Elsevier, 1984.

[2] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, Vol. 1: Introduction. Cambridge University Press, Cambridge, 1988.

[3] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, Vol. 2: Loop Amplitudes, Anomalies and Phenomenology. Cambridge University Press, Cambridge, 1988.

[4] J. Polchinski, String Theory, Vol. 1: An Introduction to the bosonic string. Cambridge University Press, Cambridge, 1998.

[5] J. Polchinski, String Theory, Vol. 2: Superstring theory and beyond. Cambridge University Press, Cambridge, 1998.

[6] W. Nahm, “Supersymmetries and their representations,” Nuclear Physics B 135 (1978) 149–166.

[7] A. Ashtekar, “New Variables for Classical and Quantum Gravity,” Physical Review Letters 57 (1986) 2244–2247.

[8] B. DeWitt, “Quantum Theory of Gravity. I. The Canonical Theory,” Physical Review 160 (1967) 1113–1148.

[9] B. DeWitt, “Quantum Theory of Gravity. II. The Manifestly Covariant Theory,” Physical Review 162 (1967) 1195–1239.

[10] J. A. Wheeler, “Superspace and the Nature of Quantum Geometrodynamics,” in Quantum Cosmology, Advanced Series in Astrophysics and Cosmology (L. Z. Fang and R. Ruffini, eds.), vol. 3, (Singapore), World Scientific, 1987.

[11] J. Barbero, “Real Ashtekar variables for Lorentzian signature space-times,” Physical Review D 51 (1995) 5507–5510, arXiv:gr-qc/9410014.

[12] T. Thiemann, “Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity,” Physics Letters B 380 (1996) 257–264, arXiv:gr-qc/9606088.

[13] T. Thiemann, “Quantum spin dynamics (QSD),” Classical and Quantum Gravity 15 (1998) 839–873, arXiv:gr-qc/9606089.

137 [14] A. Ashtekar and C. J. Isham, “Representations of the holonomy algebras of gravity and non-Abelian gauge theories,” Classical and Quantum Gravity 9 (1992) 1433–1468, arXiv:hep-th/9202053.

[15] A. Ashtekar and J. Lewandowski, “Representation Theory of Analytic Holonomy C* Algebras,” in Knots and Quantum Gravity (J. Baez, ed.), (Oxford), Oxford University Press1994. arXiv:gr-qc/9311010.

[16] A. Ashtekar and J. Lewandowski, “Differential geometry on the space of connections via graphs and projective limits,” Journal of Geometry and Physics 17 (1995) 191–230, arXiv:hep-th/9412073.

[17] A. Ashtekar and J. Lewandowski, “Projective techniques and functional integration for gauge theories,” Journal of Mathematical Physics 36 (1995) 2170–2191, arXiv:gr-qc/9411046.

[18] D. Marolf and J. M. Mour˜ao,“On the support of the Ashtekar-Lewandowski measure,” Communications in Mathematical Physics 170 (1995) 583–605, arXiv:hep-th/9403112.

[19] A. Ashtekar, J. Lewandowski, D. Marolf, J. M. Mour˜ao,and T. Thiemann, “Quantization of diffeomorphism invariant theories of connections with local degrees of freedom,” Journal of Mathematical Physics 36 (1995) 6456–6493, arXiv:gr-qc/9504018.

[20] T. Thiemann, “Quantum spin dynamics (QSD) V: Quantum Gravity as the Natural Regulator of Matter Quantum Field Theories,” Classical and Quantum Gravity 15 (1998) 1281–1314, arXiv:gr-qc/9705019.

[21] T. Thiemann, “Kinematical Hilbert spaces for Fermionic and Higgs quantum field theories,” Classical and Quantum Gravity 15 (1998) 1487–1512, arXiv:gr-qc/9705021.

[22] K. Giesel and T. Thiemann, “Algebraic quantum gravity (AQG): IV. Reduced phase space quantization of loop quantum gravity,” Classical and Quantum Gravity 27 (2010) 175009, arXiv:0711.0119 [gr-qc].

[23] M. Domagala, K. Giesel, W. Kaminski, and J. Lewandowski, “Gravity quantized: Loop quantum gravity with a scalar field,” Physical Review D 82 (2010) 104038.

[24] V. Husain and T. Pawlowski, “Time and a physical Hamiltonian for quantum gravity,” Physical Review Letters 108 (2012) 141301, arXiv:1108.1145 [gr-qc].

[25] K. Giesel and T. Thiemann, “Scalar Material Reference Systems and Loop Quantum Gravity,” arXiv:1206.3807 [gr-qc].

[26] K. Giesel and T. Thiemann, “Algebraic quantum gravity (AQG): I. Conceptual setup,” Classical and Quantum Gravity 24 (2007) 2465–2497, arXiv:gr-qc/0607099.

[27] K. Giesel and T. Thiemann, “Algebraic quantum gravity (AQG): II. Semiclassical analysis,” Classical and Quantum Gravity 24 (2007) 2499–2564, arXiv:gr-qc/0607100.

[28] K. Giesel and T. Thiemann, “Algebraic quantum gravity (AQG): III. Semiclassical perturbation theory,” Classical and Quantum Gravity 24 (2007) 2565–2588, arXiv:gr-qc/0607101.

