Loop Quantum Gravity - Indico

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Loop Quantum GravityCarlo RovelliString 08Genève, August 20081i. Loop Quantum Gravity-The problem addressedWhy loops ?ii. The theory-Kinematics:Dynamics:spin networks and quanta of spacespinfoamsiii. Physics-Cosmology, black holes, singularities, low-energy limitiv. Conclusion-Carlo RovelliWhat has been achieved? What is missing?FutureLoop Quantum GravityString082

i. Loop Quantum Gravity-Develops since the late 80s.About 200 people, 30 research groups.Several books (CR: “Quantum Gravity”):The problem addressed:How to describethe fundamental degrees of freedomwhen there is no fixed background spaceCarlo RovelliLoop Quantum GravityString083Hypotheses- A radical conceptual change in our concept of space and time is required.- The problem can be addressed already in the context of current physical theory: generalrelativity, coupled with the standard model.- In gravity, (unrenormalizable) UV divergences are consequences of a perturbation expansionaround a wrong vacuum. ! Confirmed a posteriori in LQG.- Guiding principle: a symmetry: Diffeomorphism invariance.Consistency with quantum mechanics and (in the low-energy limit) General Relativity, and fulldiff-invariance are extremely strong constraints on the theory.Main result! Definition of Diffeomorphisms invariant quantum field theory (for gauge fields plus fermions),in canonical and in covariant form.Carlo RovelliLoop Quantum GravityString084

A comment on general relativity !GR is: i) A specific field theory for the gravitational field gµν (x) : !ii) A general modification of our understanding of spacetime :spacetimeS[g] !d4x g(R λ)gravitational fieldPre GR:After GR:spacetimefieldfieldfieldThis modification is expressed by the invariance of the theory under the active action of the groupof the diffeomorphisms on all the fields of the theory.Carlo RovelliLoop Quantum GravityString085Why Loops ?Carlo RovelliLoop Quantum GravityString086

“Old” nonperturbative quantum gravityCanonical (Wheeler 64, DeWitt 63 .) States:!(q).q: 3d metric of a t 0 surface 3d diff:!(q) !(q’) if there is a diff: q " q’ Dynamics: Wheler-DeWitt eq H!(q) 0 .difficulties: Which states !(q)? Which scalar product !, # ? Operator H badly defined and UV divergent. No calculation possible.Covariant (Misner 57, Hawking 79 .) Z !Dg eiS[g]difficulties: Integral very badly defined Perturbative calculations bring UV divergences back.Carlo RovelliLoop Quantum GravityString087An old idea : Gauge fields are naturally “made up of lines”(Polyakov, Mandelstan, Wilson, Migdal, . Faraday )E(x)xFaraday lines: -In vacuum:loopsThe gravitational field too, can be described as a gauge field, with a connection as mainvariable. (Cartan, Weyl, Swinger, Utyama, ., Ashtekar)Carlo RovelliLoop Quantum GravityString088

Can we describe a quantum gauge-field-theory in terms of these lines?Yes, on the lattice. !Wilson, Kogut, Susskind, . "Variable: Ue in the gauge group GState: !(Ue)in H L2[G(number of edges), d" Haar]Operators: Magnetic field BPlaquette: (Ul U2 U3 U4)Electric field: Ee -i ! "/" Ue (Left invariant vectorfield)Dynamics: H B2 E2#:loop on the latticeLoop state: !#(Ue) Tr ( Ul . Ue . UN ) Ue # ! # is an eigenstate of El with eigenvalue having support on # . # is a “quantum excitation of a single Faraday line”.Carlo RovelliLoop Quantum GravityString089Generalization: spin-networks and spin-network statesj1Spin network S ( , jl , in )graph spins jl on linksintertwiners in on nodesi1j2j3i2nodei1j1j3j2i2linkspin network on the lattice: Sspin network SSpin network state: !S(Ue) R(j1) (Ul) . R(jL) (UL) . i1 . iN Ue S ! The spin network states S form an othonormal basis in H (Peter-Weyl).Carlo RovelliLoop Quantum GravityString0810

Can we use these loop states as abasis of states in the continuum ?%&No ! % % !The loop states % in the continuum are too singular % & 0 LPlanckfor & infinitely close to %States are “too many”, as a basis of the Hilbert space. An infinitesimal displacement in space yields a different loop stateCarlo RovelliLoop Quantum GravityString0811So far, just difficulties:- “Old” nonperturbative quantum gravity does not work.- A loop-state formalism in the continuum for Yang Mills does not work.But the two ideas provide the solution to each other’s stumbling blocks:A loop formulation of gravitysolves both sets of difficultiesCarlo RovelliLoop Quantum GravityString0812

