Black Holes in Loop Quantum Gravity Microstates and Hawking - PowerPoint PPT Presentation

Black Holes in Loop Quantum Gravity Microstates and Hawking radiation Karim NOUI Laboratoire de Math´ ematiques et de Physique Th´ eorique, TOURS Astro Particules et Cosmologie, PARIS Based on PhysRevLett.105 (2009) 031302 - Phys.Rev. D82 (2010) 044050 - JHEP 1105 (2011) 016 arXiv :1212.4060 - JHEP 1305 (2013) 139 - arXiv :1309.4563 With many collaborators A.Perez (Marseille), M. Geiller (Penn State), A. Gosh (India), J. Engle (Florida) E. Frodden (Chile), D. Pranzetti (AEI) IHES - november 2013 Karim NOUI Black Holes in LQG 1/20

Introduction From Loop Quantum Gravity... Context : Loop Quantum Gravity At the classical level ⊲ Hamiltonian quantization of gravity : locally M = Σ × T ⊲ Formulation : Ashtekar-Barbero first order gravity ⊲ Partial gauge fixing (similar to ADM) : SL (2 , C ) reduced to SU (2) At the quantum level ⊲ Hypothesis : states are one-dimensional excitations ⊲ Consequences : non standard quantization but Diff-invariance ⊲ Kinematical theory : the geometry (area and volume) is discrete ⊲ Physical consequences : minimal length (UV cut-off), singularities resolution, and also statistical description of Black Holes Open questions ⊲ Quantum Dynamics ? Spin-Foams from TQFT... ⊲ Barbero-Immirzi parameter γ ? Relevance at the quantum level... ⊲ Vacuum ? How classical geometry emerges from LQG ? IHES - november 2013 Karim NOUI Black Holes in LQG 2/20

Introduction ... To Quantum Black Holes Relation to Chern-Simons theory Classical correspondence between CS and (spherical) BH ⊲ Symplectic geometry is those of a Chern-Simons theory ⊲ SU (2) Gauge group and the level (coupling constant) k ∝ a H ⊲ Manifold : a two-sphere with arbitrary number of punctures Quantization is very well-known ⊲ Hilbert space of quantum states from quantum group U q ( su (2)) ⊲ Dimension is finite and explicit (rather simple) formula Thermodynamics of Black Holes ⊲ Black Hole entropy : S = a H / 4 − 3 / 2 log a H in Planck units ⊲ Problems : γ fixed at quantum level and distinguishable punctures ! ⊲ No Hawking radiation, no temperature... Up to recent results Our recent results ⊲ γ is no more relevant : γ = ± i and SL (2 , C ) gauge group ⊲ Quantum version of Hawking (local) radiation IHES - november 2013 Karim NOUI Black Holes in LQG 3/20

Overview 1. Loop Quantum Gravity in a nut shell • Why does ADM canonical quantization fail ? • From Ashtekar gravity... • ... To kinematical quantum states • Physical interpretation : discrete geometry 2. Black Holes in LQG: a quick review • Heuristic picture • Relation to Chern-Simons theory 3. Complex variables and Hawking radiation • Back to complex variables • The new Black Hole partition function • Hawking radiation IHES - november 2013 Karim NOUI Black Holes in LQG 4/20

Loop Quantum Gravity in a nut shell Why does ADM canonical quantization fail ? Lagrangian formulation : M is the 4D space-time ⊲ Einstein-Hilbert action : functional of the metric g � d 4 x � S EH [ g ] = | g | R Hamiltonian formulation : M = Σ × T (’61) ⊲ ADM variables : ds 2 = N 2 dt 2 − ( N a dt + h ab dx b )( N a dt + h ac dx c ) ⊲ ADM action : ( h , π ) canonical variables � � d 3 x (˙ S ADM [ h , π ; N , N a ] = h π + N a H a [ h , π ] + NH [ h , π ]) dt ⊲ Constraints H = 0 = H a generate the diffeomorphisms What about the quantization ? ⊲ Highly non linear constraints : quantum ambiguities and no solutions ⊲ Huge symmetry group : how to take it into account ? IHES - november 2013 Karim NOUI Black Holes in LQG 5/20

Loop Quantum Gravity in a nut shell From Ashtekar gravity... Starting point : first order formulation of gravity ⊲ A tetrad e I µ (4 × 4 matrix) such that g µν = e I µ e J ν η IJ ⊲ a so (3 , 1) spin-connection ω IJ µ related to Levi-Civitta connection ⊲ First order Hilbert-Palatini action � S HP [ e , ω ] = � ⋆ ( e ∧ e ) ∧ F ( ω ) � ⊲ Canonical analysis leads to second class constraints : problematic ! The Ashtekar variables (’86) ⊲ Restrict ω to be (anti) self-dual : ⋆ω ± = ± i ω ± and S A = S HP [ e , ω ± ] ⊲ No more second class constraints : right number of d.o.f. ⊲ Classically equivalent to Einstein-Hilbert theory ⊲ Complex variables ( γ = ± i ) : E a = ǫ abc e b × e c and A i a = ω i a + γω 0 i a ⊲ Pair of canonical variables : j δ 3 ( x , y ) { A i a ( x ) , E b j ( y ) } = (8 πγ G ) δ b a δ i IHES - november 2013 Karim NOUI Black Holes in LQG 6/20

