Loop Quantum Gravity Olivier Piguet Departamento de F´ısica,Universidade Federal de Vi¸cosa,Brasil STARS2013, 2nd Caribbean Symposium on Cosmology, Gravitation, Nuclear and Astroparticle Physics May 4-6, 2013, La Habana, Cuba 1 / 35 Abstract The purpose of this talk is to give a short general introduction to Loop Quantum Gravity (LQG), beginning with some motivations for quantizing General Relativity, listing various attempts and then focusing on the case of LQG. 2 / 35 Summary • Why quantize gravitation? • Einstein Theory • Perturbative Quantum Gravity • Loop Quantum Gravity • A simple exemple • Prospects 3 / 35 Why quantize gravitation? p 3 35 Planck's length lP = ~G=c ≈ 1:6 × 10− m 43 Planck's time tP = lP=c ≈ 0:5 × 10− s 19 2 Planck's mass mP = ~=lPc ≈ 10 GeV=c 4 / 35 p 5 43 At Planck's time tP = ~G=c ≈ 0:5 × 10− , p 3 35 or Planck's scale lP = ~G=c ≈ 1:6 × 10− , enter in interplay: - Gravity (G); - Relativity (c) ) Relativistic gravity = General Relativity (GR); - Quantum Mechanics (~) ) Quantum General Relativity = "Quantum Gravity". 19 15 Note: Planck's energy EP ≈ 10 GeV ≈ 10 E (LHC) 7 ≈ 10 E (Ultra High Energy Cosmic Rays) 5 / 35 Einstein Theory Space-time Geometry defined by the metric gµν(x)(µ = 0; ··· ; D − 1): 2 µ ν ds = gµν(x)dx dx Einstein equation with cosmological constant Λ: 1 8πG R − g R + g Λ = T ; µν 2 µν µν c4 µν •R µν and R (defined from the Riemann curvature tensor) are expressions in derivatives of the gravitational field gµν, up to the second order; • Tµν is the energy-momentum tensor, source of the gravitational field. 6 / 35 Perturbative Gravity 1. Consider a small perturbation hµν around the flat (Minkovsky) metric ηµν = diagonal (−1; 1; 1; 1): gµν = ηµν + hµν ; jhj << 1 ; and introduce it in the action, or in the field equations: hµν + non-linear interactions (hµν;@ρhµν;@ρ@σhµν) / Tµν : At zeroth order, without matter: hµν = 0 ! Gravitational waves. 7 / 35 LIGO detector of gravitational waves (2 sites, in Louisiana and Washington states, USA) 8 / 35 2. Try to quantize this theory perturbatively (Feynmann graphs expansion), considering hµν as just another field (but of spin 2), like the electromagnic field Aµ, etc., propagating in flat Minkowsky space. Result: Non-renormalizable theory! O que fazer? Proposal 1 Look for a more fundamental theory, the Standard Model and General Relativity being `low" energy approximations of it. Exemple: String Theory. Proposal 2 Find a non-perturbative framework, hopefully free of ultraviolet singularities, too. Exemple: Loop Quantum Gravity. More proposals Delta Gravity (Alfaro); Horava Gravity; etc. Alternative: Gravity as an "emergent"phenomenon { Induced Gravity. (From Andrei Sakharov to AdS-CFT correspondence) 9 / 35 First Order Formalism of Einstein Theory Tangent space at point x: µ Vector basis eI = eI @µ,(I = 0; ··· ; D − 1), µ ν Pseudo-orthogonal: eI · ej := gµνeI ej = ηIJ = diag(−1; 1; ··· ; 1). I I µ I µ 1 Dual basis (vielbein): 1-forms e = eµdx , with (eµ) = (eI )− and I J ηIJ eµeν = gµν. IJ JI IJ µ Lorentz connection: ! = −! = !