Spin foams with timelike surfaces

Transcription

1 Spin foams with timelike surfaces Florian Conrady Perimeter Institute ILQG seminar April 6, 2010 FC, Jeff Hnybida, arxiv: [gr-qc] FC, arxiv: [gr-qc] Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

2 Outline 1 Motivation 2 Coherent states 3 Three ways to simplicity 4 Spin foams 5 Summary Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

3 Motivation Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

4 Main innovations of the last years EPRL master constraint Engle, Livine, Pereira, Rovelli, Nucl.Phys.B799,2008 EPRL model correct coupling between 4 simplices relation to canonical LQG Coherent states Livine, Speziale, Phys.Rev.D76:084028,2007 simplicity constraints on expectation values geometric understanding of intertwiners FK model Freidel, Krasnov, Class.Quant.Grav.25:125018,2008 Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

5 Restriction of triangulations In the Lorentzian EPRL model, normals U of tetrahedra are timelike. timelike U All tetrahedra are Euclidean and triangles can be only spacelike. Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

6 What we did We extended the EPRL model to also include spacelike normals U. timelike U spacelike U Lorentzian tetrahedra are allowed and triangles can be spacelike or timelike. Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

7 Three cases normal U timelike U spacelike U spacelike U triangle spacelike spacelike timelike } {{ } EPRL Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

8 Covariant perspective extension natural a priori no reason to forbid Lorentzian tetrahedra permits timelike boundaries restriction could lead to artifacts Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

9 Canonical perspective Are restricted triangulations preferred from a Hamiltonian point of view? In the examples I know of the transition from space to spacetime leads to a 4d lattice with timelike (or null) edges. causal dynamical triangulations evolution schemes for Lorentzian Regge calculus Ambjorn, Jurkiewicz, Loll Barrett, Galassi, Miller, Sorkin, Tuckey, Williams Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

10 Canonical perspective Are restricted triangulations preferred from a Hamiltonian point of view? In the examples I know of the transition from space to spacetime leads to a 4d lattice with timelike (or null) edges. causal dynamical triangulations evolution schemes for Lorentzian Regge calculus Ambjorn, Jurkiewicz, Loll Barrett, Galassi, Miller, Sorkin, Tuckey, Williams The Hamiltonian approach to Lorentzian spin foams creates a sequence of 3d spatial lattices. cannot (yet) be directly compared with 4d triangulations discussed here Han, Thiemann Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

11 Coherent states Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

12 Little groups normal U timelike U spacelike U spacelike U gauge fix U = (1, 0, 0, 0) U = (0, 0, 0, 1) U = (0, 0, 0, 1) little group SO(3) SO(1,2) SO(1,2) triangle spacelike spacelike timelike } {{ } EPRL Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

13 Little groups normal U timelike U spacelike U spacelike U gauge fix U = (1, 0, 0, 0) U = (0, 0, 0, 1) U = (0, 0, 0, 1) little group SU(2) SU(1,1) SU(1,1) triangle spacelike spacelike timelike } {{ } EPRL Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

14 Representation theory SL(2, C) SU(2) generators J i, K i J 1, J 2, J 3 Casimirs C 1 = J 2 K 2 C 2 = 4 J K J 2 unitary irreps H (ρ,n) D j ρ R, n Z + j Z + /2 C 1 = 1 2 (n2 ρ 2 4) C 2 = ρn J 2 = j(j + 1) Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

15 Representation theory SU(2) SU(1,1) generators J 1, J 2, J 3 J 3, K 1, K 2 Casimirs J 2 Q = (J 3 ) 2 (K 1 ) 2 (K 2 ) 2 unitary irreps discrete series D j D ± j C ǫ s continuous series j Z + /2 j = 1 2, 1, j = 1 2 +is, 0 < s < J 2 = j(j + 1) Q = j(j 1) Q = s Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

16 Notation J J 1 J 2 J 3 and K K 1 K 2 K 3 transform like Euclidean 3 vectors under SU(2). Similarly, F J 3 K 1 K 2 and G K 3 J 1 J 2 transform like Minkowksi 3 vectors under SU(1,1). Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

