Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
An introduction to Lattice Quantum Gravity
Marco Beria Sissa, Trieste
April 15, 2011
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Outline
1
Introduction
2
Lattice Quantum Gravity
3
Numerics
4
Applications
5
Conclusions
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Outline
1
Introduction
2
Lattice Quantum Gravity
3
Numerics
4
Applications
5
Conclusions
An introduction to Lattice Quantum Gravity M.Beria
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
An introduction to Lattice Quantum Gravity M.Beria
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
An introduction to Lattice Quantum Gravity M.Beria
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.
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Outline
1
Introduction
2
Lattice Quantum Gravity
3
Numerics
4
Applications
5
Conclusions
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Path Integral Formulation Pure Gravity
ST [g ] =
Z
d
√
d x g
T
1 R −λ 16πG
Minkowski/Euclidean Path integral Gauge fields can be added
Z= Meaning of D[g , T ]:
Z
D[g , T ]e −ST [g ]
Sum over Topologies T (ill defined in 2D ∼ genus!, higher D probably worst) Sum over Metrics g (at fixed Topology) (D ∝ (d − 2)L) An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Path Integral Formulation Pure Gravity
ST [g ] =
Z
d
√
d x g
T
1 R −λ 16πG
Minkowski/Euclidean Path integral Gauge fields can be added
Z= Meaning of D[g , T ]:
Z
D[g , T ]e −ST [g ]
Sum over Topologies T (ill defined in 2D ∼ genus!, higher D probably worst) Sum over Metrics g (at fixed Topology) (D ∝ (d − 2)L) An introduction to Lattice Quantum Gravity M.Beria
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 is subjected 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).
An introduction to Lattice Quantum Gravity M.Beria
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 is subjected 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).
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Generalities on Simplicial Complexes A d-Simplex is a generalization of the concept of Triangle in d-dimensions
Point, segment, triangle, tetrahedron...
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Generalities on Simplicial Complexes
Each d-simplex contains
(d+1)! (d−k)!(k+1)!
k-simplicies. An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Generalities on Simplicial Complexes
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Lattice Geometry Simplicial Approximation Idea: Regge 1960 Triangulation/Simplicial Approximation of Smooth Manifolds
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Lattice Geometry Simplicial Approximation
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Lattice Path Integral Sites in a d-simplex, s, labeled by i (
d(d+1) ) 2
lij2 = ηµν [xi − xj ]µ [xi − xj ]ν Vertexes 0, 1, 2, . . . 12 l0j (s) + l0i2 (s) − lij2 (s) 2 q 1 Vd (s) = det gij (s) d! