PoS(QGQGS 2011)001 http://pos.sissa.it/ ce. ve on loop quantum gravity for d at Zakopane, Poland in March tudents and the second to young tum geometry and quantum gravity , PA 16802, USA ive Commons Attribution-NonCommercial-ShareAlike Licen † ∗ [email protected] Speaker. This article is based on the opening lecture at the third quan school sponsored by the European Science Foundation and hel 2011. The goal ofyoung the researchers. lecture was The to first present part a is broad perspecti addressed to beginning s researchers who are already working in quantum gravity. ∗ † Copyright owned by the author(s) under the terms of the Creat c

Institute for Gravitational Physics and Geometry, Physics Department, 104 Davey, Penn State, University Park E-mail: Abhay Ashtekar Introduction to Loop Quantum Gravity 3rd Quantum Gravity and Quantum GeometryFebruary School, 28 - March 13, 2011 Zakopane, Poland PoS(QGQGS 2011)001 on µν F Abhay Ashtekar odify not only Maxwellian y Einstein already in a 1916 do unto gravity as one would y by Bronstein, Rosenfeld and other. In the gravitational case, , atoms would have to radiate not sented by a tensor field general developments since then uantizes the two radiative modes matical arena on which the field omagnetic waves. inkowski space-time, say Maxwell’s y in tiny amounts. As this is hardly pace-like three-planes, and analyze 1)-decomposition, and the quantum ns. Attempts were made to extend • ed into two parts. The first provides urfaces determine those on any other ical approach, electric and magnetic te: de and a half. In this section, I will ight cones and the notion of causality. ruct physical quantities such as fluxes r with quantum gravity can/should go ey physical and conceptual problems of e forefront, and quantum states naturally etic case the two methods are completely al role of the space-time metric. al on a spatial three-slice. In the covariant generally refer to books and review articles which sum- f quantum gravity. References to original papers can be 2 The electromagnetic field had been successfully quantized 1 . This is not accidental. The reason is deeply rooted in one of profound electrodynamics but also the new theory of gravitation. only electromagnetic but also gravitational energy, if onl Nevertheless, due to the inneratomic movement of electrons true in Nature, it appears that quantum theory would have to m Since this introduction is addressed to non-experts, I will The necessity of a quantum theory of gravity was pointed out b First, there was the beginning: exploration. The goal was to This section, addressed to beginning researchers, is divid To appreciate this point, let us begin with field theories in M 1 • directly to section 2; there will be no loss of continuity. 1.1 Development of Quantum Gravity: A Bird's Eye View paper in the Preussische Akademie Sitzungsberichte. He wro Minkowski space. Thepropagates. space-time The geometry background, provides Minkowskian metric the providesWe l kine can foliate this space-timehow by the a values of one-parameter electric family and magnetic ofsurface. fields s on The one isometries of of these the s Minkowskiof metric energy, let momentum, us and const angular momentum carried by electr marize the state of thefound in art these at reviews. various stages of development o quantum gravity. Researchers who are already quite familia using two approaches: canonical and covariant. In the canon arise as gauge-invariant functionals of the vector potenti a broad historical perspective and the second illustrates k Papers on the subjectPauli. began to However, detailed appear work inloosely began the represent only four 1930s in stages, most thepresent each notabl a sixties. spanning sketch roughly these The developments. a deca do unto any other physical field [9]. fields obeying Heisenberg’s uncertainty principle are at th approach on the on theof other the Maxwell hand, field one in firststates space-time, isolates without naturally and carrying arise out then as a q these (3+ elements techniques of to the general relativity. Fock Inequivalent. space the of electromagn Only photo the emphasishowever, the changes difference in is going from one to an Introduction to LQG 1. Introduction the essential features of general relativity, namely the du theory to be specific. Here, the basic dynamical field is repre PoS(QGQGS 2011)001 , 0 ) t ial , = x ( ~ E Ψ t a precise Abhay Ashtekar ious. To see this, recall first ravitational waves. Since there cally different attitudes to face generated by the Hamiltonian is 16]. The basic canonical variable l canonical commutation relations ion of causality. The emphasis is absence of a space-time geometry, ators on the fixed (‘spatial’) three- nalogous to the equation Div lativity is well-defined and attempts nalogous to the Minkowski metric in eometry. The space-time metric itself he Hamiltonian formulation, general of general relativity, its elegance and anifestations of these difficulties. It is age of the program, completed in the s causality, time, scattering states, and t is so difficult to analyze singularities usality, time, and evolution, one must he kinematical arena and dynamics, in e formation of a black hole, one must , in spite of the conceptual difficulties ides light cones, defines causality, and ions naturally decompose into two sets: ity, on retaining the compelling fusion ructure determined by that solution, ask f general relativity. It is this feature that ven in the formulation of basic physical ledge of the causal structure of the entire er, this feature also brings with it a host heory was worked out in detail by Dirac, relativistic quantum mechanics, particles sic dynamical entity of the theory, similar produces a probability amplitude, tself). On the other hand it is the analog of within that space-time is its event horizon. ) is the entire space-time. If not, space-time + + I a space-time. As an extreme example, consider ( I J 3 construct . Similarly, in quantum gravity, even after evolving an init ) t ( x of its future infinity ) of the Maxwell theory. This dual role of the metric is in effec + µν I F ( − J In general relativity, by contrast, there is no background g In quantum theory the problems become significantly more ser The canonical and the covariant approaches adopted dramati statement of the equivalence principle that is at the heart o is largely responsible for the powerful conceptual economy of problems. We see already in the classical theory several m its aesthetic beauty, its strangeness in proportion. Howev because there is no background geometry, forof example, the that theory i and tois define no the a energy priori and space-time, momentumfirst to carried solve introduce by the notions g dynamical as equations basic and as ca contains a black hole and theThus, future because boundary there of is no longer athe clean classical separation theory between t substantial care and effort is needed e rather than a specific trajectory, Introduction to LQG black holes, whose traditional definition requires the know questions. is the fundamental dynamical variable. OnMaxwell’s the theory; one it hand it determines is space-time a dictates geometry, the prov propagation of all physical fieldsthe (including Newtonian i gravitational potential and therefore thein ba this respect to the space-time. To find if thefirst given obtain initial their maximal conditions evolution lead and, toif using the th the causal st causal past that, because of the uncertainty principle,do not already have in well-defined trajectories; non- time-evolution only state, one would not behow left is one with to a introduce specific even habitualblack space-time. physical holes? notions In such a the was the 3-metric on a spatialfour slice. constraints The ten on Einstein’s the equat metricof and electrodynamics) its and conjugate six momentum (a evolution equations. Thus, in t Bergmann, Arnowitt, Deser and Misner and others [2, 5, 13, 6, these problems. In the canonicalmentioned approach, above, one the notices Hamiltonian that formulationto of use general it re asare a stepping to stone lead to usto quantization. to be the The thought basic fundamenta of asmanifold uncertainty time commute principle. evolution. is supposed The Theon to fact motion preserving capture that the the certain geometrical oper appropriateof character not gravity of and general geometry relativ thatearly Einstein 1960s, created. the Hamiltonian In formulation of the the first classical st t PoS(QGQGS 2011)001 µν g theoretic tech- Abhay Ashtekar rture for quantum theory by etric tensor are now split. The achinery of perturbation theory ions of the quantum theory were lementary particles were to arise hat the quanta have spin two and e number of degrees of freedom. cesses are to give radiative correc- m geometrodynamics continued to ry in the canonical program “drove tering matrices are to be computed, definite program to compute ampli- particular, in quantum geometrody- he time. In 1963 Feynman extended is to split the space-time metric lties associated with the presence of d [4, 16]. Wheeler also launched an ary particles." eem to melt away. Thus, in the tran- eler, DeWitt and others in the limited ativity was first reduced to a quantum e covariant quantization program was lve the quantum Einstein’s equations. iolating energy conditions. Sociologi- quantum field theory. Indeed, with this rt Lagrangian tells us how they interact eared to be tamed and forced to fit into ty and geometry is a moderate price to echniques which had been so successful -geometries. Wheeler therefore baptized and particle physics was to be recast as could not be resolved without additional eometrodynamics program became stag- mewhat misleading because the introduction of a , the deviation of the physical metric from the is used mainly to emphasize that this approach does not µν is to be a background, kinematical metric, often . If the background is in fact chosen to be flat, one 4 h µν µν η η that is quantized. Quanta of this field propagate on the in the Wheeler DeWitt equation remain unresolved. Fur- µν h , where µν Gh √ + approach [6, 8, 10] the emphasis is just the opposite. Field- µν [3, 4]. η is Newton’s constant, and 2 G = µν g In the context of quantum gravity, the term ‘covariant’ is so In the second stage, this framework was used as a point of depa In the covariant 2 infinite number of degrees of freedom geometrodynamics niques are put at the forefront. The first step in this program splitting most of the conceptualsition problems to discussed the above quantum s theory it is only more in tune with the mainstream developments in physics at t background metric violates diffeomorphisminvolve covariance. a 3+1 decomposition It of space-time. pay for ushering-in the powerful machinery of perturbative the mold created to describe other forces of Nature. Thus, th chosen background, the dynamicaloverall field. attitude The is two that roles this sacrifice of of the the m fusion of gravi classical background space-time with metric can use the Casimir operatorsrest of mass the zero. Poincaré These are group thewith and gravitons. one show The another. t Einstein-Hilbe Thus, in thisfield program, theory quantum general in rel Minkowski space.that had One been so could successful apply intudes to particle for it physics. various One all scattering now the had processes. m a Unruly gravity app in two parts, chosen to be flat, Introduction to LQG relativity could be interpreted as the dynamicalit theory of 3 Bergmann, Komar, Wheeler DeWitt andwritten others. down The and basic several equat importantambitious questions program were in addresse which thefrom internal non-trivial, quantum microscopic numbers topological of configurations e Therefore, after an initial burst ofnant. activity, Interesting the results quantum have g beencontext obtained of by Misner, quantum Whe cosmology whereHowever, even one in freezes this all special but case, a the finit initial singularity ‘chemistry of geometry’. However, most of the work in quantu thermore, even at the formal level, is has been difficult to so ‘external’ inputs into the theory, such as the use of matterin v quantum electrodynamics appeared to play no role here. In remain formal; indeed, even todayan the field theoretic difficu cally, the program faced another limitation: concepts and t namics, it is hard to seehow how Feynman gravitons diagrams are are to to emerge, dictatetions. how dynamics To scat use and a virtual well-known pro phrasea [7], wedge the between emphasis general on relativity geomet and the theory of element PoS(QGQGS 2011)001 Abhay Ashtekar This led to a success- 3 an with great enthusiasm and ffects are taken into account for siasm was first generated by the lectroweak interactions, the parti- matter [17]. To appreciate the sig- es at high energies which improve vitational case. In the case of weak w energy phenomena could be ex- amental theory, it has no predictive calculations were performed to esti- ormalizable (if the masses of gauge t ignores important processes at high d matter quanta. He showed that the f photons and electrons. This theory rs. Although one can still use it as an ity. A few years later DeWitt carried ve corrections