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QUANTUM GRAVITY: WHERE ARE WE? • troductory literature, institutions where loop grav- ity is studied, web pages, and other information This is a non-technical section in which I illustrate the that may be of use to students and researchers. problem of quantum gravity in general, its origin, its im- portance, and the present state of our knowledge in this Section V, “Main ideas and physical inputs”, regard. • discusses the physical and mathematical ideas on The problem of describing the quantum regime of the which loop quantum gravity is based, at a rather gravitational field is still open. There are tentative theo- technical level. ries, and competing research directions. For an overview, The actual theory is described in detail in section see [121]. The two largest research programs are string • VI, “The formalism”. theory and loop quantum gravity. But several other di- rections are being explored, such as twistor theory [154], Section VII, “Results”, is devoted to the results noncommutative geometry [68], simplicial quantum grav- • that have been derived from the theory. I have di- ity [6,65,53,1], Euclidean quantum gravity [104,107], the vided results in two groups. First, the “technical” Null Surface Formulation [85–87] and others. results (VII A), namely the ones that have impor- String theory and loop quantum gravity differ not only tance for the construction and the understanding because they explore distinct physical hypotheses, but of the theory itself, or that warrant the theory’s also because they are expressions of two separate commu- consistency. Second, the “physical” results (VII B): nities of scientists, which have sharply distinct prejudices, what the theory says about the physical world. and view the problem of quantum gravity in surprisingly different manners. In section VIII, “Open problems and current • lines of investigation”, I illustrate what I con- sider the main open problems, and the main cur- A. What is the problem? The view of a high energy rently active research lines. physicist In section IX, “Short summary and conclu- • sion”, I summarize very briefly the state and the High energy physics has obtained spectacular successes result of the theory, and present (necessarily very during this century, culminated with the (far from linear) preliminary!) conclusions. establishment of quantum field theory as the general form of dynamics and with the comprehensive success of the At the price of several repetitions, the structure of this SU(3) SU(2) U(1) Standard Model. Thanks to this review is very modular: sections are to a large extent in- success,× now a few× decades old, physics is in a condition dependent from each other, have different style, and can in which it has been very rarely: there are no experimen- be combined according to the interest of the reader. A tal results that clearly challenge, or clearly escape, the reader interested only in a very brief overview of the the- present fundamental theory of the world. The theory we ory and its results, can find this in section IX. Graduate have encompasses virtually everything – except gravita- students or persons of general culture may get a general tional phenomena. From the point of view of a particle idea of what goes on in this field and its main ideas from physicist, gravity is then simply the last and weakest of sections II and VII. If interested only in the technical as- the interactions. It is natural to try to understand its pects of the theory and its physical results, one can read quantum properties using the strategy that has been so sections VI and VII alone. Scientists working in this field successful for the rest of microphysics, or variants of this can use section VI and VII as a reference, and I hope they strategy. The search for a conventional quantum field will find sections II, III and V and VIII stimulating. . . theory capable of embracing gravity has spanned several I will not enter technical details. However, I will point decades and, through an adventurous sequence of twists, to the literature where these details are discussed. I have moments of excitement and disappointments, has lead to tried to be as complete as possible in indicating all rel- string theory. The foundations of string theory are not evant aspects and potential difficulties of the issues dis- yet well understood; and it is not yet entirely clear how cussed. a supersymmetric theory in 10 or 11 dimensions can be The literature in this field is vast, and I am sure that concretely used for deriving comprehensive univocal pre- there are works whose existence or relevance I have failed dictions about our world.∗ But string theory may claim to recognize. I sincerely apologize with the authors whose contributions I have neglected or under-emphasized and I strongly urge them to contact me to help me making this review more complete. The “living reviews” are con- ∗I heard the following criticism to loop quantum gravity: stantly updated and I should be able to correct errors and “Loop quantum gravity is certainly physically wrong, be- omissions. cause: (1) it is not supersymmetric, (2) is formulated in four dimensions”. But experimentally, the world still insists

2 extremely remarkable theoretical successes and is today nally consistent manner, but we do not have a novel con- the leading and most widely investigated candidate the- ceptual foundation capable of supporting both theories. ory of quantum gravity. This is why we do not yet have a theory capable of pre- In string theory, gravity is just one of the excitations of dicting what happens in the physical regime in which a string (or other extended object) living over some back- both theories are relevant, the regime of Planck scale 33 ground metric space. The existence of such background phenomena, 10− cm. metric space, over which the theory is defined, is needed General relativity has taught us not only that space for the formulation and for the interpretation of the the- and time share the property of being dynamical with the ory, not only in perturbative string theory, but in the rest of the physical entities, but also –more crucially– recent attempts of a non-perturbative definition of the that spacetime location is relational only (see section theory, such as M theory, as well, in my understanding. V C). Quantum mechanics has taught us that any dy- Thus, for a physicist with a high energy background, the namical entity is subject to Heisenberg’s uncertainty at problem of quantum gravity is now reduced to an aspect small scale. Thus, we need a relational notion of a of the problem of understanding what is the mysterious quantum spacetime, in order to understand Planck scale nonperturbative theory that has perturbative string the- physics. ory as its perturbation expansion, and how to extract Thus, for a relativist, the problem of quantum gravity information on Planck scale physics from it. is 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 this B. What is the problem? The view of a relativist synthesis, the notions of space and time need to be deeply reshaped, in order to keep into account what we have For a relativist, on the other hand, the idea of a funda- learned with both our present “fundamental” theories. mental description of gravity in terms of physical excita- Unlike perturbative or nonperturbative string theory, tions over a background metric space sounds physically loop quantum gravity is formulated without a back- very wrong. The key lesson learned from general rela- ground spacetime, and is thus a genuine attempt to grasp tivity is that there is no background metric over which what is quantum spacetime at the fundamental level. Ac- physics happens (unless, of course, in approximations). cordingly, the notion of spacetime that emerges from the The world is more complicated than that. Indeed, for a theory is profoundly different from the one on which con- relativist, general relativity is much more than the field ventional quantum field theory or string theory are based. theory of a particular force. Rather, it is the discovery that certain classical notions about space and time are inadequate at the fundamental level; they require mod- C. Strings or loops? ifications which are possibly as basics as the ones that quantum mechanics introduced. One of such inadequate Above, I have emphasized the radically distinct cul- notions is precisely the notion of a background metric tural paths leading to string theory and loop quantum space (flat or curved), over which physics happens. This gravity. Here, I attempt a comparison of the actual profound conceptual shift has led to the understanding achievements that the two theories have obtained so far of relativistic gravity, to the discovery of black holes, to for what concerns the description of Planck scale physics. relativistic astrophysics and to modern cosmology. Once more, however, I want to emphasize here that, From Newton to the beginning of this century, physics whatever prejudices this or that physicist may have, both has had a solid foundation in a small number of key no- theories are tentative: as far as we really know, either, tions such as space, time, causality and matter. In spite or both, theories could very well turn out to be physi- of substantial evolution, these notions remained rather cally entirely wrong. And I do not mean that they could stable and self-consistent. In the first quarter of this cen- be superseded: I mean that all their specific predictions tury, quantum theory and general relativity have mod- could be disproved by experiments. Nature does not al- ified this foundation in depth. The two theories have ways share our aesthetic judgments, and the history of obtained solid success and vast experimental corrobora- theoretical physics is full of enthusiasm for strange the- tion, and can be now considered as established knowl- ories turned into disappointment. The arbiter in science edge. Each of the two theories modifies the conceptual are experiments, and not a single experimental result sup- foundation of classical physics in a (more or less) inter- ports, not even very indirectly, any of the current theories that 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- on looking four-dimensional and not supersymmetric. In my opinion, people should be careful of not being blinded by their own speculation, and mistaken interesting hypotheses (such as supersymmetry and high-dimensions) for established truth. †For a detailed discussion of this idea, see [180] and [197].

3 ativity (proton decay, supersymmetric particles, exotic sional objects – call them loops or strings. I understand particles, solar system dynamics) have for the moment that in another living review in this issue, Lee Smolin been punctually falsified by the experiments. Comparing explores the possible relations between string theory and this situation with the astonishing experimental success loop gravity [191]. of the Standard Model and classical general relativity should make us very cautious, I believe. Lacking exper- iments, theories can only be compared on completeness III. HISTORY OF LOOP QUANTUM GRAVITY, and aesthetic criteria – criteria, one should not forget, MAIN STEPS that according to many favored Ptolemy over Coperni- cus, at some point. The following chronology does not exhaust the liter- The main merit of string theory is that it provides a su- ature on loop quantum gravity. It only indicates the perbly elegant unification of known fundamental physics, key steps in the construction of the theory, and the first and that it has a well defined perturbation expansion, fi- derivation of the main results. For more complete refer- nite order by order. Its main incompleteness is that its ences, see the following sections. (In the effort of group- non-perturbative regime is poorly understood, and that ing some results, something may appear a bit out of the we do not have a background-independent formulation of chronological order.) the theory. In a sense, we do not really know what is the theory we are talking about. Because of this poor under- 1986 Connection formulation of classical general standing of the non perturbative regime of the theory, relativity Planck scale physics and genuine quantum gravitational Sen, Ashtekar. phenomena are not easily controlled: except for a few Loop quantum gravity is based on the formulation computations, there is not much Planck scale physics de- of classical general relativity, which goes under the rived from string theory so far. There are, however, two name of “new variables”, or “Ashtekar variables”, sets of remarkable physical results. The first is given by or “connectio-dynamics” (in contrast to Wheeler’s some very high energy scattering amplitudes that have “geometro-dynamics”). In this formulation, the been computed (see for instance [2–5,209,199]). An in- field variable is a self-dual connection, instead of triguing aspect of these results is that they indirectly the metric, and the canonical constraints are sim- suggest that geometry below the Planck scale cannot be pler than in the old metric formulation. The idea probed –and thus in a sense does not exist– in string of using a self-dual connection as field variable and theory. The second physical achievement of string the- the simple constraints it yields were discovered by ory (which followed the d-branes revolution) is the recent Amitaba Sen [190]. Abhay Ashtekar realized that derivation of the Bekenstein-Hawking black hole entropy in the SU(2) extended phase space a self-dual con- formula for certain kinds of black holes [198,111,110,109]. nection and a densitized triad field form a canon- The main merit of loop quantum gravity, on the other ical pair [7,8] and set up the canonical formalism hand, is that it provides a well-defined and mathemati- based on such pair, which is the Ashtekar formal- cally rigorous formulation of a background-independent ism. Recent works on the loop representation are non-perturbative generally covariant quantum field the- not based on the original Sen-Ashtekar connection, ory. The theory provides a physical picture and quanti- but on a real variant of it, whose use has been intro- tative predictions on the world at the Planck scale. The duced into Lorentzian general relativity by Barbero main incompleteness of the theory regards the dynam- [39–42]. ics, formulated in several variants. So far, the theory has lead to two main sets of physical results. The first is the 1986 Wilson loop solutions of the hamiltonian derivation of the (Planck scale) eigenvalues of geometri- constraint cal quantities such as areas and volumes. The second is Jacobson, Smolin. the derivation of black hole entropy for “normal” black Soon after the introduction of the classical Ashtekar holes (but only up to the precise numerical factor). variables, Ted Jacobson and Lee Smolin realized Finally, strings and loop gravity, may not necessarily in [128] that the Wheeler-DeWitt equation re- be competing theories: there might be a sort of com- formulated in terms of the new variables admits plementarity, at least methodological, between the two. a simple class of exact solutions: the traces of This is due to the fact that the open problems of string the holonomies of the Ashtekar connection around theory regard its background-independent formulation, smooth non-selfintersecting loops. In other words: and loop quantum gravity is precisely a set of techniques the Wilson loops of the Ashtekar connection solve for dealing non-perturbatively with background indepen- the Wheeler-DeWitt equation if the loops are dent theories. Perhaps the two approaches might even, smooth and non self-intersecting. to some extent, converge. Undoubtedly, there are sim- 1987 The Loop Representation ilarities between the two theories: first of all the obvi- Rovelli, Smolin. ous fact that both theories start with the idea that the The discovery of the Jacobson-Smolin Wilson loop relevant excitations at the Planck scale are one dimen-