µ dx . IJ 1 IJ µ ν IJ IK J Curvature: R = 2 R µνdx ^ dx = d! + ! ^ !K . I I I K Torsion: T = de + ! K ^ e . Palatini-Holst action (for pure D = 4 gravity): 1 Z 1 S = " eI ^ eJ ^ RKL − eI ^ eJ ^ R : 8πG IJKL γ IJ I J ρ I Field equations: T = 0 ; Rµν := eµeI R Jρν = 0. Independent of the Barbero-Immirzi parameter γ! 10 / 35 Loop Quantum Gravity (LQG): The Main Issues • Rewrite classical General Relativity as a Yang-Mills gauge theory with connection (gauge field) Ai (x) for a compact Lie group G (usually SU(2)) (Ashtekar, 1986). (GR gauge group is Lorentz SO(1,3), non-compact. Do partial gauge fixing ! residual symmetry SO(3) or SU(2).) Ai is the Yang-Mills gauge vector field for the group G = SO(3) or SU(2). • Dirac canonical formalism: - Assume space-time manifold splitted as M4 = IR × M3 = times × space. No background metric! 11 / 35 Coordinates chosen such that T (P )= x0 t £ h£ ≡ hz and q M3(t3) (xa, a = 1, 2, 3) be the coordinates of M3(t). (in order to simplify) £ `£ `z q M3(t2) 2 x 1 £ x `£ `z q M3(t1) 1 12 / 35 • Dirac canonical formalism: 3 X α - Hamiltonian H[A; P] = λ Cα[A; P] α=1 i Pi (x) is the conjugate momentum of A (x). Vierbein eI and Lorentz connection !IJ are functions of components of A and P. - Constraints Cα ≈ 0( ) H ≈ 0) • Quantization { formal: Wave functional Ψ[A] δ Operators A^, P^: A^i (x)Ψ = Ai (x)Ψ, P^ (x)Ψ = −i Ψ. j ~δAj (x) ^ • Apply the Constraints: CαΨ[A] = 0. 13 / 35 Note: Constraints $ Gauge invariance: - Yang-Mills G-invariance ∼ local SU(2)(⊂ Lorentz) invariance. - Space diffeomorphisms - Time diffeomorphisms (\coordinate time" evolution is a gauge transformation!) 14 / 35 Hamiltoniano H ≈ 0: Where time has gone? 15 / 35 Compare with QED (free theory) 1. Defined on a metric space. 2. Hamiltonian is: Z 1 H = d3x (E2 + B2)+ A rE 2 t = energy+ constraint ; where the electric potencial At $ Lagrange multiplier for the Gauss constraint rE ≈ 0. 16 / 35 \Loop" Quantization (Ashtekar, Rovelli, Smolin, etc.) Configuration space = space of connections A is very big and difficult to handle. How to define an integration measure DA −! a scalar product Z < Ψ1jΨ2 >= DA Ψ1[A]Ψ2[A] ? Instead, restrict on the values of A(x) on finite sets of lines (graphs): 17 / 35 Graph Γ On each line γn: an holonomy R γn hγn = P e A; P means path ordering. hγn 2 G is an element of the gauge group. Gauge transformation g(x) 2 G: 1 hγ0 n = g(xfin)hγn g − (xin) (Roger Penrose) xin, xfin: initial and final points of γn. Consider all possible graphs! 18 / 35 Ψ[A] ! Γ(hγ1 ; ··· ; hγN ) (\holonomy functions") ; where Γ = fγ1; ··· ; γN g (\graph"), γn = curve in space Σ, and R γn hγn = P e A (\holonomy"); P means path ordering : Scalar product: Z Z Γj Γ0 0 = δΓΓ0 dµ(g1) ··· dµ(gN ) (g1; ··· gN ) 0 (g1; ··· gN ) ; where gn 2 G, dµ(gn) is the Haar integration measure on the group G { which must be compact!. −! Kinematical Hilbert Space Hkin (non-separable!) Orthonormal basis: Γ;j1; jN = gauge invariant combinations of ··· products Rj1 ··· RjN , where Rjn = matrix elements of the α1β1 αN βN αnβn spin jn representation of hγn 2 G −! 