Can the supercomputer provide new insights into quantum gravity?

Transcription

1 EU ENRAGE Network Random Geometries Can the supercomputer provide new insights into quantum gravity? David Rideout (in collaboration with S. Zohren; S. Major, S. Surya; J. Brunnemann; development team) Theoretical Physics Group Imperial College London Loops 07, 30 June 2007

2 Plan of this Talk 1 What is?

3 Plan of this Talk 1 What is?

4 Plan of this Talk 1 What is?

5 Plan of this Talk 1 What is?

6 Plan of this Talk 1 What is?

7 Outline 1 What is? 2 Review of : Fundamentally Discrete for Entropy Bound Spherically Symmetric Hyperboloidal Slices 3 Spatial Computing Spatial from Simulations 4 5 Spatial

8 The Computational Framework Spatial

9 The Computational Framework Began as open source environment for numerical relativity Properly designed abstractions & interfaces Physics code unchanged as technologies change underneath Collaboration: passing parameter files, web interface to running codes Cooperation: inherit developments of others, including computer scientists Community: build up community code base for addressing difficult problems in QG Spatial abstract ugly CS issues away from physics

10 The Computational Framework What features are useful for quantum gravity computations? Not just for solving PDEs on a fixed lattice... Modularity! collaboration & community building Language independence Separate computational details from physics Code sharing develop community code base Leverage developments of others, e.g. students, no continual reinventing the wheel Pass around parameter files Works well for numerical relativity, causal set QG: Do same for, spin foams, CDTs? Portability Automatic parallelism Spatial Automatic detection and linking to numerical libraries

11 Outline 1 What is? 2 Review of : Fundamentally Discrete for Entropy Bound Spherically Symmetric Hyperboloidal Slices 3 Spatial Computing Spatial from Simulations 4 5 Spatial

12 ements matter Susskind Process Isolated matter system mass E, entropy Smatter Stot initial = S matter + S shell Stot final = S BH = A 4 Circumscribing sphere, area A Shell mass A 16π E Matter system stable on time scale > A Collapse shell onto system to form black hole } GSL S matter A 4 Susskind entropy bound Spatial

13 Covariant Entropy Bound Consider any spacelike co-dimension 2 surface B, of area A Consider congruence of null geodesics emanating from and orthogonal to B, which is everywhere non-expanding: a light sheet Covariant bound: entropy on light sheet A 4 Spatial infalling matter No known violations Can recover other bounds under suitable conditions

14 Holographic Principle holographic principle : Region with boundary of area A described by no more than A 4 degrees of freedom universal relation between geometry and entropy/information Can we see this emerge from discrete quantum gravity? Spatial

15 : Fundamentally Discrete Based upon two main observations: Need for discreteness Richness of causal structure y Properties of discrete causal order : irreflexive (x x) transitive (x y and y z x z) locally finite ( {y x y z < ) Some definitions: maximal elements (inextendible) antichain x Spatial

16 Spacetime Manifold as Emergent Structure Embedding order preserving map φ : C (M, g) x y φ(x) φ(y) x, y C Faithful embedding ( Sprinkling ): preserves number volume correspondence scale of geometry mean spacing of embedded change points faithful embedding (M, g) approximates C Spatial time 0.5 time ents space space

17 An entropy bound from causal sets Spherically symmetric spacelike hypersurface Σ of finite volume in strongly causal spacetime of dimension PSfrag d replacements 3. Causet C faithfully embeded into D + (Σ). D + (Σ) : Maximum entropy associated to Σ given by max(c) Claim: Leads to Susskind s entropy bound in continuum limit S max = A 4 Σ H + (Σ) B(Σ) Spatial

18 Calculate number of maximal elements Analytically: Use properties of Poisson distribution d In < n >= ρ dx d exp ( J + (x) D + (S) ) D + (S) Numerically: Use CausalSets toolkit within framework: Sprinkle via Poisson distribition Deduce order relations Count maximal elements Spatial

19 Ball in 2+1 dimensional Minkowski space # Spatial "!

20 Ball in 3+1 dimensional Minkowski space Number of maximal elements: n = N 3 F 3 ( 1 2, 1, 1; 5 4, 7 4, 2; N 8 ) Asymptotic behavior N 1: n = 3 2πN Define Poisson embedding at unit density in fundamental units: ρ = N V = 1 l 4 f N = V l 4 f Volume of cone of radius R: V = πr4 3 n = 6 l 2 f Choose a fundamental scale l f = 4 6lp Then we have: S max = 4πR2 4 = A 4 4πR 2 4 Spatial

21 Ball in 3+1 dimensional Minkowski space Spatial nts

22 Spherically Symmetric Hyperboloidal Slices Same boundary sphere t R r D + (Σ) ar Σ Spatial R

23 Spherically Symmetric Hyperboloidal Slices Same boundary sphere Spatial

24 Spherically Symmetric Hyperboloidal Slices Same boundary sphere Spatial All give same asymptotic form S max = 4πR 4, with same l f!

