Quantum Gravity: a Brief Review of the Past, a Selective Picture of the Present, a Glimpse of the Future

Quantum Gravity: A Brief Review of the Past, a Selective Picture of the Present, a Glimpse of the Future

Daniele Oriti Arnold Sommerfeld Center for Theoretical Physics Munich Center for Mathematical Philosophy LMU-Munich, Germany, EU Conference “Beyond the Standard Model: Historical-Critical Perspectives” Galileo Galilei Institute, Florence, Italy, EU 21.10.2019 Quantum Gravity: very much beyond the Standard Model one thing clearly missing from Standard Model, making it intrinsically incomplete: gravity!

but incorporating gravity in the quantum framework of Standard Model is not just like adding a new particle or a new interaction…..

….it means revising drastically our very notions of space and time, and the very foundations of our description of the universe Why we need to go beyond GR and QFT

two incompatible conceptual (and mathematical) frameworks for space, time, geometry and matter

GR QFT spacetime (geometry) is a dynamical entity itself spacetime is fixed background for fields’ dynamics there are no preferred temporal (or spatial) directions evolution is unitary (conserved probabilities) with respect to a given (preferred) temporal direction physical systems are local and locally interacting nothing can be perfectly localised everything (incl. spacetime) evolves deterministically everything evolves probabilistically all dynamical fields are continuous entities interaction and matter fields are made of “quanta” every property of physical systems (incl. spacetime) and of their interactions can be precisely determined, in every property of physical systems and their principle interactions is intrinsically uncertain, in general

so, what are, really, space, time, geometry, and matter? Why we need to go beyond GR and QFT

several open physical issues, at limits of GR and QFT or at interface (where both are expected to be relevant) • breakdown of GR for strong gravitational ﬁelds/large energy densities

spacetime singularities - black holes, big bang - quantum effects expected to be important

• divergences in QFT - what happens at high energies? how does spacetime react to such high energies?

• what happens to quantum ﬁelds close to big bang? what generates cosmological ﬂuctuations, and how? Why we need to go beyond GR and QFT

several open physical issues, at limits of GR and QFT or at interface (where both are expected to be relevant) • breakdown of GR for strong gravitational ﬁelds/large energy densities

spacetime singularities - black holes, big bang - quantum effects expected to be important

fermionic fields are added, one must generalize GR to the Einstein–Cartan theory or to the Poincarégauge theory, because spin is the source of torsion, a geometric quantity that is identically zero in GR (see e.g. Gronwald and Hehl 1996). As one• recognizesdivergences from in QFT (2), - what these happens equations at high can energies? no longer how have does ex spacetimeactly the react to such high energies? same form if the quantum nature of the fields in Tµν is taken into account. For then we have• what operators happens to in quantum Hilbert fields space close on theto big right-hand bang? what side generates and c lassicalcosmological fluctuations, and how? functions on the left-hand side. A straightforward generalization would be to replace T by its quantum expectation value, • noµν proper understanding of interaction of geometry with quantum matter, if gravity is not quantized 1 8πG R g R + Λg = Ψ Tˆ Ψ . not a consistent(4) theory µν − 2 µν µν c4 ⟨ | µν| ⟩ These ‘semiclassical Einstein equations’ lead to problems when viewed as exact equations at the most fundamental level, cf. Carlip (2008) and the references therein. They spoil the linearity of quantum theory andeven seem to be in conflict with a performed experiment (Page and Geilker 1981). They may nevertheless be of some value in an approximate way. Independent of the problems with (4), one can try to test them in a simple setting such as the Schrödinger–Newton equation; it seems, however, that such a test is not realisable in the foreseeable future (Giulini and Grossardt 2011). This poses the question of the connection between gravity and quantum theory (Kiefer 2012). Despite its name, quantum theory is not a particular theory for a particular interaction. It is rather a general framework for physical theories, whose fundamental concepts have so far exhibited an amazing universality. Despite the ongoing discussion about its interpretational foundations (which we shall address in the last section), the concepts of states in Hilbert space, and in particular the superposition principle, have successfully passed thousands of experimental tests. It is, in fact, the superposition principle that points towards the need for quantizing gravity. In the 1957 Chapel Hill Conference, Richard Feynman gave the following argument (DeWitt and Rickles 2011, pp. 250–60), see also Zeh (2011). He considers a Stern–Gerlach type of experiment in which two spin-1/2 particles are put into a superposition of spin up and spin downand is guided to two counters. He then imagines a connection of the counters to a ball of macroscopic dimensions. The superposition of the particles is thereby transferred to a superposition of the ball being simultaneously at two posi- tions. But this means that the ball’s gravitational field is in a superposition, too! In Feynman’s own words (DeWitt and Rickles 2011, p. 251): Now, how do we analyze this experiment according to quantum mechanics? We have an amplitude that the ball is up, and an

Why we need to go beyond GR and QFT

hints of disappearance of spacetime itself, more radical departure from GR and QFT

• challenges to “localization” in semi-classical GR minimal length scenarios

• spacetime singularities in GR breakdown of continuum itself?

• black hole thermodynamics black holes satisfy thermodynamic relations if spacetime itself has (Boltzmann) entropy, it has microstructure if entropy is finite, this implies discreteness • Einstein’s equations as equation of state (Jacobson et al) GR dynamics is effective equation of state for any microscopic dofs collectively described by a spacetime, a metric and some matter fields

fundamental discreteness of spacetime? is spacetime itself “emergent” from non-spatiotemporal, non-geometric, quantum building blocks (“atoms of space”)?

Why we need to go beyond GR and QFT

if spacetime (with its continuum structures, metric, matter ﬁelds, topology) is emergent,

even large scale features of gravitational dynamics can (and maybe should) have their origin in more fundamental (“atomic”) theory cannot trust most notions on which effective quantum ﬁeld theory is based (locality, separation of scales, etc)

e.g. : dark matter (galactic dynamics), dark energy (accelerated cosmological expansion) - either 95% of the universe is not known, or we do not understand gravity at large scales

e.g. cosmological constant as possible large scale manifestation of microscopic (quantum gravity) physics What has to change (in going from GR to QG)

• quantum ﬂuctuations (superpositions) of spacetime structures

• geometry (areas, distances, volumes, curvature, etc)

• causality (causal relations)

• topology?

• dimensionality?

• breakdown of continuum description of spacetime?

• fundamental discreteness? of space? of time?

• entirely new degrees of freedom - “atoms of space”?

• but then, how does usual spacetime “emerge”?

• new QG scale: Planck scale

no spacetime or geometry? how can we even talk of “scales”? total failure of effective ﬁeld theory intuition? Quantum Gravity: what happened so far

(years between 1950-2005) General strategy being followed:

quantise GR, adapting and employing standard techniques

diﬀerent research directions are born, corresponding to diﬀerent quantization techniques:

perturbative quantization, canonical quantization, covariant (path integral) quantization

all achieve key insights

all get stuck and die of starvation (or are maintained alive in a vegetative state) Quantum Gravity: (covariant) perturbative quantization

DeWitt (1950), Gupta (1952): general formulation of perturbative quantization metric perturbations

gµ⌫ = ⌘µ⌫ + hµ⌫

ﬂat metric (Minkowski) background metric provides notion of space, time and causality

linear diffeomorphisms are gauge symmetry (background breaks full symmetry)

metric perturbations are quantized analogously to other gauge interactions

“gravitons”: massless, spin-2 quanta of perturbative gravitational ﬁeld

Feynman, DeWitt,… (1962-1967, …): tree-level scattering amplitudes, 1-loop corrections to. Newton’s law, background-ﬁeld method, unitarity, gauge-ﬁxing, ghosts, ….