138 [29] T. Thiemann, “Quantum spin dynamics (QSD): 7. Symplectic structures and continuum lattice formulations of gauge field theories,” Classical and Quantum Gravity 18 (2001) 3293–3338, arXiv:hep-th/0005232. [30] T. Thiemann, “Gauge field theory coherent states (GCS): I. General properties,” Classical and Quantum Gravity 18 (2001) 2025–2064, arXiv:hep-th/0005233. [31] T. Thiemann and O. Winkler, “Gauge field theory coherent states (GCS): II. Peakedness properties,” Classical and Quantum Gravity 18 (2001) 2561–2636, arXiv:hep-th/0005237. [32] T. Thiemann and O. Winkler, “Gauge field theory coherent states (GCS): III. Ehrenfest theorems,” Classical and Quantum Gravity 18 (2001) 4629–4681, arXiv:hep-th/0005234. [33] H. Sahlmann, T. Thiemann, and O. Winkler, “Coherent states for canonical quantum general relativity and the infinite tensor product extension,” Nuclear Physics B 606 (2001) 401–440, arXiv:gr-qc/0102038. [34] T. Thiemann, “Complexifier coherent states for quantum general relativity,” Classical and Quantum Gravity 23 (2006) 2063–2117, arXiv:gr-qc/0206037. [35] N. Bodendorfer, T. Thiemann, and A. Thurn, “New Variables for Classical and Quantum Gravity in all Dimensions I. Hamiltonian Analysis,” arXiv:1105.3703 [gr-qc]. [36] N. Bodendorfer, T. Thiemann, and A. Thurn, “New Variables for Classical and Quantum Gravity in all Dimensions II. Lagrangian Analysis,” arXiv:1105.3704 [gr-qc]. [37] N. Bodendorfer, T. Thiemann, and A. Thurn, “New Variables for Classical and Quantum Gravity in all Dimensions III. Quantum Theory,” arXiv:1105.3705 [gr-qc]. [38] N. Bodendorfer, T. Thiemann, and A. Thurn, “New Variables for Classical and Quantum Gravity in all Dimensions IV. Matter Coupling,” arXiv:1105.3706 [gr-qc]. [39] N. Bodendorfer, T. Thiemann, and A. Thurn, “On the Implementation of the Canonical Quantum Simplicity Constraint,” arXiv:1105.3708 [gr-qc]. [40] N. Bodendorfer, T. Thiemann, and A. Thurn, “Towards Loop Quantum Supergravity (LQSG) I. Rarita-Schwinger Sector,” arXiv:1105.3709 [gr-qc]. [41] N. Bodendorfer, T. Thiemann, and A. Thurn, “Towards Loop Quantum Supergravity (LQSG) II. p-Form Sector,” arXiv:1105.3710 [gr-qc]. [42] N. Bodendorfer, T. Thiemann, and A. Thurn, “Towards Loop Quantum Supergravity (LQSG),” Physics Letters B 711 (2012) 205–211, arXiv:1106.1103 [gr-qc]. [43] N. Bodendorfer, A. Stottmeister, and A. Thurn, “Loop quantum gravity without the Hamiltonian contraint,” arXiv:1203.6525 [gr-qc]. [44] N. Bodendorfer, A. Stottmeister, and A. Thurn, “On a partially reduced phase space quantisation of general relativity conformally coupled to a scalar field,” arXiv:1203.6526 [gr-qc]. [45] N. Bodendorfer, T. Thiemann, and A. Thurn, “New Variables for Classical and Quantum Gravity in all Dimensions V: Isolated Horizon Boundary Degrees of Freedom,” To be published. (2012).

139 [46] R. Arnowitt, S. Deser, and C. W. Misner, “The dynamics of general relativity,” in Gravitation: An introduction to current research (L. Witten, ed.), (New York), pp. 227–265, Wiley, 1962. arXiv:gr-qc/0405109.

[47] P. Peldan, “Actions for gravity, with generalizations: A Review,” Classical and Quantum Gravity 11 (1994) 1087–1132, arXiv:gr-qc/9305011.

[48] E. Cremmer, B. Julia, and J. Scherk, “Supergravity theory in 11 dimensions,” Physics Letters B 76 (1978) 409–412.

[49] N. Narnhofer and W. Thirring, “Covariant QED Without Indefinite Metric,” Reviews in Mathematical Physics 4 (1992) 197–211.

[50] H. Gomes, S. Gryb, and T. Koslowski, “Einstein gravity as a 3D conformally invariant theory,” Classical and Quantum Gravity 28 (2011) 045005, arXiv:1010.2481 [gr-qc].

[51] S. Holst, “Barbero’s Hamiltonian derived from a generalized Hilbert-Palatini action,” Physical Review D 53 (1996) 5966–5969, arXiv:gr-qc/9511026.

[52] G. Date, R. Kaul, and S. Sengupta, “Topological interpretation of Barbero-Immirzi parameter,” Physical Review D 79 (2009) 044008, arXiv:0811.4496 [gr-qc].

[53] P. A. M. Dirac, Lectures on Quantum Mechanics. Belfer Graduate School of Science, Yeshiva University Press, New York, 1964.

[54] N. Barros e S´a,“Hamiltonian analysis of general relativity with the Immirzi parameter,” International Journal of Modern Physics D 10 (2001) 261–272, arXiv:gr-qc/0006013.

[55] S. Alexandrov, “SO (4, C )-covariant Ashtekar-Barbero gravity and the Immirzi parameter,” Classical and Quantum Gravity 17 (2000) 4255–4268, arXiv:gr-qc/0005085.

[56] M. Han, Y. Ma, Y. Ding, and L. Qin, “Hamiltonian analysis of n-dimensional Palatini gravity with matter,” Modern Physics Letters A20 (2005) 725–732, arXiv:gr-qc/0503024.

