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Loop Quantum Gravity

(Review written for the electronic journal LIVING REVIEWS)

Carlo Rovelli,Department of Physics and Astronomy,

University of Pittsburgh, Pittsburgh Pa 15260, USArovelli@pitt.edu

(September 29, 1997)

AbstractThe problem of finding the quantum theory of the gravitational field, and thus understanding what is quantum space-time, is still open. One of the most active of the current approaches is loop quantum gravity. Loop quantum gravityis a mathematically well-defined, non-perturbative and background independent quantization of general relativity,with its conventional matter couplings. The research in loop quantum gravity forms today a vast area, ranging frommathematical foundations to physical applications. Among the most significative results obtained are: (i) The compu-tation of the physical spectra of geometrical quantities such as area and volume; which yields quantitative predictionson Planck-scale physics. (ii) A derivation of the Bekenstein-Hawking black hole entropy formula. (iii) An intriguingphysical picture of the microstructure of quantum physical space, characterized by a polymer-like Planck scale dis-creteness. This discreteness emerges naturally from the quantum theory and provides a mathematically well-definedrealization of Wheeler’s intuition of a spacetime “foam”. Long standing open problems within the approach (lackof a scalar product, overcompleteness of the loop basis, implementation of reality conditions) have been fully solved.The weak part of the approach is the treatment of the dynamics: at present there exist several proposals, which areintensely debated. Here, I provide a general overview of ideas, techniques, results and open problems of this candidatetheory of quantum gravity, and a guide to the relevant literature.

I. INTRODUCTION

The loop approach to quantum gravity is ten years old.The first announcement of this approach was given at aconference in India in 1987 [182]. This tenth anniversaryis a good opportunity for attempting an assessment ofwhat has and what has not been accomplished in theseten years of research and enthusiasm.

During these ten years, loop quantum gravity hasgrown into a wide research area and into a solid andrather well-defined tentative theory of the quantum grav-itational field. The approach provides a candidate the-ory of quantum gravity. It provides a physical picture ofPlanck scale quantum geometry, calculation techniques,definite quantitative predictions, and a tool for discussingclassical problems such as black hole thermodynamics.

We do not know whether this theory is physically cor-rect or not. Direct or indirect experimental corrobora-tion of the theory is lacking. This is the case, unfor-tunately, for all present approaches to quantum gravity,due, of course, to the minuteness of the scale at whichquantum properties of spacetime (presumably) manifestthemselves. In the absence of direct experimental guid-ance, we can evaluate a theory and compare it with al-ternative theories only in terms of internal consistencyand consistency with what we do know about Nature.

Long standing open problems within the theory (suchas the lack of a scalar product, the incompleteness ofthe loop basis and the related difficulty of dealing withidentities between states, or the difficulty of implement-

ing the reality conditions in the quantum theory) havebeen solidly and satisfactorily solved. But while it isfairly well developed, loop quantum gravity is not yeta complete theory. Nor has its consistency with clas-sical general relativity been firmly established yet. Thesector of the theory which has not yet solidified is the dy-namics, which exists in several variants presently underintense scrutiny. On the other hand, in my opinion thestrength of the theory is its compelling capacity to de-scribe quantum spacetime in a background independentnonperturbative manner, and its genuine attempt to syn-thesize the conceptual novelties introduced by quantummechanics with the ones introduced by general relativity.

The other large research program for a quantum the-ory of gravity, besides loop quantum gravity, is stringtheory, which is a tentative theory as well, and is moreambitious than loop gravity, since it also aims at unifyingall known fundamental physics into a single theory. Insection II C, I will compare strengths and weaknesses ofthese two competing approaches to quantum gravity.

This “living review” is intended as a tool for orientingthe reader in the field of loop gravity. Here is the plan ofthe review:

• Section II, “Quantum Gravity: where arewe?”, is an introduction to the problem, the rea-son of its relevance, and the present state of ourknowledge.

• Section III, “History of loop quantum grav-ity”, is a short overview of the historical develop-ment of the theory.

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• Section IV, “Resources” contains pointers to in-troductory literature, institutions where loop grav-ity is studied, web pages, and other informationthat may be of use to students and researchers.

• Section V, “Main ideas and physical inputs”,discusses the physical and mathematical ideas onwhich loop quantum gravity is based, at a rathertechnical level.

• The actual theory is described in detail in sectionVI, “The formalism”.

• Section VII, “Results”, is devoted to the resultsthat have been derived from the theory. I have di-vided results in two groups. First, the “technical”results (VII A), namely the ones that have impor-tance for the construction and the understandingof the theory itself, or that warrant the theory’sconsistency. Second, the “physical” results (VII B):what the theory says about the physical world.

• In section VIII, “Open problems and currentlines of investigation”, I illustrate what I con-sider the main open problems, and the main cur-rently active research lines.

• In section IX, “Short summary and conclu-sion”, I summarize very briefly the state and theresult of the theory, and present (necessarily verypreliminary!) conclusions.

At the price of several repetitions, the structure of thisreview is very modular: sections are to a large extent in-dependent from each other, have different style, and canbe combined according to the interest of the reader. Areader interested only in a very brief overview of the the-ory and its results, can find this in section IX. Graduatestudents or persons of general culture may get a generalidea of what goes on in this field and its main ideas fromsections II and VII. If interested only in the technical as-pects of the theory and its physical results, one can readsections VI and VII alone. Scientists working in this fieldcan use section VI and VII as a reference, and I hope theywill find sections II, III and V and VIII stimulating. . .

I will not enter technical details. However, I will pointto the literature where these details are discussed. I havetried to be as complete as possible in indicating all rel-evant aspects and potential difficulties of the issues dis-cussed.

The literature in this field is vast, and I am sure thatthere are works whose existence or relevance I have failedto recognize. I sincerely apologize with the authors whosecontributions I have neglected or under-emphasized andI strongly urge them to contact me to help me makingthis review more complete. The “living reviews” are con-stantly updated and I should be able to correct errors andomissions.

II. QUANTUM GRAVITY: WHERE ARE WE?

This is a non-technical section in which I illustrate theproblem of quantum gravity in general, its origin, its im-portance, and the present state of our knowledge in thisregard.

The problem of describing the quantum regime of thegravitational field is still open. There are tentative theo-ries, and competing research directions. For an overview,see [121]. The two largest research programs are stringtheory and loop quantum gravity. But several other di-rections are being explored, such as twistor theory [154],noncommutative geometry [68], simplicial quantum grav-ity [6,65,53,1], Euclidean quantum gravity [104,107], theNull Surface Formulation [85–87] and others.

String theory and loop quantum gravity differ not onlybecause they explore distinct physical hypotheses, butalso because they are expressions of two separate commu-nities of scientists, which have sharply distinct prejudices,and view the problem of quantum gravity in surprisinglydifferent manners.

A. What is the problem? The view of a high energyphysicist

High energy physics has obtained spectacular successesduring this century, culminated with the (far from linear)establishment of quantum field theory as the general formof dynamics and with the comprehensive success of theSU(3) × SU(2) × U(1) Standard Model. Thanks to thissuccess, now a few decades old, physics is in a conditionin which it has been very rarely: there are no experimen-tal results that clearly challenge, or clearly escape, thepresent fundamental theory of the world. The theory wehave encompasses virtually everything – except gravita-tional phenomena. From the point of view of a particlephysicist, gravity is then simply the last and weakest ofthe interactions. It is natural to try to understand itsquantum properties using the strategy that has been sosuccessful for the rest of microphysics, or variants of thisstrategy. The search for a conventional quantum fieldtheory capable of embracing gravity has spanned severaldecades and, through an adventurous sequence of twists,moments of excitement and disappointments, has lead tostring theory. The foundations of string theory are notyet well understood; and it is not yet entirely clear howa supersymmetric theory in 10 or 11 dimensions can beconcretely used for deriving comprehensive univocal pre-dictions about our world.∗ But string theory may claim

∗I heard the following criticism to loop quantum gravity:“Loop quantum gravity is certainly physically wrong, be-cause: (1) it is not supersymmetric, (2) is formulated infour dimensions”. But experimentally, the world still insists

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extremely remarkable theoretical successes and is todaythe leading and most widely investigated candidate the-ory of quantum gravity.

In string theory, gravity is just one of the excitations ofa string (or other extended object) living over some back-ground metric space. The existence of such backgroundmetric space, over which the theory is defined, is neededfor the formulation and for the interpretation of the the-ory, not only in perturbative string theory, but in therecent attempts of a non-perturbative definition of thetheory, such as M theory, as well, in my understanding.Thus, for a physicist with a high energy background, theproblem of quantum gravity is now reduced to an aspectof the problem of understanding what is the mysteriousnonperturbative theory that has perturbative string the-ory as its perturbation expansion, and how to extractinformation on Planck scale physics from it.

B. What is the problem? The view of a relativist

For a relativist, on the other hand, the idea of a funda-mental description of gravity in terms of physical excita-tions over a background metric space sounds physicallyvery wrong. The key lesson learned from general rela-tivity is that there is no background metric over whichphysics happens (unless, of course, in approximations).The world is more complicated than that. Indeed, for arelativist, general relativity is much more than the fieldtheory of a particular force. Rather, it is the discoverythat certain classical notions about space and time areinadequate at the fundamental level; they require mod-ifications which are possibly as basics as the ones thatquantum mechanics introduced. One of such inadequatenotions is precisely the notion of a background metricspace (flat or curved), over which physics happens. Thisprofound conceptual shift has led to the understandingof relativistic gravity, to the discovery of black holes, torelativistic astrophysics and to modern cosmology.

From Newton to the beginning of this century, physicshas had a solid foundation in a small number of key no-tions such as space, time, causality and matter. In spiteof substantial evolution, these notions remained ratherstable and self-consistent. In the first quarter of this cen-tury, quantum theory and general relativity have mod-ified this foundation in depth. The two theories haveobtained solid success and vast experimental corrobora-tion, and can be now considered as established knowl-edge. Each of the two theories modifies the conceptualfoundation of classical physics in a (more or less) inter-

on looking four-dimensional and not supersymmetric. In myopinion, people should be careful of not being blinded by theirown speculation, and mistaken interesting hypotheses (suchas supersymmetry and high-dimensions) for established truth.

nally consistent manner, but we do not have a novel con-ceptual foundation capable of supporting both theories.This is why we do not yet have a theory capable of pre-dicting what happens in the physical regime in whichboth theories are relevant, the regime of Planck scalephenomena, 10−33 cm.

General relativity has taught us not only that spaceand time share the property of being dynamical with therest of the physical entities, but also –more crucially–that spacetime location is relational only (see sectionVC). Quantum mechanics has taught us that any dy-namical entity is subject to Heisenberg’s uncertainty atsmall scale. Thus, we need a relational notion of aquantum spacetime, in order to understand Planck scalephysics.

Thus, for a relativist, the problem of quantum gravityis the problem of bringing a vast conceptual revolution,started with quantum mechanics and with general rela-tivity, to a conclusion and to a new synthesis.† In thissynthesis, the notions of space and time need to be deeplyreshaped, in order to keep into account what we havelearned with both our present “fundamental” theories.

Unlike perturbative or nonperturbative string theory,loop quantum gravity is formulated without a back-ground spacetime, and is thus a genuine attempt to graspwhat is quantum spacetime at the fundamental level. Ac-cordingly, the notion of spacetime that emerges from thetheory is profoundly different from the one on which con-ventional quantum field theory or string theory are based.

C. Strings or loops?

Above, I have emphasized the radically distinct cul-tural paths leading to string theory and loop quantumgravity. Here, I attempt a comparison of the actualachievements that the two theories have obtained so farfor what concerns the description of Planck scale physics.

Once more, however, I want to emphasize here that,whatever prejudices this or that physicist may have, boththeories are tentative: as far as we really know, either,or both, theories could very well turn out to be physi-cally entirely wrong. And I do not mean that they couldbe superseded: I mean that all their specific predictionscould be disproved by experiments. Nature does not al-ways share our aesthetic judgments, and the history oftheoretical physics is full of enthusiasm for strange the-ories turned into disappointment. The arbiter in scienceare experiments, and not a single experimental result sup-ports, not even very indirectly, any of the current theoriesthat go beyond the Standard Model and general relativity.To the contrary, all the predictions made so far by theo-ries that go beyond the Standard Model and general rel-

†For a detailed discussion of this idea, see [180] and [197].

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ativity (proton decay, supersymmetric particles, exoticparticles, solar system dynamics) have for the momentbeen punctually falsified by the experiments. Comparingthis situation with the astonishing experimental successof the Standard Model and classical general relativityshould make us very cautious, I believe. Lacking exper-iments, theories can only be compared on completenessand aesthetic criteria – criteria, one should not forget,that according to many favored Ptolemy over Coperni-cus, at some point.

The main merit of string theory is that it provides a su-perbly elegant unification of known fundamental physics,and that it has a well defined perturbation expansion, fi-nite order by order. Its main incompleteness is that itsnon-perturbative regime is poorly understood, and thatwe do not have a background-independent formulation ofthe theory. In a sense, we do not really know what is thetheory we are talking about. Because of this poor under-standing of the non perturbative regime of the theory,Planck scale physics and genuine quantum gravitationalphenomena are not easily controlled: except for a fewcomputations, there is not much Planck scale physics de-rived from string theory so far. There are, however, twosets of remarkable physical results. The first is given bysome very high energy scattering amplitudes that havebeen computed (see for instance [2–5,209,199]). An in-triguing aspect of these results is that they indirectlysuggest that geometry below the Planck scale cannot beprobed –and thus in a sense does not exist– in stringtheory. The second physical achievement of string the-ory (which followed the d-branes revolution) is the recentderivation of the Bekenstein-Hawking black hole entropyformula for certain kinds of black holes [198,111,110,109].