’t Hooft,Veltman, …., Goroff, Sagnotti (1971-1986): divergences, non-rinormalizability without and with matter

proposed possible solutions: a) add new physical ingredients (new matter, new symmetry), b) modify gravitational dynamics, c) quantise non-perturbatively Quantum Gravity: canonical quantization

Bergmann, Dirac (1950-1959): canonical quantization of (constrained) gauge systems

Arnowit, Deser, Misner (1961): Hamiltonian formulation of General Relativity, diffeomorphism constraints

Bergmann-Komar, Peres, DeWitt, Wheeler (1962-1967): canonical quantum gravity in ADM variables

kl h (x),K (x0) (x x0) Hi(hij,Kkl)=0 ij / ik jl H(hij,Kkl)=0 spatial 3-metric extrinsic curvature

invariance under spatial and temporal diffeomorphisms (encode whole dynamics) quantum level: (h ) ij 2 H H h , (h )= 0 H h , (h )=0 i ij h ij ij h ij ✓ kl ◆ ✓ kl ◆ c b Wheeler, DeWitt, Teitelboim, Kuchar, Isham…. (1967-1987, …): properties of “superspace of 3-geometries”, problem of time, scalar product on quantum states, quantum cosmology, lots of semiclassical analyses, …. formalism too ill-deﬁned at mathematical level to constitute solid approach to QG (beyond semi-classical or “in-principle” analyses) Quantum Gravity: covariant path integral quantization

Misner, Wheeler,… (1957-): idea of sum-over-histories formulation of QG, non-perturbative transition amplitudes (and scalar product) between QG states via sum over spacetime geometries

Wheeler (1963) suggests to deﬁne it via discrete lattice (Regge) regularization —-> quantum Regge calculus S h 2 transition amplitude (or scalar product) 2 from one 3-geometry to another iS (g) h H =0 h h = ge M | i2H | i h 1| 2i D Zh1,h2 M g

probability amplitude for each sum over spacetime “history” (4-geometry), depending 4-geometries on GR action (or modiﬁed one) h S 1 1

Hawking, Hartle, Teitelboim, Halliwell,… (1978-1991, …): Euclidean continuation, covariant (no-boundary) deﬁnition of “wave function of the universe”, relation to canonical theory, implementation of diffeomorphism symmetry, covariant quantum cosmology, lots of semi-classical applications, …….. formalism too ill-deﬁned at mathematical level to constitute solid approach to QG (beyond semi-classical or “in-principle” analyses)

many results also within quantum Regge calculus (Rocek, Sorkin, Williams, Hamber, ….) main lessons

a) Quantum gravity is perturbatively non-renormalizable, as gµ⌫ = ⌘µ⌫ + hµ⌫ a QFT for the metric ﬁeld (e.g. around Minkowski space)

can stillrelativity be used as effective field theory (incorporating quantum (loop) corrections) with fixed cutoff

4 2 2 µν Sgrav = d x√g Λ + R + c1 R + c2 R R + ...+ Lmatter (26) ! " κ2 µν # J. Donoghue, C. Burgess, ….. Here the terms haveand it zero, is predictive two and (eg four graviton derivatives scattering respectiv and correctionsely. to Newtonian potential) it has toFollowing be somehow our reproduced EFT script, from we more turn tofundamental experiment theory, to determine which shouldthe parameters also explain of its this failure Lagrangian. The first term, the cosmological constant, appears to be non-zero but it is so tiny that it is not relevant on ordinary scales. The EFT treatment does not say b) we haveanything template novel (general about quantum the smallness structure, of the implementation cosmological of cons symmetries,tant - it is non-perturbative treated simply as(phase) transitionsan experimental between geometries, fact. The etc) next of term full non-perturbative is the Einstein action,theory in w theith coefficientcontinuum, determinedwhich should be realised concretely by more fundamental theory, to the extent in which continuum picture holds from Newton’s constant κ2 = 32πG.Thisistheusualstartingplaceforatreatment we haveof generalwell-defined relativity. list of conceptual The curvature issues squared (concerning terms time, yield space, effects causality, that are semi-classical tiny on normal limit, interpretationscales if of the quantum coefficients mechanics,c1,2 are etc) of that order need unity. to be In addressed. fact these are for boundedunderstanding by experiment and use of full QG [13] to be less than 10+74 -theridiculousweaknessofthisconstraintillustratesjust we havehow several irrelevant suggestions these terms of QG are corrections for normal to classical physics. phenomena So we see (also that generalnon-perturbative) coordinate invariance allows a simple energy expansion. c) At times people worry that the presence of curvature squared terms in the action will we havelead learned to instabilities how hard or is pathological the Quantum behavior. Gravity problem, Such poten mathematically,tial problems physically, have been conceptually shown to only occur at scales beyond the Planck scale [14] where yet higher order terms are also equally important. This is not a flaw of the effective fieldtheory,whichholdsonly below the Planck scale. Given the assumption of a well-behaved full theory of gravity, there is no aspect of the effective theory that needs to display a pathology.

Quantization and renormalization

The quantization of general relativity is rather like that ofYang-Millstheory.There are subtle features connected with the gauge invariance, so that only physical degrees of freedom count in loops. Feynman[15], and then DeWitt[16],didthissuccessfullyin the 1960’s, introducing gauge fixing and then ghost fields to cancel off the unphysical graviton states. The background field method employed by ’tHooft and Veltman[17] was also a beautiful step forward. It allows the expansion about a background metric (¯gµν )andexplicitlypreservesthesymmetriesofgeneralrelativity. It is then clear that quantization does not spoil general covariance and that all quantum effects respect this symmetry. The fluctuation of the metric around the backgroundisthegraviton

gµν (x)=g¯µν (x)+κhµν (x) (27)

and the action can be expanded in powers of hµν (x) (with corresponding powers of κ). The Feynman rules after gauge ﬁxing and the addition of ghostshavebeengivenin several places[1, 2, 17] and need not be repeated here. They are unremarkable aside from the complexity of the tensor indices involved. Renormalization also proceeds straightforwardly. As advertised, the divergences are local, with the one loop effect being equivalent to[17] 1 2 1 7 ΔL = R2 + R Rµν (28) 16π2 4 d "120 20 µν # − The decade closes with the main lines of the covariant and the canonical theory clearly deﬁned. It will soon become clear that neithertheoryworks.

6TheMiddleAges:1970-1983

1970 The decade of the seventies opens with a world of caution. Reviving a point made by Pauli, a paper by Zumino [35], suggests that the quantization of GR may be problematic and might make sense only by viewing GR as the low energy limit of a more general theory. 1971 Using the technology developed by DeWitt and Feynman for gravity, t’Hooft Other new thingsand Veltman we learned decide (from tosemi-classical study the renormalizability gravity) that are here of GR.to stay Alm (forost QG) as a warm up exercise, they consider the renormalization of Yang-Mills theory, and find Spacetime singularities that the theory is renormalizable – resultHawking, that Penrose, has wonGeroch, them….. this year Nobel breakdown of prizeGR for [36].strong Ingravitational a sense, fields/large one can energy say that densities the - firstinevitable physical in classical resu GRlt of the research center of blackin holes, quantum big bang gravity - quantum is effects the proof expected that to be Yang-Mills important theory is renormalizable. 1971 David Finkelstein writes his inspiring “spacetime code” series of papers [37] (which, among others ideas, discuss quantum groups). 1973 Following the program, t’Hooft finds evidence of un-renormalizable diver- gencesorigin in GR of with the black matter hole fields. (the Shortly Bekenstein-Hawking) after, t’Hooft and Ve entropy.ltman, asFor well a as Schwarzshild Black hole thermodynamicsDeserblack and hole, Van Nieuwenhuizen, this is confirm the evidence [38]. 3 Bekenstein, Bardeen,1974 Carter, Hawking (1973): a notion of entropy 1 c can be formally associated to black holes, and laws of black hole S = A (2) mechanics recast inHawking the form of announcesblack hole thermodynamics the derivation of black4 hole¯hG radiation [39]. A (macro- why finite? whyscopically) holographic? Schwarzshild black hole of mass M emits thermal radiation at the where A is the areadue to of which the microstructure? black hole surface. An influential, clarifying and if of Boltzmanntemperature type, at the same time intriguing paper¯hc3 is written two years later byBillUnruh. Hawking (1974): black holes emit thermal radiation, T = with temperature proportionalThe paper to horizon points curvature out the existence8πkGM of a general relation between accelerated Theobservers, result comes quantum as a surprise, theory, anticipated gravity only and by thermodynamics the observation by [42 Beken-]. Something deep BHs evaporate stein,awayabout …. a yearto nature earlier, should that entropy be hidden is naturally in this associated tangleto of black problems, holes, and but thuswe do not yet signals violation of some basic principle become what? what happens theyknow could what. be thought, inof spacetime some obscure physics sense, (unitarity? as “hot”locality? [40], and by the Bardeen- to information content?Carter-Hawking analysis of the analogy between laws of thermodynamics and 1975 dynamical behavior of black holes. Hawking’s result is not directly connected to quantumIt gravity becomes –it generally is a skillful accepted application that of quantum GR coupled fieldtheoryincurved to matter is notrenormal- spacetime–izable. butThe has research a very program strong impact started on with the field. Rosenfeld, It foster Fierzsanintenseand Pauli is dead. activity1976 in quantum field theory in curved spacetime, it opensanewfieldof research in “black hole thermodynamics” (for a review of the two, see [43]), and it opensAfirstattempttosavethecovariantprogramismadebySteven the quantum-gravitational problems of understanding the statistical Wein- berg, who explore the idea of asymptotic safety [44], developing earlier ideas from Giorgio Parisi [45], Kenneth Wilson and others, suggesting that non- renormalizable theories could nevertheless be meaningful. 11 1976 To resuscitate the covariant theory, even if in modified form,thepathhasal- ready been indicated: find a high energy modification of GR. Preserving general covariance, there is not much one can do to modify GR. An idea that attracts much enthusiasm is supergravity [46]: it seems that by simplycouplingaspin 3/2 particle to GR, namely with the action (in first order form)

4 1 i µνρσ S[g,Γ, ψ]= d x √ g R ϵ ψµγ5γν Dρψσ , ! − "2G − 2 # one can get a theory ﬁnite even at two loops. 1977 Another, independent, idea is to keep the same kinematics andchangethe action. The obvious thing to do is to add terms proportional tothedivergences. Stelle proves that an action with terms quadratic in the curvature

4 2 µν S = d x √ g αR + βR + γR Rµν . , ! − $ % is renormalizable for appropriate values of the coupling constants [47]. Unfor- tunately, precisely for these values of the constants the theory is bad. It has negative energy modes that make it unstable around the Minkowski vacuum and not unitary in the quantum regime. The problem becomes to ﬁnd a theory renormalizable and unitary at the same time, or to circumventnon-unitarity.

