Loop Quantum Gravity 2. The quantization : discreteness of space

Transcription

1 Loop Quantum Gravity 2. The quantization : discreteness of space Karim NOUI Laboratoire de Mathématiques et de Physique Théorique, TOURS Astro Particules et Cosmologie, PARIS Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 1/16

2 Overview of the second part Introduction Back to the classical phase space : the polymer hypothesis Loop Quantum Gravity: Quantization Kinematics : discreteness of space The question of the dynamics : from 3D to 4D Conclusion Towards physical applications Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 2/16

3 Introduction Back to the classical phase space Hamiltonian analysis of Ashtekar-Barbero gravity Classical phase space P = T A with A = {SU(2) connections} Classical variables : an su(2)-connection A i a and an electric field E a i {E a i (x), A j b (y)} = κ δa b δj i δ(x y), κ = 8πGγ Symmetries : action of S = C (Σ, SU(2)) Diff M S Fun(P) Fun(P) Physical observables are S-invariant functions Fun(P) inv Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 3/16

4 Introduction Back to the classical phase space Hamiltonian analysis of Ashtekar-Barbero gravity Classical phase space P = T A with A = {SU(2) connections} Classical variables : an su(2)-connection A i a and an electric field E a i {E a i (x), A j b (y)} = κ δa b δj i δ(x y), κ = 8πGγ Symmetries : action of S = C (Σ, SU(2)) Diff M S Fun(P) Fun(P) Physical observables are S-invariant functions Fun(P) inv Schrodinger-like quantization A and E are promoted into operators : [Â, Ê] = i κi Choice of polarization : A is the coordinate and states are Ψ(A) Â Ψ(A) = A Ψ(A), Ê Ψ(A) = i κ δψ(a) δa No measure dµ(a) on Fun(A) to define a Hilbert structure Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 3/16

5 Introduction The polymer hypothesis The space of discrete connections Let γ a graph with E edges and V vertices One restricts A to A(γ) = (A e1,, A ee ) with A e = P exp( A) SU(2) where e is an edge Space of cylindrical functions Cyl(γ) e Cyl(γ) = Fun(A(γ)) Fun(SU(2) E ) The configuration space is the direct sum Cyl(Σ) = γ Cyl(γ) Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 4/16

6 Introduction The polymer hypothesis The space of discrete connections Let γ a graph with E edges and V vertices One restricts A to A(γ) = (A e1,, A ee ) with A e = P exp( A) SU(2) where e is an edge Space of cylindrical functions Cyl(γ) e Cyl(γ) = Fun(A(γ)) Fun(SU(2) E ) The configuration space is the direct sum Cyl(Σ) = γ Cyl(γ) The discrete electric fields To each surface S on Σ and any v su(2) E v (S) = Tr(v E adj ). With a given graph γ, S are dual to edges e Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 4/16 S

7 Kinematics : discreteness of space The holonomy-flux algebra Poisson structure on the discrete phase space The Poisson structure is inherited from canonical analysis {E v (S), ϕ(a e )} = κ R v A e<x ϕ(a e<x A e>x ) with x = S e E v (S) acts as a vector field : A(e) A(e <x ) A(e >x ). Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 5/16

8 Kinematics : discreteness of space The holonomy-flux algebra Poisson structure on the discrete phase space The Poisson structure is inherited from canonical analysis {E v (S), ϕ(a e )} = κ R v A e<x ϕ(a e<x A e>x ) with x = S e E v (S) acts as a vector field : A(e) A(e <x ) A(e >x ). Action of symmetries : G Diff Σ with G = SU(2) V Gauss constraint : ϕ(a(e)) ϕ(g(s(e)) 1 A(e)g(t(e))) Diffeomorphisms : ϕ(a(e)) ϕ(a(ξ(e))) Similar action on the variables E f (S) Symmetries are automorphisms of classical Poisson algebra Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 5/16