[57] A. N. Bernal and M. Sanchez, “On Smooth Cauchy Hypersurfaces and Geroch’s Splitting Theorem,” Communications in Mathematical Physics 243 (2003) 461–470, arXiv:gr-qc/0306108.

[58] J. F. Plebanski, “On the separation of Einsteinian substructures,” Journal of Mathematical Physics 18 (1977) 2511–2520.

[59] J. W. Barrett and L. Crane, “Relativistic spin networks and quantum gravity,” Journal of Mathematical Physics 39 (1998) 3296–3302, arXiv:gr-qc/9709028.

[60] J. Engle, R. Pereira, and C. Rovelli, “Flipped spinfoam vertex and loop gravity,” Nuclear Physics B 798 (2008) 251–290, arXiv:0708.1236 [gr-qc].

[61] L. Freidel and K. Krasnov, “A new spin foam model for 4D gravity,” Classical and Quantum Gravity 25 (2008) 125018, arXiv:0708.1595 [gr-qc].

[62] L. Freidel, K. Krasnov, and R. Puzio, “BF description of higher-dimensional gravity theories,” Advances in Theoretical and Mathematical Physics 3 (1999) 1289–1324, arXiv:hep-th/9901069.

140 [63] P. Mitra and R. Rajaraman, “Gauge-invariant reformulation of an anomalous gauge theory,” Physics Letters B 225 (1989) 267–271.

[64] R. Anishetty and A. S. Vytheeswaran, “Gauge invariance in second-class constrained systems,” Journal of Physics A: Mathematical and General 26 (1993) 5613–5619.

[65] A. S. Vytheeswaran, “Gauge unfixing in second-class constrained systems,” Annals of Physics 236 (1994) 297–324.

[66] T. Thiemann, Modern Canonical Quantum General Relativity. Cambridge University Press, Cambridge, 2007.

[67] J. Engle, E. R. Livine, R. Pereira, and C. Rovelli, “LQG vertex with finite Immirzi parameter,” Nuclear Physics B 799 (2008) 136–149, arXiv:0711.0146 [gr-qc].

[68] J. Engle, R. Pereira, and C. Rovelli, “The Loop-Quantum-Gravity Vertex Amplitude,” Physical Review Letters 99 (2007) 161301, arXiv:0705.2388 [gr-qc].

[69] E. Livine and S. Speziale, “New spinfoam vertex for quantum gravity,” Physical Review D 76 (2007) 084028, arXiv:0705.0674 [gr-qc].

[70] W. Kaminski, M. Kisielowski, and J. Lewandowski, “Spin-foams for all loop quantum gravity,” Classical and Quantum Gravity 27 (2010) 095006, arXiv:0909.0939 [gr-qc].

[71] S. Gielen and D. Oriti, “Classical general relativity as BF-Plebanski theory with linear constraints,” Classical and Quantum Gravity 27 (2010) 185017, arXiv:1004.5371 [gr-qc].

[72] S. Alexandrov and E. Livine, “SU(2) loop quantum gravity seen from covariant theory,” Physical Review D 67 (2003) 044009, arXiv:gr-qc/0209105.

[73] M. Geiller, M. Lachieze-Rey, K. Noui, and F. Sardelli, “A Lorentz-Covariant Connection for Canonical Gravity,” SIGMA 7 (2011) 083, arXiv:1103.4057 [gr-qc].

[74] T. Thiemann, “Quantum spin dynamics (QSD): II. The kernel of the Wheeler-DeWitt constraint operator,” Classical and Quantum Gravity 15 (1998) 875–905, arXiv:gr-qc/9606090.

[75] T. Thiemann, “Quantum spin dynamics: VIII. The master constraint,” Classical and Quantum Gravity 23 (2006) 2249–2265, arXiv:gr-qc/0510011.

[76] Y. Yamasaki, Measures on Infinite Dimensional Spaces. World Scientific Publishing Company, 1985.

[77] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis. Academic Press, 1981.

[78] N. J. Vilenkin, Special Functions and the Theory of Group Representations. American Mathematical Society, 1968.

[79] A. Perez, “Spin foam models for quantum gravity,” Classical and Quantum Gravity 20 (2003) R43–R104, arXiv:gr-qc/0301113.

[80] S. Alexandrov and P. Roche, “Critical Overview of Loops and Foams,” Physics Reports 506 (2011) 41–86, arXiv:1009.4475 [gr-qc].

141 [81] A. Baratin and D. Oriti, “Group Field Theory with Noncommutative Metric Variables,” Physical Review Letters 105 (2010) 221302, arXiv:1002.4723 [hep-th].

[82] W. Wieland, “Complex Ashtekar variables and reality conditions for Holst’s action,” Annales Henri Poincar´e (2011) DOI: 10.1007/s00023–011–0134–z, arXiv:1012.1738 [gr-qc].

[83] F. Girelli and E. R. Livine, “Reconstructing quantum geometry from quantum information: spin networks as harmonic oscillators,” Classical and Quantum Gravity 22 (2005) 3295–3313, arXiv:gr-qc/0501075.

[84] L. Freidel and E. R. Livine, “The fine structure of SU(2) intertwiners from U(N) representations,” Journal of Mathematical Physics 51 (2010) 082502, arXiv:0911.3553 [gr-qc].

[85] L. Freidel and E. R. Livine, “U(N) Coherent States for Loop Quantum Gravity,” Journal of Mathematical Physics 52 (2011) 052502, arXiv:1005.2090 [gr-qc].

[86] M. Dupuis and E. R. Livine, “Revisiting the simplicity constraints and coherent intertwiners,” Classical and Quantum Gravity 28 (2011) 085001, arXiv:1006.5666 [gr-qc].

