Spin foam vertex and loop gravity

Spin foam vertex and loop gravity J Engle, R Pereira and C Rovelli Centre de Physique Théorique CNRS Case 907, Université de la Méditerranée, F-13288 Marseille, EU Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 1

Plan Introduction and state of art the BC model and its problems SO(4) Plebanski action Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 2

Plan Introduction and state of art the BC model and its problems SO(4) Plebanski action Classical setup Regge calculus lattice GR Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 2

Plan Introduction and state of art the BC model and its problems SO(4) Plebanski action Classical setup Regge calculus lattice GR Quantization constraints projection onto LQG states vertex: a proposal Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 2

Plan Introduction and state of art the BC model and its problems SO(4) Plebanski action Classical setup Regge calculus lattice GR Quantization constraints projection onto LQG states vertex: a proposal Conclusion Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 2

State of art and the BC model ρ BC ρ 12 23 ρ 13 BC ρ 15 ρ 25 ρ 14 ρ 35 ρ 24 BC ρ 34 BC ρ 45 BC [Barrett,Crane 97] Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 3

State of art and the BC model ρ BC ρ 12 23 ρ 13 BC ρ 15 ρ 25 ρ 14 ρ 35 ρ 24 BC ρ 34 BC ρ 45 [Barrett,Crane 97] Quantization of the classical tetrahedron imposing a set of constraints à là Dirac: [Baez,Barrett 99] Ĉ ψ = 0 unique solution of the constraints [Reisenberger 99] BC Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 3

State of art and the BC model ρ BC ρ 12 23 ρ 13 BC ρ 15 ρ 25 ρ 14 ρ 35 ρ 24 BC ρ 34 BC ρ 45 [Barrett,Crane 97] Quantization of the classical tetrahedron imposing a set of constraints à là Dirac: [Baez,Barrett 99] Ĉ ψ = 0 Boundary states do not agree with LQG states and problems in the calculation of non diagonal components of the propagator. [Alesci,Rovelli to appear] and talk by Carlo Rovelli. BC Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 3

Plebanski action and constraints S P [B, ω] = B IJ F IJ (ω) + φ IJKL B IJ B KL Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 4

Plebanski action and constraints S P [B, ω] = B IJ F IJ (ω) + φ IJKL B IJ B KL constrained BF theory Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 4

Plebanski action and constraints S P [B, ω] = B IJ F IJ (ω) + φ IJKL B IJ B KL constrained BF theory Constraints: ɛ IJKL Bµν IJ Bαβ KL = V ɛ µναβ Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 4

Plebanski action and constraints S P [B, ω] = B IJ F IJ (ω) + φ IJKL B IJ B KL constrained BF theory Constraints: ɛ IJKL Bµν IJ Bαβ KL = V ɛ µναβ Two sectors of solutions: 02] [Reisenberger 98;De Pietri,Freidel 99;Perez B = (±) e e B = (±) (e e) Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 4

Regge discretization Simplicial discretization of a 4-dimensional space time: 4-simplices (v), tetrahedra (t) and triangles (f) Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 5

Regge discretization Simplicial discretization of a 4-dimensional space time: 4-simplices (v), tetrahedra (t) and triangles (f) Curvature is concentrated on triangles Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 5

Regge discretization Simplicial discretization of a 4-dimensional space time: 4-simplices (v), tetrahedra (t) and triangles (f) Curvature is concentrated on triangles In 2d: ϕ θ Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 5

Regge discretization Simplicial discretization of a 4-dimensional space time: 4-simplices (v), tetrahedra (t) and triangles (f) Curvature is concentrated on triangles In our case: (...) t4 v34 tn f t3 vn1 v23 t1 t2 v12 Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 5

Discrete variables To each t: e(t) = e(t) I av I dx a Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 6

Discrete variables To each t: e(t) = e(t) I av I dx a To each pair (v, t): e(v) I a = (V vt ) I J e(t) J a ( Vtv := Vvt 1 ) Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 6

Discrete variables To each t: e(t) = e(t) I av I dx a To each pair (v, t): To each f = t t : e(v) I a = (V vt ) I J e(t) J a e(t ) = U t t e(t) ( Vtv := Vvt 1 ) and :B(t) IJ ab = ɛij KLe(t) K a e(t) L b B f (t) = f B(t) (B(t ) = U t t B(t)U tt ) Flatness implies: U t1 t n = V t1 v 12...V v(n 1)n t n Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 6