19 / 35 −! Spin networks, represented by \coloured" graphs: Basis vectors of Hilbert space Hkin. 20 / 35 Finally, apply the constraints (the difficult part!) ! (separable) Physical Hilbert space. ... and construct the Observables of the theory. 21 / 35 1. (2+1) { Gravity as a Chern-Simons theory with a Barbero-Immirzi parameter (Model due to Valentin Bonzom, Etera R. Livine, CQG 25 (2008) Work by R.M.S. Barbosa, C.P. Constantinidis, Z. Oporto, O. Piguet, CQG 29 (2012) ) I I µ 1 JK µ e = eµdx (Triad) ;!I = 2 "IJK ! = !I µdx (Spin connection) ; I J metric gµν(x) = ηIJ eµ(x)eν (x) ; ηIJ = diag(−1; 1; 1) (Minkowski metric) ; I ; J; ··· = 0; 1; 2 ; µ, ν; ··· = t; x; y : Cosmological constant Λ > 0 ! de Sitter gauge group SO(3,1) I Generators: J (Lorentz), PI (\translations") I J IJ K I I K K [J ; J ] = " K J ; [J ; PJ ] = " J PK ; [PI ; PJ ] = Λ"IJK J : I I SO(3,1) { connection: A = !I J + e PI . 22 / 35 D = 2 + 1 de Sitter gravity described by SO(3,1) Chern-Simons action: κ Z 2 SCS(A) = − A; dA + AA : (^ symbols omitted) 2 =IR Σ 3 M × 2 quadratic Casimir operators: 1 C = P JI ; C = η PI PJ − JI JJ (1) I (2) IJ Λ ! 2 invariant quadratic forms h ; i ! 2 actions: S(1) & S(2) 1 Total action: S = S − S : (1) γ (2) γ: Barbero-Immirzi-like parameter. N.B.: Classical field equations are independent of γ: Λ F (A) = dA+A2 = 0 ; or: R − e × e = 0 ; T ≡ de + ! × e = 0 : 2 23 / 35 A well chosen partial gauge fixing ! The theory is equivalent to a simple SU(2) Chern-Simons theory: Z κ i i 1 j k SCS(A) = − A ; dA + "ijk A A ; γ =IR Σ 3 M × where the dynamical components of the connection are: p p p i 2 2 1 1 0 0 (Ax ; i = 1; 2; 3) = ( Λex − γ!x ; − Λex + γ!x ; −!x − γ Λex ) ; p p i 2 1 0 (Ay ; i = 1; 2; 3) = Λey ; − Λey ; −!y : i j γ ij 2 Poisson brackets: fAx (x); Ay (x0)g = κ δ δ (x − x0) ; N.B. The Hamiltonian is completely constrained, with i Fxy (A) ≈ 0 being the only constraint. (Curvature constraint { the analog of the Gauss constraint of QED.) 24 / 35 Loop quantization of such a theory is known. (C.P. Constantinidis, G. Luchini and O. Piguet, CQG27 (2010) 065009 R.M.S. Barbosa, C.P. Constantinidis, O. Piguet and Zui Oporto, CQG29 (2012)) Special case: space Σ is a cylinder Coordenadas x ; y ; 0 ≤ x < 2π ; −∞ < y < +1 25 / 35 h i i γ Commutation relations: A^i (x); A^j (x ) = − ~ δij δ2(x − x ) ; x y 0 κ 0 Field operators act on wave functionals Ψ[Ax ] as ^i i ^i i~γ δ Ax (x)Ψ[Ax ] = Ax (x)Ψ[Ax ] ; Ay (x)Ψ[Ax ] = − i Ψ[Ax ] : κ δAx (x) # Dunne, Jackiw and Trugenberger, Ann.Phys. 194 (1989) 197: General solution of the (curvature) constraint is of the form: 2iπα0[ x ] inv Ψ[Ax ] = e A [Ax ] ; 2iπα0[ x ] the phase e A is a special solution of the constraint, and inv is invariant under the \x-gauge" transformations δAx = @x " + Ax × ": 26 / 35 Restriction to \Holonomy functions": 2iπα0 ΨΓ;f [Ax ] = e f (h1[Ax ]; ··· hN [Ax ]) ; where: \graph" Γ = fγ1; ··· ; γN g ; γn = closed curve at y = yn = const.