involving closed loops perturbation series can be systemati- .e., at large distances. Thus, the idea ve terms were added to the Einstein- ral relativity but only at high energies, systematic procedure is not available; g mechanism). e constant and the electron mass. Thus, they can be systematically absorbed in the Feynman rules for calculating scat- chosen judiciously, the resulting theory gh individual terms in the perturbation itnessed a renaissance of quantum field ural to hope that the situation would be on. By the early seventies, the covariant roblem was that this Fermi model led to model of Glashow, Weinberg and Salam boulis and others showed that the theory mbles the free theory in the high energy logy, would be similar to Fermi’s model. research soon ran into its first road block. nely troublesome. Put differently, quantum le in the initial stages of the extension of perturbative 5 Buoyed, however, by the success of perturbative methods in e In fact DeWitt’s quantum gravity work [8] played a seminal ro Consequently, the second stage of the covariant program beg 3 effective theory in the lowpower energy at regime, all! regarded as a fund is not only renormalizable but asymptotically free; it rese theory acquires an infinite number of undetermined paramete a non-renormalizable theory. The correct, renormalizable agrees with Fermi’s at low energies but marshals new process does in fact have a better ultraviolet behavior. Stelle, Tom mate radiative corrections. Unfortunately, however, this The theory was shown topure be non-renormalizable gravity when and two already loop atnificance e one of loop this for result, gravity letis coupled us with perturbatively return renormalizable. to the This quantum means theory that, o althou of virtual particles, these infinitiesthe values are of of free a parameters of specificby the type; renormalizing theory, the these fine parameters, structur cally individual rendered terms finite. in the Ininfinities quantum that general arise due relativity, to such radiative a corrections are genui cle physics community was reluctant tointeractions, give them it up was in known the for gra plained using some Fermi’s time simple that four-point interaction. the The observed p lo the ultraviolet behavior of the theory. It was therefore nat The fact that it is notenergies renormalizable which was are, taken to however, unimportant meanwas that at that i low the energies, correct theory i ofi.e., gravity would near differ the from gene PlanckHilbert regime. Lagrangian. With If this the aim, relative higher coupling constants derivati are theory. The enthusiasm spilled over to gravity. Courageous expansion of a physical amplitude may diverge due to radiati similar in quantum gravity. General relativity, in this ana ful theory of electroweak interactions. Particle physics w techniques from Abelian to non-Abelian gauge theories. this analysis to completion bytering systematically amplitudes formulating among gravitons andtheory between is gravitons unitary an order by order in the perturbative expansi hope. The motto was:discovery Go that forth, Yang-Mills perturb, theory and coupled expand.particles to are The generated fermions enthu by is a ren spontaneous symmetry-breakin Introduction to LQG perturbative methods from quantum electrodynamics to grav approach had led to several concrete results [8]. PoS(QGQGS 2011)001 It turned out 4 Abhay Ashtekar ory. to all orders; it does not even perstring [20, 31]. Since strings reene and Schwarz showed that ry to analyze strong interactions d apply perturbative techniques. ad a potential quantum anomaly sts and the covariant approach by ingle, new fundamental constant c Feynman diagrams are replaced finite 1-dimensional extended objects — attack. In the approaches discussed proves the ultraviolet behavior and, haracteristic of gauge theories, string rom below, and consequently the the- es of the gravitational field would be ementation of this viewpoint occurred avity is finite to four loops even though t could be settled in the next 2-3 years. lity that the most sophisticated of these y is t 2 or 3 loop order, in the string theory suggesting that the theory would not be ton. In this sense, gravity was already ing us a renormalizable quantum theory ars or so, there has been a resurgence of as soon realized that this was a blessing erwent a dramatic revision. Since it is a lativity was in fact realized. However, it unification of electromagnetic and weak ory could be consistent in certain space- ived unexpected boosts. These launched ts Hamiltonian is manifestly positive and odes of excitations of the string. Initially quantum theory has brand new elements; heory would result only when gravity is ecially among particle physicists, that supergravity tarity; probability fails to be conserved. The ty theory. For instance, in the centennial celebration of [15] suggested that end of theoretical physics was in sight 6 wo talks on quantum gravity, both devoted to supergravity. A nceton [14] —the proceedings of which were videotaped and 8 supergravity— can be employed as a genuine grand unified the = 8 supergravity was likely to be the final theory. N = N For a number of years, there was a great deal of confidence, esp By and large, the canonical approach was pursued by relativi A group of particle physicists had been studying string theo 4 it is no longer aby local world-sheet quantum diagrams. field This theory.although explicit replacement The calculations dramatically field have im been theoreti community carried it out is only widely a believed thathave the to perturbation be theor renormalized. The theory is also unitary. It has a s time dimensions —26 for a purely bosonic string and 10 for a su theory of extended objects rather than point particles, the because theory included also a spin-2, masslessin excitation. disguise: But it the w theorybuilt automatically into incorporated the a theory! gravi which However, threatened it to was make knownthere it that is an inconsistent. the anomaly theory cancelation In and h perturbative the string the mid 1980s,were G assumed to liveHowever, in in a this reincarnation, flat the background covariant approach space-time, und one coul was on the threshold ofEinstein’s providing birthday the at the complete Institute quantum ofarchived gravi Advanced for future Study, Pri historians and physicists—year there later, were in t his Lucasian Chair inaugural address Hawking there was an embarrassment: in addition to the spin-1 modes c from a novel angle. Thestrings— idea and was associate to particle-like replace states point with particles various m by the theory is unitary. However, there are several arguments ory is drastically unstable! In particular,success it of violates the uni electroweak theoryabove, suggested gravity a was considered second in line isolation. of The successful coupled with suitably chosen matter. Thein most supergravity. striking impl Here, the hopecanceled by was those that of the suitably chosen bosonic fermionicof infiniti sources, gravity. giv Much effort went intotheories the — analysis of the possibi that cancelations of infinities do occur.interest Over in the this last area five ye [55]. Itit has contains now been matter shown fields that coupled supergr to gravity. Furthermore, i renormalizable at seven loops. This is still an open issue bu Introduction to LQG limit. Thus, the initial hope ofturned ‘curing’ out quantum that general the re Hamiltonian of this theory is unbounded f interactions suggested the possibility that a consistent t the third phase in the development of quantum gravity. particle physicists. In the mid 1980s, both approaches rece PoS(QGQGS 2011)001 Abhay Ashtekar ic strings, when summed, uld be regarded as a serious if one uses superstrings instead. string represent different particles, n random surfaces due to Ambjorn vated approaches to quantum grav- ics and high temperature supercon- ight on the conceptually central is- y of everything’ on the other hand, pic dimensions. Consequently, the osmological constant but they have mensional. One can compactify the t can plead incompleteness and shift t its burden. It must face the Planck d considers space-times with higher irections cannot be microscopic; the fundamental physics; it would be the From the viewpoint of local quantum stant is positive. Considerable effort nces. In particular, it completely ig- rspective of quantum gravity, this ap- ace-times is taken to be equivalent to ere has been a long-standing hope in on theory also faces some serious lim- ly incomplete theory. It ignores many hat space-time can be approximated by 5 ve key non-perturbative structures. ould equal the cosmological radius! So s, the series is also believed to diverge. e provides a non-perturbative definition ological constant in the bulk is essential ure has little to do with the macroscopic ly, it is not even Borel-summable.) They y conditions [49]. More precisely, in this m in the perturbation series is ultra-violet ng them to make unsuspected connections g electroweak and strong interactions, this ond to the world we actually observe. Nonetheless, ions! QED, these are regarded as ‘harmless’ for calculation w vista: the concrete possibility that unification could be is issue. 7 However Gross and Periwal have shown that in the case of boson 6 But it does appear that there are infrared divergences. As in To date, none of the low energy reductions appears to corresp Unfortunately, it soon became clear that string perturbati The current and the fourth stage of the particle physics moti 6 5 brought about by a tightly woven, non-local theory. of physical effects. I thank Ashoke Sen for discussions on th string theory has provided us with a glimpse of an entirely ne macroscopic dimensions, the AdS/CFT duality is yet to shed l world we live in. Finally, even if one overlooks this issue an field theories that particle physicists have usedmathematical in structure studyin seems almost magical.the string Therefore community that th this theory‘theory would of encompass everything’. all of ity is largely devoted to unraveling such structuresbetween and gravity usi and other areas ofductivity. physics It such is as widely fluid believed dynam that the AdS/CFT conjectur were made initially to extendnot the had ideas notable to success. a Second,unwanted positive the dimensions or bulk using zero space-time n-spheres, c is but 10 thecorrespondence compactified di requires d that the radius ofif these one spheres just sh looks around,non-perturbative string one theory should defined see through these the conject large macrosco Introduction to LQG —the string tension— and, since various excited modes of the of string theory on space-times satisfying certain boundar to this correspondence while the observed cosmological con there is a built-in principle for unification of all interact itations. Perturbative finiteness would imply thatfinite. each ter the series diverges and does soalso uncontrollably. gave (Technical arguments that theIndependent support conclusion for would these arguments not has beand come from changed others. work o One mightfailure of wonder the why theory. the AfterRecall divergence all, however of in that quantum the quantum electrodynamic electrodynamics sum isprocesses sho an that inherent come into playnores at the microstructure high of space-time energies and or simplya short assumes smooth t dista continuum even below thethe Planck scale. burden of Therefore, i this infinityhas to nowhere a to more hide. complete theory. Itregime A cannot squarely. ‘theor plead If the incompleteness theory and is shif to be consistent, it must ha correspondence string theory on asymptoticallycertain AdS gauge bulk theories sp on its boundary.proach However, has from some the serious pe limitations. First, a negative cosm PoS(QGQGS 2011)001 i s 8 ). That 2 Pl nnection ℓ . However, β etric extracted Abhay Ashtekar parameter. is precisely (area/4 a real parameter rics to connections is that the rs simplifies significantly if we ing observation: the geometrody- n forgotten and connections were articularly by Gambini, Pullin and s on the interaction. In particular, auge theories [22, 26]. The ‘wedge nctions of loops on the 3-manifold. eak and strong interactions, there is ified framework to describe all four e basic object. In fact we now know ues from gravity and supergravity to ed in the strong coupling regimes of eral relativity as a theory of connec- Barbero-Immirzi nce of general relativity. One could 2]; and certain cosmological models. uations can now be visualized simply n evaporating black hole or the fate of resentation’ was introduced by Rovelli Einstein’s equations considerably. For —general relativity in 2+1 dimensions n highly successful in the quantization ticles’ that Weinberg referred to largely llel transport of vectors and found that st —and the strength— of these devel- eatures arise. These were exploited in p representation played an important role in the inating history, see [33].) However, they is called ‘connection-dynamics’. s now regarded as a dynamical theory of ch are complex. These have a direct interpretation in retation and the constraints are no longer polynomial in er, one is