[87] M. Dupuis and E. R. Livine, “Holomorphic Simplicity Constraints for 4d Spinfoam Models,” Classical and Quantum Gravity 28 (2011) 215022, arXiv:1104.3683 [gr-qc].

[88] P. Mitra and R. Rajaraman, “Gauge-invariant reformulation of theories with second-class constraints,” Annals of Physics 203 (1990) 157–172.

[89] R. Jackiw and R. Rajaraman, “Vector-Meson Mass Generation by Chiral Anomalies,” Physical Review Letters 54 (1985) 1219–1221.

[90] J. Lott and R. Rajaraman, “Degrees of freedom and the quantization of anomalous gauge theories,” Physics Letters B 165 (1985) 321–326.

[91] L. Faddeev, “Operator anomaly for the gauss law,” Physics Letters B 145 (1984) 81–84.

[92] R. Rajaraman, “Hamiltonian formulation of the anomalous chiral Schwinger model,” Physics Letters B 154 (1985) 305–309.

[93] C. Flori and T. Thiemann, “Semiclassical analysis of the Loop Quantum Gravity volume operator: I. Flux Coherent States,” arXiv:0812.1537 [gr-qc].

[94] C. Rovelli, “Zakopane lectures on loop gravity,” arXiv:1102.3660 [gr-qc].

[95] A. Higuchi, “Quantum linearization instabilities of de Sitter spacetime. II,” Classical and Quantum Gravity 8 (1991) 1983–2004.

[96] D. Giulini and D. Marolf, “On the generality of refined algebraic quantization,” Classical and Quantum Gravity 16 (1999) 2479–2488, arXiv:gr-qc/9812024.

[97] D. Giulini and D. Marolf, “A uniqueness theorem for constraint quantization,” Classical and Quantum Gravity 16 (1999) 2489–2505, arXiv:gr-qc/9902045.

[98] M. Dupuis, L. Freidel, E. R. Livine, and S. Speziale, “Holomorphic Lorentzian Simplicity Constraints,” arXiv:1107.5274 [gr-qc].

142 [99] S. Alexandrov, “Hilbert space structure of covariant loop quantum gravity,” Physical Review D 66 (2002) 024028, arXiv:gr-qc/0201087.

[100] E. R. Livine, “Projected spin networks for Lorentz connection: linking spin foams and loop gravity,” Classical and Quantum Gravity 19 (2002) 5525–5541, arXiv:gr-qc/0207084.

[101] Y. Ding and C. Rovelli, “The volume operator in covariant quantum gravity,” arXiv:0911.0543 [gr-qc].

[102] D. Freedman, P. van Nieuwenhuizen, and S. Ferrara, “Progress toward a theory of supergravity,” Physical Review D 13 (1976) 3214–3218.

[103] A. Salam and E. Sezgin, eds., Supergravities in Diverse Dimensions. North-Holland, World Scientific, 1989.

[104] S. Deser, “Nonrenormalizability of (last hope) D= 11 supergravity, with a terse survey of divergences in quantum gravities,” arXiv:hep-th/9905017.

[105] T. Thiemann, “The LQG string - loop quantum gravity quantization of string theory: I. Flat target space,” Classical and Quantum Gravity 23 (2006) 1923–1970, arXiv:hep-th/0401172.

[106] W. Fairbairn, K. Noui, and F. Sardelli, “Canonical analysis of algebraic string actions,” Journal of High Energy Physics 2010 (2010) 045, arXiv:0908.0953 [hep-th].

[107] W. J. Fairbairn, K. Noui, and F. Sardelli, “Equivalence of the self-dual and Nambu-Goto strings,” arXiv:1012.5073 [hep-th].

[108] Z. Bern, J. Carrasco, L. Dixon, H. Johansson, and R. Roiban, “Ultraviolet Behavior of N=8 Supergravity at Four Loops,” Physical Review Letters 103 (2009) 081301, arXiv:0905.2326 [hep-th].

[109] M. B. Green, J. G. Russo, and P. Vanhove, “Non-renormalisation conditions in type II string theory and maximal supergravity,” Journal of High Energy Physics 2007 (2007) 099, arXiv:hep-th/0610299.

[110] R. Kallosh, “On UV finiteness of the four loop N= 8 supergravity,” Journal of High Energy Physics 2009 (2009) 116, arXiv:0906.3495 [hep-th].

[111] S. Deser, J. Kay, and K. Stelle, “Hamiltonian formulation of supergravity,” Physical Review D 16 (1977) 2448–2455.

[112] E. Fradkin and M. A. Vasiliev, “Hamiltonian formalism, quantization and S matrix for supergravity,” Physics Letters B 72 (1977) 70–74.

[113] M. Pilati, “The canonical formulation of supergravity,” Nuclear Physics B 132 (1978) 138–154.

[114] G. Senjanovi´c,“Hamiltonian formulation and quantization of the spin-3/2 field,” Physical Review D 16 (1977) 307–312.

[115] M. Henneaux, “Hamiltonian formulation of d = 10 supergravity theories,” Physics Letters B 168 (1986) 233–238.

