Loop quantum gravity: A brief introduction.

Transcription

1 Loop quantum gravity: A brief introduction. J. Fernando Barbero G. Instituto de Estructura de la Materia, CSIC. Graduate Days Graz, March 30-31, 2017 LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

2 Summary 1 Motivation (Why canonical quantum gravity?). 2 A non-metric Hamiltonian approach: The Ashtekar variables. 3 Quantization. 4 Applications: Black hole entropy. Loop Quantum Cosmology. 5 The rest of LQG The covariant approach: spin foams. Group field theory. The continuum limit. 6 Where do we stand today? Not exhaustive. Disclaimer I will talk about some selected items. I will try to give enough details about them. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

3 Motivation Should we quantize gravity? The interplay between the quantum world and the geometry of spacetime must be understood. Quantum gravity Different approaches to the quantization of gravity: Perturbative (including strings and asymptotic safety). Canonical. Causal Dynamical Triangulations. Causal sets, shape dynamics,... The LQG choice Canonical quantization LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

4 Particle physics QG Perturbative quantum gravity: Try to use the same approach that worked so well for the other interactions. The goal is to obtain S matrix elements (containing the physics of particle interactions, giving predictions for accelerator experiments, and providing necessary information for cosmology). The use of perturbation theory for this means that one wants to obtain physical predictions as power series in G N. For this to be at all possible there are some consistency requirements (that can be met for the other interactions) such as renormalizability (the ability to absorb the infinities appearing in physical amplitudes by a redefinition of the coupling constants). Does it work? Maybe: Naive perturbative QG is non-renormalizable but the asymptotic safety scenario offers some hope and string theories contain gravity in a arguably consistent way. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

5 Relativist s QG What kind of phenomena would a relativist love to understand in a quantum theory of gravity? Singularities: In a sense QFT solves some problems associated with singularities of the classical EM field, Coulomb potential, infinite energy of a single charge. It does it without introducing extended objects but rather by changing the description of the interactions. The high curvature regime of general relativity, in particular, black holes and the Big-Bang. The origin of black hole entropy and its detailed microscopical description? Black hole evaporation, in particular solve the problems related to unitarity and information loss. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

6 Relativist s QG (goals) Define a mathematically rigorous quantum theory of the gravitational field (general relativity or something very close). To this end we need to understand how to quantize background free field theories (related to diff-invariance). Find useful quantum gravitational observables (hopefully leading to verifiable experimental or observational predictions). Solve the problem of time and, in general, the problem of spacetime covariance (time does not exist as an external object in GR). As the gravitational field is a manifestation of space-time geometry the quantization of general relativity will require us to understand the fate of geometry after quantization or, in other words, the meaning of quantized geometry. Get a detailed picture of emergence of classical geometry at large scales from the purely quantum gravity theory (the semiclassical regime and the continuum limit). LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

7 A possible choice: canonical quantization Use Dirac s method (quantization of constrained systems) The starting point is a Hamiltonian formulation for gravity. This was first obtained in the early sixties by Arnowitt, Deser, and Misner (the ADM formalism) after some pioneering work by Dirac. Obtain a quantum version of the constraints. In metric variables this leads to the famous Wheeler-DeWitt equation, apparently referred to by one of his authors as that damned equation!. Work on this issue has provided lots of interesting insights but has not produced a completely satisfactory theory of QG. An important issue: learn how to deal with diff-invariance and understand quantization in the absence of a metric background. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

8 Dirac s method in a nutshell Some important physical systems (such as EM) are singular (there is a problem to go from a Lagrangian to a Hamiltonian formulation). The fiber derivative (a.k.a. definition of conjugate momenta) is not a diffeomorphism between TQ and T Q. Hamiltonian dynamics can only be defined in a consistent way on a submanifold of the phase space T Q. This is usually described as the vanishing of some phase space functions (constraints). The physical Hilbert space is obtained by quantizing these constraints and finding their kernels (well, one tries to do that...) There is a very concrete method due to Dirac (Dirac s algorithm in the following) that consistently gives the constraints starting from any Lagrangian. A warning: one must follow it to the letter. There are other ways to arrive at the same results (GNH method) that have some conceptual advantages and are somehow simpler to use. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