25 Outline 1 What is? 2 Review of : Fundamentally Discrete for Entropy Bound Spherically Symmetric Hyperboloidal Slices 3 Spatial Computing Spatial from Simulations 4 5 Spatial

26 Spatial Structure Causal Order x z w y Temporal notions, such as proper time, are easy to extract from causet Spatial notions difficult, because of Lorentz invariance of lattice has infinite valence (nearest neighbors) Spatial

27 Spatial Structure Causal Order x z w y Temporal notions, such as proper time, are easy to extract from causet Spatial notions difficult, because of Lorentz invariance of lattice has infinite valence (nearest neighbors) Spatial Resolution Make local selection of frame inextendible antichain Edgeless spacelike hypersurface: Every element related to some element of inextendible antichain

28 Spatial Structure Causal Order Resolution Make local selection of frame inextendible antichain Edgeless spacelike hypersurface: Every element related to some element of inextendible antichain Element Causet Faithfully Embedded into SxI Spatial time space

29 Spatial Structure Causal Order Element Causet Faithfully Embedded into SxI time space Spatial Cauchy surface, but e.g. does not possess inital data... Only intrinsic information is cardinality Can we use neighboring causal structure to deduce which elements are spatial nearest neighbors?

30 Spatial Structure Causal Order Cauchy surface, but e.g. does not possess inital data... Only intrinsic information is cardinality Can we use neighboring causal structure to deduce which elements are spatial nearest neighbors? Spatial Thickened antichain A v = {x x fut(a) A and past(x) \ past(a) v},

31 Spatial Structure Causal Order Thickened antichain A v = {x x fut(a) A and past(x) \ past(a) v}, Thickened Antichain in SxI Causal Set Spatial time space

32 Spatial Structure Causal Order Thickened antichain A v = {x x fut(a) A and past(x) \ past(a) v}, embedding.asc antichain.asc Spatial

33 from Thickened Antichain Can we deduce the continuum homology from the thickened antichain? t maximal elements of thickened antichains cast shadows on minimal elements provides cover of space nerve construction of simplicial complex homology Spatial Kinematical 0 Hilbert Space y x

34 from Thickened Antichain Nerve: Assign vertex to each set U i in cover q sets in cover form a q 1-simplex if they have non-vanishing intersection COVER NERVE 2 G F 1 C 1 2 H D B E G C F H D E Spatial 1 A A B

35 Simplicial homology C k = vector space with generator for each k-simplex α α i = k 1-simplex obtained by deleting ith vertex of α boundary map k : C k C k 1 = P k i=0 ( )i α i 2 = 0 Z k = Ker( k ) = k-cycles B k = Im( k+1 ) = k-boundaries H k = Z k /B k Betti numbers b k = dim(h k ) For example: COVER 2 G F 1 2 H 1 1 D B C A 2 1 E NERVE b 0 = 1 b 1 = 1 b i = 0 i > 1 Cover and thus homology will vary with thickness v A G C F H D B E Spatial

36 Current Status: Theorem Causet C in globally hyperbolic spacetime with compact spatial slice Σ If inextendible antichain A in C in Σ with appropriate separation of scales, then... A thickened to vol n in this range has homology of space via nerve Spatial Can likely smooth out bad antichain to get good one, via smoothing (Ricci flow?) Conditions on theorem minor. We explore this numerically.