17 SU(2) decomposition of SL(2, C) irrep Canonical basis H (ρ,n) j=n/2 D j ½ (ρ,n) = j j=n/2 m= j Ψ j m Ψ j m Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

18 SU(1,1) decomposition of SL(2, C) irrep H (ρ,n) n/2 D + j ds Cs ǫ n/2 D j ds Cs ǫ j>1/2 0 j>1/2 0 ½ (ρ,n) = n/2 j>1/2 m=j + α=1,2 0 Ψ + j m ds µ ǫ (s) Ψ + j m + Ψ s (α) m ±m=ǫ n/2 j>1/2 m=j Ψ s (α) m Ψ j m Ψ j m (see chapter 7 in Rühl s book) Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

19 Non normalizability States j m in the continuous series irrep Cs ǫ are normalizable, but the corresponding states in H (ρ,n) are not. Ψ s (α) m (α Ψ ) Ψ (α) δ(s s) s m s m = µ ǫ (s) δ α α δ m m Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

20 SU(2) coherent states Definition j g D j (g) j j, g SU(2) j N D j (g( N)) j j, N S 2 SU(2)/U(1) Completeness relation ½ j = (2j + 1) dg j g j g = (2j + 1) 2 N jn SU(2) S 2 d At the level of the SL(2, C) irrep H (ρ,n) this becomes P j = (2j + 1) dg Ψ j g Ψ j g = (2j + 1) SU(2) d 2 N S 2 Ψ j N Ψ j N. Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

21 SU(1,1) coherent states discrete series Definition j g ± D j (g) j ±j, g SU(1,1) j N D j (g( N)) j ±j, N H± SU(1,1)/U(1) Upper/lower hyperboloid H ± = { N N 2 = 1,N 0 0} Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

22 SU(1,1) coherent states discrete series Definition j g ± D j (g) j ±j, g SU(1,1) j N D j (g( N)) j ±j, N H± SU(1,1)/U(1) Completeness relation ± j = (2j 1) dg ½ j g ± j g ± = (2j 1) SU(1,1) H ± d 2 N j N j N At the level of the SL(2, C) irrep H (ρ,n) this becomes P ± j = (2j 1) dg Ψ ± j g Ψ ± j g = (2j 1) d 2 N Ψ j N Ψ j N. SU(1,1) H ± Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

23 Expectation values So far the coherent states are the ones introduced by Perelomov. They have the property that j N J j N = j N, N S 2 j N F j N = j N, N H± in accordance with the fact that J 2 = j(j + 1) 0 and F 2 = j(j 1) 0. Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

24 SU(1,1) coherent states continuous series Question What are the appropriate coherent states for the continuous series? In this case Q = F 2 = (s 2 + 1/4) < 0, so the classical vector N should be spacelike. Perelomov uses the state j m = 0, resulting in a zero classical vector. Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

25 SU(1,1) coherent states continuous series Question What are the appropriate coherent states for the continuous series? In this case Q = F 2 = (s 2 + 1/4) < 0, so the classical vector N should be spacelike. F J 3 K 1 K 2 Build coherent states from eigenstates of K 1 or K 2! Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

26 Eigenstates of K 1 K 1 j λσ = λ j λσ, < λ <, σ = ± These states are not normalizable: Mukunda, Barut and Phillips, Lindblad and Nagel j λ σ j λσ = δ(λ λ)δ σ σ To obtain finite inner products, one needs a smearing with wavefunctions! Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

27 SU(1,1) coherent states continuous series Definition j g sp D j (g) j s +, g SU(1,1) j N D j (g( N)) j s +, N Hsp SU(1,1)/G 1 H sp = { N N 2 = 1} = single sheeted spacelike hyperboloid G 1 = subgroup generated by K 1 Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