The main merit of loop quantum gravity, on the otherhand, is that it provides a well-defined and mathemati-cally rigorous formulation of a background-independentnon-perturbative generally covariant quantum field the-ory. The theory provides a physical picture and quanti-tative predictions on the world at the Planck scale. Themain incompleteness of the theory regards the dynam-ics, formulated in several variants. So far, the theory haslead to two main sets of physical results. The first is thederivation of the (Planck scale) eigenvalues of geometri-cal quantities such as areas and volumes. The second isthe derivation of black hole entropy for “normal” blackholes (but only up to the precise numerical factor).

Finally, strings and loop gravity, may not necessarilybe competing theories: there might be a sort of com-plementarity, at least methodological, between the two.This is due to the fact that the open problems of stringtheory regard its background-independent formulation,and loop quantum gravity is precisely a set of techniquesfor dealing non-perturbatively with background indepen-dent theories. Perhaps the two approaches might even,to some extent, converge. Undoubtedly, there are sim-ilarities between the two theories: first of all the obvi-ous fact that both theories start with the idea that therelevant excitations at the Planck scale are one dimen-

sional objects – call them loops or strings. I understandthat in another living review in this issue, Lee Smolinexplores the possible relations between string theory andloop gravity [191].

III. HISTORY OF LOOP QUANTUM GRAVITY,MAIN STEPS

The following chronology does not exhaust the liter-ature on loop quantum gravity. It only indicates thekey steps in the construction of the theory, and the firstderivation of the main results. For more complete refer-ences, see the following sections. (In the effort of group-ing some results, something may appear a bit out of thechronological order.)

1986 Connection formulation of classical generalrelativitySen, Ashtekar.Loop quantum gravity is based on the formulationof classical general relativity, which goes under thename of “new variables”, or “Ashtekar variables”,or “connectio-dynamics” (in contrast to Wheeler’s“geometro-dynamics”). In this formulation, thefield variable is a self-dual connection, instead ofthe metric, and the canonical constraints are sim-pler than in the old metric formulation. The ideaof using a self-dual connection as field variable andthe simple constraints it yields were discovered byAmitaba Sen [190]. Abhay Ashtekar realized thatin the SU(2) extended phase space a self-dual con-nection and a densitized triad field form a canon-ical pair [7,8] and set up the canonical formalismbased on such pair, which is the Ashtekar formal-ism. Recent works on the loop representation arenot based on the original Sen-Ashtekar connection,but on a real variant of it, whose use has been intro-duced into Lorentzian general relativity by Barbero[39–42].

1986 Wilson loop solutions of the hamiltonianconstraintJacobson, Smolin.Soon after the introduction of the classical Ashtekarvariables, Ted Jacobson and Lee Smolin realizedin [128] that the Wheeler-DeWitt equation re-formulated in terms of the new variables admitsa simple class of exact solutions: the traces ofthe holonomies of the Ashtekar connection aroundsmooth non-selfintersecting loops. In other words:the Wilson loops of the Ashtekar connection solvethe Wheeler-DeWitt equation if the loops aresmooth and non self-intersecting.

1987 The Loop RepresentationRovelli, Smolin.The discovery of the Jacobson-Smolin Wilson loop

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solutions prompted Carlo Rovelli and Lee Smolin[182,163,183,184] to “change basis in the Hilbertspace of the theory”, choosing the Wilson loops asthe new basis states for quantum gravity. Quan-tum states can be represented in terms of theirexpansion on the loop basis, namely as functionson a space of loops. This idea is well known inthe context of canonical lattice Yang-Mills theory[211], and its application to continuous Yang-Millstheory had been explored by Gambini and Trias[95,96], who developed a continuous “loop represen-tation” much before the Rovelli-Smolin one. Thedifficulties of the loop representation in the contextof Yang-Mills theory are cured by the diffeomor-phism invariance of GR (see section VI H for de-tails). The loop representation was introduced byRovelli and Smolin as a representation of a classicalPoisson algebra of “loop observables”. The relationwith the connection representation was originallyderived in the form of an integral transform (an in-finite dimensional analog of a Fourier transform)from functionals of the connection to loop func-tionals. Several years later, this loop transform wasshown to be mathematically rigorously defined [12].The immediate results of the loop representationwere two: the diffeomorphism constraint is com-pletely solved by knot states (loop functionals thatdepend only on the knotting of the loops), makingearlier suggestions by Smolin on the role of knottheory in quantum gravity [195] concrete; and (suit-able [184,196] extensions of) the knot states withsupport on non-selfintersecting loops were provento be solutions of all quantum constraints, namelyexact physical states of quantum gravity.

1988 - Exact states of quantum gravityHusain, Brugmann, Pullin, Gambini, Kodama.The investigation of exact solutions of the quan-tum constraint equations, and their relation withknot theory (in particular with the Jones polyno-mial and other knot invariants) has started soonafter the formulation of the theory and continuedsince [112,57–60,157,92,94,131,89].

1989 - Model theoriesAshtekar, Husain, Loll, Marolf, Rovelli, Samuel,Smolin, Lewandowski, Marolf, Thiemann.The years immediately following the discovery ofthe loop formalism were mostly dedicated to un-derstanding the loop representation by studying itin simpler contexts, such as 2+1 general relativ-ity [13,146,20], Maxwell [21], linearized gravity [22],and, much later, 2d Yang-Mills theory [19].

1992 Classical limit: weavesAshtekar, Rovelli, Smolin.The first indication that the theory predicts Planckscale discreteness came from studying the statesthat approximate geometries flat on large scale [23].

These states, denoted “weaves”, have a “polymer”like structure at short scale, and can be viewed asa formalization of Wheeler’s “spacetime foam”.

1992 C∗ algebraic frameworkAshtekar, Isham.In [12], Abhay Ashtekar and Chris Isham showedthat the loop transform introduced in gravity byRovelli and Smolin could be given a rigorous math-ematical foundation, and set the basis for a mathe-matical systematization of the loop ideas, based onC∗ algebra ideas.

1993 Gravitons as embroideries over the weaveIwasaki, Rovelli.In [125] Junichi Iwasaki and Rovelli studied the rep-resentation of gravitons in loop quantum gravity.These appear as topological modifications of thefabric of the spacetime weave.

1993 - Alternative versionsDi Bartolo, Gambini, Griego, Pullin.Some versions of the loop quantum gravity alterna-tive to the “orthodox” version have been developed.In particular, these authors above have developedthe so called “extended” loop representation. See[80,78].

1994 - Fermions,Morales-Tecotl, Rovelli.Matter coupling were beginning to be explored in[150,151]. Later, matter’s kinematics was studiedby Baez and Krasnov [133,35], while Thiemann hasextended his results on the dynamics to the coupledEinstein Yang-Mills system in [200].

1994 The dµ0 measure and the scalar productAshtekar, Lewandowski, Baez.In [14–16] Ashtekar and Lewandowski set the ba-sis of the differential formulation of loop quantumgravity by constructing its two key ingredients: adiffeomorphism invariant measure on the space of(generalized) connections, and the projective fam-ily of Hilbert spaces associated to graphs. Usingthese techniques, they were able to give a mathe-matically rigorous construction of the state space ofthe theory, solving long standing problems derivingfrom the lack of a basis (the insufficient control onthe algebraic identities between loop states). Us-ing this, they defined a consistent scalar productand proved that the quantum operators in the the-ory were consistent with all identities. John Baezshowed how the measure can be used in the con-text of conventional connections, extended it to thenon-gauge invariant states (allowing the E operatorto be defined) and developed the use of the graphtechniques [24,28,27]. Important contributions tothe understanding of the measure were also givenby Marolf and Mourao [149].

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1994 Discreteness of area and volume eigenvaluesRovelli, Smolin.In my opinion, the most significative result of loopquantum gravity is the discovery that certain ge-ometrical quantities, in particular area and vol-ume, are represented by operators that have dis-crete eigenvalues. This was found by Rovelli andSmolin in [186], where the first set of these eigenval-ues were computed. Shortly after, this result wasconfirmed and extended by a number of authors,using very diverse techniques. In particular, Re-nate Loll [142,143] used lattice techniques to ana-lyze the volume operator and corrected a numericalerror in [186]. Ashtekar and Lewandowski [138,17]recovered and completed the computation of thespectrum of the area using the connection represen-tation, and new regularization techniques. Frittelli,Lehner and Rovelli [84] recovered the Ashtekar-Lewandowski terms of the spectrum of the area,using the loop representation. DePietri and Rov-elli [77] computed general eigenvalues of the vol-ume. Complete understanding of the precise rela-tion between different versions of the volume oper-ator came from the work of Lewandowski [139].

1995 Spin networks - solution of the overcom-pleteness problemRovelli, Smolin, Baez.A long standing problem with the loop basis wasits overcompleteness. A technical, but crucial stepin understanding the theory has been the discoveryof the spin-network basis, which solves this over-completeness. This step was taken by Rovelli andSmolin in [187] and was motivated by the work ofRoger Penrose [153,152], by analogous bases usedin lattice gauge theory and by ideas of Lewandowski[137]. Shortly after, the spin network formalismwas cleaned up and clarified by John Baez [31,32].After the introduction of the spin network basis,all problems deriving from the incompleteness ofthe loop basis are trivially solved, and the scalarproduct could be defined also algebraically [77].

1995 LatticeLoll, Reisenberger, Gambini, Pullin.Various lattice versions of the theory have appearedin [141,159,94,79].

1995 Algebraic formalism / Differential formal-ismDePietri Rovelli / Ashtekar, Lewandowski, Marolf,Mourao, Thiemann.The cleaning and definitive setting of the two mainversions of the formalisms was completed in [77]for the algebraic formalism (the direct descendentof the old loop representation); and in [18] forthe differential formalism (based on the Ashtekar-Isham C∗ algebraic construction, on the Ashtekar-Lewandowski measure, on Don Marolf’s work on

the use of formal group integration for solving theconstraints [145,147,148], and on several mathe-matical ideas by Jose Mourao).

1996 Equivalence of the algebraic and differentialformalismsDePietri.In [76], Roberto DePietri proved the equivalenceof the two formalisms, using ideas from Thiemann[205] and Lewandowski [139].

1996 Hamiltonian constraintThiemann.The first version of the loop hamiltonian con-straint is in [183,184]. The definition of theconstraint has then been studied and modifiedrepeatedly, in a long sequence of works, byBrugmann, Pullin, Blencowe, Borissov and others[112,47,60,58,57,59,157,92,48]. An important stepwas made by Rovelli and Smolin in [185] with therealization that certain regularized loop operatorshave finite limits on knot states (see [140]). Thesearch culminated with the work of Thomas Thie-mann, who was able to construct a rather well-defined hamiltonian operator whose constraint al-gebra closes [206,201,202]. Variants of this con-straint have been suggested in [192,160] and else-where.

1996 Real theory: solution of the reality condi-tions problemBarbero, Thiemann.As often stressed by Karel Kuchar, implement-ing the complicated reality condition of the com-plex connection into the quantum theory was, until1996, the main open problem in the loop approach.‡

Following the directions advocated by FernandoBarbero [39–42], namely to use the real connec-tion in the Lorentzian theory, Thiemann found anelegant elegant way to completely bypass the prob-lem.

1996 Black hole entropyKrasnov, Rovelli.A derivation of the Bekenstein-Hawking formula forthe entropy of a black hole from loop quantum grav-ity was obtained in [176], on the basis of the ideas ofKirill Krasnov [134,135]. Recently, Ashtekar, Baez,Corichi and Krasnov have announced an alterna-tive derivation [11].

1997 AnomaliesLewandowski, Marolf, Pullin, Gambini.

‡“The loop people have a credit card called reality condi-

tions, and whenever they solve a problem, they charge thecard, but one day the bill comes and the whole thing breaksdown like a card house” [136].

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These authors have recently completed an exten-sive analysis of the issue of the closure of the quan-tum constraint algebra and its departures from thecorresponding classical Poisson algebra [140,90],following earlier pioneering work in this direc-tion by Brugmann, Pullin, Borissov and others[55,61,88,50]. This analysis has raised worries thatthe classical limit of Thiemann’s hamiltonian oper-ator might fail to yield classical general relativity,but the matter is still controversial.

1997 Sum over surfacesReisenberger Rovelli.A “sum over histories” spacetime formulation ofloop quantum gravity was derived in [181,160]from the canonical theory. The resulting covari-ant theory turns out to be a sum over topologicallyinequivalent surfaces, realizing earlier suggestionsby Baez [26,27,31,25], Reisenberger [159,158] andIwasaki [124] that a covariant version of loop grav-ity should look like a theory of surfaces. Baez hasstudied the general structure of theories defined inthis manner [33]. Smolin and Markoupolou haveexplored the extension of the construction to theLorentzian case, and the possibility of altering thespin network evolution rules [144].

IV. RESOURCES

• A valuable resource for finding relevant literature isthe comprehensive “Bibliography of Publications re-lated to Classical and Quantum Gravity in terms ofConnection and Loop Variables”, organized chrono-logically. The original version was compiled byPeter Hubner in 1989. It has subsequently beenupdates by Gabriella Gonzales, Bernd Brugmann,Monica Pierri and Troy Shiling. Presently, it is be-ing kept updated by Christopher Beetle and Ale-jandro Corichi. The last version can be found onthe net in [44].

• This living review may serve as an updated intro-duction to quantum gravity in the loop formalism.More detailed (but less up to date) presentationsare listed below.

– Ashtekar’s book [9] may serve as a valuablebasic introductory course on Ashtekar vari-ables, particularly for relativists and math-ematicians. The part of the book on theloop representation is essentially an autho-rized reprint of parts of the original RovelliSmolin article [184]. For this quantum part, Irecommend looking at the article, rather thanthe book, because the article is more complete.

