Introduction Lattice Quantum Gravity Numerics Applications Conclusions
An introduction to Lattice Quantum Gravity
Marco Beria
Sissa, Trieste
April 15, 2011
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Outline
1 Introduction
2 Lattice Quantum Gravity
3 Numerics
4 Applications
5 Conclusions
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Outline
1 Introduction
2 Lattice Quantum Gravity
3 Numerics
4 Applications
5 Conclusions
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Historical Motivations
Difficulties in traditional quantization of gravity→ non renormalizability
Lattice Quantum Gravity→ Covariant path integral in quantized background
Montecarlo simulation on lattices
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Historical Motivations
Difficulties in traditional quantization of gravity→ non renormalizability
Lattice Quantum Gravity→ Covariant path integral in quantized background
Montecarlo simulation on lattices
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Applicative Motivations
Non perturbative Quantum gravity
AdS/CFT far form the classical limit
Random Surface Theory → Homology, Algebraic topology
Cosmology, Astrophysics → early universe.
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Outline
1 Introduction
2 Lattice Quantum Gravity
3 Numerics
4 Applications
5 Conclusions
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Path Integral Formulation
Pure Gravity
ST [g ] =
∫
Tddx√g
(1
16πGR − λ
)
Minkowski/Euclidean Path integral
Gauge fields can be added
Z =
∫D[g , T ]e−ST [g ]
Meaning of D[g , T ]:
Sum over Topologies T(ill defined in 2D ∼ genus!, higher D probably worst)
Sum over Metrics g (at fixed Topology)(D ∝ (d − 2)L)
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Path Integral Formulation
Pure Gravity
ST [g ] =
∫
Tddx√g
(1
16πGR − λ
)
Minkowski/Euclidean Path integral
Gauge fields can be added
Z =
∫D[g , T ]e−ST [g ]
Meaning of D[g , T ]:
Sum over Topologies T(ill defined in 2D ∼ genus!, higher D probably worst)
Sum over Metrics g (at fixed Topology)(D ∝ (d − 2)L)
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Two Sides of Gravity
Traditional quantum mediator approach (graviton)
Quantum geometry: the geometry fluctuates
The lattice Quantum Gravity assumes that the background itself issubjected to quantum fluctuation
gravity remains geometry (closer to Einstein formulation)
gauge field theory must be set on this quantized background
The topology is kept fixed (sphere).
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Two Sides of Gravity
Traditional quantum mediator approach (graviton)
Quantum geometry: the geometry fluctuates
The lattice Quantum Gravity assumes that the background itself issubjected to quantum fluctuation
gravity remains geometry (closer to Einstein formulation)
gauge field theory must be set on this quantized background
The topology is kept fixed (sphere).
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Generalities on Simplicial Complexes
A d-Simplex is a generalization of the concept of Triangle ind-dimensions
Point, segment, triangle, tetrahedron...
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Generalities on Simplicial Complexes
Each d-simplex contains (d+1)!(d−k)!(k+1)! k-simplicies.
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Generalities on Simplicial Complexes
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Lattice Geometry Simplicial Approximation
Idea: Regge 1960
Triangulation/Simplicial Approximation of Smooth Manifolds
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Lattice Geometry Simplicial Approximation
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Lattice Path Integral
Sites in a d-simplex, s, labeled by i ( d(d+1)2 )
l2ij = ηµν [xi − xj ]µ [xi − xj ]
ν
Vertexes 0, 1, 2, . . .
gij(s) =1
2
[l20j(s) + l20i (s)− l2ij (s)
]
Vd(s) =1
d!