9 Metric Hamiltonian gravity: ADM formulation Is there a Hamiltonian formulation for GR? YES Derive it from the Einstein-Hilbert action S = 1 e σgr; κ = 8πG N 2κ M c 3 by following Dirac s approach M is a 4-dim manifold M = R Σ (global hyperbolicity [Geroch]). Σ ia a smooth, orientable, closed (i.e. compact and without boundary) 3-manifold, e is a fiducial volume form and σ is the space-time signature (σ = 1 Lorentzian, σ = +1 Riemannian). Introduce a time function t defining a foliation of M by smooth 3- dim hypersurfaces Σ t diffeomorphic to Σ and a time flow direction t a (a globally defined smooth vector field such that t a a t = 1). Alternatively one can introduce a congruence of spacetime filling curves. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

10 Metric Hamiltonian gravity: ADM formulation Given a metric of signature ( +++) it defines a unit time-like normal n a on the points of each Σ t. Notation: I use Penrose notation (no coordinate charts!) (t a X(M), t a X (M), t ab X(M) X(M),...) Let us define: The induced metric h ab = g ab + n a n b (on vectors X a tangent to each Σ t we have h ab X b = g ab X b := X b, and h ab n a = 0). The lapse N := g ab t a n b = (n a a t) 1, ( a denotes a torsion-free connection on M). The shift N a := h a b tb = t a Nn a. Let us define also: The unique, torsion-free, derivative operator D a on each Σ t compatible with h ab. a This is given by D a T 1,...a k b 1...b l := h a1 d 1 h e l b l ha f d f T 1,...d k e 1...e l. The extrinsic curvature K ab := ha c c n b = 1 2N (L th ab 2D (a N b) ). LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

11 Metric Hamiltonian gravity: ADM formulation The information contained in g ab is the same as the one present in (N, N a, h ab ) but these are 3-dimensional objects. Let us rewrite the E-H action in terms of these (The fiducial, nondynamical volume form e := e abcd is chosen so that it satisfies L t e abcd = 0). Remember that K ab can be written as a time derivative of h ab and D a N b. S = M decomposition e hn[σ (3) R + K ab K ab K 2 ] := R (3) el G := Σ t L G R Here (3) e abc := t d e dabc. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

12 Metric Hamiltonian gravity: ADM formulation Canonical conjugate momenta p ab := L G ḣab = h(k ab Kh ab ) The momenta associated with the lapse N and the shift N a are zero. These become first class constraints implying that N and N a are arbitrary (a strange thing for Lagrange multipliers to do!). Hamiltonian density S = (3) eh G := H G R Σ t R [ H G = p ab ḣ ab L G = N σ h (3) R + 1 (p ab p ab 1 ] h 2 p2 ) 2N b D a (h 1/2 p ab ) LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

13 Metric Hamiltonian gravity: ADM formulation N and N a are Lagrange multipliers enforcing the secondary constraints: σ h (3) R + 1 (p ab p ab 1 h 2 p2 ) = 0 (Scalar constraint) D a (h 1/2 p ab ) = 0 (Vector constraint) They generate gauge transformations. Time evolution (!!!) and 3-dimensional diffeomorphisms respectively. For instance: V (Λ) := Λ a D a p ab, {V (Λ), q ab } = L Λ q ab, {V (Λ), p ab } = L Λ p ab. The full evolution of h ab and K ab is given by the Hamilton equations (H G denotes the Hamiltonian) ḣ ab = δh G δp ab = {H G, h ab }, ṗ ab = δh G δh ab = {H G, p ab } LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

14 ADM formulation: The final description Phase space: Γ(h ab, p ab ) Symplectic structure: Ω = Σ (3) eδh ab δp ab. Alternatively the Poisson brackets are [these should be understood as brackets between weighted versions of the configuration and momentum variables] {h ab (x), h cd (y)} = 0 {p ab (x), h cd (y)} = δ a (c δb d) δ3 (x, y) {p ab (x), p cd (y)} = 0 Constraints (first class): σ h (3) R + 1 h (p ab p ab 1 2 p2 ) = 0 (Scalar constraint) D a (h 1/2 p ab ) = 0 (Vector constraint) Two local d.o.f. (the two polarizations of the gravitational field!) LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

15 A new point of view (as of the mid eighties...) A surprising development [Ashtekar 1986]: The best way to approach the canonical quantization of gravity is to describe it as a theory of SU(2) connections (or SO(3) in some simple situations). This approach is the starting point of Loop Quantum Gravity (LQG in short). It is a canonical approach. The label non-perturbative (which has become a trade mark for LQG) refers to the fact that no splitting of a metric is used. Notice, however, that some approximation scheme (perturbation theory) may well have to be developed to obtain sensible and testable (at least in principle) physical predictions. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