4 solutions prompted Carlo Rovelli and Lee Smolin These states, denoted “weaves”, have a “polymer” [182,163,183,184] to “change basis in the Hilbert like structure at short scale, and can be viewed as space of the theory”, choosing the Wilson loops as a formalization of Wheeler’s “spacetime foam”. the new basis states for quantum gravity. Quan- tum states can be represented in terms of their 1992 C∗ algebraic framework expansion on the loop basis, namely as functions Ashtekar, Isham. on a space of loops. This idea is well known in In [12], Abhay Ashtekar and Chris Isham showed the context of canonical lattice Yang-Mills theory that the loop transform introduced in gravity by [211], and its application to continuous Yang-Mills Rovelli and Smolin could be given a rigorous math- theory had been explored by Gambini and Trias ematical foundation, and set the basis for a mathe- [95,96], who developed a continuous “loop represen- matical systematization of the loop ideas, based on tation” much before the Rovelli-Smolin one. The C∗ algebra ideas. difficulties of the loop representation in the context 1993 Gravitons as embroideries over the weave of Yang-Mills theory are cured by the diffeomor- Iwasaki, Rovelli. phism invariance of GR (see section VI H for de- In [125] Junichi Iwasaki and Rovelli studied the rep- tails). The loop representation was introduced by resentation of gravitons in loop quantum gravity. Rovelli and Smolin as a representation of a classical These appear as topological modifications of the Poisson algebra of “loop observables”. The relation fabric of the spacetime weave. with the connection representation was originally derived in the form of an integral transform (an in- 1993 - Alternative versions finite dimensional analog of a Fourier transform) Di Bartolo, Gambini, Griego, Pullin. from functionals of the connection to loop func- Some versions of the loop quantum gravity alterna- tionals. Several years later, this loop transform was tive to the “orthodox” version have been developed. shown to be mathematically rigorously defined [12]. In particular, these authors above have developed The immediate results of the loop representation the so called “extended” loop representation. See were two: the diffeomorphism constraint is com- [80,78]. pletely solved by knot states (loop functionals that depend only on the knotting of the loops), making 1994 - Fermions, earlier suggestions by Smolin on the role of knot Morales-Tecotl, Rovelli. theory in quantum gravity [195] concrete; and (suit- Matter coupling were beginning to be explored in able [184,196] extensions of) the knot states with [150,151]. Later, matter’s kinematics was studied support on non-selfintersecting loops were proven by Baez and Krasnov [133,35], while Thiemann has to be solutions of all quantum constraints, namely extended his results on the dynamics to the coupled exact physical states of quantum gravity. Einstein Yang-Mills system in [200].

1988 - Exact states of quantum gravity 1994 The dµ0 measure and the scalar product Husain, Br¨ugmann, Pullin, Gambini, Kodama. Ashtekar, Lewandowski, Baez. The investigation of exact solutions of the quan- In [14–16] Ashtekar and Lewandowski set the ba- tum constraint equations, and their relation with sis of the differential formulation of loop quantum knot theory (in particular with the Jones polyno- gravity by constructing its two key ingredients: a mial and other knot invariants) has started soon diffeomorphism invariant measure on the space of after the formulation of the theory and continued (generalized) connections, and the projective fam- since [112,57–60,157,92,94,131,89]. ily of Hilbert spaces associated to graphs. Using these techniques, they were able to give a mathe- 1989 - Model theories matically rigorous construction of the state space of Ashtekar, Husain, Loll, Marolf, Rovelli, Samuel, the theory, solving long standing problems deriving Smolin, Lewandowski, Marolf, Thiemann. from the lack of a basis (the insufficient control on The years immediately following the discovery of the algebraic identities between loop states). Us- the loop formalism were mostly dedicated to un- ing this, they defined a consistent scalar product derstanding the loop representation by studying it and proved that the quantum operators in the the- in simpler contexts, such as 2+1 general relativ- ory were consistent with all identities. John Baez ity [13,146,20], Maxwell [21], linearized gravity [22], showed how the measure can be used in the con- and, much later, 2d Yang-Mills theory [19]. text of conventional connections, extended it to the non-gauge invariant states (allowing the E operator 1992 Classical limit: weaves to be defined) and developed the use of the graph Ashtekar, Rovelli, Smolin. techniques [24,28,27]. Important contributions to The first indication that the theory predicts Planck the understanding of the measure were also given scale discreteness came from studying the states by Marolf and Mour˜ao [149]. that approximate geometries flat on large scale [23].

5 1994 Discreteness of area and volume eigenvalues the use of formal group integration for solving the Rovelli, Smolin. constraints [145,147,148], and on several mathe- In my opinion, the most significative result of loop matical ideas by Jos´eMour˜ao). quantum gravity is the discovery that certain ge- ometrical quantities, in particular area and vol- 1996 Equivalence of the algebraic and differential ume, are represented by operators that have dis- formalisms crete eigenvalues. This was found by Rovelli and DePietri. Smolin in [186], where the first set of these eigenval- In [76], Roberto DePietri proved the equivalence ues were computed. Shortly after, this result was of the two formalisms, using ideas from Thiemann confirmed and extended by a number of authors, [205] and Lewandowski [139]. using very diverse techniques. In particular, Re- 1996 Hamiltonian constraint nate Loll [142,143] used lattice techniques to ana- Thiemann. lyze the volume operator and corrected a numerical The first version of the loop hamiltonian con- error in [186]. Ashtekar and Lewandowski [138,17] straint is in [183,184]. The definition of the recovered and completed the computation of the constraint has then been studied and modified spectrum of the area using the connection represen- repeatedly, in a long sequence of works, by tation, and new regularization techniques. Frittelli, Br¨ugmann, Pullin, Blencowe, Borissov and others Lehner and Rovelli [84] recovered the Ashtekar- [112,47,60,58,57,59,157,92,48]. An important step Lewandowski terms of the spectrum of the area, was made by Rovelli and Smolin in [185] with the using the loop representation. DePietri and Rov- realization that certain regularized loop operators elli [77] computed general eigenvalues of the vol- have finite limits on knot states (see [140]). The ume. Complete understanding of the precise rela- search culminated with the work of Thomas Thie- tion between different versions of the volume oper- mann, who was able to construct a rather well- ator came from the work of Lewandowski [139]. defined hamiltonian operator whose constraint al- gebra closes [206,201,202]. Variants of this con- 1995 Spin networks - solution of the overcom- straint have been suggested in [192,160] and else- pleteness problem where. Rovelli, Smolin, Baez. A long standing problem with the loop basis was 1996 Real theory: solution of the reality condi- its overcompleteness. A technical, but crucial step tions problem in understanding the theory has been the discovery Barbero, Thiemann. of the spin-network basis, which solves this over- As often stressed by Karel Kuchaˇr, implement- completeness. This step was taken by Rovelli and ing the complicated reality condition of the com- Smolin in [187] and was motivated by the work of plex connection into the quantum theory was, until Roger Penrose [153,152], by analogous bases used 1996, the main open problem in the loop approach.‡ in lattice gauge theory and by ideas of Lewandowski Following the directions advocated by Fernando [137]. Shortly after, the spin network formalism Barbero [39–42], namely to use the real connec- was cleaned up and clarified by John Baez [31,32]. tion in the Lorentzian theory, Thiemann found an After the introduction of the spin network basis, elegant elegant way to completely bypass the prob- all problems deriving from the incompleteness of lem. the loop basis are trivially solved, and the scalar product could be defined also algebraically [77]. 1996 Black hole entropy Krasnov, Rovelli. 1995 Lattice A derivation of the Bekenstein-Hawking formula for Loll, Reisenberger, Gambini, Pullin. the entropy of a black hole from loop quantum grav- Various lattice versions of the theory have appeared ity was obtained in [176], on the basis of the ideas of in [141,159,94,79]. Kirill Krasnov [134,135]. Recently, Ashtekar, Baez, Corichi and Krasnov have announced an alterna- 1995 Algebraic formalism / Differential formal- tive derivation [11]. ism DePietri Rovelli / Ashtekar, Lewandowski, Marolf, 1997 Anomalies Mour˜ao, Thiemann. Lewandowski, Marolf, Pullin, Gambini. The cleaning and definitive setting of the two main versions of the formalisms was completed in [77] for the algebraic formalism (the direct descendent of the old loop representation); and in [18] for ‡“The loop people have a credit card called reality condi- the differential formalism (based on the Ashtekar- tions, and whenever they solve a problem, they charge the Isham C∗ algebraic construction, on the Ashtekar- card, but one day the bill comes and the whole thing breaks Lewandowski measure, on Don Marolf’s work on down like a card house” [136].

6 These authors have recently completed an exten- ideas can be found in the Rovelli’s review pa- sive analysis of the issue of the closure of the quan- per [164]. This is more oriented to a reader tum constraint algebra and its departures from the with physics background. corresponding classical Poisson algebra [140,90], – A recent general introduction to the new vari- following earlier pioneering work in this direc- ables which includes several of the recent tion by Br¨ugmann, Pullin, Borissov and others mathematical developments in the quantum [55,61,88,50]. This analysis has raised worries that theory is given by Ashtekar’s Les Houches the classical limit of Thiemann’s hamiltonian oper- 1992 lectures [10]. ator might fail to yield classical general relativity, – A particularly interesting collection of papers but the matter is still controversial. can be found in the volume [29] edited by John 1997 Sum over surfaces Baez. The other book by Baez, and Muniain Reisenberger Rovelli. [36], is a simple and pleasant introduction to A “sum over histories” spacetime formulation of several ideas and techniques in the field. loop quantum gravity was derived in [181,160] – The last and up to date book on the loop from the canonical theory. The resulting covari- representation is the book by Gambini and ant theory turns out to be a sum over topologically Pullin [93], especially good in lattice tech- inequivalent surfaces, realizing earlier suggestions niques and in the variant of loop quantum by Baez [26,27,31,25], Reisenberger [159,158] and gravity called the “extended loop representa- Iwasaki [124] that a covariant version of loop grav- tion” [80,78] (which is nowadays a bit out of ity should look like a theory of surfaces. Baez has fashion, but remains an intriguing alternative studied the general structure of theories defined in to “orthodox” loop quantum gravity). this manner [33]. Smolin and Markoupolou have – The two standard references for a complete explored the extension of the construction to the presentation of the basics of the theory are: Lorentzian case, and the possibility of altering the DePietri and Rovelli [77] for the algebraic spin network evolution rules [144]. formulation; and Ashtekar, Lewandowski, Marolf, Mourao and Thiemann (ALM 2T ) [18] for the differential formulation. IV. RESOURCES Besides the many conferences on gravity, the loop • A valuable resource for finding relevant literature is gravity community has met twice in Warsaw, in • the comprehensive “Bibliography of Publications re- the “Workshop on Canonical and Quantum Grav- lated to Classical and Quantum Gravity in terms of ity”. Hopefully, this will become a recurrent meet- Connection and Loop Variables”, organized chrono- ing. This may be the right place to go for learning logically. The original version was compiled by what is going on in the field. For an informal ac- Peter H¨ubner in 1989. It has subsequently been count of the last of these meetings (August 1997), updates by Gabriella Gonzales, Bernd Br¨ugmann, see [161]. Monica Pierri and Troy Shiling. Presently, it is be- Some of the main institutions where loop quantum ing kept updated by Christopher Beetle and Ale- • gravity is studied are jandro Corichi. The last version can be found on the net in [44]. – The “Center for Gravity and Geometry” at Penn State University, USA. I recommend This living review may serve as an updated intro- their invaluable web page [156], maintained • duction to quantum gravity in the loop formalism. by Jorge Pullin, for finding anything you need More detailed (but less up to date) presentations from the web. are listed below. – Pittsburgh University, USA [100]. – Ashtekar’s book [9] may serve as a valuable – University of California at Riverside, USA. basic introductory course on Ashtekar vari- John Baez moderates an interesting news- ables, particularly for relativists and math- group, sci.physics.research, with news from ematicians. The part of the book on the the field. See [34]. loop representation is essentially an autho- – Albert Einstein Institute, Potsdam, Berlin, rized reprint of parts of the original Rovelli Europe [117]. Smolin article [184]. For this quantum part, I recommend looking at the article, rather than – Warsaw University, Warsaw, Europe. the book, because the article is more complete. – Imperial College, London, Europe [67]. – A simpler and more straightforward introduc- – Syracuse University, USA [208]. tion to the Ashtekar variables and basic loop – Montevideo University, Uruguay.

