Ground state of quantum gravity in dynamical triangulation

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1 Ground state of quantum gravity in dynamical triangulation Jan Smit Institute for Theoretical Physics, University of Amsterdam & Utrecht University, the Netherlands 1

2 1. Euclidean dynamical triangulation 2. Curvature of emerging space-times 3. Test particles 4. Induced mass in the crumpled phase 5. Scaling 6. Conclusion Presenting unpublished ( 96) data of work with Bas V. de Bakker 2

3 Euclidean Dynamical Triangulation (EDT) Simplicial manifold build of equilateral 4-simplices N i, i = 0, 1,..., 4 χ = N 0 N 1 + N 2 N 3 + N 4 number of i-simplices Euler χ fixed: only two N i are independent Weingarten, NPB210[FS6](1982)229; Ambjørn, Jurkiewicz, PLB278(1992)42; Agishtein, Migdal, MPLA7(1992)1039 3

4 pure gravity S = d 4 x ( ) 2Λ0 R g 16πG 0 κ 2 N 2 + κ 4 N 4 κ 2 = V 2 8πG 0, κ 4 = Λ θV 2 8πG 0 θ = arccos(1/4) Regge deficit angle V i = li i + 1 i! 2 i volume of i-simplex l edge length 4

5 Z = Dg e S Z(κ 2, κ 4 ) = T e κ 2N 2 κ 4 N 4 Z(κ 2, N 4 ) = = N 4 e κ 4N 4 Z(κ 2, N 4 ) T (N 4 ) e κ 2N 2 (N 4 ) γ 3 e κc 4 N 4, N 4 - well defined for fixed topology (e.g. S 4, χ = 2) - κ 4 > κ c 4 (κ 2) controls average volume N 4 5

6 Explore system at given N 4, topology S 4 (χ = 2) canonical average O = 1 Z(κ 2, N 4 ) T (N 4 ),S 4 e κ2n2 O phase transition at κ 2 = κ c 2 (N 4) κ 2 < κ c 2, crumpled phase κ c 2, transition region > κ c 2, elongated phase 6

7 < 96 transition considered continuous, 2nd order 2nd order fixed point believed necessary for continuum limit other possibility: critical regions, evidence for scaling 96 1st order Catterall, Kogut, Renken, PLB328(1994)277; Ambjørn, Jurkiewicz, NPB451(1995)643 De Bakker, JS, NPB439(1995)239. Bialas, Burda, Krywicki, Petersson, NPB472(1996)293; De Bakker, PLB389(1996)238 7

8 problematic features - proliferation of baby universes, spikes - singular structure in crumpled phase 97 new development: AMM scenario may cure spikes - effective action incorporating conformal anomaly - central charge Q 2 in 4D analogous to c in 2D remarkable difference: - spikes suppressed for c < 1 (2D) and Q 2 > 8 (4D) Hotta, Izubuchi, Nishimura, PTP94(1995)263; Catterall, Kogut, Renken, Thorleifsson, NPB468(1996)263. Antoniadis, Mazur, Mottola, NPB388(1992)627; PLB323(1994)284. AMM, PLB394(1997)49; Jurkiewicz, Krzywicki, PLB392(1997)291. 8

9 Q 2 = 1 ( N S + 11 ) N WF + 62N V 28 + Q 2 grav Q 2 grav = 1566/ Weyl 2 action = 1411/ Einstein action N S, N WF, N V : # scalar-, Weyl fermion-, vector-fields 28/ from conformal mode Q 2 > 8 for N S > 57 or N V 1 Antoniadis, Mazur, Mottola, NPB388(1992)627 9

10 Q 2 related to γ ln Z(κ 2, N 4 ) = κ c 4 N (κ 2) + [γ(κ 2 ) 3] ln N N ( 4 γ = 2 Q Q ) 4 2 Antoniadis, Mazur, Mottola, PLB323(1994)284; PLB394(1997)49 10

11 add matter, study phase structure and compute γ 98: controversy German-Japanese groups Causal DT arrived (prohibits creation of baby universes in time direction) 00: only Japanese group continued with EDT Bilke et al.; Horata et al Ambjørn, Loll, NPB(1998)536, Ambjørn, Jurkiewicz, Loll, PRL85(2000)924 11

12 1st order phase transition line 0 1 κ 2 c 2 3 κ Crumpled phase We make observation of transion at Nv=1 Branched Polymer Smooth phase correspond to c=1 barrier in 2D QG obscure transition line Nv We expect n th order phase transition line (n > 1). connect to 4D QG similar to 2D QG for c < 1 case no sign of 1st order at X Horata, Egawa, Tsuda, Yukawa, PTP106(2001)