then faced with the task of imposing appropriate simplify uations polynomial in the basic variables. This simplicity aints. Since then this strategy has become crucial because rent, fourth phase, the name is still used to distinguish thi iemann introduced novel ideas to handle quantization of the 8 s has so far been developed only for real connections. Immirz imit is real general relativity. Barbero introduced real co However, now these are ‘spin-connections’ required 7 systems. in later papers) is referred to as the γ in the expression of the (anti-)self-dual connections with non-gravitational could be chosen so that the leading term in black hole entropy i ± β on the space of spin-connections (with respect to a natural m (which is often denoted by β geodesic motion This is the origin of the name ‘loop quantum gravity’. The loo Perhaps the most important advantage of the passage from met This reformulation used (anti-)self-dual connections whi On the relativity side, the third stage began with the follow 8 7 variables by replacing the quantum ‘reality conditions’ to ensure that the classical l is why was regarded as crucial for passage to quantum theory. Howev initial stages. Although this is no longer the case in the cur from the constraint equations). Sinceconnections, this general reincarnation relativity of i the canonical approach This led to a number of interesting andterms space-time intriguing geometry results, and also p render the constraint eq the late eighties and early nineties to solve simpler models the basic variables. But the strategy became viable after Th the rigorous functional calculus on the space of connection to parallel propagate spinors, andexample, they the turn dynamical out evolution to dictatedas by a Einstein’s eq phase-space of general relativity isbetween now general the relativity and same the as theory thatdisappears of of elementary without par g having tonow sacrifice import the into general geometrical relativity esse of techniques that gauge have theories. bee Atfundamental the interactions. kinematic level, then, Thewhile there dynamics, there is of is a a un course, backgroundnone depend space-time in geometry general in relativity. electrow Therefore, qualitatively new f [22, 23, 29]; linearized gravityTo explore clothed the as physical, a 3+1 dimensional gauge theory,and theory a Smolin. ‘loop [2 rep Here, quantum states were taken to be suitable fu now the connection does not have a natural space-timespecific interp non-polynomial terms that now feature in the constr suggested that the value of Introduction to LQG sues such as the space-time structure inside the horizon of a opments is, rather, that they enable one to use known techniq big-bang type cosmological singularities. The recent thru solve some of the difficultfield mathematical theories problems describing encounter namics program laid outregard by a Dirac, spatial Bergmann, connection —rather Wheeler thanthat, and the among othe 3-metric— others, as Einstein th andtions Schrödinger already in had the recast fifties.(For gen a brief account of this fasc the theory becomes ratherre-introduced complicated. afresh in This the episode mid had 1980s [22]. bee used the ‘Levi-Civita connection’ that features in the para PoS(QGQGS 2011)001 (and geometries was system- Abhay Ashtekar spin foams quantum , supergravity. ormal until mid-nineties. Passage ’ of the polymer-like quantum 3 To construct a well-defined theory e sum has not been systematically p quantum gravity: onnections for constructing Hilbert . It is a rigorous mathematical the- One finds that, at the Planck scale, n-point functions live is encoded in coarse-grained approximation. The venated the canonical quantization geometry effects have already been ide a sum over histories formulation to fresh, non-perturbative techniques h at this School.) nter the sum are excitations are 1-dimensional, rather xist! he long-standing problems associated ity [28]. Thus, there was rapid and does in, say, gauge theories. Rather, n an appropriate limit. Furthermore, theory. There are detailed arguments ntroduction of a background geometry ese quantities involve non-polynomial ding, e.g. to areas of surfaces and vol- that quantum general relativity (or su- rmally used in perturbative treatments. tion of the ‘transition amplitudes’ and l domain, quantum geometry provides ides the basic mathematical framework morphism invariant measures and there ed the fourth (and the current) stage in es lurking in the procedure. Indeed, such t, given a suitable ‘boundary state’, there itatively different from Minkowski back- infinite dimensional space of connections at this approach may lead to well-defined, ng field theories. To pay due respect to the 9 quantum theory of Riemannian geometry quantum general relativity, or its supersymmetric version Fortunately, the folk-theorems turned out to be incorrect. Over the last six years, another frontier has advanced in loo However, a number of these considerations remained rather f is a conceptual framework to calculate n-point functions no the associated development of group field[35, theory) 39, which 48]. prov The new elementof here a is specific that the type; historiesgeometries they that that e can emerged be in regarded thederived as canonical starting the approach. from the ‘time So classicalone evolution far uses theory th semi-heuristic as considerations one tothen generally arrive explores at physical a properties defini ofto the the resulting effect quantum thatalthough one the recovers underlying theory the is diffeomorphism Einstein invarian Hilbert action i Information about the background space-time on which these approach from others. capable of handling field theoretic issues,atically a constructed in the mid-nineties [38]. This launch pergravity) may well be non-perturbativelyshown to finite. resolve the Quantum big-bang singularity andwith solve black some holes. of (See t lectures by Giesel, Sahlmann and Sing integrals are notoriously difficult to performgeneral in covariance interacti of Einstein’s theory, onewere needed folk-theorems diffeo to the effect that such measures did not e the canonical approach. Just as differential geometry prov ground used in perturbative treatments reinforced the idea unanticipated progress in aprogram. number Since the of canonical approach directions doesor not which require use the reju of i perturbation theory, and becausefrom one gauge now theories, has in access relativity circles therenon-perturbative is a hope th to the loop representation required an integrationand over the the formal methods were insensitive to possible infiniti to formulate modern gravitationalthe theories necessary in concepts the and classica techniquesory in which the enables quantum one domain to performspaces integration of on states and the to space define ofumes geometric c of operators regions, correspon even though thefunctions classical of expressions the of th Riemannian metric.geometry has There a are definite nolike discrete infinities. polymers, structure. and the Its space-time fundamental fact continuum that arises the structure only of as space-time a at Planck scale is qual Introduction to LQG their collaborators, relating knot theory and quantum grav PoS(QGQGS 2011)001 satisfac- any Abhay Ashtekar ght us many valuable lessons. lems that are ‘internal’ to indi- vity predicts that the space-time smooth continuum at arbitrarily ricted myself to the ‘main-stream’ problems still remain. The status detail in the Euclidean context by ion to any quantum gravity theory, or the big-crunch but we expect the lf flat— gravitational fields in the g standing issues that e inadequate to construct quantum inal approaches —particularly, the and dimensional reduction in string r several decades. However, I would l constituents, is the evolution of the etry and the continuum picture only of these approaches are inter-related s particularly true of the path integral triangulations, asymptotic safety and t perturbative, field theoretic methods ld’ of ferro-magnets? If so, what are underlie the whole subject. Examples l approach. In this sub-section, I will g of a Heisenberg quantum model of a veal this structure to us. e the microstructure of space-time nor , asymptotic safety scenarios [47], dis- Baratin, Perini, Fairbairn, Bianchi and his approach have provided the point of -commutative geometry [25], causal sets 0, 40], twistor theory [11, 21, 43] and the ne, making quantum gravity indispensable s primarily a signal that the physical theory 10 : It is widely believed that the prediction of a singular- state of the universe free of singularities? General relati The Big-Bang and other singularities • Approaches to quantum gravity face two types of issues: Prob For brevity and to preserve the flow of discussion, I have rest The first three stages of developments in quantum gravity tau ity, such as the big-bang ofhas classical general been relativity, pushed i beyond the domain of its validity. A key quest tory quantum theory of gravity should address. then, is: What replaces theapproximations, big-bang? analogous Are to the the classical ‘meanthe geom (magnetization) microscopic fie constituents? What isferro-magnet? the When space-time formulated analo in terms ofquantum these fundamenta curvature must grow unboundedly as we approachquantum the effects, big-bang ignored by general relativity, to interve Perhaps the most important among themwhich is the have realization been tha so successfulgravity. in other The branches assumption ofsmall that physics scales ar space-time leads can to be inconsistencies.presuppose replaced its We can nature. by neither We a must ignor let quantum gravity itself re programs whose development can be continuously trackedlike ove to emphasize that thereEuclidean are path integral a approach number [12], of Reggecrete other approaches calculus highly [50], [24] causal orig dynamical triangulationstheory [3 of H-spaces [18], asymptotic quantization [19], non and some of them lie at theapproach foundation pioneered of other by avenues. Misner, This and i Hawking, developed Hartle, in Halliwell much and others. greater Ideasdeparture developed in for t the ongoing developments inspin causal foams. dynamical 1.2 Physical Questions of Quantum Gravity vidual programs and physical and conceptualof questions the that former are:twistor program, Incorporation mechanisms of for physical breakingtheory, —rather of and supersymmetry than issues ha of space-timefocus covariance on in the the second canonica type of issues by recalling some of the lon [37, 44] and Topos theory [51, 56]. Ideas underlying several is described in detail inKaminski the at this lectures school. by Rovelli, Speziale, Introduction to LQG the chosen ‘boundary state’. However, a number of important PoS(QGQGS 2011)001 . This ¯ h G 4 / hor Abhay Ashtekar a = nts, Bekenstein ar- in thermodynamics BH T S rd quantum mechanics, the [12, 34, 41]. However, at first at is the interplay between the S ? Can one recover the familiar f thermodynamics, provided one show that the proportionality fac- les in equilibrium obey two basic tself is dynamical. Therefore, as e fundamental, Pauli-Schrödinger istical mechanical origin of black avity? Is there an imprint of the ’ approach, does the fundamental be given by xterior curved geometry? Can one se the BCH analysis was based on ow that this low energy description er bound on curvature? How close act radiate quantum mechanically as are quite different from those of the , a background independent descrip- tions of interpretation of the quantum heir area [12, 34, 41]. About the same that correctly describes the big-bang? tum gravity theory would also be free sent in the standard perturbative quan- he three pillars of fundamental physics les and practical methods of computing ate sense, bridging the vast gap of some nics. However, the argument itself is a rmative, can one pin point why the stan- t is the essential feature which makes the l ideas, reminiscent of the Bohr theory of ? Do these semi-classical sectors provide ity? What can we say about the ‘initial [12, 27]. Using the analogy with the first ., through ‘information loss’? guiding principle? π In general relativity, there is no background 2 / κ ¯ h 11 physical = T to some multiple of the temperature κ . Two years later, using quantum field theory on a black hole ¯ h to a corresponding multiple of the entropy hor a In the early seventies, using imaginative thought experime general relativity and simple dimensional considerations Planck scale physics and the low energy world: Black holes: • There are of course many more challenges: adequacy of standa • rather hodge-podge mixture of classical and semi-classica this similarity was thought toclassical be only a formal analogy becau though they are black bodies at temperature conclusion is striking and deep—general because it relativity, quantum brings together theory t and statistical mecha atom. A natural question thentheory is: of what the is Hydrogen the atom?hole analog entropy? More of the precisely, What mor what isquantum is the degrees the nature of stat of freedom a responsible quantumderive for black the entropy hole and Hawking and the effect wh e classical