– A simpler and more straightforward introduc-tion to the Ashtekar variables and basic loop

ideas can be found in the Rovelli’s review pa-per [164]. This is more oriented to a readerwith physics background.

– A recent general introduction to the new vari-ables which includes several of the recentmathematical developments in the quantumtheory is given by Ashtekar’s Les Houches1992 lectures [10].

– A particularly interesting collection of paperscan be found in the volume [29] edited by JohnBaez. The other book by Baez, and Muniain[36], is a simple and pleasant introduction toseveral ideas and techniques in the field.

– The last and up to date book on the looprepresentation is the book by Gambini andPullin [93], especially good in lattice tech-niques and in the variant of loop quantumgravity called the “extended loop representa-tion” [80,78] (which is nowadays a bit out offashion, but remains an intriguing alternativeto “orthodox” loop quantum gravity).

– The two standard references for a completepresentation of the basics of the theory are:DePietri and Rovelli [77] for the algebraicformulation; and Ashtekar, Lewandowski,Marolf, Mourao and Thiemann (ALM2T )[18] for the differential formulation.

• Besides the many conferences on gravity, the loopgravity community has met twice in Warsaw, inthe “Workshop on Canonical and Quantum Grav-ity”. Hopefully, this will become a recurrent meet-ing. This may be the right place to go for learningwhat is going on in the field. For an informal ac-count of the last of these meetings (August 1997),see [161].

• Some of the main institutions where loop quantumgravity is studied are

– The “Center for Gravity and Geometry” atPenn State University, USA. I recommendtheir invaluable web page [156], maintainedby Jorge Pullin, for finding anything you needfrom the web.

– Pittsburgh University, USA [100].

– University of California at Riverside, USA.John Baez moderates an interesting news-group, sci.physics.research, with news fromthe field. See [34].

– Albert Einstein Institute, Potsdam, Berlin,Europe [117].

– Warsaw University, Warsaw, Europe.

– Imperial College, London, Europe [67].

– Syracuse University, USA [208].

– Montevideo University, Uruguay.

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V. MAIN IDEAS AND PHYSICAL INPUTS

The main physical hypotheses on which loop quantumgravity relies are only general relativity and quantum me-chanics. In other words, loop quantum gravity is a ratherconservative “quantization” of general relativity, with itstraditional matter couplings. In this sense, it is verydifferent from string theory, which is based on a strongphysical hypothesis with no direct experimental support(the world is made by strings).

Of course “quantization” is far from a univocal algo-rithm, particularly for a nonlinear field theory. Rather, itis a poorly understood inverse problem (find a quantumtheory with the given classical limit). More or less subtlechoices are made in constructing the quantum theory. Idiscuss these choices below.

A. Quantum field theory on a differentiable manifold

The main idea beyond loop quantum gravity is to takegeneral relativity seriously. We have learned with generalrelativity that the spacetime metric and the gravitationalfield are the same physical entity. Thus, a quantum the-ory of the gravitational field is a quantum theory of thespacetime metric as well. It follows that quantum grav-ity cannot be formulated as a quantum field theory overa metric manifold, because there is no (classical) metricmanifold whatsoever in a regime in which gravity (andtherefore the metric) is a quantum variable.

One could conventionally split the spacetime metricinto two terms: one to be consider as a background,which gives a metric structure to spacetime; the otherto be treated as a fluctuating quantum field. This, in-deed, is the procedure on which old perturbative quan-tum gravity, perturbative strings, as well as current non-perturbative string theories (M-theory), are based. Infollowing this path, one assumes, for instance, that thecausal structure of spacetime is determined by the un-derlying background metric alone, and not by the fullmetric. Contrary to this, in loop quantum gravity weassume that the identification between the gravitationalfield and the metric-causal structure of spacetime holds,and must be taken into account, in the quantum regimeas well. Thus, no split of the metric is made, and thereis no background metric on spacetime.

We can still describe spacetime as a (differentiable)manifold (a space without metric structure), over whichquantum fields are defined. A classical metric structurewill then be defined by expectation values of the grav-itational field operator. Thus, the problem of quantumgravity is the problem of understanding what is a quan-tum field theory on a manifold, as opposed to quantumfield theory on a metric space. This is what gives quan-tum gravity its distinctive flavor, so different than ordi-nary quantum field theory. In all versions of ordinaryquantum field theory, the metric of spacetime plays an

essential role in the construction of the basic theoreti-cal tools (creation and annihilation operators, canonicalcommutation relations, gaussian measures, propagators. . . ); these tools cannot be used in quantum field over amanifold.

Technically, the difficulty due to the absence of a back-ground metric is circumvented in loop quantum gravityby defining the quantum theory as a representation of aPoisson algebra of classical observables which can be de-fined without using a background metric. The idea thatthe quantum algebra at the basis of quantum gravityis not the canonical commutation relation algebra, butthe Poisson algebra of a different set of observables haslong been advocated by Chris Isham [118], whose ideashave been very influential in the birth of loop quantumgravity.§ The algebra on which loop gravity is the loopalgebra [184]. The particular choice of this algebra is notharmless, as I discuss below.

B. One additional assumption

In choosing the loop algebra as the basis for the quan-tization, we are essentially assuming that Wilson loopoperators are well defined in the Hilbert space of the the-ory. In other words, that certain states concentrated onone dimensional structures (loops and graphs) have finitenorm. This is a subtle non trivial assumptions enteringthe theory. It is the key assumption that characterizesloop gravity. If the approach turned out to be wrong,it will likely be because this assumption is wrong. TheHilbert space resulting from adopting this assumption isnot a Fock space. Physically, the assumption correspondsto the idea that quantum states can be decomposed on abasis of “Faraday lines” excitations (as Minkowski QFTstates can be decomposed on a particle basis).

Furthermore, this is an assumption that fails in con-ventional quantum field theory, because in that con-text well defined operators and finite norm states needto be smeared in at least three dimensions, and one-dimensional objects are too singular.∗∗ The fact thatat the basis of loop gravity there is a mathematical as-sumption that fails for conventional Yang-Mills quantumfield theory is probably at the origin of some of the resis-tance that loop quantum gravity encounters among somehigh energy theorists. What distinguishes gravity fromYang-Mills theories, however, and makes this assumptionviable in gravity even if it fails for Yang-Mills theory is

§Loop Quantum Gravity is an attempt to solve the last prob-lem in Isham’s lectures [118].∗∗The assumption does not fail, however, in two-dimensional

Yang-Mills theory, which is invariant under area preserv-ing diffeomorphisms, and where loop quantization techniqueswere successfully employed [19].

8

diffeomorphism invariance. The loop states are singu-lar states that span a “huge” non-separable state space.(Non-perturbative) diffeomorphism invariance plays tworoles. First, it wipes away the infinite redundancy. Sec-ond, it “smears” a loop state into a knot state, so thatthe physical states are not really concentrated in one di-mension, but are, in a sense, smeared all over the entiremanifold by the nonperturbative diffeomorphisms. Thiswill be more clear in the next section.

C. Physical meaning of diffeomorphism invariance,and its implementation in the quantum theory

Conventional field theories are not invariant under adiffeomorphism acting on the dynamical fields. (Everyfield theory, suitably formulated, is trivially invariant un-der a diffeomorphism acting on everything.) General rel-ativity, on the contrary is invariant under such transfor-mations. More precisely every general relativistic theoryhas this property. Thus, diffeomorphism invariance is nota feature of just the gravitational field: it is a feature ofphysics, once the existence of relativistic gravity is takeninto account. Thus, one can say that the gravitationalfield is not particularly “special” in this regard, but thatdiff-invariance is a property of the physical world thatcan be disregarded only in the approximation in whichthe dynamics of gravity is neglected. What is this prop-erty? What is the physical meaning of diffeomorphisminvariance?

Diffeomorphism invariance is the technical implemen-tation of a physical idea, due to Einstein. The idea is adeep modification of the pre-general-relativistic (pre-GR)notions of space and time. In pre-GR physics, we assumethat physical objects can be localized in space and timewith respect to a fixed non-dynamical background struc-ture. Operationally, this background spacetime can bedefined by means of physical reference-system objects,but these objects are considered as dynamically decou-pled from the physical system that one studies. Thisconceptual structure fails in a relativistic gravitationalregime. In general relativistic physics, the physical ob-jects are localized in space and time only with respectto each other. Therefore if we “displace” all dynamicalobjects in spacetime at once, we are not generating adifferent state, but an equivalent mathematical descrip-tion of the same physical state. Hence, diffeomorphisminvariance.

Accordingly, a physical state in GR is not “located”somewhere [180,169,177] (unless an appropriate gaugefixing is made). Pictorially, GR is not physics over astage, it is the dynamical theory of (or including) thestage itself.

Loop quantum gravity is an attempt to implementthis subtle relational notion of spacetime localization inquantum field theory. In particular, the basic quantumfield theoretical excitations cannot be localized some-where (localized with respect to what?) as, say, photons

are. They are quantum excitations of the “stage” itself,not excitations over a stage. Intuitively, one can un-derstand from this discussion how knot theory plays arole in the theory. First, we define quantum states thatcorrespond to loop-like excitations of the gravitationalfield, but then, when factoring away diffeomorphism in-variance, the location of the loop becomes irrelevant. Theonly remaining information contained in the loop is thenits knotting (a knot is a loop up to its location). Thus,diffeomorphism invariant physical states are labeled byknots. A knot represent an elementary quantum excita-tion of space. It is not here or there, since it is the spacewith respect to which here and there can be defined. Aknot state is an elementary quantum of space.

In this manner, loop quantum gravity ties the new no-tion of space and time introduced by general relativitywith quantum mechanics. As I will illustrate later on,the existence of such elementary quanta of space is thenmade concrete by the quantization of the spectra of geo-metrical quantities.

D. Problems not addressed

Quantum gravity is an open problem that has beeninvestigated for over seventy years now. When one con-templates two deep problems, one is tempted to believethat they are related. In the history of physics, there aresurprising examples of two deep problems solved by onestroke (the unification of electricity and magnetism andthe nature of light, for instance); but also many examplesin which a great hope to solve more than one problemat once was disappointed (finding the theory of stronginteractions and getting rid of quantum field theory in-finities, for instance). Quantum gravity has been asked,at some time or the other, to address almost every deepopen problem in theoretical physics (and beyond). Hereis a list of problems that have been connected to quan-tum gravity in the past, but about which loop quantumgravity has little to say:

Interpretation of quantum mechanics. Loop quan-tum gravity is a standard quantum (field) theory.Pick your favorite interpretation of quantum me-chanics, and use it for interpreting the quantumaspects of the theory. I will refer to two such inter-pretations below: When discussing the quantiza-tion of area and volume, I will use the relation be-tween eigenvalues and outcomes of measurementsperformed with classical physical apparata; whendiscussing evolution, I will refer to the histories in-terpretation. The peculiar way of describing timeevolution in a general relativistic theory may re-quire some appropriate variants of standard inter-pretations, such as generalized canonical quantumtheory [166,168,165] or Hartle’s generalized quan-tum mechanics [103]. But loop quantum gravity

9

has no help to offer to the scientists that have spec-ulated that quantum gravity will solve the measure-ment problem. On the other hand, the spacetimeformulation of loop quantum gravity that has re-cently been developed (see Section VI J) is natu-rally interpreted in terms of histories interpreta-tions [103,119,120,122,123]. Furthermore, I thinkthat solving the problem of the interpretation ofquantum mechanics might require relational ideasconnected with the relational nature of spacetimerevealed by general relativity [179,180].

Quantum cosmology. There is widespread confusionbetween quantum cosmology and quantum grav-ity. Quantum cosmology is the theory of the en-tire universe as a quantum system without externalobserver [102]: with or without gravity, makes nodifference. Quantum gravity is the theory of onedynamical entity: the quantum gravitational field(or the spacetime metric): just one field among themany. Precisely as for the theory of the quantumelectromagnetic field, we can always assume thatwe have a classical observer with classical measur-ing apparata measuring gravitational phenomena,and therefore study quantum gravity disregardingquantum cosmology. For instance, the physics ofa Planck size small cube is governed by quantumgravity and, presumably, has no cosmological im-plications.

Unifications of all interactions or “Theory of Ev-erything”. A common criticism to loop quantumgravity is that it does not unify all interactions.But the idea that quantum gravity can be under-stood only in conjunctions with other fields is aninteresting hypothesis, not an established truth.

Mass of the elementary particles. As far as I see,nothing in loop quantum gravity suggests that onecould compute masses from quantum gravity.

Origin of the Universe. It is likely that a sound quan-tum theory of gravity will be needed to understandthe physics of the Big Bang. The converse is prob-ably not true: we should be able to understand thesmall scale structure of spacetime even if we do notunderstand the origin of the Universe.

Arrow of time. Roger Penrose has argued for sometime that it should be possible to trace the timeasymmetry in the observable Universe to quantumgravity.

Physics of the mind. Roger has also speculated thatquantum gravity is responsible for the wave func-tion collapse, and, indirectly, governs the physics ofthe mind [155].

A problem that has been repeatedly tied to quantumgravity, and which loop quantum gravity might be able

to address is the problem of the ultraviolet infinities inquantum field theory. The very peculiar nonperturbativeshort scale structure of loop quantum gravity introducesa physical cutoff. Since physical spacetime itself comesin quanta in the theory, there is literally no space inthe theory for the very high momentum integrations thatoriginate the ultraviolet divergences. Lacking a completeand detailed calculation scheme, however, one cannot yetclaim with confidence that the divergences, chased fromthe door, will not reenter from the window.

