Causal structure and me in (generalized) CDT Quantum Gravity

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Causal structure and -me in (generalized) CDT Quantum Gravity
Renate Loll
triangulated model of quantum space
Schrödinger Ins.tute, 16 July 2014
Wednesday, July 16, 14
Ins$tute for Mathema$cs, Astrophysics and Par$cle Physics (IMAPP), Radboud University Nijmegen (NL) Defining quantum gravity: Feynman path integral,
“Sum over Histories”
Newton’s constant
cosmological constant
Z(GN , ⇤) =
Z
Dg e
Einstein-‐Hilbert ac.on
iS EH [g]
spacetimes
g2G
• quantum superposi$on principle, applied to space$me itself
• each “path” is a four-‐dimensional, curved space$me geometry g, which can be thought of as a three-‐dimensional, spa$al geometry developing in $me
• the challenge is to make the path integral Z(GN,Λ) into a well-‐
defined, computable quan$ty (this is not straighMorward)
( Einstein-‐Hilbert ac$on: S
EH
1
=
GN
⇥
d x
det g(R[g, ⇥g, ⇥ 2 g]
4
2 )
)
Quantum Gravity from Causal Dynamical TriangulaBons (CDT)
... is a nonperturba*ve implementa$on of the gravita$onal path integral, much in the spirit of laRce quantum field theory, but based on dynamical laRces, reflec$ng the dynamical nature of space$me geometry.
CDT is currently the only candidate quantum theory of gravity which can generate dynamically a space$me with semiclassical proper$es from pure quantum excita$ons, without using a background metric. (C)DT has also given us crucial new insights into nonperturba$ve dynamics and piMalls.
for a comprehensive exposi$on, see our recent review: J. Ambjørn, A. Görlich, J. Jurkiewicz, R.L., Phys. Rep. 519 (2012) 127
Geometric idea behind the CDT approach
to make the path integral well defined at an intermediate stage (“regulariza$on”), we approximate smooth curved space$mes by piecewise flat ones, namely, triangula$ons: typical path integral history (2d quantum gravity) approxima.ng a given classical curved surface through triangula.on
Quantum Theory: approxima.ng the space of all curved geometries by a space of triangula.ons crucial in space$me dimension 4: nonperturba$ve computa$onal tools (Monte Carlo simula$ons) to extract quan$ta$ve results
Key points of the CDT approach:
Few ingredients/priors:
quantum superposi$on principle
locality and causal structure (not Euclidean quantum gravity)
no$on of (proper) $me Wick rota$on
triangulated torus
standard tools of quantum field theory
Few free parameters (Λ, GN, Δ)
Robustness of construc$on; universality
At intermediate stage, approximate curved space$mes by triangula$ons
Crucial: nonperturb. computa$onal tools to extract quan$ta$ve results
Key results in 4d:
• dynamical “emergence” of space$me
• scale-‐dependent dimensionality (2 → 4)
• nontrivial phase structure
• second-‐order phase transi$on!
piece of causal triangula.on
Key points of the CDT approach:
Few ingredients/priors:
quantum superposi$on principle
locality and causal structure (not Euclidean quantum gravity)
no$on of (proper) $me Wick rota$on
triangulated torus
standard tools of quantum field theory
Few free parameters (Λ, GN, Δ)
Robustness of construc$on; universality
At intermediate stage, approximate curved space$mes by triangula$ons
Crucial: nonperturb. computa$onal tools to extract quan$ta$ve results
Key results in 4d:
• dynamical “emergence” of space$me
• scale-‐dependent dimensionality (2 → 4)
• nontrivial phase structure
• second-‐order phase transi$on!
piece of causal triangula.on
NonperturbaBve gravitaBonal path integral via CDT
PI becomes a “democra$c”, regularized sum over piecewise flat space$mes: Z(GN , ⇤) := lim
Newton’s constant
a!0
N !1
cosmol. constant
X
inequiv.
triangul.s
T 2Ga,N
Regge
1
iSG
,⇤ [T ]
N
e
C(T )
|Aut(T)|
Regge ac$on
edge length a = diff-‐invariant UV regulator
Each triangula$on T represents a different curved space$me, consis$ng of N simplices, which can be “Wick-‐rotated” to a Riemannian space.