from singularity first on the principles final of quantum description, quantum e.g gr issue of time, of measurement theoryframework, and the the issue associated of ques diffeomorphism invariant observab background space-time, Hawking showed that black holes in f dard ‘bottom-up’ perturbative approach fails? That is,fundamental wha description mathematically coherent but is ab and the horizon area tors must involve Planck’s constant metric, no inert stageindicated on above, which one dynamics expects unfolds. thatof Geometry a a background i fully space-time satisfactory geometry. quan tion However, of must necessity use physical conceptsfamiliar, and low mathematical energy tools physics. that Adoes arise major from challenge the then pristine, Planckian is world16 to in orders sh an appropri of magnitudetheory in admit the a ‘sufficient energy number’ scale. of semi-classicalenough states In of this a ‘top-down backgrounddescription? geometry If to the anchor answers to low these energy questions are physics in the affi tum gravity? Introduction to LQG before infinite curvatures are reached. If so, what is the upp to the singularity canconditions’, we i.e., ‘trust’ the classical quantum general stateIf relativ of they geometry have to and be matter imposed externally, is there a gued that black holes must carrytime, an Bardeen, entropy Carter proportional and to t Hawking (BCH)laws, showed which that have black the ho same formequates as the black the hole zeroth surface and gravity the first laws o law, one can then conclude that the black hole entropy should PoS(QGQGS 2011)001 is ches There is no a manifold but Abhay Ashtekar tivity stands out (or supergravity, is r y Giesel, Sahlmann, Rovelli and ral relativity seriously: gravity 9 answer is often assumed to be in ievements, challenges and oppor- ropriate mathematical language to ot understand well or those would chnical accounts in [38, 39, 45] and th us longer than any of the current een by following older monographs n structural and conceptual issues. I tions have been tested touantum an amazing electrodynamics. Therefore, scription of Nature. Nonetheless, this here should be no background metric. stions that are largely independent of ere, one first had a 4-point interaction nced undergraduate) level, see [53]. und. antum matter on a background geom- nal dynamical equations. This role is um gravity described in the rest of this . ion and S-matrices, exploration of the ects of gravity. There in detail are more basic from a physical t non-perturbative quantum gravity, but gous fashion, in the final picture one can get rid o provide glimpses of directions that have by considering ‘abstract’ spin networks in the canonical eory can be formulated combinatorially [22, 23]. While e complete handle on the underlying mathematics. 12 a comprehensive review. The choice of the material was not exist as consistent theories non-perturbatively? be ‘born quantum mechanically’. Thus, in contrast to approa both Riemannian geometry. In the classical domain, general rela Does quantum general relativity, coupled to suitable matte quantum In loop quantum gravity, one takes the central lesson of gene In 2+1 dimensions, although one begins in a completely analo Detailed treatments of the subject can be found in lectures b In classical gravity, Riemannian geometry provides the app As explained in section 1.1, in particle physics circles the In this section, I will summarize the overall viewpoint, ach 9 Singh in these proceedings and even morereferences complete and therein. more te (The development[22, of 26, the 28].) subject For can a be treatment at s a more elementary2.1 (i.e. adva Viewpoint developed by particle physicists, one doesetry not and begin use with perturbation qu theory to incorporate quantum eff of the background manifold as well. Thus, the fundamentalapproach th and 2-complexes in spin foams, one still needs a mor been and are being pursued;need and detours ii) to Avoiding explain issues strengths that and weaknesses. I do n geometry whence, in a fundamental quantum gravityGeometry theory, and t matter should no metric, or indeed any other physical fields, in the backgro as the best available theorydegree of gravity, of some accuracy, of surpassing whose even predic it the is legendary natural tests to of ask: q some steps have been taken to achieve this in 3+1 dimensions, guided by two principles: i) Choosing illustrative topics t role of topology and topology change,chapter, etc, one etc. adopts In the loop view quant that theviewpoint three because issues they discussed are rooted inthe specific general approach conceptual being que pursued.leading Indeed approaches. they have been wi 2. Loop quantum gravity tunities underlying loop quantum gravity.would The like emphasis to is emphasize o that this is formulate the physical, kinematical notionsnow as taken well by as the fi its supersymmetric generalization) implication that such a theory would beis the a final, fascinating complete and de important open question in its own right the negative, not because there isbecause concrete of the evidence analogy agains to the theory of weak interactions. Th Introduction to LQG their properties, convenient ways of computing time evolut PoS(QGQGS 2011)001 n the Abhay Ashtekar uum picture should ! Failure of the standard eneral relativity did exist as a hich fails to be renormalizable. ss useful. re such physical expectations are 1, loop quantum gravity has had propagators. Therefore, it is often quantum geometry, several of the mative. There exist quantum field s themselves, the theory would not emoved. Functional analytic issues Z h classical general relativity, while lizable theory of electro-weak inter- edom are now faced squarely. Inte- heory even though it does not unify ) in which the standard perturbation turbing around the trivial, Gaussian orces. However, just as general rel- ulations of the Fermi model but by these inconsistencies. iance do restrict the form of interac- cacci, Perini and others there is now rbative treatments may largely be due hould continue to exist in presence of tial criterion for usefulness of a theory lar in Euclidean quantum gravity. Im- t that, in the case of general relativity, of the renormalization group flow. In- ation in the classical domain, quantum may also admit a non-trivial fixed point and the fact that we do not yet have a viable on to assume that it would be the ‘final’ um general relativity points in a similar e treatment which correctly incorporates the other hand, the scale of interest is exactly soluble ± nts pre-suppose that the space-time can be ting ways [42]. ons, pushing further the existing frontiers W 13 these interactions. Put differently, such a theory would of interest to physics under consideration. This assumptio determines at all scales and there is no physical basis to pre-suppose that the contin Pl ℓ Are there any situations, outside loop quantum gravity, whe However, as indicated in the Introduction, even if quantum g From the historical and conceptual perspectives of section be valid down to that scale.to The this failure grossly of incorrect the assumption standard andthe pertu a physical non-perturbativ micro-structure of geometry may well be free of Progress occurred not byreplacing looking the model for by non-perturbative the Glashow-Salam-Weinberg form renorma actions, in which the 4-point interaction is replaced by borne out in detail mathematically?theories (such The as answer the is Gross-Neveuexpansion model in in is the three not affir dimensions renormalizable although the theory is Introduction to LQG model due to Fermi which works quite well at low energies but w assumed that perturbative non-renormalizability of quant direction. However this argument overlooks the crucial fac candidate for the grand unified theory does not make QCD any le there is a qualitatively new element. Perturbative treatme perturbation expansion can occurpoint because rather one than insists the on moreterestingly, per physical, thanks non-trivial to fixed point recent worknon-trivial and by growing Reuter, evidence Lauscher, that situation Per pressive calculations may have be shown simi that pure[47, Einstein theory 52]. Furthermore, the requirementmatter that constrains the the fixed couplings point in s non-trivial and interes mathematically consistent theory, there istheory no a of priori all reas known physics.requirements of background In independence particular, and astions general is between covar gravity the and case matter wit have fields a and built-in among principle matter which field not be a satisfactory candidateativity for has unification had of powerful all implicationsgeneral in known relativity f spite should of have this qualitativelyof new limit physics. predicti Indeed, unification does noteven appear to in be other an interactions. essen QCD,strong for interactions example, with is electro-weak ones. a powerful Furthermore, t 2.2 Advances several successes. Thanksroadblocks to encountered the by quantum systematic geometrodynamics development wererelated r of to the presence of an infinite number of degrees of fre assumed to be a continuum is safe for weak interactions. In the gravitational case, on Planck length PoS(QGQGS 2011)001 odels antum opera- oop quan- Abhay Ashtekar ible with the rotational ch matter fields are non-zero. , including the exotic singular- ) ith the big-crunch singularity in and others has shown that non- ρ differential geometry to calculate ( d past the ‘formal’ stage of devel- ian manifold. To turn these quanti- ed in this context, the spectrum of p derlying general relativity is only a antum dynamics still remain, it has ical gravitational attraction, thereby t framework have been resolved. In e situations, the operators and their nd the required operators have been zed by Rovelli and Speziale, there is well established to extract useful and lassical theory and the application of = ut compatibility between discreteness thematical precision, the Hamiltonian pled to gravity. The physical distance perties of quantum black holes, giving d matter, still described by an equation in loop quantum gravity. For instance, Planck scale but rises very quickly and te an effective repulsive force which is p such as areas of 2-surfaces and volumes olumes of physical objects, measured in ns, Singh has shown that these quantum m of the fate of the big-bang in quantum significance of this specific discreteness? ss of eigenvalues of the angular momen- sues and discuss some recent advances. as to define the surfaces and regions metry simply provides us with formulas to ome of the long standing issues about the na- physical 14 surface leads to physical results. In 2+1 dimensions, this strong curvature singularities in homogeneous isotropic m all In Friedmann-Lemaitre-Robertson-Walker (FLRW) models, l The specific quantum Riemannian geometry underlying loop qu of non-relativistic quantum mechanics is perfectly compat z ˆ J , e.g. by focusing on surfaces of black holes or regions in whi Quantum cosmology. Quantum geometry. • • ities such as the big-rip. (These can occur with non-standar gravity [54]. Work by Bojowald, Ashtekar, Pawlowski,negligible Singh when the curvature falls significantly below the invariance of that theory. tum gravity has resolved the long-standing physical proble perturbative effects originating in quantum geometry crea dramatically in the deep Planckreplacing regime the to big-bang overcome by the a class the quantum closed bounce. models. The More generally, same using is effective true equatio w geometry effects also resolve where matter sources have an equation of state of the type Introduction to LQG grals on infinite dimensional spaces are rigorously defined a ture of the big-bang, physics of the very early universe, pro systematically constructed. Thanks toand this the high spin level foam of programs ma inopment. loop quantum More gravity importantly, have although leape been key possible issues to related use the to partshighly qu of non-trivial the physical program predictions. that In are particular, s already meaning to the n-point functionsthis in sub-section, a I background will independen further clarify some conceptual is gravity predicts that eigenvalues of geometric operators — of 3-dimensional regions— are discrete. Thus, continuum un Recall first that, in thecompute classical areas theory, of differential surfaces geo and volumes of regions in a Riemann coarse grained approximation. What is the direct ties into physical observables of general relativity, one h tum operator tionally Once this is done, onevalues can of simply these use observable. the formulas Thethe supplied area situation by of is the rather isolated similar the horizon is quantum a geometry Dirac area observable formula in to the c of these eigenvalues and Lorentz invariance.no As tension was whatsoever: emphasi it suffices to recall that discretene Freidel, Noui and Perez have introduced point particles cou the rest frames. Finally sometimes questions are raised abo between these particles is againthe a length Dirac operator observable. haseigenvalues When direct correspond us to physical the meaning. ‘proper’ lengths, areas In and v all thes PoS(QGQGS 