7 V. MAIN IDEAS AND PHYSICAL INPUTS essential role in the construction of the basic theoreti- cal tools (creation and annihilation operators, canonical The main physical hypotheses on which loop quantum commutation relations, gaussian measures, propagators gravity relies are only general relativity and quantum me- . . . ); these tools cannot be used in quantum field over a chanics. In other words, loop quantum gravity is a rather manifold. conservative “quantization” of general relativity, with its Technically, the difficulty due to the absence of a back- traditional matter couplings. In this sense, it is very ground metric is circumvented in loop quantum gravity different from string theory, which is based on a strong by defining the quantum theory as a representation of a physical hypothesis with no direct experimental support Poisson algebra of classical observables which can be de- (the world is made by strings). fined without using a background metric. The idea that Of course “quantization” is far from a univocal algo- the quantum algebra at the basis of quantum gravity rithm, particularly for a nonlinear field theory. Rather, it is not the canonical commutation relation algebra, but is a poorly understood inverse problem (find a quantum the Poisson algebra of a different set of observables has theory with the given classical limit). More or less subtle long been advocated by Chris Isham [118], whose ideas choices are made in constructing the quantum theory. I have been very influential in the birth of loop quantum discuss these choices below. gravity.§ The algebra on which loop gravity is the loop algebra [184]. The particular choice of this algebra is not harmless, as I discuss below. A. Quantum field theory on a differentiable manifold

B. One additional assumption The main idea beyond loop quantum gravity is to take general relativity seriously. We have learned with general relativity that the spacetime metric and the gravitational In choosing the loop algebra as the basis for the quan- field are the same physical entity. Thus, a quantum the- tization, we are essentially assuming that Wilson loop ory of the gravitational field is a quantum theory of the operators are well defined in the Hilbert space of the the- spacetime metric as well. It follows that quantum grav- ory. In other words, that certain states concentrated on ity cannot be formulated as a quantum field theory over one dimensional structures (loops and graphs) have finite a metric manifold, because there is no (classical) metric norm. This is a subtle non trivial assumptions entering manifold whatsoever in a regime in which gravity (and the theory. It is the key assumption that characterizes therefore the metric) is a quantum variable. loop gravity. If the approach turned out to be wrong, One could conventionally split the spacetime metric it will likely be because this assumption is wrong. The into two terms: one to be consider as a background, Hilbert space resulting from adopting this assumption is which gives a metric structure to spacetime; the other not a Fock space. Physically, the assumption corresponds to be treated as a fluctuating quantum field. This, in- to the idea that quantum states can be decomposed on a deed, is the procedure on which old perturbative quan- basis of “Faraday lines” excitations (as Minkowski QFT tum gravity, perturbative strings, as well as current non- states can be decomposed on a particle basis). perturbative string theories (M-theory), are based. In Furthermore, this is an assumption that fails in con- following this path, one assumes, for instance, that the ventional quantum field theory, because in that con- causal structure of spacetime is determined by the un- text well defined operators and finite norm states need derlying background metric alone, and not by the full to be smeared in at least three dimensions, and one- metric. Contrary to this, in loop quantum gravity we dimensional objects are too singular.∗∗ The fact that assume that the identification between the gravitational at the basis of loop gravity there is a mathematical as- field and the metric-causal structure of spacetime holds, sumption that fails for conventional Yang-Mills quantum and must be taken into account, in the quantum regime field theory is probably at the origin of some of the resis- as well. Thus, no split of the metric is made, and there tance that loop quantum gravity encounters among some is no background metric on spacetime. high energy theorists. What distinguishes gravity from We can still describe spacetime as a (differentiable) Yang-Mills theories, however, and makes this assumption manifold (a space without metric structure), over which viable in gravity even if it fails for Yang-Mills theory is quantum fields are defined. A classical metric structure will then be defined by expectation values of the grav- itational field operator. Thus, the problem of quantum gravity is the problem of understanding what is a quan- §Loop Quantum Gravity is an attempt to solve the last prob- tum field theory on a manifold, as opposed to quantum lem in Isham’s lectures [118]. field theory on a metric space. This is what gives quan- ∗∗The assumption does not fail, however, in two-dimensional tum gravity its distinctive flavor, so different than ordi- Yang-Mills theory, which is invariant under area preserv- nary quantum field theory. In all versions of ordinary ing diffeomorphisms, and where loop quantization techniques quantum field theory, the metric of spacetime plays an were successfully employed [19].

8 diffeomorphism invariance. The loop states are singu- are. They are quantum excitations of the “stage” itself, lar states that span a “huge” non-separable state space. not excitations over a stage. Intuitively, one can un- (Non-perturbative) diffeomorphism invariance plays two derstand from this discussion how knot theory plays a roles. First, it wipes away the infinite redundancy. Sec- role in the theory. First, we define quantum states that ond, it “smears” a loop state into a knot state, so that correspond to loop-like excitations of the gravitational the physical states are not really concentrated in one di- field, but then, when factoring away diffeomorphism in- mension, but are, in a sense, smeared all over the entire variance, the location of the loop becomes irrelevant. The manifold by the nonperturbative diffeomorphisms. This only remaining information contained in the loop is then will be more clear in the next section. its knotting (a knot is a loop up to its location). Thus, diffeomorphism invariant physical states are labeled by knots. A knot represent an elementary quantum excita- C. Physical meaning of diffeomorphism invariance, tion of space. It is not here or there, since it is the space and its implementation in the quantum theory with respect to which here and there can be defined. A knot state is an elementary quantum of space. Conventional field theories are not invariant under a In this manner, loop quantum gravity ties the new no- diffeomorphism acting on the dynamical fields. (Every tion of space and time introduced by general relativity field theory, suitably formulated, is trivially invariant un- with quantum mechanics. As I will illustrate later on, der a diffeomorphism acting on everything.) General rel- the existence of such elementary quanta of space is then ativity, on the contrary is invariant under such transfor- made concrete by the quantization of the spectra of geo- mations. More precisely every general relativistic theory metrical quantities. has this property. Thus, diffeomorphism invariance is not a feature of just the gravitational field: it is a feature of physics, once the existence of relativistic gravity is taken D. Problems not addressed into account. Thus, one can say that the gravitational field is not particularly “special” in this regard, but that Quantum gravity is an open problem that has been diff-invariance is a property of the physical world that investigated for over seventy years now. When one con- can be disregarded only in the approximation in which templates two deep problems, one is tempted to believe the dynamics of gravity is neglected. What is this prop- that they are related. In the history of physics, there are erty? What is the physical meaning of diffeomorphism surprising examples of two deep problems solved by one invariance? stroke (the unification of electricity and magnetism and Diffeomorphism invariance is the technical implemen- the nature of light, for instance); but also many examples tation of a physical idea, due to Einstein. The idea is a in which a great hope to solve more than one problem deep modification of the pre-general-relativistic (pre-GR) at once was disappointed (finding the theory of strong notions of space and time. In pre-GR physics, we assume interactions and getting rid of quantum field theory in- that physical objects can be localized in space and time finities, for instance). Quantum gravity has been asked, with respect to a fixed non-dynamical background struc- at some time or the other, to address almost every deep ture. Operationally, this background spacetime can be open problem in theoretical physics (and beyond). Here defined by means of physical reference-system objects, is a list of problems that have been connected to quan- but these objects are considered as dynamically decou- tum gravity in the past, but about which loop quantum pled from the physical system that one studies. This gravity has little to say: conceptual structure fails in a relativistic gravitational regime. In general relativistic physics, the physical ob- Interpretation of quantum mechanics. Loop quan- jects are localized in space and time only with respect tum gravity is a standard quantum (field) theory. to each other. Therefore if we “displace” all dynamical Pick your favorite interpretation of quantum me- objects in spacetime at once, we are not generating a chanics, and use it for interpreting the quantum different state, but an equivalent mathematical descrip- aspects of the theory. I will refer to two such inter- tion of the same physical state. Hence, diffeomorphism pretations below: When discussing the quantiza- invariance. tion of area and volume, I will use the relation be- Accordingly, a physical state in GR is not “located” tween eigenvalues and outcomes of measurements somewhere [180,169,177] (unless an appropriate gauge performed with classical physical apparata; when fixing is made). Pictorially, GR is not physics over a discussing evolution, I will refer to the histories in- stage, it is the dynamical theory of (or including) the terpretation. The peculiar way of describing time stage itself. evolution in a general relativistic theory may re- Loop quantum gravity is an attempt to implement quire some appropriate variants of standard inter- this subtle relational notion of spacetime localization in pretations, such as generalized canonical quantum quantum field theory. In particular, the basic quantum theory [166,168,165] or Hartle’s generalized quan- field theoretical excitations cannot be localized some- tum mechanics [103]. But loop quantum gravity where (localized with respect to what?) as, say, photons

9 has no help to offer to the scientists that have spec- to address is the problem of the ultraviolet infinities in ulated that quantum gravity will solve the measure- quantum field theory. The very peculiar nonperturbative ment problem. On the other hand, the spacetime short scale structure of loop quantum gravity introduces formulation of loop quantum gravity that has re- a physical cutoff. Since physical spacetime itself comes cently been developed (see Section VI J) is natu- in quanta in the theory, there is literally no space in rally interpreted in terms of histories interpreta- the theory for the very high momentum integrations that tions [103,119,120,122,123]. Furthermore, I think originate the ultraviolet divergences. Lacking a complete that solving the problem of the interpretation of and detailed calculation scheme, however, one cannot yet quantum mechanics might require relational ideas claim with confidence that the divergences, chased from connected with the relational nature of spacetime the door, will not reenter from the window. revealed by general relativity [179,180].

Quantum cosmology. There is widespread confusion VI. THE FORMALISM between quantum cosmology and quantum grav- ity. Quantum cosmology is the theory of the en- Here, I begin the technical description of the basics of tire universe as a quantum system without external loop quantum gravity. The starting point of the construc- observer [102]: with or without gravity, makes no tion of the quantum theory is classical general relativity, difference. Quantum gravity is the theory of one formulated in terms of the Sen-Ashtekar-Barbero connec- dynamical entity: the quantum gravitational field tion [190,7,40]. Detailed introductions to the (complex) (or the spacetime metric): just one field among the Ashtekar formalism can be found in the book [9], in the many. Precisely as for the theory of the quantum review article [164], or in the conference proceedings [81]. electromagnetic field, we can always assume that The real version of the theory is presently the most widely we have a classical observer with classical measur- used. ing apparata measuring gravitational phenomena, Classical general relativity can be formulated in phase and therefore study quantum gravity disregarding space form as follows [9,40]. We fix a three-dimensional quantum cosmology. For instance, the physics of manifold M (compact and without boundaries) and con- a Planck size small cube is governed by quantum i sider a smooth real SU(2) connection Aa(x) and and a gravity and, presumably, has no cosmological im- ˜a plications. vector density Ei (x) (transforming in the vector repre- sentation of SU(2)) on M. We use a,b,... = 1, 2, 3 for Unifications of all interactions or “Theory of Ev- spatial indices and i,j,... = 1, 2, 3 for internal indices. erything”. A common criticism to loop quantum The internal indices can be viewed as labeling a basis in gravity is that it does not unify all interactions. the Lie algebra of SU(2) or the three axis of a local triad. But the idea that quantum gravity can be under- We indicate coordinates on M with x. The relation be- stood only in conjunctions with other fields is an tween these fields and conventional metric gravitational ˜a interesting hypothesis, not an established truth. variables is as follows: Ei (x) is the (densitized) inverse triad, related to the three-dimensional metric gab(x) of Mass of the elementary particles. As far as I see, constant-time surfaces by nothing in loop quantum gravity suggests that one ab ˜a ˜b could compute masses from quantum gravity. g g = Ei Ei , (1)

Origin of the Universe. It is likely that a sound quan- where g is the determinant of gab; and tum theory of gravity will be needed to understand i i i the physics of the Big Bang. The converse is prob- Aa(x)=Γa(x)+ γ ka(x); (2) ably not true: we should be able to understand the i small scale structure of spacetime even if we do not Γa(x) is the spin connection associated to the triad, (de- i i i i understand the origin of the Universe. fined by ∂[aeb] =Γ[aeb]j, where ea is the triad). ka(x) is the extrinsic curvature of the constant time three surface. Arrow of time. Roger Penrose has argued for some In (2), γ is a constant, denoted the Immirzi parameter, time that it should be possible to trace the time that can be chosen arbitrarily (it will enter the hamilto- asymmetry in the observable Universe to quantum nian constraint) [114–116]. Different choices for γ yield gravity. different versions of the formalism, all equivalent in the Physics of the mind. Roger has also speculated that classical domain. If we choose γ to be equal to the imag- inary unit, γ = √ 1, then A is the standard Ashtekar quantum gravity is responsible for the wave func- − tion collapse, and, indirectly, governs the physics of connection, which can be shown to be the projection of the mind [155]. the selfdual part of the four-dimensional spin connection on the constant time surface. If we choose γ = 1, we ob- A problem that has been repeatedly tied to quantum tain the real Barbero connection. The hamiltonian con- gravity, and which loop quantum gravity might be able straint of Lorentzian general relativity has a particularly