√det gij(s)
Deficit angle and curvature at the hinge h
δ(h) = 2π −∑
s 6=s′⊃hcos−1
(ns · ns′|ns ||ns′ |
)R(h) ∝ δ(h)
ACh(h)
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Lattice Path Integral
δgij(s) =1
2
[δl20j + δl20i − δl2ij
]
Substitution∫Dg →
∫Dl2
∫Dl2 →
∫ ∞
ε
∏
s
V σd (s)
∏
ij
dl2ijΘ(l2ij )
Pure Gravity Action
S(l2) = λ∑
s
Vd(s)− k∑
h
Vd(h)R(h)
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Outline
1 Introduction
2 Lattice Quantum Gravity
3 Numerics
4 Applications
5 Conclusions
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Fluctuating Manifolds
start with an initial conditions (e.g. classical solution)
approximation via simplicial complexes
identify a set of elementary moves (Alexander moves) on simplicialgluing that covers the space of T -manifolds in an ergodic way(discrete version of diffeomorphism invariance)
give a probability for such moves → Random Surface Theory
study the statistical mechanics of such an ensemble.
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Fluctuating Manifolds
start with an initial conditions (e.g. classical solution)
approximation via simplicial complexes
identify a set of elementary moves (Alexander moves) on simplicialgluing that covers the space of T -manifolds in an ergodic way(discrete version of diffeomorphism invariance)
give a probability for such moves → Random Surface Theory
study the statistical mechanics of such an ensemble.
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Fluctuating Manifolds
start with an initial conditions (e.g. classical solution)
approximation via simplicial complexes
identify a set of elementary moves (Alexander moves) on simplicialgluing that covers the space of T -manifolds in an ergodic way(discrete version of diffeomorphism invariance)
give a probability for such moves → Random Surface Theory
study the statistical mechanics of such an ensemble.
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Random Surface Theory
∫Dl2e−S(l2) → Partition Function
e.g. 3D − 4D the discretized action turns to be the linearcombination
S3D = k3N3 − k0N0 , S4D = k4N4 − k2N2
Sum over simplicial complexes made of Ni i-simplicies and Nj
j-simplicies
Z (ki , kj) =∑
Ni ,Nj
W (Ni ,Nj)e−S
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Random Surface Theory
∫Dl2e−S(l2) → Partition Function
e.g. 3D − 4D the discretized action turns to be the linearcombination
S3D = k3N3 − k0N0 , S4D = k4N4 − k2N2
Sum over simplicial complexes made of Ni i-simplicies and Nj
j-simplicies
Z (ki , kj) =∑
Ni ,Nj
W (Ni ,Nj)e−S
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Causal Moves
Alexander Moves
Select a subcomplex change with another complex that glues withthe rest of the manifold.
define a causal time direction
restrict moves to causal one
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Causal Moves
Data Structure (id , type, t, v1, v2, . . . , id1, id2, . . .)
2↔ 6, 4↔ 4, 2↔ 3.
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Outline
1 Introduction
2 Lattice Quantum Gravity
3 Numerics
4 Applications
5 Conclusions
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Large distance behavior
S = −2m2∑
t
Rt(At(l2), δt(l
2)) + λ∑
s
Vs(l2) ,σ = 0
GV (d) = 〈∑
s⊃v0Vs
∑
s′⊃vdVs′〉c
GR(d) = 〈∑
s⊃v0Rs
∑
s′⊃vdRs′〉c
Expected large distance functional form
GR,V (d) ∼ e−md
da∼2
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Large distance behavior3
(a) Conventional Regge gravity: ! . . .m2P = !0.0775, " . . .m2
P = !0.0785, " . . . m2P = !0.0795
-0.001
0
0.001
0.002
0.003
0.004
0.005
0.006
1 2 3 4
GV
d
!
! ! !
"
" " "
"
" " "-0.2
-0.15
-0.1
-0.05
0
0.05
1 2 3 4
GR
d
!
! ! !
"
" " "
"
" " "
-0.01
-0.005
0
0.005
0.01
1 2 3 4
GR
d
!
!!
"
""
"
""
(b) Group theoretical approach: ! . . .m2P = !0.055, " . . .m2
P = !0.0555, " . . . m2P = !0.056
-0.001
0
0.001
0.002
0.003
0.004
0.005
0.006
1 2 3 4
GV
d
!
! ! !
"
" " "
"
" " "-2
-1.5
-1
-0.5
0
0.5
1 2 3 4
GR
d
!
! ! !
"
" " "
"
" " "
-0.2-0.15-0.1
-0.050
0.050.1
0.150.2
1 2 3 4
GR
d
!