13 Grand Canonical simulation result b = Q2 2 = (3) (N S + 62N V ) (3) note striking accordance with analytic formula correct contribution of gravitons and matter fields Horata, Egawa, Yukawa, PTP108(2002)

14 expect Planck length G 1/2 = O(l) (c.f. RG studies, asymptotic safety ) but massless gravitons may emerge as N 4 EDT still excellent method for non-perturbative study of quantum-gravitational ground state Compare chiral models for NG bosons in QCD, or emergence of photons in Z(n) gauge theory, n 5. 14

15 Curvature R (Regge) has divergent terms in R need to measure curvature at larger scales for a smooth geometry in n dimensions, volume within geodesic radius r from point: V (r) curvature R found from [ ] V (r) = C n r n 1 Rr2 6(n + 2) + O(r4 ) ( ) V (r) = nc n r n 1 1 Rr2 6n +, C n = πn/2 (n/2)! 15

16 V (r) = v eff N(r), V (r) = v eff N (r) N(r) = average number of 4-simplices within geodesic distance r N (r) = N(r) N(r 1) geodesic distance r: minimum distance (number of hops ) between centers of 4-simplices r = 1 between neighbors l = 10 De Bakker, JS, NPB439(1995)239 16

17 N (r) r Number of simplices N (r) at distance r from the (arbitrary) origin at κ 2 = 0.80 (crumpled phase), 1.22 (transition region), 1.50 (elongated phase), for N 4 =

18 choose n = 4 and fit N (r) = ar 3 + br 5 for small r (but not too small) then R V 24b/a R V < 0, crumpled phase > 0, elongated phase 0, transition region 17

19 R_V k2 Curvature R V as a function of κ 2 for N 4 = 8000 and

20 R V depends on fitting range running curvature R eff (r) at distance r skip put N (r) = a(r)r 3 + b(r)r 5 N (r + 1) = a(r)(r + 1) 3 + b(r)(r + 1) 5 R eff (r ) 24b(r)/a(r) or R eff (r ) = 24(r + 1)3 r 3 N (r + 1)/N (r) (r + 1) 5 r 5 N (r + 1)/N (r). 19

21 R_eff r R eff (r) for κ 2 = 0.80,..., 1.50; N 4 = Planckian region r 5 20

22 Euclidean Robertson-Walker metric ( proper time r) ds 2 = dr 2 + a(r) 2 dω 3, dω 3 metric on S 3 vn (r) = a(r) 3 v = v eff /2π 2 R = 6 ( ä a ȧ2 a + 1 ) 2 a 2 21

23 8 "np dat" "np dat" N (r) 1/3 at κ 2 = 1.26 (crumpled phase) and 1.3 (elongated phase), N 4 =

24 N' r r N (r) at κ 2 = (crumpled phase), N 4 =

25 extrapolate a(r) = (vn (r)) 1/3 linearly to zero v such that a (0) = a (1) = 1 a r r scale factor a(r) for κ 2 = 1.266, N 4 = 64000, crumpled phase 24

26 a(0) 0: can shift a(r) horizontally such that a(0) = 0 - lattice artefact, don t bother shifting a(r) vertically downwards such that a(0) = 0 appears to give not as good results (enhances 1/a 2 term in R). 25

27 R r r RW curvature R(r) for κ 2 = 1.266, N 4 = 64000, crumpled phase 26

28 effective action S = d 4 x g r2 [ 2π 2 dr r 1 ( λ 1 16πG R + ) λa πG (aȧ2 + a) + ] 27

29 solutions of δs = 0 Gλ > 0 : a = r 0 sin r, r 0 R = 12 r0 2 = 32πGλ, S 4 Gλ < 0 : a = r 0 sinh r, r 0 R = 12 r0 2 = 32π Gλ, H 4 try fitting S 4 ( de Sitter ) in elongated phase, H 4 ( anti-de Sitter ) in crumpled phase De Bakker, JS, NPB439(1995)239; S 4 fits looked reasonable at κ κ c 2, for 6 r r max, H 4 not done at the time. 28

30 crumpled phase fit r 0 sinh[(r s 0 )/r 0 ] to a(r) e.g. in region where R < 0 (3 r 11) or R < 0 after minimum of R (6 r 11) elongated phase fit r 0 sin[(r s 0 )/r 0 ] to a(r) in region 4 r 11 29

31 a r r hyperbolic-sine fit to a(r) data at R < 0 (r = 3,..., 13), κ 2 = 1.266, r 0 = 9.7, s 0 = 2.2, crumpled phase 30