10 simple form in the γ = √ 1 formalism, while the hamil- a1a2 [α](s ,s )= − T 1 2 tonian constraint of Euclidean general relativity has a a2 a1 = Tr[Uα(s ,s )E˜ (s )Uα(s ,s )E˜ (s )], (7) − 1 2 2 2 1 1 simple form when expressed in terms of the γ = 1 real a1...aN [α](s ...sN )= connection. Other choices of γ are viable as well. In par- T 1 aN a1 ticular, it has been argued that the quantum theory based = Tr[Uα(s1,sN )E˜ (sN )Uα(sN ,sN 1) . . . E˜ (s1)] − − on different choices of γ are genuinely physical inequiva- s2 where Uα(s1,s2) exp Aa(α(s))ds is the parallel lent, because they yield “geometrical quanta” of different ∼P { s1 } magnitude [189]. Apparently, there is a unique choice of propagator of Aa along α,R defined by γ yielding the correct 1/4 coefficient in the Bekenstein- d dαa(s) Hawking formula [134,135,176,11,178,70], but the matter Uα(1,s)= Aa(α(s)) Uα(1,s) (8) is still under discussion. ds ds The spinorial version of the Ashtekar variables is given (See [77] for more details.) These are the loop observ- in terms of the Pauli matrices σi,i =1, 2, 3, or the su(2) ables, introduced in Yang Mills theories in [95,96], and i generators τi = σi, by in gravity in [183,184]. − 2 The loop observables coordinatize the phase space and a a a E˜ (x)= i E˜ (x) σi =2E˜ (x) τi, (3) have a closed Poisson algebra, denoted the loop algebra. − i i i i i This algebra has a remarkable geometrical flavor. For Aa(x)= A (x) σi = A (x) τi . (4) a −2 a a instance, the Poisson bracket between [α] and [β](s) is non vanishing only if β(s) lies overTα; if it does,T the a Thus, Aa(x) and E˜ (x) are 2 2 anti-hermitian complex result is proportional to the holonomy of the Wilson loops matrices. × obtained by joining α and β at their intersection (by The theory is invariant under local SU(2) gauge trans- rerouting the 4 legs at the intersection). More precisely formations, three-dimensional diffeomorphisms of the a a 1 [α], [β](s) = ∆ [α, β(s)] [α#β] [α#β− ] . manifold on which the fields are defined, as well as under {T T } T − T (coordinate) time translations generated by the hamilto-  (9) nian constraint. The full dynamical content of general relativity is captured by the three constraints that gen- Here erate these gauge invariances [190,9]. dαa(s) As already mentioned, the Lorentzian hamiltonian ∆a[α, x]= ds δ3(α(s), x) (10) constraint does not have a simple polynomial form if we Z ds use the real connection (2). For a while, this fact was is a vector distribution with support on α and α#β is considered an obstacle for defining the quantum hamil- the loop obtained starting at the intersection between α tonian constraint; therefore the complex version of the 1 and β, and following first α and then β. β− is β with connection was mostly used. However, Thiemann has re- reversed orientation. cently succeeded in constructing a Lorentzian quantum A (non-SU(2) gauge invariant) quantity that plays a hamiltonian constraint [206,201,202] in spite of the non- role in certain aspects of the theory, particularly in the polynomiality of the classical expression. This is the rea- regularization of certain operators, is obtained by inte- son why the real connection is now widely used. This grating the E field over a two dimensional surface S choice has the advantage of eliminating the old “reality conditions” problem, namely the problem of implement- ˜a i E[S,f]= dSa Ei f , (11) ing non-trivial reality conditions in the quantum theory. ZS where f is a function on the surface S, taking values A. Loop algebra in the Lie algebra of SU(2). In alternative to the full loop observables (5,6,7), one also can take the holonomies and E[S,f] as elementary variables [15,17]; this is more Certain classical quantities play a very important role natural to do, for instance, in the C*-algebric approach in the quantum theory. These are: the trace of the holon- [12]. omy of the connection, which is labeled by loops on the three manifold; and the higher order loop variables, ob- tained by inserting the E field (in n distinct points, or B. Loop quantum gravity “hands”) into the holonomy trace. More precisely, given a loop α in M and the points s1,s2,...,sn α we define: ∈ The kinematic of a quantum theory is defined by an

[α]= Tr[Uα], (5) algebra of “elementary” operators (such as x and i¯hd/dx, a T − a or creation and annihilation operators) on a Hilbert space [α](s)= Tr[Uα(s,s)E˜ (s)] (6) T − . The physical interpretation of the theory is based on H and, in general the connection between these operators and classical vari- ables, and on the interpretation of as the space of the H

11 quantum states. The dynamics is governed by a hamil- (14) defines a well defined scalar product on the space tonian, or, as in general relativity, by a set of quantum of these linear combinations. Completing the space of constraints, constructed in terms of the elementary oper- these linear combinations in the Hilbert norm, we obtain ators. To assure that the quantum Heisenberg equations a Hilbert space . This is the (unconstrained) quantum H have the correct classical limit, the algebra of the elemen- state space of loop gravity.†† carries a natural uni- tary operator has to be isomorphic to the Poisson algebra tary representation of the diffeomorphismH group and of of the elementary observables. This yields the heuristic the group of the local SU(2) transformations, obtained quantization rule: “promote Poisson brackets to commu- transforming the argument of the functionals. An im- tators”. In other words, define the quantum theory as portant property of the scalar product (14) is that it is a linear representation of the Poisson algebra formed by invariant under both these transformations. the elementary observables. For the reasons illustrated is non-separable. At first sight, this may seem as in section V, the algebra of elementary observables we a seriousH obstacle for its physical interpretation. But we choose for the quantization is the loop algebra, defined will see below that after factoring away diffeomorphism in section VI A. Thus, the kinematic of the quantum invariance we may obtain a separable Hilbert space (see theory is defined by a unitary representation of the loop section VI H). Also, standard spectral theory holds on algebra. Here, I construct such representation following , and it turns out that using spin networks (discussed a simple path. below)H one can express as a direct sum over finite di- We can start “`ala Schr¨odinger” by expressing quan- mensional subspaces whichH have the structure of Hilbert tum states by means of the amplitude of the connection, spaces of spin systems; this makes practical calculations namely by means of functionals Ψ(A) of the (smooth) very manageable. connection. These functionals form a linear space, which Finally, we will use a Dirac notation and write we promote to a Hilbert space by defining a inner prod- Ψ(A)= A Ψ , (15) uct. To define the inner product, we choose a particular h | i set of states, which we denote “cylindrical states” and in the same manner in which one may write ψ(x)= x Ψ begin by defining the scalar product between these. in ordinary quantum mechanics. As in that case,h how-| i Pick a graph Γ, say with n links, denoted γ1 ...γn, ever, we should remember that A is not a normalizable immersed in the manifold M. For technical reasons, we state. | i require 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 a C. Loop states and spin network states n function f(g1 ...gn) on [SU(2)] . The graph Γ and the function f determine a functional of the connection as A subspace 0 of is formed by states invariant un- follows der SU(2) gaugeH transformations.H We now define an or- thonormal basis in 0. This basis represents a very im- ψΓ,f (A)= f(U1(A),...,Un(A)). (12) portant tool for usingH the theory. It was introduced in [187] and developed in [31,32]; it is denoted spin network (These states are called cylindrical states because they basis. were introduced in [14–16] as cylindrical functions for First, given a loop α in M, there is a normalized state the definition of a cylindrical measure.) Notice that we ψ (A) in , which is obtained by taking Γ = α and can always “enlarge the graph”, in the sense that if Γ is α f(g)= TH r(g). Namely a subgraph of Γ′ we can always write − ψα(A)= TrUα(A). (16) ψΓ,f (A)= ψΓ′,f ′ (A). (13) − We introduce a Dirac notation for the abstract states, by simply choosing f independent from the U ’s of the and denote this state as α . These sates are called loop ′ i | i links which are in Γ′ but not in Γ. Thus, given any two states. Using Dirac notation, we can write cylindrical functions, we can always view them as having the 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 ††This construction of H as the closure of the space of the cylindrical functions of smooth connections in the scalar prod- uct (14) shows that H can be defined without the need of re- (ψ , ψ )= dg ...dg f(g ...g ) h(g ...g ). ∗ Γ,f Γ,h 1 n 1 n 1 n curring to C algebraic techniques, distributional connections ZSU(2)n or the Ashtekar-Lewandowski measure. The casual reader, (14) however, should be warned that the resulting H topology is different than the natural topology on the space of connec- where dg is the Haar measure on SU(2). This scalar tions: if a sequence Γn of graphs converges pointwise to a product extends by linearity to finite linear combinations graph Γ, the corresponding cylindrical functions ψΓn,f do not of cylindrical functions. It is not difficult to show that converge to ψΓ,f in the H Hilbert space topology.

12 ψα(A)= A α , (17) geometry, and who gave the first version of spin network h | i theory [152,153].) It is easy to show that loop states are normalizable. Prod- Given a spin network S, we can construct a state ucts of loop states are normalizable as well. Following ΨS(A) as follows. We take the propagator of the con- tradition, we denote with α also a multiloop, namely a nection along each link of the graph, in the representa- collection of (possibly overlapping) loops α1,...,αn, , tion associated to that link, and then, at each node, we { } and we call contract the matrices of the representation with the in- variant tensor. We obtain a state ΨS(A), which we also ψα(A)= ψα (A) . . . ψα (A) (18) 1 × × n write as a multiloop state. (Multi-)loop states represented the ψS(A)= A S . (20) main tool for loop quantum gravity before the discov- h | i ery of the spin network basis. Linear combinations of One can then show the following. multiloop states (over-)span , and therefore a generic state ψ(A) is fully characterizedH by its projections on the The spin network states are normalizable. The nor- • multiloop states, namely by malization factor is computed in [77]. They are SU(2) gauge invariant. ψ(α) = (ψα, ψ). (19) • Each spin network state can be decomposed into a The “old” loop representation was based on representing • finite linear combination of products of loop states. quantum states in this manner, namely by means of the functionals ψ(α) over loop space defined in(19). Equation The (normalized) spin network states form an or- (19) can be explicitly written as an integral transform, • thonormal basis for the gauge SU(2) invariant as we will see in section VI G. states in (choosing the basis of invariant tensors Next, consider a graph Γ. A “coloring” of Γ is given appropriately).H by the following. The scalar product between two spin network 1. Associate an irreducible representation of SU(2) to • states can be easily computed graphically and al- each link of Γ. Equivalently, we may associate to gebraically. See [77] for details. each link γi a half integer number si, the spin of the irreducible, or, equivalently, an integer number The spin network states provide a very convenient basis pi, the “color” pi =2si. for the quantum theory. The spin network states defined above are SU(2) gauge 2. Associate an invariant tensor v in the tensor prod- invariant. There exists also an extension of the spin uct of the representations s1 ...sn, to each node network basis to the full Hilbert space (see for instance of Γ in which links with spins s1 ...sn meet. An [17,51], and references therein). invariant tensor is an object with n indices in the representations s1 ...sn that transform covariantly. If n = 3, there is only one invariant tensor (up to a D. Relation between spin network states and loop multiplicative factor), given by the Clebsh-Gordon states and diagrammatic representation of the states coefficient. An invariant tensor is also called an in- tertwining tensor. All invariant tensors are given A diagrammatic representation for the states in is by the standard Clebsch-Gordon theory. More pre- very useful in concrete calculations. First, associateH to cisely, for fixed s1 ...sn, the invariant tensors form a loop state α a diagram in M, formed by the loop α a finite dimensional linear space. Pick a basis vj itself. Next, notice| i that we can multiply two loop states, is this space, and associate one of these basis ele- obtaining a normalizable state. We represent the prod- ments to the node. Notice that invariant tensors uct of n loop states by the diagram formed by the set exist only if the tensor product of the representa- of the n (possibly overlapping) corresponding loops (we tions s1 ...sn contains the trivial representation. denote this set “multiloop”). Thus, linear combinations This yields a condition on the coloring of the links. of multiloops diagrams represent states in . Represent- For n = 3, this is given by the well known Clebsh- ing states as linear combinations of multiloopsH diagrams Gordan condition: each color is not larger than the makes computation in easy. sum of the other two, and the sum of the three Now, the spin networkH state defined by the graph with colors is even. no nodes α, with color 1, is clearly, by definition, the We indicate a colored graph by Γ, ~s, ~v , or simply S = loop state α , and we represent it by the diagram α. { } The spin network| i state α, n determined by the graph Γ, ~s, ~v , and denote it a “spin network”. (It was Penrose | i {who first} had the intuition that this mathematics could without nodes α, with color n can be obtained as follows. be relevant for describing the quantum properties of the 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 linear combination of these n! diagrams, taken with alternate signs (namely with the sign determined by the parity of e2 e2 e e3 e v e3 the permutation) is precisely the state α, n . The rea- 1 1 RΦ | i son of this key result can be found in the fact that an E3