16 The SO(3) ADM formulation Let us perform now a simple change of variables. Introduce a triad i.e. three 1-forms e i a, i = 1, 2, 3 defining a frame at each point of Σ (det e 0) Write the metric as h ab = e i ae j b δ ij Introduce Ẽi a = (det e)ei a with ei ae aj = δ ij (densitized inverse triad). Define Ka i = 1 det e K abẽ j b δ ij. The new variables are canonical. We can rewrite the constraints in terms of these variables. Before we do that it is important to realize that we have now local SO(3) rotations of ea i and Ka i that do not change neither h ab nor K ab so there must be extra constraints to generate them. These can easily be found from the condition K [ab] = 0 (the 2 nd fundamental form is symmetric). LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

17 The SO(3) ADM formulation Phase space: Γ(K i a, Ẽ a i ) Symplectic structure (Poisson brackets): {Ka(x), i K j a b (y)} = {Ẽi (x), Ẽ j b (y)} = 0 {Ẽi a (x), K j b (y)} = δi j δb a δ3 (x, y) Constraints (first class): R is the scalar curvature of h ab := e i ae bi. ɛ ijk KaẼ j ak = 0 ] a D a [Ẽk Kb k δa bẽ k c K c k = 0 σ hr + 2 h Ẽ [c k Ẽ d] l K k c K l d = 0 Two local degrees of freedom (of course!) LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

18 GR in Yang-Mills phase space: Ashtekar vars. Consider the transformation γẽ a i = 1 γ Ẽ a i γa i a = Γ i a + γk i a Γ i a is a SO(3) connection that defines a covariant derivative compatible with the triad. [a e i b] + ɛi jk Γj [a ek b] = 0 can be inverted to get Γi a (a SO(3) connection). γ C is known as the Immirzi parameter. The Poisson brackets between the new variables γ A i a and γ Ẽ a i { γ A i a(x), γ A j b (y)} = { γẽ a i (x), γ Ẽ b j (y)} = 0 { γ A i a(x), γ Ẽ b j (y)} = δ i j δ b a δ 3 (x, y) So this is a canonical transformation! are LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

19 GR in Ashtekar variables. Phase space: Γ( γ A i a, γ Ẽi a ) [smooth SO(3) connections and triads on Σ, i.e. a Yang-Mills phase space] Symplectic structure (Poisson brackets): The variables γ A ai (x) and γ Ẽj b (y) are canonical; Constraints (first class): D a Ẽ a i = 0 Gauss F i abẽ b i = 0 Vector [ Ẽ [a i Ẽ b] j ɛ ijk Fab k + 2(σ ] γ2 ) γ 2 (A i a Γ i a)(a i a Γ j b ) = 0, Scalar Where D a Ẽi a = a Ẽi a the SO(3) curvature. Two local degrees of freedom + ɛ ijk A j aẽ ak, and F i ab = 2 [aa i b] + ɛijk A aj A bk is LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

20 The self-dual and Holst actions The new formulation can be derived from an action principle S = M The Holst action e I e J (ɛ IJKL Ω KL 2 γ Ω IJ) e I takes values in a 4-dimensional R-vector space (I = 0, 1, 2, 3). ω I J takes values in the Lie algebra of SO(1, 3) (Lorentzian signature) or SO(4) (Riemannian signatures). ɛ IJKL is the alternationg tensor in V. Ω I J = dωi J + ωi K ωk J is the curvature 2-form of ωi J. If γ = i (Lor. case) or γ = 1 (Riem. case) the action can be written in terms of the self-dual curvature of a self-dual connection ω I + J. This action is invariant under diffeomorphisms of M and internal gauge transformations (SO(1, 3) or SO(4) depending on the signature). LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

21 GR in Yang-Mills phase space: Ashtekar vars. Comments GR in these new variables is a background independent relative of SO(3) [or SU(2)] Yang-Mills theory. The fact that the configuration variable is a connection is a cornerstone of the formalism. What happens with γ? If σ = +1 (Riemannian signature) we can cancel the last term by choosing γ = ±1. In this case the variables A i a and Ẽi a are real and the scalar constraint takes a very simple form. If σ = 1 (Lorentzian signatures, i.e. the real thing) we face two choices: If we want to remove the ugly term we have to take γ = ±i. If we want to have real variables we have to live with complicated constraints. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

