Universita degli Studi di Torino

Facolta di Scienze MFN

Laurea Specialistica in Fisica delle Interazioni Fondamentali

Simplicial Techniques inQuantum Gravity: SpinNetworks and Spinfoams

Davide Girolami

Relatore: Dott. Lorenzo FatibeneCorrelatore: Prof. Nicolao Fornengo

17 Dicembre 2009

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Contents

I Spin Networks and Spinfoams 15

1 The road to Quantum Gravity 171.1 Classical Gravity . . . . . . . . . . . . . . . . . . . . 17

1.1.1 A revolutionary theory: General Relativity.Which interpretation? . . . . . . . . . . . . . 17

1.1.2 General Relativity and Quantum Mechanics:a difficult marriage . . . . . . . . . . . . . . . 18

1.1.3 Quantum Field Theory on curved spacetime . 181.1.4 Metric and tetrad formalism . . . . . . . . . . 191.1.5 Selfdual connection: an alternative formula-

tion of GR . . . . . . . . . . . . . . . . . . . . 201.1.6 Barbero-Immirzi connection . . . . . . . . . . 221.1.7 Plebanski formalism . . . . . . . . . . . . . . 23

1.2 Quantum Gravity . . . . . . . . . . . . . . . . . . . . 241.2.1 Canonical approach: a quantization algorithm 241.2.2 Field equations . . . . . . . . . . . . . . . . . 26

2 Loop Quantum Gravity 292.1 General overview . . . . . . . . . . . . . . . . . . . . 292.2 Classical canonical theory . . . . . . . . . . . . . . . 302.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1 Hilbert space of physical states . . . . . . . . 322.3.2 Operators . . . . . . . . . . . . . . . . . . . . 392.3.3 A final view on kinematics . . . . . . . . . . . 46

2.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 482.4.1 The quantum Hamiltonian operator . . . . . . 482.4.2 Matrix elements . . . . . . . . . . . . . . . . . 512.4.3 The propagator in LQG . . . . . . . . . . . . 532.4.4 How many vacua! . . . . . . . . . . . . . . . . 55

2.5 Coupling to matter . . . . . . . . . . . . . . . . . . . 562.5.1 Gravity–Matter theory: kinematics . . . . . . 562.5.2 Dynamics . . . . . . . . . . . . . . . . . . . . 58

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2.5.3 A possible end of the road . . . . . . . . . . . 58

3 Spinfoams 613.1 A brief presentation . . . . . . . . . . . . . . . . . . . 613.2 From knots to foams: covariantization . . . . . . . . 613.3 Models . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.1 General propagator . . . . . . . . . . . . . . . 653.3.2 BF Theories . . . . . . . . . . . . . . . . . . . 663.3.3 Simplex and Euler characteristic . . . . . . . . 683.3.4 Extension to generic dimension . . . . . . . . 713.3.5 Dynamics . . . . . . . . . . . . . . . . . . . . 763.3.6 BF dynamics and spinfoams . . . . . . . . . . 813.3.7 A way to renormalization: q–deformation . . . 833.3.8 3D and 4D BF quantization . . . . . . . . . . 843.3.9 4D Quantum Gravity . . . . . . . . . . . . . . 863.3.10 Quantum Groups . . . . . . . . . . . . . . . . 89

II Applications 93

4 Spin network and Spinfoams calculations 954.1 The main framework: SU(2) representation theory . 954.2 Spin Networks . . . . . . . . . . . . . . . . . . . . . . 984.3 Spinfoams . . . . . . . . . . . . . . . . . . . . . . . . 101

4.3.1 3j–symbol . . . . . . . . . . . . . . . . . . . . 1014.3.2 6j–symbol . . . . . . . . . . . . . . . . . . . . 1034.3.3 15j–symbol . . . . . . . . . . . . . . . . . . . 107

5 Canonical formalism: application to f(R) theories 1135.1 Introduction to f(R) theories . . . . . . . . . . . . . 1135.2 f(R) gravity field equations . . . . . . . . . . . . . . 117

5.2.1 GR recap . . . . . . . . . . . . . . . . . . . . 1175.2.2 f(R) gravity . . . . . . . . . . . . . . . . . . . 118

5.3 Extension of LQG formalism to f(R) . . . . . . . . . 119

A All we need is math 125A.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 125A.2 Tensors in brief . . . . . . . . . . . . . . . . . . . . . 125

A.2.1 Definition . . . . . . . . . . . . . . . . . . . . 125A.2.2 Techniques of calculation . . . . . . . . . . . . 126A.2.3 Levi-Civita tensor . . . . . . . . . . . . . . . . 128

A.3 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 129A.4 Notations . . . . . . . . . . . . . . . . . . . . . . . . 130

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List of Figures

2.1 Area from T operator . . . . . . . . . . . . . . . . . . 452.2 T (x, r, s, t) Operator . . . . . . . . . . . . . . . . . . 462.3 Operator H: action at a trivalent node . . . . . . . . 502.4 D+− at a trivalent node . . . . . . . . . . . . . . . . 52

3.1 A spinfoam . . . . . . . . . . . . . . . . . . . . . . . 643.2 Dualizing an octahedron we obtain a cube . . . . . . 713.3 Dual skeleton . . . . . . . . . . . . . . . . . . . . . . 723.4 Triangulation in 3D . . . . . . . . . . . . . . . . . . . 733.5 Labelling spins in 3D triangulation . . . . . . . . . . 743.6 4–node splitting . . . . . . . . . . . . . . . . . . . . . 743.7 One of the possible cutting parallelograms . . . . . . 753.8 Splitting as combinatorial sum . . . . . . . . . . . . . 753.9 3D dual skeleton intersection . . . . . . . . . . . . . . 773.10 trivial intertwiner . . . . . . . . . . . . . . . . . . . . 783.11 3D non–trivial intertwiner . . . . . . . . . . . . . . . 793.12 6j–symbol . . . . . . . . . . . . . . . . . . . . . . . . 803.13 15j–symbol . . . . . . . . . . . . . . . . . . . . . . . 823.14 Hopf algebra . . . . . . . . . . . . . . . . . . . . . . . 90

4.1 Action of H at a node . . . . . . . . . . . . . . . . . 964.2 Intertwiner at 3–valent node . . . . . . . . . . . . . . 974.3 Intertwiner at 3–valent node explicitated . . . . . . . 974.4 Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.5 A seminal example . . . . . . . . . . . . . . . . . . . 994.6 2–loop decomposition . . . . . . . . . . . . . . . . . . 994.7 Two tetravalent nodes . . . . . . . . . . . . . . . . . 1004.8 3j . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.9 4–valent node . . . . . . . . . . . . . . . . . . . . . . 1044.10 change of basis . . . . . . . . . . . . . . . . . . . . . 1054.11 6j–symbol . . . . . . . . . . . . . . . . . . . . . . . . 1064.12 explicit splitting . . . . . . . . . . . . . . . . . . . . . 1074.13 explicit basis changing . . . . . . . . . . . . . . . . . 1084.14 15j . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5

4.15 15–symbol splitting . . . . . . . . . . . . . . . . . . . 1104.16 explicit 15j–symbol . . . . . . . . . . . . . . . . . . . 111

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List of Tables

3.1 Simplexes . . . . . . . . . . . . . . . . . . . . . . . . 683.2 3D Triangulation . . . . . . . . . . . . . . . . . . . . 703.3 4D Triangulation . . . . . . . . . . . . . . . . . . . . 713.4 Dictionary of the triangulation . . . . . . . . . . . . . 81

4.1 15j resume . . . . . . . . . . . . . . . . . . . . . . . . 111

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Writing is good, Thinking is betterCleverness is good, Patience is better

The Devil is in the detail

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Acknowledgements

Thanks to: all the goodwilled readersn1Thanks to: the Guit Forum, Anna Fino, Jeanette Nelson, FrankSaueressig, Shan Majid, Carlo Rovelli, Joe Billingham, Nicolao For-nengo, Miguel Onorato, Giampiero Passarino, Luca Mercalli, Clau-dio Castellano, Sax... for different reasonsn! Thanks to: Viacheslav Belavkin and Gerardo Adesso, for thegreat opportunityen Thanks to: Lorenzo Fatibene, for an asymptotically infinite pa-tiencenn Thanks to: my family

1n > 3

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10

Introduction

The modern research of a quantum theory of the gravitational in-teraction sees many possible lines of promising developments. Apossible strategy is to follow the path determined by a conservativeapproach, starting from the assumptions that the two most reliablephysical theories, i.e. General Relativity and Quantum Mechanics,only apparently disagree. This implies to focus on the possible com-mon characteristics. From a mathematical point of view, GeneralRelativity is constructed by Differential Geometry, while QuantumMechanics is written in the language of Group Representation The-ory. It could be a good idea trying to redefine General Relativitywith the mathematical language of Groups. An insightful method-ology concerns to treating the geometrical aspects of spacetime justas intrinsically combinatorial. All the components of the spacetime,the arena of physical events, are deprived of any metric propertyand equipped with labels, a spin representation for the surfaces andan invariant tensor for volumes. The geometrical spacetime is re-placed by an abstract network of relations and connections betweenthese labels. In this sense, the spacetime is discretized, and in awider sense quantized. The building blocks of this new spacetimeare simplexes, the generalization in n dimensions of the common tri-angle. One can develop a set of techniques to make calculations onthis new framework, interpreting the gravitational interaction andthe curvature of spacetime as a well-defined combinatorial disposi-tion of labels. This is the simplicial approach to Quantum Gravity,which replaces the differential geometry working on the spacetimeinterpreted as a manifold, with the group representation theory, thelanguage of Quantum Mechanics. The classical manifold and geom-etry emerge from this combinatorial world in the classical limit (atleast one hope they can be shown to emerge!). The strategy adoptedto explain the theory and to solve exercises consists to follow twoparallel paths: algebraic and the graphic.

We summarize the contents of this thesis by chapter.

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In chapter 1, we resume the fundamental steps that lead to QuantumGravity. First, we present General Relativity in a new formalism,determining the field equations. After that, we introduce Quan-tum Mechanics and Quantum Field Theories, pointing out some ofthe reasons of the failure in applying a perturbative approach to agravitational theory. These two lines of research would cross theirdooms in quantum gravity, i.e. a theory of gravity that satisfies therequest of mathematical consistency and describes the gravitationalinteraction at every scale of energy, possibly in accordance with theStandard Model of particles and the cosmological observations.

Chapter 2 regards Loop Quantum Gravity. The aim is not toconstruct a perturbative theory of gravity, for reason explained inthe text. We want to search for a background independent frame-work, a canonical theory with a cut–off that hopefully allows tocancel divergences of General Relativity in the classical limit. Weadopt techniques of the Yang-Mills theories on lattice, with a sem-inal distinction with respect to them: the lattice of our theory isnot fixed and embedded in a manifold, but it represents graphicallythe spacetime itself as a dynamical object. We describe a generalsketch about a quantization algorithm for a theory of gravity. Inparticular, we present a glance on the main features of the LoopQuantum Gravity, a partial gauge fixing theory that interprets theWilson loops as building blocks of quantum states, i.e. the SpinNetworks, which after a ”normalization” procedure become elementof a separable Hilbert space, and take the name of Spin Knots. Themathematical environment is described together with the kinemati-cal theory, which is constructed defining area and volume operators,quantum correspondent to the classical geometrical quantities. Thedynamics is based on the extension of Quantum Mechanics objects,e.g. the propagators, to a new conceptual framework.

In chapter 3 we have present a huge class of models that representan attempt to construct a ”sum over paths” Quantum Gravity, thatis a path integral formulation for gravitational interaction. Thestrategy is trying to develop a mathematically consistent version ofthe Hawking ”sum over geometries g” path integral:

W [g, g′] =

∫

g|t=1=g;g|t′=0=g′D[gµν(x)]eiSG[g]. (1)

A possible key to solve the task is the discretization of the field vari-ables, which leads to define a set of covariant theories, the SpinfoamModels, with deep similarities with a well studied class of field the-

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ories, namely the BF theories. A Spinfoam is a sort of two dimen-sional Feynman diagram, a graphical representation of a physicaltransition from initial and final spin network states. The field vari-ables of our theory are the holonomies of the connection along edgesof a dual triangulation of the spacetime, and the line integral alongeach edge of a face of the above mentioned triangulation. These arethe discretized, and in this sense quantized, version of the variablesof the classical General Relativity. This is not a new brand ap-proach to Quantum Gravity: Ponzano and Regge [24][25] developedan equivalent discretized version of 3D General Relativity, definingpath integrals between triangulated 2D surfaces with links with aquantized length. The seminal difference is that the quantization inthe Ponzano–Regge theory is an ansatz, whilst in the new simplicialtechniques is a result. As such, it is not something chosen by theobserver, but it is the physical situation that suggests a canonicaldiscretization.Also, we stress the connection between Loop Quantum Gravity dy-namics and Spinfoam Models: a spinfoam can be interpreted as apath between two spin knots. Several spinfoam models are pre-sented, and each of them is defined by a partition function as prop-agator from two boundary states representing physical volumes, i.e.two spin knots s, s′. Its general form is

W (s, s′) =∑

σ∪σ′w(Γ(σ))Πfdim(jf )ΠeAe(jf , ie)ΠvAv(jf , ie), (2)

which is a combinatorial function of transition amplitude A associ-ated to spinfoam faces f , edges e and vertices v. From the dynamicalpoint of view this does not mean that the propagator is the sameof Loop approach: in fact, in LQG theory we are not able to obtainan ultimate Hamiltonian function and we can not calculate transi-tion amplitudes; moreover, in Spinfoam formalism, we do not knowwhich is the most physically relevant model. All that we can sayis that LQG and Spinfoams Models are theories that use the samemethods for the same problems, tough the technical details are notalways clear.

The chapter 4 is about calculations regarding transition ampli-tudes in Loop Quantum Gravity and Spinfoam Models. We adopttechniques of SU(2) representation theory to perform calculations,and show how to algebraically prove some claims and results whichare usually accepted on a graphical stance in Quantum Gravity lit-erature. Treating spinfoams compels to generalize the well–knownrecoupling theory of angular momentum used in Quantum Mechan-

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ics. The Wigner nj–symbols, generalization of the Clebsch–Gordancoefficients, are the components of transition amplitudes associatedto the spinfoam vertices. They find many applications involvingseveral branches of modern Physics, in particular the treatment ofmultispin systems.

Finally, the chapter 5 regards the f(R) theories, a huge class ofmodels born from a ”slight” modification of Hilbert action: replac-ing the Ricci scalar R with an analytic function f(R), we define atheory with the same kinematics of the standard General Relativity,but with a different dynamics. After that, we show that the samerelation holds between a theory defined by the Holst Lagrangianfunction, the starting point of Loop Quantum Gravity, and its f(R)deformation. The application of f(R) theories as toy models tofit experimental data in Cosmology and, at the same time, theirconnection with Loop Quantum Gravity, seems to indicate a newopportunity to investigate the physical reliability of the alternativetheories of gravity based on the simplicial approach.

The main general reference is the introductory book to LoopQuantum Gravity written by C.Rovelli [1]. The monumental andinsightful Gravitation[2] has strongly influenced our outlook on thistopic. Whenever a reference is not specified , the figures have beenrealized by the open source drawing software Ipe.

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Part I

Spin Networks andSpinfoams

15

Chapter 1

The road to QuantumGravity

1.1 Classical Gravity

1.1.1 A revolutionary theory: General Relativity. Whichinterpretation?

The revolutionary theory of General Relativity broke down withconventional assumptions about the absoluteness of space and time,strongly supported by our common sense. In a deeper analysis, onecan say that two interpretations arise from it: the geometrical one,that leads to consider spacetime as a dynamical object, and thealgebraic one, where there is not a spacetime as alternative frame-work where other fields live any longer, but there is a spacetime asa dynamical field that interacts with particles and gauge bosons ofStandard Model. After decades of discussions about interpretationof Quantum Mechanics, nowadays the foundational problem is of-ten considered as unphysical. However, we will follow the algebraicpoint of view, because, in order to develop a quantum theory ofgravitation, the geometrical one presents some problems. For exam-ple, it is yet unclear wheather geometry is a quantum characteristicof spacetime or it emerges at classical level. Many hints seem topoint towards a discrete quantum spacetime, where geometry needsat least to be modified. From this perspective, we have to abandonour intuitive (but wrong) conception of space and time. Localiza-tion becomes a relational instance: we do not care about positionof an object on a manifold representing spacetime, but we define itscontiguity with respect to other objects or events. Nevertheless, itis dutiful to say that this interpretation is not universally acceptedand, at this time, it does not affect practical results in Particle

17

Physics. So, we can decide to not worry about foundational ques-tions unless it is useful for measurements, or that it is worthy tryingto look into issues over merely empirical matters.

1.1.2 General Relativity and Quantum Mechanics: a dif-ficult marriage

Let us sketch to essential features of quantum theories: we focus onthe aspects in open contrast with respect to the General Relativistictheory.Quantum Mechanics is defined using external time variable t or afixed and non–dynamical background spacetime; dynamical objectscan be discretized and subjected by intrinsically non–deterministicand probabilistic laws. The mathematical framework of quantumtheories is a Gel’fand triple, also called rigged Hilbert space: S ⊂K ⊂ S ′, where K is a Hilbert space, S a subset dense in H witha weak topology and S ′ is the dual space of S. If we consider ameasure dµ on a manifold M , it is immediately defined a Gel’fandtriple with S space of fast decreasing smooth functions on M , S ′

space of the tempered distribution and K = L2[M,dµ]. Physically,K is the space of kinematical states. A complete set of commut-ing observables suffices to describe system: the quantum events aretheir eigenstates.The quantum dynamics is encoded in transition probability ampli-tude W (x, t, x′, t′) =< x, t|x′, t′ >. This means that we have lessinformations in comparison with Classical Mechanics: we can notknow (and so we do not care about this!) what happens betweenthe points x′ and x because it is not physics. The fields configu-rations must satisfy the boundary condition which concern to theinitial and final states. Unfortunately, it is obvious that W is notinvariant under diffeomorphism. In fact, in a (general) relativistictheory, one can easily show that covariant probability amplitudesmust necessarily be constant! Hence, it is necessary to build a newextended Quantum Mechanics framework that permits to constructa consistent quantized theory of gravity, that retains the classicalfeatures of General Relativity.

1.1.3 Quantum Field Theory on curved spacetime

The data set (x, t, x′, t′) in Quantum Mechanics is replaced in a fieldtheory by the couple [Σ, ϕ], where Σ is a 3-surface bounding a space-time region and ϕ is a field configuration on Σ. The field theoreticalspace is the space of all (Σ, ϕ). Quantum dynamics can be expressed

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in terms of an amplitude W [Σ, ϕ], that is a sum over field configu-rations that take the value ϕ on Σ.Now, even before trying to quantize, we observe in General Relativ-ity incurable pathologies. The theory predicts singularities in space-time texture, corresponding to Black Holes and Big Bang. The hopeis a spontaneous cure when we quantize GR. But the perturbativeapproach, typical of Quantum Field Theories, is ill-suited to treatgravitational interaction. The perturbation series is divergent: atevery order infinite short-distance divergences come up, and we cannot maintain predictability. In other words, any attempt to build thegraviton spin 2 Lagrangian leads to a non–renormalizable theory, aswe can observe considering power counting, i.e. the dimensionalityof coupling constant. A fundamental reason of this failure is re-markable considering the conceptual assumptions, axiomatized byWightman, under a perturbative QFT. This is incompatible withthe general relativistic framework: it relies on Poincare invariance,evolution in time is determined by a non–vanishing Hamiltonian op-erator, n–points functions W (xa, . . . , xn) are absurdly constant in atheory with diffeomorphism invariance, and the absence of a non–dynamical background metric does not permit to describe physics atshort distances. To sum up, considering a background independentQFT: diffeomorphism invariance implies that W is independent of Σand depends only on the boundary values of the fields ϕ. Gravitationseems to have idiosyncracy to quantization and the whole frameworkof Quantum Field Theory breaks down once we make gravitationalfield dynamical and there is no background metric structure anylonger. We cite one technical example of this situation: in the stan-dard approach to QFT, one introduces a Hilbert space to describestates. Its very basic structure depends on the background, sinceMinkowski metric is used to define the inner product of the Hilbertspace. In a general relativistic theory there is no background met-ric to be used at a kinematical level to define Hilbert space innerproduct. Thus, one is forced to adopt a non–standard approach orto abandon General Relativity.

1.1.4 Metric and tetrad formalism

In the first works about General Relativity, the field equations arewritten using the so–called metric formalism:

Rµν −1

2gµνR = 8πGTµν , (1.1)

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where Rµν is the Ricci tensor, R the Ricci scalar and Tµν is theclassical notation of stress–energy tensor. The field g selects at eachpoint a class of preferred frames in which the motion is ”inertial”.In this theory, a physical state corresponds mathematically to anequivalence class of field configurations under active diffeomorphism,i.e. the group acting on the space of metrics on a manifold M . Onthe other hand, the group of passive diffeomorphisms acts on thespace of tensor fields gµν(x) on Rn representing metrics under theassumption that we have given a coordinate system x on M . Sinceg does not permit to treat fermions, it was introduced the tetradfield eIµ, such that at each point x in the spacetime manifold M wehave

gµν = eIµeJνηIJ , (1.2)

where ηIJ is the Minkowski metric. Since now, whenever we usetetrad field, gravitational field and spacetime, we mean the samemathematical and physical entity. Obviously, we can rewrite Ein-stein equations in the new tetrad formalism:

RIµ −

1

2eIµR = 8πGT Iµ . (1.3)

You can observe that tensors are characterized by both latin thangreek indices (see appendix1); using the exterior calculus, this equa-tion becomes

εIJKL(eI ∧RJK) = 0, (1.4)

where Rjk is the curvature of spin connection ω, a one form ofso(3, 1) that is contracted in this way

ωij(x) = ωiµj(x)dxµ, (1.5)

so we have

RIJ = dωIJ + ωIK ∧ ωKJ . (1.6)

It is possible to define an action for this theory:

S[e, ω] =1

16πG

∫εIJKL(eI ∧ eJ ∧RKL). (1.7)

1.1.5 Selfdual connection: an alternative formulation ofGR

We present a more recent version of Einstein equations. It was devel-oped by Ashtekar, in order to cope with the problem of quantization

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of the Hamiltonian constraint, also called relativistic Hamiltonian[1], or Wheeler–DeWitt operator. Ashtekar rewrites General Rela-tivity with tools belonging to gauge theories. The fundamental fieldsare the connections and no longer the metric fields. This is the keyof the success of this formulation. In fact, for many years physicistsinsisted on trying to quantize GR written in metric formulations,e.g. ADM formalism over all. But this approach fails when theysearched for an effective technique to quantize the Hamiltonian con-straint. In particular, this one is written in a non polynomial andnon analytic form with respect to metric, so that it is not renor-malizable [3]. The selfdual formulation simplifies the constructionof the hamiltonian theory; in the next section we will see why.So we consider both Lorentz complex so(3, 1,C) and Euclidean su(2)algebra. The latter case is physically meaningless, but produce anuseful toy model for Quantum Gravity. Surely, we can observe that

spin(4) = su(2)× su(2) (1.8)

spin(3, 1) = su(2)⊗ C = sl(2,C), (1.9)

and we define a projection P± such that

P± : spin(4) −→ su(2) (1.10)

P± : spin(3, 1) −→ su(2)⊗ C (1.11)

If a principal connection is subjected to a principal morphism, wecan say that its image is a principal connection as well. In thiscase, if we consider a connection ω on a spin bundle, then we have aselfdual(antiselfdual) connection ±ω on the image of the bundlewith respect to the principal morphism, that are principal bundles.So we obtain

±ωiµ =± piIJωIJµ , (1.12)

where piIJ is the projector related to P± and satisfies

pijk =1

2εijk, (1.13)

pi0j = −pij0 =ε

2δij. (1.14)

In conclusion, we get

±ωkµ = ε3ω0kµ +

1

2εkijω

ijµ . (1.15)

In the Euclidean case (ε = 1) the connection is real, in the Lorentzianone (ε = i), it is complex. From now, we consider only the selfdual

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connection, leaving out the antiselfdual formulation because leads toidentical results. We can use this new connection as a new variablefor our theory. It is possible to prove that the selfdual component ofthe curvature R of the connection ω is the curvature F of its selfdualconnection +ω:

+Riµν = ε3R0i

µν +1

2εijkR

jkµν = +F i

µν . (1.16)

The selfdual Lagrangian is

L+ = k±piIJ±F i ∧ eI ∧ eJ , (1.17)

where the term k represents constant terms. Performing its variationwe find the field equations:

DeI ∧ eJpiIJ = 0+F i ∧ eIpiIJ = 0.