E4 irreducible representation of SU(2) can be obtained as RΦ e4 e4 E2 N2 N3 the totally symmetric tensor product of the fundamental N4 N5 representation with itself. For details, see [77]. N1 E5 Next, consider a graph Γ with nodes. Draw ni parallel lines along each link γi. Join pairwise the end points

e5 e5 of these lines at each node (in an arbitrary manner), in e0 e6 e0 e6 such a way that each line is joined with a line from a different link (see Figure 1). In this manner, one obtain a multiloop diagram. Now antisymmetrize the ni parallel lines along each link, obtaining a linear combination of diagrams representing a state in . One can show that H this state is a spin network state, where ni is the color of the links and the color of the nodes is determined by the pairwise joining of the legs chosen [77]. Again, simple FIG. 1. Construction of “virtual” nodes and “virtual” links SU(2) representation theory is behind this result. over an n-valent node. More in detail, if a node is trivalent (has 3 adjacent links), then we can join legs pairwise only if the total Viceversa, multiloop states can be decomposed in spin number of the legs is even, and if the number of the network states by simply symmetrizing along (real and legs in each link is smaller or equal than the sum of the virtual) nodes. This can be done particularly easily di- number of the other two. This can be immediately rec- agrammatically, as illustrated by the graphical formulae ognized as the Clebsch-Gordan condition. If these con- in [187,77]. These are standard formulae. In fact, it ditions are satisfied, there is only a single way of join- is well known that the tensor algebra of the SU(2) irre- ing legs. This corresponds to the fact that there is only ducible representations admits a completely graphical no- one invariant tensor in the product of three irreducible of tation. This graphical notation has been widely used for SU(2). Higher valence nodes can be decomposed in triva- instance in nuclear and atomic physics. One can find it lent “virtual” nodes, joined by “virtual” links. Orthogo- presented in detail in books such as [214,52,66]. The ap- nal independent invariant tensors are obtained by varying plication of this diagrammatic calculus to quantum grav- over all allowed colorings of these virtual links (compat- ity is described in detail in [77], which I recommend to ible with the Clebsch-Gordan conditions at the virtual anybody who intends to perform concrete calculations in nodes). Different decompositions of the node give differ- loop gravity. ent orthogonal bases. Thus the total (links and nodes) It is interesting to notice that loop quantum gravity coloring of a spin network can be represented by means of was first constructed in a pure diagrammatic notation, in the coloring of the real and the virtual links. See Figure [184]. The graphical nature of this calculus puzzled some, 1. and the theory was accused of being vague and strange. Only later the diagrammatic notation was recognized to be the very conventional SU(2) diagrammatic calculus, well known in atomic and nuclear physics.‡‡

‡‡The following anecdote illustrates the level of the initial confusion. The first elaborate graphical computation of the eigenvalues of the area led to the mysterious formula Ap = 1 2 2 ¯hG p + 2p, with integer p. One day, Junichi Iwasaki, at the timep a student, stormed into my office and told me to 1 rewrite the formula in terms of the half integer j = 2 p. This yields the very familiar expression for the eigenvalues of the SU(2) Casimir Aj = ¯hG j(j + 1). Nobody had previously noticed this fact. p

14 E. The representation of the states n . In doing so, all elementary opera- tors are algebraic:| i x ˆ n = 1 ( n 1 + (n + 1) n +1 ), √2 i | i | − i | i I now define the quantum operators, corresponding to pˆ n = − ( n 1 (n+1) n+1 ). Similarly, in quantum √2 the -variables, as linear operators on . These form a | i | − i− | i T H gravity we can directly construct the quantum theory in representation of the loop variables Poisson algebra. The the spin-network (or loop) basis, without ever mention- operator [α] acts diagonally T ing functionals of the connections. This representation of the theory is denoted the “loop representation”. [α]Ψ(A)= Tr Uα(A) Ψ(A). (21) T − A section of the first paper on loop quantum gravity (Recall that products of loop states and spin network by Rovelli and Smolin [184] was devoted to a detailed states are normalizable states). In diagrammatic nota- study of “transformation theory” (in the sense of Dirac) tion, the operator simply adds a loop to a (linear combi- on the state space of quantum gravity, and in particular nation of) multiloops on the relations between the loop states [α] Ψ = α Ψ . (22) ψ(α)= α ψ (25) T | i | i| i h | i Higher order loop operators are expressed in terms of and the states ψ(A) giving the amplitude for a connection the elementary “grasp” operation. Consider first the op- field configuration A, and defined by erator a(s)[α], with one hand in the point α(s). The T ψ(A)= A ψ . (26) operator annihilates all loop states that do not cross the h | i point α(s). Acting on a loop state β , it gives Here A are “eigenstates of the connection operator”, or, | i | i a 2 a 1 more precisely (since the operator corresponding to the (s)[α] β = l ∆ [β, α(s)] α#β α#β− , T | i 0 | i − | i connection is ill defined in the theory) the generalized  (23) states that satisfy

A where we have introduced the elementary length l0 by T [α] A = T r[ e α ] A . (27) | i − P R | i 16π¯hG l2 =¯hG = Newton = 16π l2 (24) However, at the time of [184] the lack of a scalar product 0 c3 P lanck made transformation theory quite involved. and ∆a and # are defined in section VI A. This action On the other hand, the introduction of the scalar prod- extends by linearity, continuity and by the Leibniz rule uct (14) gives a rigorous meaning to the loop transform. to products and linear combinations of loop states, and In fact, we can write, for every spin network S, and every to the full . In particular, it is not difficult to compute state ψ(A) its actionH on a spin network state [77]. Higher order ψ(S)= S ψ = (ψS, ψ). (28) loop operators act similarly. It is a simple exercise to h | i verify that these operators provide a representation of This equation defines a unitary mapping between the two the classical Poisson loop algebra. presentations of : the “loop representation”, in which H All the operators in the theory are then constructed one works in terms of the basis S ; and the “connection | i in terms of these basics loop operators, in the same way representation”, in which one uses wave functionals ψ(A). in which in conventional QFT one constructs all opera- The development of the connection representation fol- tors, including the hamiltonian, in terms of creation and lowed a winding path through C∗-algebraic [12] and annihilation operators. The construction of the compos- measure theoretical [14,16,15] methods. The work of ite operators requires the development of regularization Ashtekar, Isham, Lewandowski, Marolf, Mourao and techniques that can be used in the absence of a back- Thiemann has finally put the connection representation ground metric. These have been introduced in [196] and on a firm ground, and, indirectly has much clarified the developed in [186,77,18,23,138,17]. mathematics underlying the original loop approach. In the course of this development, doubts were raised about the precise relations between the connection and the loop F. Algebraic version (“loop representation”) and formalisms. Today, the complete equivalence of these differential version (“connection representation”) of two approaches (always suspected) has been firmly es- the formalism, and their equivalence tablished. In particular, the work of Roberto DePietri [76] has proven the unitary equivalence of the two for- Imagine we want to quantize the one dimensional har- malisms. For a recent discussion see also [139]. monic oscillator. We can consider the Hilbert space of square integrable functions ψ(x) on the real line, and express the momentum and the hamiltonian as differen- G. Other structures in H tial operators. Denote the eigenstates of the hamilto- nian as ψn(x) = x n . It is well known that the the- The recent developments in the mathematical founda- ory can be expressedh | entirelyi in algebraic form in terms tions of the connection representation have increased the

15 mathematical rigor of the theory, raising it to the stan- node into a trivalent tree; in this case, we have a single dards of mathematical physics. This has been obtained internal links. The expansion can be done in different at the price of introducing heavy mathematical tools, of- ways (by pairing links differently). These are related to ten unfamiliar to the average physicist, perhaps widening each other by the recoupling theorem of pg. 60 in Ref. the language gap between the quantum gravity and the [130] high energy physics community. b c The reason for searching a mathematical-physics level @ r of precision is that in quantum gravity one moves on b j c a b i ri @r r = @ (29) a very unfamiliar terrain –quantum field theory on @  c d j  Xi a d manifolds– where the experience accumulated in conven- a d tional quantum field theory is often useless and some- ab i times even misleading. Given the unlikelihood of finding where the quantities are su(2) six-j symbols direct experimental corroboration, the research can only  c d j  aim, at least for the moment, at the goal of finding a (normalized as in [130]). Equation (29) follows just from consistent theory, with the correct limits in the regimes the definitions given above. Recoupling theory provides that we control experimentally. In these conditions, high a powerful computational tool in this context. mathematical rigor is the only assurance of the consis- Since spin network states satisfy recoupling theory, tency of the theory. In the development of quantum they form a Temperley-Lieb algebra [130]. The scalar field theory mathematical rigor could be very low be- product (14) in is given also by the Temperley-Lieb H cause extremely accurate empirical verifications assured trace of the spin networks, or, equivalently by the Kauff- the physicists that “the theory may be mathematically man brackets, or, equivalently, by the chromatic evalua- meaningless, but it is nevertheless physical correct, and tion of the spin network. See Ref. [77] for an extensive therefore the theory must make sense even if we do not discussion of these relations. understand how.” Here, such an indirect experimental Next, admits a rigorous representation as an L2 H reassurance is lacking and the claim that the theory is space, namely a space of square integrable functions. To well founded can be based only on a solid mathematical obtain this representation, however, we have to extend control of the theory. the notion of connection, to a notion of “distributional One may object that a rigorous definition of quantum connection”. The space of the distributional connections gravity is a vain hope, given that we do not even have is the closure of the space of smooth connection in a cer- a rigorous definition of QED, presumably a much sim- tain topology. Thus, distributional connections can be pler theory. The objection is particularly valid from the seen as limits of sequences of connections, in the same point of view of a physicist who views gravity “just as any manner in which distributions can be seen as limits of other field theory like the ones we already understand”. sequences of functions. Usual distributions are defined But the (serious) difficulties of QED and of the other as elements of the topological dual of certain spaces of conventional field theories are ultraviolet. The physical functions. Here, there is no natural linear structure in hope supporting the quantum gravity research program the space of the connections, but there is a natural du- is that the ultraviolet structure of a diffeomorphism in- ality between connections and curves in M: a smooth variant quantum field theory is profoundly different from connection A assigns a group element Uγ(A) to every seg- the one of conventional theories. Indeed, recall that in a ment γ. The group elements satisfy certain properties. very precise sense there is no short distance limit in the For instance if γ is the composition of the two segments theory; the theory naturally cuts itself off at the Planck γ1 and γ2, then Uγ(A)= Uγ1 (A)Uγ2 (A). scale, due to the very quantum discreteness of spacetime. A generalized connection A¯ is defined as a map that Thus the hope that quantum gravity could be defined rig- assigns an element of SU(2), which we denote as Uγ(A¯) orously may be optimistic, but it is not ill founded. or A¯(γ) to each (oriented) curve γ in M, satisfying the 1 1 After these comments, let me briefly mention some of following requirements: i) A¯(γ− ) = (A¯(γ))− ; and, ii) 1 the structures that have been explored in . First of all, A¯(γ2 γ1) = A¯(γ2) A¯(γ1), where γ− is obtained from H ◦ · the spin network states satisfy the Kauffman axioms of γ by reversing its orientation, γ2 γ1 denotes the com- ◦ the tangle theoretical version of recoupling theory [130] position of the two curves (obtained by connecting the (in the “classical” case A = 1) at all the points (in 3d end of γ1 with the beginning of γ2) and A¯(γ2) A¯(γ1) is · space) where they meet. (This− fact is often misunder- the composition in SU(2). The space of such general- stood: recoupling theory lives in 2d and is associated by ized connections is denoted . The cylindrical functions A Kauffman to knot theory by means of the usual projec- ΨΓ,f (A) defined in section VI C as functions on the space tion of knots from 3d to 2d. Here, the Kauffman axioms of smooth connections extend immediately to generalized are not satisfied at the intersections created by the 2d connections projection of the spin network, but only at the nodes in Ψ (A¯)= f(A¯(γ ,..., A¯(γ )). (30) 3d. See [77] for a detailed discussion.) For instance, con- Γ,f 1 n sider a 4-valent node of four links colored a,b,c,d. The We can define a measure dµ0 on the space of general- color of the node is determined by expanding the 4-valent ized connections by A

