U(N) FRAMEWORK FOR LOOP QUANTUM GRAVITY: A PRELIMINARY BLACK HOLE MODEL

U(N) FRAMEWORK FOR LOOP QUANTUM GRAVITY: A PRELIMINARY BLACK HOLE MODEL Iñaki Garay Programa de Pós-Graduação em Física Universidade Federal do Pará In collaboration with Jacobo Díaz Polo and Etera Livine Belém. August 5, 2014.

Outline OUTLINE 1. Introduction 2. Loop quantum gravity Main concepts Black holes in LQG 3. The U(N) framework for LQG intertwiners Intertwiner space The Schwinger representation U(N) algebra and operators 4. The 2-vertex graph and the global U(N) symmetry 5. Looking for a black hole model Black hole creation operator 3 + N vertex graph 6. Generalizations and conclusions Iñaki Garay (UFPA) 2

Introduction OUTLINE 1. Introduction 2. Loop quantum gravity Main concepts Black holes in LQG 3. The U(N) framework for LQG intertwiners Intertwiner space The Schwinger representation U(N) algebra and operators 4. The 2-vertex graph and the global U(N) symmetry 5. Looking for a black hole model Black hole creation operator 3 + N vertex graph 6. Generalizations and conclusions Iñaki Garay (UFPA) 3

Introduction INTRODUCTION Why Quantum Gravity? Singularities and the incompleteness of General Relativity. The matter/geometry equivalence. Quantizing gravitational waves. Microstates for the black hole entropy. Main difficulties In GR the space-time is a dynamical object. The quantization of gravity is the quantization of the spacetime itself (we need background independence). Adapt Quantum Mechanics to a fluctuating space-time. Loop Quantum Gravity approach Canonical and non-perturbative quantization of GR. NO unification, NO supersymmetry. Strong emphasis on the background independence and diff. invariance. It is not a complete theory so far. Dynamics!! Iñaki Garay (UFPA) 4

Introduction INTRODUCTION U(N) framework for LQG New and refreshing tool to deal with the LQG problems. Based on the Schwinger representation of the SU(2) Lie group. Nice results about the implementation of the dynamics and semiclassical states for simple models. Simple models to extract physics from LQG 2-vertex model: implementation of the dynamics, nice application to cosmology and interesting black hole properties. 3+N vertex model: sensible alternative to study dynamical processes on black holes. Radiation! Tinsel model : obtaining a field theory from LQG. Iñaki Garay (UFPA) 5

Loop quantum gravity OUTLINE 1. Introduction 2. Loop quantum gravity Main concepts Black holes in LQG 3. The U(N) framework for LQG intertwiners Intertwiner space The Schwinger representation U(N) algebra and operators 4. The 2-vertex graph and the global U(N) symmetry 5. Looking for a black hole model Black hole creation operator 3 + N vertex graph 6. Generalizations and conclusions Iñaki Garay (UFPA) 6

Loop quantum gravity Main concepts OUTLINE 1. Introduction 2. Loop quantum gravity Main concepts Black holes in LQG 3. The U(N) framework for LQG intertwiners Intertwiner space The Schwinger representation U(N) algebra and operators 4. The 2-vertex graph and the global U(N) symmetry 5. Looking for a black hole model Black hole creation operator 3 + N vertex graph 6. Generalizations and conclusions Iñaki Garay (UFPA) 7

Loop quantum gravity Main concepts INTRODUCTION TO LQG Loop Quantum Gravity proposes a canonical quantization of GR. The representation is fundamentally different from the usual background dependent Fock representation. Hilbert space of LQG is constructed with spin-networks (functions over oriented graphs with edges labeled by SU(2) irreps.). Iñaki Garay (UFPA) 8

Loop quantum gravity Main concepts INTRODUCTION TO LQG ADM decomposition + Ashtekar variables. Constraints!! Kinematical sector is well described. But... Hamiltonian constraint is still an open problem. Geometrical meaning. Area and volume operators. Edges correspond to area elements. Vertices correspond to volume elements. Iñaki Garay (UFPA) 9