22 GR in Yang-Mills phase space: Ashtekar vars. This parameter shows up in the definition of the area and volume observables that are an essential ingredient of the formalism. It is not an unobservable ambiguity but, rather, has physical consequences. It plays a significant role in the description of black holes and LQC. The fact that the internal symmetry group is compact is very important in the construction of the Hilbert spaces used to quantize the theory (a good reason to use real variables). Once it was understood that even the complicated form of the constraints could be handled (more or less...) the emphasis was placed on the geometric meaning of the new variables. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

23 (Dirac) quantization Several steps: 1 Find a representation of the basic classical variables in a suitable kinematic Hilbert space H kin. 2 Represent the constraints in this Hilbert Space as self adjoint operators. 3 Find their kernels (solutions to the quantum constraints) to define physical states and a suitable scalar product in H phys. 4 In practice the process is much subtler, in particular with regards to the implementation of the vector constraint (the generator of 3-dim diffeos). Group averaging methods. 5 Find a complete set of (gauge invariant) observables and phrase the relevant physical questions in terms of them. This is highly nontrivial for background independent field theories with local degrees of freedom. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

24 Quantization A formal approach to quantization would consist in copying the standard quantization rules used in the familiar finite-dimensional quantum mechanical models. [Configuration variables Multiplication operators ˆX Ψ(x) = xψ(x)]. Momentum variables Derivative operators [ ˆP x Ψ(x) = i x Ψ(x)]. Choosing here the Ashtekar connection A as configuration variable and the triad as momentum one would be led to consider wave functionals Ψ[A] and represent the connection itself as a multiplication operator and the triad as a functional derivative iδ/δa. In order to properly define the kinematical Hilbert space in a rigorous way one introduces a different set of variables holonomies and fluxes defined in a space of wave functionals Ψ[A] which are square integrable with respect to a diff and SU(2) invariant measure known as the Ashtekar-Lewandowski measure dµ AL. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

25 Quantization dµ AL defines a kinematical inner product Ψ, Φ = Ψ[A]Φ[A]dµ AL Holonomies Given a path e in the spatial manifold Σ we define ( ) h e [A] := P exp A SU(2), where P de notes the path ordered exponential. 1 h e [A] is parametrization independent. 2 h e [A] defines a representation of the group of paths: h e [A] = h e1 [A]h e2 [A] if e = e 1 e 2. 3 Under SU(2) gauge transformations the holonomies change as h e = g(x(0)h e [A]g 1 (x(1))). 4 Under a diffeo φ it transforms as h e [φ A] = h φ 1 (e)[a]. e LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

26 Quantum geometric operators 5 Given a graph α with edges e i, i = 1,..., n we build a Hilbert space H α of cylindrical functions ψ γ,f = f (h e1 [A],..., h e1 [A]) where f : SU(2) n C is square integrable w.r.t. the Haar measure in SU(2). Fluxes Associated to smooth surfaces S (α SU(2) is a smearing field): E(S, α) := α i Ẽ i. The Poisson algebra defined by these variables admits a unique quantization in a Hilbert space with diff-inv. states (LOST&F th.). Spin networks: A very convenient orthonormal basis in this space is provided by the spin network states. These are labeled by graphs of the type shown above (consisting in edges with spin labels). LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51 S

27 Geometric operators I will start now discussing geometric operators (quantum Riemannian geometry), we are finally doing the promised quantum geometry! The length and angle operators The area operator The volume operator Curvature operator. General considerations They can all be rigourously defined in the Hilbert space used above. They have discrete spectra. The generalized spin network basis introduced before is well adapted to their description. The area operator is important in the computation of black hole entropy. The volume operator is a basic ingredient for the quantization of the scalar constraint. The curvature operator provides an alternative quantization of the scalar constraint. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

28 Area operator Consider a surface embedded in S Σ. We will require it to be closed. The densitized triad Ẽi a encodes the metric information. Hence we can write the area of a surface in terms of it. If we choose a normal n a to the points of S the area, as a function(al) of Ẽi a, takes the form A S [Ẽi a ] = (Ẽi a Ẽj b δ ij n a n b ) 1/2 S We want now to quantize the operator A S [Ẽ i a ]. This means that we have to define its action on the vectors in H. To this end we need to know its action on the elements of the orthonormal basis that we have introduced above (spin networks). A reasonable way to approach this problem is trying to express it in terms of the flux operators E[S, f ]. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