16 knots”. An s-knot s is an equivalence class of spin net- dµ0[A¯]ΨΓ,f (A¯) dg1 ...dgn f(g1 ...gn). (31) Z ≡ ZSU(2)n works S under diffeomorphisms. An s-knot is charac- terized by its “abstract” graph (defined only by the ad- In fact, one may show that (31) defines (by linear- jacency relations between links and nodes), by the col- ity and continuity) a well-defined absolutely continuous oring, and by its knotting and linking properties, as in measure on . This is the Ashtekar-Lewandowski (or knot-theory. Thus, the physical quantum states of the A Ashtekar-Lewandowski-Baez) measure [14–16,26]. Then, gravitational field turn out to be essentially classified by one can prove that = L2[ , dµ0], under the natural knot theory. H A isomorphism given by identifying cylindrical functions. There are various equivalent way of obtaining Diff It follows immediately that the transformation (19) be- from . One can use regularization techniques forH defin- tween the connection representation and the “old” loop ing theH quantum operator corresponding to the classical representation is given by diffeomorphism constraint in terms of elementary loop operators, and then find the kernel of such operator. A Equivalently, one can factor by the natural action of ψ(α)= dµ0[A¯] Tr e α Ψ(A)¯ . (32) Z P R the Diffeomorphism group thatH it carries. Namely This is the loop transform formula that was derived Diff = H . (33) heuristically in [184]; here it becomes rigorously defined. H Diff(M) Furthermore, can be seen as the projective limit of H the projective family of the Hilbert spaces Γ, associated There are several rigorous ways for defining the quotient H to each graph Γ immersed in M. Γ is defined as the of a Hilbert space by the unitary action of a group. See n H space L2[SU(2) , dg1 ...dgn], where n is the number of in particular the construction in [18], which follows the links in Γ. The cylindrical function ΨΓ,f (A) is naturally ideas of Marolf and Higuchi [145,147,148,108]. associated to the function f in Γ, and the projective In the quantum gravity literature, a big deal has been H structure is given by the natural map (13) [18,149]. made of the problem that a scalar product is not defined Finally, Ashtekar and Isham [12] have recovered the on the space of solutions of a constraint Cˆ, defined on a representation of the loop algebra by using C*-algebra Hilbert space . This, however, is a false problem. It is representation theory: The space / , where is the true that if zeroH is in the continuum spectrum of Cˆ then A G G group of local SU(2) transformations (which acts in the the corresponding eigenstates are generalized states and obvious way on generalized connections), is precisely the the scalar product is not defined between them. But Gelfand spectrum of the abelian part of the loop algebra. theH generalized eigenspaces of Cˆ, including the kernel, One can show that this is a suitable norm closure of the inherit nevertheless a scalar product from Hˆ . This can space of smooth SU(2) connections over physical space, be seen in a variety of equivalent ways. For instance, modulo gauge transformations. it can be seen from the following theorem. If Cˆ is self Thus, a number of powerful mathematical tools are adjoint, then there exist a measure µ(λ) on its spectrum at hand for dealing with nonperturbative quantum grav- and a family of Hilbert spaces (λ) such that ity. These include: Penrose’s spin network theory, SU(2) H representation theory, Kauffman tangle theoretical re- = dµ(λ) (λ), (34) coupling theory, Temperley-Lieb algebras, Gelfand’s H Z H C∗algebra spectral representation theory, infinite dimen- sional measure theory and differential geometry over in- where the integral is the continuous sum of Hilbert spaces described for instance in [101]. Clearly (0) is the kernel finite dimensional spaces. H of Cˆ equipped with a scalar product. This is discussed for instance in [162]. H. Diffeomorphism invariance There are two distinct ways of factoring away the dif- feomorphisms in the quantum theory, yielding two dis- The next step in the construction of the theory is to tinct version of the theory. The first way is to factor factor away diffeomorphism invariance. This is a key away smooth transformations of the manifold. In doing step for two reasons. First of all, is a “huge” non- so, finite dimensional moduli spaces associated with high separable space. It is far “too large”H for a quantum valence nodes appear [98], so that the resulting Hilbert field theory. However, most of this redundancy is gauge, space is still nonseparable. The physical relevance of and disappears when one solves the diffeomorphism con- these moduli parameters is unclear at this stage, since they do not seem to play any role in the quantum theory. straint, defining the diff-invariant Hilbert space Diff . This is the reason for which the loop representation,H as Alternatively, one can consistently factor away continu- defined here, is of great value in diffeomorphism invariant ous transformations of the manifold. This possibility has theories only. been explored by Zapata in [215,216], and seems to lead to a consistent theory free of the residual non separabil- The second reason is that Diff turns out to have a natural basis labeled by knots.H More precisely by “s- ity.

17 I. Dynamics and ǫ′ sufficiently small, Oˆǫ Ψ and Oˆǫ′ Ψ are diffeomor- phic equivalent. Thus, here| diffi invariance| i plays again a Finally, the definition of the theory is completed by crucial role in the theory. giving the hamiltonian constraint. A number of ap- For a discussion of the problems raised by the Thie- proaches to the definition of a hamiltonian constraint mann operator and of the variant proposed, see section have been attempted in the past, with various degrees VIII. of success. Recently, however, Thiemann has succeeded in providing a regularization of the hamiltonian con- straint that yields a well defined, finite operator. Thie- J. Unfreezing the frozen time formalism: the mann’s construction [206,201,202] is based on several covariant form of loop quantum gravity clever ideas. I will not describe it here. Rather, I will sketch below the final form of the constraint (for the A recent development in the formalism is the trans- Lapse=1 case), following [175]. lation of loop quantum gravity into spacetime covariant I begin with the Euclidean hamiltonian constraint. We form. This was initiated in [181,160] by following the have steps that Feynman took in defining path integral quan- tum mechanics starting from the Schr¨odinger canonical Hˆ s = Aǫǫ′ (pi...pn) Dˆ ′ s . theory. More precisely, it was proven in [160] that the | i i;(IJ),ǫǫ | i Xi (XIJ) ǫX= 1 ǫ′X= 1 matrix elements of the operator U(T ) ± ±

T (35) dt d3x Hˆ (x) U(T ) e 0 , (36) ≡ Here i labels the nodes of the s-knot s; (IJ) labels couples R R of (distinct) links emerging from i. p1...pn are the colors obtained exponentiating the (Euclidean) hamiltonian ˆ ′ constraint in the proper time gauge (the operator that of the links emerging from i. Di;(IJ)ǫǫ is the operator that acts on an s -knot by: (i) creating two additional generates evolution in proper time) can be expanded in nodes, one along each of the two links I and J; (ii) creat- a Feynman sum over paths. In conventional QFT each ing a novel link, colored 1, joining these two nodes, (iii) term of a Feynman sum corresponds naturally to a cer- tain Feynman diagram, namely a set of lines in spacetime assigning the coloring pI + ǫ and, respectively, pJ + ǫ′ to the links that join the new formed nodes with the node meeting at vertices (branching points). A similar natu- i. This is illustrated in Figure 2. ral structure of the terms appears in quantum gravity, but surprisingly the diagrams are now given by surfaces is spacetime that branch at vertices. Thus, one has a formulation of quantum gravity as a sum over surfaces q in spacetime. Reisenberger [158] and Baez [30] had ar- q gued in the past that such a formulation should exist, J and Iwasaki has developed a similar construction in 2+1 ˆ J D+ = q 1 1 dimensions. Intuitively, the time evolution of a spin net- − "b −"b J " b " b work in spacetime is given by a colored surface. The sur- " b " bJJ "" r pbb "" p+1 bb faces capture the gravitational degrees of freedom. The r p formulation is “topological” in the sense that one must ˆ ′ FIG. 2. Action of Di;(IJ)ǫǫ . sum over topologically inequivalent surfaces only, and the contribution of each surface depends on its topology only. The coefficients Aǫǫ′ (pi...pn), which are finite, can be This contribution is given by the product of elementary expressed explicitly (but in a rather laborious way) in “vertices”, namely points where the surface branches. terms of products of linear combinations of 6 j symbols The transition amplitude between two s-knot states − of SU(2), following the techniques developed in detail si and sf in a proper time T is given by summing in [77]. Some of these coefficients have been explicitly |overi all (branching,| i colored) surfaces σ that are bounded computed [51]. The Lorentzian hamiltonian constraint by the two s-knots si and sf is given by a similar expression, but quadratic in the Dˆ operators. sf U(T ) si = [σ](T ). (37) h | | i A The operator defined above is obtained by introduc- Xσ ∂σ=si sf ing a regularized expression for the classical hamiltonian ∪ constraint, written in terms of elementary loop observ- The weight [σ](T ) of the surface σ is given by a product ables, turning these observables into the corresponding A operators and taking the limit. The construction works over the n vertices v of σ: rather magically, relying on the fact, first noticed in [188], (- iT )n ˆ ˆ [σ](T )= Av(σ). (38) that certain operator limits Oǫ O turns out to be fi- A n! nite on diff invariant states, thanks→ to the fact that for ǫ Yv

18 The contribution Av(σ) of each vertex is given by the matrix elements of the hamiltonian constraint operator between the two s-knots obtained by slicing σ immedi- 6 s ately below and immediately above the vertex. They turn 3 f out to depend only on the colors of the surface compo- 8 Σ f nents immediately adjacent the vertex v. The sum turns out to be finite and explicitly computable order by order. q As in the usual Feynman diagrams, the vertices de- scribe the elementary interactions of the theory. In par- 7 5 ticular, here one sees that the complicated structure of 3 6 1 s1 the Thiemann hamiltonian, which makes a node split into 8 three nodes, corresponds to a geometrically very simple vertex. Figure 3 is a picture of the elementary vertex. p Notice that it represents nothing but the spacetime evo- lution of the elementary action of the hamiltonian con- 5 s straint, given in Figure 2. 3 i Σ i 7

FIG. 4. A term of second order.

The sum over surfaces version of loop quantum grav- ity provides a link with certain topological quantum field theories and in particular with the the Crane-Yetter model [71–75], which admit an extremely similar rep- resentation. For a discussion on the precise relation between topological quantum field theory and diffeo- morphism invariant quantum field theory, see [160] and [171,124,83]. FIG. 3. The elementary vertex. The idea of expressing the theory as a sum over sur- faces has been developed by Baez [33], who has stud- An example of a surface in the sum is given in Figure ied the general form of generally covariant quantum field 4. theories formulated in this manner, and by Smolin and Markopoulou [144], who have studied how to directly capture the Lorentzian causal structure of general rela- tivity modifying the elementary vertices. They have also explored the idea that the long range correlations of the low 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 theoretical description of percolation.