IMPORTANT: the causal structure of CDT is essen$al! This does not work in Euclidean signature (DT) -‐ no sensible classical limit (~mid-‐90s).
In CDT, “causality” is enforced through a preferred slicing by simplicial spa$al hypermanifolds, with an associated preferred “proper 0me”.
Dynamical emergence of spaceBme in CDT
For suitable bare coupling constants, CDT quantum gravity produces a “quantum space$me”, that is, a ground state, whose macroscopic scaling proper$es are four-‐dimensional and whose macroscopic shape is that of a well known cosmology, de Si0er space.
Evidence: Measuring the expecta$on value of the spa$al three-‐volume <V3(t)> of the universe as a func$on of (discrete, proper) $me t, we find a “volume profile” characteris$c of de Siper space.
three-‐volume V3(t)
proper .me t
What is the role of the disBnguished foliaBon and the associated proper Bme?
Does the preferred $me/
folia$on affect the results?
Not a gauge choice (there are no coordinates).
Con$nuum interpreta$on of “t” on large scales only, in the context of de Siper universe.
proper .me t+1
proper .me t
We have introduced a generalized version of CDT quantum gravity (“gCDT” in the following), where the causal structure and the preferred $me are dissociated (in fact, there is no preferred $me). We have repeated the analysis of the phase structure and the volume profiles previously performed for 2+1 CDT quantum gravity. (S. Jordan and R.L., Phys. Lep. B 724 (2013) 155; Phys. Rev. D 88 (2013) 044055)
Nonfoliated, gCDT quantum gravity in 2+1 dimensions: findings
By introducing a generalized set of simplicial building blocks, we got rid of the preferred $me and associated folia$on, while maintaining a well-‐behaved local causal (= light cone) structure everywhere. In the “nice”, physical phase of the model the simplicial geometry shows some tendency to “foliate”, depending on the coupling.
W.r.t. a no$on of “cosmological” proper $me introduced a posteriori on the geometries we find an excellent matching of volume profiles to a Euclidean de Siper universe throughout.
This strongly suggests that the dis$nguished folia$on is not an essen$al part of CDT’s “background structure”, and that gCDT quantum gravity lies in the same universality class as CDT.
The addi$onal computa$onal complexi$es of gCDT are significant. Wednesday, July 16, 14
gCDT = CDT quantum gravity without dis0nguished foliaBon standard path integral formula$on needs a $me t; propagator G(gin,gout;t) sa$sfies X
G(gin , gout ; t) =
G(gin , g; t1 )G(g, gout ; t2 ), t = t1 +t2
g
proper *me is a natural geometric choice; in standard CDT, slices of constant proper $me t=0, 1, 2, 3, ... coincide with simplicial submanifolds, consis$ng of purely spa$al (d-‐1)-‐simplices
timelike
edges
t=3
t=2
t=1
spacelike edge
building block of standard 1+1 CDT (with light cone)
t=0
building causal space$mes from proper-‐$me strips in standard CDT quantum gravity
NEW: CDT quantum gravity without dis0nguished foliaBon standard path integral formula$on needs a $me t; propagator G(gin,gout;t) sa$sfies X
G(gin , gout ; t) =
G(gin , g; t1 )G(g, gout ; t2 ), t = t1 +t2
g
proper *me is a natural geometric choice; in standard CDT, slices of constant proper $me t=0, 1, 2, 3, ... coincide with simplicial submanifolds, consis$ng of purely spa$al (d-‐1)-‐simplices
timelike
edges
t=3
t=2
t=1
spacelike edge
building block of standard 1+1 CDT (with light cone)