2011)001 s of quan- Abhay Ashtekar general relativity, non-singular, s long been a highly non-trivial that the homogeneous modes of ces where an infinite number of s in the isotropic case, effective ter fields that violate energy con- ce has accumulated in support of he underlying quantum geometry ticular they show that the matter odels —the 1-polarization Gowdy n loop quantum cosmology? The mology, even with an inflationary which is) faster than, say, the dust s initiated by Mena, Garay, Martin- awking and others? It does so be- ore recent singularity theorems due ularity is again resolved: any time a ole entropy problem is complemen- ft hand side of Einstein’s equations, 6 ensity approaches the Planck regime, y correct. One might therefore hope re better and better approximated by herefore assume that the universe bas s were studied extensively in the early ity in classical general relativity, ‘the to Belinskii, Khalatnikov and Lifshitz a ds satisfy all energy conditions. How aining spatial derivatives, so that the ghts at first. But a careful examination . l cases the singularity had persisted. A hese Bianchi models is also important / n the singularity is avoided because the p quantum cosmology has opened a door c shears dominate in Einstein’s equations e added even as a perturbation to a FLRW ties are resolved by the quantum geometry 15 refer to field equations at all. How are these evaded? not Loop quantum cosmology illuminates dynamical ramification ). ) ρ ( p = p Quantum Horizons. I will conclude with the discussion of a conceptual point. In Finally, the simplest type of (non-linear) inhomogeneous m • A proper treatment of anisotropies (i.e. Bianchi models) ha space-times— have also been analyzed in detail.quantum These model gravity literature, prior to thesystematic advent study of in LQC. the In context al ofBenito, loop quantum Vehlino cosmology and wa others bythe making model an correspond astute to use aresolves the of Bianchi big-bang I the singularity. space-time. fact Once again, t effects underlying loop quantum gravity. issue in general bouncing scenarios because thein anisotropi the contracting phase before the bounce, diverging (as 1 issue turned out to be quite subtleby and Ashtekar, there Wilson-Ewing were and some others oversi hasshear shown scalar that —a the potential sing for the Weylthe curvature— or repulsive matter d force ofequations quantum can geometry again grows be todensity used dilute is it. to again bounded gain above. A from physical a Singularity insights. more resolution general consideration. in In There t (BKL) is par a that conjecture says due that asterms one approaches containing a time space-like derivatives singular dominatedynamics over of those the gravitational cont fieldthe at dynamics any of one Bianchi spatial models’.the point BKL a By conjecture now and considerable itthat eviden is the widely singularity resolution believed in to the be Bianchito models essentiall showing in that loo all strong curvature, space-like singulari bouncing models can be and haveditions. been In constructed loop by quantum using cosmology, mat can by the contrast, matter theory fiel then evadecause singularity the quantum theorems geometry of effects Penrose,whence modify these H the theorems geometric, are le inapplicable.to However Borde, there are Guth m and Vilenkin which do tum geometry but within thedegrees of context freedom of are mini frozen. and The midi application superspa to the black h Introduction to LQG of state model, they tend to grow unboundedly. What is the situation i solutions violate a key assumption of these theorems as well or radiation matter density. Therefore, if anisotropies ar These theorems were motivated by inflationary scenariobeen and eternally t undergoing an expansion.potential, the In pre-bounce loop branch is quantum contracting. cos Thus agai PoS(QGQGS 2011)001 ust one is given N even by the Abhay Ashtekar huge 1. ) is accounted for by of large black holes? + , of the area operator in ) j ) 2 Pl . To probe its properties, 1 ℓ 4 + / j ( omagala, Meissner and others j hor o provide such an analysis. The end on non-gravitational charges? type of isolated horizon for this a the correct numerical coefficient First, while Wheeler’s ideas hold the elementary cells do not have p cosmological one in the de Sitter e Chern-Simons theory on a punc- y question in the subject has been: this entropy? This relation implies e total number of states al to the area of the isolated hori- ce of the isolated horizon boundary stic picture, which he christened ‘It = ( from classical general relativity and astrophysical interest which may be ∼ le considerations involving mapping ic sub-leading correction, whose co- ) The final result has two significant alysis from first principles? ls, each with one Planck unit of area, opic in a little more detail because it ences of quantum geometry which are es reside? To answer these questions, incorporate cosmological horizons for BH ’ in general relativity [41] and use it in uires a quantum horizon. Second, basic ny one cell is 2 m, litative features of Wheeler’s picture are S ualitative picture is simple and attractive. h arbitrary mass and angular momentum : i) one does not require near-extremality; quantum horizon quantum states, a number that is 16 4, one has to fix the Barbero-Immirzi parameter of 77 / is the number of elementary cells, whence entropy is given by ) is a half integer; ii) individual cells carry much more than j 2 Pl j /ℓ hor a = ( n . Thus, apart from a numerical coefficient, the entropy (‘It’ hor a where n ∼ 2 = N Why does this value of the Barbero-Immirzi parameter not dep Nonetheless, a careful counting of states by Lewandowski, D Ashtekar, Baez, Corichi and Krasnov used quantum geometry t As explained in the Introduction, since mid-seventies, a ke ln N and assign to each cell two microstates, or one ‘Bit’. Then th = 2 Pl loop quantum gravity, where ‘bit’ of information; the number of states associated with a has shown that thezon. number of To microstates get is theloop again exact quantum proportion numerical gravity factor to of a—e.g. 1 specific value. the One spherically can symmetricspace-time. use one a with Once specific zero the charge, valuein or of the the leading the order parameter contribution is formoments, isolated fixed, charge, horizons one etc. wit gets (Oneefficient also does obtains not a depend precise on logarithm differences the with Barbero-Immirzi respect parameter. to the stringone theory can calculations handle ordinary 4-dimensionaldistorted black and/or holes rotating; of direct and,which ii) thermodynamics one considerations also can apply simultaneously [12]. This important property can be traced back to a key consequen standards of statistical mechanics.in Where the do early all 1990s these Wheelerfrom stat had Bit’. suggested Divide the the following black heuri ℓ hole horizon into elementary cel for any 2-surface, the loop quantumfeatures gravity calculation of req both ofPlanck Wheeler’s area; arguments values undergo of their a area change: are dictated i) by the spectru borne out, geometry of a quantum horizon is much more subtle. What is the statisticalWhat mechanical are origin the of microscopic the degreesthat of entropy a freedom solar that mass account black for hole must have exp10 by S first step was to analyzeconjunction the to structure quantum of geometry ‘isolated to horizons one define has an to isolated combine thequantum isolated Riemannian horizon geometry boundary of conditions loop quantumtured gravity sphere, with the th theory ofclass a groups. non-commutative This torus detailed and analysis subt showed that, while qua Introduction to LQG tary in that one considers the full theory but probes consequ But can these heuristic ideas be supported by a systematic an not sensitive to full quantumwas dynamics. not covered I in any will of discuss the this main t lectures at this school. assigning two states (‘Bit’) to each elementary cell. This q PoS(QGQGS 2011)001 at terms: -manifold . Instead, one Abhay Ashtekar N space-time The challenge of quantum dy- ambiguities in the definition of est in the subject, thanks to the oncrete progress could be made enues have been pursued to con- ity based on coherent states [36]. n infinite number of Hamiltonian ically natural can nonetheless lead perators representing the Hamilto- nstraint equations and endow these ble quantum theories. However, in zon states. These techniques have quantum horizon geometry through l-understood for the Gauss and dif- nd diffeomorphism constraints [45]. enstein entropy depends only on the st is the ‘Master constraint program’ rom general relativity in completely re work is needed in the full theory. asenor, Agullo, Borja, Diaz-Polo and nt is due to Gambini, Pullin and their tes associated with matter. This pro- urface can be modeled by an isolated n the limited context of loop quantum ct interpretation in ctures have only volume terms. (Fur- uire detailed knowledge of how quan- resolved, the principal strength of the ely, we do not understand the physical philosophy is similar to that advocated the Hamiltonian constraint. Non-trivial ra, Kaul, Majumdar and others. er) charges. (For more detailed accounts antum dynamics: finding the physically tational part of the symplectic structure the value of the cosmological constant.) t space. The general consensus in the loop 17 theory, and, ii) restriction to large black holes implies th , each smeared by a so-called ‘lapse function’ x 3 full d ) x ( itself in an appropriate sense and then integrates it on the 3 S ) ) x x ( ( N S R )= N ( S Quantum Einstein’s equations in the canonical framework. Over the last three years there has been a resurgence of inter To summarize, as in other approaches to black hole entropy, c • The current status can be summarized as follows. Four main av . In simple examples, this procedure leads to physically via M appropriate scalar product on physical states. The general squares the integrand loop quantum gravity the procedurethe does Hamiltonian not constraint. remove Rather, anystrategy if of lies the in the its ambiguities potential are to complete the last stepby in John qu Klauder over theA years second in strategy his to approach solve to the quantum quantum grav Hamiltonian constrai Introduction to LQG conditions: detailed calculations show thathas only a surface the term gravi at the horizon; the matter symplectic stru thermore, the gravitational surface termConsequently, is there insensitive are to no independentvides surface a quantum natural sta explanation ofhorizon the area and fact is that independent of the electro-magneticof Hawking-Bek (or these oth results, see [38, 41].) impressive use of number theory techniquesto by sharpen Barbero, Vill and veryopened significantly new extend avenues the to further countingcontributions explore of by the Perez, hori microstates Engle, of Noui, the Pranzetti, Ghosh, Mit in loop quantum gravity because: i)tum the dynamics analysis does is not implemented req in the Hawking radiation is negligible, whencehorizon the [41]. black hole The s statesthey responsible refer for to entropy the have geometry of a the dire quantum, isolated horizon. namics in the full theoryphysical is states to with find the structure solutions of to anquantum the appropriate gravity Hilber quantum community co is thatfeomorphism while constraints, it the is situation far is fromdevelopment wel being due definitive to for Thiemann isnian that constraint well-defined exist candidate on o theHowever space there of are solutions several ambiguities to [38]meaning the and, of Gauss choices unfortunat a made to resolve them.cosmology has Detailed shown analysis that i choices which appearto to unacceptable be mathemat physical consequencestame such situations as with low departures curvature f [54]. Therefore, much mo struct and solve the quantumintroduced Hamiltonian by constraint. Thiemann The [45]. fir Theconstraints idea here is to avoid using a PoS(QGQGS 2011)001 intersect- Abhay Ashtekar rrive at spin-foam diffeomorphisms. (Until this h, as discussed below, provide ravity [38, 39, 45]. By mimick- an be used as ‘rods and clocks’ h furthermore has a potential of usain and Pawlowski have used In the mid 1990s Brown, Kuchaˇr hat the starting point in canonical the definition of the Hamiltonian ur years, there has been extensive ster Constraint program, issues as- latively new invariants of formulation of quantum mechanics, solutions to all constraints. Because tails can be found in the lectures by directions converge to the same type jan, Laddha, Henderson, Tomlin and h integrals can be used to restrict the spin foam program. Indeed, currently analyzed this issue especially in 2+1 cal theory. This is a very promising a diffeomorphism constraint, one has of loop quantum gravity. n of this theory. This work launched f loop quantum cosmology that lies at he Hamiltonian constraint one is led to e domain on which matter fields serve nd quantum gravity. As our knowledge iale, Baratin, Perini, Fairbairn, Bianchi, loop quantum gravity theories for these terized field theories. Now the emphasis nstraints in the classical theory. Giesel, rplay between quantum gravity