VII. PHYSICAL RESULTS

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

19 A. Technical A remarkable result in this context is that ultra- violet divergences do not seem to appear, strongly Solutions of the hamiltonian constraints. supporting the expectation that the natural cut off • One of the most surprising results of the theory introduced by quantum gravity might cure the ul- is that it has been possible to find exact solu- traviolet difficulties of conventional quantum field tions of the hamiltonian constraint. This follows theory. from the key result that the action of the hamilto- Application to other theories. The loop repre- nian constraints is non vanishing only over nodes • sentation has been applied in various other contexts of the s-knots [183,184]. Therefore s-knots with- such as 2+1 gravity [13,146,20] (on 2+1 quantum out nodes are physical states that solve the quan- gravity, in the loop and in other representations, tum Einstein dynamics. There is an infinite num- see [62]) and others [21]. ber of independent states of this sort, classified by conventional knot theory. The physical interpreta- Lattice and simplicial models. A number tion of these solutions is still rather obscure. Vari- • of interesting discretized versions of the theory are ous other solutions have been found. See the re- being studied. See in particular [141,159,94,79]. cent review [82] and reference therein. See also [113,131,56–59,157,94,129]. In particular, Pullin has studied in detail solutions related to the Chern- B. Physical Simon term in the connection representation and to the Jones polynomial in the loop representation. Planck scale discreteness of space According to a celebrated result by Witten [212], • the two are the loop transform of each other. The most remarkable physical result obtained from loop quantum gravity is, in my opinion, evidence Time evolution. Strong field perturbation for a physical (quantum) discreteness of space at • expansion. “Topological Feynman rules”. the Planck scale. This is manifested in the fact Trying to describe the temporal evolution of the that certain operators corresponding to the mea- quantum gravitational field by solving the hamilto- surement of geometrical quantities, in particular nian constraint yields the conceptually well-defined area and volume, have discrete spectrum. Accord- [168], but notoriously non-transparent frozen-time ing to the standard interpretation of quantum me- formalism. An alternative is to study the evolu- chanics (which we adopt), this means that the the- tion of the gravitational degrees of freedom with ory predicts that a physical measurement of an area respect to some matter variable, coupled to the or a volume will necessarily yield quantized results. theory, which plays the role of a phenomenologi- Since the smallest eigenvalues are of Planck scale, cal “clock”. This approach has lead to the tenta- this implies that there is no way of observing areas tive definition of a physical hamiltonian [188,48], or volumes smaller than Planck scale. Space comes and to a preliminary investigation of the possibil- in “quanta” in the same manner as the energy of ity of transition amplitudes between s-knot states, an oscillator. The spectra of the area and volume order by order in a (strong coupling) perturbative operators have been computed with much detail expansion [175]. In this context, diffeomorphism in loop quantum gravity. These spectra have a invariance, combined with the key result that the complicated structure, and they constitute detailed hamiltonian constraint acts on nodes only, implies quantitative physical predictions of loop quantum that the “Feynman rules” of such an expansion are gravity on Planck scale physics. If we had experi- purely topological and combinatorial. mental access to Planck scale physics, they would allow the theory to be empirically tested in great Fermions. Fermions have been added to the detail. • theory [150,151,35,207]. Remarkably, all the im- portant results of the pure GR case survive in the A few comments are in order. GR+fermions theory. Not surprisingly, fermions – The result of the discreteness of area and vol- can be described as open ends of “open spin net- ume is due to Rovelli and Smolin, and ap- works”. peared first in reference [186]. Later, the re- Maxwell and Yang-Mills. The extension of sult has been recovered by alternative tech- • the theory to the Maxwell field has been studied in niques and extended by a number of authors. [132,91]. The extension to Yang-Mills theory has In particular, Ashtekar and Lewandowski [17] been recently explored in [200]. In [200], Thiemann have repeated the derivation, using the con- shows that the Yang-Mills term in the quantum nection representation, and have completed hamiltonian constraint can be defined in a rigorous the computation of the spectrum (adding the manner, extending the methods of [206,201,202]. sector which was not computed in [186].) The Ashtekar-Lewandowski component of the

20 spectrum has then been rederived in the loop operator. Now, as first realized in [172], it is representation by Frittelli Lehner and Rovelli plausible that that the operator A is, mathe- in [84]. Loll has employed lattice techniques matically, the same operator as the pure grav- to point out a numerical error in [186] (cor- ity area operator. This is because we can rected in the Erratum) in the eigenvalues of gauge fix the matter variables, and use mat- the volume. The analysis of the volume eigen- ter location as coordinates, so that non-diff- values has been performed in [77], where gen- invariant observables in the pure gravity the- eral techniques for performing these calcula- ory correspond precisely to diff-invariant ob- tions are described in detail. The spectrum servables in the matter+gravity theory. Thus, of the volume has then been analyzed also discreteness of the spectrum of the area opera- in [203]. There are also a few papers that tor is likely to imply discreteness of physically have anticipated the main result presented measurable areas, but it is important to em- in [186]. In particular, Ashtekar Rovelli and phasize that this implication is based on some Smolin have argued for a physical discreteness additional hypothesis on the relation between of space emerging from the loop representa- the pure gravity and the gravity+matter the- tion in [23], where some of the eigenvalues of ories. the area already appear, although in implicit form. The first explicit claim that area eigen- The discreteness of area and volume is derived as values might in principle be observable (in the follows. Consider the area A of a surface Σ. The presence of matter) is by Rovelli in [172]. physical area A of Σ depends on the metric, namely on the gravitational field. In a quantum theory of – The reader will wonder why area and volume gravity, the gravitational field is a quantum field seem to play here a role more central than operator, and therefore we must describe the area length, when classical geometry is usually de- of Σ in terms of a quantum observable, described scribed in terms of lengths. The reason is that by an operator Aˆ. We now ask what is the quan- the length operator is difficult to define and tum operator Aˆ in nonperturbative quantum grav- of more difficult physical interpretations. For ity. The result can easily be worked out by writing attempts in this direction, see [204]. Whether the standard formula for the area of a surface, and this is simply a technical difficulty or it reflects replacing the metric with the appropriate function some deep fact, is not clear to me.§§ of the loop variables. Promoting these loop vari- – Area and volume are not gauge invariant oper- ables to operators, we obtain the area operator Aˆ. ators. Therefore, we cannot directly interpret The actual construction of this operator requires them as representing physical measurements. regularizing the classical expression and then tak- Realistic physical measurements of areas and ing the limit of a sequence of operators, in a suitable volumes always refer to physical surfaces and operator topology. For the details of this construc- spatial regions, namely surfaces and spatial re- tion, see [186,77,84,51]. An alternative regulariza- gions determined by some physical object. For tion technique is discussed in [17]. The resulting instance, I can measure the area of the surface area operator Aˆ acts as follows on a spin network of a certain table. In the dynamical theory state S (assuming here for simplicity that S is a that describes the gravitational field as well spin network| i without nodes on Σ): as the table, the area of the surface of the ta- ble is a diffeomorphism invariant quantity A, 2 l0 which depends on gravitational as well as mat- Aˆ[Σ] S =  pi(pi + 2) S (39) | i 2 | i ter variables. In the quantum theory, A will i XS Σ p be represented by a diffeomorphism invariant  ∈{ ∩ }  where i labels the intersections between the spin network S and the surface Σ, and pi is the color of the link of S crossing the i th intersection. §§I include here a comment on this issue I received from John − This result shows that the spin network states (with Baez: “I believe there is a deep reason why area is more fun- a finite number of intersection points with the sur- damental than length in loop quantum gravity. One way to say it is that the basic field is not the tetrad e but the 2-form face and no nodes on the surface) are eigenstates E = e ∧ e. Another way to say it is that the loop representa- of the area operator. The corresponding spectrum tion is based on the theory of quantized angular momentum. is labeled by multiplets ~p = (p1, ..., pn) of positive Angular momentum is not a vector but a bivector, so it cor- half integers, with arbitrary n, and given by responds not to an arrow but to a oriented area element.” On 2 this, see Baez’s [33]. On the relation between E field and area l0 A~p [Σ] = pi(pi + 2). (40) see in particular [170]. 2 Xi p

21 Shifting from color to spin notation reveals the theory. Notice that in a general relativistic context SU(2) origin of the spectrum: the Minkowski solution does not have all the prop- erties of the conventional field theoretical vacuum. 2 A~j [Σ] = l0 ji(ji + 1), (41) (In gravitational physics there is no real equivalent Xi p of the conventional vacuum, particularly in the spa- tially compact case.) One then expects that flat where j1, ..., jn are half integer. For the full spec- space is represented by some highly excited state trum, see [17] (connection representation) and [84] in the theory. States in that describe flat space (loop representation). when probed at low energyH (large distance) have A similar result can be obtained for the volume been studied in [23,217,49,99]. These have a dis- [186,142,143,77,139]. Let us restrict here for sim- crete structure at the Planck scale. Furthermore, plicity to spin networks S with nondegenerate four- small excitations around such states have been con- sidered in [125], where it is shown that contains valent nodes, labeled by an index i. Let ai,bi,ci, di H be the colors of the links adjacent to the i th node all “free graviton” physics, in a suitable approxi- − mation. and let Ji label the basis in the intertwiner space. ˆ The volume operator V acts as follows The Bekenstein-Mukhanov effect. • 3 Recently, Bekenstein and Mukhanov [46] have sug- l0 (i) Vˆ S = iW (ai,bi,ci, di) S (42) | i 2 q| || i gested that the thermal nature of Hawking’s radia- Xi tion [105,106] may be affected by quantum prop- erties of gravity (For a review of earlier sugges- i where W is an operator that acts of the finite di- tions in this direction, see [193]). Bekenstein and mensional space of the intertwiners in the i th − Mukhanov observe that in most approaches to node, and its matrix elements are explicitly given quantum gravity the area can take only quantized (in a suitable basis) by (ǫ = ) ± values [97]. Since the area of the black hole sur- a+b+c+d face is connected to the black hole mass, black hole i t+ǫ 2 W (a,b,c,d)t ǫ = ǫ( 1) mass is likely to be quantized as well. The mass of − − − × 1 a + b + t +3 c + d + t +3 the black hole decreases when radiation is emitted. 4t(t + 2) 2 2 Therefore emission happens when the black hole makes a quantum leap from one quantized value 1+ a + b t 1+ a + t b 1+ b + t a − − − (43) of the mass (energy) to a lower quantized value, 2 2 2 1 very much as atoms do. A consequence of this pic- 1+ c + d t 1+ c + t d 1+ d + t c 2 ture is that radiation is emitted at quantized fre- − − − . 2 2 2  quencies, corresponding to the differences between energy levels. Thus, quantum gravity implies a dis- cretized emission spectrum for the black hole radi- See [51]. The volume eigenvalues vi are obtained by diagonalizing these matrices. For instance, in ation. the simple case a = b, c = d = 1, we have This result is not physically in contradiction with Hawking’s prediction of a continuous thermal spec- (¯hG)3 v = a(a + 2); (44) trum, because spectral lines can be very dense a 2 p in macroscopic regimes. But Bekenstein and Mukhanov observed that if we pick the simplest if d = a + b + c, we have ansatz for the quantization of the area –that the Area is quantized in multiple integers of an elemen- (¯hG)3 v = abc(a + b + c). (45) tary area A0–, then the emitted spectrum turns out a,b,c 2 p to be macroscopically discrete, and therefore very different from Hawking’s prediction. I will denote For more details, and the full derivation of these this effect as the kinematical Bekenstein-Mukhanov formulas, see [51,203] effect. Unfortunately, however, the kinematical Classical limit. Quantum states representing Bekenstein-Mukhanov effect disappears if we re- • flat spacetime. Weaves. Discrete small scale place the naive ansatz with the spectrum (41) com- structure of space. puted from loop quantum gravity. In loop quantum gravity the eigenvalues of the area become exponen- The s-knot states do not represent excitations of tially dense for a macroscopic black hole, and there- the quantum gravitational field over flat space, but fore the emission spectrum can be consistent with rather over “no-space”, or over the gµν = 0 solu- Hawking’s thermal spectrum. This is due to the tion. A natural problem is then how flat space (or details of the spectrum (41) of the area. A detailed any other smooth geometry) might emerge from the

22 discussion of this result is in [43], but the result was renormalization effects may affect the relation be- already contained (implicitly, in the first version) in tween the bare and the effective Newton constant, [17]. It is important to notice that the density of and this may be reflected in γ. For discussion of the eigenvalues shows only that the simple kinemat- the role of γ in the theory, see [189]. On the issue ical argument of Bekenstein and Mukhanov is not of entropy in loop gravity, see also [194]. valid in this theory, and not that their conclusions is necessarily wrong. As emphasized by Mukhanov, a discretization of the emitted spectrum could be VIII. MAIN OPEN PROBLEMS AND MAIN still be originated dynamically. CURRENT LINES OF INVESTIGATION