29 Area operator The idea is to decompose S in N two dimensional cells S I of small coordinate size. Use the three Lie algebra vectors τ i (in place of the α i ) and consider the flux variables E[S I, τ i ] on each cell. Let us take A N [S] := γ l=1 N (E[S I, τ i ]E[S I, τ j ]η ij ) 1/2. This is an approximate expression for the area ( Riemann sum ) in the sense that if the number of cells goes to infinity in such a way that their coordinate size goes uniformly to zero we recover the area in the limit N. To quantize we take advantage of the fact that in each cell E[S I, τ i ]E[S I, τ j ]η ij is a positive self adjoint operator on H (it then has a well defined square root). LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

30 Area operator The action of this operator on an element of H α for a fixed graph is straightforward to obtain. The idea is to refine the partition so that every elementary cell has, at most, one transverse intersection with the graph. In this case the only terms contributing come from the S I that intersect α. Once this point is reacher further refinements do nothing. The resulting operator can be written as the following sum over the vertices of α that lie on S Â S,α = 4πγl 2 P ( S,v,α ) 1/2 where S,v,α is an operator given by a quadratic combination of operators associated with each edge leaving or arriving at the v s appearing in the previous sum. v LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

31 Area operator: Comments The previous expression is defined on a Hilbert space H α for a fixed graph. In order to see if it is defined on the whole Hilbert space H one has to check some consistency requirements related to the fact that a function may be cylindrical w.r.t. different graphs. It is possible to prove that this is always possible. The previous operator can be extended as a self-adjoint operator to the full Hilbert space H. It is SU(2) invariant and diff-covariant. The eigenvalues of the area operator are given in general by finite sums of the form 4πγl 2 P [2j (u) (j (u) + 1) + 2j (d) (j (d) + 1) j (u+d) (j (u+d)+1 )] 1/2 α S where the j (u), j (d), and j (u+d) are half integers (eigenvalues of the angular momentum operator for the edges in the expression of the area operator subject to some inequality constraints). LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

32 Area operator: Comments It is possible to obtain simple expressions for the area operator if we restrict ourselves to the (internal) gauge invariant subspace of H. For example, if the intersections of α and S are just 2-degree vertices the spectrum of the area operator takes the simple form 8πγl 2 P (j I (j I + 1)) 1/2 I Notice that the Immirzi parameter γ appears in all these expressions. This means that it is not an irrelevant arbitrariness in the definition of some canonical transformations but, rather, shows up in (eventually) observable magnitudes such as areas. A final pictorial interpretation of this is the following. The quantum excitations of geometry are 1-dimensional and carry a flux of area. Any time a graph pierces a surface it endows it with a quantum of area. This picture is completely different from the one in Fock spaces (quantum excitations as particles). LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

33 Volume operator The strategy to define it is similar to the one for the area operator. The volume of a 3-dim region B (a certain open subset of Σ) is classically given by V B = h This can be expressed in terms of the triad as V B = B B 1 3! ɛ abcɛ ijk Ẽi a As before we want to rewrite this expression in terms of flux operators. To this end we divide B in cells of a small coordinate volume. In each cell we introduce three surfaces such that each of them splits the cell in two disjoint pieces. This defines the so called internal regularization (others are possible) Ẽ b j Ẽ c k 1/2. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

34 Volume operator An approximate expression for the volume in terms of flux operators is then (8πγl 2 P )3 1/2 ɛ ijk η abc E[S a, τ i ]E[S b, τ j ]E[S c, τ k ] 6 cells When the coordinate size of the cells goes to zero this gives the volume of the region B. As in the case of the area operator we define a family of operators for each graph α (satisfying similar consistency conditions). The resulting operator is given by ˆV B,α := α v (8πγl 2 P )3 48 ɛ ijk ɛ(e 1, e 2, e 3 ) ˆ i e 1,e 2,e 3 1/2 J (v,e 1) J ˆ (v,e 2) ˆ j J (v,e 3) k where α is an undetermined constant and ɛ(e 1, e 2, e 3 ) is the orientation factor of the family of edges (e 1, e 2, e 3 ). LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