12 GR from local horizon thermodynamics

GR from local horizon thermodynamics OtherEinstein new equation things of state:we learnedANALOGY that are here to stay (for QG)

Einstein equation of state: ANALOGY spacetime thermodynamics S(E, V ) => dS = (S/E)dE + (S/V )dV BH thermodynamicsQ = dE + pdV generalised to cosmological horizons, similarQ for surfaces in flat space (Unruh effect) 1 T = (S/E) S(E, V ) => isdS any= equilibrium(region(S/ of) E spacetime)dE + ( a thermodynamic.S/V )dV system? => Q = dE + pdVentropy balance Q = T dS p = T (S/V ) T. Jacobson (1995),Q ….., T. Padmanabhan, Einstein’s equations as equation1 of state T = (S/equationE)…… of state equilibrium Q = T dS => GR GRfrom dynamics local is eff horizonective equation thermodynamics of state for any microscopic dofs entropy balance p = T (collectivelyS/V ) described by a spacetime, a metric and some matter fields

equation of state geometric entropy local matter-energy Einstein eq. as crucial: “holographic” behaviour IDEA + => functional perturbationsassume constantequation of state => G(g) ∝ T(φ, g) ⇥S = ⇥A A = H ⇥ d⇤ geometricanalogue entropy gravity inlocal condensed matter-energy matter systemsEinstein eq. as IDEA + => functional perturbations C.equilibrium Barcelo,equation S. Liberati, of M.= state Visser,Rindler ‘05 horizon bifurcation surface ⇥ effective curved metric (from background fluid) and quantum matter fields (describing excitations over fluid) from non-geometric atomic theory (quantum⇥S = liquids,⇥ A =0 => p =0 optical systems, ordinary fluids, …) Q Unruh, Parentani, Visser, Weinfurtner, Jacobson,perturbation … (1981-…) via heat flux

+ ⇥ d + (⇥2) by Raychaudhuri p d⇥ p O B_p => dS = ⇥˜d⌅[⇤ ⌅(1/2 ⇤2 + ⇧ 2 + R la lb)] ⇥ ⇥ ab p H equilibrium recovery via entropy balance law Q = TdS

⇧ 2 2 a b T dS = ⇤˜d⌃[⌅ ⌃(1/2 ⌅ + + Rabl l )]p = 2⌥ H ⇥ ⇥ =0 = ⇤˜d⌃ ( ⌃⇧)T lalb = ⇥Q ab A has local and non-local contributions H GR from local horizon thermodynamics

GR from local horizon thermodynamics OtherEinstein new equation things of state:we learnedANALOGY that are here to stay (for QG)

+Is gravity⇥ d an emergent+ phenomenon?(⇥2) by Raychaudhuri Are spacetimep and dﬁelds⇥ p just collectiveO emergent entities? B_p => dS = ⇥˜d⌅[⇤ ⌅(1/2 ⇤2 + ⇧ 2 + R la lb)] ⇥ ⇥ ab p H equilibrium recovery via entropy balance law Q = TdS

⇧ 2 2 a b T dS = ⇤˜d⌃[⌅ ⌃(1/2 ⌅ + + Rabl l )]p = 2⌥ H ⇥ ⇥ =0 = ⇤˜d⌃ ( ⌃⇧)T lalb = ⇥Q ab A has local and non-local contributions H While straightforward approaches loose momentum, and new insights come from other corners of (semi-classical) gravitational physics …..

…. new QG approaches are developed, gain traction, achieve results, oﬀer further insights

• some are (or at least start as) continuations of previous attempts in diﬀerent form

• sometimes the new ingredients/hypothesis have radical, unexpected consequences

• similar mathematical structures end up being shared by several formalisms

• stages of development, languages, but also priors and goals of diﬀerent approaches vary greatly While straightforward approaches loose momentum, and new insights come from other corners of (semi-classical) gravitational physics …..

…. new QG approaches are developed, gain traction, achieve results, oﬀer further insights

• some are (or at least start as) continuations of previous attempts in diﬀerent form

• sometimes the new ingredients/hypothesis have radical, unexpected consequences

• similar mathematical structures end up being shared by several formalisms

• stages of development, languages, but also priors and goals of diﬀerent approaches vary greatly

several sub-communities form, with sometimes diﬃcult relationships String theory (and related) (…… , a lot of people, …..)

starting idea: quantum theory of strings, interacting and propagating on given spacetime background string excitations: infinite particles of any spin/ mass; incl. graviton consistent (around flat space) and finite perturbation theory in 10d background spacetime satisfies GR equations many different (consistent) versions (different matter content, different symmetries) - all require supersymmetry and spacetime dimension > 4

central result: spacetime as seen by strings, as opposed to point particles/ﬁelds, has very different topology and geometry; e.g. distances smaller than minimal string length cannot be probed

many non-perturbative aspects; extended (d>1) conﬁgurations (branes) as fundamental as strings, and interacting with them (Polchinski, …., 1994 - ) String theory (and related) (…… , a lot of people, …..)

dualities between various string theories and supergravity: different aspects of same underlying fundamental theory (M-theory)?

dualities show that spacetime topology and dimension are themselves dynamical

AdS/CFT correspondence: a (gauge) QFT with conformal invariance on 4d ﬂat space could fully encode the physics of a gravitational theory in 5d (with AdS boundary); viceversa, semiclassical GR (with extra conditions) could describe the physics of a peculiar many-body quantum system in different dimension is the world holographic? are gravity and gauge theories equivalent? many results and new directions

large number of mathematical results and radical generalisation of quantum ﬁeld theory QG as QFT - Supergravity

Freedman, Ferrara, van Nieuwenheuzen, Zumino, Julia, Wess, DeWitt, Nicolai, deWit, … (1976 - )

one way out of non-renormalizability of perturbative gravity: new symmetry: supersymmetry motivated also by extensions of Standard Model of particle physics (for any interaction a new matter ﬁeld) SUGRA is supersymmetric extension of GR with supersymmetric group replacing the local Lorentz group

“gravitino” is super partner of “graviton”) as QFT, SUGRA is better defined, perturbatively, that are gravity ….. recently, more evidence of nice cancellation of divergences …. a perturbativela well-defined field theory of QG? in 11 spacetime dimensions it emerges as low energy limit of string theory QG as QFT - Lattice Quantum Gravity

Basic idea: covariant quantisation of gravity as sum over “discrete geometries”

Continuum spacetime manifold replaced by simplicial lattice; metric data encoded in edge lengths

Gravitational action is discretised version of Einstein-Hilbert action (Regge action)

Quantum Regge calculus T. Regge, R. Williams, H. Hamber, B. Dittrich, B. Bahr, …. Path integral of discrete geometries: ﬁxed simplicial lattice, sum over edge length variables S ( Le ) Z = lim dµ( Le ) e R { } continuum limit via lattice reﬁnement !1 { } Z (Causal) Dynamical Triangulations J. Ambjorn, J. Jurkiewicz, R. Loll, D. Benedetti, A. Goerlich, T. Budd, …

Path integral of discrete geometries: S ( Le=a ) sum over all possible (causal) simplicial lattices Z = lima 0 µ(a, ) e R { } ! (fixed topology), fixed edge lengths continuum limit via sum over finer and finer lattices evidence of nice geometricX (deSitter-like) continuum phase 10 R. Percacci contradict the notion that the UV limit is defined by k .Thepoint →∞ is that only statements about dimensionless quantities are physically meaningful, and the statement “k ”ismeaninglessuntilwespecify →∞ the units. In a fundamental theory one cannot refer to any external “absolute” quantity as a unit, and any internal quantity which is chosen as a unit will be subject to the RG flow. If we start from low energy (k′ 1) and we increase k, k′ will initially increase at the same rate, ≪ ′ because in this regime ∂tG 0; however, when k 1wereachtheFP 2≈ ′ ≈ regime where G(k) G˜∗/k and therefore k stops growing. ≈ The second consequence concerns the graviton anomalous dimension, which in d dimensions is ηg = ∂t log Zg = ∂t log Z˜g + d 2. Since we − have argued that ∂tZ˜g =0atagravitationalFP,ifZ˜g∗ =0wemust ̸ have ηg∗ = d 2. The propagator of a field with anomalous dimension − η behaves like p−2−η,sooneconcludesthatatanontrivialgravitational FP the graviton propagator behaves like p−d rather than p−2,aswould follow from a naive classical interpretation of the Einstein-Hilbert action. Similar behaviour is known also in other gauge theories away from the critical dimension, see e.g. Kazakov (2003).