Black Hole Entropy from Loop Quantum Hamiltonian constraint. The kinematic of the theory • Gravity is well understood, both physically (quanta of area Indirect arguments [105,106,45,210] strongly sup- and volume, polymer-like geometry) and from the port the idea that a Schwarzschild black hole of mathematical point of view ( , s-knot states, area (macroscopic) area A behaves as a thermodynam- and volume operators). TheH part of the theory ical system governed by the Bekenstein-Hawking which is not yet fully under control is the dynamics, entropy which is determined by the hamiltonian constraint. A plausible candidate for the quantum hamiltonian k S = A (46) constraint is the operator introduced by Thiemann 4¯hGNewton [206,201,202]. The commutators of the Thiemann operator with itself and with the diffeomorphism (k is the Boltzmann constant; here I put the speed constraints close, and therefore the operator de- of light equal to one, but write the Planck and New- fines a complete and consistent quantum theory. ton constants explicitly). A physical understanding However, doubts have been raised on the physical and a first principles derivation of this relation re- correctness of this theory, and some variants of the quires quantum gravity, and therefore represents a operator have been considered. challenge for every candidate theory of quantum The doubts originate from various considera- theory. A derivation of the Bekenstein-Hawking tions. First, Lewandowski, Marolf and others have expression (46) for the entropy of a Schwarzschild stressed the fact the quantum constraint algebra black hole of surface area A via a statistical me- closes, but it is not isomorphic to the classical chanical computation, using loop quantum gravity, constraint algebra of GR [140]. Recently, a de- was obtained in [134,135,176]. tailed analysis of this problem has been completed This derivation is based on the ideas that the en- by Marolf, Lewandowski, Gambini and Pullin [90]. tropy of the hole originates from the microstates The failure to reproduce the classical constraint of the horizon that correspond to a given macro- algebra has been disputed, and is not necessarily scopic configuration [213,63,64,37,38]. Physical ar- a problem, since the only strict requirement on guments indicate that the entropy of such a system the quantum theory, besides consistency, is that is determined by an ensemble of configurations of its gauge invariant physical predictions match the the horizon with fixed area [176]. In the quantum ones of classical general relativity in the appropri- theory these states are finite in number, and can ate limit. Still, the difference in the algebras may be counted [134,135]. Counting these microstates be seen as circumstantial evidence (not a proof) for using loop quantum gravity yields the failure of the classical limit. The issue is tech- nically delicate and still controversial. I hope I will c k S = A. (47) be able to say something more definitive in the next γ 4¯hGNewton update of this review. Second, Br¨ugmann [54] and Smolin [192] have (An alternative derivation of this result has been pointed out a sort of excess “locality” in the form announced from Ashtekar, Baez, Corichi and Kras- of the operator, which, intuitively, seems in contra- nov [11].) γ is defined in section VI, and c is a real diction with the propagation properties of the Ein- number of the order of unity that emerges from the stein equations. Finally, by translating the Thie- combinatorial calculation (roughly, c 1/4π). If ∼ mann operator into a spacetime covariant four- we choose γ = c, we get (46) [189,70]. Thus, the dimensional formalism, Reisenberger and Rovelli theory is compatible with the numerical constant in have noticed a suspicious lack of manifest 4-d co- the Bekenstein-Hawking formula, but does not lead variance in the action of the operator [160], a fact to it univocally. The precise significance of this fact pointing again to the possibility of anomalies in the is under discussion. In particular, the meaning of γ quantum constraint algebra. is unclear. Jacobson has suggested [127] that finite

23 Motivated by these doubts, several variants of Thie- would be of great interest. mann’s operator have been suggested. The original Ultimately, the final tests of any proposal for the Thiemann’s operators is constructed using the vol- hamiltonian constraint operator must be consis- ume operator. There are two versions of the vol- tency and a correct classical limit. Thus, the so- ume operator in the literature: VRL, introduced lution of the hamiltonian constraint puzzle is likely in [186] and VAL, introduced in [14,16,15]. See to be subordinate to the solution of the problem [139] for a detailed comparison. Originally, Thie- of extracting the classical limit (of the dynamics) mann thought that using VRS in the hamiltonian from the theory. constraint would yield difficulties, but it later be- come clear that this is not the case [140]. Both Matter. The basics of the description of matter in versions of the volume can be used in the defini- the loop formalism have been established in tion, yielding two alternative versions of the hamil- [150,151,133,26,207,200]. Work needs to be done tonian [140]. Next, in its simplest version the oper- in order to develop a full description of the basic ator is non-symmetric. Since the classical hamilto- matter couplings. In particular there are strong re- nian constraint constraint is real (on SU(2) gauge curring indications that the Planck scale discrete- invariant states), one might expect a correspond- ness naturally cuts the traditional quantum field ing self-adjoint quantum operator.∗∗∗ Accordingly, infinities off. In particular, in [200], Thiemann ar- several ways of symmetrizing the operator have gues that the Hamiltonian constraint governing the been considered (see a list in [140]). Next, Smolin coordinate time evolution of the Yang-Mills field is has considered some ad hoc modifications of the a well defined operator (I recall that, due to the constraint in [192]. Finally, the spacetime covari- ultraviolet divergences, no rigorously well defined ant formalism in [160] naturally suggest a “covari- Hamiltonian operator for conventional Yang-Mills antisation” of the operator, described in [160] un- theory is known in 4 dimensions.) If these indica- der the name of “crossing symmetry”. This co- tions are confirmed, the result would be very re- variantisation amount to adding to the vertex de- markable. What is still missing are calculational scribed in Figure 3 the vertices, described in Figure techniques that could allow us to connect the well- 5, which are simply obtained by re-orienting Figure defined constraint with finite observables quantities 3 in spacetime. such as scattering amplitudes. Spacetime formalism. In my view, the development of continuous spacetime formalisms, [181,160,33,144], is one of the most promising areas of development of the theory, because it might be the key for addressing most of the open problems. First, a spacetime formalism frees us from the obscurities of the frozen time formalism, and allows an intu- itive, Feynman-style, description of the dynamics of quantum spacetime. I think that the classical limit, the quantum description of black holes, or graviton-graviton scattering, just to mention a few examples, could be addressed much more easily in FIG. 5. The crossing symmetric vertices. the covariant picture. Second, it allows the general ideas of Hartle [103] and Isham [119,120,122,123] A full comparative analysis of this various proposals on the interpretation of generally covariant quan- tum theories to be applied in loop quantum gravity. This could drastically simplify the complications of the canonical way of dealing with general covariant

∗∗∗ observables [169,167]. Third, the spacetime formal- There is an argument which is often put forward against ism should suggest solutions to the problem of se- the requirement of self-adjointness of the hamiltonian con- lecting the correct hamiltonian constraint: it is usu- straint H: let H be self-adjoint and O be any operator of ally easier to deal with invariances in the lagrangian the form O = [H,A] , where A is any operator (many op- erators that we do not expect could vanish have this form). rather than in the hamiltonian formalisms. The Then the expectation value of O vanishes on physical states spacetime formalism is just born, and much has to |ψi from hψ|O|ψi = hψ|[H,A]|ψi = 0. The mistake in this be done. See the original papers for suggestions argument is easily detected by replacing H with a nonrela- and open problems. tivistic free particle quantum hamiltonian, A with x and |ψi Black holes. The derivation of the Bekenstein-Hawking with an eigenstate of the momentum: the error is to neglect entropy formula is a major success of loop quan- the infinities generated by the use of generalized states. tum gravity, but much remains to be understood.

24 A clean derivation from the full quantum theory field? Presumably, such general technique should is not yet available. Such a derivation would re- involve some kind of expansion, since we could not quire us to understand what precisely is the event hope to solve the theory exactly. Attempts to de- horizon in the quantum theory. In other words, fine physical expansions have been initiated in [175] given a quantum state of the geometry, we should and, in different form, in [160]. Ideally, one would be able to define and “locate” its horizon (or what- want a general scheme for computing transition am- ever structure replaces it in the quantum theory). plitudes in some expansion parameter around some To do so, we should understand how to effectively state. Computing scattering amplitudes would be deal with the quantum dynamics, how to describe of particular interest, in order to make connection the classical limit (in order to find the quantum with particle physics language and to compare the states corresponding to classical black hole solu- theory with string predictions. tions), as well as how do describe asymptotically flat quantum states. Classical limit. Finally, to prove that loop quantum gravity is a valuable candidate for describing quan- Besides these formal issues, at the roots of the black tum spacetime, we need to prove that its classical hole entropy puzzle there is a basic physical prob- limit is GR (or at least overlaps GR in the regime lem, which, to my understanding, is still open. The where GR is well tested). The traditional connec- problem is to understand how we can use basic ther- tion between loop quantum gravity and classical modynamical and statistical ideas and techniques GR is via the notion of weave, a quantum state that in a general covariant context.††† To appreciate the “looks semiclassical” at distances large compared to difficulty, notice that statistical mechanics makes the Planck scale. However, the weaves studied so heavy use of the notion of energy (say in the def- far [23,99] are 3d weaves, in the sense that they are inition of the canonical or microcanonical ensem- eigenstates of the three dimensional metric. Such bles); but there is no natural local notion of energy a state corresponds to an eigenstate of the position associated to a black hole (or there are too many for a particle. Classical behavior is recovered not of such notions). Energy is an extremely slippery by these states but rather by wave packets which notion in gravity. Thus, how do we define the sta- have small spread in position as well as in momen- tistical ensemble? Put in other words: to compute tum. Similarly, the quantum Minkowski spacetime the entropy (say in the microcanonical) of a normal should have small spread in the three metric as well system, we count the states with a given energy. In as in its momentum – as the quantum electromag- GR we should count the states with a given what? netic vacuum has small quantum spread in the elec- One may say: black hole states with a given area. tric and magnetic field. To recover classical GR But why so? We do understand why the number of from loop quantum gravity, we must understand states with given energy governs the thermodynam- such states. Preliminary investigation in this di- ical behavior of normal systems. But why should rection can be found in [126,125], but these papers the number of states with given area govern the are now several years old, and they were written thermodynamical behavior of the system, namely before the more recent solidification of the basics govern its heat exchanges with the exterior? A ten- of the theory. Another direction consists in the di- tative physical discussion of this last point can be rect study of coherent states in the state space of found in [178]. the theory. How to extract physics from the theory. As these brief notes indicate, the various open prob- Assume we pick a specific hamiltonian constraint. lems in loop quantum gravity are interconnected. In a Then we have, in principle, a well defined quantum sense, loop quantum gravity grew aiming at the nonper- theory. How do we extract physical information turbative regime and the physical results obtained so far from it? Some physical consequences of the theory, are in this regime. The main issue is then to recover the such as the area and volume eigenvalues, or the en- long distance behavior of the theory. That is, to study its tropy formula, have been extracted from the the- classical limit and the dynamics of the low energy exci- ory by various ad hoc methods. But is there a gen- tations over a semiclassical background. Understanding eral technique, say corresponding to the traditional this aspect of the theory would assure us that the theory QFT perturbation expansion of the S matrix, for we are dealing with is indeed a quantum theory of the describing the full dynamics of the gravitational gravitational field, would allow us to understand quan- tum black holes, would clarify the origin of infinities in the matter hamiltonians and so on. Still in other words,

††† what mostly needs to be understood is the structure of A general approach to this problem and an idea about its the (Minkowski) vacuum in loop quantum gravity. connection with the “problem of time” in quantum gravity has been developed by Connes and Rovelli in [173,174,69].

25 IX. SHORT SUMMARY AND CONCLUSION that quantum gravity was solved (I think it was the turn of supergravity). The list of theories that claimed to be In this section, I very briefly summarize the state of final and have then ended up forgotten or superseded loop quantum gravity and its main results. The mathe- is a reason of embarrassment for the theoretical physics matics of the theory is solidly defined, and is understood community, in my opinion. from several alternative points of view. Long standing In my view, loop quantum gravity is the best we can problems such as the lack of a scalar product, the diffi- do so far in trying to understand quantum spacetime, cult of controlling the overcompletness of the loop basis from a nonperturbative, background-independent point and the problem of implementing the reality condition of view. Theoretically, we have reasons to suspect that in the quantum theory have been successfully solved or this approach could represent a consistent quantum the- sidestepped. The kinematics is given by the Hilbert space ory with the correct classical limit, but there are also , defined in Section VI B, which carries a representa- some worrying contrary indications. The theory yields a tionH of the basic operators: the loop operators (22-23). definite physical picture of quantum spacetime and def- A convenient orthonormal basis in is provided by the inite quantitative predictions, but a systematic way of spin network states, defined in SectionH VI C. The diffeo- extracting physical information is still lacking. Experi- morphism invariant states are given by the s-knot states, mentally, there is no support to the theory, neither direct and the structure and properties of the (diff-invariant) nor indirect. The spectra of area and volume computed quantum states of the geometry are quite well under- in the theory could or could not be physically correct. I stood (Section VI H). These states give a description of wish I could live long enough to know! quantum spacetime in terms of polymer-like excitations of the geometry. More precisely, in terms of elementary excitations carrying discretized quanta of area. ACKNOWLEDGMENTS The dynamics is coded into the hamiltonian constraint. A well defined version of this constraint exists (see equa- I am especially indebted with Michael Reisenberger, tion (35)), and thus a consistent theory exists, but a Roberto DePietri Don Marolf, Jerzy Lewandowski, John proof that the classical limit of this theory is classical Baez, Thomas Thiemann and Abhay Ashtekar for their general relativity is still lacking. Alternative versions of accurate reading of the manuscript, their detailed sug- the hamiltonian constraint have been proposed and are gestions and their ferocious criticisms. Their inputs have under investigation. In all these cases the hamiltonian been extensive and their help precious. I am grateful has the crucial properties of acting on nodes only. This to all my friends in the community for the joy of do- implies that its action is naturally discrete and combina- ing physics together. This work was supported by NSF torial. This fact is possibly at the roots of the finiteness Grant PHY-95-15506. of the theory. A large class of physical states which are exact solutions of the dynamics are given by s-knots with- out nodes; other exact states are related to knot theory invariants (Section VII A). The theory can be extended to include matter, and there are strong indications that ultraviolet divergences do not appear. 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