28 SU(1,1) coherent states continuous series Definition j g sp dλ 1 δ f δ (λ s)d j (g) j λ+, g SU(1,1) j N λ D j (g( N)) j λ+, N Hsp SU(1,1)/G 1 H sp = { N N 2 = 1} = single sheeted spacelike hyperboloid G 1 = subgroup generated by K 1 1, x δ/2 f δ (x) = 0, x > δ/2 Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

29 SU(1,1) coherent states continuous series Definition j g sp dλ 1 δ f δ (λ s)d j (g) j λ+, g SU(1,1) j N λ D j (g( N)) j λ+, N Hsp SU(1,1)/G 1 Completeness relation ½ ǫ j = dg j g δ sp j g δ sp = SU(1,1) H sp d 2 N dλ 1 δ f δ(λ s) j N λ j N λ Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

30 Smearing in s At the level of the SL(2, C) irrep H (ρ,n), one also needs a smearing in s. We define a smeared projector onto the irrep with spin j = 1/2 + is: P ǫ s (δ) α=1,2 ±m=ǫ 0 ds µ ǫ (s )f δ (s s) Ψ (α) s m Ψ (α) s m Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

31 SU(1,1) coherent states continuous series Definition Ψ (α) s g δ ds µ ǫ (s )f δ (s s) dλ 1 δ f δ (λ s)d (ρ,n) (g) Ψ (α) s λ + 0 Completeness relation P ǫ s (δ) = α=1,2 SU(1,1) dg Ψ (α) s g δ Ψ (α) s g δ Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

32 Three ways to simplicity Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

33 Three ways to simplicity Derivation of extended simplicity constraints by three methods } 1. weak imposition of constraints 2. master constraint (as advocated by EPRL) 3. restriction of coherent state basis (inspired by FK model) Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

34 Classical tetrahedron Describe a tetrahedron by four bivectors J = B + 1 γ B, where B is constrained to be simple. Simplicity constraint: unit four vector U such that U B = 0. Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

35 Classical simplicity constraints Express B in terms of the total bivector J: B = (J γ2 γ 2 1γ ) + 1 J Starting point for quantization: U (J 1γ J ) = 0 Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

36 First case: normal U timelike timelike U In the gauge U = (1,0,0,0), the simplicity constraint takes the form J + 1 γ K = 0 The little group is SU(2), so we use states of the SU(2) decomposition! Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

37 Spacelike U spacelike U In the gauge U = (0,0,0,1), the simplicity constraint becomes F + 1 γ G = 0 The little group is SU(1,1), so we use states of the SU(1,1) decomposition! Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

38 Basic example of second class constraints Phase space and constraints (q i,p i ), i = 1,2 q 1 q 2 = 0 {q i,p i } = δ ij p 1 p 2 = 0 Change of variables q ± = 1 2 (q 1 ± q 2 ) q = p = 0 p ± = 1 2 (p 1 ± p 2 ) a ± = 1 2 (p ± iq ± ) a = 0 Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

39 Weak imposition of constraints Impose a ψ = 0 on physical states, implying ϕ a ψ = ϕ a ψ = 0 ϕ, ψ H phys Physical Hilbert H phys space is spanned by n + 0, n + N 0. Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

40 Restriction of coherent state basis Overcomplete basis of coherent states α + α, a ± α ± = α ± α ± : ½ H = 1 π 2 dα + dα α + α + α α Restrict basis to states whose expectation values satisfy the constraint, i.e. to labels α = 0. Projector on H phys P phys 1 π dα + α + α Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

41 Master constraint Master constraint operator M = a a = 1 ( p q ) Define H phys as the subspace of states with minimal eigenvalue w.r.t. M. H phys spanned by n + 0, n + N 0. Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

42 From classical to quantum simplicity J + 1 γ K = 0 F + 1 γ G = 0 4γC 3 = ρn B IJ ( B) IJ = 0 (ρ γn) ( ) ρ + n γ = 0 B IJ ( B) IJ = 0 is the diagonal simplicity constraint. C 3 is the Casimir of the little group determined by the normal U. Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