1.4 The Gravitational Fixed Point Iwillnowdescribesomeoftheevidencethathasaccumulatedin favor of anontrivialgravitationalFP.Earlyattemptsweremadeinthe context of the ϵ–expansion around two dimensions (ϵ = d 2), which yields − 8 R. Percacci ˜ ˜2 QG as QFTβ G˜- =AsymptoticϵG qG . Safety Scenario(1.4.1) 8 1.3 The case of− gravity R. Percacci ˜ 38 Thus there is a UV–attractive FP at G∗ = ϵ/q.Theconstantq = 3 for pureWeQuantum shall gravity use gravity (Weinberg a derivative is perturbatively (1979), expansion Kawai ofnon-renormalizableΓk: & Ninomiya1.3 The (1990),as QFT case seeof the ofAida metric gravity & gµ⌫ = ⌘µ⌫ + hµ⌫ ∞ Kitazawa (1997) for two–loop(n) results).(n) Unfortunately,(n) for a while it was Can it makeΓ sensek(gµν ; non-perturbatively?gi )= gi (k) i (gµν ) , S. Weinberg,(1. M.3.1) Reuter, C. Wetterich, H. Gies, D. Litim, We shall use a derivativeO expansion of Γ : not clear whether one could trustn!=0 ! thei continuation ofR. this Percacci, result D. Benedetti, tok four A. Eichhorn, …. dimensions(n) (ϵ =2). (n) (n) ∞ where = ddx √g and are polynomials in the curvature MostEffectiveO ofi the action recent progressMi inM thisi approach( hasn) come from the(nappli-) (n) tensor and its" derivatives containing 2Γnkderivatives(gµν ; g of)= the metric; i isg an (k) (gµν ) , (1.3.1) cation(~ tocovariant gravity path of the integral) ERGE. It was showni by Wetterich (1993i)thatOi index that labels different operators with the same numbern of!=0derivatives.!i the effective action(nΓ)k defined in (1.2.2) satisfies the equation Thedefined dimension as ofsolutiongi is todn non-perturbative= d 2n.Thefirsttwopolynomialsarejust RG equations (e.g. Wetterich eqn) (0) (1) (n) − d (n) (1) (n) =1, where= R.Thecorrespondingcouplingsare=2 d x g −1 andg = Zg are= polynomials in the curvature 1 i δ Γk √ i i M 1 (0)M O ABM BA M− look for non-Gaussian UV fixed points 16πG , g =2∂tΓkZg=Λ, ΛSTrbeing the cosmological+ Rk constant.∂tRk , Newton’s(1.4.2) − tensor2 and!δφ itsAδφ" derivativesB " containing 2n derivatives of the metric; i is an constant G appears in Zg,whichinlinearizedEinsteintheoryisthewave index that labels different operators with14 the same numberR. Percacci of derivatives. wherefunctionif STrtheory renormalization is has a trace non-trivial over of momentaUV the fixed graviton. point as well then Neglecting as it is over "asymptotically tota particlelderivatives, spec safe”ies and and could be fundamental (n) (2) " anyone spacetime can choose or asThe internal terms dimension with indices, four derivatives including of g ofais sign thedn metric= -1 ford fe2rmionicn.Thefirsttwopolynomialsarejust= C2 fields i M1 G (0) (1) (2) 2 − (1) (the square of the Weyl tensor)=1, and 2= =RR.Thecorrespondingcouplingsare.Wealsonotethatthe g = Z = necessarily studied in various truncationsM (+ matter fields etc) g coupling constantsM of1 higher derivative(0)M gravity are not the coefficients 1 − (2) , g =2(2) Z−1gΛ, Λ being(2) −1 the cosmological constant. Newton’s gi buteg rather Einstein-Hilbert their16 inversesπG truncation 2λ =(g ) and ξ =(g ) .Thus, − 1 2 0.8 constant G appears in Zg,whichinlinearizedEinsteintheoryisthewave (n≤2) d 1 2 1 2 Γ = d x √g 2ZgΛ ZgR + C + R . (1.3.2) 0.6 k #function$ renormalization− 2λ ofξ the% graviton. Neglecting totalderivatives, one can choose as terms with four derivatives of the metric0.4 (2) = C2 As in any other QFT, Zg can be eliminated from the action by a rescaling M1 of the field. Under constant rescalings of g ,ind dimensions, (2) 2 0.2 accumulating (theevidence square for existence of the of Weyl µUVν fixed tensor) point of and R^2 type2 = R .Wealsonotethatthe" M ! (n) −2 d−2n (n) -0.2 -0.1 0.1 0.2 Γcouplingk(gµν ; gi )= constantsΓbk(b gµν ; ofb highergi ) . derivative(1.3.3) gravity are not the coefficients (2) (2) −1 (2) −1 This relation is thegi analogbut of rather (1.2.9) their for the inverses metric, but 2λ als=(ocoincidesgFig.1 1.2.) Theand flow inξ the=( Einstein–Hilbertg2 ) .Thus, truncation, see Eq.(1.4.9-10). The nontrivial FP at Λ˜ =0.171, G˜ =0.701 is UV–attractive with eigenvalues with (1.2.3), the invariance at the basis of dimensional analysis; fixing−1.69 ± it2.49i.TheGaußianFPisattractivealongtheΛ˜–axis with eigenvalue amounts to a choice of unit(n of≤ mass.2) Thisd is where gravity differs−2andrepulsiveinthedirection(0 from 1 2 1 2 .04, 1.00) with eigenvalue 2. Γk = d x √g 2ZgΛ ZgR + C + R . (1.3.2) any other field theory (Percacci & Perini# (2004), Percacci$ (2−007)). In 2λ ξ % usual QFT’s such as (1.2.8), one can exploit the two invarianceslet (1.2.3) us consider pure gravity in the Einstein–Hilbert truncation, i.e. ne- 2 As in any other QFT, Zg can be eliminatedglecting terms from with then 2. action In a suitable by a gauge rescaling the operator δ Γk is and (1.2.9) to eliminate simultaneously k and Z from the action. In the ≥ δgµν δgρσ afunctionof 2 only. Then, rather than taking as ∆ the whole lin- case of pure gravity there is only one such invariance and one has to −∇ of the field. Under constant rescalingsearized of wavegµν operator,,ind asdimensions, we did before, we use (1.4.4) with ∆ = 2. −∇ make a choice. In this way we retain explicitly the dependence on Λ and R.Usingthe (n) (n) If we choose k as unit of mass, we can define the effective action,optimized−2 cutodff−,withgaugeparameter12n /α = Z,theERGEgives Γk(gµν ; gi )=Γbk(b gµν ; b gi ) . (1.3.3) 2 3 Γ˜(˜g ; Z˜ , Λ˜,...)=Γ (˜g ; Z˜ , Λ˜,...)=Γ (g ; Z , Λ,...) , (1.3.4) ˜ 2 ˜ 36−41Λ˜+42Λ˜ −600Λ˜ ˜ 467−572Λ˜ ˜2 µν g 1 µν g k µν g 2(1 2Λ) Λ + 72π G + 288π2 G β˜ =− − (1.4.9) This relation is the analog of (1.2.9) forΛ the metric, but˜ 2 als29−ocoincides9Λ˜ ˜ 2 Zg 1 Λ (1 2Λ) 72π G whereg ˜µν = k gµν , Z˜g = 2 = , Λ˜ = 2 ,etc..Thereisthen − − k 16πG˜ k ˜ ˜ 2 with (1.2.3), the invariance at the basis of2(1 dimensional2Λ˜)2G˜ 373−654 anaΛ+600lysis;Λ G˜2 fixing it no freedom left to eliminate Zg.Physicallymeasurablequantitieswill 72π βG˜ = − − ˜ (1.4.10) ˜ (1 2Λ˜)2 29−9Λ G˜ depend explicitlyamounts on Zg,sobytheargumentsofsection1.2,wehaveto to a choice of unit of mass. This is where− − gravity72π differs from ˜ ˜ impose that ∂tZgany=0,orequivalently other field theory∂tG =0,ataFP. (Percacci & PeriniThis flow is (2004), shown in Figure Percacci 2. (2007)). In Lauscher & Reuter (2002a), Reuter & Saueressig (2002) have stud- usual QFT’s such as (1.2.8), one canied exploit the gauge– the and cuto twoff–dependence invarian of theces FP (1.2.3) in the Einstein–Hilbert and (1.2.9) to eliminate simultaneouslytruncation.k and TheZ dimensionlessfrom the quantity action.Λ′ = ΛG In(the the cosmological constant in Planck units) and the critical exponents have a reassuringly case of pure gravity there is only oneweak suchdependence invariance on these parameters. and one This hashas been to taken as a sign that the FP is not an artifact of the truncation. Lauscher & Reuter make a choice. (2002b) have also studied the ERGE including a term R2 in the truncation. They find that in the subspace of ˜ spanned by Λ˜, G,˜ 1/ξ,the If we choose k as unit of mass, we can define the effective action,Q