and knot a closes, so that one is assured that there could provide increasingly significant in- f finding Dirac observables and makes it rmulation of quantum gravity which does infinitesimal diffeomorphisms in the kinematical setup.) Very simultaneous 18 finite Four different avenues to quantum gravity have been used to a Spin foams: In this discussion I have focused primarily on pure gravity. • The first avenue is the Hamiltonian approach to loop quantum g The third approach comes from spin-foam models [35, 39] whic knots discovered by Vassiliev. This is a novel approach whic work, the focus was onrecently, the this program action has only witnessed of promisingdefine advances. shares T qualitative features ofthe ‘improved foundation dynamics’ of o the most significant advances in that area. systems. Deparametrization greatly facilitates the task o the Poisson bracket between twoto Hamiltonian constraints find is a viable expression of the operator generating and Romano had introduced frameworksthereby in providing which a matter natural fields ‘de-parametrization’ c of the co these considerations as the point of departure to construct easier to interpret the quantumsociated theory. with However, quantization as ambiguities in stillas the good remain Ma clocks and and th rodsGiesel still and needs Salhmann. to be clarified. Further de ing the procedure that led FeynmanRovelli [1] and to a Reisenberger sum proposed over a histories the space-time spin-foam formulatio program. The second route stems from the fact t Introduction to LQG collaborators. It builds on their extensivetheory work [28]. on the The inte more recent of these developments use the re is no obstruction to obtaining a large number of Thiemann, Tambornino, Domagala, Kaminski, Lewandowski, H ing a path integral approachwork to in quantum this area, gravity. discussed in Overand the the Kaminski articles in last by this Rovelli, fo volume. Spez choice Transition amplitudes of from the pat Hamiltoniandirection constraint and operator Freidel, in Noui, the Perez,dimensions. canoni Rovelli The and idea others in the have others, fourth is approach, to due use insights to gained Varadara from the analysis of parame constraint by requiring that the quantum constraint algebr models (SFMs). The fact that ideasof from structures seemingly and unrelated models has provided athis strong is impetus the to most the active area on the mathematical physics side is on drastically reducing the large freedom in the choice of enhancing the relation between topologicalof field invariants theories of a intersecting knots deepens,sights. this approach In particular, it has thenot potential refer of even leading to to a a background fo manifold (see footnote 9). PoS(QGQGS 2011)001 Abhay Ashtekar nsional gravity that inspired implicial decomposition of the iven 2-simplex, the sum is also es or to take an appropriate ‘con- se there is a non-zero area gap— distinct paths to quantum gravity amental theory based on abstract ath integral approaches have been y, thanks to the generalizations by ould provide detailed dynamics in However, there does not yet exit a , and Freidel-Krasnov, put forward ecome an active research area (see, final result would be finite in either osely related to ‘vertex expansions’ use: i) there do exist semi-heuristic d to a spin foam. These three routes n a matrix. The Boulatov model was rett-Crane model which cured some theory [32, 35]. The well-controlled r the physically interesting values of imensions —the Boulatov model — rrett, Hellmann, Dowdall, Fairbairn, uses holonomies and discrete areas of e the correct classical limit; and, iv) o include a cosmological constant by -time geometry. Examples are matrix hole entropy considerations), the two s natural to begin with the path integral ometry using edge lengths and deficit ences. More recently, Fairbairn, Meus- nd avenue and led to the SFM of Barrett ourth avenue starts from approaches to o Einstein gravity, and in higher dimen- ensional gravity [39, 48]. The perturba- natural candidate is the topological B-F bove, it can be arrived at from four dif- ativity that emphasizes connections over mulation of gauge theories. Nonetheless from classical general relativity, say, along 19 The third route comes from the Ponzano-Regge model of 3-dime Four years ago, two groups, Engle-Livine-Pereira-Rovelli However, the issue of whether to sum over distinct 2-complex Introduction to LQG loop quantum gravity is a rewritingmetrics of [38]. classical general Therefore rel in the passage to quantum theory it i theory because in 3 space-time dimensions it is equivalent t ferent avenues; and iii) Detailed asymptotic analysis by Ba formulation of appropriate gauge theories. A particularly sions general relativity can bepath regarded integral formulation as of a the BF constrained theory BF and provided the Crane. seco Regge calculus in higherspace-time dimensions. manifold, describes Here its oneangles discrete begins and Riemannian constructs with a ge path a integral s loop in terms quantum of gravity them. in If place one ofare edge inspired lengths, by one various is aspects againquantum of le gravity general in relativity. which The gravity isstructures f to that, emerge to from begin a with,models more have fund nothing for to 2-dimensional do gravity with andwhere space the their basic extension object is to a 3-d further field generalized on to a a group group manifold field rathertive theory tha tailored expansion to of 4-dim this groupin field SFMs. theory Thus turned the out SFMs bemeet. lie very at Through cl a contributions junction of where manye.g., four researchers [35, apparently it 39]). has now b precise proposals for the sumloop quantum over gravity. quantum geometries The motivations thatthe were c Babero-Immirzi different parameter but (selected, fo e.g.,proposals by agree. the This black is anof improvement the over problems the faced earlier byKaminski, Bar that Kisielowski model. and Perhaps Lewandowski, more thebrought importantl canonical closer and to p one another:systematic ‘derivation’ they leading use to the this proposal same starting kinematics. considerations motivating the passage; ii) as I indicated a Pereira and others stronglyBecause indicates of that the use these of models quantumthis hav geometry sum —more over quantum precisely geometries becau has noberger, ultra-violet Han diverg and others havea extended these natural considerations use t ofinfra-red quantum finite. groups. It is then argued that,tinuum for limit’ a is g still debated and it is not known whether the the lines used in textbooks tothe arrive at program the has path attracted integral for a large number of researchers beca PoS(QGQGS 2011)001 to extract Hamiltonian Abhay Ashtekar he lectures by Rovelli, . Interestingly, under seem- nse of the extent to which tely many degrees of freedom, l-controlled: under a single as- spatial topology for definiteness, dels that has been analyzed in de- orating the inhomogeneous modes zero, the resulting quantum theory 3 ie algebra, where the aminski and Lewandowski. In loop nalog of the sum over 2-complexes s. Even if these gauge fixing strate- stem admit space-like foliations on till obtain a quantum theory tailored wdy models, this will likely involve mes. Finally, effective equations for while others provide challenges and T e by providing an illustrative list of o a non-perturbative and background ‘internal clock’ with respect to which inhomogeneous modes. If one were ase space from a well-established Hamiltonian sfully used in the Gowdy models by d in part by the success of loop quan- l have to be non-trivially generalized sless scalar field. This is a particularly take over this strategy to full quantum nd Romano showed that one can rear- s been shown to converge, and further- at the reader is familiar with basic ideas ngularity resolution (which we expect to ki at this school. purely perturbative super-string theory, where there is evidence rolled. 20 In section 2.2 I outlined four strategies that are being used as a relational time variable. With φ one can use is constant. Consequently, even though the system has infini In the cosmological mini-superspaces, the situation is wel φ Hamiltonian Theory: 10 Mathematically, this situation is somewhat reminiscent to Developments summarized so far should suffice to provide a se • A much more detailed discussion of spin foams can be found in t 10 constraints Poisson commute with each other oningly the mild entire assumptions ph one canwhich show that solutions of thisas sy in LQC one can decompose all fields into homogeneous and that each term in the expansion is finite but the sum is not cont Speziale, Baratin, Perini, Fairbairn, Bianchi, and Kamins 2.3 Challenges and Opportunities advances in loop quantum gravity already provideindependent an formulation avenue t of quantumthe gravity. open I issues. will Some of conclud opportunities these for are further currently work. driving This the discussion field assumes th quantum dynamics. I will now sketchtum another cosmology, avenue, that inspire has been proposed byquantum Domagala, cosmology, Giesel, a K massless scalar field oftenobservables serves as of an physical interest evolve [54].gravity by The focusing idea on is general to relativityinteresting coupled with system a because, mas already inrange the the 1990s, constraints Kuchaˇra of this system so that they form a true L to truncate the system bywould setting be the precisely inhomogeneous the modes loop to quantumtail cosmology by of Bianchi Ashtekar I and mo Wilson-Ewing.using One the might imagine ‘hybrid’ incorp quantization schemeMena, that Martin-Benito, has Pawlowski and been others, succes to although handle it the wil fact that theresome are gauge no fixing Killing of fields. the diffeomorphism Asgies and in do Gauss the not constraint work Go globally onto the a full ‘non-linear phase space, neighborhood’ one of should FLRWthis s or system Bianchi would also I provide space-ti valuable insights into the si Introduction to LQG case. behind current research in loop quantum gravity. 2.3.1 Foundations sumption that a sum and(called an integral the can vertex be expansion interchanged, in themore, the a spin converge to foam literature) the ha ‘correct’theory result [54]. that is already known PoS(QGQGS 2011)001 gnificant ad- Abhay Ashtekar e BF theory, they inherit e should actually sum over r fields and their couplings to g of the ‘vertex expansion’ that ndow for this analysis. A second y specific matter content. Recent sources are included, it continues gestion of Oriti that the coupling -complex). It is then natural to ask lysis by Ashtekar, Campiglia and 9, 48] but the physical meaning of ncorporated using quantum groups work by Varadarajan, Laddha, and avity. bly restricted. For scalar fields, in is analysis will enable one to place ular, is there a systematic physical it is now appropriate to invest time amline this analysis and provide the general relativistic singularity, found mber of terms? In group field theory l theory, Fairbairn, Meusberger, Han pproach to explore this issue in detail. couplings (beyond the quadratic term er the sum and experts in rigorous field le is invoked, the number of terms may . Recent work on group field theory by e carried out to a satisfactory conclu- ed that there is a precise sense in which xplain’ the triviality of such theories in coming from loop quantum gravity? e constraint algebra in quantum theory. roup approach has provided interesting ansion), or take an appropriately defined full theory. evidence for a non-trivial fixed point for it and finiteness of the sum over histories ion as a perturbation series in a parameter ar, one would be able to compare and con- 21 As discussed in section 2.2, the spin foam program has made si Spin foams: A second important open issue is to find restrictions on matte • First, we need a better understanding of the physical meanin Second, as I mentioned in section 2.2, the issue of whether on Finally, because the EPRL and FK models are motivated from th Introduction to LQG persist in an appropriate, well-defined sense).trast In their particul prediction with the simple BKL behavior near the by Andersson and Rendall forloop this quantum system. cosmology in More the generally, setting th of full loop quantum gr gravity for which thission. non-perturbative quantization Supersymmetry, can for b example,work is by known Bodendorfer, Thiemann to and allow Thurn only haspossibility opened ver is a suggested fresh by wi theWhen analysis of it the is closure extended of toTomlin th referred allow to for in matter section2.2 couplings, couldFinally, provide the as a recent mentioned promising a in sectionhints. 