Loop quantum gravity Main concepts INTRODUCTION TO LQG Motivation The Challenge of Quantum Gravitaty Classical Formulation Problem of Time Secured Land Quantum Kinematics Geometrical meaning. Unknown Area Territory and volume operators. Open Questions and Summary Kinematical Coherent States Edges correspond to area elements. Spin Vertices Network correspond Basis T γ,j,i to volume H j Hermite elements. Polynomials Kinematical spatial geometry operators v 8,I 8 e 15,j 15 e 19,j 19 v 10,I 10 e 18,j 18 e 22,j 22 e 17,j 17 e 20,j 20 v 7,I 7 e 16,j 16 e 14,j 14 v 6,I 6 vm,im e 21,j 21 v 9,I 9 e 8,j 8 e 12,j 12 e 13,j 13 en,jn e 11,j 11 v 4,I 4 e 10,j 10 v 5,I 5 v 1,I 1 e 1,j 1 e 2,j 2 e 3,j 3 e 6,j 6 e 7,j 7 e 5,j 5 e 9,j 9 e 4,j 4 v 3,I 3 v 2,I 2 Thomas Thiemann Elements of Loop Quantum Gravity Iñaki Garay (UFPA) 9

Iñaki nimation: Garay (UFPA) 9 Loop quantum gravity Main concepts Unknown Territory Open Questions and Summary Kinematical spatial geometry operators Kinematical Coherent States INTRODUCTION TO LQG.. or on Faces of Dual Cell Complex (Triangulation)

Loop quantum gravity Black holes in LQG OUTLINE 1. Introduction 2. Loop quantum gravity Main concepts Black holes in LQG 3. The U(N) framework for LQG intertwiners Intertwiner space The Schwinger representation U(N) algebra and operators 4. The 2-vertex graph and the global U(N) symmetry 5. Looking for a black hole model Black hole creation operator 3 + N vertex graph 6. Generalizations and conclusions Iñaki Garay (UFPA) 10

oach: Ignore black hole interior Loop quantum gravity n inner boundary Black holes in LQG BLACK HOLES IN LQG d by spin-networks ced by edges orizon requirements on the boundary: Isolated horizon on surface given by els (as usual in LQG) rvature concentrated Fernando Barbero Effective approach: Ignore black hole interior Locally impose horizon requirements: Isolated horizon Iñaki Garay (UFPA) 11

Loop quantum gravity Black holes in LQG orizon requirements on the boundary: Isolated horizon BLACK d by spin-networks HOLES IN LQG ced by edges on surface given by els (as usual in LQG) rvature concentrated Fernando Barbero Bulk: described by spin-networks. Horizon: pierced by edges. Fix the value of the macroscopic area of the horizon Compute the number of compatible horizon states: S = log N(A) The Bekenstein formula is obtained: S = 1 A 4l 2 P Iñaki Garay (UFPA) 11

Loop quantum gravity n inner boundary Black holes in LQG orizon requirements on the boundary: Isolated horizon BLACK HOLES IN LQG d by spin-networks ced by edges on surface given by els (as usual in LQG) rvature concentrated Fernando Barbero But... It is a kinematical description. No description of radiation processes!! Iñaki Garay (UFPA) 11

The U(N) framework for LQG intertwiners OUTLINE 1. Introduction 2. Loop quantum gravity Main concepts Black holes in LQG 3. The U(N) framework for LQG intertwiners Intertwiner space The Schwinger representation U(N) algebra and operators 4. The 2-vertex graph and the global U(N) symmetry 5. Looking for a black hole model Black hole creation operator 3 + N vertex graph 6. Generalizations and conclusions Iñaki Garay (UFPA) 12