37 S I H 0 H 1 H 2 H 3 H 4 H 5 βi Spatial nts N = 2000 cosmic scale at v = 433 v

38 T 2 I H 0 H 1 H 2 βi Spatial nts v N = 5793

39 T 2 I Kaluza Klein 5 4 H 0 H 1 H 2 H 3 H 4 H 5 H 6 βi Spatial nts v b = a/4

40 Transitive Percolation N = 2000; p = H 0 H 1 H 2 H 3 25 nts βi v Spatial

41 Outline 1 What is? 2 Review of : Fundamentally Discrete for Entropy Bound Spherically Symmetric Hyperboloidal Slices 3 Spatial Computing Spatial from Simulations 4 5 Spatial

42 Consider γ Σ v 4 e e 4 3 v 2 e 1 e2 v 3 e v 8 7 e e 6 v 5 1 v7 e v 6 8 e 9 v 5 Canonical variables: holonomies h ei, fluxes E i(s) E(γ)... edge set of γ V (γ)... vertex set of γ Kinematical Hilbert space H kin,γ = L 2 (Ā γ, dµ γ) Spin Network Functions T γ j m n (A) := Y e E(γ) p 2je + 1[π je (h(a))] men e Basis of H kin,γ (Peter & Weyl): 2j + 1 [ π j (h e ) ] h m e j m e ; n e SNF en e Can replace 2 Ej j me ; n e = J j j me ; n e Action of operators only containing E j can be expressed as action of usual angular momentum operators acting on a spin system

43 Volume : Structure Classical Volume Expression V (R)= d 3 x det q(x) = d 3 x det E (x) R R = d 3 1 x 3! εijk ε abc Ei a(x)eb j (x)ec k (x) R Structure of the volume operator ˆV γ (R) f γ l 3 P ɛ(i, J, K) ˆq IJK f γ v V (γ) IJK Spatial

44 Volume : Structure Classical Volume Expression V (R)= d 3 x det q(x) = d 3 x det E (x) R R = d 3 1 x 3! εijk ε abc Ei a(x)eb j (x)ec k (x) Structure of the volume operator ˆV γ (R) f γ l 3 P ɛ(i, J, K) R v V (γ) IJK graph structure ɛ(i, J, K) = sgn ( det(ė I, ė J, ė K )(v) ) ˆq IJK fγ Spatial

45 Volume : Structure Classical Volume Expression V (R)= d 3 x det q(x) = d 3 x det E (x) R R = d 3 1 x 3! εijk ε abc Ei a(x)eb j (x)ec k (x) R Structure of the volume operator ˆV γ (R) f γ l 3 P ɛ(i, J, K) ˆq IJK fγ v V (γ) IJK edge spin ˆq IJK := [ (J IJ ) 2 ], (J }{{} JK)2 εijk JI ij j J J K k (J I +J J ) 2 Spatial

46 Volume Edge Spins & Recoupling Theory ments e 1 e N e 8 e 2 e 3 v e 7 e 4 e5 e 6 Spatial

47 Volume Edge Spins & Recoupling Theory ments j 1 j N j 8 j 2 j 3 v j 7 j 4 j5 j 6 Spatial

48 Volume Edge Spins & Recoupling Theory ments j 1 j 3 j 2 j 4 j5 j N v j 8 j 6 j 7 PSfrag replacements Tensor Basis v j 1 j 2 j 3 j 4 j N T v j m n = N j k m k ; n k > k=1 Spatial

49 Volume Edge Spins & Recoupling Theory ments j 1 j 3 j 2 j 4 j5 j N v j 8 j 6 Decomposition π j1 π j2 = j 7 j 1 +j 2 π j12 j 12 = j 1 j 2 PSfrag replacements Tensor Basis v j 1 j 2 j 3 j 4 j N T v j m n = N j k m k ; n k > k=1 Spatial

50 Volume Edge Spins & Recoupling Theory ments j 1 j 3 j 2 j 4 j5 j N v j 8 j 6 Decomposition π j1 π j2 = j 7 j 1 +j 2 π j12 j 12 = j 1 j 2 PSfrag replacements Tensor Basis v j 1 j 2 j 3 j 4 j N T v j m n = N j k m k ; n k > k=1 Recoupling Basis PSfrag replacements j 7 j 8 j 1 j 2 j 3 j 4 j N a 2 a 3 T v I J M j n = a J M ; j n > J Spatial