Diffeomorphism invariance : S and S’ are gauge equivalent if S can be transformed into S’by a diffeomorphism !S’ S S” S ! S’ ! S” ! for any !.! States are determined only by an abstract graph ' with j’s and i’si1s s-knot states s ', j, i , wherej2j5j1j4i4j6j3i2j3Carlo RovelliLoop Quantum GravityString0813ii. The theory.- Start from GR, or GR standard model, or any other diff-invariant theory, in a formulation wherethe field is decribed by a connection A (Ashtekar).- Do a canonical quantization of the theory, using a basis of spin network states and operators actingon these.- Impose diffeomorphism invariance on the states.- Study the Wheeler deWitt equation.Result:! A (separable) Hilbert space H of states, and an operator algebra A .! Basis of H: abstract spin network states: graph labelled by spins and intertwiners.! A well defined UV-finite dynamics.Carlo RovelliLoop Quantum GravityString0814

H:H ext : (norm-closure of the) space of the (cylindrical) functionals Ψγi,f [A] f (Uγ1 [A], ., Uγn [A])where Uγ [A] e!γA, equipped with the scalar product!(Ψγi,f , Ψγi,g ) dU f (U1, ., Un) g(U1, ., Un)SU (2)n! This product is SU(2) and diff invariant. Hence H ext carries a unitary representation of Diffand local SU(2). The operation of factoring away the action of these groups is well-defined, anddefines H H ext/(Diff and local SU(2)).!Uγ [A]Σ!(A:!!Aγ! Operators well-defined on H ext, and self-adjoint: Uγ [A] eand EΣ EΣwhere E is the variable conjugate to A, smeared on a two-surface Σ . This acts as an SU(2)Left-invariant vector field on the SU(2) of each line that intersects Σ .Carlo RovelliLoop Quantum GravityString0815The LOST uniqueness theorem(Fleishhack 04; Lewandowski, Okolow, Sahlmann, Thiemann 05)- (Hext, A) provides a representation of the classical poisson algebra of theobservables Uγ [A] and EΣ , carrying a unitary representation of Diff,- ! this representation is unique.(cfr.: von Neumann theorem in nonrelativistic QM.)Carlo RovelliLoop Quantum GravityString0816

Interpretation of the spin network states S SVolume: V(R): function of the gravitational fieldV(R) !R "g !R "EEE!V(R) operator! V(R) is a well-defined self-adjoint operator in H ext,! It has discrete spectrum. Eigenstates: spin networks state S .R- Eigenvalues receive a contribution for each node of S inside RNode “Chunk of space”with quantized volumeCarlo RovelliLoop Quantum GravityString0817SArea: A(() A(() ( "g ( "EE!A(() operator! A(() well-defined selfadjoint operator in H ext .! The spectrum is discrete.! Area gets a contribution for each link of S that intersects (.A 8π !G γArea eigenvalues:(! "ji(ji 1)i( γ Immirzi parameter)Link “Quantum of surface”jwith quantized area(Carlo RovelliLoop Quantum GravityString0818

i1j5j1j4i1j2i4i4 j6i3i2i3i2j3 s ' , jl , in quantum numbersof volumes-knot stateconnectivity between the elementaryquantum chunks of spacequantum numbersof area! Spin network states represent discrete quantum excitations of spacetimeCarlo RovelliLoop Quantum GravityString0819 Spin networks are not excitations in space: they are excitations of space.! Background independent QFT! Discrete structure of space at the Planck scale - in quantum senseFollows from:standard quantum theory (cfr granularity of oscillator’s energy) standard general relativity (because “space is a field”).Carlo RovelliLoop Quantum GravityString0820

Loops & strings: a cartoon comparisonIf a string is:a closed lines in space, thatforms matter and forces,a loop is:a closed line that forms space itselfas well as matter and forces.Carlo RovelliLoop Quantum GravityString0821III. The theory - dynamicsH(x) AxsGiven by a Wheeler-deWitt operator H in H: H ! 0s’ H is defined by a regularization of the classical Hamiltonian constraint. In the limit in whichthe regularization is removed.!H is a well defined self-adjoint operator, UV finite on diff-invariant states.H% (x)Carlo Rovellix %#0Loop Quantum GravityString0822

%#0Carlo RovelliLoop Quantum GravityString0823%#0Carlo RovelliLoop Quantum GravityString0824

%#0Carlo RovelliLoop Quantum GravityString0825%#0Carlo RovelliLoop Quantum GravityString0826

%#0The limit % # 0 is trivial becausethere is no short distance structure at all in the theory ! The theory is naturally ultraviolet finiteCarlo RovelliLoop Quantum GravityString0827Matter YM, fermionsSame techniques: The gravitational field is not special! UV finiteness remainsYM and fermions on spin networks on a Planck scale lattice !Notice: no lattice spacing to zero ! Matter from braiding ?inquantum numbersof matter(jl , kn) s ' , jl , in , kl 'Carlo RovelliLoop Quantum GravityString0828

III. The theory - covariant dynamics: spinfoamsProjector P !(H) on the kernel of H:"""!s P s# !s δ(H) s# !s !DN eiN H s# c0!s" s# c1 !s" H s# c2 !s" H 2 s# .sf63!fsfsf!f8!fqp 3567 .s118p!isisi!i!s" P s# !i! (σ,jf ,in) (s s")Carlo Rovelli"si53"dim(jf )7Av (jf , in)vfLoop Quantum GravityString0829Hxs As’s’Two-complex, colored with spins and intertwiners spinfoamiajabVertex amplitude: A(j,i)s!A(jab, ia) ! i a ,iaCarlo Rovelli15j"(1 γ)jab2#"# 1 γ )jab , i 15j,i a fii a,i aaa a2Loop Quantum GravityString0830