The U(N) framework for LQG intertwiners Intertwiner space OUTLINE 1. Introduction 2. Loop quantum gravity Main concepts Black holes in LQG 3. The U(N) framework for LQG intertwiners Intertwiner space The Schwinger representation U(N) algebra and operators 4. The 2-vertex graph and the global U(N) symmetry 5. Looking for a black hole model Black hole creation operator 3 + N vertex graph 6. Generalizations and conclusions Iñaki Garay (UFPA) 13

The U(N) framework for LQG intertwiners Intertwiner space INTERTWINER SPACES FOR THE LIE GROUP SU(2) Intertwiners between N irreducible representations of spin j 1,.., j N : H j1,..,j N Inv[V j 1.. V j N ]. Space of intertwiners with N legs and fixed total area J = i j i : H (J) N H j1,..,j N. i j i =J Hilbert space of N-valent intertwiners: H N {j i } H j1,..,j N = J N H (J) N. Iñaki Garay (UFPA) 14

The U(N) framework for LQG intertwiners The Schwinger representation OUTLINE 1. Introduction 2. Loop quantum gravity Main concepts Black holes in LQG 3. The U(N) framework for LQG intertwiners Intertwiner space The Schwinger representation U(N) algebra and operators 4. The 2-vertex graph and the global U(N) symmetry 5. Looking for a black hole model Black hole creation operator 3 + N vertex graph 6. Generalizations and conclusions Iñaki Garay (UFPA) 15

The U(N) framework for LQG intertwiners The Schwinger representation THE SCHWINGER REPRESENTATION Representation of the su(2) Lie algebra in terms of harmonic oscillators. 2N oscillators with creation operators a i, b i (with i = 1,..., N): [a i, a j ] = [b i, b j ] = δ ij, [a i, b j ] = 0. The generators of the SU(2) on each leg of the intertwiner: J z i = 1 2 (a i a i b i b i), J + i = a i b i, J i = a i b i, E i = (a i a i+b i b i). Standard commutation algebra (E i is the Casimir operator): [J z i, J ± i ] = ±J ± i, [J + i, J i ] = 2J z i, [E i, J i ] = 0. Correspondence with the standard j, m basis of SU(2) reps.: m i = 1 2 (na i n b i ), j i = 1 2 (na i + n b i ). Iñaki Garay (UFPA) 16

The U(N) framework for LQG intertwiners The Schwinger representation Iñaki Garay (UFPA) 17 of SU(2) in terms of two harmonic oscillators, we replace all lines SCHWINGER REPRESENTATION ON THE EDGES j a b

The U(N) framework for LQG intertwiners U(N) algebra and operators OUTLINE 1. Introduction 2. Loop quantum gravity Main concepts Black holes in LQG 3. The U(N) framework for LQG intertwiners Intertwiner space The Schwinger representation U(N) algebra and operators 4. The 2-vertex graph and the global U(N) symmetry 5. Looking for a black hole model Black hole creation operator 3 + N vertex graph 6. Generalizations and conclusions Iñaki Garay (UFPA) 18

The U(N) framework for LQG intertwiners U(N) algebra and operators U(N) ALGEBRA Intertwiner states are, by definition, invariant under the global SU(2) action generated by: J z = N Ji z, i=1 J ± = i J ± i. The simplest family of invariant operators was identified E ij : H (J) N H(J) N E ij = a i a j + b i b j, E ij = E ji. These operators are invariant under global SU(2) transformations and form a u(n) algebra: [ J, E ij ] = 0, [E ij, E kl ] = δ jk E il δ il E kj. All operators E ij commute with E = i E i, thus they leave the total area J invariant: [E ij, E] = 0. Iñaki Garay (UFPA) 19