51 Edge Spins & Recoupling Theory < a ˆq IJK a > = 1 4 ( 1)j K +j I +a I 1 +a K ( 1) a I a I ( 1) P J 1 n=i+1 jn ( 1) P K 1 p=j+1 jp X(j I, j J) 1 2 X(j J, j K) 1 2 " n o J 1 ai 1 j I a Y I 1 a I j I " K 1 " Y n=j+1 n=i+1 q(2a I + 1)(2a I + 1) q (2a J + 1)(2a J + 1) p n # (2a n + 1)(2a n + 1)( 1) a n 1 +a n 1+1 j n a n 1 a o n 1 a n a n 1 p n # (2a n + 1)(2a n + 1)( 1) a n 1 +a n 1+1 j n a n 1 a o n o n ak j K a K 1 1 a n a n 1 1 a K 1 j K n ( 1) a J +a J 1 aj j J a o n J 1 a J 1 j J a o J 1 a J 1 j J 1 a J j J n ( 1) a J +a J 1 a J j J a o j ff# J 1 a J 1 j J a J 1 a J 1 j J 1 a J j J Y I 1 δ ana n n=2 NY n=k δ ana n where X(a, b) = 2a(2a + 1)(2a + 2)2b(2b + 1)(2b + 2)

52 Graph Structure e 1 e 2 e 3 e N e 8 e 7 v e 4 e 6 e 5 N edges at vertex v Spatial

53 Graph Structure e 3 e 2 e 1 ė1 ė2 e N ėn ė8 e 8 e 7 e 4 ė3 ė7 v ė6 ė4 ė5 e 6 e 5 N edges at vertex v with tangent vectors Spatial

54 Graph Structure ėn ė1 ė2 ė8 ė7 ė3 v ė6 ė4 ė5 N edges at vertex v with tangent vectors Spatial

55 Graph Structure ėn ė1 ė2 ė8 ė3 ė7 v ė6 ė4 (135) ė5 N edges at vertex v with tangent vectors edge triple e 1, e 3, e 5 Spatial

56 Graph Structure ėn ė1 ė2 ė8 ė3 ė7 v ė6 ė4 (135) ė5 N edges at vertex v with tangent vectors edge triple e 1, e 3, e 5 ɛ(1 3 5) := sgn ( det( ė 1, ė 3, ė 5 ) ) Spatial

57 Graph Structure ė2 ė1 ėn ė8 ė3 ė7 v ė6 ė4 (135) ė5 N edges at vertex v with tangent vectors edge triple e 1, e 3, e 5 ɛ(1 3 5) := sgn ( det( ė 1, ė 3, ė 5 ) ) General: ɛ(l M N) := sgn ( det( ė L, ė M, ė N ) ) Spatial

58 Graph Structure ė2 ė1 ėn ė8 ė3 ė7 v ė6 ė4 (135) ė5 N edges at vertex v with tangent vectors edge triple e 1, e 3, e 5 ɛ(1 3 5) := sgn ( det( ė 1, ė 3, ė 5 ) ) General: ɛ(l M N) := sgn ( det( ė L, ė M, ė N ) ) ( ) System of N 3 inequalities (assume ɛ(lmn) = ±1): Spatial

59 Graph Structure ė2 ė1 ėn ė8 ė3 ė7 v ė6 ė4 (135) ė5 N edges at vertex v with tangent vectors edge triple e 1, e 3, e 5 ɛ(1 3 5) := sgn ( det( ė 1, ė 3, ė 5 ) ) General: ɛ(l M N) := sgn ( det( ė L, ė M, ė N ) ) ( ) System of N 3 inequalities (assume ɛ(lmn) = ±1): 0 < ɛ(l M N) det( ė L, ė M, ė N ) ) Spatial

60 Graph Structure ė2 ė1 ėn ė8 ė3 ė7 v ė6 ė4 (135) ė5 N edges at vertex v with tangent vectors edge triple e 1, e 3, e 5 ɛ(1 3 5) := sgn ( det( ė 1, ė 3, ė 5 ) ) General: ɛ(l M N) := sgn ( det( ė L, ė M, ė N ) ) ( ) System of N 3 inequalities (assume ɛ(lmn) = ±1): 0 < ɛ(l M N) det( ė L, ė M, ė N ) ) What sign combinations will occur at all? Spatial

61 Graph Structure Edge Spins! Gauge Invariance: J N = N 1 J L L=1 Spatial

62 Graph Structure Edge Spins! Gauge Invariance: J N = N 1 J L L=1 This implies for ˆV γ acting on gauge invariant spin networks: ˆV γ =: I,J,K<N I,J,K<N [ ɛ(ijk) ɛ(jkn) + ɛ(ikn) ɛ(ijn) ] ˆqIJK σ(ijk) ˆq IJK where 4 σ(ijk) 4 Spatial