1.1, the Specifically, Reuter renormalization et g al have presented significant vacuum general relativity in 4 dimensions [47]. When matter to exist only when theparticular, Percacci matter and content Perini and have couplingsin found the are that action) suita are polynomial ruled out,Minkowski space-times an [42]. intriguing Are result there that similar may constraints ‘e vances over the last four years. Results on the classical lim for a fixed 2-complex are especially encouraging. Therefore and effort on key foundational issues. results when one sumsapproximation over that arbitrary lets 2-complexes. us terminate Ineach the term sum partic is after a multiplied finite bythis nu a coupling power of constant the in couplingHenderson space-time constant in terms [3 the is not cosmologicalconstant known. context is bears Ana related out to anand the early others cosmological sug have constant. shown that(which In the also cosmological the makes constant the ful spin can foam be sumif i infrared there is finite a for precise a sense fixed in 2 interpreting the vertex expans physically related to the cosmological constant also in the various 2-complexes (i.e., add up all terms in the vertex exp continuum limit is still open. Rovelli and Smerlak have argu the two procedures coincide. But so fartheory there have is expressed no the control concern ov that, unlessgrow a uncontrollably new as princip one increases theOriti, number of Rivasseau, vertices Gurau, Krajewski andnecessary others mathematical control. may help stre PoS(QGQGS 2011)001 ck- ng as they mes are at Abhay Ashtekar mportant role in the physics observations. It is important to ind. In the standard treatment of ceptual issues. logy and focuses just on a finite have been introduced [45]. These ans are expected to be manifestly solated in the polymer framework ation, normal ordered operators in ions. Here, a number of interesting es it is dictated by a Hamiltonian. d on an effective solution, we have pace-times are special? These con- e fixed background space-time, and e from infra-red issues which can be es are now being analyzed in detail erators? Can one ‘explain’ why the lable. Even in the ‘deparameterized orks with conformal or proper time, i and Lewandowski have shown that arization procedure routinely used in awn due attention in general relativity antum field theory in flat space-times, e variable. In quantum geometry state h normal ordering. For quantum field e fact that, in the inflationary scenario, this theory can be arrived at by starting an be systematically derived from loop o analyze the questions on the quantum ea due to Oriti that one should only sum metries; we do not have a single, well- ruction of viable semi-classical states of re put off by the idea. As a consequence, respond to general relativity. In addition, um gravity, dynamics is teased out of the 22 Because the initial motivations for inflation are not as stro In low energy physics one uses quantum field theory on given ba that plays the role of internal time; proper and conformal ti scalar field Low energy physics: The very early universe: • • Since quantum field theory in FLRW space-times plays such an i Introduction to LQG certain (‘Plebanski’) sectors which classicallyanalysis do of not cosmological cor spin foams re-enforces an early id over ‘time oriented’ quantum geometries.by Engle Some and of others. these But issu more work is needed on these basic con ground space-times. Therefore one is naturallyfrom led loop to quantum ask gravity if and makingchallenges systematic appear approximat to be within[38] and reach. elements of Fock quantum field statesdevelopments theory have lead on to been quantum concrete geometry i questions. Forthe example, Hamiltonian in qu and other operatorstheory are on regularized quantum throug geometry, onfinite the [45, other 38]. hand, Can the onethe Hamiltoni then Minkowski continuum show arise that, naturally inso-called from a Hadamard these suitable states finite approxim of op quantumsiderations field could theory also in provide curved valuable s hints for the const of the early universe, it isquantum especially gravity. important A to number know of ifquantum obstacles c fields immediately on come cosmological to space-times, m makes one a typically heavy w use of thediscusses causal dynamics structure as made an available unitary by evolutionof th in the loop chosen quantum tim cosmology, nonepicture’ of these it structures is are a avai best operators. Even when theonly quantum a state probability is distribution sharply for peake various space-time geo constraint while in quantumThese field obstacles theory seem in formidable at curvedthey first, space-tim can Ashtekar, be Kaminsk overcome ifnumber of one modes works of the with test spatially field.faced The compact later first while topo assumption the frees on second restrictiononly was a motivated by finite th numberremove of these restrictions modes and of use the perturbationsgravity resulting origin are framework t of relevant Hadamard to statescosmology. and of the adiabatic regul 2.3.2 Applications are often portrayed to be, several prominent relativists we quantum geometry. defined, classical causal structure. Finally, in loop quant recent developments in the inflationary paradigm have not dr PoS(QGQGS 2011)001 Abhay Ashtekar quantum cosmo- ich the inflationary under consideration k ∆ ne analyzes CMB inhomo- data. Numerical simulations ty imply the existence of tiny family illeteau, Grain and Mielczarek; field theory on background space-time together : two types: Particle Physics issues as made notable advances in fac- omy of thought it is nonetheless that is observed today. Although ig bang is replaced by a quantum t a phase of accelerated expansion ds describing perturbations were in tuation can be regarded as a classical um at the onset of inflation so as not ig bang singularity. For reasons dis- rable interest to attempt to provide a adigm seriously: it predicted the main gime and the corresponding extension ound (CMB) which were subsequently phase of slow roll inflation compatible e small deviations which could be ob- ly relevant wave numbers are in a finite he large scale structure of the universe antum state at the bounce, when evolved tivity in domains where it is not applica- eral relativity and theorems due to Borde, re. Will a generic homogeneous, isotropic ons. y, if we evolve the Fourier modes of interest 23 to adequately handle them. . Using this fact, we can write down the four assumptions on wh k ∆ Let me first explain this point in some detail. Note first that o The issues that are left open by this standard paradigm are of Then QFT on FLRW space-times and classical general relativi Recent work by Ashtekar, Sloan, Agullo, Nelson; Barreau, Ca Introduction to LQG circles. In my view, there is a compelling case to take the par features of inhomogeneities in the cosmic microwaveobserved backgr and which serve as seeds for structure formation. geneities in terms of their Fourierrange, modes say and observational scenario is based: 1) Some time in itsduring which very the early Hubble history, parameter H the2) was universe During nearly underwen constant. this phase thewith linear universe perturbations. is well-described by3) a A FLRW few e-foldings before the longest wave length mode in the exited the Hubble radius, thesethe Fourier Bunch-Davis modes vacuum. of quantum fiel 4) Soon after a mode exited theperturbation Hubble radius, and its evolved quantum via fluc linearized Einstein’s equati inhomogeneities in CMB which have beenshow seen by that the these 7 year seeds WMAP the grow assumptions to are yield by theimpressive. no large means In scale particular, compelling, structure in thelies this overall just paradigm, econ in the vacuum origin fluctuations!quantum of gravity t completion Therefore, of it this is paradigm. of conside ble. Therefore, one needs a viable treatment of the Planck re and Quantum Gravity issues. Let me1) focus Initial on singularity: the The second paradigm assumes for classical now gen cussed in section 1, this is an artifact of using generalof rela the inflationary paradigm. 2) Probability of inflation:bounce. In So it loop is quantum natural to cosmology,initial introduce the state initial for b conditions the the background, whenwith evolved, the encounter seven a year WMAP data? 3) Trans-Planckian issues: In classical generalback relativit in time, they become trans-Planckian. We need a quantum to contradict the observations.served More in future importantly, missions? are ther and Fernandez-Mendez, Mena-Marugan,Olmedo and Vehlino, h Guth and Vilenkin then imply that space-time had an initial b forward in time agrees sufficiently with the Bunch Davis vacu logical space-times 4) Observations: The question then is whether the initial qu PoS(QGQGS 2011)001 Abhay Ashtekar ck hole to shrink background space-time; fixed their interaction? What are the ntum gravity completion of the ctured by Hawking, is shown in ue is avoided and quantum field se from the bounce to the surface oration. It also led to some new ns now have physical frequencies on a arities with the general loop quan- aporation process and shed light on y analytical tools to extract physics try is nearly flat and quantum fields s well described by a homogeneous on of the quantum horizon geometry hase of the evaporation. The natural ne must use to get a sufficiently long digm and potentially observable pre- s. This fact, coupled with the absence he expression of the entropy is again ulations is that perturbations originate s: i) Hawking’s original calculation of ince the general paradigm has several cillates around the minimum of the po- tate. This idea that the entire evolution l general relativity. This changed with , Brandenberger and others have shown or other research that will provide both until relatively recently, there were very shown that the first law can be extended out having to assume that non-linearities ics will still remain. In particular: What ctrum of scalar and tensor modes in the Black hole thermodynamics was initially elating modes in the pre-bounce epoch to o investigate whether loop quantum cos- Therefore, it would be healthy to look also hat have been advocated by Barndenberger smaller than that in the inflationary scenario, 24 much [41]. Hawking radiation will cause the horizon of a large bla 2 Pl ℓ 4 / hor a Black hole evaporation: The issue of the final state: below the Planck scale. Therefore, the trans-Planckian iss slowly, whence it is reasonable to expect that the descripti • The space-time diagram of the evaporating black hole, conje Finally, even if loop quantum gravity does offer a viable qua given by to these time-dependent situations and the leading term in t the advent of dynamical horizonsfrom which numerical provide simulations the necessar ofinsights black on the hole fundamental formation side. in and particular, evap it was also can be neglected throughout the pre-bounce phase. few analytical results on dynamical black holes in classica much theory in curved space-times should beof viable for singularity, enables these one mode to calculatethose a in transfer the matrix post-bounce epoch. r Under suitablethat assumptions this relation gives risepost-bounce to phase. a But nearly the scale underlying premisein invariant in spe the these distant calc past ofrepresenting the perturbations contracting are branch taken where to be thefrom in geome the their distant vacuum past s inmodel the with contracting small phase perturbations to isattractive the not features, bounce at it i all would realistic. bemology of bounces But allow considerable s similar alternatives interest to t inflation with developed in the context of stationary black holes. Indeed, the left-hand drawing in Fig. 1.black It hole is radiance, based in on two the ingredient framework of quantum field theory at the bounce, modes of direct interest to the CMB observatio slow roll? Is therecouplings only that one produce inflaton the known or particles many? astential the at If inflaton the os many, end what of inflation are (thefor so-called alternatives ‘reheating’)? to inflation. Indeed, in the alternatives t tum cosmology paradigm. Becauseof the last expansion scattering of in the the univer post-bounce branch is very can be extended from isolated toquestion dynamical then horizons is: in Can this p onethe issue describe of in information detail loss? the black hole ev is the physical origin of the inflaton field? Of the potential o and others involve bouncing models and therefore have simil a viable quantum gravity completiondictions. of the inflationary para inflationary paradigm, open issues related to particle phys Introduction to LQG ing these questions [54] but there are ample opportunities f PoS(QGQGS 2011)001 on not Abhay Ashtekar to persist in the full quantum 0 i this argument is incorrect in two roduces a domain in which there is analysis is not as complete or n is teleological and refers to the s in full quantum theory. st. If so, the space-time does ravity effects during the last stages e context of loop quantum gravity aningful; it is simply ‘transcended’ s a very long time for a large black + ive conclusion that the singularity is on of this semi-classical picture is its J e black hole has shrunk to Planck size ty which serves as a new boundary to mation. Intuitively, the lost information ions that have been lost due to thermal − al process depicted in this figure should hat the radius of the event horizon must t just at the end point of evaporation but . Using ideas from quantum cosmology, he singularity resolution in LQC. What forms J awking. Information is lost because part of the I H 25 of this larger space-time.



