VI. THE FORMALISM

Here, I begin the technical description of the basics ofloop quantum gravity. The starting point of the construc-tion of the quantum theory is classical general relativity,formulated in terms of the Sen-Ashtekar-Barbero connec-tion [190,7,40]. Detailed introductions to the (complex)Ashtekar formalism can be found in the book [9], in thereview article [164], or in the conference proceedings [81].The real version of the theory is presently the most widelyused.

Classical general relativity can be formulated in phasespace form as follows [9,40]. We fix a three-dimensionalmanifold M (compact and without boundaries) and con-sider a smooth real SU(2) connection Ai

a(x) and and a

vector density Eai (x) (transforming in the vector repre-

sentation of SU(2)) on M . We use a, b, . . . = 1, 2, 3 forspatial indices and i, j, . . . = 1, 2, 3 for internal indices.The internal indices can be viewed as labeling a basis inthe Lie algebra of SU(2) or the three axis of a local triad.We indicate coordinates on M with x. The relation be-tween these fields and conventional metric gravitationalvariables is as follows: Ea

i (x) is the (densitized) inversetriad, related to the three-dimensional metric gab(x) ofconstant-time surfaces by

g gab = Eai E

bi , (1)

where g is the determinant of gab; and

Aia(x) = Γi

a(x) + γ kia(x); (2)

Γia(x) is the spin connection associated to the triad, (de-

fined by ∂[aeib] = Γi

[aeb]j, where eia is the triad). ki

a(x) is

the extrinsic curvature of the constant time three surface.In (2), γ is a constant, denoted the Immirzi parameter,

that can be chosen arbitrarily (it will enter the hamilto-nian constraint) [114–116]. Different choices for γ yielddifferent versions of the formalism, all equivalent in theclassical domain. If we choose γ to be equal to the imag-inary unit, γ =

√−1, then A is the standard Ashtekar

connection, which can be shown to be the projection ofthe selfdual part of the four-dimensional spin connectionon the constant time surface. If we choose γ = 1, we ob-tain the real Barbero connection. The hamiltonian con-straint of Lorentzian general relativity has a particularly

10

simple form in the γ =√−1 formalism, while the hamil-

tonian constraint of Euclidean general relativity has asimple form when expressed in terms of the γ = 1 realconnection. Other choices of γ are viable as well. In par-ticular, it has been argued that the quantum theory basedon different choices of γ are genuinely physical inequiva-lent, because they yield “geometrical quanta” of differentmagnitude [189]. Apparently, there is a unique choice ofγ yielding the correct 1/4 coefficient in the Bekenstein-Hawking formula [134,135,176,11,178,70], but the matteris still under discussion.

The spinorial version of the Ashtekar variables is givenin terms of the Pauli matrices σi, i = 1, 2, 3, or the su(2)generators τi = − i

2 σi, by

Ea(x) = −i Eai (x)σi = 2Ea

i (x) τi, (3)

Aa(x) = − i

2Ai

a(x)σi = Aia(x) τi . (4)

Thus, Aa(x) and Ea(x) are 2×2 anti-hermitian complexmatrices.

The theory is invariant under local SU(2) gauge trans-formations, three-dimensional diffeomorphisms of themanifold on which the fields are defined, as well as under(coordinate) time translations generated by the hamilto-nian constraint. The full dynamical content of generalrelativity is captured by the three constraints that gen-erate these gauge invariances [190,9].

As already mentioned, the Lorentzian hamiltonianconstraint does not have a simple polynomial form if weuse the real connection (2). For a while, this fact wasconsidered an obstacle for defining the quantum hamil-tonian constraint; therefore the complex version of theconnection was mostly used. However, Thiemann has re-cently succeeded in constructing a Lorentzian quantumhamiltonian constraint [206,201,202] in spite of the non-polynomiality of the classical expression. This is the rea-son why the real connection is now widely used. Thischoice has the advantage of eliminating the old “realityconditions” problem, namely the problem of implement-ing non-trivial reality conditions in the quantum theory.

A. Loop algebra

Certain classical quantities play a very important rolein the quantum theory. These are: the trace of the holon-omy of the connection, which is labeled by loops on thethree manifold; and the higher order loop variables, ob-tained by inserting the E field (in n distinct points, or“hands”) into the holonomy trace. More precisely, givena loop α in M and the points s1, s2, . . . , sn ∈ α we define:

T [α] = −Tr[Uα], (5)

T a[α](s) = −Tr[Uα(s, s)Ea(s)] (6)

and, in general

T a1a2 [α](s1, s2) =

= −Tr[Uα(s1, s2)Ea2(s2)Uα(s2, s1)E

a1(s1)], (7)

T a1...aN [α](s1 . . . sN ) =

= −Tr[Uα(s1, sN )EaN (sN )Uα(sN , sN−1) . . . Ea1(s1)]

where Uα(s1, s2) ∼ Pexp{∫ s2

s1Aa(α(s))ds} is the parallel

propagator of Aa along α, defined by

d

dsUα(1, s) =

dαa(s)

dsAa(α(s)) Uα(1, s) (8)

(See [77] for more details.) These are the loop observ-ables, introduced in Yang Mills theories in [95,96], andin gravity in [183,184].

The loop observables coordinatize the phase space andhave a closed Poisson algebra, denoted the loop algebra.This algebra has a remarkable geometrical flavor. Forinstance, the Poisson bracket between T [α] and T a[β](s)is non vanishing only if β(s) lies over α; if it does, theresult is proportional to the holonomy of the Wilson loopsobtained by joining α and β at their intersection (byrerouting the 4 legs at the intersection). More precisely

{T [α], T a[β](s)} = ∆a[α, β(s)][

T [α#β] − T [α#β−1]]

.

(9)

Here

∆a[α, x] =

∫

dsdαa(s)

dsδ3(α(s), x) (10)

is a vector distribution with support on α and α#β isthe loop obtained starting at the intersection between αand β, and following first α and then β. β−1 is β withreversed orientation.

A (non-SU(2) gauge invariant) quantity that plays arole in certain aspects of the theory, particularly in theregularization of certain operators, is obtained by inte-grating the E field over a two dimensional surface S

E[S, f ] =

∫

S

dSa Eai f i, (11)

where f is a function on the surface S, taking valuesin the Lie algebra of SU(2). In alternative to the fullloop observables (5,6,7), one also can take the holonomiesand E[S, f ] as elementary variables [15,17]; this is morenatural to do, for instance, in the C*-algebric approach[12].

B. Loop quantum gravity

The kinematic of a quantum theory is defined by analgebra of “elementary” operators (such as x and ihd/dx,or creation and annihilation operators) on a Hilbert spaceH. The physical interpretation of the theory is based onthe connection between these operators and classical vari-ables, and on the interpretation of H as the space of the

11

quantum states. The dynamics is governed by a hamil-tonian, or, as in general relativity, by a set of quantumconstraints, constructed in terms of the elementary oper-ators. To assure that the quantum Heisenberg equationshave the correct classical limit, the algebra of the elemen-tary operator has to be isomorphic to the Poisson algebraof the elementary observables. This yields the heuristicquantization rule: “promote Poisson brackets to commu-tators”. In other words, define the quantum theory asa linear representation of the Poisson algebra formed bythe elementary observables. For the reasons illustratedin section V, the algebra of elementary observables wechoose for the quantization is the loop algebra, definedin section VI A. Thus, the kinematic of the quantumtheory is defined by a unitary representation of the loopalgebra. Here, I construct such representation followinga simple path.

We can start “a la Schrodinger” by expressing quan-tum states by means of the amplitude of the connection,namely by means of functionals Ψ(A) of the (smooth)connection. These functionals form a linear space, whichwe promote to a Hilbert space by defining a inner prod-uct. To define the inner product, we choose a particularset of states, which we denote “cylindrical states” andbegin by defining the scalar product between these.

Pick a graph Γ, say with n links, denoted γ1 . . . γn,immersed in the manifold M . For technical reasons, werequire the links to be analytic. Let Ui(A) = Uγi

, i =1, . . . , n be the parallel transport operator of the connec-tion A along γi. Ui(A) is an element of SU(2). Pick afunction f(g1 . . . gn) on [SU(2)]n. The graph Γ and thefunction f determine a functional of the connection asfollows

ψΓ,f (A) = f(U1(A), . . . , Un(A)). (12)

(These states are called cylindrical states because theywere introduced in [14–16] as cylindrical functions forthe definition of a cylindrical measure.) Notice that wecan always “enlarge the graph”, in the sense that if Γ isa subgraph of Γ′ we can always write

ψΓ,f (A) = ψΓ′,f ′(A). (13)

by simply choosing f ′ independent from the Ui’s of thelinks which are in Γ′ but not in Γ. Thus, given any twocylindrical functions, we can always view them as havingthe same graph (formed by the union of the two graphs).Given this observation, we define the scalar product be-tween any two cylindrical functions [137,14–16] by

(ψΓ,f , ψΓ,h) =

∫

SU(2)n

dg1 . . . dgn f(g1 . . . gn)h(g1 . . . gn).

(14)

where dg is the Haar measure on SU(2). This scalarproduct extends by linearity to finite linear combinationsof cylindrical functions. It is not difficult to show that

(14) defines a well defined scalar product on the spaceof these linear combinations. Completing the space ofthese linear combinations in the Hilbert norm, we obtaina Hilbert space H. This is the (unconstrained) quantumstate space of loop gravity.†† H carries a natural uni-tary representation of the diffeomorphism group and ofthe group of the local SU(2) transformations, obtainedtransforming the argument of the functionals. An im-portant property of the scalar product (14) is that it isinvariant under both these transformations.H is non-separable. At first sight, this may seem as

a serious obstacle for its physical interpretation. But wewill see below that after factoring away diffeomorphisminvariance we may obtain a separable Hilbert space (seesection VI H). Also, standard spectral theory holds onH, and it turns out that using spin networks (discussedbelow) one can express H as a direct sum over finite di-mensional subspaces which have the structure of Hilbertspaces of spin systems; this makes practical calculationsvery manageable.

Finally, we will use a Dirac notation and write

Ψ(A) = 〈A|Ψ〉, (15)

in the same manner in which one may write ψ(x) = 〈x|Ψ〉in ordinary quantum mechanics. As in that case, how-ever, we should remember that |A〉 is not a normalizablestate.

C. Loop states and spin network states

A subspace H0 of H is formed by states invariant un-der SU(2) gauge transformations. We now define an or-thonormal basis in H0. This basis represents a very im-portant tool for using the theory. It was introduced in[187] and developed in [31,32]; it is denoted spin networkbasis.

First, given a loop α in M , there is a normalized stateψα(A) in H, which is obtained by taking Γ = α andf(g) = −Tr(g). Namely

ψα(A) = −TrUα(A). (16)

We introduce a Dirac notation for the abstract states,and denote this state as |α〉. These sates are called loopstates. Using Dirac notation, we can write

††This construction of H as the closure of the space of thecylindrical functions of smooth connections in the scalar prod-uct (14) shows that H can be defined without the need of re-curring to C∗ algebraic techniques, distributional connectionsor the Ashtekar-Lewandowski measure. The casual reader,however, should be warned that the resulting H topology isdifferent than the natural topology on the space of connec-tions: if a sequence Γn of graphs converges pointwise to agraph Γ, the corresponding cylindrical functions ψΓn,f do notconverge to ψΓ,f in the H Hilbert space topology.

12

ψα(A) = 〈A|α〉, (17)

It is easy to show that loop states are normalizable. Prod-ucts of loop states are normalizable as well. Followingtradition, we denote with α also a multiloop, namely acollection of (possibly overlapping) loops {α1, . . . , αn, },and we call

ψα(A) = ψα1(A) × . . .× ψαn

(A) (18)

a multiloop state. (Multi-)loop states represented themain tool for loop quantum gravity before the discov-ery of the spin network basis. Linear combinations ofmultiloop states (over-)span H, and therefore a genericstate ψ(A) is fully characterized by its projections on themultiloop states, namely by

ψ(α) = (ψα, ψ). (19)

The “old” loop representation was based on representingquantum states in this manner, namely by means of thefunctionals ψ(α) over loop space defined in(19). Equation(19) can be explicitly written as an integral transform,as we will see in section VI G.

Next, consider a graph Γ. A “coloring” of Γ is givenby the following.

1. Associate an irreducible representation of SU(2) toeach link of Γ. Equivalently, we may associate toeach link γi a half integer number si, the spin ofthe irreducible, or, equivalently, an integer numberpi, the “color” pi = 2si.

2. Associate an invariant tensor v in the tensor prod-uct of the representations s1 . . . sn, to each nodeof Γ in which links with spins s1 . . . sn meet. Aninvariant tensor is an object with n indices in therepresentations s1 . . . sn that transform covariantly.If n = 3, there is only one invariant tensor (up to amultiplicative factor), given by the Clebsh-Gordoncoefficient. An invariant tensor is also called an in-tertwining tensor. All invariant tensors are givenby the standard Clebsch-Gordon theory. More pre-cisely, for fixed s1 . . . sn, the invariant tensors forma finite dimensional linear space. Pick a basis vj

is this space, and associate one of these basis ele-ments to the node. Notice that invariant tensorsexist only if the tensor product of the representa-tions s1 . . . sn contains the trivial representation.This yields a condition on the coloring of the links.For n = 3, this is given by the well known Clebsh-Gordan condition: each color is not larger than thesum of the other two, and the sum of the threecolors is even.

We indicate a colored graph by {Γ, ~s, ~v}, or simply S ={Γ, ~s, ~v}, and denote it a “spin network”. (It was Penrosewho first had the intuition that this mathematics couldbe relevant for describing the quantum properties of the

geometry, and who gave the first version of spin networktheory [152,153].)