The U(N) framework for LQG intertwiners U(N) algebra and operators CHANGING THE AREA Annihilation and creation ops. to move between the spaces H (J) N : F ij = (a i b j a j b i ), F ji = F ij. F ij : H (J) N H(J 1) N F ij : H (J) N H(J+1) N Invariant under global SU(2) transformations, but they do not commute anymore with the total area operator E. Closed algebra together with the operators E ij : [E ij, E kl ] = δ jk E il δ il E kj [E ij, F kl ] = δ il F jk δ ik F jl, [E ij, F kl ] = δ jkf il δ jl F ik, [F ij, F kl ] = δ ike lj δ il E kj δ jk E li + δ jl E ki + 2(δ ik δ jl δ il δ jk ), [F ij, F kl ] = 0, [F ij, F kl ] = 0. Iñaki Garay (UFPA) 20

The 2-vertex graph and the global U(N) symmetry OUTLINE 1. Introduction 2. Loop quantum gravity Main concepts Black holes in LQG 3. The U(N) framework for LQG intertwiners Intertwiner space The Schwinger representation U(N) algebra and operators 4. The 2-vertex graph and the global U(N) symmetry 5. Looking for a black hole model Black hole creation operator 3 + N vertex graph 6. Generalizations and conclusions Iñaki Garay (UFPA) 21

s in two vertices α and β, linked by N edges all oriente The 2-vertex graph and the global U(N) symmetry s i = 1..N. We now have U(N) operators acting at each THE 2-VERTEX GRAPH i. matching conditions to ensure that we are working with The simplest non-trivial graph for spin network states in LQG: a graph with two vertices linked by N edges. e 1 e 2 e3 e 4 α e N 3 e N 2 β e N 1 e N Iñaki Garay (UFPA) 22

Finally, the matching conditions to ensure that we are working with true spin n The 2-vertex graph and the global U(N) symmetry for THE all edges 2-VERTEX i. GRAPH e 1 e 2 e3 α e 4 e N 3 e N 2 β e N 1 e N Each edge must carry a unique SU(2) representation, thus the FIG. 3: The 2-vertex graph with vertices α and β and the N edges linking them. spin j i seen from α or β must be the same. Matching conditions: E a basic loop consisting in two edges (ij). Then i E (α) we apply i E (β) our i = 0. conjectured formula The Hilbert space of spin network states for this 2-vertex graph is: 1 = Ei + 1 (α) (F ij F (β) ij + E (α) ij E j + 1 E(β) ij + E (α) ji E (β) ji + F (α) ij F (β) ij 1 2 ) H H (α) Ei + 1 j 1,..,j N H (β) j 1,..,j N. {ji } same expression as we have derived in the earlier work [7] apart from the global Iñaki Garay (UFPA) 22

The 2-vertex graph and the global U(N) symmetry GLOBAL u(n) ALGEBRA We can introduce the operators: E ij E (α) ij E (β) ji They form a u(n) algebra: [E ij, E kl ] = δ jk E il δ il E kj. E k are part of this larger u(n) algebra. Looking for a U(N) invariant subspace The subspace of spin network states invariant under the U(N)-action: [ ] 2 H inv Inv 2 U(N) H = Inv U(N) [H 2 ] = Inv U(N) N H (J β) N H (J) N J α,j β H (Jα) are irreducible U(N)-representations [L.Freidel, E.Livine]. U(N)-invariance J α = J β. There exists a unique invariant vector J H (J) N 2 H inv = ( J N C J J (f ) J 0 = ij F (α) ij H(J) N. F ) (β) J ij 0 Iñaki Garay (UFPA) 23

Looking for a black hole model OUTLINE 1. Introduction 2. Loop quantum gravity Main concepts Black holes in LQG 3. The U(N) framework for LQG intertwiners Intertwiner space The Schwinger representation U(N) algebra and operators 4. The 2-vertex graph and the global U(N) symmetry 5. Looking for a black hole model Black hole creation operator 3 + N vertex graph 6. Generalizations and conclusions Iñaki Garay (UFPA) 24