9 Kinematics : discreteness of space Quantization of the phase space Construction of the algebra of quantum operators Phase space P : a = (ϕ, v) with ϕ Cyl(Σ) and v in su(2) a are promoted into operators â They form a non-commutative algebra P q It is associative, unital and the product is defined by â 1 â 2 = a 1 ˆa 2 + i {a 1 ˆ, a 2 } + o( ) 2 G Diff Σ acts as automorphisms on P q Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 6/16

10 Kinematics : discreteness of space Quantization of the phase space Construction of the algebra of quantum operators Phase space P : a = (ϕ, v) with ϕ Cyl(Σ) and v in su(2) a are promoted into operators â They form a non-commutative algebra P q It is associative, unital and the product is defined by â 1 â 2 = a 1 ˆa 2 + i {a 1 ˆ, a 2 } + o( ) 2 G Diff Σ acts as automorphisms on P q Representation theory : GNS framework Any representation of P q is a direct sum of cyclic representations A cyclic representation is characterized by a positive state ω P q The representation : (H, π, Ω) with Ω cyclic : π(p q )Ω dense in H ω is required to be invariant under G Diff Σ LOST Theorem : The representation is unique! Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 6/16

11 Kinematics : discreteness of space Kinematical states The Ashtekar-Lewandowski measure Generalization of the Wilson loops : elements of Cyl(γ) Hilbert structure defined from Ashtekar-Lewandowski measure : dµ AL = dµ(a e ) With dµ(a e ) the SU(2) Haar measure The completion of Cyl defines the Hilbert space H 0 e Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 7/16

12 Kinematics : discreteness of space Kinematical states The Ashtekar-Lewandowski measure Generalization of the Wilson loops : elements of Cyl(γ) Hilbert structure defined from Ashtekar-Lewandowski measure : dµ AL = dµ(a e ) With dµ(a e ) the SU(2) Haar measure The completion of Cyl defines the Hilbert space H 0 Properties of the representation : Irreducible unitary infinite dimensional representation Action of Diff Σ is not weakly continuous Stone : Infinitesimal generators of Diff Σ do not exist e Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 7/16

13 Kinematics : discreteness of space Kinematical states The Ashtekar-Lewandowski measure Generalization of the Wilson loops : elements of Cyl(γ) Hilbert structure defined from Ashtekar-Lewandowski measure : dµ AL = dµ(a e ) With dµ(a e ) the SU(2) Haar measure The completion of Cyl defines the Hilbert space H 0 Properties of the representation : Irreducible unitary infinite dimensional representation Action of Diff Σ is not weakly continuous Stone : Infinitesimal generators of Diff Σ do not exist Kinematical states : invariance under G Diff Σ SU(2) invariance at the nodes of the graphs : H inv Identify states related by a diffeomorphism States of H diff are labelled by knots Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 7/16 e

14 Kinematics : discreteness of space Spin-Network basis Harmonic analysis on SU(2) Unitary irreducible representations of SU(2) are labelled by spins j ϕ(g) = j Tr(π j (g) ˆϕ j ) where π j : SU(2) V j for any g SU(2) and dim(v j ) = 2j + 1 Invariant tensor (intertwiner) ι : n j=1 V j C s.t. ι (n) (g) = ι Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 8/16

15 Kinematics : discreteness of space Spin-Network basis Harmonic analysis on SU(2) Unitary irreducible representations of SU(2) are labelled by spins j ϕ(g) = j Tr(π j (g) ˆϕ j ) where π j : SU(2) V j for any g SU(2) and dim(v j ) = 2j + 1 Invariant tensor (intertwiner) ι : n j=1 V j C s.t. ι (n) (g) = ι Spin-networks form an orthonormal basis of H inv n1 j1 j2 j3 n2 j n x Harmonic analysis on SU(2) : e irreps and n intertwiners A spin-network S is a colored graph (γ; j, ι) Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 8/16

