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Loop Quantum Gravity Loop Quantum Gravity Carlo Rovelli String 08 Genève, August 2008 1 i. Loop Quantum Gravity - The problem addressed - Why loops ? ii. The theory - Kinematics: spin networks and quanta of space - Dynamics: spinfoams iii. Physics - Cosmology, black holes, singularities, low-energy limit iv. Conclusion - What has been achieved? What is missing? - Future Carlo Rovelli Loop Quantum Gravity String08 2 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 describe the fundamental degrees of freedom when there is no fixed background space Carlo Rovelli Loop Quantum Gravity String08 3 Hypotheses - 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: general relativity, coupled with the standard model. - In gravity, (unrenormalizable) UV divergences are consequences of a perturbation expansion around 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 full diff-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 Rovelli Loop Quantum Gravity String08 4 A comment on general relativity g (x) S[g]= d4x √ g (R + λ) •! GR is: i) A specific field theory for the gravitational field µ ν : − ! •! ii) A general modification of our understanding of spacetime : spacetime gravitational field Pre GR: After GR: field field spacetime field This modification is expressed by the invariance of the theory under the active action of the group of the diffeomorphisms on all the fields of the theory. Carlo Rovelli Loop Quantum Gravity String08 5 Why Loops ? Carlo Rovelli Loop Quantum Gravity String08 6 “Old” nonperturbative quantum gravity Canonical (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 ...) iS[g] • Z = Dg e ! difficulties: • Integral very badly defined • Perturbative calculations bring UV divergences back. Carlo Rovelli Loop Quantum Gravity String08 7 An old idea : Gauge fields are naturally “made up of lines” (Polyakov, Mandelstan, Wilson, Migdal, ... Faraday ) E(x) x Faraday lines: + - In vacuum: loops The gravitational field too, can be described as a gauge field, with a connection as main variable. (Cartan, Weyl, Swinger, Utyama, ..., Ashtekar) Carlo Rovelli Loop Quantum Gravity String08 8 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 G (number of edges) State: !(Ue) in H = L2[G , d" Haar] Operators: Magnetic field BPlaquette: (Ul U2 U3 U4) Electric field: Ee = -i ! "/" Ue (Left invariant vector field) Dynamics: H = B2 + E2 #: loop on the lattice Loop 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 Rovelli Loop Quantum Gravity String08 9 Generalization: spin-networks and spin-network states Spin network S = ($ , jl , in ) graph $ j1 i1 spins jl on links intertwiners in on nodes j2 j3 i 2 node i1 j1 j2 j3 i 2 link spin network on the lattice: S spin network S (j ) (j ) . Spin network state: !S(Ue) = R 1 (Ul) ... R L (UL) i1 ... iN = < Ue | S > ! The spin network states | S > form an othonormal basis in H (Peter-Weyl). Carlo Rovelli Loop Quantum Gravity String08 10 Can we use these loop states as a % basis of states in the continuum ? & No ! • < % | % > = ! The loop states | % > in the continuum are too singular << LPlanck • < % | & > = 0 for & infinitely close to % States are “too many”, as a basis of the Hilbert space. • An infinitesimal displacement in space yields a different loop state Carlo Rovelli Loop Quantum Gravity String08 11 So 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 gravity solves both sets of difficulties Carlo Rovelli Loop Quantum Gravity String08 12 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’s i1 j2 j5 j1 s-knot states | s > = |', j, i > , where s = j6 j4 i4 j3 i2 j3 Carlo Rovelli Loop Quantum Gravity String08 13 ii. The theory. - Start from GR, or GR+standard model, or any other diff-invariant theory, in a formulation where the field is decribed by a connection A (Ashtekar). - Do a canonical quantization of the theory, using a basis of spin network states and