Given a spin network S, we can construct a stateΨS(A) as follows. We take the propagator of the con-nection along each link of the graph, in the representa-tion associated to that link, and then, at each node, wecontract the matrices of the representation with the in-variant tensor. We obtain a state ΨS(A), which we alsowrite as

ψS(A) = 〈A|S〉. (20)

One can then show the following.

• The spin network states are normalizable. The nor-malization factor is computed in [77].

• They are SU(2) gauge invariant.

• Each spin network state can be decomposed into afinite linear combination of products of loop states.

• The (normalized) spin network states form an or-thonormal basis for the gauge SU(2) invariantstates in H (choosing the basis of invariant tensorsappropriately).

• The scalar product between two spin networkstates can be easily computed graphically and al-gebraically. See [77] for details.

The spin network states provide a very convenient basisfor the quantum theory.

The spin network states defined above are SU(2) gaugeinvariant. There exists also an extension of the spinnetwork basis to the full Hilbert space (see for instance[17,51], and references therein).

D. Relation between spin network states and loopstates and diagrammatic representation of the states

A diagrammatic representation for the states in H isvery useful in concrete calculations. First, associate toa loop state |α〉 a diagram in M , formed by the loop αitself. Next, notice that we can multiply two loop states,obtaining a normalizable state. We represent the prod-uct of n loop states by the diagram formed by the setof the n (possibly overlapping) corresponding loops (wedenote this set “multiloop”). Thus, linear combinationsof multiloops diagrams represent states in H. Represent-ing states as linear combinations of multiloops diagramsmakes computation in H easy.

Now, the spin network state defined by the graph withno nodes α, with color 1, is clearly, by definition, theloop state |α〉, and we represent it by the diagram α.The spin network state |α, n〉 determined by the graphwithout nodes α, with color n can be obtained as follows.Draw n parallel lines along the loop α; cut all lines at an

13

arbitrary point of α, and consider the n! diagrams ob-tained by joining the legs after a permutation. The linearcombination of these n! diagrams, taken with alternatesigns (namely with the sign determined by the parity ofthe permutation) is precisely the state |α, n〉. The rea-son of this key result can be found in the fact that anirreducible representation of SU(2) can be obtained asthe totally symmetric tensor product of the fundamentalrepresentation with itself. For details, see [77].

Next, consider a graph Γ with nodes. Draw ni parallellines along each link γi. Join pairwise the end pointsof these lines at each node (in an arbitrary manner), insuch a way that each line is joined with a line from adifferent link (see Figure 1). In this manner, one obtaina multiloop diagram. Now antisymmetrize the ni parallellines along each link, obtaining a linear combination ofdiagrams representing a state in H. One can show thatthis state is a spin network state, where ni is the colorof the links and the color of the nodes is determined bythe pairwise joining of the legs chosen [77]. Again, simpleSU(2) representation theory is behind this result.

More in detail, if a node is trivalent (has 3 adjacentlinks), then we can join legs pairwise only if the totalnumber of the legs is even, and if the number of thelegs in each link is smaller or equal than the sum of thenumber of the other two. This can be immediately rec-ognized as the Clebsch-Gordan condition. If these con-ditions are satisfied, there is only a single way of join-ing legs. This corresponds to the fact that there is onlyone invariant tensor in the product of three irreducible ofSU(2). Higher valence nodes can be decomposed in triva-lent “virtual” nodes, joined by “virtual” links. Orthogo-nal independent invariant tensors are obtained by varyingover all allowed colorings of these virtual links (compat-ible with the Clebsch-Gordan conditions at the virtualnodes). Different decompositions of the node give differ-ent orthogonal bases. Thus the total (links and nodes)coloring of a spin network can be represented by means ofthe coloring of the real and the virtual links. See Figure1.

v

e0

e e3

4

56

2 e

e

ee

1

E5

2N4

2

E3

EN

5N

3

N4

N1

E

e e

e

e

e

e0 e

RΦ

RΦ

6

12

3

4

5

FIG. 1. Construction of “virtual” nodes and “virtual” linksover an n-valent node.

Viceversa, multiloop states can be decomposed in spinnetwork states by simply symmetrizing along (real andvirtual) nodes. This can be done particularly easily di-agrammatically, as illustrated by the graphical formulaein [187,77]. These are standard formulae. In fact, itis well known that the tensor algebra of the SU(2) irre-ducible representations admits a completely graphical no-tation. This graphical notation has been widely used forinstance in nuclear and atomic physics. One can find itpresented in detail in books such as [214,52,66]. The ap-plication of this diagrammatic calculus to quantum grav-ity is described in detail in [77], which I recommend toanybody who intends to perform concrete calculations inloop gravity.

It is interesting to notice that loop quantum gravitywas first constructed in a pure diagrammatic notation, in[184]. The graphical nature of this calculus puzzled some,and the theory was accused of being vague and strange.Only later the diagrammatic notation was recognized tobe the very conventional SU(2) diagrammatic calculus,well known in atomic and nuclear physics.‡‡

‡‡The following anecdote illustrates the level of the initialconfusion. The first elaborate graphical computation of theeigenvalues of the area led to the mysterious formula Ap =12hG

√

p2 + 2p, with integer p. One day, Junichi Iwasaki, atthe time a student, stormed into my office and told me torewrite the formula in terms of the half integer j = 1

2p. This

yields the very familiar expression for the eigenvalues of theSU(2) Casimir Aj = hG

√

j(j + 1). Nobody had previouslynoticed this fact.

14

E. The representation

I now define the quantum operators, corresponding tothe T -variables, as linear operators on H. These form arepresentation of the loop variables Poisson algebra. Theoperator T [α] acts diagonally

T [α]Ψ(A) = −TrUα(A) Ψ(A). (21)

(Recall that products of loop states and spin networkstates are normalizable states). In diagrammatic nota-tion, the operator simply adds a loop to a (linear combi-nation of) multiloops

T [α] |Ψ〉 = |α〉|Ψ〉. (22)

Higher order loop operators are expressed in terms ofthe elementary “grasp” operation. Consider first the op-erator T a(s)[α], with one hand in the point α(s). Theoperator annihilates all loop states that do not cross thepoint α(s). Acting on a loop state |β〉, it gives

T a(s)[α] |β〉 = l20 ∆a[β, α(s)][

|α#β〉 − |α#β−1〉]

,

(23)

where we have introduced the elementary length l0 by

l20 = hG =16πhGNewton

c3= 16π l2Planck (24)

and ∆a and # are defined in section VI A. This actionextends by linearity, continuity and by the Leibniz ruleto products and linear combinations of loop states, andto the full H. In particular, it is not difficult to computeits action on a spin network state [77]. Higher orderloop operators act similarly. It is a simple exercise toverify that these operators provide a representation ofthe classical Poisson loop algebra.

All the operators in the theory are then constructedin terms of these basics loop operators, in the same wayin which in conventional QFT one constructs all opera-tors, including the hamiltonian, in terms of creation andannihilation operators. The construction of the compos-ite operators requires the development of regularizationtechniques that can be used in the absence of a back-ground metric. These have been introduced in [196] anddeveloped in [186,77,18,23,138,17].

F. Algebraic version (“loop representation”) anddifferential version (“connection representation”) of

the formalism, and their equivalence

Imagine we want to quantize the one dimensional har-monic oscillator. We can consider the Hilbert space ofsquare integrable functions ψ(x) on the real line, andexpress the momentum and the hamiltonian as differen-tial operators. Denote the eigenstates of the hamilto-nian as ψn(x) = 〈x|n〉. It is well known that the the-ory can be expressed entirely in algebraic form in terms

of the states |n〉. In doing so, all elementary opera-tors are algebraic: x|n〉 = 1√

2(|n − 1〉 + (n + 1)|n + 1〉),

p|n〉 = −i√2(|n−1〉−(n+1)|n+1〉). Similarly, in quantum

gravity we can directly construct the quantum theory inthe spin-network (or loop) basis, without ever mention-ing functionals of the connections. This representationof the theory is denoted the “loop representation”.

A section of the first paper on loop quantum gravityby Rovelli and Smolin [184] was devoted to a detailedstudy of “transformation theory” (in the sense of Dirac)on the state space of quantum gravity, and in particularon the relations between the loop states

ψ(α) = 〈α|ψ〉 (25)

and the states ψ(A) giving the amplitude for a connectionfield configuration A, and defined by

ψ(A) = 〈A|ψ〉. (26)

Here |A〉 are “eigenstates of the connection operator”, or,more precisely (since the operator corresponding to theconnection is ill defined in the theory) the generalizedstates that satisfy

T [α] |A〉 = −Tr[Pe∫

αA] |A〉. (27)

However, at the time of [184] the lack of a scalar productmade transformation theory quite involved.

On the other hand, the introduction of the scalar prod-uct (14) gives a rigorous meaning to the loop transform.In fact, we can write, for every spin network S, and everystate ψ(A)

ψ(S) = 〈S|ψ〉 = (ψS , ψ). (28)

This equation defines a unitary mapping between the twopresentations of H: the “loop representation”, in whichone works in terms of the basis |S〉; and the “connectionrepresentation”, in which one uses wave functionals ψ(A).

The development of the connection representation fol-lowed a winding path through C∗-algebraic [12] andmeasure theoretical [14,16,15] methods. The work ofAshtekar, Isham, Lewandowski, Marolf, Mourao andThiemann has finally put the connection representationon a firm ground, and, indirectly has much clarified themathematics underlying the original loop approach. Inthe course of this development, doubts were raised aboutthe precise relations between the connection and the loopformalisms. Today, the complete equivalence of thesetwo approaches (always suspected) has been firmly es-tablished. In particular, the work of Roberto DePietri[76] has proven the unitary equivalence of the two for-malisms. For a recent discussion see also [139].

G. Other structures in H

The recent developments in the mathematical founda-tions of the connection representation have increased the

15

mathematical rigor of the theory, raising it to the stan-dards of mathematical physics. This has been obtainedat the price of introducing heavy mathematical tools, of-ten unfamiliar to the average physicist, perhaps wideningthe language gap between the quantum gravity and thehigh energy physics community.

The reason for searching a mathematical-physics levelof precision is that in quantum gravity one moves ona very unfamiliar terrain –quantum field theory onmanifolds– where the experience accumulated in conven-tional quantum field theory is often useless and some-times even misleading. Given the unlikelihood of findingdirect experimental corroboration, the research can onlyaim, at least for the moment, at the goal of finding aconsistent theory, with the correct limits in the regimesthat we control experimentally. In these conditions, highmathematical rigor is the only assurance of the consis-tency of the theory. In the development of quantumfield theory mathematical rigor could be very low be-cause extremely accurate empirical verifications assuredthe physicists that “the theory may be mathematicallymeaningless, but it is nevertheless physical correct, andtherefore the theory must make sense even if we do notunderstand how.” Here, such an indirect experimentalreassurance is lacking and the claim that the theory iswell founded can be based only on a solid mathematicalcontrol of the theory.

One may object that a rigorous definition of quantumgravity is a vain hope, given that we do not even havea rigorous definition of QED, presumably a much sim-pler theory. The objection is particularly valid from thepoint of view of a physicist who views gravity “just as anyother field theory like the ones we already understand”.But the (serious) difficulties of QED and of the otherconventional field theories are ultraviolet. The physicalhope supporting the quantum gravity research programis that the ultraviolet structure of a diffeomorphism in-variant quantum field theory is profoundly different fromthe one of conventional theories. Indeed, recall that in avery precise sense there is no short distance limit in thetheory; the theory naturally cuts itself off at the Planckscale, due to the very quantum discreteness of spacetime.Thus the hope that quantum gravity could be defined rig-orously may be optimistic, but it is not ill founded.

After these comments, let me briefly mention some ofthe structures that have been explored in H. First of all,the spin network states satisfy the Kauffman axioms ofthe tangle theoretical version of recoupling theory [130](in the “classical” case A = −1) at all the points (in 3dspace) where they meet. (This fact is often misunder-stood: recoupling theory lives in 2d and is associated byKauffman to knot theory by means of the usual projec-tion of knots from 3d to 2d. Here, the Kauffman axiomsare not satisfied at the intersections created by the 2dprojection of the spin network, but only at the nodes in3d. See [77] for a detailed discussion.) For instance, con-sider a 4-valent node of four links colored a, b, c, d. Thecolor of the node is determined by expanding the 4-valent

node into a trivalent tree; in this case, we have a singleinternal links. The expansion can be done in differentways (by pairing links differently). These are related toeach other by the recoupling theorem of pg. 60 in Ref.[130]

a

b

d

c

�@ �

@

jr r =

∑

i

{

a b ic d j

}

a

b

d

c

�

@�

@ir

r

(29)

where the quantities

{

a b ic d j

}

are su(2) six-j symbols

(normalized as in [130]). Equation (29) follows just fromthe definitions given above. Recoupling theory providesa powerful computational tool in this context.

Since spin network states satisfy recoupling theory,they form a Temperley-Lieb algebra [130]. The scalarproduct (14) in H is given also by the Temperley-Liebtrace of the spin networks, or, equivalently by the Kauff-man brackets, or, equivalently, by the chromatic evalua-tion of the spin network. See Ref. [77] for an extensivediscussion of these relations.

Next, H admits a rigorous representation as an L2

space, namely a space of square integrable functions. Toobtain this representation, however, we have to extendthe notion of connection, to a notion of “distributionalconnection”. The space of the distributional connectionsis the closure of the space of smooth connection in a cer-tain topology. Thus, distributional connections can beseen as limits of sequences of connections, in the samemanner in which distributions can be seen as limits ofsequences of functions. Usual distributions are definedas elements of the topological dual of certain spaces offunctions. Here, there is no natural linear structure inthe space of the connections, but there is a natural du-ality between connections and curves in M : a smoothconnection A assigns a group element Uγ(A) to every seg-ment γ. The group elements satisfy certain properties.For instance if γ is the composition of the two segmentsγ1 and γ2, then Uγ(A) = Uγ1

(A)Uγ2(A).