We stress again that selfdual formulation is equivalent to GR. Itis remarkable that antiselfdual connection is left unconstrained. Infact, the tetrad field is a field on Spin(4) or Spin(3, 1), while theselfdual connection is defined on SU(2) or SU(2) × C. We haveneglected half of degrees of freedom from the theory, since we canignore antiselfdual connection.

1.1.6 Barbero-Immirzi connection

For the Lorentz signature we can consider complex valued selfdualconnection:

Ai = ωi + iω0i, (1.18)

but it is difficult to impose reality condition on the measure in quan-tum theory. We want to avoid complex connection. Barbero con-sidered real valued connection also for the Lorentz signature. ThenImmirzi discovered that is possible to develop Hamiltonian formu-lation for all values of a parameter for both Euclidean and Lorentzcase. We define

γAiµ = γω0iµ +

1

2εijkω

jkµ , (1.19)

with γ ∈ R − 0. Unfortunately, except for γ = ±1 in Euclideansector, that leads again to selfdual and antiselfdual connection, thecoefficients of this connection do not transform as a SU(2) con-nection. The problem is solved, in Euclidean case, thanks to the

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construction of a new bundle for Spin(4) but reduced to the groupSU(2): on this bundle we define a new connection θ, with curvatureΘ, so we have:

Aiµ = γθ0iµ +

1

2εijkθ

jkµ . (1.20)

A similar method is adopted in Lorentz signature. In both cases, wehave defined a real field with correct transformation laws. The realparameter γ is the Barbero-Immirzi parameter, whose physi-cal interpretation is not clear, because it does not exist in classicalphysics. Perez and Rovelli demonstrate in [4] that γ is absorbedby coupling constant of fermionic theory. This formulation has thesame good properties of selfdual one, with the rather importantadvantage of describing Lorentz GR, i.e. the real physics. The La-grangian takes the form associated to Holst, that is

Lγ = 4kγΘIJ ∧ eK ∧ eLεIJKL = (1.21)

= 2k(ΘIJ ∧ eK ∧ eLεIJKL +2

γΘIJ ∧ eI ∧ eJ) (1.22)

where γΘIJ = 12(ΘIJ + ε2

2γε IJKL.. ΘKL). From this, we can derive the

following field equations:∇[µe

[Iρ e

J ]σ] + γ

2ε IJKL.. ∇[µe

[Kρ e

L]σ] = 0

ΘIJ[µνeIρ] + γ

2ε IJKL.. ΘKL

[µν eIρ] = 0.(1.23)

We leave out to prove the equivalence between General Relativity,selfdual and Barbero-Immirzi formulations. One can see the fewpassages in [5].

1.1.7 Plebanski formalism

The Plebanski formalism is derived from the selfdual one. Let usdefine a selfdual two forms:

Σi = P iIJe

I ∧ eJ , (1.24)

where P is the selfdual projector such that:

Aiµ = P iIJω

IJµ . (1.25)

It is satisfied

DΣi = dΣi + AIJ ∧ ΣJ = 0. (1.26)

23

Vice versa, one can prove that any two form Σi, which is selfdualand such that DΣi = 0, is in the form Σi = P i

IJeI ∧ eJ . We can

write the GR action as

S[Σ, A] =i

16πG

∫Σi ∧ F i + λΣjΣ

j. (1.27)

We cite this formalism because it also has applications in Spinfoamsmodels.

1.2 Quantum Gravity

During last eighty years many physicists have been engaged in con-structing an effective quantum gravity theory. We can say that theirefforts have followed three main lines of research: the covariant, thecanonical and the sum over histories one [1]. We treat in this sectionthe canonical approach, but when we will talk about spinfoams, wehave to cross the Feynman sums over all possible paths. In the firstsubsection we present the algorithm summarized by [3] that leadsto quantization of a canonical theory. After that, we outline the re-sults obtained. At the end of section, we introduce the fundamentalissues concerning the birth of Loop Quantum Gravity.

1.2.1 Canonical approach: a quantization algorithm

First of all, we need to define a suitable framework where it is possi-ble to develop a classical covariant theory. It is necessary to extendthe powerful formalism of symplectic geometry in order to treat mul-tidimensional space, considering the time variable (and other nar-row dimensions depicted by exotic theories) at the same onthologicallevel of spatial coordinates. We adopt the so called multisymplecticgeometry ([6] and [7]). Let us initially construct a space with n fieldvariables, the associated momenta and an energy variable: this isthe multiphase space. We equip it with a tensorial (n + 1)–form,the multisymplectic form, obtaining a multisymplectic manifold andgeneralized Poisson brackets, called Poisson forms.Now, we are able to develop an heuristic procedure that leads to aquantum theory. Given a multisymplectic manifold, its process ofquantization consists of the following steps

• Polarization: the space phase coordinates are divided in con-figuration and momentum variables. It implies that the wavefunctions depend only on half of original elementary variables.

24

• Quantum configuration space: the phase space is extended fromsmooth fields to distributional configurations, that defines thequantum space. The choice of the quantum space is a stochasticprocess.

• Kinematical measures: the quantum space is equipped with ameasure such that we can integrate functions of an infinite num-ber of degrees of freedom, and the space itself is a irreduciblerepresentation of the canonical commutation relation.

• Constraint operator: after that operators are defined, we con-struct constraints over them in classical theory.

• Imposing the constraints: constructing a Gel’fand triple, it ispossible solve the dynamics of the system, i.e. the constraintsin quantum theory.

• Quantum anomalies: the raise of problems related to orderingin self-adjoint operator products leads to the definition of othernon–classical constraints.

• Physical scalar product: the complete construction of a Hilbertspace introducing an Hermitean, sesquilinear non-negative prod-uct.

• Observables: the definition of self–adjoint operators respondingto classical gauge invariants. The problem is that even classicalobservables are not known! Upon Rovelli and others, evolutionof a physical quantity in a background independent theory canonly be study relative to another one. However, this point isnot cleared at all.

The main characteristics of a Canonical General Relativity theoryare the background independence and a non–perturbative formula-tion. The spacetime is broken in a 3 + 1 form and presented in aHamiltonian formulation. If we impose invariance under extendeddiffeomorphisms, that can be singular in some points of the mani-fold, we obtain a Hamiltonian constraint H that defines the Hilbertspace of the states such that

Hψ = 0; (1.28)

this is the Wheeler-DeWitt operator, i.e. the canonical evolutionof the Schrodinger equation. When we have a Lagrangian invariantunder diffeomorphism, which is the case of General Relativity, wecan apply the Legendre transformations and we get H = 0. Thisis the Hamiltonian constraint, which we want to quantize, and it

25

corresponds to the third field equation obtained in the next subsec-tion. But its quantization is a serious matter in metric formalism.So we have to use Ashtekar formulation. In fact a self-dual con-nection and a triad field form a canonical pair: we can develop thecanonical formalism based on such pair, which is the Ashtekar for-malism. Wheeler-DeWitt equation, reformulated in terms of thenew variables, admits a simple class of exact solutions: the tracesof the holonomies of the Ashtekar connection around smooth simpleloops. The Wilson loops of the Ashtekar connection are classes ofsolutions of the Wheeler-DeWitt equation if the loops are smoothand non self-intersecting [3]. The more recent works on the looprepresentation are not based on the original Sen-Ashtekar connec-tion because it is complex and imposing the condition of reality isdifficult, but on a real variant of it, whose use has been introducedinto Lorentzian case by Barbero. Now we see the concrete stepsconcerning this process.

1.2.2 Field equations

We start from the field equations obtained in previous sections andconstruct a Hamiltonian formulation of General Relativity. Let usconsider a 3d–surface in coordinate space coordinatized by τ . Defin-ing on this surface Aia(τ) and a momentum field Ea

i [τ ] such that, ina gauge with e0(τ) = 0:

Eai (τ) =

1

2εijkε

abcejb(τ)ekc (τ), (1.29)

dei + Γij[E] ∧ ej = 0, (1.30)

Aia + Aia = Γia[E], (1.31)

where Γ[E] is the Levi–Civita connection associated to E; thenAia(τ) is the projection of the 4d selfdual connection Aiµ(x). Varyingthe surface, we have a class of solutions of Einstein equations, andconsequently a class of Aia(τ), that defines a physical state. In brief,Lagrangian defines a Poincar–Cartan form with which we can definecovariant Hamiltonian function S = S[A], that depends on gaugeclass of the connection. The Hamilton function S[A] obeys to

δS[A]

δAia[τ ]= Ei

a[τ ]. (1.32)

26

We can project the field equations on the 3-surface, and we get theHamilton-Jacobi equations of the system:

DaEai = 0

F iabE

ai = 0

F ijabE

ai E

bj = 0,

where F is the curvature two–form. The first equation representsthe SU(2) gauge invariance. The second one is related to invari-ance under extended diffeomorphisms on Σ. The last one is theHamiltonian constraint (that it will be the ikonal approximation ofSchrodinger equation). These are the equations for selfdual connec-tion. To deal with Barbero-Immirzi one needs to follow an identicalprocess. The equations obtained are:

DaEai = 0

F iabE

ai = 0

εijk FkabE

ai E

bj − 2(ε2 − γ2)k

[iak

j]b E

ai E

bj = 0,

where γkia = Aia − Γia and ε = 1 in the Euclidean case or ε = iin the Lorentzian one. For complete evaluation of Barbero–Immirziformulation of the Holst Lagrangian, we refer to [8] Now there aretwo possibilities: to solve field equations and trying to quantize thesolutions, or to build a Hilbert space of physical states and to imple-ment step by step the equations as operators acting on this space.Loop Quantum Gravity follows the second path. We treat in detailthe construction of the physical states space and quantization innext sections.In next chapter we will outline the construction of the Loop Quan-tum Gravity emphasizing the merging of two lines of thinking re-garding Quantum Mechanics and General Relativity: the construc-tion of a background independent theory allows to consider loopstates as the basis of Hilbert states; at the same time, the use ofloop basis makes the control of diffeomorphism invariants possible.From this, the Loop Quantum Gravity was born: a theory basedon background independence and without the necessity to adopt aperturbative approach.

27

28

Chapter 2

Loop Quantum Gravity

2.1 General overview

In this chapter we treat about an attempt to construct a quan-tum theory of gravity. We will present a formulation of QuantumMechanics sufficiently general to deal with General Relativity, inwhich there is no more any global time variable. Divergences, whichmake impossible a consistent perturbative theory, are cured witha spontaneous discretization. Physics of small distances is ”natu-rally” cut off by the theory. In Loop Quantum Gravity we buildthe states space as done Yang–Mills lattice theories. Loop statesform a Hilbert space. The difference with lattice gauge theories isthat the lattice in LQG is a dynamical object itself and have a min-imum scale not to be scaled down to zero. This point is the keyto solve the well known problem regarding loop states: they arenot normalizable in classical theories, and, even if they naturallyrecall to Faraday intuitive definition of the fundamental concept offield, we are not used to consider them as good physical states infield theories [1]. In spite of this, Y–M gauge theories techniques,and the request of background independence, lead to LQG, a the-ory where loop states are the building blocks of normalized physicalstates. Technically, in QFT spacetime is continuous, and loop statesnecessary to span it are too many. Conversely, in LQG spacetimeitself is formed by loop-like states. A localization of a loop state ispossible only in relation to other loops states, not with respect toa fixed background: if we move infinitesimally a state we obtain adifferent physical state only if it intersect another loop state, other-wise we obtain only another element in the gauge equivalence classof the initial loop state. It is the invariance under extended diffeo-morphisms that reduces the number of loop states, and these havean intrinsic Planck thickness. This is a key point that deserves to

29

be stressed. We will adopt techniques typical of lattice theories butLoop Quantum Gravity is absolutely not a lattice theory! The needsto work in a discretized environment is not initially postulated, butit is an internal kinematical request. Moreover, a quantum gravitystate is not graphically represented by a state on a lattice, but bya quantum superposition of states defined on a lattice. These areimportant technical remarks, but the core of the question is anotherone. LQG is intrinsically different from a QFT on lattice. If wevary the size of the lattice, the theory does not change! Decreas-ing the lattice size, there is not approximation of the continuousconventional spacetime, and the accuracy of the theory does notimprove, but surprisingly the value of the eigenvalues of geometricaloperators as Area and Volume increases. The (dual of) lattice itselfis the spacetime: the theory does not provide physical interpreta-tion for lattice size at magnitude orders smaller than Planck length.Thus, LQG does not accept a smooth geometry and a continuousspacetime, and measures of geometrical quantities, i.e. measuresof the gravitational field, are subjected to quantum probabilisticlaws. Spacetime, whose geometry is expressed by the gravitationalfield, is a dynamical field, and each dynamical field is quantized atsmall scales and manifests itself in discrete quanta whose evolutionis governed by combinatorial laws.

2.2 Classical canonical theory

First of all, we search for a mathematical framework to describe bothclassical mechanics and General Relativity. This is the dynamicalsystem:

Definition 2.2.1 a dynamical system (C,Γ, f) is characterized by

• a relativistic configuration space C, mapped by partial observ-ables, arena of the events

• a relativistic phase space Γ, where states live

• an evolution equation f=0, where f : C × Γ −→ V , where V isa space vector.

A state in Γ determines a relation between observables, expressedas a k–dimensional surface γ in C. This is a physical motion. Ifk > 1 the system presents a gauge invariance. In order to constructa physical theory consistent to General Relativity, we must, first ofall, find the observables that coordinatizes C and their relationshipwith measurable quantity: this is the kinematics. After that, we

30

determine phase space of physical states Γ, i.e. of the possible cor-relations between observables. The function f expresses all of theserelations and makes possible physical prediction with the evolutionequation. This second step enables to define the dynamics of thesystem. Why we can define evolution equation as H = 0? To answerthis question, we observe that Hamiltonian H generates the evolu-tion parameter τ in the action functional of our theory. If action isinvariant under reparametrization of τ , then this is physically mean-ingless and it is not an observable. So τ is a gauge-type parameterand his generator is a constraint and vanishes on a surface Σ of thecotangent space T ∗C.Let us sketch this heuristic procedure in the case of General Relativ-ity. States are equivalence classes under local Lorentz and extendeddiffeomorphisms transformations of 4–D field configurations eiµ thatare solutions of Einstein equations. The space of these equivalenceclasses is the phase space Γ. Considering a 3–D surface σ with-out boundaries in coordinate space, let Aia(~τ) to be the connectionon σ induced by the selfdual connection Aiµ(x): a state selects agauge family of this connections compatible between themselves.We rewrite the equations presented above:

DaEai = 0

F iabE

ai = 0

F ijabE

ai E

bj = 0.

(2.1)

Reminding that:

Eai = (det e)eai (2.2)

Da = ∂a − εijkAja (2.3)

F iab = ∂Aib − ∂Aia + εijkA

jaA

kb . (2.4)

The Hamilton equation of the system is

δS[A]

δAia[τ ]= Ea

i [τ ]; (2.5)

the constraint equations can be rewritten as:

DaδS[A]δAi

a[τ ]= 0,

F iab

δS[A]δAi

a[τ ]= 0,

F iab

δS[A]

δAja[τ ]

δS[A]

δAkb [τ ]εjk.i = 0.

(2.6)

31

2.3 Kinematics

2.3.1 Hilbert space of physical states

Now it is time to follow the path we have generally described inlast section and which leads to Loop Quantum Gravity. We startdefining the space of kinematical states K. Before to do it, we needto introduce some basic ingredients. The main field of the theory isthe principal connection A on a manifold M for a group G. One ofthe most important geometrical quantity in quantum gravity is theholonomy of a connection along a curve γ:

Definition 2.3.1 Given a connection A in a group G on a manifoldM , and a curve γ in M , the holonomy U [A, γ] of A along γ is anelement of G such that:

U [A, γ] = ρ exp

∫ 1

0

dsγµ(s)Aiµ(γ(s))τi = ρ exp

∫ 1

0

dsA(γ(s)) =

=∞∑

n=0

∫ 1

0

ds1

∫ s1

0

ds2 . . .

∫ sn−1

0

dsnA(γ(sn)) . . . A(γ(s1)).

Let us stress that the element U [A, γ] ∈ G so defined depends onthe trivialization (hence on the gauge) when γ is an open curve,whilst it is gauge covariant ( though not invariant) when γ is closed,i.e. a loop. This will become irrelevant since below we shall seethat only holonomies along closed curves will enter the framework.Vectors τi form a basis of Lie algebra of G. It is remarkable that pathordered ρ defined by series expansion can be interpreted as productof holonomies on infinitesimal differentiable path. Holonomy is themathematical expression of the gauge classes of the connection A.In selfdual GR we have SU(2) connections Aia on a 3D–surface Σ,which is boundary of a spacetime region. In our manifold we canembed a geometrical substructure: a lattice.

Definition 2.3.2 An abstract oriented lattice L is defined by

• a set of n nodes N ,

• a set of l links L,

• a function f : Z2 × L −→ N .

An oriented morphism between two lattices L,L′ is a map Φ :L −→ L′ that is made of two maps φ(ϕ, ψ) where:

• ϕ : N −→ N ′

32

• ψ : L −→ L′,

and it preserves the function f , i.e.

f(s, α) = n⇒ f ′(s, ϕ(α)) = ψ(n). (2.7)

Instead of considering a lattice embedded in a manifold, we tryto represent quantum gravity on the lattice itself, viewing it as adynamical object. Parallel transport between nodes can be writtenon abstract lattices as a map Pα : G −→ G, where α ∈ L is a linkthat goes from a node n to a node m. Hence, we can label the link,which completely encodes the parallel transport along α, with anelement gα of a Lie group G.Now, let us introduce an embedded lattice Γ with l links and nnodes on M and a function f : Gl −→ R. We define a functionalΨΓ,f [A] = f(U1 . . . Ul): Uα are the holonomies of A along a link γα.Holonomies label links with an irreducible representation j of SU(2).The functionals ΨΓ,f [A] are called cylindrical functions and formthe space K. Two cylindrical functions can always be defined onthe same lattice. In fact, one can canonically extend a cylindricalfunctional on finer lattices by adding trivial arguments along thelinks to be added. The set of all cylindrical functions, when f andΓ vary, is the kinematical space K. It is possible to equip K with acanonical scalar product:

< ΨΓ,f ,ΨΓ,g >=

∫

SU(2)

dU1 . . . dUlf(U1 . . . Ul)g(U1 . . . Ul), (2.8)

where dU is the SU(2) Haar measure. Let us stress that this in-ner product is defined without using any background metric on Σ.Therefore, K is a Hilbert space, and consequently we can define theGel’fand triple S ⊂ K ⊂ S ′, where S is the test functions subsetthat is dense in K, and S ′ is the space of the distributions, i.e. thedual of S. Now, we want to find a basis of K. Let us fix a partic-ular lattice Γ. The set of the cylindrical functions on Γ is denotedby KΓ, and K ′Γ = L2[Gl] is the configuration space of the latticetheory on Γ. The Peter–Weyl theorem claims that a set of finite–dimensional irreducible representations of Gl forms an orthonormalbasis of functional space L2[Gl]. Thus, we can define elements ofthis basis as

|Γ, j1 . . . jl, α1 . . . αl, β1 . . . βl >= |Γ, jl, αl, βl >,(2.9)

where Γ identifies a lattice, Rji , with ji 6= 0, are the non–trivialirreducible representations of Gl and αi, βi terms span on the range

33

of ji. A basis of K is obtained varying lattice Γ. One can prove thatomitting the trivial representation j = 0 is necessary and sufficientto avoid redundancy due to the fact that when γ ⊂ γ′ then Kγ ⊂Kγ′ . If |A > is a state of K, we have

< A|Γ, j1 . . . jl, α1 . . . αl, β1 . . . βl >= Rj1(Uα1β1

) . . . Rjl(Uαlβl

). (2.10)

These states are neither normalized nor countable. If we want apredictive quantum theory, we have to search for vectors that satisfythese two features. First, we look for normalized states in K. Theseare the loop states, concerning to graph formed by closed simplecurves:

Ψα[A] = TrU(α,A), (2.11)

|Ψα|2 =

∫

SU(2)

dU |TrU |2 = 1. (2.12)

The functional on loop states is the loop transform:

Ψ[α] =< Ψα|Ψ > . (2.13)

It is remarkable that adding an extra link at a lattice does notmodify the information on parallel transport, because it would belabelled by j = 0. This fact causes redundancy between states of K.We must consider only states |Γ, jα, αα, βα >, with jα 6= 0. These

vectors generates subspace KΓ of K, such that

K =⊕

Γ

KΓ. (2.14)

The mathematical meaning of this result is that states |Γ, jα, αα, βα >are an orthonormal basis of K, but they form an uncountable set ofvectors; then, K is a non–separable space. An infinitesimal pertur-bation of the lattice sends a state in an orthogonal one. This factof having an uncountable basis does not warrant to use K as spaceof physical states. However, we still have the constraint equation tosolve. if we consider a subspace K0 ⊂ K formed by SU(2)–invariantstates, we discover that loop states can generate it. We implementfirst field equation in K: states in K ′ are just solutions of the firstequation of our dynamical system, namely:

Daδ

δAai= 0. (2.15)

We have reduced K to its subspace of functionals invariant undergauge transformations. We construct a second triple S0 ⊂ K0 ⊂ S ′0.In K0 the multiloop states live:

Ψα[A] = Πiψαi[A] = ΠiTrU(αi, A). (2.16)

34

This is a state consisting of a family of n loop states α = (α1 . . . αn).The loop state are normalizable in a Yang Mills theory on lattice,but not in a continuous theory, because in this case a finite dis-placement of the loop always sends the physical system in a newstate. Nevertheless, it is not a problem because of decomposition ofK as space spanned by KΓ. Loop representation is a sort of Fouriertransform of connection; a connection A can be written as:

Ψ[α] =

∫dµTrU(A,α)Ψ[A]. (2.17)

Multiloop states are a complete set of vectors, but they are notlinearly independent. We want a basis with elements linearly inde-pendent between each other. For this reason, it is useful to considersuperposition of states |Γ, j, α, β >:

Definition 2.3.3 A spin network state is defined as

|S >= |Γ, jα, in >, (2.18)

where Γ is an oriented lattice embedded in a manifold, jα labelsirreducible j–representation (j 6= 0) associated to each link α, andin is an independent rule of permuting, i.e. an intertwiner, assignedat each node n.