Γ˜(˜gµν ; Z˜g, Λ˜,...)=Γ1(˜gµν ; Z˜g, Λ˜,...)=Γk(gµν ; Zg, Λ,...) , (1.3.4)

2 Zg 1 Λ whereg ˜ = k g , Z˜ = 2 = , Λ˜ = 2 ,etc..Thereisthen µν µν g k 16πG˜ k no freedom left to eliminate Zg.Physicallymeasurablequantitieswill depend explicitly on Z˜g,sobytheargumentsofsection1.2,wehaveto impose that ∂tZ˜g =0,orequivalently∂tG˜ =0,ataFP. 4 4 ab ab where (ab) (G ) identifies a spin network functional labelled by a closed graph with rep- where (ab) (G ) identifies a spin network functional labelled by a closed graph with rep- ij (,J ,◆ ) ij (,J(ij) ,◆i) (ij) i (ab) resentations J (ab) associated to the di↵erent edges linking two vertices i and j, and intertwiners ◆ resentations J(ij) associated to the di↵erent edges linking two vertices i and j, and intertwiners ◆i (ij) i associated to its vertices; g (resp. g )(witha, b =1,...,d) are group elements being the argu- associated to its vertices; gia (resp. gjb)(witha, b =1,...,d) are group elements being the argu- ia jb ments of the field associated to the vertex i (resp. j), so that a pair of indices (a, b) denotes each ments of the field associated to the vertex i (resp. j), so that a pair of indices (a, b) denotes each of the edges connecting two vertices i and j. The bosonic statistics implies a symmetrisation of of the edges connecting two vertices i and j. The bosonic statistics implies a symmetrisation of with respect to permutations of the vertex labels. These observables act on the Fock vacuum with respect to permutations of the vertex labels. These observables act on the Fock vacuum creating a spin network state associated to a graph . creating a spin network state associated to a graphLoop. Quantum Gravity (and spin foam models) A. Ashtekar, C. Rovelli, L. Smolin, T. Thiemann,GFT J. as Lewandowski, 2nd quantised J. Pullin, H. reformulation Sahlmann, B. Dittrich, of …… the LQG kinematics - We now discuss in more GFT as 2nd quantised reformulation ofCanonical the LQG quantization kinematics of GR - asWe gauge now theory discuss (connection in more variables):detail in what sense GFTi providesb a 2nd1 quantisedb formalism for spin networks and how one can (Aa ,Ei = peei ) detail in what sense GFT provides a 2nd quantisedfocus on formalism gravity, matter for spin coupled networks but not and central how one can link (a certain version of) canonical LQG and GFT directly, without passing through the spin foam Quantization of Systems with Constraints iHamiltonian formulationj of GR a b a k a k link (a certain version of) canonical LQG and GFT directly,Two dynamical models without for full LQG passing through the spin foam formulation, but providing in turn a clear link between the latter and canonical LQG. More details A Relational(x),A Formalism:( Observablesy) = & EvolutionE (x),E (y) =0 E (x),A (y) (x, y) formulation, but providing in turn a clear linkquantum betweenOutlook states the and Work latter of in Progress “space”{ anda canonicalare graphsb } LQG.labeled{ More iby algebraic detailsj (group-theoretic)}can be found{ in jdata: [16] spin . b networks}/ b j Basis of kin can be found in [16] . H By ‘LQG2 kinematical Hilbert space’ we intend, here, a Hilbert space constructed out kinematical Hilbert space of quantum states: 2 =lim H = L ¯ By ‘LQG kinematical Hilbert space’ weSpin intend, network functions here,[Ashtekar, Isham, a Lewandowski,Hilbert Rovelli, Smolin space ’90] constructedH out of states associatedA to closed graphs and such that, for each graph ,wehave = j15 S ⇡ H 2 E V E Haar of states associated to closed graphs and such that, for eachj graph ,wehavej14 = L G /G ,dµ= dµ (hereG=G SU(2)= SU(2)), where e are the links of the graph (E is their 19 H e=1 e 2 E V E Haar j18 Haar L G /G ,dµ= dµ (here G = SU(2)), wherej22e are the linksj17 of the graph (E is their total⇣ number), with a graph-based⌘ scalar product defined the Haar measure on each link µ . e=1 e j j Q e 20 16 spin networks can be understood as (generalised) j8 The same Hilbert space can be represented also in the flux basis, via the non-commutative Fourier total⇣ number), with a graph-based⌘ scalar product defined the Haar measure on each link µHaar. j21 epiecewise-flat discrete geometries Q j12 j13 transform [21, 22], in terms of functions of Lie algebra elements, that are the natural ‘momen- The same Hilbert space can be represented also in the flux basis, via thej11 non-commutative Fourier E j23 underlyingtum’ graphs variables are fordual the to (simplicial classical LQG lattices) phase space on a given graph: [ ⇤G]⇥ (before constraints). transform [21, 22], in terms of functionsGeometric operators of Lie algebra elements, thatj10 are the natural ‘momen- T j The union for all graphs of such Hilbert spaces is, of course, not a Hilbert space. In the LQG 2 j 3 j6 E j7 j9 tum’ variables for the classical LQG phase space onj a given graph:j [ G]⇥ (before constraints). 1 5 T ⇤ and spin foam literature, one finds di↵erent ways in which these graph-based Hilbert spaces can j4 The union for all graphs of such Hilbertin terms of spin operators at vertices: spaces is, of course, notfJ~e aj Hilberte at vg space. In the LQG be organised to define the Hilbert space of the theory. One is to simply consider the direct sum 1 and spin foam literature, one finds di↵erent waysGeometric in which observables these graph-basedcorrespond to operators; Hilbert spaces some of can over all possible graphs: LQG = . Another, corresponding to the canonical construction Kristina Giesel Dynamics of LQG H H be organised to define the Hilbert space of thethem theory. have discrete One is spectrum to simply: discretization consider the of quantum direct sum in thej continuum, is to define appropriate equivalence classes for states over di↵erent graphs and 1 geometry! (Rovelli, Smolin, Ashtekar, Lewandowski, 1995-1997) 2 over all possible graphs: LQG = . Another, corresponding to the canonical construction then take the projective limit of infinitely refined graphs: LQG =lim [ H . Of course, the H H H !1 ⇡ in the continuum, is to define appropriate equivalence classes for states over di↵erent graphs and two spaces are very di↵erent. The GFT Hilbert space can be understood as a di↵erent proposal 2 to define a Hilbert space out of a union of the graph-based Hilbert spaces, by ‘decomposing them then take the projective limit of infinitely refined graphs:j LQG =lim [ H .j Of course, the H !1 rigorous implementation of spatial diffeomorphism invariance two spaces are very di↵erent. The GFT Hilbert space can be understood as a⇡ di↵erent proposal into elementary building blocks’. loop quantum cosmology: singularity resolution consistent implementationThe basic idea of isHamiltonian to consider constraint; any wave some function in ,where is a graph with V nodes, as an to define a Hilbert space out of a union of the graph-based Hilbert spaces, by ‘decomposingsolutions them of it; on-shell anomaly-free algebra H element of = L2 (G d/G) V ,dµ= V d dµv , satisfying special restrictions. The into elementary building blocks’. HV ⇥ ⇥ v=1 i=1 Haar,i Volume, angle, and length similar. latter space can be understood as the space of V spin network vertices, each possessing d outgoing The basic idea is to consider any wave function in ,where is a graph with V nodes, as an ⇣ Q Q ⌘ H open links, and the extra restrictions enforce the gluing of suitable pairs of such open links to form element of = L2 (G d/G) V ,dµ= V d dµv , satisfying special restrictions. The HV ⇥ ⇥ v=1 i=1 Haar,i the links of the graph . In group space, these extra restrictions are conditions of invariance under latter space can be understood as the space of V spin network vertices, each possessing d outgoing ⇣ Q Q ⌘ the group action, which can be enforced through projectors. A function can be obtained from open links, and the extra restrictions enforce the gluing of suitable pairs of such open links to form a wavefunction V V as the links of the graph . In group space, these extra restrictions are conditions of invariance under 2 H ab ab ab ab 1 the group action, which can be enforced through projectors. A function can be obtained from (Gij )= d↵ij V (...,gia ↵ij ,...,gjb↵ij ,...)= (gia(gjb) ) , (5) G a wavefunction V V as [(ia),(jb)] Z 2 H Y with the same notation as in 4. This defines an embedding of elements of into . The same ab ab ab ab 1 H HV (Gij )= d↵ij V (...,gia ↵ij ,...,gjb↵ij ,...)= (gia(gjb) ) , (5) construction can be phrased in the flux and spin representations. Moreover, the scalar product of G [(ia),(jb)] Z two quantum states in associated to the same graph agrees with the one computed in (i.e. Y HV H the scalar product in , once restricted by gluing conditions associated to the graph ,reduces with the same notation as in 4. This defines an embedding of elements of into . The same HV V to the one in ). This means that is embedded faithfully in . Obviously also contains H H H H HV HV construction can be phrased in the flux and spin representations. Moreover, the scalar product of states associated to open graphs, that is graphs with some links ending up in 1-valent vertices, i.e. two quantum states in associated23.05.2011 to the same graph agrees with the onequantum geometry computed in (i.e. 8 HV H with links of open spin network vertices not glued to any other. the scalar product in , once restricted by gluing conditions associated to the graph ,reduces HV The physical picture behind V is that of a ‘many-atom’ Hilbert space, with each ‘quantum to the one in ). This means that is embedded faithfully in . Obviously also contains H H H HV HV states associated to open graphs, that is graphs with some links ending up in 1-valent vertices, i.e. with links of open spin network vertices not glued to any other. The physical picture behind is that of a ‘many-atom’ Hilbert space, with each ‘quantum HV 2-complex bordered by the graphs of γ and γ′ respectively, a collection of spins j associated J { f } with faces f and a collection of intertwiners ιe associated to edges e . Both spins and intertwiners of∈ J exterior faces and edges match the{ boundary} values defined by∈ theJ spin networks s and s′ respectively. Spin foams : s s′ and ′ : s′ s′′ can be composed into ′ : s s′′ by gluing together the two correspondingF → 2-complexesF at→s′. A spin foam model is anFF assignment→ of amplitudes A[ ] which is consistent with this composition rule in the sense that F A[ ′]=A[ ]A[ ′]. (74) FF F F Transition amplitudes between spin network states are defined by