35 Volume operator: Comments The removal of the regulator (i.e. of the auxiliary partition used to define the volume operator) is non-trivial now because the volume operator obtained by just taking the limit keeps some memory of the details of the partition. Nevertheless there are ways to handle this issue. The orientation function is zero if the tangent vectors to the edges e 1, e 2, e 3 are linearly dependent at the point where they meet. This means, in particular, that the volume operator is zero when acting on state vectors defined on planar graphs i.e. graphs such that at each vertex the tangent vectors are contained in a plane. The volume operator is SU(2) gauge invariant and diffeomorphism covariant as the area operator. The total volume operator (i.e. ˆV Σ ) is diff-invariant. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

36 Volume operator The volume operator is zero also when acting on gauge invariant states if the vertices are at most of degree 3 (tri-valent). The eigenvalues of the volume operator are real and discrete. They are not known in general but can be computed in many interesting cases. In particular when the vertices are four-valent. The volume operator plays a central role in the implementation of the quantum constraints because the quantum version of the scalar constraint can be written by using Poisson brackets of the total volume operator and the basic canonical variables. Other regularizations are possible, for example the so called external regularization obtained by considering the faces of the cells used in the approximation of the volume operator in the process of writing it in terms of the flux operators. The volume operators built by using these different approaches have different properties. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

37 Applications: Black hole entropy Black holes in LQG are modeled by considering general relativity in the presence of isolated horizons. These are inner boundaries of the spacetime where the induced metric satisfies some conditions: Non expanding horizons : null, 3-dim submanifolds of (M, g ab ) such that 1 They have the topology of S 2 R. 2 The expansion of any null normal q ab a l b vanishes. 3 The Einstein field equations hold on with T ab satisfying the (mild) condition that T a b lb is a future directed, causal vector. (This is true for minimally coupled matter fields). This definition guarantees that the area of is constant, there is no matter flux through, and the horizon geometry (q ab, D) is time-independent. Isolated horizons: Non-expanding horizons with an essentially unique null normal l that is a symmetry of the horizon geometry (L l q ab = 0, [L l, D] = 0). LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

38 Applications: Black hole entropy 4 Isolated horizons capture important features of BH physics (for instance, 0 th and 1 st laws of BH thermodynamics). 5 They are appropriate to model black holes in equilibrium without requiring that the exterior geometry be stationary. 6 They can model rotating black holes or black holes with distorted horizons. 7 A reduction of general relativity consisting of spacetimes with isolated horizons as inner boundaries admits a Hamiltonian description. This is a key first step towards quantization. 8 This idea is similar in spirit to the study of the quantization of mini and midisuperspace models. A subset of the gravitational field configurations is selected by imposing restrictions on the metrics. Mini and midisuperspaces Symmetry requirement on the metrics. Black holes The allowed metrics must have an isolated horizon that is also an inner boundary of spacetime. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

39 Applications: Black hole entropy LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

40 Applications: Black hole entropy Quantization is carried out by combining LQG methods with ideas borrowed form Chern-Simons models [H = H bulk H surface ]. The entropy is obtained from a maximally mixed (thermal) density matrix (tracing over bulk states). In all the different proposals the entropy computations can be phrased as concrete combinatorial problems in terms on the spin labels associated with the edges of spin networks piercing the BH horizon. The Bekenstein-Hawking law (S = A/4) is reproduced in all of them, furthermore, logarithmic corrections can be found. A caveat: in some old proposals γ must be chosen to get 1/4 factor! Approach γ Logarithmic correction ABCK-DL γ DL = /2 log(a/l 2 P ) GM γ GM = /2 log(a/l 2 P ) ENP γ ENP = γ GM 3/2 log(a/l 2 P ) LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

41 Applications: Black hole entropy LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

42 Applications: LQC 1 Loop quantum cosmology (LQC) is the quantization of cosmological models (hence, most of the physical degrees of freedom are frozen from the start) using LQG-inspired methods, in particular, polymer quantizations where the Hilbert space shares some features with those of full LQG (non-separability and the existence of an area gap a 0 the difference between the two lowest area eignevalues). 2 Discreteness of geometry plays a fundamental role close to the Big Bang. One of the consequences of this is the fact that difference equations (discrete) play a fundamental role in the description of the quantum dynamics of the system. 3 In FLRW models, these effects can be incorporated in the dynamics through the modified equation for the scale factor (ȧ a ) 2 = 8πG ) N ρ (1 ρρ0 k 3 a 2 + Λ ( ) 1 3, with ρ 8πGN c = γ 2 a 0 3 LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