We will dedicate a subsection to intertwiners below, but at thismoment we can briefly define them as elements of a fixed basis ofinvariant space of tensor product of irreducible representations car-ried by links converging at a node. Spin network states are relatedto kinematical states in K in this way:

|Γ, jα, in >=∑

α∈L

|Γ, jα, α, β > Πn∈N(in)β1...βnα1...αn

. (2.19)

It is worthwhile to spend few words for intertwiners. At each nodewe have incoming and outgoing links. Commutation relationshipsbetween links adjacent at the same node are defined by the choiceof an intertwiner. Group representation of SU(2) are labeled byan integer j. If j = 0, the representation is called trivial, becauseall vectors on which the group acts are sent in the null vector. Weknow that it is possible to factorize a tensorial product of represen-tations with a sum of irreducible representations. Indicating Rj anirreducible j–representation, we have:

⊗

ji

Rji =⊕

k

Rjk . (2.20)

35

We are interested in cases when a trivial representation jk = 0appears in this sum, corresponding to the existence of an invariantsubspace of the vector space that we are considering. For example:

R12 ⊗R 1

2 = R0 ⊕R1. (2.21)

This occurrence is related to the satisfaction of the Clebsch–Gordanconditions, mathematical relations involving the indices j of therepresentations. Considering n irreducible representations Rji , eachof them acting on a vector space V i. A basis of V =

⊗i V

i ise1...n =

⊗i ei, where every ei is a basis of V i. If the integers ji sat-

isfy Clebsch–Gordan conditions, then V has an invariant subspaceW . An element of a fixed basis of W is called an intertwiner in. Welabel each vertex of a spin network with an intertwiner: it carriesthe information about commuting relationships between incomingand outgoing links of the node, i.e. between the representations ji.First, we determine invariant vectors for each possible commuta-tion, and therefore we have multiloop states, that were found to benot linearly independent; then, we define a basis of W , and so wefind intertwiners and a spin network, that are linearly independent.Which are these invariant vectors? For example, the only two in-variant vectors under spin 1

2SU(2) transformations are εAB and δAB.

In a next chapter we will present brief calculation of spin networksfunctionals. The index of each intertwiner is contracted with theindices of representations of adjacent links. We can rewrite spinnetworks as functionals of connections:

Ψs[A] =< A|s >≡⊗

l

Rjl(U(A, γl)) ·⊗

n

in. (2.22)

Spin networks form an orthonormal basis of K0. They are finitelinear combinations of multiloop states. In fact, representations jcan be seen as symmetrized product of 2j fundamental representa-tion. Loop states are nothing but spin network states representedby graphs with no nodes and labelled by the fundamental represen-tation j = 1

2so that gravitational field is just on the loop and null

otherwhere. In this basis, intertwiners are reduced to linear combi-nations of εAB and δAB, that graphically join links to closed curves,i.e. to loops. Thus, spin networks are linear combinations of mul-tiloop states, represented as traces of product of holonomies alongclosed lines. See Chapter 4 for examples and calculations.Now, we observe that spin networks basis is still uncountable. Loopstates in Yang Mills lattice gauge theory lose normalizability whenwe send the theory to the continuum limit. This problem is over-

36

come in LQG by constructing states invariant for extended diffeo-morphisms. We remark that in lattice Yang–Mills theories there isnothing like diffeomorphism invariance to be required and the the-ories need a cut off to be imposed to obtain normalized states. Thecut off is an alien ingredient, necessary because of technical reasons:in principle, one should show that every physical result is indepen-dent of the cut off chosen. This is nothing like what happens inLQG as we shall discuss below.Spin networks are not invariant under diffeomorphisms. This oper-ation can act on |Γ, jl, in > modifying graph, orientation or orderingof links. Under gauge transformations, a connection A transformsin not homogeneous way; on the other hand, its holonomy alonga generic path γ (in this case along a link) does it. If the path isa loop, then holonomies transforms with their adjoint representa-tion. Moreover, we must remember that intertwiners are invariantvectors. Now, we take a gauge transformation on Σ, Φ : Σ −→ Σ.Connection A transforms as Φ∗ : A −→ Φ∗A, and functionals Φ onS ′0 as Φ −→ UψΦ. Naively speaking, this is an extended diffeomor-phism. There are some technical issues about singularities allowedin Φ. An extended diffeomorphsm can be singular in finitely manypoints identified with nodes. This has important consequences onthe functional spaces appearing in the sequel, ensuring that they areenough big but not too much. Since the previous considerations, wecan claim that we have gauge invariant states if:

< Φ∗A|Γ, jα, in >=< A|Γ, jα, in >, . (2.23)

and they solve the second field equation of our theory:

F iab

δΨ

δAai= 0. (2.24)

In S ′0 diffeomorphism invariance is expressed by

(UψΦ)(Ψ) = Φ(Uψ−1)(Ψ), (2.25)

Φ(UψΨ) = Φ(Ψ). (2.26)

Diffeomorphism invariant states on S ′0 form the space Kdiff . A keypoint is maintaining not too strict conditions for the states. Ex-tended diffeomorphism is a map that could be singular at a finitenumber of points. But the most crucial fact is that there exist twotypes of diffeomorphism: the one that change the graph Γ, the othermodifies only ordering or orientation. The latter come in finite num-ber of results, so there is a discrete group of states obtained by theinitial spin network |S >. If the graph Γ is not deformed by this map,

37

then a spin network state could be send only in a finite number ofstates, precisely in the same spin network except for ordering or ori-entation: its imagine is a discrete set. Conversely, a graph deformedimplies that the spin network is mapped in a state orthogonal to it.If we chose diffeomorphism as it is used in physics, we would have anon–separable Kdiff , because we have a continuous, uncountable setof states, one for every possible deformation of graph. There is notenet that deny this possibility, but we will construct operators, andconsequently we will depict a physical framework, which they are notaffected by deformation of graphs, but are sensitive to the colouringof nodes and links. For these reason, in LQG we work with extendeddiffeomorphisms. Spacetime is discrete and non–continuous, so, an-alytically, must be a distributional and non–smooth object, and wemust use extended diffeomorphisms. It is possible to quotient spaceK0 with respect to extended diffeomorphisms. An equivalence classis called knot and it represents an unoriented graph. In the sub-space KK of Kdiff formed by states of the same equivalence class,i.e. the same knot, we can distinguish states each other by colours oflinks and nodes, that is, indices jl and in. We write as |s >= |K, c >the states in a knot K characterized by a colour c of links and nodes.These states form an orthonormal basis of KK , and are called spinknots states. The most important property by far of knots is thatform a discrete set, and so Kdiff admits a countable orthonormalbasis formed by |s > states: it is a separable Hilbert space. Knotswith nodes results to be a discrete set because we have chosen ex-tended diffeomorphisms. All this procedure is about a process ofdiscretization! A graph is initially embedded in a manifold. Thediffemorphism invariance enables to construct class of equivalencesof graphs obtained deforming into each other, so there is a reduc-tion of state and the collapse of the manifold in the graph itself.Equivalence classes of embedded spin networks are spin knots, anda quantum state of space is determined by a superposition of spinknots. Divergences are intrinsically removed by the theory itself. Innext section, we will observe that there is a physical reason underthis wise choice of the gauge group. To summarize, we have startedwith kinematical space, viewed as set of possible boundary statesof our theory, i.e. the set of possible initial and final data, or in afield theory, of the value of fields configurations on the boundarysurface Σ; the implementation of the fields equations has restrictedinitial space in a set of vectors as solutions of two field equations ofthe system, that is, in a separable Hilbert space. Now, it is time todefine operators in Loop Quantum Gravity.

38

2.3.2 Operators

An operator is the mathematical expression of a measurable quan-tity of our theory. The study of a physical system consists of thedefinition of a complete set of commuting observables [9] and howthey act on solutions of field equations. In quantum gravity, it isnatural to identify as observables common geometrical quantities.This is the lesson of General Relativity: geometry of the system isnot a priori [2], but it is related to the interaction of system withgravitational field. The geometry of spacetime is not a static in-stance, but it is a dynamical one, because gravitational field itself isa dynamical object.Considering a 3d–surface σ in the configuration space, parametrizedby ~τ , we look for the area of a generic 2–surface S in σ. We re-call definition of Plebanski form and the gravitational electric fieldassociated:

Σi = P iIJe

I ∧ eJ , (2.27)

Σi(~τ) = Σiabdτ

a ∧ dτ b, (2.28)

Eai(~τ) = εabcΣibc(~τ). (2.29)

The area of S is the flux of the norm of E across S:

A(S) =

∫

S

|E|; (2.30)

immediately, we get the volume of a 3d–region R:

V (R) =

∫

R

d3τ√|detE|. (2.31)

Our task is to connect these geometrical quantities with the observ-ables of the theory, represented as operators on the Hilbert spaceKdiff introduced above.

Operators in K

The two fundamental fields in loop quantum gravity are the connec-tion Aia(τ) and its conjugate momentum Ei

a(τ) defined in previoussection, and related to the gravitational field. Quantum Mechanicsis constructed assuming that each classical physical quantity admitsa relative quantum observable obtained in view of the correspon-dence principle. We can apply this procedure in this case, anddefining two field operators on quantum gravity states, i.e. func-

39

tionals Ψ[A], such that

Aia(τ)Ψ[A] = Aia(τ)Ψ[A], (2.32)

Eai (τ)Ψ[A] = −i~ δ

δAia(τ)Ψ[A]; (2.33)

I have left out dimensional terms, considering relations in naturalunits. A painful problem is that the action of these operators sendsfunctionals out of space K! In order to have a consistent kinemat-ical theory, we need to take suitable functions of these naive fieldoperators. Instead of Aia, we have an operator Tα = TrU(A,α) suchthat

Tα|S >= |S ∪ α >; (2.34)

therefore Tα acts on spin network states and adds a loop α in thefundamental representation j = 1

2. In general , any cylindrical func-

tion of the connection is an operator well defined in K.Unfortunately, defining a good operator starting from the conjugatemomenta E is much more difficult. I resume the main steps of theprocess, referring to [1] for a beautiful graphical explanation. Letus remind that:

U(A, γ) = ρ exp

∫ 1

0

dsγµ(s)Aiµ(γ(s))τi, (2.35)

it is possible to show that

δ

δAia(x)U(A, γ) =

∫ 1

0

dsγa(s)δ3(γ(s), x)[U(A, γ1)τiU(A, γ2)],(2.36)

where s is a parametrization of the loop γ, the factor γa(s) arecoordinate expression for the loop and γa(s) is the tangent vectorat s to the loop; the point x cuts the loop in two segments γ1

and γ2. Action of E on holonomies produces a two dimensionalsmeared distribution; thus, we look for a two dimensional operatorcontaining E. Now, we take a 2D surface S in Σ parametrized by:(σ1, σ2) −→ xa(σ1, σ2). Let us consider the operator

Ei(S) ≡ −i∫

S

dσ1dσ2εabc∂xb(~σ)

∂σ1

∂xc(~σ)

∂σ2

δ

δAia(x(~σ)). (2.37)

40

How it acts on holonomies? If we assume that γ intersects S onlyat a point p, separating γ in two segments γ1 e γ2, we obtain

Ei(S)U(A, γ) =

∫

S

dσ1dσ2εabc∂xa(~σ)

∂σ1

∂xb(~σ)

∂σ2

δ

δAic(x(~σ))U(A, γ) =

= −i∫

S

∫

γ

dσ1dσ2dsεabc∂xa(~σ)

∂σ1

∂xb(~σ)

∂σ2

∂xc(~σ)

∂s×

× δ3(x(~σ), x(s))U(A, γ1)τiU(A, γ2).

This integral always vanishes except if there is intersection betweenγ and S. We notice that under integration there is a Jacobian thattransforms integration variables. Geometrically, it is a wedge prod-

uct, so if vectors ∂x(~σ)∂σ

are parallel each other it goes to zero: thisis the case of a curve γ laying on S. This integral is coordinateindependent and can result in the values ±1, discriminating the twopossible orientations of γ. Accordingly, we simply obtain, reintro-ducing ~:

Ei(S)U(A, γ) = ±i~U(A, γ1)τiU(A, γ2). (2.38)

For an arbitrary number of intersections and a generic SU(2) rep-resentation, we have

Ei(S)Rj(U(A, γ)) =∑

P∈(S∩γ)

±i~RjU(A, γ1)jτiRj(U(A, γ2)).(2.39)

The key point here is that, while Tα defined by (2.37) is a ”onedimensional operator”, Ei(τ) is mathematically a vector density, atwo form that it must be integrated over a surface, and this time itis its surface integral to be a well defined operator on K:

Ei(S) =

∫

S

Ei. (2.40)

Area and Volume operators

We have to reduce the Hilbert space where our operators act to K0.Noticing that a cylindrical function defines an operator, it is easyconclude that gauge invariant cylindrical functions give operatorson gauge invariant states. In this way we can affirm that Tα is welldefined on K0. Unfortunately, Ei(S) is not gauge invariant. We tryto solve this problem contracting the i index, that is responsible oflack of gauge invariance:

E2(S) ≡∑

i

Ei(S)Ei(S). (2.41)

41

Here the Cartan–Killing metric on su(2) is used to contract; it existssince SU(2) is a semisimple Lie group. We can consider the actionof E2(S) on spin networks |S >, when the surface S intersects thegraph of |S > (with the prescription that there are no nodes of thespin network on S):

E2(S)|S >= −j(τi)j(τi)|S >= ~2j(j + 1)|S > . (2.42)

If we had more than one intersection between the spin networkand the surface, we would loss gauge invariance. However, we candivide S in N small surfaces Sn such that they become smaller andsmaller increasing N . From this assumption, and by observing that∪nSn = S, and we can define a new operator:

A(S) ≡ limN−→∞

∑

n

√E2(Sn)|S >= −j(τi)j(τi)|S >= ~

√j(j + 1)|S > .(2.43)

The trick consists of considering operators that acts on infinitesimalsurfaces Sn. We note that in general one cannot define the squareroot of operators on the Hilbert space, because of the lost of lin-earity; here we are using the fact that we have operators which areexplicitly diagonal on the basis of spin networks. We can say thatthere is at most one intersection between the graph Γ of |S > andSn. In the limit N → ∞, we identify infinitesimal surfaces withthe points of intersection. Remembering the action of E(S) on spinnetworks given by (2.42), we have:

A(S)|S >= ~∑

P∈(S∪Γ)

√jp(jp + 1)|S >, (2.44)

where jp is the representation assumed on the link that crosses Sin p. Therefore, we have found an SU(2) invariant and self–adjointoperator A(S), diagonalized by spin network states. This is thequantum operator associated to the classical quantity

A(S) =

∫

S

d2σ√nanbEa

i Ebi . (2.45)

The spectrum of A(S) is discrete: for n arbitrary and i = 1, . . . , n,the main sequence is

A = ~∑

i

√ji(ji + 1). (2.46)

This is an amazing result: the physical area is quantized and canassume values allowed by the spectrum of the self–adjoint operator

42

A(S). Inserting dimensional constants and generalizing our theorywith the Barbero–Immirzi parameter γ, we have:

A(S) = 8πγ~Gc−3∑

i

√ji(ji + 1). (2.47)

The lowest Area eigenvalue, i.e. the smallest measurable value of anarea, finds the natural location in Planck scale:

lPlanck ∼ 10−33cm, (2.48)

Aj= 12∼ APlanck = l2Planck ∼ 10−66cm2. (2.49)

In [1] a complete calculation of the spectrum is performed, includ-ing the presence of nodes on the surface S. We only sketch thisgeneralization. We can restrict to nodes on the surface; if a linkintersects the surface, we can always add a trivial node by splittingthe link into two points. We then distinguish links up that leavetransversely the nodes from one side of the surface, links down, thatleave transversely the nodes from the other side of the surface, andlinks tangent, with adjacent links leaving the node tangentially inthe surface; we assume respectively representations ju, jd, jt. Thecomplete spectrum of Area operator is defined by action on spinnetworks, and in this notation we have

A|S >= 4πγ~Gc−3∑

u,d,t

√2ju(ju + 1) + 2jd(jd + 1)− 2jt(jt + 1)|S > .(2.50)

We stress that having an eigenvalue in the degenerate sector meansto consider area of surface that cuts link precisely at the node, carry-ing a geometrical information under the Planck scale. It is not clearat this time, but maybe this point tells us that degenerate sector isnot physical, and can be removed by a different regularization of theArea operator [1]. In Loop Quantum Gravity we have a minimumnon vanishing value of geometrical quantities as area and volume,naturally removing problems related to high energy (and thus smalldistances) physics.There is another way to construct an effective SU(2)–invariant Areaoperator. We mention this algorithm because it is suitable to ex-tension to higher dimensions. We embed a ”straight” path γrs, withend points r and s, in S. We can introduce the operator

T ab(r, s) = T abγrs= Ea

i (r)Ebj (s)R

1(U(A, γ))ij, (2.51)

then we define another operator

E2(S) =

∫

S

d2σd2σ′na(σ)nb(σ′)T ab(σ, σ′). (2.52)

43

In the limit of small surface, this operator is equivalent to the E2(S)defined above. Graphically, T abγrs

adds a node at each intersection ofthe spin network with S, in this case at points r and s, and a link γrsin representation j = 1. The normalized intertwiner outputs from

j(τi)αβ = nj(ii)

αβ , (2.53)

(nj)2 = Tr(j(τ i)j(τi)) = j(j + 1)(2j + 1). (2.54)

We resume the last steps in Figure 2.1, postponing to the Chapter4 further details about calculations. We can extend these discussionat the case with three points of intersection r, s, t plus a point x onS connected with them (Figure 2.2):

T abc(x, r, s, t) =1

3!εijkE

al (r)Eb

m(s)Ecn(t)R1(U(A, γxr))

il ××R1(U(A, γxs))

jmR1(U(A, γxt))kn (2.55)

E3(S) =

∫

S

d2σd2σ′d2σ′′na(σ)nb(σ′)nc(σ

′′)T abc(σ, σ′, σ′′). (2.56)

In the limit such that the points of intersection collapse in x, weobserve that

T abc(x, r, s, t) = 2εijkEai(x)Ebj(x)Eck(x) = 2εabc... detE(x). (2.57)

Now, we adopt again the procedure of quantization used by Areaoperator. We consider the volumes Vε of infinitesimal regions Rε asa partition of a three dimensional region R. The Volume operatorV(R) is defined by

V(R) = limε→0

Vε(Rε). (2.58)

It is remarkable that topology where the limit is defined is the samethat rules convergence of functionals of smooth connections A. Wecan also note that a small region can have three points of intersectionwith a spin network only if it contains a vertex of the spin networkitself, and volume operator can only act on it depending on theinterwiner associated to the vertex:

V(R)|S > = limε→0

∑

n∈(S∩R)

Vε(Rε)|S >,

V(R)|Γ, jl, i1, . . . , iN > = (16π~G)32R

i′nin|Γ, jl, i1, . . . , i′n, . . . , iN > .

To generate a non–vanishing volume, the nodes need to be at leasttetravalent [10]. But it is not a surprise, because a polyhedron musthave at least four faces! This point will be clear when we will treat

44

link:=

j

Ei(r)

r s

j = 1T (r, s) = Ei(r)1RijEj(s) :=Ei(r) := jτi

j

= r

j

T (rs)

j

= h2jτijτj

r

j j

j = 1

If r = s then we can approximate

j

r s

j = 1

=r = s

j

1s

==

j

j

j

1

Finally, we have

E2(S)

j

= Tr(jτijτj) = h2j(j + 1)(2j + 1)

1

j

j

j

j

1 1j

j

j

= h2j(j + 1)

j

and

A2(S)

j

= const.h2j(j + 1)

j

h jτi r

j

r

Figure 2.1: Area from T operator

45

r s

t

x

Figure 2.2: T (x, r, s, t) Operator

the spinfoams. The Volume operator is diagonal in a suitable basisof intertwiners and it has discrete spectrum.In conclusion, if one has to believe that spin networks representsspacetime at Planck scale, then physical space is discretized, par-titioned in Planck scale cells. The graph Γ is the dual lattice to aspacetime triangulation: each node on Γ corresponds to a quantaof volume, each link to a quanta of area. Two nodes connected byl links are the graphical representation of two volumes of physicalspace separated by a surface whose area is given by

A = 8πc−3~G∑

l

√jl(jl + 1). (2.59)

The quantum numbers of Area operator are the colours jl of thelinks; on the same hand, the intertwiners associated to nodes arethe quantum numbers of the Volume operator. In literature thereis another equivalent formulation of Volume operator, realized dis-criminating nodes with or without links on the same plane. In thisformulation, there is invariance under diffeomorphisms, but not un-der extended diffeomorphisms [3].