Looking for a black hole model Black hole creation operator OUTLINE 1. Introduction 2. Loop quantum gravity Main concepts Black holes in LQG 3. The U(N) framework for LQG intertwiners Intertwiner space The Schwinger representation U(N) algebra and operators 4. The 2-vertex graph and the global U(N) symmetry 5. Looking for a black hole model Black hole creation operator 3 + N vertex graph 6. Generalizations and conclusions Iñaki Garay (UFPA) 25

Looking for a black hole model Black hole creation operator BLACK HOLE CREATION OPERATOR Physical interpretation of our states J (f ) J 0. Reduced density matrix on the system α: ρ J J (f ) J 0 0 f J, ρ α Tr β ρ Tr H (J)(β) N ρ Theorem The reduced density matrix on α is the totally mixed state on the Hilbert space of N-valent intertwiners with fixed total area J: ρ α = 1 I (J) D H. N,J N Iñaki Garay (UFPA) 26

Looking for a black hole model Black hole creation operator BLACK HOLE CREATION OPERATOR Reduced density matrix on the system α: ρ J J (f ) J 0 0 f J, ρ α Tr β ρ Tr (J)(β) H ρ N Theorem The reduced density matrix on α is the totally mixed state on the Hilbert space of N-valent intertwiners with fixed total area J: Proof ρ α = 1 I (J) D H. N,J N The full density matrix is by definition invariant under the U(N)-action acting on both vertices: ρ = (U Ū) ρ (U 1 Ū 1 ). Uρ α U 1 = Tr β (U I)ρ(U 1 I) = Tr β (I Ū 1 )ρ(i Ū) = Tr β ρ = ρ α Normalization makes the rest. Iñaki Garay (UFPA) 26

Looking for a black hole model Black hole creation operator BLACK HOLE CREATION OPERATOR Reduced density matrix on the system α: ρ J J (f ) J 0 0 f J, ρ α Tr β ρ Tr (J)(β) H ρ N Theorem The reduced density matrix on α is the totally mixed state on the Hilbert space of N-valent intertwiners with fixed total area J: ρ α = 1 I (J) D H. N,J N Mixed state with maximal entropy on the system α. Our initial pure state is maximally entangled. Maximal ignorance state for each of the two sub-systems. (f ) J as black hole creation operators? Hilbert space of states for a (quantum) black hole in LQG correspond to the intertwiners with the same fixed total area. Iñaki Garay (UFPA) 26

Looking for a black hole model Black hole creation operator LIMITATIONS AND GENERALIZATIONS Limitations The two vertices are strongly glued with the matching conditions. They grow or shrink together, so there is no freedom for different evolutions for the two vertices. Generalizations Change the underlying graph and use a more complicated spin network setting. Iñaki Garay (UFPA) 27

Looking for a black hole model 3 + N vertex graph OUTLINE 1. Introduction 2. Loop quantum gravity Main concepts Black holes in LQG 3. The U(N) framework for LQG intertwiners Intertwiner space The Schwinger representation U(N) algebra and operators 4. The 2-vertex graph and the global U(N) symmetry 5. Looking for a black hole model Black hole creation operator 3 + N vertex graph 6. Generalizations and conclusions Iñaki Garay (UFPA) 28

ing a third vertex to the graph, which will be part of the out region. This new Looking for a black hole model 3 + N vertex graph aph by N edges, each one meeting one of the old edges at a new trivalent vertex n3 figure + N 4. VERTEX The interior GRAPH region continues to be the vertex α and the boundary trivalent vertices are considered as part of the exterior region, but can also be cal boundary between the in and out regions. ertices α, Eβ, ij andωf are ij operators linked by Nacting edgesover and we thehave three N additional vertices. (auxiliary) Besides, trivalent operators E a i ab and F a i ab (i = 1,..., N and a, b = α, β, γ) acting on the N trivalent vertices. n the new vertices in analogy to the ones we already have for the simpler 2-vertex ) and F (Ω) ij acting on the vertex Ω as well as E (ai) jk and F (ai) jk, with i = 1,..., N rivalent vertices. Iñaki it isgaray possible (UFPA) to shrink the area surrounding the in region without resulting29 in