Discrete variables To each t: e(t) = e(t) I av I dx a To each pair (v, t): To each f = t t : e(v) I a = (V vt ) I J e(t) J a e(t ) = U t t e(t) ( Vtv := Vvt 1 ) and :B(t) IJ ab = ɛij KLe(t) K a e(t) L b B f (t) = f B(t) (B(t ) = U t t B(t)U tt ) (...) t4 v34 tn U(13) t3 f V(v23t3) vn1 t1 v12 t2 v23 V(t2v23) Flatness implies: U t1 t n = V t1 v 12...V v(n 1)n t n V(t1v12) V(v12t2) Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 6

Discrete variables To each t: e(t) = e(t) I av I dx a To each pair (v, t): To each f = t t : e(v) I a = (V vt ) I J e(t) J a e(t ) = U t t e(t) ( Vtv := Vvt 1 ) and :B(t) IJ ab = ɛij KLe(t) K a e(t) L b B f (t) = f B(t) (B(t ) = U t t B(t)U tt ) Flatness implies: U t1 t n = V t1 v 12...V v(n 1)n t n Curvature is given by the holonomy around a link: U f (t 1 ) := V t1 v 12...V vn1 t 1 Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 6

Discrete constraints Closure: f t B(t) f = 0 Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 7

Discrete constraints Closure: f t B(t) f = 0 Simplicity constraints: C ff := B f B f = 0 C ff := B f B f = 0 Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 7

Discrete constraints Closure: f t B(t) f = 0 Simplicity constraints: C ff := B f B f = 0 C ff := B f B f = 0 the solutions to these constraints correspond to the two sectors of Plebanski theory. Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 7

Boundary variables tetrahedra (t) nodes (n) faces (f) links (l) Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 8

Boundary variables tetrahedra (t) nodes (n) faces (f) links (l) Boundary variables: B f B l U tt U l Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 8

Boundary variables tetrahedra (t) nodes (n) faces (f) links (l) Boundary variables: B f B l U tt U l phase space of a SO(4) Yang-Mills lattice gauge theory (!!) Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 8

Dynamics I: bulk S bulk = f T r[b f (t)u f (t)] Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 9

Dynamics I: bulk S bulk = f T r[b f (t)u f (t)] approximates GR in the continuum limit Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 9

Dynamics I: bulk S bulk = f T r[b f (t)u f (t)] approximates GR in the continuum limit clear relation with the Regge action (!!): S Regge = f A f sin ɛ f Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 9

Dynamics II: boundary U1 f t t U2 Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 10

Dynamics II: boundary U1 t f t U2 Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 10

Dynamics II: boundary U1 t f t U2 S = f T r[b f (t)u tt ] Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 10

Summary of the classical theory S LGR = T r[b f (t)u f (t)] + T r[b f (t)u tt ] f int f Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 11

Summary of the classical theory S LGR = T r[b f (t)u f (t)] + T r[b f (t)u tt ] f int f f t B f (t) = 0, C ff = B f (t) B f (t) = 0, C ff = B f (t) B f (t) = 0. Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 11

Summary of the classical theory S LGR = T r[b f (t)u f (t)] + T r[b f (t)u tt ] f int f f t B f (t) = 0, C ff = B f (t) B f (t) = 0, C ff = B f (t) B f (t) = 0. Boundary variables : {(B l, U l )} Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 11

Plan Introduction and state of art the BC model and its problems SO(4) Plebanski action Classical setup Regge calculus lattice GR Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 12

Quantization: constraints ««N N ψ + j l j l i+ n i n (g+ l,g l )= l D(j+ l ) (g + l ) Nn i+ n l D(j l ) (g l ) Nn i n Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 13

Quantization: constraints ««N N ψ + j l j l i+ n i n (g+ l,g l )= l D(j+ l ) (g + l ) Nn i+ n l D(j l ) (g l ) Nn i n For C ff, no problem: j + = j Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 13

Quantization: constraints ««N N ψ + j l j l i+ n i n (g+ l,g l )= l D(j+ l ) (g + l ) Nn i+ n l D(j l ) (g l ) Nn i n For C ff, no problem: j + = j The off diagonal simplicity constraints do not form a closed algebra under poisson brackets: [C 12, C 13 ] U [Baez,Barrett 99] Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 13

Quantization: constraints ««N N ψ + j l j l i+ n i n (g+ l,g l )= l D(j+ l ) (g + l ) Nn i+ n l D(j l ) (g l ) Nn i n For C ff, no problem: j + = j The off diagonal simplicity constraints do not form a closed algebra under poisson brackets: [C 12, C 13 ] U [Baez,Barrett 99] Solution: impose them weakly. χ, ψ H phys χ C ff ψ = 0 Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 13