SpinfoamsZ ! "σjf ivA(jab, ia) dim(jf )15j i a ,iaAv (jf , iv )vf!""(1 γ)jab2#"# 1 γ )jab , i 15j,i a fii a,i aaa a2 Can be directly derived from a discretization of the action of general relativity, ona variable lattice.version of Hawking’s “integral over geometries” Can be interpreted as a discrete!Z Dg eiS[g] ! Z is generated as the Feynman expansion of an auxiliary field theory definedon a group manifold. This is a 4d generalization of the matrix models (à laBoulatov, Ooguri). ! In 3d, it gives directly the old Ponzano Regge model (A 6j). (With cosmologicalconstant: Turaev-Viro state sum model.)Carlo RovelliLoop Quantum GravityString0831III. PhysicsCarlo RovelliLoop Quantum GravityString0832

Loop cosmology Discrete cosmological time ! Big Bang singularity removed (from Planck scale non-locality) ! Evolution “across” the big bang (robust) ! (Super-) inflationary behavior at small a(t) ! scale invariant spectrum with the observed spectral index ns.classical evolutiona(t)inflationary phasetfinite minimal volumeCarlo RovelliLoop Quantum GravityString0833Black holes ! Entropy finite, proportional to the area ! Physical black holes S A/4 if a dimensionless free parameter(Immirzi parameter) is fixedend of black hole evaporationsingularityhorizonCarlo RovelliLoop Quantum GravityString0834

Black holes ! Entropy finite, proportional to the area ! Physical black holes S A/4 if a dimensionless free parameter(Immirzi parameter) is fixed ! R 0 singularity under control:singularity removed:end of black hole evaporationnon-classical spacetimehorizonCarlo RovelliLoop Quantum GravityString0835background-independent n-point functions ! low-energy limitW (x, y) ϕy!Dφ φ(x)φ(y)eiS[φ]φintx where is independent from x and y if D) andS[)] are invariant under Diff.Choose a closed 3-surface where x and y lie, and rewrite W as!W (x, y) Dϕ ϕ(x)ϕ(y) W [ϕ] Ψ[ϕ]W [ϕ] !φint Σ ϕDφint eiS[φint]Instead of being determined by boundary conditions at infinity, let the boundary statebe a state picked on a given boundary geometry q.!W [x, y; q] Dϕ ϕ(x)ϕ(y) W [ϕ] Ψq [ϕ]This expression is meanigful in a Diff-invariant theory, and reduces to flat space npoint function for appropriate boundary state.Carlo RovelliLoop Quantum GravityString0836

Boundary values of the gravitational field geometry of box surface distance and time separation of measurementsParticle detectors field measurementsSpacetime regionDistance and time measurements gravitational field measurmentsIn GR, distance and time measurementsare field measurements like the other ones:they are part of the boundary data of the problem.Carlo RovelliLoop Quantum GravityString0837The graviton two-point function in LQC reads!0 g ab(x)g cd(y) 0" !W [s] g ab(x)g cd(y)Ψq [s]sAnd for large distances this is give at first order byW [s] λ5!!"n m#dim(jnm)Avertex(jnm)! The asymptotic expansion of the vertex gives the low-energy behavior of the theory.! Premiminary results yield: - free graviton propagator; - 3 point function; (henceNewton law); - first order corrections to the free graviton propagator. Calculations inprogress.38

Mathematical developments Diffeomorphism invariant measures (Ashtekar-Lewandowski measure) C* algebraic techniques Category theory “Quantum geometry” Uniqueness of the representationCarlo RovelliLoop Quantum GravityString0839IV. Summary Loop quantum gravity is a technique for defining Diff-invariant QFT. It offers a radically newdescription of space and time by merging in depth QFT with the diff-invariance introduced by GR. It provides a quantum theory of GR plus the standard model in 4d, which is naturally UV finite andhas a discrete structure of space at Planck scale. Has applications in cosmology, black hole physics, astrophysics; it resolves black hole and big bangsingularities.- Unrelated to a natural unification of the forces (we are not at the “end of physics”).- Different versions of the dynamics exist.- Low-energy limit still in progress. Fundamental degrees of freedom explicit. The theory is consistent with today’s physics. No need of higher dimensions (high-d formulation possible). No need of supersymmetry (supersymmetric theories possible). Consistent with, and based on, basic QM and GR insights.Carlo RovelliLoop Quantum GravityString0840

Carlo Rovelli Loop Quantum Gravity String08 Loop cosmology Discrete cosmological time ! Big Bang singularity removed (from Planck scale non-locality) ! Evolution “across” the big bang (robust) ! (Super-) inflationary behavior at smal

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