A generalized connection A is defined as a map thatassigns an element of SU(2), which we denote as Uγ(A)or A(γ) to each (oriented) curve γ in M , satisfying thefollowing requirements: i) A(γ−1) = (A(γ))−1; and, ii)A(γ2 ◦ γ1) = A(γ2) · A(γ1), where γ−1 is obtained fromγ by reversing its orientation, γ2 ◦ γ1 denotes the com-position of the two curves (obtained by connecting theend of γ1 with the beginning of γ2) and A(γ2) · A(γ1) isthe composition in SU(2). The space of such general-ized connections is denoted A. The cylindrical functionsΨΓ,f(A) defined in section VI C as functions on the spaceof smooth connections extend immediately to generalizedconnections

ΨΓ,f (A) = f(A(γ1, . . . , A(γn)). (30)

We can define a measure dµ0 on the space of general-ized connections A by

16

∫

dµ0[A] ΨΓ,f(A) ≡∫

SU(2)n

dg1 . . . dgn f(g1 . . . gn). (31)

In fact, one may show that (31) defines (by linear-ity and continuity) a well-defined absolutely continuousmeasure on A. This is the Ashtekar-Lewandowski (orAshtekar-Lewandowski-Baez) measure [14–16,26]. Then,one can prove that H = L2[A, dµ0], under the naturalisomorphism given by identifying cylindrical functions.It follows immediately that the transformation (19) be-tween the connection representation and the “old” looprepresentation is given by

ψ(α) =

∫

dµ0[A] TrPe

∫

αAΨ(A). (32)

This is the loop transform formula that was derivedheuristically in [184]; here it becomes rigorously defined.

Furthermore, H can be seen as the projective limit ofthe projective family of the Hilbert spaces HΓ, associatedto each graph Γ immersed in M . HΓ is defined as thespace L2[SU(2)n, dg1 . . . dgn], where n is the number oflinks in Γ. The cylindrical function ΨΓ,f(A) is naturallyassociated to the function f in HΓ, and the projectivestructure is given by the natural map (13) [18,149].

Finally, Ashtekar and Isham [12] have recovered therepresentation of the loop algebra by using C*-algebrarepresentation theory: The space A/G, where G is thegroup of local SU(2) transformations (which acts in theobvious way on generalized connections), is precisely theGelfand spectrum of the abelian part of the loop algebra.One can show that this is a suitable norm closure of thespace of smooth SU(2) connections over physical space,modulo gauge transformations.

Thus, a number of powerful mathematical tools areat hand for dealing with nonperturbative quantum grav-ity. These include: Penrose’s spin network theory, SU(2)representation theory, Kauffman tangle theoretical re-coupling theory, Temperley-Lieb algebras, Gelfand’sC∗algebra spectral representation theory, infinite dimen-sional measure theory and differential geometry over in-finite dimensional spaces.

H. Diffeomorphism invariance

The next step in the construction of the theory is tofactor away diffeomorphism invariance. This is a keystep for two reasons. First of all, H is a “huge” non-separable space. It is far “too large” for a quantumfield theory. However, most of this redundancy is gauge,and disappears when one solves the diffeomorphism con-straint, defining the diff-invariant Hilbert space HDiff .This is the reason for which the loop representation, asdefined here, is of great value in diffeomorphism invarianttheories only.

The second reason is that HDiff turns out to have anatural basis labeled by knots. More precisely by “s-

knots”. An s-knot s is an equivalence class of spin net-works S under diffeomorphisms. An s-knot is charac-terized by its “abstract” graph (defined only by the ad-jacency relations between links and nodes), by the col-oring, and by its knotting and linking properties, as inknot-theory. Thus, the physical quantum states of thegravitational field turn out to be essentially classified byknot theory.

There are various equivalent way of obtaining HDiff

from H. One can use regularization techniques for defin-ing the quantum operator corresponding to the classicaldiffeomorphism constraint in terms of elementary loopoperators, and then find the kernel of such operator.Equivalently, one can factor H by the natural action ofthe Diffeomorphism group that it carries. Namely

HDiff =H

Diff(M). (33)

There are several rigorous ways for defining the quotientof a Hilbert space by the unitary action of a group. Seein particular the construction in [18], which follows theideas of Marolf and Higuchi [145,147,148,108].

In the quantum gravity literature, a big deal has beenmade of the problem that a scalar product is not definedon the space of solutions of a constraint C, defined on aHilbert space H. This, however, is a false problem. It istrue that if zero is in the continuum spectrum of C thenthe corresponding eigenstates are generalized states andthe H scalar product is not defined between them. Butthe generalized eigenspaces of C, including the kernel,inherit nevertheless a scalar product from H . This canbe seen in a variety of equivalent ways. For instance,it can be seen from the following theorem. If C is selfadjoint, then there exist a measure µ(λ) on its spectrumand a family of Hilbert spaces H(λ) such that

H =

∫

dµ(λ)H(λ), (34)

where the integral is the continuous sum of Hilbert spacesdescribed for instance in [101]. Clearly H(0) is the kernel

of C equipped with a scalar product. This is discussed forinstance in [162].

There are two distinct ways of factoring away the dif-feomorphisms in the quantum theory, yielding two dis-tinct version of the theory. The first way is to factoraway smooth transformations of the manifold. In doingso, finite dimensional moduli spaces associated with highvalence nodes appear [98], so that the resulting Hilbertspace is still nonseparable. The physical relevance ofthese moduli parameters is unclear at this stage, sincethey do not seem to play any role in the quantum theory.Alternatively, one can consistently factor away continu-ous transformations of the manifold. This possibility hasbeen explored by Zapata in [215,216], and seems to leadto a consistent theory free of the residual non separabil-ity.

17

I. Dynamics

Finally, the definition of the theory is completed bygiving the hamiltonian constraint. A number of ap-proaches to the definition of a hamiltonian constrainthave been attempted in the past, with various degreesof success. Recently, however, Thiemann has succeededin providing a regularization of the hamiltonian con-straint that yields a well defined, finite operator. Thie-mann’s construction [206,201,202] is based on severalclever ideas. I will not describe it here. Rather, I willsketch below the final form of the constraint (for theLapse=1 case), following [175].

I begin with the Euclidean hamiltonian constraint. Wehave

H|s〉 =∑

i

∑

(IJ)

∑

ǫ=±1

∑

ǫ′=±1

Aǫǫ′(pi...pn) Di;(IJ),ǫǫ′ |s〉.

(35)

Here i labels the nodes of the s-knot s; (IJ) labels couplesof (distinct) links emerging from i. p1...pn are the colors

of the links emerging from i. Di;(IJ)ǫǫ′ is the operatorthat acts on an s -knot by: (i) creating two additionalnodes, one along each of the two links I and J ; (ii) creat-ing a novel link, colored 1, joining these two nodes, (iii)assigning the coloring pI + ǫ and, respectively, pJ + ǫ′ tothe links that join the new formed nodes with the nodei. This is illustrated in Figure 2.

""

"""

bb

bbbr

q

p

D+−

JJJJJ

""

"""

bb

bbb

=

r p

q

1q−1

p+1

FIG. 2. Action of Di;(IJ)ǫǫ′ .

The coefficients Aǫǫ′(pi...pn), which are finite, can beexpressed explicitly (but in a rather laborious way) interms of products of linear combinations of 6− j symbolsof SU(2), following the techniques developed in detailin [77]. Some of these coefficients have been explicitlycomputed [51]. The Lorentzian hamiltonian constraint

is given by a similar expression, but quadratic in the Doperators.

The operator defined above is obtained by introduc-ing a regularized expression for the classical hamiltonianconstraint, written in terms of elementary loop observ-ables, turning these observables into the correspondingoperators and taking the limit. The construction worksrather magically, relying on the fact, first noticed in [188],

that certain operator limits Oǫ → O turns out to be fi-nite on diff invariant states, thanks to the fact that for ǫ

and ǫ′ sufficiently small, Oǫ|Ψ〉 and Oǫ′ |Ψ〉 are diffeomor-phic equivalent. Thus, here diff invariance plays again acrucial role in the theory.

For a discussion of the problems raised by the Thie-mann operator and of the variant proposed, see sectionVIII.

J. Unfreezing the frozen time formalism: thecovariant form of loop quantum gravity

A recent development in the formalism is the trans-lation of loop quantum gravity into spacetime covariantform. This was initiated in [181,160] by following thesteps that Feynman took in defining path integral quan-tum mechanics starting from the Schrodinger canonicaltheory. More precisely, it was proven in [160] that thematrix elements of the operator U(T )

U(T ) ≡ e

∫

T

0dt

∫

d3x H(x), (36)

obtained exponentiating the (Euclidean) hamiltonianconstraint in the proper time gauge (the operator thatgenerates evolution in proper time) can be expanded ina Feynman sum over paths. In conventional QFT eachterm of a Feynman sum corresponds naturally to a cer-tain Feynman diagram, namely a set of lines in spacetimemeeting at vertices (branching points). A similar natu-ral structure of the terms appears in quantum gravity,but surprisingly the diagrams are now given by surfacesis spacetime that branch at vertices. Thus, one has aformulation of quantum gravity as a sum over surfacesin spacetime. Reisenberger [158] and Baez [30] had ar-gued in the past that such a formulation should exist,and Iwasaki has developed a similar construction in 2+1dimensions. Intuitively, the time evolution of a spin net-work in spacetime is given by a colored surface. The sur-faces capture the gravitational degrees of freedom. Theformulation is “topological” in the sense that one mustsum over topologically inequivalent surfaces only, and thecontribution of each surface depends on its topology only.This contribution is given by the product of elementary“vertices”, namely points where the surface branches.

The transition amplitude between two s-knot states|si〉 and |sf 〉 in a proper time T is given by summingover all (branching, colored) surfaces σ that are boundedby the two s-knots si and sf

〈sf |U(T )|si〉 =∑

σ∂σ=si∪sf

A[σ](T ). (37)

The weight A[σ](T ) of the surface σ is given by a productover the n vertices v of σ:

A[σ](T ) =(- i T )n

n!

∏

v

Av(σ). (38)

18

The contribution Av(σ) of each vertex is given by thematrix elements of the hamiltonian constraint operatorbetween the two s-knots obtained by slicing σ immedi-ately below and immediately above the vertex. They turnout to depend only on the colors of the surface compo-nents immediately adjacent the vertex v. The sum turnsout to be finite and explicitly computable order by order.

As in the usual Feynman diagrams, the vertices de-scribe the elementary interactions of the theory. In par-ticular, here one sees that the complicated structure ofthe Thiemann hamiltonian, which makes a node split intothree nodes, corresponds to a geometrically very simplevertex. Figure 3 is a picture of the elementary vertex.Notice that it represents nothing but the spacetime evo-lution of the elementary action of the hamiltonian con-straint, given in Figure 2.

FIG. 3. The elementary vertex.

An example of a surface in the sum is given in Figure4.

q

p

5

5

f

i

6

7

8

8

1

3

7

63

3

s

1

s f

i

s

Σ

Σ

FIG. 4. A term of second order.

The sum over surfaces version of loop quantum grav-ity provides a link with certain topological quantum fieldtheories and in particular with the the Crane-Yettermodel [71–75], which admit an extremely similar rep-resentation. For a discussion on the precise relationbetween topological quantum field theory and diffeo-morphism invariant quantum field theory, see [160] and[171,124,83].

The idea of expressing the theory as a sum over sur-faces has been developed by Baez [33], who has stud-ied the general form of generally covariant quantum fieldtheories formulated in this manner, and by Smolin andMarkopoulou [144], who have studied how to directlycapture the Lorentzian causal structure of general rela-tivity modifying the elementary vertices. They have alsoexplored the idea that the long range correlations of thelow energy regime of the theory are related to the exis-tence of a phase transition in the microscopic dynamics,and has found intriguing connections with the theoreticaldescription of percolation.

VII. PHYSICAL RESULTS

In section VI, I have sketched the basic structure ofloop quantum gravity. This structure has been devel-oped in a number of directions, and has been used toderive a number of results. Without any ambition ofcompleteness, I list below some of these developments.

19

A. Technical

• Solutions of the hamiltonian constraints.One of the most surprising results of the theoryis that it has been possible to find exact solu-tions of the hamiltonian constraint. This followsfrom the key result that the action of the hamilto-nian constraints is non vanishing only over nodesof the s-knots [183,184]. Therefore s-knots with-out nodes are physical states that solve the quan-tum Einstein dynamics. There is an infinite num-ber of independent states of this sort, classified byconventional knot theory. The physical interpreta-tion of these solutions is still rather obscure. Vari-ous other solutions have been found. See the re-cent review [82] and reference therein. See also[113,131,56–59,157,94,129]. In particular, Pullinhas studied in detail solutions related to the Chern-Simon term in the connection representation andto the Jones polynomial in the loop representation.According to a celebrated result by Witten [212],the two are the loop transform of each other.

• Time evolution. Strong field perturbationexpansion. “Topological Feynman rules”.Trying to describe the temporal evolution of thequantum gravitational field by solving the hamilto-nian constraint yields the conceptually well-defined[168], but notoriously non-transparent frozen-timeformalism. An alternative is to study the evolu-tion of the gravitational degrees of freedom withrespect to some matter variable, coupled to thetheory, which plays the role of a phenomenologi-cal “clock”. This approach has lead to the tenta-tive definition of a physical hamiltonian [188,48],and to a preliminary investigation of the possibil-ity of transition amplitudes between s-knot states,order by order in a (strong coupling) perturbativeexpansion [175]. In this context, diffeomorphisminvariance, combined with the key result that thehamiltonian constraint acts on nodes only, impliesthat the “Feynman rules” of such an expansion arepurely topological and combinatorial.