2.3.3 A final view on kinematics

In this subsection we collect some discussions about physical inter-pretation of the framework introduced in LQG kinematics and somecritics which have been moved to it. These are not necessary to fol-low and are here briefly sketched without the aim of being complete.The arguments are heuristic both because we are just touch on themand because of their nature.In LQG a physical state is a superpositions of spin networks as theelectromagnetic field is a superposition of n–photons states. Quantaof area and volume correspond to photons energy and the spectrum

46

of area and volume to the momentum of photons. The passagefrom spin network to spin knot involves invariance under diffeo-morphisms: physically, with spin knots we neglect localization aspositioning of the graph on a 3D–manifold, and we regard to lo-calization as relational concept: links are quantum excitations ofthe space, that determine spatial relation between quanta of volumerepresented by nodes. A question can be asked at this point: areaand volume are not gauge invariant quantities: how is it possiblethat the Area and Volume operator are well defined in quantumgravity? The answer is found following the analogy with classicalphysics. Indeed, a surface in General Relativity is not only definedby the analytic expression S : (σ1σ2) −→ xa(σ1σ2), but by a couple(S, g): if we do not define a metric g on S, we are not able to deter-mine a diffeomorphism invariant geometrical object called the ”areaof S”. Relativistic diffeomorphism invariant observables are notdefined as instantaneous operators, describing states for a certainvalue of the parameter of evolution or, in covariant physics, of thespacetime metric. On the contrary, they are operators that encodetheir evolution, i.e. just as defined in Heisenberg picture, resultingdiffeomorphism invariants. However, we treat an area and surfacesin a naive non–invariant formulation because it does not affect thespectral properties of geometrical operators and physical meaning ofthe theory. In conclusion, if we fix S and calculate (S, g), we do notexpect the same result for calculation of (φ∗S, g), but for (φ∗S, φ∗g).We observe that in LQG spin knots substitute the geometries andspin networks replace the metrics. It is possible to demonstrate thatthe geometrical classical quantities as surfaces and volumes can beinterpreted as the low energy limit of LQG quantum operator. Onthe other hand, in the infrared region an increased density of thelattice not only does not improve the accuracy of the theory, butalso it happens exactly the contrary! Unfortunately, without a solidphenomenology we ignore whether the relational interpretation andits implications are correct and physically meaningful.Lastly, we consider the problem of Lorentz invariance. Spin net-works states are not invariant under Lorentz transformations, butit is possible that physical states, that are quantum superpositionof spin networks, are. Moreover, we can note that geometrical mea-sures of lengths, areas and volumes are Lorentz invariant in a quan-tum theory. If an observer measures angular momentum Lz of asystem, a second observer, rotated by an angle α with respect tothe first one, measures an angular momentum L′z, and he finds thesame eigenvalues of Lz, but with a probability distribution withmean value shifted by cosα with respect to Lz. Obviously, it is not

47

an accident that this transformation affects the quantity related toclassical physics. The rotation changes the expectation value, notthe single measure. On the same hand, this line of reasoning allowsone to say that the result of a measure of a geometrical observableby an observer is not boosted with respect to the result of a measuredone by a boosted observer, because the observers are consideringdifferent operators. In particular, they are considering the samegeometric operator but at different events separated by a timelikeinterval. Quantum field theory teaches that in this case the twooperators do not commute and the two observers are consideringoperators equipped with different eigenstates, even if with the samespectrum. For further details about this see [15]

2.4 Dynamics

2.4.1 The quantum Hamiltonian operator

It is now time to discuss dynamics in Loop Quantum Gravity. Wehave seen that the dynamics of a system in a relativistic Hamiltoniantheory is governed by the Hamiltonian constraints. Indeed, we havedealt with kinematics considering solutions of the first two fieldsequation related to SU(2) and extended diffeomorphism invariance;dynamics is then described in terms of solutions of the third fieldequation:

εjk. . iFiab

δΨ

δAja

δΨ

δAkb= 0. (2.60)

The first step consists to define a suitable Hamiltonian operatorH. This strategy is difficult to be implemented; technically, it isquite hard to define the square of the operator δ

δAi in the first place.Hence, we start on Kdiff representing the Hilbert space and thenconsider spin knots in its kernel with respect to the classical fieldequations. Parametrizing with ~τ the usual 3D–surface embedded ina 4D–manifold coordinatized by x, we have:

F ijab(~τ)Ea

i (~τ)Ejb (~τ) = H(~τ) = 0. (2.61)

It is convenient to rewrite H in a different form, by the way aswe already did for the area to be quantized. Since Ea

i (~τ) is theconjugate momentum of the connection Aia(~τ), we have the Poissonbracket

Aia(~τ), Eai (~τ ′) = const.δbaδ

ijδ

3(~τ , ~τ ′). (2.62)

48

Recalling that the volume is defined as

V =

∫d3x√detE(x), (2.63)

we obtain

V,Aia(x) = const.3Eb

j (x)Eck(x)εabcε

ijk

2detE(x). (2.64)

This result leads, after few calculations, to rewrite Hamiltonian con-straint H as

H[N ] =

∫NTr(F ∧ V,A) = 0, (2.65)

where N is a lapse function. Note that the determinant at thedenominator has been removed. This is how to treat the selfdualformulation, and consequently the Euclidean unphysical Hamilto-nian constraint. In case of the Lorentzian physical operator, onehas

HLorentz = const.T r([Ka, Kb][Ea, Eb])√

detE(x)−H[N ] (2.66)

where Kia = Aia−Γia. Thereafter, we shall consider Euclidean version

of the operator only, because it is technically simpler and the Lorentzcase can be treated similarly. For a quantization of HLorentz see [3].Applying the correspondence principle, in the quantum theory wehave

H[N ] =i

~

∫NTr(F ∧ [V, A]) = 0. (2.67)

where the commutator replaces Poisson bracket and V is the quan-tum Volume operator; this result holds ∀N . We can represent curva-ture F and the connection A by quantum operators, written in termsof holonomies. For this, we expand holonomies in series. Consider-ing a point x in a 3D–surface S, a tangent vector u at x, and a pathγxu of length ε starting in x and tangent to u. The correspondentholonomies can be expanded as

U(A, γxu) = 1 + εuaAa(x) +©(ε2). (2.68)

Now, we insert another tangent vector v at x of length ε. Holonomyon the triangular loop αxuv is

U(A,αxuv) = 1 +1

2ε2uavbFab(x) +©(ε3). (2.69)

49

By using the notation hγ = U(A, γ), by partitioning the surface inm infinitesimal regions R of volume ε3 where acts a well definedquantum Volume operator V (Rm) and by considering that Nm =N(xm), we write the quantum Hamiltonian constraint as

H[N ] = − i~

limε→0

∑

m

NmεijkTr

(hγ−1

xmukhαxmuiuj

[V(Rm), hγxmuk

]).(2.70)

We have to choose in each region Rm a point xm, three tangentvectors ui, the path γxuk

and the loop αxuiuj, in order to have an op-

erator H which turns out to be not trivial and invariant under gaugetransformations and diffeomorphisms. We remark that volume op-erator vanishes at n–valent nodes with n < 4, while hγxmuk

sendsa n–valent node in an n + 1 one. Therefore, the sum over regionsRm reduces to a sum over regions containing nodes with n ≥ 3, i.eto a sum over regions Rn where nodes of spin networks states arelocated. Defining Hε the operator defined using this reduced sum:we have

H|S >= limε→0

Hε|S > . (2.71)

The key point is that this limit exists in the subspace of diffeo-morphism invariant states! Once again the regularization, i.e. thecontrol of the theory at small distances, is obtained thanks to dif-femorphism invariance. Let us in fact consider a three valent nodeat xn: the vectors ul, ul′ul′′ tangent to the three links l, l′, l′′, thepath γxul

and the loop constructed with two tangent vectors at l′, l′′

and a straight coordinates line that connects l′, l′′. If Ψ ∈ Kdiff thelimit is reduced to

(HΨ)|S >= Ψ(H|S >) = limε→0

Ψ(Hε|S >). (2.72)

Graphically, H adds to the graph two sides of length ε and an ex-ternal arc (see Figure 2.3). For the values ε < εm, where εm is the

12

j

j′

j′′

nD+−

n′′

n′

j′ + 12

j′′ − 12

j′′

nj

j′

=

Figure 2.3: Operator H: action at a trivalent node

value that implies that added paths cross other links and nodes on

50

the graph, the limit is independent of the value of ε. It does notmatter the dimension of the triangle superimposed, it suffices thatit remains a triangle! We can transform a triangle in another onewith different size, i.e. a value ε in another value of ε, with a dif-feomorphism. Hence, this transformation is internal at a class ofdiffeomorphisms and it does not change the state: the limit is finiteand well-defined:

limε→0

Ψ(Hε|S >) = Ψ(H|S >). (2.73)

There is no possibility of divergences at small scales, and the theoryis simply finite; anything is reduced to finite combinatorics. If wesend the regulator ε from a value ε < εm to a value ε′ < ε, we makea gauge non–physical transformation.Finally, let us present a topological remark. Regarding nodes withn > 4, there are different coordinate-dependent ways in which theexternal arc connects links. These ways are the homotopy classes oflines going from the north to the south pole of a n−2 sphere: in orderto obtain full covariance, we have to add sum over these classes inthe definition of H, choosing for each of r-th class a representativeelement αrx,l,l′,l′′ , with r = 1, . . . , Nn. Thus, the final form of thequantum Hamiltonian operator acting on a spin network state is

H|S >= − i~

∑

n∈S;l,l′,l′′,r

Nnεll′l′′Tr(hγ−1

xnlhαr

xnl′l′′[V(Rn), hγxnl

])|S > .(2.74)

It is important to notice that H is not diffeomorphism invariant, andH|S > is not in Kdiff . But we do not care about it: the solutionsof HΨ=0, i.e. the physical states, are in Kdiff , because they belongto the kernel of H, which, how we have seen, is a finite operatorin Kdiff . This situation is identical to classical physics, where H isnot diffeomorphism invariant (it is not even a Lorentz invariant!) butH = 0 works as equation for diffeomorphism invariant equivalenceclasses of solutions.

2.4.2 Matrix elements

Quantum field theory teaches us that the physical information isencoded in elements of a scattering matrix. We follow an iter thatwill lead us to an analogue situation in quantum gravity.We have seen that the physical states are the spin knots |s >. Con-sidering the definition of H in terms of the curvature F and gravita-tional electric field E, we observe that H grasps two spin networksand create a holonomy, implying the insertion of a loop next to the

51

nodes. In general, H action is restricted to nodes, is discrete andcombinatorial, changing topology of graph, sending a node in threenodes and multiplying the state by a number depending on quan-tum numbers around the node. In particular, the action of H on aspin knots produces the following results:

• a sum of terms for each knot, one for each node;

• a sum of terms for each node, one for each triple of links arrivingat the nodes;

• for each triple, a term for each permutation of the links l.l′.l′′,which we call operator Dn,l′,l′′,r,±,±, and it acts at a trivalent asdepicted in Figure 2.4.

12

j

j′

j′′

n)

n′′

n′

j′ + 12

j′′ − 12

j′′

nj

j′

=D+−(

Figure 2.4: D+− at a trivalent node

This operator acts at each node n in this way:

• it creates two nodes n′, n′′ at a finite distance from n along linksl′, l′′;

• it creates a new link labelled by j = 12

connecting n′, n′′: thisis the arc;

• it sends colours j′, j′′ of l′, l′′ in j′− 12, j′′+ 1

2(in Lorentz theory

it is possible to outcome also different shifts of j′, j′′);

• it modifies intertwiner of n, substituting it with an intertwinerfitted for the new colours of links.

We write

H|S >=∑

n∈S

Nn

∑

l,l′,l′′,r

∑

ε,′ε′′=±

Hn,l′,l′′,ε,′ε′′Dn,l′,l′′,r,ε,′ε′′ |S >, (2.75)

52

and the matrix elements for a transition from |S > to |S ′ > is givenby

< S ′|H|S >=∑

n∈S

Nn

∑

|Ψ>=UΨ|S′>

∑

l,l′,l′′

∑

ε,′ε′′=±

< Ψ|Hn,l′,l′′,ε,′ε′′Dn,l′,l′′,,ε,′ε′′ |S > .(2.76)

In Chapter 4 we present a detailed calculation of the action of H ata trivalent node.We are not satisfied by operator H because it adds arcs but does notcancel them. Since the scalar product is well-defined on Kdiff , wecan introduce the adjoint H† and consider the symmetric operatorHs = 1

2(H + H†), that adds and removes arcs, and so it could be

a more suitable operator for our theory (though in [3] reasons areprovided to support the opposite point of view!).The path we have followed is not the unique way towards the quan-tization of the Hamiltonian constraint. This issue is still under dis-cussion. There are several alternative definitions of H with slightdifferences between them, and everyone lead to the same classicallimit. Only experimental data and a further development of thetheoretical framework of LQG will be able to select the best suitedH to describe the physical dynamics.

2.4.3 The propagator in LQG

With Quantum Mechanics in mind, we can encode the dynamics ofa physical system in the transition probability amplitude. In QFTthe propagator is W (Σ, φ). We can consider a surface α in configura-tion space, such that α = [xµ(~τ), ϕ(~τ)], where xµ(~τ) is a Minkowski3-surface and ϕ(x(~τ)) = φ(~τ) are the value of fields on this surface.In the Hilbert space K we construct functionals Ψ[α], a representa-tion where fields operator are diagonal, and dynamics is describedby HΨ[α] = 0.In Quantum Gravity W is independent of Σ because of diffemor-phism invariance. Propagator W depends only on boundary valueof fields, i.e values of fields on the boundary 3D–surface of a space-time region. We stress that there is not any physical reason anylonger to suppose a relation between the propagator and the choiceof Σ. Dependance of W on fields suffices to encode relative distancesand time separation of the components of the measuring apparatus,since gravitational field carries the information on geometry of thespacetime. The reason is simple: there is no more a spacetime asarena of physics, but only spacetime as dynamical field that en-codes geometry of the our system: W = W [A]. Hence, it suffices toconsider only fields on the surface, but not the localization of the

53

surface in spacetime. In LQG we do not measure position. Our datasets regard on measures on spin knots |s >. Eigenvectors of gravi-tational field are spin knots: the dynamics of gravitational field canbe described giving amplitude W (s) for spin knot states. If we wantto express the transition amplitude for a process involving an initialstate |sin > going to a final state |sout >, we have W = W (sout, sin).The propagator is related to a solution of the Hamilton-Jacobi equa-tions by:

W ∼ e−i~S[A]. (2.77)

Thus, S[A], a classical quantity, is viewed as the phase of the quan-tum functional Ψ[A], and Hamilton–Jacobi theory is the ikonal ap-proximation of Quantum Mechanics.We can construct a new Hilbert space resulting as product of the twoHilbert space K ′ = K∗diff ⊗Kdiff . It contains all possible couples ofinitial and final states for a system. Locally, we have K ′ ∼ Kdiff .We can define s = sin∪sout as states in K ′. Operator H acts only onnodes: spin networks without nodes are solutions of the Wheeler–DeWitt equation, belonging to the kernel of H. We can constructan operator P such that

P =

∫dτ exp (− i

τH), (2.78)

P : Kdiff −→ kerH, (2.79)

where ker H is the subspace of Kdiff of the solutions of Wheeler–DeWitt equation. The operator P is a linear functional that sendsspin knot states into solutions of hamiltonian constraint, namelyinto another Hilbert space. In general we write

W =< sout| exp−i~Hτ |sin > . (2.80)

Solutions of the Hamiltonian constraint H form a linear space withan inner product which is mathematically derived by the well definedinner product in Kdiff :

W (s, s′) =< s|P |s′ >Kdiff=< s|s′ >H . (2.81)

What is the difference with respect to the inner product into thekinematical space? We have to consider that Wheeler–DeWitt op-erator implies the existence of the eigenvalue zero in the spectrum ofHamiltonian constraint, and ker H is the correspondent eigenspace.If the zero belongs to the discrete spectrum of H, then there is nodifference between the two products. On the contrary, in case it is

54

a generalized eigenvalues being part of the continuum spectrum ofH, then eigenvectors belong to the distributional space K ′′diff andare not normalized states. This forces us to define the above innerproduct on ker H, written as matrix element of the P operator.

2.4.4 How many vacua!

The propagator W can be expressed in relation to a particular state|0t > of the tensorial space H∗ ⊗ H:

W (sout, sin) =< s|0t >, (2.82)

where |0t > is the covariant vacuum. In this sense, the prop-agator of our theory can be seen as a measure of the correlationbetween a couple of measure, one at the time zero, and anotherone at the time t: the dynamics says the probability that this cou-ple of measurements are physically correlated. The term ”initial”and ”final”, under this interpretation, are deprived of the commonmeaning. Considering a state in Heisenberg picture |α, t >, where αencloses all variable except the time, the propagator can be writtenas

W (α, t;α′, t′) = < α, t|α′, t′ >= x < α|e−i(t−t′)H |α′ >=

=∑

n

Hn(α)H†n(α′)e−i(t−t0)En , (2.83)

where H|Hn >= En|Hn >. Performing the analytic continuation inimaginary time, we obtain:

W (α,−it;α′, 0) = < α,−it|α′, 0 >=< α|e−Ht|α′ >=

=∑

n

Hn(α)H†n(α′)e−Ent. (2.84)

The covariant vacuum is related to the lowest energy state, i.e. theMinkowski vacuum |0M > that satisfy H0(α) =< α|0M >, by therelation

limt→∞

eE0t|0−it >= |0M > ⊗ < 0M |. (2.85)

Thanks to this new object, we can rewrite dynamics of the system ina covariant form: W =< s|0t >. It is impressive that the dynamicscan be completely expressed in terms of the covariant vacuum.There is a third type of vacuum: the geometrical vacuum |∅ >. Itis the kinematical state representing absence of spacetime. At firstsight, it could be a philosophical problem, since physicists do not

55

spend their time to study vacuum issues as these ones. But we havesaid that spacetime is not an environment where particles and fieldslive, but it is a dynamical field itself. We can connect geometricalvacuum with a physical state thanks to the P operator:

P |∅ >= |0t > . (2.86)

Geometrical vacuum is sent in the covariant vacuum state belongingto space ker H. It is not surprising: the Hamiltonian H vanishes inour theory because it is the generator of time translations, that aregauge transformations. We can say that all quantum gravity statesare vacuum states. Finally, we calculate the propagator associatedto the state of lowest non–vanishing volume |1v >. Recalling that:

V(R)|S > = limε→0

∑

n∈(S∩R)

Vε(Rε)|S >, (2.87)

V(R)|Γ, jl, i1, . . . , iN > = 16π~G32R

i′nin|Γ, jl, i1, . . . , i′n, . . . , iN >,

immediately we have

V |1v >= 16π~G32 |1v >, (2.88)

and the associated propagator is

W (0, 1v) =< 0|1v >=< 0|V > . (2.89)

2.5 Coupling to matter

It is now time to couple to matter the vacuum theory that we havedefined above. The term matter denotes anything that is not grav-itational field: fermions, Yang–Mills fields and Higgs scalar fields.However, we have claimed several times that in Quantum Gravitythe spacetime is identified with gravitational field, and it must betreated as any other dynamical field. This leads us to naturallyextend the previous construction of kinematics and dynamics forgravitational field to other physical particles and fields predicted bythe Standard Model, with some particular care.

2.5.1 Gravity–Matter theory: kinematics

Starting from quantum gravity theory, we make the following as-sumptions:

• we consider as main field the connection A = (Agrav, AYM),where Agrav is the well known selfdual (or Barbero–Immirzi)

56

connection of our theory without matter, and AYM is the con-nection for the Yang–Mills groupGYM = SU(3)×SU(2)×U(1).Hence, A is the connection for the group G = SU(2)×GYM ;

• holonomies of Y–M fields are operator on the Hilbert space Kas gravitational field;

• spin knots become knotted graphs with links coloured by ir-reducible representations of G, that are given by products ofrepresentations of SU(2) and GYM : each link is equipped witha jl and an irreducible representation kl of GYM ; on the samehand, we have at each node a new GYM–intertwiner wn.

• surfaces represented by links have area which is defined as theflux of the gravitational electric field and are crossed by thegauge electric field (the curvature derived from electric field isthe magnetic field).

Now, we want to construct a consistent scalar product, insertingmatter terms. Provided a fermion field ψ and scalar field ϕ, a graphΓ with L paths γl, a finite number N of nodes xn and a functionf : GL × SN × GN −→ R, we define a cylindrical functional of theconnection A:

ΨΓ,f [A,ψ, ϕ] =

= f(U(A, γ1), . . . , U(A, γL), ψ(x1), . . . , ψ(xN), eϕ(x1), . . . , eϕ(xN )).

We generalize the scalar product on cylindrical functions on thesame graph:

< ΨΓ,f |ΨΓ,g >=

=

∫

GL

dUl

∫

SN

dχN

∫

GN

dU ′nf(U1, . . . , UL, χ1, . . . , χN , U ′1, . . . , U′N)×

×g(U1, . . . , UL, χ1, . . . , χN , U′1, . . . , U

′N). (2.90)

We have extended the space K in order to enclose fermions andscalars. Fermions fields η(x), redefined as ξ ≡ η

√|detE|, are ob-

tained contracting holonomies indices with intertwiners: fermionslive at nodes, and they are sources of a gravitational field. At eachnode, we introduce a parameter Fn that determines the number offermions in the volume associated to the node. A multiplet of scalarfields ϕ(x) transforms as the representation k of GYM : the vectorspace where the representation acts is not compact. This problemis coped with the definition of a new field U(x) = eϕ(x) and selectingthe adjoint representation of GYM . This implies that U(x) takes

57

value in the compact group GYM . Like it happen with fermions,the scalars of the theory live at nodes, and we identify their numberwith the parameter Sn. In conclusion, the most general definitionof a quantum gravity state is given by a spin knot state written interms of quantum numbers as

|s >= |Γ, jl, kl, Fn, Sn, in, wn > . (2.91)

2.5.2 Dynamics

The interaction of spacetime with matter defines an additive decou-pled dynamics. Calling Hgrav the operator found in previous section,the generalized Hamiltonian constraint for a system is simply

H = Hgrav +HYM +Hf +Hs (2.92)

Hgrav = F ijabE

ai E

jb

HYM =1

2g2YMe

3tr(EaEb)Tr(ε

aεb +BaBb)

Hf =1

2eEai (iπτ iDaχ+Da(πτ

iχ) +i

2Kiaπχ+ c.c.)

Hs =1

2e(p2 + tr(EaEb)Tr(DaϕDbϕ) + e2V (ϕ2)),

where we used: E, ε, π, p as the conjugated momentums; F , curva-ture of connection A, and the magnetic field B, curvature of theY–M potential; D as the SU(2) × GYM–covariant derivative; tr as

trace in SU(2), Tr as trace in GYM ; e ≡√|detE|; V is the Higgs po-

tential; p2 = Tr(pp) and ϕ2 = Tr(ϕϕ); in the end, gYM is the Y–Mcoupling constant. The steps that lead to this formulation, and theregularization process in terms of well defined quantum operators,are described in [12]. These global operator H works on states ofthe space K extended to matter coupling. The normalizability ofthis operator holds as far as we deal with the discrete spacetime,and breaks down when we consider expression of gravitational fieldas a smooth background field instead of operators.

2.5.3 A possible end of the road

Let us comment about the main consequences of background inde-pendence. In canonical Quantum Gravity we need to abandon thetraditional interpretation of spacetime as a continuous entity, math-ematically represented by smooth manifolds, and develop a theoryfor the discretized spacetime, a field whose dynamics is quantizedand subjected to probabilistic laws. The continuous spacetime is,

58

in analogy with respect to what happens in Quantum Mechanics,the classical limit of the discrete one, and its properties are relatedto spectral characteristics of quantum operators that describe in-teractions between gravitational field, i.e. the spacetime, and theother fields. Surely, this theory is far from being confirmed, maybefrom a technological point of view is almost non–falsificable: it mustbe considered as a physical theory in the sense that at least thereis a clear direct link between theory and what one could observe atPlanck scales. Just for pure speculation, let us suppose that is reallya physical theory. We want to stress that this approach could besuggested by nature. How is it possible? The physics is an empiri-cal subject: a measure is an association of a number and a unit ofmeasure. The second one is often established upon our comfort andsuitable to describe the world at our scale (we measure the lengthof a football field in meters, certainly not in light years!). But thereare units that are derived from nature and in fact, they have notremarkable effects on our everyday life: the velocity of light c, thePlanck constant ~, and the universal gravitational constant G. Wecan rewrite them in relation to ”our” units: Meter, Second, Kilo-gram, and we see that the number associated is extremely big (c)or little (~, G). On the other hand, if we develop a theory on alattice, we automatically define a preferred units of measure scale,the lattice itself. And the construction of Hilbert space in LQG issimilar to that in lattice theories. In Quantum Gravity we try todefine physics at the Planck scale, so we have Planck length, time,energy and so forth. Almost citing, we didn’t order this! We areobliged to consider this scale. This is not an anthropogenic choice,but it is established by nature. So, if we construct a theory whosefundamental scale is a natural scale, we think that is a relevant fact.Despite of this, the physics at Planck scale is technologically a sortof hic sunt leones, to which access is denied at this time.

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60

Chapter 3

Spinfoams

3.1 A brief presentation

It is time to consider the extension of three dimensional, purelyspatial, formulation of Quantum Gravity that we have seen in theformer chapter. This is realized adopting elegant mathematical tech-niques known for long time but developed massively in this fieldonly in the last fifteen years. We are talking about the SU(2) (andSO(4)) Representation Theory, the main tool to treat multispinsystems. The result is a 4D covariant theory, whose dynamics isconstructed with transition amplitudes as sum over paths, the spin-foams, inspired by Hawking’s Euclidean sum over geometries [13].We stress that in this chapter we will talk about quantization assynonym of discretization. This pretty geometrically point of youlead us to define a geometrical object, the simplex, that in dimen-sion 4 we can interpret as a geometry, i.e. an equivalence class ofmetrics under diffeomorphism.It is possible to demonstrate a correspondence between spinfoammodels and the so called GFT, i.e. the Group Field Theories, thatare quantum field theories where spacetime coordinates are replacedby group elements. We refer to [1] for details, but we omit to discussthis issue.Good references for Spinfoam models are [14] and [15]

3.2 From knots to foams: covariantization

We have treated the Hamiltonian formulation of Quantum Gravity.In this chapter we use Lagrangian formalism, in which our theoryis manifestly covariant. Remembering the quantum propagator asmatrix elements of the sui generis projector from the kinematical

61

space to subspace of dynamics equation solutions, we write

W (x, t;x′, t′) =< x, t|P |x′, t′ >K∼∫

x(t)=x;x(t′)=x′D[x(t)]eiS[x], (3.1)

with S[x] =∫ tt′dtL(x(t), x′(t)) as action of the classical system. This

definition holds for a generic system that evolves from an event(x′, t′) to an event (x, t). A naive propagator in quantum gravitywould heuristically be

W [g, g′] =

∫

g|t=1=g;g|t′=0=g′D[gµν(x)]eiSG[g]. (3.2)

The sum is over 4D–metrics, since the choice of the time boundaryvalues is established irrelevant, in view of diffeomorphism invariance.This is to be understood as the agenda of what we want to definemore than a definition. In fact, this formulation has been studiedfor long time, but with unsatisfactory results: there is not a consis-tent non–perturbative definition of D[gµν(x)], and the perturbativeapproach leads to non–renormalizable divergences. Following thelesson of Feynman, we think the propagator in Quantum Mechan-ics as sum of matrix elements of infinitesimal evolution operators

e−iH0t−t′N along N infinitesimal paths, with N →∞, dt = t−t′

N→ 0:

∫

x(t)=xx(t′)=x′D[x(t)]eiS[x] ≡ lim

N→∞

∫dx1 . . . dxN−1

< x|e−iH0t−t′N |xN−1 >< xN−1|e−iH0

t−t′N |xN−2 > . . .