′ s, s phys = A[ ], (75) ⟨ ⟩ ′ F F!:s→s where the notation anticipates the interpretation of such amplitudes as defining the physical scalar Loop Quantum Gravityproduct. (and The spin domain foam of the previousmodels) sum is left unspecified at this stage. We shall discuss this question further in Section V. This last equation is the spin foam counterpart of equation (73). M. Reisenberger, C. Rovelli, J. Baez, J. Barrett, L. Crane, A. Perez, E. Livine, DO, S. Speziale, …… This definition remains formal until we specify what the set of allowed spin foams in the sum are evolution of spin networks involves and define the“histories” corresponding (dynamical amplitudes. interaction processes) are also changes in combinatorial structure purely algebraic and combinatorial: spin foams and in algebraic labels l l j p j p q n s n s o m o k m k q n s m l l p j j q p o k k → q l purely algebraic and combinatorial o j k “path integral for quantum gravity” l l j j

k k

iS(g) (j, i) (j0,i0) = Figurew() 5: A typical path in a path(J, integral I) version” of loop quagentum gravity” is given by a series of h | 0 i transitions through differentA spin-network states⇡ representingD a state of 3-geometries. Nodes and ,0 J , I j,j0,i,i0 Z X| links in the{ } spin{X}| network evolve into 1-dimensional edges and faces. New links are created and spin networks/spin foams can be understoodspins as are (generalised) reassigned at vertexes (emphasized on the right). The ‘topological’ structure is provided piecewise-flat discrete geometries by the underlying 2-complex whileLots of the results geometric on quantum degrees of geometry freedom areand encoded in the labeling of its elements with irreducible representationsmathematics of and quantum intertw gravitationaliners. field; underlying graphs and 2-complexes are dual to (simplicial) lattices inspiring models of quantum black holes The background-independentand quantum character cosmology of spin foams is manifest. The 2-complex can be correct discrete semi-classical limit in termsthought of Regge of as representingcalculus ‘space-time’ while the boundary graphs as representing ‘space’. They do not carry any geometrical information in contrast with the standard concept of a lattice. Geometry is encoded in the spin labelings which represent the degrees of freedom of the gravitational field. In standard quantum mechanics the path integral is used to compute the matrix elements of the evolution operator U(t). It provides in this way the solution for dynamics since for any kinematical state Ψ the state U(t)Ψ is a solution to Schrödinger’s equation. Analogously, in a generally covariant theory the path integral provides a device for constructing solutions to the quantum constraints. Transition amplitudes represent the matrix elements of the so-called generalized ‘pro- ′ ′ jection’ operator P (i.e., s, s phys = sP, s recall the general discussion of Sections 2.2) such that P Ψ is a physical state⟨ for⟩ any kinematical⟨ ⟩ state Ψ. As in the case of the vector constraint

30 Matrix models (Migdal, Kazakov, David, Duplantier, Ambjorn, Kawai, Di Francesco, Zuber, Brezin, .....)

• discrete 2d GR on each 2d triangulation

used to deﬁne world sheet theory of strings in large-N limit: control over topologies and dominance of planar surfaces, emergent continuum theory is 2d Liouville quantum gravity continuum limit and phase. transition to theory of continuum surfaces Matrix models (Migdal, Kazakov, David, Duplantier, Ambjorn, Kawai, Di Francesco, Zuber, Brezin, .....)

Abstract theories of matrices which give quantum 2d spacetime as (statistical) superposition of discrete surfaces

• discrete 2d GR on each 2d triangulation

used to deﬁne world sheet theory of strings in large-N limit:

control over topologies and dominance of planar surfaces, emergent continuum theory is 2d Liouville quantum gravity continuum limit and phase. transition to theory of continuum surfaces GFT ROOTS GFT OVERVIEW OF RESULTS CONCLUSIONS

(Migdal, Kazakov, David, Duplantier, Ambjorn, Kawai, Di Francesco, Zuber, Brezin, .....) FIRST ROOT:MATRIX MODELSMatrix models

(quantum) 2d spacetime as a (statistical)Abstract theories superposition of matrices of discrete which give surfaces quantum 2d spacetime as (statistical) superposition of discrete surfaces i building block of space: M j i, j = 1,...,NN N hermitian matrix i × microscopic dynamics: i k 1 i j λ i j k S(M, λ)= M j M i M j M k M i 2 − √ j j N 1 2 − S(M) λ VΓ χΓ Z = Me = ZΓ = λ N Z D „√ « XΓ N XΓ

 

−1   simplicial intepretation: M K jkli ij   i M M ij ji ` ´ j i j

k M V jk jmknli M   ki   • discrete 2d GR on each 2d triangulation Γ 2d simplicial complex ∆ (triangulation) ≃ 2d discrete spacetime used to deﬁne world sheet theory of strings ≃ in large-N limit: fundamental building blocks are 1d simplices with no additional data; microscopic dynamics: nocontrol GR, pure over 2d topologies combinatorics and dominance & geometry of planar surfaces, emergent continuum theory is 2d Liouville quantum gravity

continuum limit and phase. transition to theory of continuum5/42 surfaces GFT ROOTS GFT OVERVIEW OF RESULTS CONCLUSIONS

(Migdal, Kazakov, David, Duplantier, Ambjorn, Kawai, Di Francesco, Zuber, Brezin, .....) FIRST ROOT:MATRIX MODELSMatrix models

(quantum) 2d spacetime as a (statistical)Abstract theories superposition of matrices of discrete which give surfaces quantum 2d spacetime as (statistical) superposition of discrete surfaces i building block of space: M j i, j = 1,...,NN N hermitian1 matrix2 g 3 1 i jl k g i m k jnl i ×S(M)= trM trM = M jK kiM l M jM nM l V mki microscopic dynamics: 2 pN 2 pN i k 1 i j λ i j k S(M, λ)= M j M i M j M k M i 2 − √ j j N 1 2 − S(M) λ VΓ χΓ Z = Me = ZΓ = λ N Z D „√ « XΓ N XΓ

 

−1   simplicial intepretation: M K jkli ij   i M M ij ji ` ´ j i j

continuum limit and phase. transition to theory of continuum5/42 surfaces GFT ROOTS GFT OVERVIEW OF RESULTS CONCLUSIONS