• Fermions. Fermions have been added to thetheory [150,151,35,207]. Remarkably, all the im-portant results of the pure GR case survive in theGR+fermions theory. Not surprisingly, fermionscan be described as open ends of “open spin net-works”.

• Maxwell and Yang-Mills. The extension ofthe theory to the Maxwell field has been studied in[132,91]. The extension to Yang-Mills theory hasbeen recently explored in [200]. In [200], Thiemannshows that the Yang-Mills term in the quantumhamiltonian constraint can be defined in a rigorousmanner, extending the methods of [206,201,202].

A remarkable result in this context is that ultra-violet divergences do not seem to appear, stronglysupporting the expectation that the natural cut offintroduced by quantum gravity might cure the ul-traviolet difficulties of conventional quantum fieldtheory.

• Application to other theories. The loop repre-sentation has been applied in various other contextssuch as 2+1 gravity [13,146,20] (on 2+1 quantumgravity, in the loop and in other representations,see [62]) and others [21].

• Lattice and simplicial models. A numberof interesting discretized versions of the theory arebeing studied. See in particular [141,159,94,79].

B. Physical

• Planck scale discreteness of space

The most remarkable physical result obtained fromloop quantum gravity is, in my opinion, evidencefor a physical (quantum) discreteness of space atthe Planck scale. This is manifested in the factthat certain operators corresponding to the mea-surement of geometrical quantities, in particulararea and volume, have discrete spectrum. Accord-ing to the standard interpretation of quantum me-chanics (which we adopt), this means that the the-ory predicts that a physical measurement of an areaor a volume will necessarily yield quantized results.Since the smallest eigenvalues are of Planck scale,this implies that there is no way of observing areasor volumes smaller than Planck scale. Space comesin “quanta” in the same manner as the energy ofan oscillator. The spectra of the area and volumeoperators have been computed with much detailin loop quantum gravity. These spectra have acomplicated structure, and they constitute detailedquantitative physical predictions of loop quantumgravity on Planck scale physics. If we had experi-mental access to Planck scale physics, they wouldallow the theory to be empirically tested in greatdetail.

A few comments are in order.

– The result of the discreteness of area and vol-ume is due to Rovelli and Smolin, and ap-peared first in reference [186]. Later, the re-sult has been recovered by alternative tech-niques and extended by a number of authors.In particular, Ashtekar and Lewandowski [17]have repeated the derivation, using the con-nection representation, and have completedthe computation of the spectrum (adding thesector which was not computed in [186].)The Ashtekar-Lewandowski component of the

20

spectrum has then been rederived in the looprepresentation by Frittelli Lehner and Rovelliin [84]. Loll has employed lattice techniquesto point out a numerical error in [186] (cor-rected in the Erratum) in the eigenvalues ofthe volume. The analysis of the volume eigen-values has been performed in [77], where gen-eral techniques for performing these calcula-tions are described in detail. The spectrumof the volume has then been analyzed alsoin [203]. There are also a few papers thathave anticipated the main result presentedin [186]. In particular, Ashtekar Rovelli andSmolin have argued for a physical discretenessof space emerging from the loop representa-tion in [23], where some of the eigenvalues ofthe area already appear, although in implicitform. The first explicit claim that area eigen-values might in principle be observable (in thepresence of matter) is by Rovelli in [172].

– The reader will wonder why area and volumeseem to play here a role more central thanlength, when classical geometry is usually de-scribed in terms of lengths. The reason is thatthe length operator is difficult to define andof more difficult physical interpretations. Forattempts in this direction, see [204]. Whetherthis is simply a technical difficulty or it reflectssome deep fact, is not clear to me.§§

– Area and volume are not gauge invariant oper-ators. Therefore, we cannot directly interpretthem as representing physical measurements.Realistic physical measurements of areas andvolumes always refer to physical surfaces andspatial regions, namely surfaces and spatial re-gions determined by some physical object. Forinstance, I can measure the area of the surfaceof a certain table. In the dynamical theorythat describes the gravitational field as wellas the table, the area of the surface of the ta-ble is a diffeomorphism invariant quantity A,which depends on gravitational as well as mat-ter variables. In the quantum theory, A willbe represented by a diffeomorphism invariant

§§I include here a comment on this issue I received from JohnBaez: “I believe there is a deep reason why area is more fun-damental than length in loop quantum gravity. One way tosay it is that the basic field is not the tetrad e but the 2-formE = e ∧ e. Another way to say it is that the loop representa-tion is based on the theory of quantized angular momentum.Angular momentum is not a vector but a bivector, so it cor-responds not to an arrow but to a oriented area element.” Onthis, see Baez’s [33]. On the relation between E field and areasee in particular [170].

operator. Now, as first realized in [172], it isplausible that that the operator A is, mathe-matically, the same operator as the pure grav-ity area operator. This is because we cangauge fix the matter variables, and use mat-ter location as coordinates, so that non-diff-invariant observables in the pure gravity the-ory correspond precisely to diff-invariant ob-servables in the matter+gravity theory. Thus,discreteness of the spectrum of the area opera-tor is likely to imply discreteness of physicallymeasurable areas, but it is important to em-phasize that this implication is based on someadditional hypothesis on the relation betweenthe pure gravity and the gravity+matter the-ories.

The discreteness of area and volume is derived asfollows. Consider the area A of a surface Σ. Thephysical area A of Σ depends on the metric, namelyon the gravitational field. In a quantum theory ofgravity, the gravitational field is a quantum fieldoperator, and therefore we must describe the areaof Σ in terms of a quantum observable, describedby an operator A. We now ask what is the quan-tum operator A in nonperturbative quantum grav-ity. The result can easily be worked out by writingthe standard formula for the area of a surface, andreplacing the metric with the appropriate functionof the loop variables. Promoting these loop vari-ables to operators, we obtain the area operator A.The actual construction of this operator requiresregularizing the classical expression and then tak-ing the limit of a sequence of operators, in a suitableoperator topology. For the details of this construc-tion, see [186,77,84,51]. An alternative regulariza-tion technique is discussed in [17]. The resulting

area operator A acts as follows on a spin networkstate |S〉 (assuming here for simplicity that S is aspin network without nodes on Σ):

A[Σ] |S〉 =

l202

∑

i∈{S∩Σ}

√

pi(pi + 2)

|S〉 (39)

where i labels the intersections between the spinnetwork S and the surface Σ, and pi is the color ofthe link of S crossing the i− th intersection.

This result shows that the spin network states (witha finite number of intersection points with the sur-face and no nodes on the surface) are eigenstatesof the area operator. The corresponding spectrumis labeled by multiplets ~p = (p1, ..., pn) of positivehalf integers, with arbitrary n, and given by

A~p [Σ] =l202

∑

i

√

pi(pi + 2). (40)

21

Shifting from color to spin notation reveals theSU(2) origin of the spectrum:

A~j [Σ] = l20∑

i

√

ji(ji + 1), (41)

where j1, ..., jn are half integer. For the full spec-trum, see [17] (connection representation) and [84](loop representation).

A similar result can be obtained for the volume[186,142,143,77,139]. Let us restrict here for sim-plicity to spin networks S with nondegenerate four-valent nodes, labeled by an index i. Let ai, bi, ci, di

be the colors of the links adjacent to the i−th nodeand let Ji label the basis in the intertwiner space.The volume operator V acts as follows

V |S〉 =l302

∑

i

√

|iW (i)(ai, bi, ci, di)||S〉 (42)

where W i is an operator that acts of the finite di-mensional space of the intertwiners in the i − thnode, and its matrix elements are explicitly given(in a suitable basis) by (ǫ = ±)

W i(a, b, c, d)t+ǫt−ǫ = −ǫ(−1)

a+b+c+d2 ×

[

1

4t(t+ 2)

a+ b + t+ 3

2

c+ d+ t+ 3

2

1 + a+ b− t

2

1 + a+ t− b

2

1 + b+ t− a

2(43)

1 + c+ d− t

2

1 + c+ t− d

2

1 + d+ t− c

2

]12

.

See [51]. The volume eigenvalues vi are obtainedby diagonalizing these matrices. For instance, inthe simple case a = b, c = d = 1, we have

va =(hG)3

2

√

a(a+ 2); (44)

if d = a+ b+ c, we have

va,b,c =(hG)3

2

√

abc(a+ b+ c). (45)

For more details, and the full derivation of theseformulas, see [51,203]

• Classical limit. Quantum states representingflat spacetime. Weaves. Discrete small scalestructure of space.

The s-knot states do not represent excitations ofthe quantum gravitational field over flat space, butrather over “no-space”, or over the gµν = 0 solu-tion. A natural problem is then how flat space (orany other smooth geometry) might emerge from the

theory. Notice that in a general relativistic contextthe Minkowski solution does not have all the prop-erties of the conventional field theoretical vacuum.(In gravitational physics there is no real equivalentof the conventional vacuum, particularly in the spa-tially compact case.) One then expects that flatspace is represented by some highly excited statein the theory. States in H that describe flat spacewhen probed at low energy (large distance) havebeen studied in [23,217,49,99]. These have a dis-crete structure at the Planck scale. Furthermore,small excitations around such states have been con-sidered in [125], where it is shown that H containsall “free graviton” physics, in a suitable approxi-mation.

• The Bekenstein-Mukhanov effect.

Recently, Bekenstein and Mukhanov [46] have sug-gested that the thermal nature of Hawking’s radia-tion [105,106] may be affected by quantum prop-erties of gravity (For a review of earlier sugges-tions in this direction, see [193]). Bekenstein andMukhanov observe that in most approaches toquantum gravity the area can take only quantizedvalues [97]. Since the area of the black hole sur-face is connected to the black hole mass, black holemass is likely to be quantized as well. The mass ofthe black hole decreases when radiation is emitted.Therefore emission happens when the black holemakes a quantum leap from one quantized valueof the mass (energy) to a lower quantized value,very much as atoms do. A consequence of this pic-ture is that radiation is emitted at quantized fre-quencies, corresponding to the differences betweenenergy levels. Thus, quantum gravity implies a dis-cretized emission spectrum for the black hole radi-ation.

This result is not physically in contradiction withHawking’s prediction of a continuous thermal spec-trum, because spectral lines can be very densein macroscopic regimes. But Bekenstein andMukhanov observed that if we pick the simplestansatz for the quantization of the area –that theArea is quantized in multiple integers of an elemen-tary area A0–, then the emitted spectrum turns outto be macroscopically discrete, and therefore verydifferent from Hawking’s prediction. I will denotethis effect as the kinematical Bekenstein-Mukhanoveffect. Unfortunately, however, the kinematicalBekenstein-Mukhanov effect disappears if we re-place the naive ansatz with the spectrum (41) com-puted from loop quantum gravity. In loop quantumgravity the eigenvalues of the area become exponen-tially dense for a macroscopic black hole, and there-fore the emission spectrum can be consistent withHawking’s thermal spectrum. This is due to thedetails of the spectrum (41) of the area. A detailed

22

discussion of this result is in [43], but the result wasalready contained (implicitly, in the first version) in[17]. It is important to notice that the density ofthe eigenvalues shows only that the simple kinemat-ical argument of Bekenstein and Mukhanov is notvalid in this theory, and not that their conclusionsis necessarily wrong. As emphasized by Mukhanov,a discretization of the emitted spectrum could bestill be originated dynamically.

• Black Hole Entropy from Loop QuantumGravity

Indirect arguments [105,106,45,210] strongly sup-port the idea that a Schwarzschild black hole of(macroscopic) area A behaves as a thermodynam-ical system governed by the Bekenstein-Hawkingentropy

S =k

4hGNewton

A (46)

(k is the Boltzmann constant; here I put the speedof light equal to one, but write the Planck and New-ton constants explicitly). A physical understandingand a first principles derivation of this relation re-quires quantum gravity, and therefore represents achallenge for every candidate theory of quantumtheory. A derivation of the Bekenstein-Hawkingexpression (46) for the entropy of a Schwarzschildblack hole of surface area A via a statistical me-chanical computation, using loop quantum gravity,was obtained in [134,135,176].

This derivation is based on the ideas that the en-tropy of the hole originates from the microstatesof the horizon that correspond to a given macro-scopic configuration [213,63,64,37,38]. Physical ar-guments indicate that the entropy of such a systemis determined by an ensemble of configurations ofthe horizon with fixed area [176]. In the quantumtheory these states are finite in number, and canbe counted [134,135]. Counting these microstatesusing loop quantum gravity yields

S =c

γ

k

4hGNewton

A. (47)

(An alternative derivation of this result has beenannounced from Ashtekar, Baez, Corichi and Kras-nov [11].) γ is defined in section VI, and c is a realnumber of the order of unity that emerges from thecombinatorial calculation (roughly, c ∼ 1/4π). Ifwe choose γ = c, we get (46) [189,70]. Thus, thetheory is compatible with the numerical constant inthe Bekenstein-Hawking formula, but does not leadto it univocally. The precise significance of this factis under discussion. In particular, the meaning of γis unclear. Jacobson has suggested [127] that finite

renormalization effects may affect the relation be-tween the bare and the effective Newton constant,and this may be reflected in γ. For discussion ofthe role of γ in the theory, see [189]. On the issueof entropy in loop gravity, see also [194].