143 [116] A. Diaz, “Hamiltonian formulation of eleven-dimensional supergravity,” Physical Review D 33 (1986) 2801–2808. [117] C. Teitelboim, “Supergravity and Square Roots of Constraints,” Physical Review Letters 38 (1977) 1106–1110. [118] M. Henneaux, “Poisson brackets of the constraints in the Hamiltonian formulation of tetrad gravity,” Physical Review D 27 (1983) 986–989. [119] A. Diaz, “Constraint algebra in eleven-dimensional supergravity,” Physical Review D 33 (1986) 2809–2812. [120] M. Sawaguchi, “Canonical formalism of N = 1 supergravity with the real Ashtekar variables,” Classical and Quantum Gravity 18 (2001) 1179–1186. [121] R. de Azeredo Campos and J. M. L. Fisch, “Hamiltonian formulation of the N=1 D=10 supergravity coupled to super Yang-Mills theory,” International Journal of Modern Physics A 3 (1988) 2371–2400. [122] T. Jacobson, “New variables for canonical supergravity,” Classical and Quantum Gravity 5 (1988) 923–935. [123] G. F¨ul¨op,“About a superAshtekar-Renteln ansatz,” Classical and Quantum Gravity 11 (1994) 1–10, arXiv:gr-qc/9305001. [124] N. N. Gorobey and A. S. Lukyanenko, “The Ashtekar complex canonical transformation for supergravity,” Classical and Quantum Gravity 7 (1990) 67–71. [125] H.-J. Matschull, “About loop states in supergravity,” Class.Quant.Grav. 11 (1994) 2395–2410, arXiv:gr-qc/9403034. [126] D. Armand-Ugon, R. Gambini, O. Obr´egon, and J. Pullin, “Towards a loop representation for quantum canonical supergravity,” Nuclear Physics B 460 (1996) 615–631, arXiv:hep-th/9508036. [127] Y. Ling and L. Smolin, “Supersymmetric spin networks and quantum supergravity,” Physical Review D 61 (2000) 044008, arXiv:hep-th/9904016. [128] Y. Ling and L. Smolin, “Holographic formulation of quantum supergravity,” Physical Review D 63 (2001) 064010, arXiv:hep-th/0009018. [129] Y. Ling, “Eleven-dimensional supergravity as a constrained topological field theory,” Nuclear Physics B 601 (2001) 191–208, arXiv:hep-th/0003285. [130] L. Smolin, “Chern-Simons theory in eleven-dimensions as a nonperturbative phase of M theory,” arXiv:hep-th/9703174. [131] L. Smolin, “A quantization of topological M theory,” Nuclear Physics B 739 (2006) 169–185, arXiv:hep-th/0503140. [132] P. J. Arias, C. di Bartolo, X. Fustero, R. Gambini, and A. Trias, “Second Quantization of the Antisymmetric Potential in the Space of Abelian Surfaces,” International Journal of Modern Physics A 7 (1992) 737–753. [133] S. Sengupta and R. K. Kaul, “Canonical supergravity with Barbero-Immirzi parameter,” Physical Review D 81 (2010) 024024, arXiv:0909.4850 [hep-th].

144 [134] S. Melosch and H. Nicolai, “New canonical variables for d=11 supergravity,” Physics Letters B 416 (1998) 91–100, arXiv:hep-th/9709227. [135] H. Matschull, “Canonical quantum supergravity in three dimensions,” Nuclear Physics B 411 (1994) 609–646, arXiv:gr-qc/9306018. [136] A. D’Adda, J. Nelson, and T. Regge, “Covariant canonical formalism for the group manifold,” Annals of Physics 165 (1985) 384–397.

[137] J. Nelson and T. Regge, “Covariant canonical formalism for gravity,” Annals of Physics 166 (1986) 234–249.

[138] A. Lerda, J. Nelson, and T. Regge, “Covariant canonical formalism for supergravity,” Physics Letters B 161 (1985) 294–296.

[139] A. Foussats and O. Zandron, “Canonical covariant formalism for gravity and supergravity in more than four dimensions,” Classical and Quantum Gravity 5 (1988) 605–617.

[140] A Foussats and O Zandron, “Hamiltonian formalism for higher curvature supergravity in D=5 dimensions,” Classical and Quantum Gravity 6 (1989), no. 8 1165–1170.

[141] A. Foussats and O. Zandron, “Hamiltonian formalism for N=1, D=10 Yang-Mills coupled supergravity,” Annals of Physics 191 (1989) 312–334.

[142] A. Foussats, C. Repetto, O. P. Zandron, and O. S. Zandron, “Second order Hamiltonian formalism and constraints algebra for non-polynomial supergravities,” Annalen der Physik 504 (1992) 505–530.

[143] A. Foussats and O. Zandron, “Algebra of constraints for supersymmetric Yang-Mills theory coupled to supergravity,” Physical Review D 43 (1991) 1883–1891.

[144] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems. Princeton University Press, 1994.

[145] D. M. Gitman and I. V. Tyutin, Quantization of Fields with Constraints. Springer, 1990.

[146] T. Thiemann and O. Winkler, “Gauge field theory coherent states (GCS): IV. Infinite tensor product and thermodynamical limit,” Classical and Quantum Gravity 18 (2001) 4997–5053, arXiv:hep-th/0005235. [147] F. Giani and M. Pernici, “N=2 supergravity in ten dimensions,” Physical Review D 30 (1984) 325–333.

[148] A. Chamseddine, “N = 4 supergravity coupled to N = 4 matter and hidden symmetries,” Nuclear Physics B 185 (1981) 403–415.

[149] C. Rovelli, Quantum Gravity. Cambridge University Press, Cambridge, 2004.

[150] J. C. Baez and J. Huerta, “An Invitation to Higher Gauge Theory,” arXiv:1003.4485 [hep-th]. [151] J. C. Baez, A. Baratin, L. Freidel, and D. K. Wise, “Infinite-Dimensional Representations of 2-Groups,” arXiv:0812.4969 [math.QA]. [152] L. Crane and M. D. Sheppeard, “2-categorical Poincare Representations and State Sum Applications,” arXiv:math/0306440 [math.QA].