. . . < x2|e−iH0t−t′N |x1 >< x1|e−iH0

t−t′N |x′ > . (3.3)

We have passed from the canonical propagator to a covariant sumover paths. Now we want to make this step for quantum gravity.It does not make sense use x or the 3D–metrics g as continuum la-bels, especially for variables with discrete spectrum. The evolutionof space geometry must be in terms of spin networks. Accordingly,we expect to have W = W (s). The integral on infinitesimal pathsbecomes a sum of variations of the three geometry represented byspin knots. The novelty with respect to continuous Quantum Grav-ity is that spin knots come in a discrete set. We rewrite evolutionoperator P as

P = limt→∞

e−Ht, (3.4)

and the propagator found in second chapter is

W =< s|P |s′ >Kdiff . (3.5)

62

The limit for the evolution operator can be removed because ofdiffeomorphism invariance. The quantity is definitely constant. Ina basis |n >, which diagonalizes H with eigenvalues En, we have

P =∑

n

|n > e−Ent < n| =∑

n

δ0,En|n >< n|, (3.6)

the operator H depends on spatial coordinates, therefore

P = Πxe−H(x)t = e−

Rd3xH(x)t. (3.7)

Hence, one can attempt to define the propagator in Quantum Grav-ity as

W (s, s′) =< s|e−R 10 dt

Rd3xH(x)|s′ >Kdiff

= limN→∞

∑

s1,...,sN

< s|e−Rdtd3xH(x)|sN >Kdiff

< sN |e−Rdtd3xH(x)|sN−1 >Kdiff

. . .

. . . < s2|e−Rdtd3xH(x)|s1 >Kdiff

< s1|e−Rdtd3xH(x)|s′ >Kdiff

. (3.8)

Adopting the Feynman’s point of view, the evolution of a physicalsystem is decomposed in infinitesimal evolutions between interme-diate states. We stop interpreting the structural impossibility to as-sociate a deterministic trajectory to a physical motion as the proofthat dynamics is probabilistic; the new outlook lead us to consider ahistory of the system σ = (s′, s1, . . . , sN , s), i.e. a discrete sequenceof shifts of spinknots. The propagator is explicitated by a weightedsum over all the possible deterministic ”trajectories”, i.e. the admis-sible histories from a 3–geometry s′ to a 3–geometry s, representedin Figure 3.1. In analogy with Quantum Field Theories, we caninterpret this spinfoam as a decay of a quantum of space in threequanta of space. The weights in this sum are the amplitudes definedas A(σ). Labelling with ν the steps of each history, we write

W (s, s′) =∑

σ

A(σ) =∑

σ

ΠνAν(σ). (3.9)

The amplitudes Aν(σ) are determined by the matrix element be-

tween two spin networks in a history: < si|e−Rdtd3xH(x)|si−1 >. Ac-

tion of the operator H selects the admissible paths and acts trans-forming nodes, so initial and final spin knots |si > and |si−1 > ofeach infinitesimal evolution must differ at a node.Finally, we should pay due attention to the possible histories of asystem. Let us first remark that: (superpositions of) spin knots rep-resent the 3D–physical space by 2D–graph (leaving out eigenstates

63

Figure 3.1: A spinfoam

related to degenerate sector of Area operator). Consequently, a his-tory of spin knots, that is a 4D–spacetime, should be representedby 3D-graph: it is the worldsheet swept by the initial spin networkto the final one, along an external dimension coordinatized by the”time” variable. Let us define these new geometrical objects:

• the faces f are the world surfaces spanned by the spin networkslinks

• the edges e are the world lines tracked by nodes;

• at each step of this evolution H acts on a nodes sending itin three nodes: this happens at points along the edges, calledvertices v, where worldlines branch;

To sum up: along the worldsheet there are many faces f separatedby edges that meet each other at vertex |a >; This graphical rep-resentation and all the combinatorial relations between f, e, v arecalled a two-complex. As this occur with spin knots, the two-complexes will be coloured:

Definition 3.2.1 A spinfoam σ is a history of spin knots repre-sented by a two–complex Γ with a representation jf associated toeach face f and an intertwiner ie assigned to each edge e:

σ = (s′, sN , . . . , s1, s) = (Γ, jf , ie). (3.10)

64

Actually, the map sending a history sequence into the two–complexis not injective; there are many different sequences representing thesame two–complex. This physically correspond to different observerscan judge events (the branches) spatially separates to happen in dif-ferent order. In this sense the two–complex are better (more conve-nient ) representations of histories as sequences. The Feynman–Kacsum over paths becomes a sum over spinfoams whose amplitudesis the product of contributions from each vertex: these amplitudesdepend on colours of faces and edges adjacent to the vertex, andthey are denoted by Aν = Aν(jf , ie).

3.3 Models

3.3.1 General propagator

In the previous chapter we have stressed that Hamiltonian operatorH is far from being unanimously recognized as written in a definiteform. Therefore, it does not surprise that in current research severaltheories different in their dynamics are considered and developed.

Definition 3.3.1 A spinfoam model is a mathematical frame-work characterized by a choice of

• a set of two–complexes Γ;

• a set of representations j and intertwiners i;

• vertex amplitudes Aν(jf , ie), which determine the form of H,face amplitudes Af (jf ) and edge amplitudes Ae(jf , ie).

In some sense this is the quantum counterpart of relativistic theory,which neglecting the details from different dynamics dictates whatthe structure of a meaningful theory should be. The most generalpropagator that assures background independence and covarianceis given by a sum over spinfoams σ, defined by Γ weighted by fac-tors w(Γ), representations jf , and intertwiners ie. Considering thatthe boundary of a spinfoam is a spin network, we have: ∂σ = s.Moreover, we can set Af (jf ) = dim(jf ). A model is defined by thepropagator over spinfoams with fixed spin knots of the boundaries:

W (s, s′) =∑

σ∪σ′w(Γ(σ))Πfdim(jf )ΠeAe(jf , ie)ΠvAv(jf , ie). (3.11)

We can say that this is the quantum gravity version of the Feynman’sn–points functions, vacuum expectation values of a time orderedproducts of field operators. In fact, if we define a field operator

65

ϕs : |0 >−→ ϕs|0 >= |s >, that sends covariant vacuum in a spinnetwork, we have

W (s, s′) =< s|P |s′ >=< s|s′ >H=< 0|ϕ†sϕs|0 > . (3.12)

The equivalence between spinfoam formalism and LQG can de provedconsidering the Gelfand–Naimark–Segal theorem, whose allows tofind the Hilbert space ker H of quantum gravity states, starting froma positive functional W (s) over an algebra of linear combination ofspin networks weighted with complex coefficients. We refer to [16]and [17] for an exhaustive treatment. The relevant advantage of thespinfoam formalism is the possibility to calculate amplitude transi-tions at each order. Provided many mathematically strict models,probably the experimental data only may select the most reliablerepresentation of the physical world.

3.3.2 BF Theories

It is fine initially to consider the application of spinfoams to a par-ticular class of theories called BF Theories. They are given by an–dimensional oriented smooth manifold M , a principal G–bundleP over M , where G is a Lie group which algebra is equipped withan invariant non degenerate bilinear form. The fundamental fieldsare a connection A on P and an ad(P )–valued (n − 2)–form E onM , where ad(P ) is the vector bundle associated to P throughoutthe adjoint action of G on its Lie algebra. Let F be the curvature ofA, that is an ad(P )–valued 2–form on M . The Lagrangian densityfor a general BF model is:

L = tr(E ∧ F ). (3.13)

The field equations are obtained by the standard variational calcu-lus:

δ

∫

M

L = 0. (3.14)

In particular we have:∫

M

L = tr(δE ∧ F + E ∧ δF ) =

= tr(δE ∧ F + E ∧ dAδA) = (3.15)

= tr(δE ∧ F + (−1)n−1dAE ∧ δA) + divergences,

where we have introduced dA, that is the gauge covariant derivativewith respect to A, and applied covariant integration by parts. No-tice one can define a torsionless connection on M to obtain covariant

66

derivative of δA, but contributions from this term cancel for sym-metry properties. Moreover, we have considered the group G to becompact, so boundary terms vanish. The reason is that our treat-ment refers to Riemannian theories and not to Lorentzian, becausegroups as SO(n, 1), are not compact and, at this time, impossibleto handle in spinfoam models. In the end, since we have that thevariation is independent from δE and δA, the field equations are:

F = 0

dAE = 0.(3.16)

The solutions of BF theory are the same locally, i.e. are the same ina local Lorentz frame. The first one implies that the connection A isflat; the second one is the Gauss law. Implementing these equationsin the cotangent bundle of the configurations space of the BF theory,that is the space of the connections of our theory, we construct thephysical states space of the theory. The observables are self–adjointoperators on this space: they are functions of A and E, that, sincethey are distributions. must be integrated over suitable regions ofspace. As we have seen in Loop Quantum gravity, Wilson loops arethe simplest gauge–invariant function of A, and spin networks area state basis for the gauge invariant space, consisting of functionof holonomies around loops. In case G = SU(2), 3–dimensionalBF theory is equivalent to Riemannian General Relativity in threedimensions. The gauge invariant quantity, since G is non–abelian,is not ∫

Σ

E, (3.17)

where Σ is a (n − 2)–manifold, and in this case a curve, but theintegral

l(Σ) =

∫

Σ

|E|, (3.18)

that is the length of the curve. In four dimensions this quantitycorresponds to the area of a surface. There is another covariant,geometric view of the quantized version of this function of E: if aspin network Ψ intersects transversely Σ, calling l(Σ) the quantumoperator corresponding to the classical integral above, we have:

l(Σ)Ψ =∑

i

C(Ri)12 Ψ, (3.19)

where i’s are the points where the spin knot intersects Σ and C isthe Casimir operator of the representation labelling the edges: a

67

spin j representation has a Casimir operator equal to j(j + 1), so

the contribution of each edge to the sum is√j(j + 1). This is not

a surprise for us! So this operator measures the flux of E alongΣ, that in four dimensions is, following LQG, the quantized areaof the surface Σ. It is absolutely crucial to stress the connectionthat we have described above between representation theory andclassical field theory. Spin knots are the objects that render this re-lation manifest: the traditional approach in quantum gravity basedon differential geometry is related and, in last instance, substituted,by a purely abstract and combinatorial vision. This is not a recentidea, nor one appeared with LQG: it is the continuation of effortsaccomplished in this direction by Penrose [18]. To sum up, equiv-alence classes of spin networks are elements of a basis for quantumfield theory and, at the same time, graphical description of grouprepresentation theory. The two points of view meet with each otherif we consider spin networks as the building blocks of n–skeleton oftriangulated manifolds. The topological properties of the space arecompletely inherited by the dual skeleton: a further terrific scenarioarises when we consider theories with local degrees of freedom ase.g. General Relativity.

3.3.3 Simplex and Euler characteristic

At this point, it is necessary to introduce the essential geometricalobjects that allows us to treat spinfoams: the simplexes, a partic-ular class of polytopes. Heuristically speaking, a polytope is thegeneralization to n dimensions of a polygon and a polyhedron, andin particular a n–simplex is the generalization to n dimensions of atriangle (see Table 3.1). More specifically, referring to [19],

Table 3.1: Simplexes

Dimension Name0 point1 segment2 triangle3 tetrahedron4 pentachoron...

...n n–simplex

Definition 3.3.2 Considering an Euclidean space Rn+1, the stan-dard n–simplex ∆n is the set of all points p0, p1, . . . , p, n ∈ Rn+1

68

such that

pi > 0,n∑

i=0

pi = 1. (3.20)

We observe that there are (n+ 1) maps φi : ∆n−1 7−→ ∆n defined as

φi(p0, p1, . . . , pi, pi+1 . . . , p, n) = (p0, p1, ..., pi, 0, pi+1 . . . , p, n)(3.21)

These maps are called faces of ∆n. In fact, the map φi outputs thei–th face of the n–simplex. Now, given a differentiable manifold M ,we define a continuous map

σn : ∆n 7−→M ; (3.22)

this map is called singular n–simplex (or simplex) of M . The i–thface of σn is given by

σn φi : ∆n−1 7−→M. (3.23)

An n–dimensional surface can be completely covered by gluing to-gether a combination of n–simplexes, that is called simplicial com-plex. A triangulation ∆ of a manifold M is a simplicial complexK, homeomorphic to M , together with a homeomorphism

h : K −→M. (3.24)

We observe that a theory of gravity is a theory about curvatureof spacetime. In Riemannian geometry, curvature is a local con-cept, while topology provides global information on a manifold. Acylinder is flat as a plane, but topologically is quite different withrespect to a plane. Vice versa, a sphere surface with a point removedcontinues to have constant positive curvature, but topologically ishomeomorphic to a plane, e.g. by stereographic projection. Thereare theorems that connect the global curvature and the topologyproperties of the manifolds. For example, the Gauss–Bonnet theo-rem [20]: given a compact surface Σ with Gaussian total curvatureK, we have

∫

Σ

Kds = 2πχ(Σ) (3.25)

where χ is the so called Euler characteristic, a topological in-variant quantity, which will be defined hereafter. Given a subsetof Xn ⊂ Rn, and the compact set Y ⊂ X, we say that X is apolyhedron if every point p ∈ X has a neighborhood in X of theform

ax+ by : a, b > 0, a+ b = 1, y ∈ Y . (3.26)

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A n–cell is a compact convex polyhedron for which the smallestaffine space containing itself is n–dimensional [15]. Finally, we definethe Euler characteristic (given a triangulation of Σ) as

χ = number of (n− 1)–cells− number of (n− 2)–cells +

+ number of (n− 3)–cells− . . .+ number of 0–cells.(3.27)

For example, in two dimensions we have χ = F (faces)−E(edges) +V (vertices) [22]. Accordingly, it could be fine that our graphicalabstract representation, that we want to use to represent a (curved)spacetime, would be defined so to have relevant topological rela-tionship with the physical spacetime. We note that in dimensionthree the only regular polyhedra, the five Platonic solids, have thesame Euler characteristic, precisely χ = 2. Also, we stress thatthe Platonic solids determine the only discrete subgroups of SO(3),and we are treating quantum gravity with representation theory ofSU(2) ∼ SO(3)⊗ Z2! In fact, let us take the transformation groupleaving invariant each Platonic solid. We note that the invariancetransformation group of the cube is the same of the octahedron, andthe invariance group of the icosahedron is the same of the dodecahe-dron, whilst the tetrahedron has an invariance group on your own.Thus, we can distinguish three groups, which are SO(3) subgroups:the octahedral, the tetrahedral and the icosahedral one. Now, weconsider a polyhedron, and we call its combinatorial dual1 thepolyhedron constructed associating at every (n− 1)–cell of the ini-tial polyhedron a 0–cell, at every (n−2)–cell a (1)–cell, and so forth(see Table 3.2 and 3.3). Then, we can observe that there is an iso-

Table 3.2: 3D Triangulation

3D Triangulation 3D Dualpoint (vertex) tetrahedronsegment (edge) triangle (face)triangle (face) edgetetrahedron vertex

morphism between transformation groups of a polyhedron and ofits dual. For example, in dimension three the cube is the dual ofthe octahedron and vice versa (see Figure 3.2 taken by [22]), theicosahedron is the dual of the dodecahedron and vice versa, whilethe tetrahedron is selfdual, i.e. it is dual by itself. Selfduality is a

1There is also a metric dualization, which treat the case we consider metric properties ofthe polyhedra, as the length of the edges.

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Table 3.3: 4D Triangulation

4D Triangulation 4D Dualpoint pentachoron

segment (edge) tetrahedrontriangle face

tetrahedron edgepentachoron vertex

peculiarity of every n–simplex. Therefore, spacetime triangulationand its dual abstract spinfoam are both n–simplex, tetrahedra in 3Dand pentachora in 4D, having in the same transformation group.

Figure 3.2: Dualizing an octahedron we obtain a cube

3.3.4 Extension to generic dimension

Let us consider an (n − 1)–manifold S, that is the traditional con-tinuous physical space. Adopting a general triangularization of S,a dual 1–skeleton is a graph having a vertex at the centre of each(n− 1)–simplex of S and an edge intersecting each (n− 2)–simplex.Our physical states are linear combinations of spin knots whosegraphs are just these dual skeletons labelled with colours at links andnodes. For example, we can apply this process to 3D–Riemannianquantum gravity, that corresponds to a 3D triangularization. Inthis case, the gauge group is SU(2), whose representations satisfythe Clebsch–Gordan decomposition:

j1 ⊗ j2 = |j1 − j2| ⊕ . . .⊕ (j1 + j2). (3.28)

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The orthonormal basis of intertwiners have only at most one ele-ment, because we work with triangles and each node is trivalent.Thus, we do not need to explicit intertwiners, but only to labeledges of dual skeleton with spins (see Figure 3.3 taken by [15]).The related triangulation of the manifold is therefore depicted in

Figure 3.3: Dual skeleton

Figure 3.4 and 3.5 taken by [15], where the length of each edge

is√j(j + 1); spin combinations must respect the trivial inequality

|j1 − j2| ≤ j3 ≤ j1 + j2, and the sum of two spins has to be aninteger.Now, we consider a triangulated 3D manifold. The dual 1–skeleton

is built drawing a vertex at the center of each tetrahedron and anedge intersecting each triangular face. The basis of intertwiners isformally written as: i = j1⊗ j2 → j3⊗ j4 From what we said in pre-vious chapters, we have to split this 4–valent node in two 3–valentones linked by a virtual edge (Figure 3.6)This procedure corresponds on the manifold to insert a parallelo-

gram that cuts in half the tetrahedron (see Figure 3.7)Summing over all the possible cuts, we obtain the area of each pos-

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Figure 3.4: Triangulation in 3D

sible parallelogram from spins. We observe that there are three pos-sible ways to cut the tetrahedron with different parallelograms, but,since these parallelograms intersect each other, their related areasdo not commute as quantum operators. Thus, a basis of quantumstates of tetrahedron is characterized by five parameters, that arethe areas of the four triangles and the choice of one of the possi-ble parallelograms (on the contrary, in classical case we need of sixparameters). The matrix that spans over all the bases is called 6j–symbol. Graphically, we represent the calculation in Figure 3.8.In particular, we call Ei, i = 1, 2, 3, 4 the vectors in R3 normal tothe faces of the tetrahedron, with lengths corresponding to their ar-eas. They satisfy the geometrical constraint

∑iEi = 0, that is the

Gauss law dAE = 0 in its discrete formulation. Moreover, these areelements of so(3)∗, and obey the algebra of angular momentum:

Ja, J b = εabcJc. (3.29)

Considering now the ”vector” (E1, . . . , E4): it is an element of (so(3)∗)4.Reducing this space to the subset of vectors obeying Gauss law, we

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Figure 3.5: Labelling spins in 3D triangulation

j4

j3

j1

j2

j

Figure 3.6: 4–node splitting

get the the phase space of tetrahedron geometries. We can quantizethis space in a geometrical sense: the Hilbert space H associated toso(3)∗ is the direct sum of all irreducible representations of SU(2),

74

Figure 3.7: One of the possible cutting parallelograms

j4j3

j5

j4

j3

j1j2

j6

j1

j2

=∑

j6(2j6 + 1)

j1 j2 j5j3 j4 j6

Figure 3.8: Splitting as combinatorial sum

with operators Ja satisfying

[Ja, J b] = iεabcJc. (3.30)

These Ei are nothing but operators acting on the Hilbert space ofthe quantum tetrahedron, but also a space of a ”quantum vector”

75

with components Ja. Consequently, a quantization of (so(3)∗)⊗4

gives us the Hilbert space H⊗4 where the following operators act:

Ea1 = Ja ⊗ 1⊗ 1⊗ 1 (3.31)

Ea2 = 1⊗ Ja ⊗ 1⊗ 1 (3.32)

Ea3 = 1⊗ 1⊗ ja ⊗ 1 (3.33)

Ea4 = 1⊗ 1⊗ 1⊗ Ja. (3.34)

From now , we leave out hat for quantum operators, since there isno possibility of misunderstanding. After the geometrical quanti-zation, we have the operators Ai =

√EiEi, related to the areas of

tethraedron faces, and the operators Aij =√

(Ei + Ej)(Ei + Ej),that correspond to the area of the parallelogram. We observe thatAij = εklijAkl: as a result, there are only three independent parallelo-grams. Since their areas do not commute, we can construct our setof commuting observables taking the triangular faces areas and theareas of one of the parallelograms:

Aiψ =√ji(ji + 1)ψ, i = 1, . . . , 4 (3.35)

A12 =√j6(j6 + 1). (3.36)

3.3.5 Dynamics

We want to construct a consistent dynamics for the BF theories.We start from the partition function for a closed 4D–manifold M .Recalling the BF Lagrangian presented in 3.13, we write

Z(M) =

∫ ∫DADE expi

∫

M

tr(E ∧ F ) =

∫DAδ(F ). (3.37)

We have to integrate over the space of flat connections: the resultwould be a sui generis volume of this space M . To perform thiscalculation, we need to discretize the integral, reducing it to a sum,that corresponds to a triangulation ofM . Furthermore, this requeststo consider the flat connections on the dual skeleton, not any longeron M . We list below the dual correspondences:

• at the centre of each n–simplex on the manifold there is a vertexof the skeleton;

• each edge intersects a (n− 1)–simplex;

• a polygonal face intersects a (n− 2)–simplex.