FIRST ROOTGFT:MROOTS ATRIX MODELSGFT OVERVIEW OF RESULTS CONCLUSIONS

(quantum) 2d spacetime as a (statistical) superposition of discrete surfaces FIRST ROOT:MATRIX Mi ODELS building block of space: M j i, j = 1,...,NN N hermitian matrix × microscopic(quantum) dynamics: 2d spacetime as a (statistical) superposition of discrete surfaces i building block of space: M j i, j = 1,...,NN N hermitian matrix × microscopic dynamics: 1 i j λ i j k S(M, λ)= M j M i M j M k M i 2 − √ 1 i j N λ i j k S(M, λ)= M j M i 1 M j M k M i 2 λ − √2 N − S(M) 1 VΓ χΓ GFT ROOTS Z = GFTMe OVERVIEW OF RESULTS= CONCLUSIONS Z2 Γ = λ N − S(M) λ VΓ χΓ Z DZ = Me = „√N « ZΓ = λ N Matrix models (Migdal,XΓ Kazakov, David, Duplantier, Ambjorn, Kawai, DiX Francesco,Γ Zuber, Brezin, .....) FIRST ROOT:MATRIX MODELS Z D „√ « XΓ N XΓ  

−1   simplicial intepretation: M K jkli Γ 2d simplicial complex ∆ (triangulation)ij   i M M ij ji ` ´ ≃ j i Γ 2d simplicial complex ∆ (triangulation)j 2d discrete spacetime k M V jk jmknli M   ki ≃ ≃   2d discretefundamental spacetime building blocks are 1d simplices• discrete with 2d no GR additi on each 2donal triangulation data; Γ 2d simplicial complex ∆ (triangulation) ≃ ≃ 2d discretemicroscopic spacetime dynamics: no GR, pure 2d combinatoricsused to deﬁne world & sheet geomet theory ofry strings ≃ in large-N limit: fundamentalfundamental building building blocks are 1d blocks simplices with are no additi 1donal simplices data; with no additional data; microscopicmicroscopic dynamics: dynamics: nocontrol GR, pure over 2d notopologies combinatorics GR, and dominance pure & geometry 2dof planar combinatorics surfaces, emergent & geomet continuum theoryry is 2d Liouville quantum gravity 5/42 continuum limit and phase. transition to theory of continuum5/42 surfaces

5/42 GFT ROOTS GFT OVERVIEW OF RESULTS CONCLUSIONS

FIRST ROOTGFT:MROOTS ATRIX MODELSGFT OVERVIEW OF RESULTS CONCLUSIONS

(quantum) 2d spacetime as a (statistical)Abstract theories superposition of matrices of discrete which give surfaces quantum 2d spacetime as (statistical) superposition of discrete surfaces i building block of space: M j i, j = 1,...,NN N hermitian1 matrix2 g 3 1 i jl k g i m k jnl −1 i ×S(M)= trMsimplicialtrM = intepretation:M jK kiM l M jM nM l V mki −1   simplicial intepretation: microscopic dynamics: 2 pN 2 pN M K ij M K ij jkli jkli i   k   1 i j λ i j k i i , λ M M S(M )= M j M i M j M k M i M ij M ji ` ´ ij j ji ` ´ 2 √ j i −j N i j j j 1 k M V 2 k jk jmknli M λ  ki M V − S(M)  VΓ χΓ jk jmknli M Z = Me  =  ZΓ = λ N ki Z D Quantum dynamics:„√ «  XΓ N XΓ 1   g 2   S(M,g) V Z = Mij e = Z = g N −1 simplicial intepretation:   D pN M K ij jkli Γ 2d simplicialZ complex ∆ (triangulation) ✓ ◆   i X X M M ij ji ` ´ ≃ j i Γ 2d simplicial complex ∆ (triangulation)j 2d discrete spacetime k M V jk jmknli M   ki ≃ ≃   2d discretefundamental spacetime building blocks are 1d simplices• discrete with 2d no GR additi on each 2donal triangulation data; Γ 2d simplicial complex ∆ (triangulation) ≃ ≃ 2d discretemicroscopic spacetime dynamics: no GR, pure 2d combinatoricsused to deﬁne world & sheet geomet theory ofry strings ≃ in large-N limit: fundamentalfundamental building building blocks are 1d blocks simplices with are no additi 1donal simplices data; with no additional data; microscopicmicroscopic dynamics: dynamics: nocontrol GR, pure over 2d notopologies combinatorics GR, and dominance pure & geometry 2dof planar combinatorics surfaces, emergent & geomet continuum theoryry is 2d Liouville quantum gravity 5/42 continuum limit and phase. transition to theory of continuum5/42 surfaces

5/42 GFT ROOTS GFT OVERVIEW OF RESULTS CONCLUSIONS

FIRST ROOTGFT:MROOTS ATRIX MODELSGFT OVERVIEW OF RESULTS CONCLUSIONS

(quantum) 2d spacetime as a (statistical)Abstract theories superposition of matrices of discrete which give surfaces quantum 2d spacetime as (statistical) superposition of discrete surfaces i building block of space: M j i, j = 1,...,NN N hermitian1 matrix2 g 3 1 i jl k g i m k jnl −1 i ×S(M)= trMsimplicialtrM = intepretation:M jK kiM l M jM nM l V mki −1   simplicial intepretation: microscopic dynamics: 2 pN 2 pN M K ij M K ij jkli jkli i   k   1 i j λ i j k i i , λ M M S(M )= M j M i M j M k M i M ij M ji ` ´ ij j ji ` ´ 2 √ j i −j N i j j j 1 k M V 2 k jk jmknli M λ  ki M V − S(M)  VΓ χΓ jk jmknli M Z = Me  =  ZΓ = λ N ki Z D Quantum dynamics:„√ «  XΓ N XΓ 1   g 2   S(M,g) V Z = Mij e = Z = g N −1 simplicial intepretation:   D pN M K ij jkli Γ 2d simplicialZ complex ∆ (triangulation) ✓ ◆   i X X M M ij ji ` ´ ≃ j i Feynman diagram = 2d simplicial complex Γ 2d simplicial complex ∆ (triangulation)j 2d discrete spacetime k M V jk jmknli M   ki ≃ ≃   2d discretefundamental spacetime building blocks are 1d simplices• discrete with 2d no GR additi on each 2donal triangulation data; Γ 2d simplicial complex ∆ (triangulation) ≃ ≃ 2d discretemicroscopic spacetime dynamics: no GR, pure 2d combinatoricsused to deﬁne world & sheet geomet theory ofry strings ≃ in large-N limit: fundamentalfundamental building building blocks are 1d blocks simplices with are no additi 1donal simplices data; with no additional data; microscopicmicroscopic dynamics: dynamics: nocontrol GR, pure over 2d notopologies combinatorics GR, and dominance pure & geometry 2dof planar combinatorics surfaces, emergent & geomet continuum theoryry is 2d Liouville quantum gravity 5/42 continuum limit and phase. transition to theory of continuum5/42 surfaces