t=0
removing $melike links exhibits the strictly foliated structure of CDT quantum gravity
Relax the strict slicing while retaining causality introduce addi$onal elementary building blocks, e.g. in 1+1 dimensions
timelike
edges
spacelike
edges
new:
timelike
edge
spacelike edge
“<s triangle”
“sst triangle”
impose local causality condi$ons at ver$ces:
building causal space$mes in generalized CDT from these two building blocks
Examples of noncausal verBces in 2D
timelike
edges
spacelike
edges
timelike
edge
new:
spacelike edge
“<s triangle”
1 light cone ✘
“sst triangle”
3 light cones ✘
2 light cones ✔
timelike
edges
spacelike
edges
new:
timelike
edge
spacelike edge
“<s triangle”
“sst triangle”
building causal space$mes in generalized CDT from these two building blocks
.melike
edges
spacelike edges
new:
.melike
edge
spacelike edge
“<s triangle”
“sst triangle”
removing $melike links results in a structure with branches and “bubbles”
Local causal structure global causal structure? Fig. 1
“global causal structure” = absence of closed $melike curves (CTCs)
a $melike curve in our context is an oriented chain of future-‐poin$ng $melike links (Fig. 1) the implica$on local global holds in standard
CDT in 1+1 dimensions, i.e. imposing the local vertex structure of Fig. 2 ensures the existence of a global $me slicing, for both open and close spa$al slices (D. BenedeR, J. Henson, Phys. Lep. B 678 (2009) 222) Fig. 2
the rela$on local-‐global is interes$ng to understand from an abstract mathema$cal point of view (c.f. the hierarchy of no$ons of causality for differen$able Lorentzian manifolds), for a descrip$on as a matrix model, and in concrete numerical implementa$ons
What is the situaBon in generalized CDT in 1+1d? it depends on the boundary condi$ons/topology:
On a simply connected triangula$on of gCDT type with an ini$al and final spacelike boundary, and with internal and boundary ver$ces of the form
one can introduce a consistent $me orienta$on, without any CTCs. (R. Hoekzema, master thesis Utrecht U. 2013) the proof proceeds by contradic$on, assuming the existence of an oriented $melike curve beginning and ending on the ini$al boundary
Counterexample on a 1+1d cylinder CTC (winding number 1)
To obtain a cylinder with ini$al and final spacelike boundaries, a, b and c should be iden$fied pairwise. Local vertex condi$ons of gCDT are obeyed everywhere, but there is a closed $melike curve along the fat red links. (due to B. Ruijl) Wednesday, July 16, 14
Are CTCs a problem in the path integral? not necessarily; they could be sufficiently “rare” in the sense of not affec$ng the con$nuum limit; at any rate, there is no classical theory in 2d
Does 2d gCDT quantum gravity lie in one of the two universality classes for dynamically triangulated models, that of Euclidean DT or Causal DT?
solving the model can be viewed in terms of the combinatorics of “bubbles” and their “decora$on” with standard CDT building blocks:
a bubble
a general gCDT configura$on is obtained by stacking bubbles: Wednesday, July 16, 14
Solving gCDT quantum gravity in 2d?
so far, no exact solu$on known (N.B.: no preferred “$me”, unlike CDT) to get an idea of the con$nuum behaviour of the model, we have studied some observables of gCDT on a cylinder with the help of Monte Carlo simula$ons, for simplicity with $me compac$fied ( torus)
a toroidal configura/on in CDT (note fixed number of /me slices!)