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Figure 3.9: 3D dual skeleton intersection

If we have a 3D manifold, its intersection with the dual skeletonis represented in Figure 3.9. The dual faces f have an arbitrarynumber of dual edges, say n, and we label them as e1f, . . . , enf andthe vertices with v1f, . . . vnf . This choice select an automatically anorientation of the face and its edges. As we have seen many times, aconnection label edges with group elements ge, or g−1

e if we invert theorientation of the path with respect to the one fixed by the generalorientation of the face. The constraint of flatness imposes:

ge1 · · · gen = 1. (3.38)

The partition function is now defined by

Z(M) =

∫

G

ΠedgeΠfδ(ge1 · · · gen), (3.39)

where the integral is taken considering the Haar measure on G. Weuse the relation

δ(g) =∑

ρ∈Irrep(G)

dim(ρ)tr(ρ(g)). (3.40)

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That is the key passage! The integral over a continous variableoutputs a delta function. But since the delta is over functions on acompact interval, i.e. the compact space of the holonomies, we canexpress it as a sum, and in general, we can replace all the integralswith sums. Thus, we obtain

Z(M) =∑

ρ∈Irrep(G)

∫

G

ΠedgeΠf dim(ρf )tr(ρf (ge1f· · · genf

)). (3.41)

From the geometrical point of view, the connection A labels dualedges with group elements, F labels dual faces by holonomies aroundthem, and E assigns representations ρf . This formulation holdsfor arbitrary dimension of the manifold M . However, we want tocalculate explicitly the suitable formula for specific cases. If Mis connected and of dimension two, we have dual edges that areshared by two dual faces, so that each edge appears two times inthe integral. Tensor product of representation is defined as

(ρ1 ⊗ ρ2)(g) = ρ1(g)⊗ ρ2(g) (3.42)

where i : ρ1 ⊗ ρ2 → C. Let us project ρ1(g)⊗ ρ2(g) in the subspaceof vectors transforming in the trivial representation. In [15] there isa fine graphical representation, that we recall in Figure 3.10. Thus,

= gg g

ρ2ρ1 ρ2 ρ1

Figure 3.10: trivial intertwiner

we obtain∫dgρ1(g)⊗ ρ2(g) =

ii∗

dim(ρ1)if ρ1 = ρ∗2

0 otherwise. (3.43)

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This integral vanishes unless ρf = ρ for all representations of allfaces. When it does not vanish then we can write

Z(M) =∑

ρ∈Irrep(G)

dim(ρ)χ(M), (3.44)

where χ(M) = vertex − edges + faces is the Euler characteristicof M (or equivalently of its triangulation). If χ(M) ≤ 0 the sumis convergent and the dynamics of BF theories is well-defined andindependent of the triangulation. This property is common bothto 3D GR and Loop Quantum Gravity but not to 4D GR, sincethis is a theory with local degrees of freedom. In three dimensionalmanifolds each dual edge belongs to three dual faces, so we have

i : ρ1 ⊗ ρ2 ⊗ ρ3 → C (3.45)∫dgρ1(g)⊗ ρ2(g)⊗ ρ3(g) =

∑i ii∗. (3.46)

Thus, we have a sum over elements of basis of intertwiners andtr(i1i

∗2) = δi1i2 . Graphically we represent it in Figure 3.11, remembering

δρ1ρ∗2

dim(ρ1)

∫dg

ρ2

ρ1 ρ2

ρ1

=g

ρ1 ρ2

i∗

i

Figure 3.11: 3D non–trivial intertwiner

that we have to apply this calculation to each face of the tetrahe-dron, we obtain a dual tetrahedron, i.e. a tetraspin knot. This is nota surprise, recalling that tetrahedron is dual to itself and belongsto the one–element tetraehdral group, that is one of the discrete

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subgroups of SO(3), and we can write the partition function of a3–dimensional BF theory as

Z(M) =∑

ρ∈Irrep(G)

∑

i

Πfdim(ρf )Πv6j, (3.47)

where 6j is graphically represented in Figure 3.12Each dual vertex has six dual faces incident to it and four edges such

ρ1

ρ2

ρ5

ρ6

ρ3

ρ4

Figure 3.12: 6j–symbol

that it is defined by six spins and four intertwiners. If G = SU(2),the intertwiners are completely determined, because each edge is as-sociated to three faces and we can explicit the value of a tetrahedralspin networks with the 6j–symbol. In the end, this calculation hasto be performed for all the tetrahedra of the dual skeleton.Finally, let us face the construction of a 4D theory. We can imme-diately write

i : ρ1 ⊗ ρ2 ⊗ ρ3 ⊗ ρ4 → C (3.48)∫dgρ1(g)⊗ ρ2(g)⊗ ρ3(g)⊗ ρ4(g) =

∑i ii∗, (3.49)

The building block of the dual skeleton is now a 4–simplex. Fourdual faces share each dual edge, labelled by the representations of the

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ten dual faces incident to the vertex in the triangulated manifold,and dual vertices are labelled by intertwiners assigned to the fivedual edges incident to the vertex. Transforming the 4-valent vertices

Table 3.4: Dictionary of the triangulation

Dimension Spin network Spinfoam Triangulation0 node vertex point1 link edge segment2 x face triangle3 x x tetrahedron4 x x pentachoron

in 3–valent ones, we obtain a spin networks characterized by 15spins. With G = SU(2), we call it 15j–symbol. The partitionfunction is then

Z(M) =∑

ρ∈Irrep(G)

∑

i

Πfdim(ρf )Πv15j, (3.50)

where 15j–symbol picture is presented in Figure 3.13. The conver-gence of the series presented in this subsection is restricted to thefollowing cases:

• the G group has a finite number of elements without restrictionregarding the dimension of M ,

• M is a 0D or 1D–manifold,

• M is a 2D–manifold with χ(M) < 0.

3.3.6 BF dynamics and spinfoams

We have presented some examples of partition function for BF theo-ries: these are combinatorial sums over quantum numbers that labelspin networks. Now we can talk about an equivalent treatment ofdynamics in terms of two–complexes. In fact, we have seen that apartition function could be also considered as sum over spinfoamhistories, starting from a spatial slice of spacetime and ending onto another one. In particular, we consider a connected spacetimeM as a map such that M : S −→ S ′, where S and S ′ are n − 1compact oriented manifolds. This construction is inspired to a stan-dard one in category theory. There one defines a “dual category”which objects (spin foams) are morphisms (histories) of the originalcategory. We triangulate M : this induces a triangulation on S, S ′,

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ρ3

i5

i3i4

i2

ρ1

ρ6

ρ10

ρ7

ρ2

ρ5

i1

ρ4

ρ8ρ9

Figure 3.13: 15j–symbol

that we call γ, γ′. Also, we have two gauge–invariant Hilbert spaceK0, K

′0, provided by a basis of spin networks, whose graphs are the

dual 1–skeleton of triangulations on S and S ′. As we have seenin the previous sections, the evolution between two states Ψ,Ψ′ isgiven by a sum over spinfoams F : Ψ −→ Ψ′:

< Ψ′|Z(M)Ψ >=∑

F

Z(F ) (3.51)

We can reduce the sum to foams having as underlying graph, i.e. thetwo–complex, the dual 2–skeleton of M . We remove from the cal-culation of partition function edges and vertices that belong to theinitial and final state, since they do not describe evolution. Then,calling M ′ : S ′ −→ S ′′ and F ′ : Ψ′ −→ Ψ′′, we want that partition

82

function preserves compositions

Z(M ′M) = Z(M ′)Z(M) (3.52)

Z(FF ′) = Z(F )Z(F ′), (3.53)

where M ′M : S −→ S ′′ and F ′F : Ψ −→ Ψ′′. This imposes to usethe square root of the edge amplitude of dual edges and dual faceswhich respective vertex and edges lies in Ψ or Ψ′. It is absolutelyimportant to clarify that the operator F is not totally equivalent,from a physical point of view, to a spinfoam. The relationship isthe same one that exists in Quantum Field Theory between an op-erator on Fock space and the related Feynman diagram. The lattercontains much more information about the amplitude transition,because it describes how the interaction occurs adding virtual in-teractions connecting external worldlines. Even though this infor-mation is not physical, it is useful to account efficiently of variouscontributions to the total amplitude by fixing about what happensin the intermediate configurations. The quantum history of the sys-tem is described by the diagram, and in our case by the spinfoam.In spite of this, it must be stressed that we are talking about statesin the Heisenberg picture, which temporal evolution is encoded inthemselves. Here it is not the evolution of the state that we are talk-ing about, but the evolution of the system from a state to anotherone. Every Feynman diagram encodes temporal evolution in itself.Spinfoams work in the similar way: in our combinatorial picture,quantum numbers associated to faces give us the area, edges arerelated to volume and internal vertices to 4–volumes of spacetime.

3.3.7 A way to renormalization: q–deformation

Thanks to spinfoam models we have a family of examples of unionbetween representation theory, essential tool to treat Quantum Me-chanics, and differential geometry, the language of General Rela-tivity. The main problem with this situation can be summarized:spinfoam models lack of mathematics consistency. Indeed, the spaceof flat connections have a natural Haar measure at most when Mhas dimension two. This implies that we are not be able neither toconstruct the Hilbert space nor to calculate transition amplitudesin general situations. Infrared divergences appear in theories fromdimension three: we have none control on spins, that can be arbi-trarily high, so that the partition function diverges. In 3D and 4DBF theories, we can remove divergences adding an extra term to the

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Lagrangian function: in the 3D case we have

L = tr(E ∧ F +Λ

6E ∧ E ∧ E), (3.54)

while the 4D one is

L = tr(E ∧ F +Λ

12E ∧ E). (3.55)

The term Λ is called cosmological constant. These new form ofthe Lagrangian implies that representation used in our calculationsis not related to a gauge group any longer, but to a quantum group.The Hilbert space become finite dimensional and the partition func-tion is a convergent series. This procedure is called q–deformation,since the quantum group is characterized by a parameter q, whichis function of Λ. When q = 1, then Λ = 0, and we obtain againthe original group, that is SU(2) in our case. The deep motivationfor these assumptions can be developed in the context of particularbackground–free gauge theory, the Chern–Simons theory. We avoidtreating in detail this point, referring to [15]. However, the key pointis that local action of the Chern–Simons theory is

SCS(A) =k

4π

∫

M

tr(A ∧ dA+2

3A ∧ A ∧ A) (3.56)

This theory admits a consistent quantization whenever G is a com-pact group. Calculation of quantum numbers are encoded in the rep-resentation theory of the quantum group UqG, where q = exp 2πi

k+h,

being h the value of Casimir operator in the adjoint representationof G and k the coupling constant in Lagrangian (3.55). It is suffi-cient to know this and that each representation of a gauge group Ginvolves a representation of the quantum group UqG, that is an alge-braic structure, which is defined below following [23]. We can labeledges of spin networks with representations of the quantum groups,but we have to keep in mind that only a finite number of irreduciblerepresentations of G induces irreducible representations of quantumgroups with suitable algebraic properties. In case G = SU(2), theonly representations satisfying these constraints are j = 0, . . . , k

2,

where k is the constant appearing in SCS.

3.3.8 3D and 4D BF quantization

In this subsection we derive field equations for BF spinfoam modelswith extra terms related to Quantum Gravity. Let us start with 3D

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BF theory, which has Lagrangian

L = tr(E ∧ F +Λ

6E ∧ E ∧ E). (3.57)

The field equations obtained from it are

F +Λ

2E ∧ E = 0, dAE = 0. (3.58)

Since the term tr(E ∧ E ∧ E) is proportional to the classic volumeinduced by the metric associated to E, if G = SO(3) this theory isequivalent to 3D Riemannian General Relativity with cosmologicalconstant. The crucial point here is that we can make a correspon-dence between this theory and the Chern–Simons one. In fact, letus see

A± = A±√

ΛE; (3.59)

we can recast the action modulo boundary terms as∫

M

tr(E ∧ F +Λ

6E ∧ E ∧ E) = SCS(A+)− SCS(A−), (3.60)

with k = 4π√Λ

in SCS. We have obtained the BF action as difference

of two Chern–Simons actions with a fixed value of k. One can seefrom the value of k as a function of Λ the reason why we haveintroduced the cosmological constant. In fact, the case without Λis degenerate in the sense that precisely the correspondence withCS theory fails. Since CS theories can be quantized, also the BFaction is. The Hilbert space for BF theory is the tensor product ofspaces for CS, so we can describe our BF theory with representationstheory of quantum groups! Since suitable representations of UqG arefinite by many, the sum over spinfoams is finite and triangulation–independent. A spinfoam model of this type is the Turaev–Viroone. Taking G = SU(2), only spins j = k

2are associated to good

representations of the quantum group. This algebraic constraintimposes a geometrical one: in the theory naturally appears a cut offon the lengths of the edges in the triangulation! Planck’s constant isdifferent from zero, so we avoid UV divergences and fix a minimumlength, and on the other hand an energy scale, we have a maximumlength related to the cosmological constant: when Λ goes to zero,then the maximum length tends to infinity.Now let us consider the 4D theory with extra term:

L = tr(E ∧ F +Λ

12E ∧ E). (3.61)

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The field equations satisfied by physical states are

F +Λ

6E = 0, dAE = 0. (3.62)

where F is the curvature of A. Following the canonical quantization,replacing A and E with correspondent operators, we obtain

(F aij +

Λ

6iεijk

δ

δAka)ψ = 0. (3.63)

If cosmological term does not vanish, the only solution is the socalled Chern–Simons state:

ψ(A) = exp−3i

Λ

∫

S

tr(A ∧ dA+2

3A ∧ A ∧ A). (3.64)

If group G is simple, connected and simply connected, and traceoperator is defined with the Cartan–Killing form, this is a gaugeinvariant state if the parameter k = 12π

Λis an integer. Hence, the

Hilbert space is one dimensional. For this theory there is not aspinfoam model, but the quantum group deformation of the spinfoam model for 4D BF theory without extra term seems to workwell as model for 4D BF theory with cosmological constant.

3.3.9 4D Quantum Gravity

According to ensembles theory in statistical mechanics, evolutionof the physical system is seen as a combinatorial and probabilisticissue. The principle is the same for spinfoams models, that are dualto path integrals of Yang–Mills lattice theory. Now the question is:are these models reliable quantum gravity theories? Local degreesof freedom are not considered by a BF Theory: this is the maindifficult we have to face in order to associate General Relativity toBF theory formalism. Quantizing the BF theories, we discover thatspin networks are the basis for Hilbert space of physical states. Therules for calculate functionals are presented in Chapter 4. The keypoint of all our treatment is that spin networks are an instrument torepresent graphically groups representation theory. BF theories arequantum field theories where groups representations are encoded ina insightful way.Now, we briefly construct Palatini formulation of General Relativity.The Lagrangian is

L = tr(e ∧ e ∧ F ), (3.65)

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field equations are

e ∧ F = 0, dA(e ∧ e) = 0. (3.66)

We find in this formulation the physics of General Relativity. As-suming e as one–to–one 1–form on manifold, we can write

dA(e ∧ e) = e ∧ dAe = dAe = 0. (3.67)

Pullbacking A to Γ in the tangent bundle, the constraint tells thatΓ is the Levi–Civita connection of metric associated to e. We canalso observe that e∧F is proportional to Einstein tensor. Moreover,we see that if E = e ∧ e, then General Relativity is equivalent toBF theory. The problem is that this relation does not work in anycase, and therefore field equations are slightly different. Anotherquestion to treat is that even if E = e∧ e we are not able to removean ambiguity on the sign of e, and to establish a one–to–one relationbetween e and E. Therefore, a spinfoam model for quantum gravitycould be built taking the model for BR theory and imposing addi-tional constraints, quantum equivalent of the classical one E = e∧e.We have to restrict this process only to compact gauge groups, be-cause BF theories at this time are not well defined, so we can obtainonly a model for Riemannian quantum gravity. Furthermore, theindependence from the triangulation chosen, obvious and necessarycharacteristic for a consistent physical theory, is possible thanks tothe lack of local degrees of freedom in BF theories: as we know, thisdoes not hold in General Relativity. This problem has not alreadysolved, and we will not treat it, but it needs to be specified what wemean with local degree of freedom. Obviously, we want to respectcausality determined by Relativistic theories, that is consequence ofthe constancy of light speed. Thus, the quantity that we measurein a spacetime region R, determined by the possible timelike paths,depends on only quantities measured within R. We have not inter-est on global behaviour of the system: a local degree of freedom is aquantity that we measure in a arbitrary small portion of spacetime.It is natural to suppose that changing the density of the triangula-tion can modify the measure of local degrees of freedom. In spite ofthis, we forget this problem and we consider spinfoams on a fixeddual 2–skeleton of a fixed triangulation.Let us look for the constraints such that E = e ∧ e. We chooseG = Spin(4), so locally E has values in so(4) algebra. Knowingthat so(4) ∼= so(3) ⊕ so(3), we can consider E = E+ + E−, whereE+,− have values in so(3). If the field equation holds, then for allvectors v, w ∈M we have

|E+(v, w)| = |E−(v, w)|, (3.68)

87

where the norm is defined by the Killing form on so(3). This relationrestricts the possible values of E to this forms: e ∧ e,−e ∧ e, ∗(e ∧e),−∗(e ∧ e), where ∗ is the Hodge operator. Now the problem ishow to implement this constraint in a spinfoam model. We remem-ber that the building block of triangulation, the tetrahedron, wasdefined only with spins associated to surfaces of faces and one of theparallelograms that cut it. But we also recall that spins determinethe integral of |E| over the surfaces. Since we have established thatG = Spin(4) = SU(2)⊗SU(2), the irreducible representations of Gare of the form j+ ⊗ j−. Therefore, we can label each dual surfacewith such a couple of spins, that also describes integrals of |E+|, |E−|over the surface. Hence, the constraint imposes us to consider pairof equal spins j ⊗ j. Moreover, the tetrahedron is labelled by anintertwiner

∑j cjij⊗ ij, where ij : j1⊗ j2 → j3⊗ j4 with these spins

that label the four faces of the tetrahedron. Since we want that foreach possible parallelogram P works in this way

∫

P

|E+| =∫

P

|E−|, (3.69)

the intertwiner must maintain its form for every permutation givenby the 6j symbol. The only one intertwiner that respects this con-straint is ([24] and [25])

i =∑

j

(2j + 1)(ij ⊗ ij). (3.70)

The partition function for the spinfoam model of the 4D RiemannianQuantum Gravity, called Barrett–Crane model([26]), is

Z(M) =∑

j= k2

Πf (2jf + 1)Πv15j (3.71)

(3.72)

Implementing other constraints ([1] for details) on intertwiners weobtain a refinement of this model, whose partition function is

Z(M) =∑

j= k2

Πf (2jf + 1)Πv10j. (3.73)

The sum is divergent, but is finite in q–deformed formulation, thathas the flaw to be dependent of the triangulation. We also notethat the bivector equation above mentioned is nothing but the dis-cretized version of the Plebanski constraint defined in Chapter 2.Once again, our theory needs to discretization of the physical space-time to ensure consistency. The spin networks are the building

88

blocks of the quantum states describing the physical space. Thespinfoams are their natural extension to the quantum spacetime.

3.3.10 Quantum Groups

Following mainly [23], we present a brief introduction to quantumgroups. The name of this mathematical object is very misleading.The attribute quantum has to be intended in a pretty mathematicalmeaning, with no reference to physics. But the problem is thatwe are discussing about application in quantum physics! So wesubstitute in our treatment the term ”quantum” with ”deformed”.We can turn our attention to the definition of deformed groups. Westart with the definition of a Hopf algebra:

Definition 3.3.3 A Hopf algebra is

• an unital algebra (H, ·, 1) over a field K

• a counital algebra (H,∆, ε) over K; the product ∆ : H →H ⊗H such that ∆ : h 7−→ h⊗ h and the unit map ε : H → Kare algebra homomorphisms and satisfy

(∆⊗ id)∆ = (id⊗∆)∆ (3.74)

(ε⊗ id)∆ = (id⊗ ε)∆ = id (3.75)

• there is a map S : H → H called antipode such that

·(id⊗ S)∆ = ·(S ⊗ id)∆ = 1ε (3.76)

We can depict the network of relationships of a Hopf algebra thanksto a commutative hexagon created by Ryan Reicher (Figure 3.14)Coordinates algebra of an algebraic group form a Hopf algebra, andthe functions K(G) on a finite group G with values in K are a Hopfalgebra as well. For all a ∈ K[G], we define

(∆a)(x, y) = a(x, y) (3.77)

(Sa)(x) = a(x−1) (3.78)

ε(a) = a(e), (3.79)

where e is the group unit element and x, y arbitrary element of G.The group structure is encoded in Hopf algebra axioms. At thesame time, we see a deep relation between the Hopf algebra and thegroup considering G ⊆ Kn, i.e. described as subset of polynomials.Coordinates of the algebra k[G] are consequently polynomial func-tions on Kn. If the field k is closed in an algebraic sense, we have a

89

HH Kηε

∇∆

∆ ∇

Id⊗ S

S ⊗ Id

H ⊗HH ⊗H

H ⊗HH ⊗H

Figure 3.14: Hopf algebra

correspondence between G and commutative algebras with a finiteset of generators. We can associate the map G×G→ G to ∆ andso forth for the other elements of the Hopf algebra. The majorityof common complex Lie groups are defined by polynomial equationsand algebra coordinate over fields with identical relations.We have to remark that the most general Hopf algebra is not commu-tative. We can consider deformed algebras, presenting noncommu-tative coordinates. This perspective led to by considering noncom-mutative geometry. As example, we construct differential 1–forms(⊕1Ω1, d) over an algebra H. The derivative d : H → Ω1 satisfiesthe Leibniz rule

d(ab) = (da)b+ a(db), ∀a, b ∈ H, (3.80)

and Ω1 = HdH. Even if H is commutative, we have not requiredthat commutation holds between 1–forms and elements of H. Incase H is a Hopf algebra, it is possible demand that Ω is invariantby translation. These differential structures (Ω1, d) can be labelled

90

by a parameter λ ∈ C and they assume the form

da(x) =a(x+ λ)− a(x)

λdx, (3.81)

dxa(x) = a(x+ λ)dx, ∀a ∈ C(x). (3.82)

We immediately observe that only when λ vanishes the dx commuteswith the functions. The expert reader can recognize the geometry ofthe affine line with the coproduct ∆x = x⊗1 + 1⊗x, that respondsto an addiction in K. At the same time, the circle has coordinatealgebra of the type K(t, t−1), with coproduct ∆t = t⊗t associated tomultiplication in K∗. The translation invariant differential 1–formsfor C(t, t−1) are labelled by a parameter q ∈ C∗, taking the form

da(t) =a(t)− a(qt)

(1− q)t dt, (3.83)

dta(t) = a(qt)dt, ∀a ∈ C(t, t−1). (3.84)

It is possible to construct the q–deformed version for all compactLie groups, including the complexified ones: we call them Cq[G].From the previous example of the circle, we define the q–deformedquantum group Cq[SL2]. The relations between generators a, b, c, dare the following:

ba = qbc, bc = cb

ca = qac, dc = qcd, db = qbd (3.85)

da = ad+ (q − q−1)bc, ad− q−1bc = 1

∆

(a bc d

)=

(a bc d

)⊗(a bc d

)(3.86)

Another remark leading to the definition of quantum group is thatthe group algebra over a field k of any group, i.e. the vector spacewith basis the elements of the group, is a Hopf algebra. We write

∆x = x⊗ x, εx = 1, Sx = x−1,∀x ∈ G. (3.87)

Also, the enveloping algebra of any Lie algebra is a Hopf algebra,too. We take the algebra (g, [, ]): its enveloping algebra U(g) isdefined taking a basis of g and, for all elements v, w ∈ g, definingthe associative algebra with the basis above as set of generators andrelations vw − wv = [v, w]. We have ∆v = v ⊗ 1 + 1⊗ v.In both the cases mentioned above, the coproduct ∆ is symmetricand the related coalgebra is said co–commutative. The actions ofthe group algebra and the enveloping algebra, extended to tensor

91

products thanks to ∆, correspond to linear action of the underlyinggroup and Lie algebra. A Hopf algebra H is such that each elementh ∈ H is equipped with actions ∆h in the tensor product. We usethis result when we want to establish whether an algebra is covariantunder H. As example, we present Uq(sl2), whose generators aree, f, qh, q−h. Imposing q2 6= 1, we write

qheq−h = q2e, qhfq−h = q−2f

ef − fe =qh − q−hq − q−1

∆e = e⊗ qh + 1⊗ e (3.88)

∆f = f ⊗ 1 + q−h ⊗ f, ∆qh = qh ⊗ qh.

Lastly, Hopf algebra are viewed as structures admitting Fouriertransform. There is a connection with first quantization and ”co-ordinate” and ”symmetry” algebra. For instance, we mention thePoincare quantum group of noncommutative spacetime algebra Cλ[R1,3].This deformed group is a possible description of a quantum particlemoving in a curved spacetime with black hole-type features. In theend, the key point is the following: we can describe algebraic groupwith its standard Hopf algebra of regular functions; at the samemanner, deformed Hopf algebras could describe new objects, calledquantum groups, deformed version of the standard algebraic ones.