Quantization: constraints ««N N ψ + j l j l i+ n i n (g+ l,g l )= l D(j+ l ) (g + l ) Nn i+ n l D(j l ) (g l ) Nn i n For C ff, no problem: j + = j The off diagonal simplicity constraints do not form a closed algebra under poisson brackets: [C 12, C 13 ] U [Baez,Barrett 99] Solution: impose them weakly. χ, ψ H phys χ C ff ψ = 0 Motivation: Gupta-Bleur, doubled system. Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 13

Solution A state ψ in H e := Inv ( H (j1,j 1 )... H (j4,j 4 )) can be written (in a given pairing) as: ψ = c i+ i i+ i Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 14

Solution A state ψ in H e := Inv ( H (j1,j 1 )... H (j4,j 4 )) can be written (in a given pairing) as: ψ = c i+ i i+ i A solution to the constraints imposed weakly is given by symmetric states: c i + i = c i i + Question: Is there a subspace of H e that matches the space of a SO(3) intertwiner (as in LQG)? Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 14

Solution A state ψ in H e := Inv ( H (j1,j 1 )... H (j4,j 4 )) can be written (in a given pairing) as: ψ = c i+ i i+ i A solution to the constraints imposed weakly is given by symmetric states: c i + i = c i i + Question: Is there a subspace of H e that matches the space of a SO(3) intertwiner (as in LQG)? Answer: Yes (!!). It is given by a suitable projection H e Inv (H 2j1... H 2j4 ) Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 14

Projection Spinor notation for I H e : I (A 1...A 2j 1 )(A 1...A 2j 1 )... (D 1...D 2j4 )(D 1...D 2j 4 ) =: I AA... DD = i A... D + i A... D Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 15

Projection Spinor notation for I H e : I (A 1...A 2j 1 )(A 1...A 2j 1 )... (D 1...D 2j4 )(D 1...D 2j 4 ) =: I AA... DD = i A... D + i A... D Projection by symmetrization: π : I AA...DD I (AA )...(DD ) =: i a... d The projection chooses the highest SU(2) irreducible in the decomposition: j j = 0... 2j Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 15

Projection Spinor notation for I H e : I (A 1...A 2j 1 )(A 1...A 2j 1 )... (D 1...D 2j4 )(D 1...D 2j 4 ) =: I AA... DD = i A... D + i A... D Projection by symmetrization: π : I AA...DD I (AA )...(DD ) =: i a... d The projection chooses the highest SU(2) irreducible in the decomposition: j j = 0... 2j The projection of the complete hilbert space of SO(4) s-nets can be defined and it matches the hilbert space of SO(3) s-nets (!) Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 15

Embedding To every projection there is an embedding. The image of a SO(3) intertwiner i, f(i) can be expanded in the SO(4) basis: f(i) = i +,i f i i+i i +, i Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 16

Embedding To every projection there is an embedding. The image of a SO(3) intertwiner i, f(i) can be expanded in the SO(4) basis: f(i) = i +,i f i i+i i +, i Its components can be seen as the evaluation of a 15j symbol: j 1 j 2 j 1 j 2 2j 1 2 j 2 i+ i i j 3 j 4 j 3 j 4 2j 3 2j 4 Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 16

Embedding To every projection there is an embedding. The image of a SO(3) intertwiner i, f(i) can be expanded in the SO(4) basis: f(i) = i +,i f i i+i i +, i The image of the embedding f [Inv (H 2j1... H 2j4 )] =: K e can be interpreted as a subspace of H e obtained by imposing a geometrical constraint. Attention: It depends on a different choice of quantization: B f J f. Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 16

Dynamics I: setup Consider one 4-simplex. The amplitude is given by : A(U ab ) = a dv a δ(v a U ab V 1 b ) Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 17

Dynamics I: setup Consider one 4-simplex. The amplitude is given by : A(U ab ) = a dv a δ(v a U ab V 1 b ) The quantum amplitude for a given boundary state ψ is given by [Oeckl 03]: A(ψ) := du ab A(U ab ) ψ(u ab ) Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 17

Dynamics II: vertex For our model the vertex amplitude is given by: A({j ab }, {i a }) := A(ψ {jab },{i a }) = i a + ia 15j (( jab 2, j ) ) ab ; (i a 2 +, i a ) f ia i a + ia Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 18

Dynamics II: vertex For our model the vertex amplitude is given by: A({j ab }, {i a }) := A(ψ {jab },{i a }) = i a + ia 15j (( jab 2, j ) ) ab ; (i a 2 +, i a ) f ia i a + ia The final amplitude is obtained by gluing 4-simplices: Z = j f,i e f (dim j f 2 ) 2 v A(j f, i e ) Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 18