Loop Quantum Gravity in a nut shell The Barbero-Immirzi parameter The Constraints become polynomials of A and E ⊲ Gauss constraint G = D a E a : complex SL (2 , C ) gauge symmetry ⊲ Vectorial constraint H a = E b · F ab : space diffeomorphisms ⊲ Scalar constraint H = E a × E b · F ab : time reparametrizations ⊲ BUT... No one knows how to deal with complex variables The Immirzi-Barbero parameter γ ⊲ Real γ : parametrizes a family of canonical transformations ⊲ Now an SU (2) connection : Ashtekar-Barbero connection ⊲ Everything formally unchanged but H is no more a polynomial H = E a × E b · ( F ab + ( γ 2 + 1) R ab ) ⊲ Lagrangian formulation : the Holst action � � ⋆ ( e ∧ e ) ∧ F ( ω ) � + 1 S HP [ e , ω ] = γ � e ∧ e ∧ F ( ω ) � ⊲ Kind of ”Wick” rotation : gauge group becomes compact SU (2) IHES - november 2013 Karim NOUI Black Holes in LQG 7/20

Loop Quantum Gravity in a nut shell Polymer states hypothesis Classical phase space of Ashtekar gravity : ⊲ Phase space : P = T ∗ ( A ) with A = { SU (2) connections } ⊲ Holonomy-flux algebra associated to edges e and surfaces S � � A ( e ) = P exp( A ) and E f ( S ) = Tr( f ⋆ E ) . e S ⊲ Cylindrical functions : f ∈ Cyl is a function of A ( e ) with e ⊂ γ ⊲ E f ( S ) acts as a vector field on f if S ∩ γ � = 0. Action of symmetries : S = G ⋉ Diff (Σ) with G = C ∞ (Σ , SU (2)) → f ( g ( s ( e )) − 1 A ( e ) g ( t ( e ))) ⊲ Gauss constraint : f ( A ( e )) �− ⊲ Diffeomorphisms : f ( A ( e )) �− → f ( A ( ϕ ( e ))) ⊲ Similar action for the variables E f ( S ) ⊲ Symmetries are automorphisms of classical algebra IHES - november 2013 Karim NOUI Black Holes in LQG 8/20

Loop Quantum Gravity in a nut shell Unicity of a (space) Diff-invariant representation Construction of the quantum algebra A ⊲ Elements of A cl are a = ( f , u ) : f ∈ Cyl and u derivatives ⊲ Ideal I of the algebra : a 1 a 2 − a 2 a 1 − i � { a 1 , a 2 } ⊲ A = A cl / I with action of automorphisms S Representation theory of A : GNS framework ⊲ Any representation of A is a direct sum of cyclic representations ⊲ A cyclic representation is characterized by a positive state ω ∈ A ∗ ⊲ The representation : ( H , π, Ω) with Ω cyclic : π ( A )Ω dense in H ⊲ ω is required to be invariant under S ⊲ LOST Theorem : The representation is unique ! Properties of the representation : ⊲ Irreducible unitary infinite dimensional representation ⊲ Hilbert space non separable : non countable basis ⊲ Action of Diff (Σ) is not weakly continuous ⊲ Stone : Infinitesimal generators of Diff (Σ) do not exist IHES - november 2013 Karim NOUI Black Holes in LQG 9/20

Loop Quantum Gravity in a nut shell Spin-Network basis Kinematical states : basis of spin-networks ⊲ They are generalizations of Wilson loops with nodes ℓ 1 ℓ i are oriented links ℓ 2 n 1 n 2 n i are nodes ℓ 3 ⊲ Harmonic analysis on SU (2) : ℓ → irreps and n → intertwiners Geometric operators : area and volume become operators ⊲ Area acts on edges and Volume on vertices Γ A ( S ) | S � = 8 πγ � G � � j P ( j P + 1) | S � P ∈S∩ Γ S c 3 ⊲ The spectra are discrete : existence of a minimal length IHES - november 2013 Karim NOUI Black Holes in LQG 10/20

Loop Quantum Gravity in a nut shell Picture of space at the Planck scale From the kinematics, Space is discrete... ⊲ Edges carry quanta of area, nodes carry quanta of volume IHES - november 2013 Karim NOUI Black Holes in LQG 11/20

Black Holes in LQG : a quick review Heuristic picture : model for real γ a H = 8 πγℓ 2 � � j ( j + 1) P j Edges crossing spherical BH Only spins 1 / 2 contribute to the area √ ⊲ Number of edges : a H = 8 πγℓ 2 3 P × N × 2 ⊲ Number of states : number of singlets in (1 / 2) ⊗ N = ⇒ Ω ∼ 2 N ⊲ Bekenstein-Hawking formula for the entropy when a H ≫ ℓ 2 P 2 log(2) ⇒ γ = log(2) S = log(Ω) ∼ N log(2) = √ a H = √ . 8 πγℓ 2 3 π 3 P Refined models : all spins contribute ⊲ The value of γ changes. Why Is γ relevant at the quantum level ? IHES - november 2013 Karim NOUI Black Holes in LQG 12/20

Black Holes in LQG : a quick review Hamiltonian formulation of Black Holes The Horizon considered as a boundary n o i t i d F n o r e c e y M d r a a d 2 t a n u ∆ o n b o H i t a I i d a M 1 r i o ⊲ Geometric conditions : null-surface, no expansion, F ∝ E ⊲ Restriction (here) : spherically symmetric black holes Sympletic structure with a horizon boundary ⊲ Requirement : conservation of symplectic structure ⊲ In terms of Ashtekar-Barbero variables � a H � δ [1 E i ∧ δ 2] A i − δ 1 A i ∧ δ 2 A i 8 π G γ ω ( δ 1 , δ 2 ) = π (1 − γ 2 ) M H ⊲ Symplectic structure of SU (2) Chern-Simons theory with k ∝ a H IHES - november 2013 Karim NOUI Black Holes in LQG 13/20