145 [153] L. Crane and D. N. Yetter, “Measurable Categories and 2-Groups,” Applied Categorical Structures 13 (2005) 501–516, arXiv:math/0305176 [math.QA].

[154] D. N. Yetter, “Measurable Categories,” Applied Categorical Structures 13 (2005) 469–500, arXiv:math/0309185 [math.CT].

[155] J. C. Baez and U. Schreiber, “Higher Gauge Theory,” in Categories in Algebra, Geometry and Mathematical Physics (A. Davydov et al, ed.), (Providence, Rhode Island), pp. 7–30, AMS2007. arXiv:math/0511710 [math.DG].

[156] P. van Nieuwenhuizen, “Supergravity,” Physics Reports 68 (1981) 189–398.

[157] R. Haag, Local Quantum Physics: Fields, Particles, Algebras. Springer, 2nd ed., 1996.

[158] J. Campbell, “On a Law of Combination of Operators bearing on the Theory of Continuous Transformation Groups,” Proc. London Math. Soc. s1-28 (1896) 381–390.

[159] J. Campbell, “On a Law of Combination of Operators (Second Paper),” Proc. London Math. Soc. s1-29 (1897) 14–32.

[160] H. Baker, “Further Applications of Metrix Notation to Integration Problems,” Proc. London Math. Soc. s1-34 (1901) 347–360.

[161] H. Baker, “On the Integration of Linear Differential Equations,” Proc. London Math. Soc. s1-35 (1902) 333–378.

[162] H. Baker, “Alternants and Continuous Groups,” Proc. London Math. Soc. s2-3 (1905) 24–47.

[163] F. Hausdorff, “Die symbolische Exponentialformel in der Gruppentheorie,” Ber. Verh. Saechs. Akad. Wiss. Leipzig 58 (1906) 19–48.

[164] W. Magnus, “On the exponential solution of differential equations for a linear operator,” Comm. Pure Appl. Math. 7 (1954) 649–673.

[165] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1: C*- and W*-Algebras. Symmetry Groups. Decomposition of States. Springer, 2nd ed., 1987.

[166] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2: Equilibrium States. Models in Quantum Statistical Mechanics. Springer, 2nd ed., 2003.

[167] A. Ashtekar, J. Lewandowski, and H. Sahlmann, “Polymer and Fock representations for a scalar field,” Classical and Quantum Gravity 20 (2003) L11–L21, arXiv:gr-qc/0211012.

[168] W. Kaminski, J. Lewandowski, and M. Bobienski, “Background independent quantizations - the scalar field: I,” Classical and Quantum Gravity 23 (2006) 2761–2770, arXiv:gr-qc/0508091.

[169] W. Kaminski, J. Lewandowski, and A. Okol´ow,“Background independent quantizations - the scalar field: II,” Classical and Quantum Gravity 23 (2006) 5547–5585, arXiv:gr-qc/0604112.

[170] J. Brown and K. Kuchar, “Dust as a standard of space and time in canonical quantum gravity,” Physical Review D 51 (1995) 5600–5629, arXiv:gr-qc/9409001.

146 [171] B. Dittrich and T. Thiemann, “Are the spectra of geometrical operators in loop quantum gravity really discrete?,” Journal of Mathematical Physics 50 (2009) 012503, arXiv:0708.1721 [gr-qc].

[172] A. D. Rendall, “Constant Mean Curvature Foliations in Cosmological Spacetimes,” Helvetica Physica Acta 69 (1996) 490–500.

[173] A. Lichnerowicz, “L’int´egrationdes ´equationsde la gravitation relativiste et le probl`eme des n corps,” Journal de Math´ematiquesPures et Appliqu´ees 23 (1944) 37–63.

[174] J. York, “Gravitational Degrees of Freedom and the Initial-Value Problem,” Physical Review Letters 26 (1971) 1656–1658.

[175] H. Gomes and T. Koslowski, “Coupling Shape Dynamics to Matter Gives Spacetime,” General Relativity and Gravitation 44 (2012) 1539–1553, arXiv:1110.3837 [gr-qc].

[176] C. Wang, “Conformal geometrodynamics: True degrees of freedom in a truly canonical structure,” Physical Review D 71 (2005) 124026, arXiv:gr-qc/0501024.

[177] C. Wang, “Towards conformal loop quantum gravity,” Journal of Physics: Conference Series 33 (2006) 285–290, arXiv:gr-qc/0512023.

[178] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, 2nd ed., 1983.

[179] L. Ford and T. Roman, “Classical scalar fields and the generalized second law,” Physical Review D 64 (2001) 024023.

[180] J. Lewandowski, A. Okol´ow, H. Sahlmann, and T. Thiemann, “Uniqueness of Diffeomorphism Invariant States on Holonomy-Flux Algebras,” Communications in Mathematical Physics 267 (2006) 703–733, arXiv:gr-qc/0504147.

[181] A. Ashtekar and A. Corichi, “Non-minimal couplings, quantum geometry and black-hole entropy,” Classical and Quantum Gravity 20 (2003) 4473–4484, arXiv:gr-qc/0305082.

[182] A. Ashtekar, A. Corichi, and K. Krasnov, “Isolated Horizons: the Classical Phase Space,” Advances in Theoretical and Mathematical Physics 3 (2000) 419–478, gr-qc/9905089.