92

Part II

Applications

93

Chapter 4

Spin network andSpinfoams calculations

We present rules of calculation of functionals related to spin net-works and spinfoams, and apply them to simple but exhaustive ex-amples. The necessary steps to perform manipulation of these ob-jects recall Feynman rules to treat spacetime diagrams contributesto scattering matrix. In general, this combinatorial techniques arederived from SU(2) representation theory and are necessary to thedevelopment of a simplicial theory in Quantum Gravity. However,the results are general and find applications in Nuclear, Chemicaland Molecular Physics, and generally in all the disciplines that treatmultispin systems. We refer mainly to [1], [5] for spin networks sec-tion and [27, 28, ?, 31, 32, 33] for spinfoams.

4.1 The main framework: SU(2) representationtheory

We list briefly some useful results:

• the spin networks are finite linear combination of multiloopstates;

• SU(2) irreducible representations can be decomposed as a sym-metrized tensor product of 2j fundamental representations ψA,A = 0, 1;

• in this basis we can consider intertwiners as combinations ofthe two SU(2) invariant tensors, i.e. δ and ε: whenever anintertwiner connects links with concourdant orientation, a δappears; on the other hand, two links with different orientationare connected by an ε;

95

• there is a powerful graphical representation of spin networks de-composition in multiloop states: we substitute each link colouredby representation j with 2j segments and perform combinato-rial calculations at each node, defining the intertwiner in thesymmetrized tensor product basis.

As relevant example, which better than words can explain this pro-cedure, we treat the trivalent intertwiner, which can represent theresult of the action of Hamiltonian operator at a trivalent node. Ev-ery n–node can be splitted in 3–valent nodes couplings. Graphicallywe have the Figure 4.1 Numbers a, b, c represent the number of seg-

) a b

c

∼

j1

j2

j3

j2

j1 j3

H(

H

Figure 4.1: Action of H at a node

ments at each edge of the triangle: they satisfy the Clebsch–Gordanconditions

2j1 = a+ c (4.1)

2j2 = a+ b (4.2)

2j3 = b+ c (4.3)

j1 + j2 + j3 = n ∈ N (4.4)

|j1 − j2| 6 j3 6 |j1 + j2| (4.5)

The invariant tensor is

iB1...B2j2A1...,A2j1

C1...C2j3= δB1

A1. . . δBa

AaδBa+1

C1. . . δ

Ba+b

CbεCb+1Aa+1 . . . εCb+cAa+c .(4.6)

96

Now we applicate this procedure to the intertwiner with respect toFigure 4.2 and 4.3, while in figure we have explicitated the ”internal”links. We have

a = 3 b = 2

c = 2

j2 = 52

j1 = 52

j3 = 2

Figure 4.2: Intertwiner at 3–valent node

iB1...B5A1...,A5C1...C4

= δB1A1δB2A2δB3A3δB4C1δB5C2εC3A4εC4A5 (4.7)

Now we are able to perform calculations on quantum states of Loop

3

2

2

2

52

52

B2 B3 B5

A4

B4

A1

C2

A3

A2

A5 C4

C3

C1

B1

Figure 4.3: Intertwiner at 3–valent node explicitated

Quantum Gravity and Spinfoam models.

97

4.2 Spin Networks

We recall from Chapter 2 that a functional on the space of connec-tions takes the form

ΨS[A] =< A|s >≡⊗

i

Rjl(U(A, γl))⊗

n

in. (4.8)

First, we consider the vacuum: trivially we have

Ψ∅[A] =< A|∅ >= 1. (4.9)

Then, let us consider a loop α coloured by the fundamental repre-sentation j = 1

2(Figure 4.4). Since the node at A ≡ B is 2–valent,

12

A

B

Figure 4.4: Loop

the corresponding intertwiner is trivial, and one has

Ψα[A] =< A|α, j 12, id >= δBAU

AB = TrU(α,A). (4.10)

Let us consider a more complicated state, presented in Figure 4.5.We can solve the exercise recalling the QFT calculation regarding tocross section of Moeller scattering. We are three linearly dependentdiagrams where the propagator is the square of each of the Mandel-stam variables. We can just calculate transition amplitude takingthe sum of two of these, adding a sign minus at a diagram that dif-fers from the other one for the exchange of two fermionic lines. Forinstance, our choice is graphically represented in Figure 4.6. Hence,the functional is the sum of two terms, one for each diagram:

98

12

12

1 A′1A1 B′

1

C1

B1

C ′1B2 B′

2

Figure 4.5: A seminal example

12

1

12

12

12

= 12

12

12

1’

α

1’

γ

β

α

β ββ

β

γ γ

α

12 +

Figure 4.6: 2–loop decomposition

ΨS[A] =1

2[δA1B1δC1B2δB′1A′1δB′2C′1U(α,A)

A′1A1U(β,A)B1

B′1U(β,A)B2

B′2U(γ,A)

C′1C1

+

+ δA1B1δC1B2δB′1A′1δB′1A′1U(α,A)

A′1A1U(β,A)B1

B′2U(β,A)B2

B′1U(γ,A)

C′1C1

] =

=1

2[U(α,A)

A′1B1U(β,A)B1

A′1U(β,A)C1

B′2U(γ,A)

B′2C1

+

+ U(α,A)B′1A1U(β,A)A1

B′2U(β,A)B2

C1U(γ,A)C1

B′1] =

(4.11)

99

=1

2Tr[U(α,A)U(β,A)]Tr[U(β,A)U(γ,A)] +

+ Tr[U(α,A)U(β,A)U(β,A)U(γ,A)]. (4.12)

We can also calculate the eigenvalue of the Area quantum operatoracting on this spin network, which is

A = 8πγ~Gc−3∑

jl

√jl(jl + 1) = 8πγ~Gc−3(

√3 + 2

√2) (4.13)

Finally, let us determine the functional of a spin network with loopsintersecting at two tetravalent nodes (see Figure 4.7):

B2

A1

D1D2

C1 C2

B1

A2

12

12

12

12

i2

i1

α

δ

β

γi1 i2

=⇒

=⇒

D2 C2

B2A2

1

C1

D1

A1 B1

1

12

+ 12

B2 A2

D2

C2

Figure 4.7: Two tetravalent nodes

ΨS[A] = N1

2[δA1B1δD1C1δB2A2δC2D2U(α,A)A2

A1U(β,A)B1

B2U(γ,A)C1

C2U(δ, A)D2

D1+

+ δA1B1δD1C1δC2A2δB2D2U(α,A)A2

A1U(β,A)B1

C2U(γ,A)C1

B2U(δ, A)D2

D1] =

=1

2[U(α,A)A2

B1U(β,A)B1

A2U(γ,A)D1

C2U(δ, A)C2

D1+

+ U(α,A)B2A1U(β,A)A1

C2U(γ,A)C1

D1U(δ, A)D1

B2] =

=1

2Tr[U(α,A)U(β,A)]Tr[U(γ,A)U(δ, A)] +

+ Tr[U(α,A)U(β,A)U(γ,A)U(δ, A)] (4.14)

We have omitted to explicit splitting and normalization N . See thenext section for details. In this case, the Area takes the values

A = 8πγ~Gc−3∑

jl

√jl(jl + 1) = 8πγ~Gc−3(2

√3) (4.15)

100

4.3 Spinfoams

4.3.1 3j–symbol

We are interested in calculating transition amplitudes of our spin-foam models. In the previous chapter we have seen that these arerepresented with the Wigner nj–symbols. We briefly recall theirrelationship with Clebsch–Gordan coefficients. Given two spins j1

and j3 with representations m1 and m3 going to a spin j2 with rep-resentation m2, we obtain

< j1m1j3m3|j2m2 > = (−1)j1−j3+m2√

2j2 + 1

(j1 j3 j2

m1 m3 −m2

)

2j1 = a+ c

2j2 = a+ b

2j3 = b+ c

j1 + j2 + j3 = n ∈ N|j1 − j3| 6 j2 6 |j1 + j3|m1 +m3 = m2

The 3j–symbol can be also expressed by the Racah formula [36]:(j1 j3 j2

m1 m3 m2

)= (−1)j1−j3−m2

√∆(j1j3j2)(j1 +m1)!(j1 −m1)!×

×√

(j3 +m3)!(j3 −m3)!(j2 +m2)!(j2 −m2)!

∑t(−1)t

X,

where t indicates all the integers for which we calculate non–vanishingfactorial term, while the triangulation ∆, and the variable X are de-fined in this way

∆ (abc) =(a+ b− c)!(a− b+ c)!(−a+ b+ c)!

(a+ b+ c+ 1)!, (4.16)

X = t!(j2 − j3 + t+m1)!(j2 − j1 + t−m3)!(j1 + j3 − j2 − t)!××(j1 − t−m1)!(j3 − t+m3)! (4.17)

For example, we have

j1 = j3 =1

2; j2 = 1

m1 = m3 =1

2;m2 = 1

(12

12

112

12−1

)= − 1√

3∼ −0.57735. (4.18)

101

Moreover, we can interpret the 3j–symbol as a normalized inter-twiner at a 3–node. We call αk the representation indices of thespin jk in an orthonormal basis of each jk, and N a normalizingfunction. We have

3j = Niα2α1α3

=

(j1 j3 j2

α1 α3 α2

), (4.19)

N2 iα1α3α2

iα2α1α3

= 1, (4.20)

and we represent it in Figure 4.8. In the previous example we have

norm.pdf

3j = Niα2α1α3

=

α2

α3

α1

α2

α3

α1

α2

= 1N2iα2α1α3

iα1α3α2

=

α3

α1

Figure 4.8: 3j

explicited some values for outgoing and incoming representations atthe 3–node. In particular, there are the same of the spin networkswhose functional has been calculated in riferimento equaz. Main-taining the same notations and the same choice regarding the spincoupling, remarking that now we have the norm of the same inter-twiner, i.e. αk = α′k, we can immediately determine the function

102

N :

N2iα1α3α2

iα2α1α3

= N2(δA1B1δC1B2δB1A1δB2C1

+ δA1B1δC1B2δB1A1δB1A1

) = 1

N2(TrId2×2TrId2×2 + TrId2×2) = 1

N2(4 + 2) = 1

N =1√6. (4.21)

4.3.2 6j–symbol

The 6j–symbol represents the vertex amplitudes related to couplingof three spins, associating to a decay of a quantum of volume inthree quanta. However, in this section we are interested to performcalculations rather than speculations.Let us consider a 4–valent node. This can be splitted into two 3–valent nodes joined by a new ”virtual” link. There are three possibleways to perform a splitting. It is immediate the analogy with theMoeller scattering in Quantum Electrodynamics: a divergent inter-action vertex is replaced by a Feynman graph where the ”virtual”photonic propagator, i.e. the virtual link, is coloured by a repre-sentation preserving the conservation of the total momentum, andgraphically joining the external fermionic lines associated to the ini-tial and final particles (see Figure 4.9). The transition amplitudeis given by the sum of two out of the three possible diagrams as-sociated to the Mandelstam variables. Therefore, we can write theintertwiner for a 4–valent node in a chosen basis as product of twointertwiner for two 3–valent nodes:

iα3α4α1α2

= iα6α1α2

iα3α4α6

(4.22)

The virtual internal link is coloured by the representation j6. Wesee that we have to sum over all the possible j6 that satisfy Clebsch–Gordan conditions.Changing the basis over the intertwiner space means coupling inanother way the spins. Now, supposing to call the new virtual linkj5 connecting j1, j4 with j2, j3, we write

iα3α4α1α2

= iα4α1α5

iα3α5α2

, (4.23)

The change of basis is so defined:

iα6α1α2

iα3α4α6

=∑

j5

(2j5 + 1)

j1j2j5

j3j4j6

iα4α1α5

iα3α5α2

(4.24)

103

=⇒

Figure 4.9: 4–valent node

where we have used the Wigner 6j–symbol

6j =

j1j2j5

j3j4j6

(4.25)

We represent graphically these algebrities in Figure ?. As an ex-ercise, we calculate the 6j–symbol associated to a general tetra-hedron(see figure?). First, we expand the 6j–symbol performingindices contractions in the basis changing formula. Following thepicture, we have:

iα6α1α2

iα3α4α6

= [iα6α1α2

iα3α4α6

iα2α3α5

iα1α5α4

]iα4α1α5

iα3α5α2

. (4.26)

We note that in square parenthesis are contracted. Hence, we candefine the 6j– symbol as sum over all the representations of theproduct of the four 3j–symbol associated to the vertices of the tetra-hedron in Figure 4.11. The formula is the following:

6j =

j1 j2 j5

j3 j4 j6

=

∑

α1...α6

(j2 j1 j6

α2 α1 α6

)(j4 j3 j6

α4 α3 α6.

)×

×(j3 j5 j2

α2 α5 α2

)(j5 j4 j1

α5 α4 α1

)(4.27)

104

=

j4

j1

j5

j3j2j3

j4j1

j2

j3 j2

j5

j1

j4

j6

j3

j4j1

j2j6

j1

j3 j2

j5

j4

j3

j4j1

j2j6

=∑

j6(2j6 + 1)6j =

∑j6

(2j6 + 1)

j1j2j5j3j4j6

j1j2j5j3j4j6

=

j1

j2

j5

j6

j3

j4

Figure 4.10: change of basis

Thus, we have to perform a contraction over the 3j–symbols asso-ciated to the four vertices of the tetrahedron. Thanks to the Racahformula [28] we have

j1 j2 j5

j3 j4 j6

=√

∆(j1j2j5)∆(j1j4j6)∆(j3j2j6)∆(j3j4j5)∑

t

(−1)t(t+ 1)!

f(t)(4.28)

where t and the triangulation ∆ are as said above, and the functionf(t) is defined as

f(t) = (t− j1 − j2 − j5)!(t− j1 − j4 − j6)!(t− j3 − j5 − j4)!(t− j2 − j3 − j6)!

(j1 + j2 + j3 + j4 − t)!(j2 + j5 + j4 + j6 +−t)!(j5 + j1 + j6 + j3 − t)! (4.29)

105

j1

j2

j5

j6

j3

j4

Figure 4.11: 6j–symbol

As exercise, let us calculate the 6j–symbol for the chosen spins.After few steps, we have

j1 =1

2, j2 = 1, j5 =

3

2

j3 =1

2, j4 = 1, j6 =

1

2.

12

1 32

12

1 12

= −1

3. (4.30)

From our point of view, it is interesting to calculate the explicitintertwiners as invariant tensors products. First, following the no-tation of Figure 4.12, let us split the tetravalent vertex. We have:

iα3α4α1α2

=∑

α6

iα6α1α2

iα3α4α6

εA1B1δD2B2εD1C1 =

∑

F1

1

(2j6 + 1)εA1B1δ

D2F1δF1B2εD1C1 (4.31)

Now, we verify the basis changing, for that choice of coupling rep-resentations, i.e. one of the 3j-symbol products to sum in order to

106

j3 = 12

j2 = 1

j4 = 1j1 = 12

A1

B1

B2C1

D2

D1

A1

B1

B2

C1

D2

D1

=⇒

A1

B1

B2

=

C1

D2

D1

∑F1

1(2j6+1)

F1

F1

Figure 4.12: explicit splitting

obtain the 6j–symbol. Translating in algebra the graph in Figure4.13, we obtain

∑F1

1

(2j6 + 1)εA1B1δ

D2F1δF1B2εD1C1 =

[∑

F1

1

(2j6 + 1)

∑

E1,E2,E3

(2j5 + 1)3εA1B1δD2F1δF1B2×

× εD1C1εE3D1εE2D2δA1E1δE3C1δE2B2δE1B1

] ∑

E1,E2,E3

1

(2j5 + 1)3δE1A1εE2D2δC1

E3εE3D1εB1E1εB2E2

∑F1

1

(2j6 + 1)εA1B1δ

D2F1δF1B2εD1C1 =

∑

F1

1

(2j6 + 1)εA1B1δ

D2F1δF1B2εD1C1 q.e.d..(4.32)

4.3.3 15j–symbol

Now we could extend this result to the coupling of an higher numberof spins. If we have a 5–valent node, the splitting can be done byinserting two virtual links, and the basis changing is algebraically

107

A1

B1

B2C1

D2

D1

F1

F1

=

A1

C1

D2

D1

F1

F1

B1

B2

D2

D1

F1

F1E1E2E3

B1

B2

B1

B2

A1A1A1A1

D2

D1

C1C1

D2

D1A1

B2

B1

B2C1

E1 E2E3

Figure 4.13: explicit basis changing

represented by the 9j–symbol, because of we have a network withnine spins, the five initial ones plus four virtual. In particular, thebasis changing between two splittings of a 7–valent node, consistedto seven initial links plus eight virtual. But there is another situationwhere the 15j is important. As we have seen in chapter 3, in someSpinfoam models is the combinatorial amplitude obtained from thecontraction of five four–valent intertwiners. We represent the 15jwith a projection of a coloured pentachoron (see Figure ?). Making achoice about the splitting of each tetravalent node, we get the pictureof Figure 4.15 Calculations of 15j–symbol are very complicated, butthey do not introduce nothing but merely technical issues to what wehave already discussed. Referring to [37] for details, we determinethe splitting of the 15j–symbol depicted in Figure ?, where we assignthe values to the links expressed in Table 4.1

Following the graphical notation of the Figure 4.16, we have thisalgebra: the invariant tensor for ten spins is

iα1α2α3α4α5α6α7α8α9α10 = iα9α1α5α8

iα10α2α1α6

iα8α3α2α7

iα4α6α3α9

iα5α7α4α10

. (4.33)

Therefore, we have

15j =∑

α1α2...α14α15

iα1α8α11

iα11α9α5

iα10α12α1

iα2α12α6

iα8α13α2

iα3α13α7

iα6α14α3

iα4α14α9

iα7α15α4

iα5α15α10

=

=∑

j1j2...j13j14j15

δAH1εH2Kδ

IE1δKE2

δJ1A ε

J2LδB1L δB2

F δH1B2δM1B1εM2Gε

M2H2δCM1εFN2 ×

× δN1C δDN1

εN2IδO1D εO2GεO2J2δ

E1J1δE2O1, (4.34)

108

j3

i5

i3i4

i2

j1

j6

j10

j7

j2

j5

i1

j4j8j9

Figure 4.14: 15j

where, as usual, the sum is over all the spin representation of thevirtual links that satisfy the Clebsch–Gordan conditions.

109

j3

j15

j1j5

j8

j1

j11

j10

j1

j9

j6j7

j2

j14

j12

j13

Figure 4.15: 15–symbol splitting

110

Table 4.1: 15j resume

link representation labelj1

12 A

j2 1 B1, B2

j312 C

j412 D

j5 1 E1, E2

j612 F

j712 G

j8 1 H1, H2

j912 I

j10 1 J1, J2

j1112 K

j1212 L

j13 1 M1, M2

j14 1 N1, N2

j15 1 O1, O2

O1

O2

E1

B1

F

H1

K

N2

N1

M2

M1

I

D

B2G

L

J1

E2

A

H2

C

J2

Figure 4.16: explicit 15j–symbol

111

112

Chapter 5

Canonical formalism:application to f (R) theories

5.1 Introduction to f(R) theories

In this chapter we commit ourself to the study of a particular class oftheories: the f(R) theories. They are modified version of classicalGeneral Relativity, where the Ricci scalar R in the Einstein-Hilbertaction is replaced by a generic analytic function of R, just the abovementioned f(R). What is the reason behind this choice? We try toexplain it just below (the main reference for this section is [34]).Last cosmological data provide evidences of an acceleration of theuniverse. This can be expressed within the General Relativity frame-work with extra source terms in the Lagrangian function. At thismoment, we are not able to construct a solid theoretical structurethat describes all experimental information, so we rely on phenome-logical models: the most reliable one seems to be the Concordancemodel. It is in accordance with the observed supernovae Ia, cosmicmicrowave background anisotropies, large scale structure formalismbaryon oscillation and weak lensing. It is also called Λ–CDM model,because it imposes to add a cosmological constant term and it is suit-able to describe a universe dominated by cold dark matter. How-ever, there are several issues related to fine tuning that suggest usto look for other possible paths. There are models that foretell aninitial inflationary phase and a phase of accelerated expansion: thisfits our phenomological knowledge about dark energy. The gravita-tional actions is equipped with non linear terms or combination ofderivatives of Ricci scalar R. The field equations result to be in aninteresting form: we can arrange them in order to write correctionsas part of the righthand side of the field equations, i.e. as energy

113

momentum tensor contribution, a source term of geometrical origin.Then the function f can be fixed so that the acceleration of universecan be associated to these geometrical source. The main problem ofsuch theories is that field equations are very difficult to be solved.However, aided by systems theory, in the last years it has becomepossible to study in a more detailed way their dynamics. A classof f(R) theories admits a matter–dominant decelerated expansion,followed by an accelerated one. But these theories have more fun-damental and purely speculative basis.A relevant question that arose since the first year after General Rel-ativity birth was: is the Einstein theory the unique reliable andconsistent gravitational theory? Born as a merely theoretical diver-tissement, modifying the Einstein–Hilbert action with the introduc-tion of higher order invariants became a serious attempt to fit thetheory to new experimental data and, moreover, higher order cor-rection of the curvature term permitted to build a renormalizablegravitational theory at 1–loop order (even if not unitary) [35],[36].A new class of GR modifications was born: the so called high–ordertheories of gravity, characterized by the introduction in the action ofhigher–order curvature invariants with respect to the Ricci scalar,whose effects would be appreciable only at high energy scales, inparticular at the Planck scale, that is the main topics of our work.Such theories are developed in relation with cosmology. More recentdata depict a Universe composed by roughly 76% of Dark Energy,20% of Dark Matter and 4% of ordinary Barionic Matter. The DarkEnergy is associated to a cosmological constant parameter, and itslarge contribution leads to consider the present Universe in an ac-celerated expansion phase. In fact, the second Friedmann equation,derived from GR under assumptions of isotropy and homogeneity,is

a

a= −4πG

3(ρ+ 3P ), (5.1)

where a is the scale factor and ρ and P are density and pressure ofthe cosmological fluid. The constraint in order to have accelerationis (ρ + 3P ) < 0, satisfied only by Dark Energy. As we have saidbefore, the Λ–CDM model is the best fit phenomenological modelto describe Universe, but it is not be able to explain all the im-plication related to the theoretical consistency, as for example themagnitude and coincidence problem [37][38]. A massive effort isbeing performed to create alternative theories and models, to solveall the open questions and fit the experimental data. One pecu-liar perspective of the problem is the following: all troubles with

114

our description of the Universe derive from a not adequate descrip-tion of the gravitational interaction. This is the dominant forcethat rules the evolution of Universe and General Relativity turnsout not sufficiently effective to describe it. So we can avoid darkcomponents and other exotic (and not directly detected) stuff, if weslightly modify the theoretical framework built on GR sources. Oneshould remind that the existence of an unobserved planet affectingthe Mercury’s orbit was supposed in order to explain the anomalousprecession of Mercury perihelion, before of GR birth. Many possi-ble path are followed and have led to a wide landscape of theories:one particular class contains the so called f(R) theories. Given theEinstein-Hilbert action

SEH =1

16πG

∫d4x√−gR + SM , (5.2)

we generalize it in this way

S =1

16πG

∫d4x√−gf(R) + SM , (5.3)

where SM is the matter action, left in implicit form.The theories associated to this action have the fine property to bequiet simple to handle, englobing the main properties of higher–order theories but not being affected by their peculiar pathologyknown as Ostrogradski instability, that is a linear instability in theHamiltonian functions related to Lagrangians containing higher thanfirst order derivatives that can not be eliminated by partial integra-tion [39]. Even if we do not consider them as consistent alternativeto GR, they can be treated as useful toy models, interesting specu-lative tools in order to reach a better understanding of GR and ingeneral about fundaments of a consistent gravitational theory.A very interesting feature of actions related to f(R) theories isthat, applying the two traditional variational principles, the met-ric and the Palatini’s one, their behaviour differ from the Einstein–Hilbert action. Indeed, if the Lagrangian function is not linear inR, the two paths methods lead to two not equivalent set of fieldequations. So we must distinguish theories belonging to three sub-classes: the metric f(R) gravity, the Palatini f(R) gravity and themetric–affine f(R) gravity ones, where the latter is obtained ap-plying Palatini method but with matter action SM dependent onthe connection. A detailed analysis of the field equations of thesetheories reveals that the metric–affine formulation is the most gen-eral f(R) gravity theories. It foretells fermion fields associated to anon–vanishing torsion term, and it reduces to GR in vacuum con-dition or in presence of conformally invariant types of matter, for

115

example the electromagnetic field. The further constraint f ′′ 6= 0reduces Palatini and metric theories, tough by means of Legendretransformation, to scalar–tensor theories called Brans–Dicke theo-ries. Indeed, quadratic corrections to Einstein–Hilbert actions intro-duce massive scalar field added to the massless graviton. This set oftheories is extensively used in cosmology. This is another evidencethat leads us to consider with attention f(R) theories. A theorythat fits the last cosmological observations must satisfy some prop-erties: having the correct dynamics and describing the behaviourof gravitational perturbations and cosmological effects imposed bythe Cosmic Microwave Background. An arbitrary f(R) gravity isin accord to the conventional assumption that Universe has a highdegree of isotropy and therefore it is characterized by a Friedmann–Robertson–Walker spacetime. Nevertheless, it is necessary to act afine–tuning (fixing the value of Λ) in order to find a theory with theproperties mentioned above. We do not investigate over, but thestudy of cosmological perturbations is the key point to discriminatebetween such theories. After that, a good gravitational theoriesmust present the correct weak limit at Newtonian theory and avoid-ing the presence of non–physical fields as ghosts. The constraintf ′′ ≥ 0 guarantees the absence of massive fields with negative normviolating unitarity, i.e. ghosts. For Palatini formulation, it has beenshowed that the Cauchy problem and therefore the predictabilityof the theory are denied by higher derivatives of the matter fieldsin the field equations. This theory results non–physical: this state-ment is corroborated by the conflict with Standard Model due tonon–perturbative corrections of field equations and strong couplingbetween gravity and matter in local reference frame at low energies.These pathologies are caused by differential structure of its fieldequations. In metric theory the Cauchy problem is always well–defined, providing us more possible solutions with respect to GR:for example, Schwarzschild metric is no longer the only sphericallysymmetric solution! It can be considered as a possible alternativeto dark energy. Metric–affine gravity at this time has not suffi-ciently developed to claim about its consistence. Another approachis expressed stating that f(R) theories are effective theories. Extraterms are the contribution in an expansion of which we are consid-ering only the leading terms. Extra degrees of freedom associated tohigh–order derivatives may have been eliminated if we would con-sider expansion at every orders [40].Now it is the time to apply all the the knowledge gained during thistreatment to a particular problem: the connection of f(R) theoriesand Loop Quantum Gravity.