n figure 4. The interior region continues to be the vertex α and the boundary Looking for a black hole model 3 + N vertex graph trivalent vertices are considered as part of the exterior region, but can also be cal boundary between the in and out regions. 3 + N VERTEX GRAPH ertices α, Eβ, ij andωf are ij operators linked by Nacting edgesover and we thehave three N additional vertices. (auxiliary) Besides, trivalent operators E The action of the a i ab and radiation F a i ab (i = 1,..., N and a, b = α, β, γ) acting on operator on the left and the the N trivalent vertices. round the vertices α and Ω without affecting the spins l n the new Evaporation vertices in analogy of the interior to the ones region we using already radiation have for the like simpler operators: 2-vertex ) and F (Ω) ij acting on the vertex Ω as well as E (ai) jk and F (ai) jk, with i = 1,..., N rivalent vertices. R ij = F (α) ij E (a i ) Ωα it is possible to shrink the area surrounding the F (Ω) ij E (a j ) Ωα in region without resulting in One can even think of the in region totally disappearing, and the out region Iñaki Garay (UFPA) 29 framework it is possible to introduce radiation-like operators. As a first attempt of the boundary by radiating some spin units to t

n figure 4. The interior region continues to be the vertex α and the boundary Looking for a black hole model 3 + N vertex graph trivalent vertices are considered as part of the exterior region, but can also be cal boundary between the in and out regions. 3 + N VERTEX GRAPH ertices α, Eβ, ij andωf are ij operators linked by Nacting edgesover and we thehave three N additional vertices. (auxiliary) Besides, trivalent operators E a i ab The action of the and radiation F a i ab (i = 1,..., N and a, b = α, β, γ) acting on the N trivalent vertices. operator on the left and the round n the new Evaporation the vertices vertices in analogy of theαinterior and to the ones region Ω without we using already radiation affecting have for the like the simpler operators: spins l 2-vertex ) and F (Ω) ij acting on the vertex R Ω as well as E (ai) jk and F (ai) ij = F (α) jk, with i = 1,..., N ij E (a i ) Ωα rivalent vertices. F (Ω) ij E (a j ) Ωα it is possible The additional to shrink the third area vertex surrounding give us the enough in freedom region without to rotate resulting the in One caninterior even think withof respect the in toregion the exterior totally region. disappearing, and the out region Iñaki framework Garay (UFPA) it is possible to introduce radiation-like operators. As a first attempt 29 of the boundary by radiating some spin units to t

Generalizations and conclusions OUTLINE 1. Introduction 2. Loop quantum gravity Main concepts Black holes in LQG 3. The U(N) framework for LQG intertwiners Intertwiner space The Schwinger representation U(N) algebra and operators 4. The 2-vertex graph and the global U(N) symmetry 5. Looking for a black hole model Black hole creation operator 3 + N vertex graph 6. Generalizations and conclusions Iñaki Garay (UFPA) 30

Generalizations and conclusions TINSEL GRAPH z i L z i 1 z i R z i 2 z i+1 L z i+1 1 z i+1 R z i+1 2 Local degrees of freedom Associate a field theory in a possible continuum limit? Iñaki Garay (UFPA) 31

Generalizations and conclusions CONCLUSIONS Nowadays there is not a complete and tested quantum description of gravity. LQG represents a background independent non-perturbative quantization of General Relativity. But the dynamics is not implemented yet. The U(N) framework represents a new way to study the open problems of LQG. Simple models to gain intuition about the theory and obtain physical results. 2-vertex graph: successful implementation of the dynamics, nice cosmological results and preliminary results about black hole models within LQG. 3+N graph: promising model for Hawking radiation in LQG. Tinsel graph... Iñaki Garay (UFPA) 32