63 Graph Structure Edge Spins! Gauge Invariance: J N = N 1 J L L=1 This implies for ˆV γ acting on gauge invariant spin networks: ˆV γ =: I,J,K<N I,J,K<N [ ɛ(ijk) ɛ(jkn) + ɛ(ikn) ɛ(ijn) ] ˆqIJK σ(ijk) ˆq IJK where 4 σ(ijk) 4 We have a sum of hermitian matrices with varying prefactors. What does that imply? Spatial

64 Graph Structure Edge Spins! Gauge Invariance: J N = N 1 J L L=1 This implies for ˆV γ acting on gauge invariant spin networks: ˆV γ =: I,J,K<N I,J,K<N [ ɛ(ijk) ɛ(jkn) + ɛ(ikn) ɛ(ijn) ] ˆqIJK σ(ijk) ˆq IJK where 4 σ(ijk) 4 We have a sum of hermitian matrices with varying prefactors. What does that imply?? What sign configurations can be realized at all? Spatial

65 Graph Structure Edge Spins! Gauge Invariance: J N = N 1 J L L=1 This implies for ˆV γ acting on gauge invariant spin networks: ˆV γ =: I,J,K<N I,J,K<N [ ɛ(ijk) ɛ(jkn) + ɛ(ikn) ɛ(ijn) ] ˆqIJK σ(ijk) ˆq IJK where 4 σ(ijk) 4 We have a sum of hermitian matrices with varying prefactors. What does that imply?? What sign configurations can be realized at all?? What consequences does gauge invariance have? Spatial

66 Graph Structure Edge Spins! Gauge Invariance: J N = N 1 J L L=1 This implies for ˆV γ acting on gauge invariant spin networks: ˆV γ =: I,J,K<N I,J,K<N [ ɛ(ijk) ɛ(jkn) + ɛ(ikn) ɛ(ijn) ] ˆqIJK σ(ijk) ˆq IJK where 4 σ(ijk) 4 We have a sum of hermitian matrices with varying prefactors. What does that imply?? What sign configurations can be realized at all?? What consequences does gauge invariance have? Contact: recoupling of spins properties of space Spatial

67 Graph Structure Edge Spins! Gauge Invariance: J N = N 1 J L L=1 This implies for ˆV γ acting on gauge invariant spin networks: ˆV γ =: I,J,K<N I,J,K<N [ ɛ(ijk) ɛ(jkn) + ɛ(ikn) ɛ(ijn) ] ˆqIJK σ(ijk) ˆq IJK where 4 σ(ijk) 4 N ( ) N 3 N (max) N ɛ N ɛ ɛ N (max) ɛ N σ N σ= , , ,324, ,912, Spatial

68 Graph Structure Edge Spins! Gauge Invariance: J N = N 1 J L L=1 This implies for ˆV γ acting on gauge invariant spin networks: ˆV γ =: I,J,K<N I,J,K<N [ ɛ(ijk) ɛ(jkn) + ɛ(ikn) ɛ(ijn) ] ˆqIJK σ(ijk) ˆq IJK where 4 σ(ijk) 4 N ( ) N 3 N (max) N ɛ N ɛ ɛ N (max) ɛ N σ N σ= , , Spatial Zero Volume states property independent from spins

69 Gauge Invariant 5-Vertex: Largest Eigenvalue λ nts σ j max

70 Gauge Invariant 5-Vertex: Largest Eigenvalue: σ-dependence λ σ j max #σ σ(123) σ(124) σ(134) σ(234) have same absolute values

71 Gauge Invariant 5-Vertex: Smallest Eigenvalue λ λ ts j max PSfrag replacements σ 2 j max σ 170

72 Gauge Invariant 5-Vertex: Smallest Eigenvalue: σ-dependence λ j max σ #σ σ(123) σ(124) σ(134) σ(234) have same absolute values

73 Gauge Invariant 5-Vertex: Histograms for each sigma config frequency vertex; sigmas = /2 3/2 4/2 5/2 6/2 7/2 8/2 9/2 10/2 11/2 12/2 13/2 14/2 15/2 16/2 17/2 18/2 19/2 20/2 21/2 22/2 23/2 24/2 25/2 26/2 27/2 28/2 29/2 30/2 31/2 32/2 33/2 34/2 35/2 36/ ts λ

74 Gauge Invariant 5-Vertex: Histograms for each sigma config frequency vertex; sigmas = /2 3/2 4/2 5/2 6/2 7/2 8/2 9/2 10/2 11/2 12/2 13/2 14/2 15/2 16/2 17/2 18/2 19/2 20/2 21/2 22/2 23/2 24/2 25/2 26/2 27/2 28/2 29/2 30/2 31/2 32/2 33/2 34/2 35/2 36/ ts λ