43 Weak imposition of constraints Consider the case U = (1, 0, 0, 0) and suppose that H phys = j J D j H (ρ,n), where J is a subset of the total set of spins {j j n/2}. Require that ϕ C ψ = 0 ϕ, ψ H phys. Unless H phys is trivial, this implies, that for some j n/2, j m J γ K3 j m = j m J ± + 1 γ K ± j m = 0 for all admissible m, m. Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

44 Weak imposition of constraints By using K ± = ±[K 3,J ± ] and K 3 j m = (...) j + 1m ma j j m + (...) j 1m, A j = ρn 4j(j + 1), one obtains A j = γ or 4γC 3 = 4γj(j + 1) = ρn. In conjunction with the constraint B B = 0, this gives 4j(j + 1) = n 2 if ρ = γn and 4j(j + 1) = ρ 2 if n = γρ. Approximate solution ρ = γn, j = n/2 H phys = D n/2 H (γn,n) Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

45 Weak imposition of constraints Next consider U = (0,0,0,1). Suppose first that the weak constraint holds for some irrep D j ± of the discrete series: j m F γ G0 j m = j m F ± + 1 γ G ± j m = 0 m,m, with F ± F 2 if 1 and G ± G 2 ig 1. According to Mukunda K 3 j m = (...) j + 1m mãj jm + (...) j 1m, Ã j = and G ± = ±[G 0,F ± ], which leads to Ãj = γ or 4γj(j 1) = ρn. Approximate solution ρ = γn, n 2, j = n/2 ρn 4j(j 1), Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

46 Weak imposition of constraints For an irrep Cs ǫ of the continuous series, the equations are the same except that à j is replaced by A j = ρn 4j(j + 1) = ρn 4(s 2 + 1/4). A solution exists only when ρ = n/γ < 1 and then Overall result for U = (0, 0, 0, 1) s 2 + 1/4 = ρ2 4 = n2 4γ 2. ρ = γn, n 2 or ρ = n/γ < 1 H phys = D + n/2 D n/2 H phys = C ǫ 1 2 n 2 /γ 2 1 Cǫ 1 2 n 2 /γ 2 1 Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

47 Restriction of coherent state basis Case U = (1,0,0,0). Resolve the identity on H (ρ,n) in terms of SU(2) coherent states, ½ (ρ,n) = (2j + 1) j=n/2 d 2 N S 2 Ψ j N Ψ j N, and require that Ψ j N J + 1 γ K Ψ j N = 0. This implies A j = γ, that is, 4γj(j + 1) = ρn. Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

48 Restriction of coherent state basis The second condition is either obtained from B B = 0 or alternatively from the requirement of minimal uncertainty in K. From γ = A j it follows that so that J 2 = 1 γ 2 K 2 + O( J ) and J 2 = 1 γ J K, ( K) 2 = 1 γ (1 γ2 ) J K 1 2 C 1 + O( J ) = γ [(1 1γ ) 4 2 C ] γ C 1 + O( J ) = γ 4 B B + O( J ). Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

49 Restriction of coherent state basis Projector on H phys P phys = (n + 1) d 2 N S 2 Ψ n/2 N Ψ n/2 N Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

50 Master constraint For the case U = (0,0,0,1) the Master constraint reads M = ( F + 1 γ G) 2 = (1 + 1γ 2 ) F 2 1 2γ 2C 1 1 2γ C 2 = 0 The diagonal constraint B B = 0 is equivalent to (1 1γ 2 ) C γ C 2 = 0. By combining the two one arrives at the desired second condition 4γ F 2 = ρn. Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

51 Master constraint In the case of the discrete series, the constraints are therefore ( )( ρ γn ρ + n ) γ = 0 4γj(j 1) = ρn Approximate solution ρ = γn, n 2 j = n/2 Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

52 Master constraint For states of the continuous series, ( )( ρ γn ρ + n ) = 0 γ ( 4γ s ) = ρn 4 Solution ρ = n γ < γ s2 = 1 4 ( ) n 2 γ 2 1 Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