32 a r ^ r the fit to a(r) 3 31

33 500 N' r r N (r) for κ 2 = 1.3, N 4 = 64000, elongated phase 32

34 100 N' r r close-up 33

35 a r r scale factor a(r) 34

36 R r r curvature R(r) 35

37 a r r sine fit to a(r) data at r = 4, 5,..., 11, r 0 = 11.0, s 0 =

38 a r ^ r the fit to a(r) 3 37

39 Test fields and particles - do not back react on geometry Scalar field S = S g + S φ 1 S g = d 4 x g (2Λ 0 R) 16πG 0 S φ = d 4 x ( 1 g 2 gµν µ φ ν φ + 1 ) 2 m2 0φ 2 Z = Dg Dφ e S O = 1 Dg Dφ e S O Z 38

40 interested in O(x)O (y) d(x,y)=r e.g. O(x) = R(x), φ(x), φ(x) 2,... d(x, y) geodesic distance depends on g implement as d 4 x g O(x)O (y) δ[d(x, y) r] d 4 x g δ[d(x, y) r] or (better) d 4 x g O(x)O (y) δ[d(x, y) r] d 4 x g δ[d(x, y) r] - independent of y - non-local observables 39

41 φ test field: quenched approximation Z = Dg e S g [det( + m2 0 )] 1/2 Z g = Dg e S g Laplace-Beltrami operator e.g. two-point function G(r) = φ(x)φ(y) d(x,y)=r G(x, y) d(x,y)=r = 1 Dg e S g Z G(x, y) d(x,y)=r g g G(x, y) = [ ( + m 2 0 ) 1] x,y 40

42 Binding energy near transition quenched approximation tentative comparison with positronium : E b = α 2 m/4, α α G = Gm 2 for m 0 = this gives α G = , l P r 0 8πG r 0 = De Bakker, JS, NPB484(1997)476 r , κ 2 = , N 4 = 32000, near transition on elongated side 41

43 Test in crumpled phase massless minimally coupled scalar acquires effective mass on space with constant negative curvature a(r) = r 0 sinh(r/r 0 ), R = 12/r 2 0 ( d G(r) = dr + 3 coth r ) d G(r) = 0, r > 0 r 0 r 0 dr G(r) 1 r 0 4πr 2 0 r results in G(r) = 1 3πr 2 0 e 3r/r 0 + O(e 5r/r 0 ), r 43

44 effective mass m = 3 r 0 = 3 4 R compute G(r) in crumpled phase: fit G(r) = ce mr + c 44

45 exponential fits to G(r), for N 4 = and κ 2 = 1.240,...,

46 m κ 2 measured masses vs κ 2 for N 4 = 32000,

47 R V measured R V 47 κ 2

48 good fits in 6 r 20! m correlated with R V expect m 0 when curvature 0 (no additive mass renormalization) consider power fit to minimum of RW curvature m = c( 3R min /4) b In binding-energy computation, mass renormalization can also be fitted by power behavior, m 2 1.5(m 2 0 )

49 0.15 m R 4 fit of c( 3R min /4) b to induced-mass data, b = 3.7, c = 4.35, N 4 =

50 Scaling try scaling ρ(x; τ) = r m N 4 N (r; κ 2, N 4 ), x = r r m N (r) is maximal at r = r m τ = shape label, e.g. τ = ρ x=1 or τ = κ 2 at standard N 4 De Bakker, JS, NPB439(1995)239 50

51 N' r r N (r) for {κ 2, N 4 } = {1.17, 8000} (blue), {1.21, 16000} (red), {1.240, 32000} (brown) and {1.266, 64000} (green) 51

52 r m N' x r m N x example of scaling in crumpled phase: r m N (xr m )/N 4 for {κ 2, N 4 } = {1.17, 8000}, {1.21, 16000}, {1.240, 32000} and {1.266, 64000} 52

53 scaling dimension d s : r m (κ 2, N 4 ) N 1/d s 4 for pairs κ 2, N 4 belonging to the same scaling sequence (same τ) d s 5.6 similar for r m r av = r rn (r)/n 4 Sometimes identified with Hausdorff dimension, Ambjørn, Jurkiewicz, NPB451(1995)643 Neglecting κ 2 dependence of r m or r av suggests d s, Catterall, Kogut, Renken, PLB328(1994)277; A &J. 53

54 scaling is approximate: deviations in small x region r m not r 0 : r 0 decreases as r m increases 54

55 Conclusion - DT gives approximate continuum results at finite lattice spacing, G = O(l), Λ = O(l 1 ) - scaling is large-scale phenomenon, does not imply l/ G 0, property of ground state - crumpled phase: negative curvature H 4 is infinite, finite volume singular structure? 55

56 - elongated phase: positive curvature thick branched polymers condensation of black holes? monsters? - crucially important to resume study of ground states and Q 2 in EDT with matter, N V 1 56

57 embedded Euclidean black hole each point on the surface is an S 2 57