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Ground state of quantum gravity in
dynamical triangulation
Jan Smit
Institute for Theoretical Physics,University of Amsterdam & Utrecht University,
the Netherlands
1
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∗Presenting unpublished (’96) data of work with Bas V. de Bakker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
Euclidean Dynamical Triangulation (EDT)∗
Simplicial manifold build of equilateral 4-simplices
Ni, i = 0,1, . . . ,4 number of i-simplices
χ = N0 −N1 + N2 −N3 + N4 Euler
χ fixed: only two Ni are independent
∗Weingarten, NPB210[FS6](1982)229; Ambjørn, Jurkiewicz,
PLB278(1992)42; Agishtein, Migdal, MPLA7(1992)1039
3
‘pure gravity’
S =
∫d4x
√g
(2Λ0 −R
16πG0
)
→ −κ2N2 + κ4N4
κ2 =V2
8πG0, κ4 =
Λ0 + 10θV2
8πG0
θ = arccos(1/4) Regge deficit angle
Vi =`i√
i + 1
i!√
2ivolume of i-simplex
` edge length
4
Z =
∫Dg e−S →
Z(κ2, κ4) =∑
Teκ2N2−κ4N4
=∑
N4
e−κ4N4Z(κ2, N4)
Z(κ2, N4) =∑
T (N4)
eκ2N2
∼ (N4)γ−3 eκc
4N4, N4 →∞
- well defined for fixed topology (e.g. S4, χ = 2)
- κ4 > κc4(κ2) controls average volume 〈N4〉
5
Explore system at given N4, topology S4 (χ = 2)
‘canonical average’
〈O〉 =1
Z(κ2, N4)
∑
T (N4),S4
eκ2N2 O
• phase transition at κ2 = κc2(N4)
κ2 < κc2, crumpled phase
' κc2, transition region
> κc2, elongated phase
6
< ’96 transition considered continuous, 2nd order∗
2nd order fixed point believed necessary for continuumlimit
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
problematic features
- proliferation of baby universes, ‘spikes’
- singular structure in crumpled phase∗
& ’97 new development: AMM scenario∗∗ may curespikes∗∗∗
- effective action incorporating conformal anomaly
- ‘central charge’ Q2 in 4D analogous to c in 2D
remarkable difference:
- spikes suppressed for c < 1 (2D) and Q2 > 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
Q2 =1
180
(NS +
11
3NWF + 62NV − 28
)+ Q2
grav
Q2grav = 1566/360 ' 8.7 Weyl2 action
= 1411/360 ' 7.8 Einstein action∗
NS, NWF, NV : # scalar-, Weyl fermion-, vector-fields
−28/180 ' −0.16 from conformal mode
Q2 > 8 for NS > 57 or NV ≥ 1
∗Antoniadis, Mazur, Mottola, NPB388(1992)627
9
Q2 related to γ
lnZ(κ2, N4)
N4= κc
4(κ2) + [γ(κ2)− 3]lnN4
N4+ · · ·
γ = 2− Q2
4
(1 +
√1− 8
Q2
)
Antoniadis, Mazur, Mottola, PLB323(1994)284; PLB394(1997)49
10
add matter, study phase structure and compute γ
& ’98: controversy German-Japanese groups∗
Causal DT arrived∗∗ (prohibits creation of baby universesin 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
κ20
Nv
κ2c
1
2 3
1st order phase transition line
We expectn th order phasetransition line(n > 1).
Crumpled phase
Branched Polymer
Smooth phase
correspond to c=1barrier in 2D QG
similar to 2D QGfor c < 1 case
1
2
3
connect to 4D QG
obscure transition line
We make observationof transionat Nv=1
no sign of 1st order at X
Horata, Egawa, Tsuda, Yukawa, PTP106(2001)1037
12
‘Grand Canonical’ simulation result∗
b =Q2
2= 0.0030(3) (NS + 62NV ) + 3.98(3)
note
1
360' 0.0028
−28 + 1566
360' 4.27
−28 + 1411
360' 3.84
striking accordance with analytic formula≈ correct contribution of gravitons and matter fields
∗Horata, Egawa, Yukawa, PTP108(2002)1171
13
• expect Planck length G−1/2 = O(`)(c.f. RG studies, ‘asymptotic safety’)
• but massless gravitons may emerge∗ as N4 →∞•EDT still excellent method for non-perturbative studyof quantum-gravitational ground state
∗ Compare chiral models for NG bosons in QCD, or emergence of
photons in Z(n) gauge theory, n ≥ 5.