16 Kinematics : discreteness of space Geometrical operators have discrete spectrum ( 95) Area operator A(S) acting on H k : A(S) = S n a Ei an bei b d 2 σ Quantum area operator : S = N n S n A(S) = lim Ei (S n )E i (S n ) with E i (S n ) = E i N S n Spectrum and Quanta of area : A(S) S = 8πγl 2 P P S Γ jp (j P + 1) S G n S x t Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 9/16

17 Kinematics : discreteness of space Geometrical operators have discrete spectrum ( 95) Area operator A(S) acting on H k : A(S) = S n a Ei an bei b d 2 σ Quantum area operator : S = N n S n A(S) = lim Ei (S n )E i (S n ) with E i (S n ) = E i N S n Spectrum and Quanta of area : A(S) S = 8πγl 2 P P S Γ jp (j P + 1) S G n S x t Volume operator V(R) acting on H 0 Classical volume on a domain R : V(R) = R d 3 x It acts on the nodes of S : discrete spectrum Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 9/16 ɛ abc ɛ ijk E ai E bj E ck 3!

18 Kinematics : discreteness of space Picture of space at the Planck scale From the kinematics, Space is discrete... Edges carry quanta of area, nodes carry quanta of volume Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 10/16

19 Question of the dynamics From the hamiltonian constraint to Spin-Foam models Difficulty with the Hamiltonian constraint The Hamiltonian constraint in the continuum H = E a E b (F ab (1 + γ 2 )K a K b ) where K = γγ(e) + A How to promote it into a quantum operator Ĥ : Γ(E) is non-linear How to regularize it in terms of holonomies A e and fluxes E v (S)? Requirements : the constraint algebra must close : no anomalies! Proposal : Thiemann (master) constraint... under construction Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 11/16

20 Question of the dynamics From the hamiltonian constraint to Spin-Foam models Difficulty with the Hamiltonian constraint The Hamiltonian constraint in the continuum H = E a E b (F ab (1 + γ 2 )K a K b ) where K = γγ(e) + A How to promote it into a quantum operator Ĥ : Γ(E) is non-linear How to regularize it in terms of holonomies A e and fluxes E v (S)? Requirements : the constraint algebra must close : no anomalies! Proposal : Thiemann (master) constraint... under construction The idea of Spin-Foam models : extracting physical scalar product We need the physical scalar product between kinematical states S 1 S 2 phys = S 1 δ(ĥ) S 2 kin Gravity path integral would exactly do that in the continuum How to make sense of the path integral where metric is replaced by combinatorial structures? Spin-Foam models Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 11/16

21 Question of the dynamics Classical gravity in three dimensions 3D Gravity is a Topological QFT The dynamics governed by a BF-action on M S[e, ω] = e F (ω) Λ 3 e e e M With Lie algebra su(2) or su(1, 1) and, the Killing form Classical solutions are locally homogeneous geometries : S 3, H 3, E 3 Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 12/16

22 Question of the dynamics Classical gravity in three dimensions 3D Gravity is a Topological QFT The dynamics governed by a BF-action on M S[e, ω] = e F (ω) Λ 3 e e e M With Lie algebra su(2) or su(1, 1) and, the Killing form Classical solutions are locally homogeneous geometries : S 3, H 3, E 3 3D Gravity is Chern-Simons theory Gauge group of gravity is the isometry group of classical solutions : SU(2) SU(2), SL(2, C), SU(2) R 3 BF-action is equivalent to Chern-Simons theory with k = Λ S[A] = 4π Tr (A da + 23 ) k A A A M Solutions are flat connections : topological degrees of freedom Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 12/16

23 Question of the dynamics Path integral of 3D gravity The path integral is a well-defined topological invariant In the case of Euclidean gravity with a positive Λ The Witten-Reshetikhin-Turaev invariant when G = SU(2) Z WRT (O, M) = [DA]O(A) exp(is[a]) When O is a Wilson loop, Z WRT (O, M) is the Jones polynomial Underlying quantum group U q (su(2)) with q = exp(2π/(k + 2)) Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 13/16