!
!
"
"
"
""
"
(c) Gauge fields coupled to gravity: ! . . . (m2P = !0.0025, ! = 1.6), " . . . (m2
P = !0.005, ! = 1.0)
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
1 2 3 4
GV
d
!
!!
!
"
" " "
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
1 2 3 4
GR
d
!
! ! !
"
" " "
-0.1
-0.05
0
0.05
0.1
1 2 3 4
GR
d
!
!
!
" "
"
Figure 1. Volume (left plots) and curvature (middle with magnification in the right plots) correlationfunctions for (a) conventional Regge gravity, (b) the group theoretical approach, and (c) the system ofnon-Abelian gauge fields coupled to Regge gravity. Error bars not explicitely drawn are in the size of thesymbols. The curves correspond to fits with the function (8).
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Phase transition
S = kiNi − kjNj , ki = Kcrit(kj)
Figure 9. A pictorial description of the smooth (left) and rough (right) phases of four-dimensional lattice
quantum gravity.
There are a number of ways by which the critical exponents can be determined accurately from
numerical simulations, but it is beyond the scope of this review to go into details. For example,
one way to extract the critical exponent ! is to fit the average curvature to the form [see Eq. (195)]
R(k) !k!kc
"AR (kc " k)! . (197)
Using this general set of procedures one obtains eventually
kc = 0.0636(11) ! = 0.335(9) , (198)
which suggests ! = 1/3 for pure quantum gravity. Note that at the critical point the gravitational
coupling is not weak, Gc # 0.626 in units of the ultraviolet cuto!.
Often it can be advantageous to express results obtained in the cuto! theory in terms of physical
(i.e. cuto! independent) quantities. By the latter one means quantities for which the cuto! depen-
dence has been re-absorbed, or restored, in the relevant definition. As an example, an expression
equivalent to Eq. (195), relating the vacuum expectation value of the local scalar curvature to the
physical correlation length " , is
<!
dx$
g R(x) >
<!
dx$
g >!
G!Gc
const."l2P
#(d"2"1/")/2$
1
"2
%(d"1/")/2
, (199)
which is obtained by substituting Eq. (192) into Eq. (195). The correct dimensions have been
restored in this last equation by supplying appropriate powers of the Planck length lP = G1/(d"2)phys ,
which involves the ultraviolet cuto! ". Then for ! = 1/3 the result of Eq. (199) becomes particularly
simple<
!dx
$g R(x) >
<!
dx$
g >!
G!Gc
const.1
lP "(200)
Note that a naive estimate based on dimensional arguments would have suggested the incorrect
result ! 1/l2P . Instead the above expression actually vanishes at the critical point. This shows that
51
dH = 5, 4, 2
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Phase transition
dH = 5, 4, 2
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Baby Universes and Fractality
Tree-like fractal structure ⇒ Baby Universes
Euclidean analogue of Black-Holes. Islands connected to the restof the manifold by narrow necks.
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Baby Universes and Fractality
Self-Generating Universe
Fractal decomposition of the Universe in Baby Universes.
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Outline
1 Introduction
2 Lattice Quantum Gravity
3 Numerics
4 Applications
5 Conclusions
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
Conclusions
Lattice Quantum Gravity: approximate quantum with fluctuatingsimplicial lattice.
Random Surface Theory.
Large distance behavior, phase transition, Baby Universes andfractality.
M.Beria
An introduction to Lattice Quantum Gravity
Introduction Lattice Quantum Gravity Numerics Applications Conclusions
References
Hamber: Quantum Gravity on the Lattice
Zhang: Causal Dynamical Triangulation in 3D
Krzywicki: Perspectives in Lattice Gravity
Ambjorn, Jain, Jurkiewicz, Kristjansen: Observing 4d baby universesin quantum gravity
Hagura, Tsuda, Yukawa: Fractal Structures of the 3d simplicialgravity
Beril, Hauke, Homolka, Markum, Riedler: Correlation function inlattice formulation of quantum gravity
Rosen: Self-Generating Universe and Many Worlds
M.Beria
An introduction to Lattice Quantum Gravity