14
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) = Cnrn
[1− Rr2
6(n + 2)+O(r4)
], Cn =
πn/2
(n/2)!
V ′(r) = nCnrn−1
(1− Rr2
6n+ · · ·
)
15
→ V (r) = veffN(r), V ′(r) = veffN ′(r)
N(r) = average number of 4-simplices withingeodesic 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 ↔ ` =√
10
De Bakker, JS, NPB439(1995)239
16
0
500
1000
1500
2000
2500
3000
0 10 20 30 40 50 60 70 80 90 100
N’(r
)
r
0.801.221.50
Number of simplices N ′(r) at distance r from the (arbitrary) ori-
gin at κ2 = 0.80 (crumpled phase), 1.22 (transition region), 1.50
(elongated phase), for N4 = 16000.
choose n = 4 and fit N ′(r) = ar3 + br5 for small r(but not too small)
then RV ≡ −24b/a
RV < 0, crumpled phase> 0, elongated phase≈ 0, transition region
17
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.7 0.8 0.9 1 1.1 1.2 1.3
R_V
k2
800016000
Curvature RV as a function of κ2 for N4 = 8000 and 16000.
18
RV depends on fitting range
• running curvature Reff(r) at distance r
skip
put
N ′(r) = a(r)r3 + b(r)r5
N ′(r + 1) = a(r)(r + 1)3 + b(r)(r + 1)5
Reff(r +1
2) ≡ −24b(r)/a(r)
or
Reff(r + 12) = 24
(r + 1)3 − r3N ′(r + 1)/N ′(r)(r + 1)5 − r5N ′(r + 1)/N ′(r)
.
19
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
R_e
ff
r
0.801.001.201.221.231.50
Reff(r) for κ2 = 0.80, . . . , 1.50; N4 = 16000
‘Planckian region’ r . 5
20
•Euclidean Robertson-Walker metric (‘proper time’ r)
ds2 = dr2 + a(r)2dΩ3, dΩ3 metric on S3
vN ′(r) = a(r)3 v = veff/2π2
R = 6
(−a
a− a2
a2+
1
a2
)
21
1
2
3
4
5
6
7
8
0 2 4 6 8 10
"np3+1.300.dat""np3+1.260.dat"
N ′(r)1/3 at κ2 = 1.26 (crumpled phase) and 1.3 (elongated phase),N4 = 64000
22
20 40 60 80 100r
500
1000
1500
2000
2500
N'Hr L
N ′(r) at κ2 = 1.266 (crumpled phase), N4 = 64000
23
extrapolate a(r) = (vN ′(r))1/3 linearly to zero
v such that a′(0) = a′(1) = 1
0 5 10 15 20 25r
5
10
15
20
25
30aHrL
scale factor a(r) for κ2 = 1.266, N4 = 64000, crumpled phase
24
a(0) 6= 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/a2 term in R).
25
10 15 20 25 30r
-0.15
-0.10
-0.05
0.05
RHrL
RW curvature R(r) for κ2 = 1.266, N4 = 64000, crumpled phase
26
effective action
S =
∫d4x
√g
(λ− 1
16πGR + · · ·
)
→ 2π2
∫ r2
r1
dr
[λa3 − 6
16πG(aa2 + a) + · · ·
]
27
solutions of δS = 0
Gλ > 0 : a = r0 sinr
r0, R =
12
r20
= 32πGλ, S4
Gλ < 0 : a = r0 sinhr
r0, R =
−12
r20
= −32π|Gλ|, H4
try fitting∗ S4 (‘de Sitter’) in elongated phase,H4 (‘anti-de Sitter’) in crumpled phase
∗De Bakker, JS, NPB439(1995)239; S4 fits looked reasonable at
κ ≈ κc2, for 6 . r . rmax , H4 not done at the time.