24 Question of the dynamics Path integral of 3D gravity The path integral is a well-defined topological invariant In the case of Euclidean gravity with a positive Λ The Witten-Reshetikhin-Turaev invariant when G = SU(2) Z WRT (O, M) = [DA]O(A) exp(is[a]) When O is a Wilson loop, Z WRT (O, M) is the Jones polynomial Underlying quantum group U q (su(2)) with q = exp(2π/(k + 2)) TheTuraev-Viro invariant Path integral of Euclidean gravity with positive Λ Z TW (M) = Z WRT (M) 2 Explicit non-perturbative calculation of the invariant 2 Z TW (S 3 ) = k + 2 sin( π k + 2 ) Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 13/16

25 Question of the dynamics The Turaev-Viro Spin-Foam model Based only on combinatorial structures A colored (spins) triangulation (tetrahedron) of M Z TV (M) = K #v {j e} [2j e + 1] q ( ) j t i jt i { 1 j2 t j3 t j4 t j5 t j6 t e T t T Based on representations and recouplings of quantum groups [x] q = qx q x q q 1 K = (q q 1 ) 2 2( Λ + 2) } q Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 14/16

26 Question of the dynamics The Turaev-Viro Spin-Foam model Based only on combinatorial structures A colored (spins) triangulation (tetrahedron) of M Z TV (M) = K #v {j e} [2j e + 1] q ( ) j t i jt i { 1 j2 t j3 t j4 t j5 t j6 t e T t T Based on representations and recouplings of quantum groups [x] q = qx q x q q 1 K = (q q 1 ) 2 2( Λ + 2) Relation to the physical scalar product of LQG Ponzano-Regge model with Λ = 0 : classical group Total agreement between physical scalar product and PR invariant Realization of the idea that path integral is scalar product between spin-networks Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 14/16 } q

27 Question of the dynamics A glimpse to Spin-Foam models in 4D Gravity is almost topological in 4D The Plebanski formulation of gravity : BF-action with constraints S[B, ω] = B F (ω) + Λ B B + Φ(B) B 2 M Φ are Lagrange multipliers : B is simple i.e. e e or e e Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 15/16

28 Question of the dynamics A glimpse to Spin-Foam models in 4D Gravity is almost topological in 4D The Plebanski formulation of gravity : BF-action with constraints S[B, ω] = B F (ω) + Λ B B + Φ(B) B 2 M Φ are Lagrange multipliers : B is simple i.e. e e or e e From Crane-Yetter invariant to Quantum Gravity path integral Path integral of BF-action in terms of combinatorial structures Z(M) = W f (j) W t (ι) (15j) s (j f, ι t ) j f,ι t t s f Representation theory of U q (so(4)) in Euclidean theory with Λ > 0 Contrary to 3D the invariant is trivial Adapt the simplicity constraints to state sum : Spin-Foams! Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 15/16

29 Conclusion Towards physical predictions Brief summary of Loop Quantum Gravity Kinematics is totally well-defined Diffeomorphism invariance : unicity of representation Quantum geometries are combinatorial structures : spin-networks Geometric kinematical operators have discrete spectra Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 16/16

30 Conclusion Towards physical predictions Brief summary of Loop Quantum Gravity Kinematics is totally well-defined Diffeomorphism invariance : unicity of representation Quantum geometries are combinatorial structures : spin-networks Geometric kinematical operators have discrete spectra Dynamics under construction Well-understood in 3D : relation to Topological invariants Constraining 4D State sums to reproduce quantum gravity amplitudes Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 16/16

31 Conclusion Towards physical predictions Brief summary of Loop Quantum Gravity Kinematics is totally well-defined Diffeomorphism invariance : unicity of representation Quantum geometries are combinatorial structures : spin-networks Geometric kinematical operators have discrete spectra Dynamics under construction Well-understood in 3D : relation to Topological invariants Constraining 4D State sums to reproduce quantum gravity amplitudes Physical predictions : look for simpler situations Isotropic models : cosmology Static Black Holes (at equilibrium) : entropy and radiation... Clermont - Ferrand ; january 2014 Karim NOUI LQG : 2.Quantization 16/16