operators acting on 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 Rovelli Loop Quantum Gravity String08 14 H : H ext : (norm-closure of the) space of the (cylindrical) functionals Ψγi,f [A]=f(Uγ1[A], ..., Uγn[A]) A where U γ [ A ]= e γ , equipped with the scalar product ! (Ψγi,f , Ψγi,g)= dU f(U1, ..., Un) g(U1, ..., Un) n !SU(2) ! This product is SU(2) and diff invariant. Hence H ext carries a unitary representation of Diff and local SU(2). The operation of factoring away the action of these groups is well-defined, and defines H = H ext/(Diff and local SU(2)). ! ! Uγ[A] Σ ( A : γA ! Operators well-defined on H ext, and self-adjoint: U γ [ A ]= e and EΣ = E ! !Σ where 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 Rovelli Loop Quantum Gravity String08 15 The LOST uniqueness theorem (Fleishhack 04; Lewandowski, Okolow, Sahlmann, Thiemann 05) - (H ext, A) provides a representation of the classical poisson algebra of the observables U γ [ A ] and E Σ , carrying a unitary representation of Diff, - ! this representation is unique. (cfr.: von Neumann theorem in nonrelativistic QM.) Carlo Rovelli Loop Quantum Gravity String08 16 Interpretation of the spin network states | S > S Volume: V(R): function of the gravitational field ! V(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 >. - Eigenvalues receive a contribution for each node of | S > inside R R Node = “Chunk of space” with quantized volume Carlo Rovelli Loop Quantum Gravity String08 17 S Area: 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 (. Area eigenvalues: A =8π !G γ ji(ji + 1) i ! " ( γ = Immirzi parameter) Link = “Quantum of surface” j with quantized area ( Carlo Rovelli Loop Quantum Gravity String08 18 i1 i1 j2 j5 j1 = i4 j4 j6 i4 i i2 3 i2 i3 j3 | s > = | ' , jl , in > quantum numbers s-knot state of volume connectivity between the elementary quantum numbers quantum chunks of space of area ! Spin network states represent discrete quantum excitations of spacetime Carlo Rovelli Loop Quantum Gravity String08 19 • 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 sense Follows from: standard quantum theory (cfr granularity of oscillator’s energy) + standard general relativity (because “space is a field”). Carlo Rovelli Loop Quantum Gravity String08 20 Loops & strings: a cartoon comparison If a string is: a closed lines in space, that forms matter and forces, a loop is: a closed line that forms space itself as well as matter and forces. Carlo Rovelli Loop Quantum Gravity String08 21 III. The theory - dynamics H(x) = A x s s’ Given by a Wheeler-deWitt operator H in H: H ! = 0 • H is defined by a regularization of the classical Hamiltonian constraint. In the limit in which the regularization is removed. ! H is a well defined self-adjoint operator, UV finite on diff-invariant states. H (x) % x ~ % # 0 Carlo Rovelli Loop Quantum Gravity String08 22 % # 0 Carlo Rovelli Loop Quantum Gravity String08 23 % # 0 Carlo Rovelli Loop Quantum Gravity String08 24 % # 0 Carlo Rovelli Loop Quantum Gravity String08 25 % # 0 Carlo Rovelli Loop Quantum Gravity String08 26 % # 0 The limit % # 0 is trivial because there is no short distance structure at all in the theory ! • The theory is naturally ultraviolet finite Carlo Rovelli Loop Quantum Gravity String08 27 Matter • YM, fermions • Same techniques: The gravitational field is not special • ! UV finiteness remains • YM and fermions on spin networks = on a Planck scale lattice ! Notice: no lattice spacing to zero ! • Matter from braiding ? i n quantum numbers of matter (jl , kn) | s > = | ' , jl , in , kl > ' Carlo Rovelli Loop Quantum Gravity String08 28 III. The theory - covariant dynamics: spinfoams Projector P = !(H) on the kernel of H: iNH s" P s = s" δ(H) s = s" DNe s ! | | # ! | | # ! | | # ! 2 = c s" s + c s" H s + c s" H s + ... 0! | # 1 ! | | # 2 ! | | # 6 s 3 f 8 sf ! f sf !f !f q p 7 5 3 6 1 s1 + ..
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