5/42 GFT ROOTS GFTGFT ROOTS OVERVIEW OFGFT RESULTS OVERVIEWCONCLUSIONS OF RESULTS CONCLUSIONS

FIRST ROOT:MATRIX MODELSIRST ROOT ATRIX ODELS F GFT:MROOTS M GFT OVERVIEW OF RESULTS CONCLUSIONS

(quantum) 2d spacetime as a (statistical)(quantum) superposition 2d spacetime of asdiscrete a (statistical) surfaces superposition of discrete surfaces IRST ROOT ATRIX ODELS i F :M Mi building block of space: M j i, jbuilding= 1,..., blockNN of space:N hermitianM j i, matrixj = 1,...,NN N hermitian matrix × × microscopic dynamics: microscopic(quantum) dynamics: 2d spacetime as a (statistical) superposition of discrete surfaces i building block of space: M j i, j = 1,...,NN N hermitian matrix × 1 i j microscopicλ i dynamics:j k 1 i j λ i j k S(M, λ)= M M M S(MM,Mλ)= M j M i M j M k M i j i j k i 2 − √ 2 − √N 1 i j N λ i j k S(M, λ)= M j M i M j M k M i 1 2 − √1 2 λ 2 N − S(M) λ − S(M) VΓ χΓ 1 VΓ χΓ GFT ROOTS Z = GFTMe OVERVIEW OF RESULTS= CONCLUSIONS Z2 Γ = λ N Z = Me = ZΓ = λ − S(NM) λ VΓ χΓ Z DZ = Me = „√N « ZΓ = λ N Z D „√N « Matrix models (Migdal,XΓ Kazakov, David, Duplantier, Ambjorn, Kawai, DiX Francesco,Γ Zuber, Brezin, .....) FIRSTXΓ ROOT:MATRIX MODELS ZXΓD „√ « XΓ N XΓ     (quantum) 2d spacetime as a (statistical)Abstract theories superposition of matrices of discrete which give surfaces quantum 2d spacetime as (statistical) superposition of discrete surfaces i building block of space: M j i, j = 1,...,NN N hermitian1 matrix2 g 3 1 i jl k g i m k jnl −1 i ×S(M)= trMsimplicialtrM = intepretation:M jK kiM l M jM nM l V mki −1 simplicialmicroscopic dynamics:−1 intepretation: 2  p simplicial2 intepretation: p   N N M K ij M K ij i M K jkli jkli ij jkli   λk     1 i j i j k i i , λ M M S(M )= M j M i M j M k M i M ij M ji ` i ´ ij j ji ` ´ 2 − √ j i M N i j M ij j ji j ` ´ j j i1 k M V 2 j k jk jmknli M ki M − S(M) λ   VΓ χΓ jk V k jmknli M Z = Me =  ZΓ = λ N M ki  jk V   jmknli Z D QuantumM dynamics:„√N «   XkiΓ XΓ 1   g 2   S(M,g) V   Z = Mij e = Z = g N −1 simplicial intepretation:   D pN M K ij jkli Γ 2d simplicialZ complex ∆ (triangulation) ✓ ◆   i X X M M ij ji ` ´ ≃ j i Feynman diagram = 2d simplicial complex Γ 2d simplicial complex ∆ (triangulation)j 2d discrete spacetime k M V jk jmknli M Γ 2d simplicial complex ∆ (triangulation)   ki ≃ ≃   ≃ 2d discretefundamental spacetime building blocks are 1d simplices• discrete with 2d no GR additi on each 2donal triangulation data; Γ 2d simplicial complex ∆ (triangulation) 2d discrete spacetime ≃ ≃ ≃ 2d discretemicroscopic spacetime dynamics: no GR, pure 2d combinatoricsused to deﬁne world & sheet geomet theory ofry strings fundamental≃ buildingin large-N blocks limit: are 1d simplices with no additional data; fundamental building blocks are 1d simplicesfundamental building with blocks no are 1d additi simplicesonal with no additi data;onal data; microscopicmicroscopic dynamics: dynamics: nocontrol GR, pure over 2d notopologies combinatorics GR, and dominance pure & geometry 2dof planar combinatorics surfaces, emergent & geomet continuum theoryry is 2d Liouville quantum gravity 5/42 microscopic dynamics: no GR, pure 2d combinatoricscontinuum & limit geomet and phase. transitionry to theory of continuum surfaces 5/42

5/42 5/42 Tensor models (Ambjorn, Jonsson, Durhuus, Sasakura, Gross, ... )

Abstract theories of tensors to give quantum spacetime as (statistical) superposition of simplicial complexes

e.g. d=3

Feynman diagrams are stranded graphs dual to 3d simplicial complexes

relation to discrete gravity on equilateral triangulations issues: no large-N limit, thus no control over topologies or continuum limit GFT ROOTS GFT OVERVIEW OF RESULTS CONCLUSIONS

GOING UP IN DIMENSION:TENSOR MODELS generalizeTensor in (combinatorial) models dimension (Ambjorn, from 1dJonsson, objects Durhuus, (edges) Sasakura, to 2d objectsGross, ... ) (triangles) - from 2d simplicial complexes as FD to 3d ones Abstract theories of tensors to give quantum spacetime as (statistical) superposition of simplicial complexes 

 i 

M j Tijk N N N tensor  → × ×  e.g. d=3 i 1 2 4 1 S(T)= 2 trT λ trT = 2 i,j,k TijkTkji λ ijklmn TijkTklmTmjnTnli −S−(i T) VΓ − Z = Te = λ PZΓ P D k Γ FeynmanR diagrams areP dual to 3d simplicial complexes j j   

           

  

quantum spacetime fromFeynman sum over diagrams all simplicial are stranded complexes graphs (mdualanifolds to 3d simplicial and complexes pseudo-manifolds, any topology)?

8/42 relation to discrete gravity on equilateral triangulations issues: no large-N limit, thus no control over topologies or continuum limit GFT ROOTS GFT OVERVIEW OF RESULTS CONCLUSIONS

 i 1  M j Tijk N N N tensor  → × × S(T )=  TijkTkji TijkTklmTmjnTnli 2 p 3 e.g. d=3 i,j,k 4! N ijklmn i 1 2 4 1 X X S(T)= 2 trT λ trT = 2 i,j,k TijkTkji λ ijklmn TijkTklmTmjnTnli −S−(i T) VΓ − Z = Te = λ PZΓ P D k Γ FeynmanR diagrams areP dual to 3d simplicial complexes j j   

           

  

quantum spacetime fromFeynman sum over diagrams all simplicial are stranded complexes graphs (mdualanifolds to 3d simplicial and complexes pseudo-manifolds, any topology)?

8/42 relation to discrete gravity on equilateral triangulations issues: no large-N limit, thus no control over topologies or continuum limit GFT ROOTS GFT OVERVIEW OF RESULTS CONCLUSIONS

 i 1  M j Tijk N N N tensor  → × × S(T )=  TijkTkji TijkTklmTmjnTnli 2 p 3 e.g. d=3 i,j,k 4! N ijklmn i 1 2 4 1 X i' l' n' X S(T)= 2 trT λ trT = 2 i,j,k TijkTkji λ ijklmn TijkTklmTmjnTnli −S−(i T) VΓ − Z = Te = λ PZΓ P D k Γ FeynmanR diagrams areP dual to 3d simpliciali complexesi' i n j j' j j' k k' k m' j j   

            k' l m   

quantum spacetime fromFeynman sum over diagrams all simplicial are stranded complexes graphs (mdualanifolds to 3d simplicial and complexes pseudo-manifolds, any topology)?

8/42 relation to discrete gravity on equilateral triangulations issues: no large-N limit, thus no control over topologies or continuum limit GFT ROOTS GFT OVERVIEW OF RESULTS CONCLUSIONS

            k' l m Quantum dynamics:   V V S(T,) F 3 V Z = Te = Z = N 2 D sym() sym() Z X X quantum spacetime fromFeynman sum over diagrams all simplicial are stranded complexes graphs (mdualanifolds to 3d simplicial and complexes pseudo-manifolds,All topological any manifolds topology)? as well as pseudo-manifolds included in perturbative sum Construction can be generalized to d spacetime dimension (d-tensors....) 8/42 relation to discrete gravity on equilateral triangulations issues: no large-N limit, thus no control over topologies or continuum limit Group ﬁeld theories (Boulatov, Ooguri, De Pietri, Freidel, Krasnov, Rovelli, Perez, DO, Livine, Baratin, ……)

Quantum ﬁeld theories over group G, enriching tensor models with group-theory data d ' : G⇥ C for gravity models, G = local gauge group of gravity (e.g. Lorentz! group)

1 S(', ')= [dg ]'(g ) (g )'(g )+ [dg ]'(g )....'(¯g ) (g , g¯ )+c.c. 2 i i K i i D! ia i1 iD V ia iD Z Z

single ﬁeld “quantum”: spin network vertex

Quantization of Systems with Constraints Hamiltonian formulation of GR Two dynamical models for full LQG Relational Formalism: Observables & Evolution Outlook and Work in Progress or tetrahedron

Basis of kin H

Spin network functions [Ashtekar, Isham, Lewandowski, Rovelli, Smolin ’90]

j15

j19 j14

j18 j22 j17 generic quantum state: arbitrary collection of j20 j16

j spin network vertices (including glued ones) 21 j12 j13 j11 j23 or tetrahedra (including glued ones) j10

j 2 j 3 j6 j7 j9 j1 j5

Kristina Giesel Dynamics of LQG Group ﬁeld theories (Boulatov, Ooguri, De Pietri, Freidel, Krasnov, Rovelli, Perez, DO, Livine, Baratin, ……)

Quantum ﬁeld theories over group G, enriching tensor models with group-theory data d ' : G⇥ C for gravity models, G = local gauge group of gravity (e.g. Lorentz! group)

1 S(', ')= [dg ]'(g ) (g )'(g )+ [dg ]'(g )....'(¯g ) (g , g¯ )+c.c. 2 i i K i i D! ia i1 iD V ia iD Z Z

quantum states are 2nd quantised spin networks/simplices single ﬁeld “quantum”: spin network vertex

Quantization of Systems with Constraints Hamiltonian formulation of GR Two dynamical models for full LQG Relational Formalism: Observables & Evolution Outlook and Work in Progress or tetrahedron

Basis of kin H

Spin network functions [Ashtekar, Isham, Lewandowski, Rovelli, Smolin ’90]

j15

j19 j14

j18 j22 j17 generic quantum state: arbitrary collection of j20 j16

j spin network vertices (including glued ones) 21 j12 j13 j11 j23 or tetrahedra (including glued ones) j10

j 2 j 3 j6 j7 j9 j1 j5

Kristina Giesel Dynamics of LQG Group ﬁeld theories

Feynman perturbative expansion around trivial vacuum N = ' 'eiS(',') = Z D D sym() A Z X Group ﬁeld theories

Feynman perturbative expansion around trivial vacuum N = ' 'eiS(',') = Z D D sym() A Z X Feynman diagrams (obtained by convoluting propagators with interaction kernels) =