28
crumpled phase
fit r0 sinh[(r − s0)/r0] 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 r0 sin[(r − s0)/r0] to a(r) in region 4 ≤ r ≤ 11
29
0 5 10 15 20r
10
20
30
40
50aHrL
hyperbolic-sine fit to a(r) data at R < 0 (r = 3, . . . ,13), κ2 = 1.266,
r0 = 9.7, s0 = −2.2, crumpled phase
30
50 100 150 200 250 300r
100
200
300
400
500
N'Hr L
N ′(r) for κ2 = 1.3, N4 = 64000, elongated phase
32
Test fields and particles
- do not ’back react’ on geometry
Scalar field
S = Sg + Sφ
Sg =1
16πG0
∫d4x
√g (2Λ0 −R)
Sφ =
∫d4x
√g
(1
2gµν∂µφ∂νφ +
1
2m2
0φ2
)
Z =
∫Dg Dφ e−S
〈O〉 =1
Z
∫Dg Dφ e−S O
38
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⟨∫
d4x√
g O(x)O′(y) δ[d(x, y)− r]∫d4x
√g δ[d(x, y)− r]
⟩
or (better)⟨∫
d4x√
g O(x)O′(y) δ[d(x, y)− r]⟩
⟨∫d4x
√g δ[d(x, y)− r]
⟩
- independent of y- non-local observables
39
φ test field: quenched approximation
Z =
∫Dg e−Sg [det(−¤ + m2
0)]−1/2
→Zg =
∫Dg e−Sg
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
⟩g
=1
Zg
∫Dg e−Sg G(x, y)|d(x,y)=r
G(x, y) =[(−¤ + m2
0)−1
]x,y
40
Binding energy near transition∗
quenched approximation
tentative comparison with positronium∗∗:
Eb = α2m/4, α → αG = Gm2
for m0 = 0.316 this gives
αG = 0.72− 0.60,`P
r0≡√
8πG
r0= 0.53− 0.47
∗De Bakker, JS, NPB484(1997)476∗∗r0 ≈ 13− 14, κ2 = 1.255− 1.259, N4 = 32000, near transition on
elongated side
41
Test in crumpled phase
massless minimally coupled scalar acquires effective masson space with constant negative curvature
a(r) = r0 sinh(r/r0), R = −12/r20
¤G(r) =
(d
dr+
3
r0coth
r
r0
)d
drG(r) = 0, r > 0
G(r) → 1
4πr2r → 0
→ 0 r →∞results in
G(r) =1
3πr20
e−3r/r0 +O(e−5r/r0), r →∞
43
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 10 20 30
1.2401.2451.2501.252
exponential fits to G(r), for N4 = 32000 and κ2 = 1.240, . . . , 1.252
45
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
1.24 1.25 1.26 1.27
m
κ2
3200064000
‘measured’ masses vs κ2 for N4 = 32000, 64000
46
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
1.2 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.3
RV
κ2
3200064000
‘measured’ RV
47
good fits in 6 ≤ r ≤ 20!
m correlated with√
RV
expect m → 0 when curvature → 0(no additive mass renormalization)
consider power fit∗ to minimum of RW curvature
m = c(√−3Rmin/4)b
∗In binding-energy computation, mass renormalization can also be
fitted by power behavior, m2 ≈ 1.5(m20)
0.65
48
æ
æ
æ
æ
0.37 0.38 0.39 0.40-3 R 4
0.11
0.12
0.13
0.14
0.15
m
fit of c(√−3Rmin/4)b to induced-mass data, b = 3.7,
c = 4.35, N4 = 64000
49
Scaling
try scaling∗
ρ(x; τ) =rm
N4N ′(r;κ2, N4), x =
r
rm
N ′(r) is maximal at r = rm
τ = shape label, e.g. τ = ρ|x=1 or τ = κ2 at standard N4
∗De Bakker, JS, NPB439(1995)239
50
0 20 40 60 80 100r
500
1000
1500
2000
2500
N'Hr L
N ′(r) for κ2, N4 = 1.17,8000 (blue), 1.21,16000 (red),
1.240,32000 (brown) and 1.266,64000 (green)
51
0 1 2 3 4x
0.2
0.4
0.6
0.8
1.0r_m N'Hx r_mLN
example of scaling in crumpled phase: rmN ′(xrm)/N4 for κ2, N4 =
1.17,8000, 1.21,16000, 1.240,32000 and 1.266,64000
52
scaling dimension∗ ds: rm(κ2, N4) ∝ N1/ds
4
for pairs κ2, N4 belonging to the same scaling sequence(same τ)∗∗
ds ≈ 5.6
similar for rm → rav =∑
r rN ′(r)/N4
∗Sometimes identified with Hausdorff dimension, Ambjørn, Jurkiewicz,
NPB451(1995)643∗∗Neglecting κ2 dependence of rm or rav suggests ds →∞, Catterall,
Kogut, Renken, PLB328(1994)277; A &J.
53
Conclusion
- DT gives approximate continuum results at finitelattice spacing,
√G = O(`),
√Λ = O(`−1)
- scaling is large-scale phenomenon, does not imply`/√
G → 0, property of ground state
- crumpled phase: negative curvature
H4 is infinite, finite volume → ‘singular structure’?
55
- elongated phase: positive curvature
thick branched polymers
condensation of black holes? monsters?
- crucially important to resume study of ground statesand Q2 in EDT with matter, NV ≥ 1
56
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