− I . Quantum space-time is larger and the incoming information H falls into the part of the future singularity that is assumed − + J I − J is adequately recovered on the Conjectured space-time diagrams of evaporating black hole − structure of space-time. Resolution of the singularity int all along what is depicted as the final singularity I However, loop quantum gravity considerations suggest that















































































the interior of theby Schwarzschild Ashtekar, horizon Bojowald, was Modesto, analyzedrefined Vandersloot in as and th that others. in theresolved Th cosmological due context. to quantum Buthave geometry the a effects qualitat singularity is as likely itsdepiction to final of be boundary. the robu The event secondglobal horizon. limitati The notion ofis no an classical event space-time, horizo whence the notion ceases to be me in fact (a) Left figure: Standard paradigm, originally proposed by H gravity theory. (b) Right figure:and New evaporates paradigm is motivated by a t dynamicalthe horizon full and quantum gravity effects becomehole important. to shrink Since to it this take of size, evaporation one will then not argues beradiation that sufficient over the such to quantum a restore long g the period. correlat Thusis there ‘absorbed’ is loss by of the infor ‘left-overspace-time. piece’ of the final singulari respects. First, the semi-classical picture breaks down no and ii) heuristics of back-reaction effects which suggest t incoming state on shrink to zero. It isbe generally reliable argued until that the the very semi-classic late stages of evaporation when th Figure 1: Introduction to LQG PoS(QGQGS 2011)001 n ing? Abhay Ashtekar -point functions in a n descends from the boundary spects and their full implica- is to develop it into a detailed r on a firm footing by Ashtekar, d FK models. As Perini’s talks at that the 2-point function falls off lli and others, this issue has been ere isn’t a canonical one represent- cially because its action and equa- alyze the mean field approximation on assumptions regarding effects of l relativity fails if the theory exists oles first introduced by Callen, Gid- tance setting. The very considerable work f the sub-leading terms. These terms l, spherically symmetric black holes rd perturbation theory in Minkowski of considerable interest to extend all ed into a detailed theory of quantum ve non-trivial led theory of black hole evaporation ity of the extended space-time. Thus, Bojowald introduced a new paradigm kar, Pretorius and Ramazanoglu have lin, Bojowald and others will serve as uate. d in the right hand drawing of Fig. 1: gliaro and Perini strongly indicate that, functional form and tensorial structure on of quantum radiation, and the initial ssical considerations would not simply re incorrect. This analysis further rein- s and under certain approximations. But rther research. Indeed the key challenge eds a better handle on contributions from t function brought out some limitations of the 26 point functions and scattering, that lie at the heart of Spin foam models provide a convenient arena to discuss n -point functions, one needs to introduce a boundary state (i n between the two points has no diffeomorphism invariant mean r when the distance Contact with low energy physics: n r • However, these calculations can be improved in a number of re However, this is still only a paradigm and the main challenge / which the expectation values are taken) and the notion of dis forces the paradigm of the figure on the right. It is therefore issues such as the graviton propagator, theory. Just as Wheeler’s ‘Ithorizon from geometry, Bit’ it ideas should were bebased transform possible on to this construct paradigm. aTaveras and detai Varadarajan More in recently, the case thisdings, of paradigm Harvey 2-dimensional was black and h put Strominger.tions The of model motion is closely interesting mimicformed those espe by governing the 4-dimensiona gravitational collapseused of a a combination scalar of field. analyticalin and Ashte complete numerical detail. methods to It an explicitlyincluding shows back that reaction, some discussed of in the the comm last paragraph, a this analysis to four dimensionson in spherically the symmetric loop midi-superspaces quantum by gravity Gambini,a Pul point of departure for this analysis. perturbative treatments. Atdiffeomorphism first, invariant it theory. seems impossible Indeed,as to how 1 ha could oneThanks say to a carefulsatisfactorily conceptual resolved. set-up To by speak Oeckl, of Colosi, Rove state. Interestingly, a detailed calculation of theBarrett-Crane 2-poin model and provided new impetusthis for school showed, the these EPRL calculations an by Bianchi,to Ding, the Ma leading order, a gravitonwill propagator arise with from the these correct models. tions have yet to be properly digested.2-complexes In with particular, large one numbers ne of verticesseem and to the be physics o sensitive toing the Minkowski choice space. of Therefore, the comparison boundaryspace with state is the and difficult. standa th This is a fertile and important area for fu Introduction to LQG in quantum theory. Using these considerations Ashtekar and in this area is to ‘explain’ why perturbative quantum genera for black hole evaporation inNow, loop it quantum is the gravity, dynamical depicte horizon thatpure evaporates with state emissi evolves to a finalthere pure is state no on information the loss. futuredismissed; null In they infin this would be paradigm, valid the in certain semi-cla space-time region for fundamental conceptual issues, they would not be inadeq PoS(QGQGS 2011)001 fixed Abhay Ashtekar non-Gaussian evel, there is already ake the basic ideas seriously and round space-time. Loop quantum t because, as developments in the es, which were not present in the possibility is to use the ‘emergent theory has two a priori elements: quantum gravity has witnessed sig- he n-point functions on the choice t would be a mistake if a significant perfluids and find rotons; consider s every few months, making a first hysics derived from loop quantum in absence of a detailed theory, the d technical open issues. Examples nalyzing the renormalization group ults in the main stream development calculate S-matrix from spin foams gravity, the micro-state representing eral relativity is the same as in gauge ctions that have already been opened to interesting dynamics that includes e of geometry, peaked at Minkowski approximation even below the Planck quantum geometries to sum over, and ions. But the non-trivial issue is that trings. Excitations of these strings, in -scale structure. The basic entities will of a non-trivial vacuum. For example, ns and applications. It is therefore im- al fields. It may also serve as a bridge ntum gravity. ch must be woven to create the classical pond not only to gravitons but also to r explorations of the role of supersym- nificantly narrow the ambiguities in the lly the failure can be traced back to the n anti-symmetric tensor. These ‘emergent malization group has a 27 Finally, there is the issue of unification. At a kinematical l Unification. • From examples discussed in this section it is clear that loop Introduction to LQG non-perturbatively. As mentioned in sectioninsistence that 1, the heuristica continuum space-time geometry is a good definition of the Hamiltonian constraint;metry initiated extending by furthe the Erlangen group; sharpening the set of scale. But a moreasymptotically detailed safe answer scenarios indicate is [47, needed. 52], thepoint? renor For example, is i an unification because the quantum configuration spacetheories of which gen govern the strongof and dynamics. electro-weak To interact conclude,phenomena’ let us scenario consider where a new speculation. degreesinitial of Lagrangian, One emerge freedom when or one particl one considers can excitations begin with solidssuperconductors and and arrive discover cooper at pairs. phonons;Minkowski In start space-time loop will with have quantum su a highlybe non-trivial Planck 1-dimensional and polymer-like. Onefluctuations can of argue these that, even 1-dimensionalother entities particles, should including a corres spin-1 particle,states’ a are scalar and likely a togravity. play A an detailed important study role ofnot in these only Minkowskian excitations gravity may p but well also lead between a loop select family quantum of gravity non-gravitation unexcited and strings string which carry theory. nogravity quantum For, suggests numbers string and that a both backg could(or, arise de Sitter) from space. the The quantum polymer-like quantumground stat threads state whi geometries could beturn, interpreted may as provide unexcited interesting matter s couplings for loop qua 2.3.3 Some Final Remarks nificant advances over the last decade,portant both for the in community its to foundatio makeup. sustained Of progress course on one dire constantly needsfraction an influx of of the new ideas. community But focusesstab i on and then constructing passing new on model to theof next loop model. quantum The cumulative gravity res nowcontinue carry to sufficient develop weight them for byof us attacking such the to issues hard t are: conceptual Finding an principles and strategies to sig addressing the problem of convergence inflows spin in foam group models; field a theory;of understanding the the boundary dependence state; of developing t approximation methods to PoS(QGQGS 2011)001 , 20 Abhay Ashtekar . Developing for- Gravitation: An Relativity, Groupos and School. Parts of this overview ned through discussions with a ry, Holst action and asymptotic theories that rose to prominence thematics underlying loop quan- ral relativity, in ) cience Foundation through its net- ore, in this work, it is important to R l state terature on spin networks, quantum ads into other areas of mathematical will rest. reidel and Fleischhack are applying ( at the subject is now sufficiently ma- lo Rovelli for their comments on the nversely, some loop quantum gravity cedures. But it is not an end in itself. f cle [46]. I would like to thank Miguel op quantum gravity and loop quantum ionary paradigm in the Planck regime; fail; fully incorporating matter fields in verse; ... The list is long enough to keep er, Dahlen, Aastrup and Grimstrup from direct physical interest ng standing physical questions. mechanics, non-commutative geometry, . In my view, this ‘outward bound’ spirit ons, it is important to focus on problems tive field theories to adequately describe ples over the space of generalized connec- quantum mechanics, Rev. Mod. Phys. 28 , NSF grant PHY 0090091 and the Eberly research , (Academic Press, New York) ed Witten L (John Wiley, New York) Geometrodynamics eds DeWitt C M and DeWitt B S (Gordon and Breach, New York) Quantum Geometry and Quantum Gravity 367-387 (1948) introduction to current research Topology My understanding of classical and quantum gravity has deepe Finally, there is a complementary direction. Because the ma [1] R. P. Feynman, Space-time approach to non-relativistic [2] Arnowitt R, Deser S and Misner C W 1962 The dynamics of gene [3] Wheeler J A 1962 [4] Wheeler J A 1964 Geometrodynamics and the issue of the fina ture to have applications to other areas. In these explorati gauge theories and geometry. Thus there is ample evidence th that other communities consider as important inis their the areas second pillar on which further development of the field References tum gravity is rigorous, the subjectphysics is and now begun mathematics to itself. maketopology inro For and example, computing, there spin is foamssafety, now loop and quantum li non gravity and commuting topos geomet theory, spectraltions tri etc. from researchers like Kauffmann,outside Marcolli, Reut the traditional loop quantumresearchers gravity such community. as Co Ma,techniques Gambini, developed Pullin, in the Singh, field Dittrich,in to cosmology, F other generalized areas quantum such mechanics, as statistical the large number of colleagues; manyare of updated whom versions were of at some the ofCampiglia, Zakopane the Kristina material Giesel, in Alok author’s Laddha arti manuscript. and This especially work Car was supported inwork part by the European S Acknowledgements: funds of The Pennsylvania State University. spin foams, particularly scalarlow fields; energy constructing physics; effec finding the detailedcosmology; relation constructing between a lo detailed completionexploring of its the observable inflat consequences in the veryyoung early researchers uni busy and happykeep for focus quite on a physical while! issues and Furtherm try to solve problems of Introduction to LQG and pin-pointing why the standard perturbative treatments malism is important because itIndeed, streamlines it the is ideas of and little pro use unless it leads to answers to the lo PoS(QGQGS 2011)001 fter 51–139 (Cambridge 71 Gravitation and Abhay Ashtekar (Cambridge UP, 1211-1256 (Cambridge UP, Isham C J, Penrose ss 37 Phys. 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