53 Table of constraints timelike U normal U spacelike U spacelike U triangle spacelike spacelike timelike little group SU(2) SU(1,1) SU(1,1) relevant irreps D j D ± j C ǫ s constr. on (ρ, n) ρ = γn ρ = γn, n 2 n = γρ > γ constr. on irreps j = n/2 j = n/2 s 2 + 1/4 = ρ 2 /4 area spectrum γ j(j + 1) γ j(j 1) γ s 2 + 1/4 = n/2 } {{ } EPRL Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

54 Spin foams Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

55 Spin foam theory Complex: simplicial complex : 4 simplex σ, tetrahedron τ, triangles t,... dual complex : vertex v, edge e, face f,... Variables (same as in EPRL): connection g e SL(2, C) irrep label n f Z + Additional variables: U e = (1,0,0,0) or (0,0,0,1): normal of tetrahedron dual to e ζ f = ±1: spacelike/timelike triangle dual to f Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

56 Uniform notation To cover the different cases we introduce a uniform notation. Little group H(ζ,U) SU(2), if ζ = 1, U = (1,0,0,0), SU(1,1), if ζ = ±1, U = (0,0,0,1),, if ζ = 1, U = (1,0,0,0). Spin j = n/2, if ζ = 1, U = (1,0,0,0), n/2, if ζ = 1, U = (0,0,0,1), i 2 n 2 /γ 2 1, if ζ = 1, U = (0,0,0,1). Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

57 Uniform notation Coherent states Ψ (α) j h δ = Ψ j h, if ζ = 1, U = (1,0,0,0), Ψ ± j h, if ζ = 1, U = (0,0,0,1), Ψ (α) j h δ, if ζ = 1, U = (0,0,0,1). Projector P j (ζ,u,δ) = d j (ζ,u) α H(ζ,U) dh Ψ (α) j h δ Ψ (α) j h δ Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

58 Vertex amplitude e e f g ev g ve v A v ((ρ f,n f );h ef,α ef,δ) = dg ev Ψ (α ef ) SL(2,C) e f j ef h ef δ D (ρ f,n f ) (g ev g ve ) Ψ (α e f ) j e f h e f δ Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

59 Spin foam sum Partition function Z = n f f ζ f =±1 U e α ef H(U e,ζ f ) dh ef d jf (U e,ζ f ) ) (1 + γ 2ζ f )nf 2 lim A v ((ζ f γ ζ f n f,n f );h ef,α ef,δ δ 0 v Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

60 Compatibility of path integral and operator formalism So far we were mainly guided by the path integral picture and defined the spin foam sum by simply summing over all choices of the normal U. From the canonical perspective, however, one would also require that the sum over intermediate states on a triangulation s boundary is a projector. For a give choice of U, this is certainly the case. Is it also true when we sum over U? Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

61 Compatibility of path integral and operator formalism normal U timelike U spacelike U spacelike U triangle spacelike spacelike timelike (ρ, n) ρ = γn ρ = γn, n 2 n = γρ > γ } {{ } EPRL Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

62 Compatibility of path integral and operator formalism, 0, 0, = 0, = 0 The sum over tetrahedral states is a projector if we exclude. P = + red>0 The boundary Hilbert space is genuinely larger than the EPRL one. Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

63 Summary Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

64 Summary extension of EPRL model tetrahedra can be Euclidean and Lorentzian triangles can be spacelike and timelike larger boundary Hilbert space discrete area spectrum of timelike surfaces definition of associated spin foam model coherent states for timelike triangles Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60

65 Outlook regions with timelike boundaries! Rovelli et al., Oeckl extension of results on EPRL model? asymptotics, graviton propagator...? canonical LQG on general boundaries? comparison with previous work on timelike surfaces Perez, Rovelli Alexandrov, Vassilevich Alexandrov, Kadar Florian Conrady (PI) Spin foams with timelike surfaces ILQGS April / 60