measure spectral and Hausdorff dimensions numerically! for reference: -‐ (E)DT has dS = 2, dH = 4 (J. Ambjørn, J. Jurkiewicz, Y. Watabiki, Nucl. Phys. B 454 (1995) 313)
-‐ CDT has dS ≤ 2, dH = 2 “almost surely” (B. Durhuus, T. Jonsson, J. Wheater, J. Stat. Phys. 139 (2010) 859)
The simulaBons: some results
The path integral a{er Wick rota$on is
X 1
Z=
e
CT
SE (T )
,
p
SE = ( 4↵
T
cosmological constant
p
1 Ntts + ↵(4
↵) Nsst )
where α:= 𝑙t2/𝑙s2 (1/4 < α < 4), so that α = 1 describes equilateral triangles.
the results presented here are from Monte Carlo simula$ons at fixed volume N ≡ N<s + Nsst ≤ 400.000 and α = 1 even for large volumes, many configura$ons contain closed $melike curves (winding around the spa$al direc$on of the torus), e.g. at N = 100.000, 30% have CTCs
we did not eliminate configura$ons with CTCs (which cannot be done with local checks) (R.L. and B. Ruijl, to appear)
The spectral dimension of gCDT in d=2
for the return probability P(σ) of a random walker a{er σ steps, recall that on flat Rd we have P(σ) σ -‐d/2 and define
dS :=
d ln(P ( ))
2
d ln( )
(∗)
2.00
measurements taken at N = 100.000
from σ ≥ 1000, obtain dS = 1.9937 ± 0.0023
same analysis for CDT gives dS = 2.0163 ± 0.0025
1.90
1.85
1.80
ds
blue and green curves are two different discre$za$ons of the RHS of (∗) 1.95
1.75
1.70
1.65
1.60
1.55
0
500
1000
1500
2000
The Hausdorff dimension of gCDT in d=2
extract dH from measuring shell volumes
n(r, i0) is the number of triangles at distance r from i0 1 X
n(r, i0 )
n(r) =
N i
0
shell volumes should scale according to N
1/d
H 1 n(r)
as func$on of a rescaled distance
N 1/dH r
integers label shells at distance r Wednesday, July 16, 14
The Hausdorff dimension of gCDT in d=2
0.8
n = 100k
n = 200k
n = 300k
n = 400k
0.7
1+1/2.7
0.5
0.4
n
n(r)
0.6
0.3
0.2
0.1
0.0
0.0
0.5
1.0
1.5
2.0
n
1/2.7
2.5
3.0
3.5
4.0
r
finite-‐size scaling: best fit from space$me volumes N ≡ n ≤ 400.000 yields dH = 2.7 ± 0.2; same analysis for CDT gives dH = 2.2 ± 0.2
quality of 2d data worse than in higher dimensions! quantum fluctua$ons and finite-‐size effects remain large at large volumes
2d quantum gravity (and gCDT) is very quantum
250
Extent
200
150
100
50
0
0
10
20
30
40
t
50
60
70
80
volume profile of a typical gCDT configura$on, with respect to an a posteriori iden$fied, averaged proper $me t
Summary and conclusions
The presence of a causal structure seems to play a crucial role in nonperturba$ve path integrals of gravity in higher dimensions, and leads to semiclassical proper$es in dynamically triangulated models. We have introduced a new model of “locally causal dynamical triangula$ons”, which unlike CDT does not possess a preferred $me folia$on, and have studied some of its proper$es in dimension 2.
We could show that on a simply connected 2d triangula$on in gCDT the local causal structure implies a global causal structure.
The Monte Carlo measurement of the spectral dimension dS ≈ 2 is compa$ble with both DT and CDT, but that of the Hausdorff dimension is at this stage compa$ble with neither; shape of shell profiles n(r) is unstable.
By contrast, in d = 2+1, there is good evidence from MC simula$ons that gCDT and CDT lie in the same universality class w.r.t. their “nice” phase. The results in dimension 1+1 are at this stage inconclusive -‐ the challenge of solving the model analy$cally remains! Wednesday, July 16, 14
Causal structure and -me in (generalized) CDT Quantum Gravity
triangulated model of quantum space
Schrödinger Ins.tute, 16 July 2014
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Causal structure and me in (generalized) CDT Quantum Gravity

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