[183] M. Nadalini, L. Vanzo, and S. Zerbini, “Thermodynamical properties of hairy black holes in n spacetime dimensions,” Physical Review D 77 (2008) 024047, arXiv:0710.2474 [hep-th].

[184] C. Mart´ınez,R. Troncoso, and J. Staforelli, “Topological black holes dressed with a conformally coupled scalar field and electric charge,” Physical Review D 74 (2006) 044028, arXiv:hep-th/0512022.

[185] A. Ashtekar, A. Corichi, and D. Sudarsky, “Non-minimally coupled scalar fields and isolated horizons,” Classical and Quantum Gravity 20 (2003) 3413–3425, arXiv:gr-qc/0305044.

[186] A. Strominger and C. Vafa, “Microscopic origin of the Bekenstein-Hawking entropy,” Physics Letters B 379 (1996) 99–104, arXiv:hep-th/9601029.

[187] R. Emparan and H. S. Reall, “Black Holes in Higher Dimensions,” Living Reviews in Relativity 11 (2008) arXiv:0801.3471 [hep-th].

147 [188] L. Smolin, “Linking Topological Quantum Field Theory and Nonperturbative Quantum Gravity,” Journal of Mathematical Physics 36 (1995) 6417–6455, arXiv:gr-qc/9505028.

[189] K. Krasnov, “Geometrical entropy from loop quantum gravity,” Physical Review D 55 (1997) 3505 – 3513, arXiv:gr-qc/9603025.

[190] C. Rovelli, “Black Hole Entropy from Loop Quantum Gravity,” Physical Review Letters 77 (1996) 3288–3291, arXiv:gr-qc/9603063.

[191] A. Ashtekar, C. Beetle, and S. Fairhurst, “Mechanics of isolated horizons,” Classical and Quantum Gravity 17 (2000) 253–298, arXiv:gr-qc/9907068.

[192] A. Ashtekar, S. Fairhurst, and B. Krishnan, “Isolated horizons: Hamiltonian evolution and the first law,” Physical Review D 62 (2000) 104025, arXiv:gr-qc/0005083.

[193] A. Ashtekar, J. Baez, and K. Krasnov, “Quantum Geometry of Isolated Horizons and Black Hole Entropy,” Advances in Theoretical and Mathematical Physics 4 (2000) 1–94, arXiv:gr-qc/0005126.

[194] J. Engle, K. Noui, and A. Perez, “Black Hole Entropy and SU(2) Chern-Simons Theory,” Physical Review Letters 105 (2010) 031302, arXiv:0905.3168 [gr-qc].

[195] I. Agullo, J. Fernando Barbero G., E. F. Borja, J. Diaz-Polo, and E. J. S. Villasenor, “Detailed black hole state counting in loop quantum gravity,” Physical Review D 82 (2010) 084029, arXiv:1101.3660 [gr-qc].

[196] J. Diaz-Polo and D. Pranzetti, “Isolated Horizons and Black Hole Entropy In Loop Quantum Gravity,” arXiv:1112.0291 [gr-qc].

[197] F. Tangherlini, “Schwarzschild field in n dimensions and the dimensionality of space problem,” Il Nuovo Cimento 27 (1963) 636–651.

[198] J. Lewandowski and T. Pawlowski, “Quasi-local rotating black holes in higher dimension: geometry,” Classical and Quantum Gravity 22 (2005) 1573–1598, arXiv:gr-qc/0410146.

[199] M. Korzynski, J. Lewandowski, and T. Pawlowski, “Mechanics of multidimensional isolated horizons,” Classical and Quantum Gravity 22 (2005) 2001–2016, arXiv:gr-qc/0412108.

[200] J. Engle, K. Noui, A. Perez, and D. Pranzetti, “Black hole entropy from an SU(2)-invariant formulation of Type I isolated horizons,” arXiv:1006.0634 [gr-qc].

[201] M. Nakahara, Geometry, Topology and Physics. Taylor & Francis, 2nd ed., 2003.

[202] M. Ba˜nados,L. Garay, and M. Henneaux, “Existence of local degrees of freedom for higher dimensional pure Chern-Simons theories,” Physical Review D 53 (1996) R593–R596.

[203] A. Ashtekar, C. Beetle, and S. Fairhurst, “Isolated horizons: a generalization of black hole mechanics,” Classical and Quantum Gravity 16 (1999) L1–L7, arXiv:gr-qc/9812065.

[204] A. Ashtekar, T. Pawlowski, and C. V. D. Broeck, “Mechanics of higher dimensional black holes in asymptotically anti-de Sitter spacetimes,” Classical and Quantum Gravity 24 (2007) 625–644, arXiv:gr-qc/0611049.

148 [205] C. Crnkovic and E. Witten, “Covariant Description of Canonical Formalism in Geometrical Theories,” in Three Hundred Years of Gravitation (W. Israel and S. Hawking, eds.), (Cambridge), pp. 676–684, Cambridge University Press, 1987.

[206] J. Lee and R. Wald, “Local symmetries and constraints,” Journal of Mathematical Physics 31 (1990) 725–743.

[207] A. Ashtekar, L. Bombelli, and O. Reula, “Covariant Phase Space of Asymptotically flat Gravitational Fields,” in Mechanics, Analysis and Geometry: 200 Years after Lagrange (M. Francaviglia and D. Holm, eds.), (Amsterdam), North Holland, 1990.

[208] A. Ashtekar, J. Engle, and C. Van Den Broeck, “Quantum horizons and black-hole entropy: inclusion of distortion and rotation,” Classical and Quantum Gravity 22 (2005) L27, arXiv:gr-qc/0412003.