VIII. MAIN OPEN PROBLEMS AND MAINCURRENT LINES OF INVESTIGATION

Hamiltonian constraint. The kinematic of the theoryis well understood, both physically (quanta of areaand volume, polymer-like geometry) and from themathematical point of view (H, s-knot states, areaand volume operators). The part of the theorywhich is not yet fully under control is the dynamics,which is determined by the hamiltonian constraint.A plausible candidate for the quantum hamiltonianconstraint is the operator introduced by Thiemann[206,201,202]. The commutators of the Thiemannoperator with itself and with the diffeomorphismconstraints close, and therefore the operator de-fines a complete and consistent quantum theory.However, doubts have been raised on the physicalcorrectness of this theory, and some variants of theoperator have been considered.

The doubts originate from various considera-tions. First, Lewandowski, Marolf and others havestressed the fact the quantum constraint algebracloses, but it is not isomorphic to the classicalconstraint algebra of GR [140]. Recently, a de-tailed analysis of this problem has been completedby Marolf, Lewandowski, Gambini and Pullin [90].The failure to reproduce the classical constraintalgebra has been disputed, and is not necessarilya problem, since the only strict requirement onthe quantum theory, besides consistency, is thatits gauge invariant physical predictions match theones of classical general relativity in the appropri-ate limit. Still, the difference in the algebras maybe seen as circumstantial evidence (not a proof) forthe failure of the classical limit. The issue is tech-nically delicate and still controversial. I hope I willbe able to say something more definitive in the nextupdate of this review.

Second, Brugmann [54] and Smolin [192] havepointed out a sort of excess “locality” in the formof the operator, which, intuitively, seems in contra-diction with the propagation properties of the Ein-stein equations. Finally, by translating the Thie-mann operator into a spacetime covariant four-dimensional formalism, Reisenberger and Rovellihave noticed a suspicious lack of manifest 4-d co-variance in the action of the operator [160], a factpointing again to the possibility of anomalies in thequantum constraint algebra.

23

Motivated by these doubts, several variants of Thie-mann’s operator have been suggested. The originalThiemann’s operators is constructed using the vol-ume operator. There are two versions of the vol-ume operator in the literature: VRL, introducedin [186] and VAL, introduced in [14,16,15]. See[139] for a detailed comparison. Originally, Thie-mann thought that using VRS in the hamiltonianconstraint would yield difficulties, but it later be-come clear that this is not the case [140]. Bothversions of the volume can be used in the defini-tion, yielding two alternative versions of the hamil-tonian [140]. Next, in its simplest version the oper-ator is non-symmetric. Since the classical hamilto-nian constraint constraint is real (on SU(2) gaugeinvariant states), one might expect a correspond-ing self-adjoint quantum operator.∗∗∗ Accordingly,several ways of symmetrizing the operator havebeen considered (see a list in [140]). Next, Smolinhas considered some ad hoc modifications of theconstraint in [192]. Finally, the spacetime covari-ant formalism in [160] naturally suggest a “covari-antisation” of the operator, described in [160] un-der the name of “crossing symmetry”. This co-variantisation amount to adding to the vertex de-scribed in Figure 3 the vertices, described in Figure5, which are simply obtained by re-orienting Figure3 in spacetime.

FIG. 5. The crossing symmetric vertices.

A full comparative analysis of this various proposals

∗∗∗There is an argument which is often put forward againstthe requirement of self-adjointness of the hamiltonian con-straint H : let H be self-adjoint and O be any operator ofthe form O = [H,A] , where A is any operator (many op-erators that we do not expect could vanish have this form).Then the expectation value of O vanishes on physical states|ψ〉 from 〈ψ|O|ψ〉 = 〈ψ|[H,A]|ψ〉 = 0. The mistake in thisargument is easily detected by replacing H with a nonrela-tivistic free particle quantum hamiltonian, A with x and |ψ〉with an eigenstate of the momentum: the error is to neglectthe infinities generated by the use of generalized states.

would be of great interest.

Ultimately, the final tests of any proposal for thehamiltonian constraint operator must be consis-tency and a correct classical limit. Thus, the so-lution of the hamiltonian constraint puzzle is likelyto be subordinate to the solution of the problemof extracting the classical limit (of the dynamics)from the theory.

Matter. The basics of the description of matter inthe loop formalism have been established in[150,151,133,26,207,200]. Work needs to be donein order to develop a full description of the basicmatter couplings. In particular there are strong re-curring indications that the Planck scale discrete-ness naturally cuts the traditional quantum fieldinfinities off. In particular, in [200], Thiemann ar-gues that the Hamiltonian constraint governing thecoordinate time evolution of the Yang-Mills field isa well defined operator (I recall that, due to theultraviolet divergences, no rigorously well definedHamiltonian operator for conventional Yang-Millstheory is known in 4 dimensions.) If these indica-tions are confirmed, the result would be very re-markable. What is still missing are calculationaltechniques that could allow us to connect the well-defined constraint with finite observables quantitiessuch as scattering amplitudes.

Spacetime formalism. In my view, the development ofcontinuous spacetime formalisms, [181,160,33,144],is one of the most promising areas of developmentof the theory, because it might be the key foraddressing most of the open problems. First, aspacetime formalism frees us from the obscuritiesof the frozen time formalism, and allows an intu-itive, Feynman-style, description of the dynamicsof quantum spacetime. I think that the classicallimit, the quantum description of black holes, orgraviton-graviton scattering, just to mention a fewexamples, could be addressed much more easily inthe covariant picture. Second, it allows the generalideas of Hartle [103] and Isham [119,120,122,123]on the interpretation of generally covariant quan-tum theories to be applied in loop quantum gravity.This could drastically simplify the complications ofthe canonical way of dealing with general covariantobservables [169,167]. Third, the spacetime formal-ism should suggest solutions to the problem of se-lecting the correct hamiltonian constraint: it is usu-ally easier to deal with invariances in the lagrangianrather than in the hamiltonian formalisms. Thespacetime formalism is just born, and much has tobe done. See the original papers for suggestionsand open problems.

Black holes. The derivation of the Bekenstein-Hawkingentropy formula is a major success of loop quan-tum gravity, but much remains to be understood.

24

A clean derivation from the full quantum theoryis not yet available. Such a derivation would re-quire us to understand what precisely is the eventhorizon in the quantum theory. In other words,given a quantum state of the geometry, we shouldbe able to define and “locate” its horizon (or what-ever structure replaces it in the quantum theory).To do so, we should understand how to effectivelydeal with the quantum dynamics, how to describethe classical limit (in order to find the quantumstates corresponding to classical black hole solu-tions), as well as how do describe asymptoticallyflat quantum states.

Besides these formal issues, at the roots of the blackhole entropy puzzle there is a basic physical prob-lem, which, to my understanding, is still open. Theproblem is to understand how we can use basic ther-modynamical and statistical ideas and techniquesin a general covariant context.††† To appreciate thedifficulty, notice that statistical mechanics makesheavy use of the notion of energy (say in the def-inition of the canonical or microcanonical ensem-bles); but there is no natural local notion of energyassociated to a black hole (or there are too manyof such notions). Energy is an extremely slipperynotion in gravity. Thus, how do we define the sta-tistical ensemble? Put in other words: to computethe entropy (say in the microcanonical) of a normalsystem, we count the states with a given energy. InGR we should count the states with a given what?One may say: black hole states with a given area.But why so? We do understand why the number ofstates with given energy governs the thermodynam-ical behavior of normal systems. But why shouldthe number of states with given area govern thethermodynamical behavior of the system, namelygovern its heat exchanges with the exterior? A ten-tative physical discussion of this last point can befound in [178].

How to extract physics from the theory.Assume we pick a specific hamiltonian constraint.Then we have, in principle, a well defined quantumtheory. How do we extract physical informationfrom it? Some physical consequences of the theory,such as the area and volume eigenvalues, or the en-tropy formula, have been extracted from the the-ory by various ad hoc methods. But is there a gen-eral technique, say corresponding to the traditionalQFT perturbation expansion of the S matrix, fordescribing the full dynamics of the gravitational

†††A general approach to this problem and an idea about itsconnection with the “problem of time” in quantum gravityhas been developed by Connes and Rovelli in [173,174,69].

field? Presumably, such general technique shouldinvolve some kind of expansion, since we could nothope to solve the theory exactly. Attempts to de-fine physical expansions have been initiated in [175]and, in different form, in [160]. Ideally, one wouldwant a general scheme for computing transition am-plitudes in some expansion parameter around somestate. Computing scattering amplitudes would beof particular interest, in order to make connectionwith particle physics language and to compare thetheory with string predictions.

Classical limit. Finally, to prove that loop quantumgravity is a valuable candidate for describing quan-tum spacetime, we need to prove that its classicallimit is GR (or at least overlaps GR in the regimewhere GR is well tested). The traditional connec-tion between loop quantum gravity and classicalGR is via the notion of weave, a quantum state that“looks semiclassical” at distances large compared tothe Planck scale. However, the weaves studied sofar [23,99] are 3d weaves, in the sense that they areeigenstates of the three dimensional metric. Sucha state corresponds to an eigenstate of the positionfor a particle. Classical behavior is recovered notby these states but rather by wave packets whichhave small spread in position as well as in momen-tum. Similarly, the quantum Minkowski spacetimeshould have small spread in the three metric as wellas in its momentum – as the quantum electromag-netic vacuum has small quantum spread in the elec-tric and magnetic field. To recover classical GRfrom loop quantum gravity, we must understandsuch states. Preliminary investigation in this di-rection can be found in [126,125], but these papersare now several years old, and they were writtenbefore the more recent solidification of the basicsof the theory. Another direction consists in the di-rect study of coherent states in the state space ofthe theory.

As these brief notes indicate, the various open prob-lems in loop quantum gravity are interconnected. In asense, loop quantum gravity grew aiming at the nonper-turbative regime and the physical results obtained so farare in this regime. The main issue is then to recover thelong distance behavior of the theory. That is, to study itsclassical limit and the dynamics of the low energy exci-tations over a semiclassical background. Understandingthis aspect of the theory would assure us that the theorywe are dealing with is indeed a quantum theory of thegravitational field, would allow us to understand quan-tum black holes, would clarify the origin of infinities inthe matter hamiltonians and so on. Still in other words,what mostly needs to be understood is the structure ofthe (Minkowski) vacuum in loop quantum gravity.

25

IX. SHORT SUMMARY AND CONCLUSION

In this section, I very briefly summarize the state ofloop quantum gravity and its main results. The mathe-matics of the theory is solidly defined, and is understoodfrom several alternative points of view. Long standingproblems such as the lack of a scalar product, the diffi-cult of controlling the overcompletness of the loop basisand the problem of implementing the reality conditionin the quantum theory have been successfully solved orsidestepped. The kinematics is given by the Hilbert spaceH, defined in Section VI B, which carries a representa-tion of the basic operators: the loop operators (22-23).A convenient orthonormal basis in H is provided by thespin network states, defined in Section VI C. The diffeo-morphism invariant states are given by the s-knot states,and the structure and properties of the (diff-invariant)quantum states of the geometry are quite well under-stood (Section VI H). These states give a description ofquantum spacetime in terms of polymer-like excitationsof the geometry. More precisely, in terms of elementaryexcitations carrying discretized quanta of area.

The dynamics is coded into the hamiltonian constraint.A well defined version of this constraint exists (see equa-tion (35)), and thus a consistent theory exists, but aproof that the classical limit of this theory is classicalgeneral relativity is still lacking. Alternative versions ofthe hamiltonian constraint have been proposed and areunder investigation. In all these cases the hamiltonianhas the crucial properties of acting on nodes only. Thisimplies that its action is naturally discrete and combina-torial. This fact is possibly at the roots of the finitenessof the theory. A large class of physical states which areexact solutions of the dynamics are given by s-knots with-out nodes; other exact states are related to knot theoryinvariants (Section VII A).

The theory can be extended to include matter, andthere are strong indications that ultraviolet divergencesdo not appear. A spacetime covariant version of the the-ory, in the form of a topological sum over surfaces isunder development (Section VI J).

The main physical results derived so far from the the-ory is given by the explicit computation of the eigen-values of area and volume, some of which are given inequations (41-45), and a derivation of the black hole en-tropy formula (Section 41). The two main (related) openproblems are to understand the description of the low en-ergy regime within the theory and to choose the correctversion of the hamiltonian constraint.

A. Conclusion

The history of quantum gravity is a sequel of momentsof great excitement followed by bitter disappointments. Idistinctively remember, as a young student, listening to avery famous physicist announcing at a major conference

that quantum gravity was solved (I think it was the turnof supergravity). The list of theories that claimed to befinal and have then ended up forgotten or supersededis a reason of embarrassment for the theoretical physicscommunity, in my opinion.

In my view, loop quantum gravity is the best we cando so far in trying to understand quantum spacetime,from a nonperturbative, background-independent pointof view. Theoretically, we have reasons to suspect thatthis approach could represent a consistent quantum the-ory with the correct classical limit, but there are alsosome worrying contrary indications. The theory yields adefinite physical picture of quantum spacetime and def-inite quantitative predictions, but a systematic way ofextracting physical information is still lacking. Experi-mentally, there is no support to the theory, neither directnor indirect. The spectra of area and volume computedin the theory could or could not be physically correct. Iwish I could live long enough to know!

ACKNOWLEDGMENTS

I am especially indebted with Michael Reisenberger,Roberto DePietri Don Marolf, Jerzy Lewandowski, JohnBaez, Thomas Thiemann and Abhay Ashtekar for theiraccurate reading of the manuscript, their detailed sug-gestions and their ferocious criticisms. Their inputs havebeen extensive and their help precious. I am gratefulto all my friends in the community for the joy of do-ing physics together. This work was supported by NSFGrant PHY-95-15506.

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