43 Applications: LQC 4 When the equation is integrated one sees that the Big-Bang is replaced by a Big-Bounce in which, after a contracting phase, the universe reaches a minimum size and starts expanding again. 5 Quantum effects act as the source of a quantum repulsion not unlike the one responsible for the stability of compact stelar objects such as white dwarfs and neutron stars. 6 Other interesting results that have been found in this setting are related to inflation. By studying the effective equations for a it is possible to see that there is a natural inflationary regime in these models and also that, in the presence of scalar fields, inflation is generic ( no inflaton potential engineering i.e. no fine tuning of the initial conditions is required to have an inflationary epoch with the right duration). 7 In order to have a fully fledged cosmological model one has to deal with all the gravitational (and matter) degrees of freedom. 8 There are many interpretational issues... LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

44 Applications: LQC Bounce of the wave function of the universe Ψ for a FLRW spacetime coupled to a scalar field Φ. v is the volume of the universe in Planck units. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

45 Applications: LQC LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

46 The rest of LQG: Spin foams k l... n j... m... A covariant point of view similar in spirit to the Feynman path integral approach. Feynman path integrals become discrete combinatorial sums over spin network amplitudes. They represent transition amplitudes from a spin network state at some time to another spin network state at a later time. Faces carry spin labels and edges carry intertwiner [invariant SU(2) tensors] labels. Boundary graphs represent space whereas the spin network itself represents spacetime. The relation with the canonical approach must be properly understood. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

47 The rest of LQG: Group field theory A development of tensor models (themselves derived from matrix models, which are relevant in string theory). They are tensor models containing algebraic data encoding quantum geometric information (discrete gravity d.o.f.) They have relations to both LQG and CDT. For instance, spin networks can be used to span the Hilbert spaces of GFT models. Spacetime becomes similar to a condensed matter system. GFT s are QFT s for spin networks endowed with covariant dynamics LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

48 The rest of LQG: The continuum limit The construction of a suitable continuum limit is necessary to comple LQG. Use iterative coarse graining methods to construct physical states. This leads to an understanding of the dynamics of LQG at different scales. In a sense, this idea replaces the renormalization flow of the standard QFT s (formulated with the help of a geometric background). It is necessary to understand the role of diff-invariance for discrete systems and how it reflects on the continumm limit. The starting point is the inductive definition of the Hilbert space of LQG (the way to get the full Hilbert space H from the H α ). LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

49 Conclusions and comments Where do we stand today? 1 The framework provided by the Ashtekar variables provides a tantalizing point of view that brings gravity close to Yang-Mills theories. 2 The quantization of the model has provided interesting mathematical constructions (geometric operators). The discreteness of the area spectrum gives a glimpse of the microstructure of space at the Plank length. 3 There is a nice interplay between the covariant picture (spin foams) and the canonical one (only partially understood). 4 Despite the fact that both BH and LQC models are not yet complete they provide independent evidence on the soundness of the approach: The Bekenstein-Hawking law is recovered. Also logarithmic corrections of the expected type are found. The LQC models offer a very appealing picture of the physics of space-time close to the Big Bang. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

50 Conclusions and comments What is left to do? 1 The semiclassical and continuum limits: recovering general relativity. 2 The problem with the Hamiltonian constraint: consistent quantum dynamics. Anomalies and full quantum space-time covariance. 3 Can the spin foam models be derived from the canonical approach? 4 Extending the results provided by the present models (black holes and LQC) to full general relativity. 5 How is the problem of the non-renormalizability of general relativity explained away? 6 Does LQG have anything to say about the conceptual issues in quantum gravity? 7 Physical predictions. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51

51 Bibliography Introductory texts R. Gambini and J. Pullin, A first course in Loop Quantum Gravity. Oxford University Press. C. Rovelli and F. Vidotto, Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory, Cambridge Monographs on Mathematical Physics. Intermediate level reviews A. Ashtekar and J. Lewandowski, Background Independent Quantum Gravity: A Status Report, Class. Quant. Grav. 21: R53, A. Ashtekar and J. Pullin Eds. 100 years of general relativity, Vol.4. Loop quantum gravity, the first 30 years, World Scientific. Advanced texts T. Thiemann, Modern Canonical Quantum General Relativity, Cambridge Monographs on Mathematical Physics. LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, /51