Conclusion Lattice (covariant) formulation of GR Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 19

Conclusion Lattice (covariant) formulation of GR Clear analysis of the constraints Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 19

Conclusion Lattice (covariant) formulation of GR Clear analysis of the constraints Weak constraining given by the projection agreement with LQG states Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 19

Conclusion Lattice (covariant) formulation of GR Clear analysis of the constraints Weak constraining given by the projection agreement with LQG states Connection with other approaches: 07;Alexandrov 07] [Livine,Speziale Geometrical interpretation forces quantization with the dual SO(4) generators Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 19

Conclusion Lattice (covariant) formulation of GR Clear analysis of the constraints Weak constraining given by the projection agreement with LQG states Connection with other approaches: 07;Alexandrov 07] [Livine,Speziale Geometrical interpretation forces quantization with the dual SO(4) generators Next step: semiclassical analysis and propagator. Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 19

Doubled system C (T R T R) f ((q 1, p 1 ), (q 2, p 2 )) and {q a, p b } = δ ab Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 20

Doubled system C (T R T R) f ((q 1, p 1 ), (q 2, p 2 )) and {q a, p b } = δ ab q 1 q 2 = 0 p 1 p 2 = 0 Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 20

Doubled system C (T R T R) f ((q 1, p 1 ), (q 2, p 2 )) and {q a, p b } = δ ab q 1 q 2 = 0 p 1 p 2 = 0 Redefining: q ± = (q 1 ± q 2 )/2 and p ± = (p 1 ± p 2 )/2 q = p = 0 and {q, p } = 1 Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 20

Doubled system C (T R T R) f ((q 1, p 1 ), (q 2, p 2 )) and {q a, p b } = δ ab q 1 q 2 = 0 p 1 p 2 = 0 Redefining: q ± = (q 1 ± q 2 )/2 and p ± = (p 1 ± p 2 )/2 q = p = 0 and {q, p } = 1 Yet, z = (p iq )/ 2 and z = (p + iq )/ 2 [ ẑ, ˆ z ] = 1 and ẑ = ˆ z = 0 Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 20

Doubled system C (T R T R) f ((q 1, p 1 ), (q 2, p 2 )) and {q a, p b } = δ ab q 1 q 2 = 0 p 1 p 2 = 0 Redefining: q ± = (q 1 ± q 2 )/2 and p ± = (p 1 ± p 2 )/2 q = p = 0 and {q, p } = 1 Yet, z = (p iq )/ 2 and z = (p + iq )/ 2 [ ẑ, ˆ z ] = 1 and ẑ = ˆ z = 0 Solution: impose half of them (!) : ẑ ψ = 0 φ ˆ z ψ = 0 Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 20

Complete proj. and emb. π : D (j,j) (g + l, g l )AA BB D (j,j) (g l, g l ) (AA ) (BB ) = D (2j) (g l ) a b Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 21

Complete proj. and emb. π : D (j,j) (g + l, g l )AA BB D (j,j) (g l, g l ) (AA ) (BB ) = D (2j) (g l ) a b f : ( l D(j l) (g l )) ( n i n) SO(4) N e dv e ( l D( j l 2, j l 2 ) ( V s(l) (g + l, g l )) 1 )Vt(l) ( n e(i) n) Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 21

Projection as a geometrical constraint Classically, the constraints C ff force the B fi to span a 3d space. Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 22

Projection as a geometrical constraint Classically, the constraints C ff force the B fi to span a 3d space. Choose a fixed vector n I = δ I 0 normal to the tetrahedron. Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 22

Projection as a geometrical constraint Classically, the constraints C ff force the B fi to span a 3d space. Choose a fixed vector n I = δ I 0 normal to the tetrahedron. Associate to each B f the dual of a SO(4) generator J f. Then 4C 4 = Jf IJJ f IJ = J ij f J ij f = 2C 3 The quantum constraint reads, up to quantization ambiguities: C = C 3 + 2 4 2C 4 + 2 + 2 = 0 Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 22

Projection as a geometrical constraint Classically, the constraints C ff force the B fi to span a 3d space. Choose a fixed vector n I = δ I 0 normal to the tetrahedron. Associate to each B f the dual of a SO(4) generator J f. Then 4C 4 = Jf IJJ f IJ = J ij f J ij f = 2C 3 The quantum constraint reads, up to quantization ambiguities: C = C 3 + 2 4 2C 4 + 2 + 2 = 0 This constraint selects the highest SU(2) irreducible. Roberto Pereira, Loops 07 Morelia 25-30 June 2007 p. 22