[209] A. Perez and D. Pranzetti, “Static isolated horizons: SU(2) invariant phase space, quantization, and black hole entropy,” Entropy 13 (2011) 744–777, arXiv:1011.2961 [gr-qc].

[210] C. Beetle and J. Engle, “Generic isolated horizons in loop quantum gravity,” Classical and Quantum Gravity 27 (2010) 235024, arXiv:1007.2768 [gr-qc].

[211] E. Witten, “Quantum field theory and the Jones polynomial,” Communications in Mathematical Physics 121 (1989) 351–399.

[212] K. Krasnov and C. Rovelli, “Black holes in full quantum gravity,” Classical and Quantum Gravity 26 (2009) 245009, arXiv:0905.4916 [gr-qc].

[213] E. Bianchi, “Entropy of Non-Extremal Black Holes from Loop Gravity,” arXiv:1204.5122 [gr-qc].

[214] S. Liberati and G. Pollifrone, “Entropy and topology for gravitational instantons,” Physical Review D 56 (1997) 6458–6466, arXiv:hep-th/9708014.

[215] T. Liko and I. Booth, “Supersymmetric isolated horizons,” Classical and Quantum Gravity 25 (2008) 105020, arXiv:0712.3308 [gr-qc].

[216] I. Booth and T. Liko, “Supersymmetric isolated horizons in ADS spacetime,” Physics Letters B 670 (2008) 61–66, arXiv:0808.0905 [gr-qc].

[217] T. Liko, “Isolated horizons, p-form matter fields, topology, and the black-hole/string correspondence principle,” Physical Review D 79 (2009) arXiv:0901.1121 [gr-qc].

[218] M. Ba˜nados,R. Troncoso, and J. Zanelli, “Higher dimensional Chern-Simons supergravity,” Physical Review D 54 (1996) 2605–2611.

[219] M. Bojowald, G. Calcagni, and S. Tsujikawa, “Observational test of inflation in loop quantum cosmology,” Journal of Cosmology and Astroparticle Physics 2011 (2011) 046–046, arXiv:1107.1540 [gr-qc].

[220] P. Moniz, “Supersymmetric Quantum Cosmology — Shaken not stirred.,” International Journal of Modern Physics A 11 (1996) 4321–4382.

[221] M. Bojowald, “Absence of a Singularity in Loop Quantum Cosmology,” Physical Review Letters 86 (2001) 5227–5230, arXiv:gr-qc/0102069.

149 [222] A. Ashtekar, T. Pawlowski, and P. Singh, “Quantum nature of the big bang: Improved dynamics,” Physical Review D 74 (2006) 084003, arXiv:gr-qc/0607039. [223] M. Demianski, Z. A. Golda, M. Heller, and M. Szydlowski, “The dimensional reduction in a multi-dimensional cosmology,” Classical and Quantum Gravity 3 (1986) 1199–1205. [224] J. Petersen, “Introduction to the Maldacena conjecture on AdS / CFT,” International Journal of Modern Physics A 14 (1999) 3597–3672, [hep-th/9902131]. [225] L. Freidel, “Reconstructing AdS/CFT,” arXiv:0804.0632v1 [hep-th]. [226] A. Laddha and M. Varadarajan, “The diffeomorphism constraint operator in loop quantum gravity,” Classical and Quantum Gravity 28 (2011) 195010, arXiv:1105.0636 [gr-qc]. [227] P. D’eath and D. Hughes, “Supersymmetric mini-superspace,” Physics Letters B 214 (1988) 498–502. [228] M. Varadarajan, “Fock representations from U(1) holonomy algebras,” Physical Review D 61 (2000) 104001, arXiv:gr-qc/0001050. [229] M. Varadarajan, “Photons from quantized electric flux representations,” Physical Review D 64 (2001) 104003, arXiv:gr-qc/0104051. [230] M. Varadarajan, “Gravitons from a loop representation of linearized gravity,” Physical Review D 66 (2002) 024017, arXiv:gr-qc/0204067. [231] A. Ashtekar, J. Lewandowski, D. Marolf, J. Mour˜ao,and T. Thiemann, “Coherent State Transforms for Spaces of Connections,” Journal of Functional Analysis 135 (1996) 519–551, arXiv:gr-qc/9412014. [232] M. Hormeß and G. Junker, “More on coupling coefficients for the most degenerate representations of SO(n),” Journal of Physics A: Mathematical and General 32 (1999) 4249–4256, arXiv:math-ph/9903035. [233] S. Alisauskas, “Coupling coefficients of SO(n) and integrals involving Jacobi and Gegenbauer polynomials,” Journal of Physics A: Mathematical and General 35 (2002) 7323–7345, arXiv:math-ph/0201048. [234] G. Junker, “Explicit evaluation of coupling coefficients for the most degenerate representations of SO(n),” Journal of Physics A: Mathematical and General 26 (1993), no. 7 1649–1661. [235] S. Alisauskas, “Some coupling and recoupling coefficients for symmetric representations of SOn,” Journal of Physics A: Mathematical and General 20 (1987) 35. [236] G. Girardi, A. Sciarrino, and P. Sorba, “Generalized Young tableaux and Kronecker products of SO(n) representations,” Physica A: Statistical Mechanics and its Applications 114 (1982) 365–369. [237] G. Girardi, A. Sciarrino, and P. Sorba, “Kronecker products for SO(2p) representations,” Journal of Physics A: Mathematical and General 15 (1982) 1119–1129. [238] V. Agrawala and J. Befinfante, “Graphical formulation of recoupling theory for any compact group,” Annals of Physics 49 (1968) 130–170.

150