75 Gauge Invariant 5-Vertex: Histograms for each sigma config frequency 1.6e e e+06 1e vertex; sigmas = /2 3/2 4/2 5/2 6/2 7/2 8/2 9/2 10/2 11/2 12/2 13/2 14/2 15/2 16/2 17/2 18/2 19/2 20/2 21/2 22/2 23/2 24/2 25/2 26/2 27/2 28/2 29/2 30/2 31/2 32/2 33/2 34/2 35/2 36/ ts λ

76 Gauge Invariant 5-Vertex: Histograms for each sigma config frequency vertex; sigmas = /2 3/2 4/2 5/2 6/2 7/2 8/2 9/2 10/2 11/2 12/2 13/2 14/2 15/2 16/2 17/2 18/2 19/2 20/2 21/2 22/2 23/2 24/2 25/2 26/2 27/2 28/2 29/2 30/2 31/2 32/2 33/2 34/2 35/2 36/ ts λ

77 Gauge Invariant 5-Vertex: Histograms for each sigma config Histograms for all sigma configs together 4.5e+07 4e e+07 3e+07 frequency 2.5e+07 2e e+07 1e+07 ts 5e λ

78 Gauge Invariant 5-Vertex: Full Spectrum Cumlative histogram for each j max 4.5e+09 4e e+09 3e+09 frequency 2.5e+09 2e e+09 1e+09 ts 5e λ

79 Gauge Invariant 5-Vertex: Full Spectrum Cumlative histogram for each j max 4.5e+09 4e e+09 3e+09 frequency 2.5e+09 2e zeros eigenvalues 1.5e+09 1e+09 ts 5e λ

80 Gauge Invariant 5-Vertex: Full Spectrum Cumlative histogram for each j max 1.8e e e e+09 frequency 1e+09 8e+08 6e+08 4e+08 2e+08 0 λ ts 11 Expand region with λ<11

81 Gauge Invariant 5-Vertex: Full Spectrum Cumlative histogram for each j max for λ < 11 4e e+08 3e e+08 frequency 2e e zeros eigenvalues 1e+08 ts 5e λ

82 Gauge Invariant 6-Vertex: Largest Eigenvalue λ nts σ j max

83 Gauge Invariant 6-Vertex: Smallest Eigenvalue nts λ σ j max

84 Gauge Invariant 6-Vertex: Full Spectrum Cumlative histogram for each j max frequency 1.6e e e+10 1e+10 8e+09 6e zeros eigenvalues 1/2 2/2 3/2 4/2 5/2 6/2 7/2 8/2 9/2 10/2 11/2 4e+09 2e+09 ts λ

85 Outline 1 What is? 2 Review of : Fundamentally Discrete for Entropy Bound Spherically Symmetric Hyperboloidal Slices 3 Spatial Computing Spatial from Simulations 4 5 Spatial

86 : Discrete QG may lead to explanation for origin of entropy bounds Susskind bound may arise via counting maximal elements Spatial

87 : Discrete QG may lead to explanation for origin of entropy bounds Susskind bound may arise via counting maximal elements : Can extract spatial homology from causal set useful tool for identifying manifoldlike orders Build suite of tools to extract macroscopic properties Spatial

88 : Discrete QG may lead to explanation for origin of entropy bounds Susskind bound may arise via counting maximal elements : Can extract spatial homology from causal set useful tool for identifying manifoldlike orders Build suite of tools to extract macroscopic properties : Important interplay: graph embedding spin recoupling Spatial discreteness of not completely decided: smallest non-zero eigenvalue Volume operator accessible to full computational analysis Start to performing computations in full Loop Quantum Spatial

89 : Discrete QG may lead to explanation for origin of entropy bounds Susskind bound may arise via counting maximal elements : Can extract spatial homology from causal set useful tool for identifying manifoldlike orders Build suite of tools to extract macroscopic properties : Important interplay: graph embedding spin recoupling Spatial discreteness of not completely decided: smallest non-zero eigenvalue Volume operator accessible to full computational analysis Start to performing computations in full Loop Quantum QG Supercomputing & : Numerical computing useful for gaining insight into QG Spatial is excellent tool to facilitate this Develop community code base to address problems in QG