116

5.2 f(R) gravity field equations

5.2.1 GR recap

We determine field equations of f(R) General Relativity with arough calculation. The main fields of General Relativity are:

• the tetrad field eIµ

• the metric gµν

• the Levi–Civita connection Γαµν

• the curvature expressed by Rαβµν and its contractions

• the spin connection ωIJµ

Starting from the Einstein–Hilbert Lagrangian

LEH =√gR, (5.4)

we define the classical field equations (in vacuum)

δSEH = 0 =⇒ δLEH = 0 =⇒=⇒ Rµν −

1

2gµνR = 0 (5.5)

We can also obtain an equivalent formulation with tetrad fields anddifferential forms. Omitting the indices, i.e. considering the tracedequation, recalling that

SEH =

∫R ∧ e ∧ e

LEH = R ∧ e ∧ eDe = de+ ω ∧ eR = dω + ω ∧ ω = Dω, (5.6)

we have

δSEH = 0 =⇒ δ(R ∧ e ∧ e) = 0

δR ∧ e ∧ e+ 2R ∧ δe ∧ e = 0

δD︸︷︷︸=Dδ

ω ∧ e ∧ e+ 2R ∧ δe ∧ e = 0

D(δω ∧ e ∧ e)− 2δω ∧D(e ∧ e) + 2R ∧ δe ∧ e = 0

(5.7)

Now we calculate the variations

δL

δe= 2R ∧ e, δL

δω= −2D(e ∧ e). (5.8)

117

Considering that D(e∧e) ∝ De∧e and that non degenerate metricssatisfy De = 0, we have the GR field equations

R ∧ e = 0D(e ∧ e) = 0.

(5.9)

5.2.2 f(R) gravity

Now we replace the Ricci scalar R with f(R), where f is an analyticfunction. We maintain f without an explicit form. Let us define

Sf =

∫d4x√gf(R). (5.10)

We have

Sf =

∫d4x√gf(R) (5.11)

δL = δ(√g)f(R) + δf(R)

√g (5.12)

(5.13)

Variation of fields g and f(R) = f(R(µν)gµν) are so defined

δ√g = −

√g

2gαβδg

αβ (5.14)

δf(R) =∂f

∂RδR = f ′(R)δR = (5.15)

= f ′(R)δR(µν)gµν +

∂f

∂RR(µν)δg

µν (5.16)

Hence, we write

δL = −√g

2gµνδg

µνf(R) + [δR(µν)gµν +R(µν)δg

µν ]√gf ′(R) =

= [R(µν)f′(R)]− 1

2gµνf(R) + δR(µν)f

′(R)]√gδgµν = 0(5.17)

The first two terms give the field equations

R(µν)f′(R)− 1

2gµνf(R) = 0; (5.18)

the third term needs a further evaluation. Recalling that

R(µν) = Rλλµν = ∂λΓ

λµν − ∂νΓλµλ + ΓλσλΓ

σµν − ΓλσνΓ

σµλ (5.19)

δRµν = ∇λδΓλµν −∇νδΓ

λµλ + 2Γσ[νλ]δΓ

λµσ (5.20)

118

we obtain

√ggµνf ′(R)δR(µν) =

√ggµνδ[Rλ

λµν ]f′(R) (5.21)

√ggµνf ′(R)[∇λδΓ

λµσ −∇νδΓ

λµλ + 2Γσ[νλ]δΓ

λµσ] (5.22)

Now, we see that the third term vanishes because of boundary con-ditions. We make an integration by parts over the first two terms,erasing divergences, and we obtain

−2∇λ[√ggµνf ′(R)][δΓλµν − δλ(µΓαν)α] + 2∇λ[

√ggµνf ′(R)][δΓλµν − δλ(µΓαν)α]

The final step is making variation with respect to the metric andthe connection: we obtain the second field equation:

∇λ[√ggµνf ′(R)] = 0 (5.23)

5.3 Extension of LQG formalism to f(R)

Let us consider the Holst Lagrangian, starting point of LQG, pre-sented in Chapter 1:

LH =1

4k(Rab ∧ ec ∧ edεabcd −

2

γRab ∧ ea ∧ eb) (5.24)

where k is a constant and γ is the Barbero–Immirzi parameter. Werecast it as

LH =1

8k(Rab

µνecρedσεabcd −

2

γRabµνeaρebσ)εµνρσds =

=e

2k(Rab

µνeµaeνb −

1

2γRabµνe

µaeνdεcd . .ab )ds =

=e

2kβRds, (5.25)

where ds is the volume induced by coordinates. Defining β = − 12γ

,

we have

βR = Rabµνe

µaeνb + βRab

µνecµedνεcdab. (5.26)

Now, we consider

L = ef(βR); (5.27)

119

we determine its field equations:

δL = e[−feaµδeµa + f ′(2∇µδΓabν e

µaeνb + 2Rb

νδeνb +

+ 2β∇µδΓabν e

cµedνεcdab + 2abµνecµδedνεcdab)] =

= 2e(f ′Raµ −

1

2eaµ + βRcd

µνeνb ε

abcd ..)δe

µa +

− 2∇µ(ef ′eµc eνd)(δ

c[aδ

db] + βεcd. .ab)δΓ

abν + 2∇µ[ef ′eµc e

νd(δ

c[aδ

db] +

+ βεcd. .ab)Γabν ]

imposing the boundary condition δΓabµ = 0 we obtain the field equa-tions

f ′Ra

µ − 12eaµ + βRcd

µνeνb ε

abcd .. = 0

∇µ(ef ′eµc eνd)(δ

c[aδ

db] + βεcd. .ab) = 0

(5.28)

where the second equation can be also written in the equivalentform:

∇µ(ef ′eµ[ceνd]) = 0. (5.29)

By considering a new frame

eaµ =√|f ′|eaµ, (5.30)

one can show that all the fields and their combination in the newframe labelling them with the tilde, for example

Rabµν = Rα

λµν gλβ eaαe

bβ (5.31)

Now, we can show that:

R = εf ′RβR = R (5.32)

Rcdµνe

νb εabcd .. = 0.

Since that, we recast the field equations:f ′Rµν − 1

2fgµν = 0

∇µ(eeµ[ceνd]) = 0,

(5.33)

and immediately we recognize the equivalence to standard metric–affine formalism of f(R). Now we consider again the selfdual for-mulation of GR (we have presented it in chapter 2):

L+ =1

2kF, (5.34)

120

that returns the following field equationspabi F

iµνe

cρεµνρσεabcd = 0,

pabi ∇µ(eaνebρ)ε

µνρσ = 0(5.35)

remembering that we have previously defined a manifold M whosecoordinate determines naturally the differential forms. If we con-sider a boundary surface

i : S −→M : kA 7−→ xµ(k) (5.36)

we can construct coordinates xµ = (t, kA) adapted to S such that

i : kA 7−→ kA, (5.37)

∂Axµ = δµA. (5.38)

Projecting the field equations on S we obtain the following con-straints:

∇AE

Ai = 0

F iABE

Ai = 0

εjki FiABE

Aj E

Bk = 0.

(5.39)

that are the equations from which we have started to describe LoopQuantum Gravity.Now, we repeat the same procedure substituting in the Lagrangianfunction F with f(F ). The field equations are

f ′pabi F

iµνe

µa − 1

2febν = 0,

pabi ∇µ(ef ′eµaeνb ) = 0

(5.40)

We make the change of basis used above to construct the new frameeaµ =

√|f ′|eaµ, Tracing the first equation, we obtain the so–called

master equation: each of its zeros determine a sector of the quantumtheory and we have to sum over all of them. In case there is only onezero, we define it as F = ρ, and we also define σ = signum(f ′(ρ)).It is possible verify that F = σ

f ′F . The fields equations become

pabi ∇µ(eeµa eνb ) = 0

f ′F − 2f = 0 =⇒ F = ρ

pabi Fiµν e

µa − 1

4F ebν = 0.

(5.41)

The first and the second equations are equivalent to the the firstand second equation of the standard Holst Lagrangian. In spite ofthis, the third equation is different. In fact, we have

∇AEAi = 0

F iABE

Ai = 0

εjki FiABE

Aj E

Bk = σ

f ′ρE

(5.42)

121

Following [41], we have verified that the Holst Lagrangian, whichis dynamically equivalent to standard GR, admits the same defor-mation by substitution of βR with f(βR), and this theory is dy-namically equivalent to the deformation of GR by substitution ofR with f(R). Since f(R) theories are models that provide differ-ent dynamics with respect to GR, and such differences are testableat cosmological scales, this deep relationship at the level of mas-ter equations with Loop Quantum Gravity seems to indicate newpossibilities to explore the physical consistency of the theory. In-serting matter terms in field equations, the equivalence continuesto be satisfied [42]. It is possible to show that this result holds atquantum level as well. In particular, let us consider the three fieldequations of LQG introduced in chapter 1. The first and the secondassures the invariance under gauge transformations and diffeomor-phism. Since f(R) theories maintain these properties, we concludethat the kinematical theory is the same for the two approaches.The discretization of the space and the construction of geometricaloperators as Area and Volume are not affected by the choice of aparticular function f . The differences arise when we try to definethe dynamics, i.e. when we consider the third field equation, thatis the Wheeler–DeWitt one: this involves the choice of the Hamil-tonian function. This is very interesting because selecting an f(R)dynamics is just what we do when we make predictions in cosmology.

122

Conclusions

In the end, we make some final considerations about this work.In the first part, we have presented and explained the key aspectsof two of the modern Quantum Gravity class of theories, the LoopQuantum Gravity and the Spinfoam models, stressing their rele-vant similarities and mentioning the differences. Both of them arefounded on the simplicial approach to Quantum Gravity, and a setof mathematical techniques derived from combinatorial calculus ap-plicated to Group Representation Theory. It implies a possible per-spective on Quantum Gravity that favours the merging betweenGeneral Relativity, the most reliable theory of gravity, and Quan-tum Mechanics, the fundamental cornerstone of the modern Physics,that is constructed and interpreted just as a not deterministic (insome sense!) and probabilistic theory. In recent years, there havebeen many interesting developments. Maybe, the most importantis the creation of the so–called EPR model, at this time the mostadvanced and promising theory based on spinfoam approach (see[43]). On the other hand, some recent experiments (see [44]), eventough whose results needs further analysis, seem to indicate thatGR continues to describe the physical world very well.The second part has concerned the applications. In chapter 4, wehave performed calculations about transition amplitude in LQG andSpinfoam models, supporting the reader with graphical tools. Inchapter 5, we have briefly introduced the f(R) theories, toy modelswith large applications in Physics, and presented the possible con-nections with Loop Quantum Gravity, that could permit to exploreits physical consistence.

123

124

Appendix A

All we need is math

We refer to [2][45] for this appendix, where we collect some usefultechnicalities for calculations in Quantum Gravity and generally inPhysics.

A.1 Matrices

In the huge literature about this topic we often face the problemto represent the components of a square matrix, its inverse andthe transpose. Since we think that it’s particularly effective, fora generic 2 × 2 matrix in this thesis we prefer adopt the notationbelow:

A = Aµν

A−1 = A νµ

tA = A µν

tA−1 = Aνµ.

A.2 Tensors in brief

A.2.1 Definition

Definition A.2.1 Given a vector space V and a field K, the tensorspace Tp

q is the set of multilinear functions such thatT ∈ Tp

q(V) : V ⊗ . . .⊗V⊗︸ ︷︷ ︸q times

V∗ ⊗ . . .⊗V∗︸ ︷︷ ︸p times

−→ K

A generic element of Tpq(V ) is T = T

ν1...νpµ1...µqe

µ1...µqν1...νp . Given vectors

vi = vµieµi∈ V and covectors αj = ανj

eνj ∈ V ∗, where eµiis a basis

125

of V and eνj is a basis of V ∗, we have

T (vi, . . . , vq, α1, . . . , αp) = vµ1 . . . vµqαν1 . . . ανpTν1...νpµ1...µq

.

The indices of T written above are called contravariants, the onesbelow are covariants. A tensor T could be considered as a multi-linear operator that starting from q vectors and p covectors gives ascalar. The number of components of a tensor is nr, where n is thedimension of the spaces V and V∗ (it is always the same if n is afinite number, because in this case the two spaces are isomorphic)and r is its rank, i.e. r = p + q. A tensor with rank 0 is triviallya scalar, one of rank 1 corresponds to an n-dimensional (co)vectorand with rank 2 we have n× n matrices.

A.2.2 Techniques of calculation

We list the principal rules of transformation for a tensor.

• Linearity of operator⊗

. Given m tensorial spaces Tpiqi, i =

1, . . . ,m we have ⊗ : ×iTpiqi−→ T

Pi piPi qi

. A generic element

of this space is∏

iTiν1...νpiµ1...µqi

⊗i eµ1...µqiν1...νpi

This operation allows todefine the tensorial space as an algebraic structure.

• Tensors having the same rank can be added:

(aT + bS)(vµ1 . . . vµqαν1 . . . ανp) = aT (vµ1 . . . vµqαν1 . . . ανp)+

+ bS(vµ1 . . . vµqαν1 . . . ανp)

with a, b ∈ K. It derives from the possibility of raising andlowering indices: in the Minkowski space we have

Tαβγ = ηβµTαµγ

Tαβγ = ηβµTαµγ

Since we talk about quantum gravity, we need to define rais-ing and lowering indices for an arbitrary Riemannian space.Considering a metric g, we adopt the following notation:

Tαβγ = gβµTαµγ

Tαβγ = gβµTαµγ

• Knowing the components of a tensor T , we can obtain thecomponents in another frame with this law of transformation:

Tα′

β′γ′ = T µνρΛα′

µ Λνβ′Λ

ργ′

where Λ matrix links a frame with the other one.

126

Now we take into account operations that starting from a tensorgive us another tensor.

• We consider the permutation of two element of a tensor T be-longing to T 0

p (or T p0 ). If we have

T (vσ1 , . . . , vσp) = T (v1, . . . , vp)

for all the permutations (σ1, . . . σp), then T is a symmetric ten-sor. In case of

T (vσ1 , . . . , vσp) = (−1)n(σ)T (v1, . . . , vp),

where n(σ) is the sign of the permutation, the tensor T is calledantisymmetric. We can define two projectors S and A thatstarting from a generic T gives us its symmetric and antisym-metric part:

(ST )(v1, . . . , vp) =1

p!

∑

σ

T (vσ1 , . . . , vσp)

(AT )(v1, . . . , vp) =1

p!

∑

σ

(−1)n(σ)T (vσ1 , . . . , vσp).

A tensor of rank 2, i.e. a square matrix, can be decomposedinto the sum of its symmetric and antisymmetric part; on theother hand, this is impossible for rank 3 tensors1. In this thesiswe use the following notation:

ST (v1, . . . , vp) = T(v1,...,vp)

AT (v1, . . . , vp) = T[v1,...,vp].

• The gradient of a tensor, for example a (11) tensor, is written as

∇T . Its components are Tαβ;γ = Tαβ,γ + ΓαµγTµβ − ΓµβγT

αµ , whereΓ

is the Levi–Civita connection. In this notation comma is usedfor gradient in flat spacetime and the semicolon for calculationsin a generic space.

• The divergence of a (11) tensor is written as ∇ · T . Its compo-

nents are Tαβ;β = Tαβ,β+ΓαµβTµβ−ΓµββT

αµ . Gradient and divergence

increase the rank of a tensor.

• We can reduce the rank of a tensor with contraction: gµνTµναβ =

Tαβ, where g is a metric tensor.

1Other types of symmetries are remarkable thanks the use of Young diagrams.

127

A.2.3 Levi-Civita tensor

The Levi-Civita tensor2 ε is a completely antisymmetric object. Ina n–dimensional space it is a n–rank tensor with components

εµ1...µn = 1

εµ1...µn = 0, unless all the indices are different

εσ(µ1)...σ(µn) =

+1 for even permutations−1 for odd permutations.

We can raise and lower indices with the relations

εα1...αn =n∏

i=1

gαiβiεβ1...βn. ... . =

n∏

i=1

gαiβidet(g)εβ1...βn .

The Levi-Civita tensor allows to construct particular tensors usedin combinatorial calculus. In index notation we have

δαβγµνρ = −εαβγλεµνρλ =

+1 if αβγ is an even permutation of µνρ−1 if αβγ is an odd permutation of µνρ0 otherwise

δαβµν = δαµδβν − δαν δβµ =

=1

2δαβγµνγ = −1

2εαβγλεµνγλ =

+1 if αβ is an even permutation of µν−1 if αβ is an odd permutation of µν0 otherwise

δαµ =1

3δαβµβ =

1

6δαβγµβγ = −1

6εαβγλεµβγλ =

+1 if α = µ0 otherwise.

From these results we immediately get the following important re-lation

εα1...αnεβ1...βn = n!δ[αi

β1. . . δ

αn]βn.

So in the most general n–dimensional space we have

δα1...αn

β1...βn=

+1 if (α1 . . . αn) is an even permutation of (β1 . . . βn)−1 if (α1 . . . αn) is an odd permutation of (β1 . . . βn)0 otherwise.

Another transformation is related with ε: the duality ∗. We cantransform vectors, second and third-rank antisymmetric tensor in

2a mathematician would affirm that it is a tensorial density, but this distinction is notrelevant for our aims.

128

this way:

∗Tλ =1

3!Tαβγε ·

αβγλ

∗Tγλ =1

2Tαβε · ·

αβγλ

∗Tβγλ = Tαε · · ·αβγλ

A.3 Forms

Now we define maybe the most important operation on tensor: thewedge product ∧. Given two forms T and S we obtain a new formT ∧ S. A form is an antisymmetric tensor. It’s possible to developa formalism suitable for theoretical physics based on forms: this isthe exterior calculus. We list the most relevant formulas of it3.

• A general p–form is an antisymmetric tensor of rank (p0) (or(0p)), and can be expanded in terms of wedge product:

ω =1

p!ωi1pe

i1 ∧ . . . ∧ eip =∑

i1<...<ip

ωi1pei1 ∧ . . . ∧ eip

≡ ω|i1p|ei1 ∧ . . . ∧ eip .

• If we have a p–form α and a q–form β, their wedge productfollows this law:

α ∧ β = (−1)pqβ ∧ α.

• Exterior derivative

d(α ∧ β) = dα + (−1)pα ∧ dβ,

and dα is a (p+ 1)–form.

• Integration of a p–form α over a p–dimensional surface S. Firstly,we rewrite α = α|i1...ip|e

i1 ∧ . . . ∧ eip using a parametrization ofthe surface: ek(λ1 . . . λp). So we have

α = a(λj)dλ1 ∧ . . . ∧ dλp∫

S

α =

∫

S

a(λj)dλ1 ∧ . . . ∧ dλp.

3Exterior derivative and integration are applicable to differential orientable manifolds.

129

Another important result is the generalised Stokes Theorem:∫

∂Σ

α =

∫

Σ

dα,

where Σ is a general (p+ 1)–surface and ∂Σ is its p–boundary.

• Norm of a p–form

‖α2‖ = α|i1...ip|αi1...ip .

• The dual of a p–form in a n–dimensional space is a (n−p)–formwith components

(∗α)k1...kn−p = α|i1...ip|ε · ... ·i1...ipk1...kn−p.

• In the end, since forms are antisymmetric tensors, we easilyfind the components of the exterior product of p vectors:

(v1 ∧ . . . ∧ vp)α1...αp = p!v[α1

1 . . . ∧ vαp]p .

A.4 Notations

In this section we resume the notations used for mathematical ob-jects that come up in this thesis many times. We decide to followthe same notation of [1], that is the most common in literature.

• For R3, we use latin indices i, j, . . . = 1, 2, 3. This ones areraised and lowered with the R3 metric δij.

• Capital latin letters represent 4D Lorentz tangent indices: I, J =0, 1, 2, 3. A Lorentz vector is written by vI = (v0, vi). TheMinkowsky metric ηIJ raises and lowers this indices.

• µ, ν, . . . = 0, 1, 2, 3 are 4D spacetime tangent indices, and a, b, . . . =1, 2, 3 are 3D tangent indices. So we can define on a 4D man-ifold the coordinates x, whose components are xµ = (t, ~x) =(x0, xa).

• The metric tensor is written as

gµν(x) = ηIJeIµ(x)eJν (x),

where eIµ(x) is the tetrad field, with√− det g = | det e|. For a

generic 2–rank tensor T we have

TIJeIµeJν = Tµν

∗Tαβ =1

2| det e|εαβµνT µν =

1

2ε · ·αβµνT

µν .

130

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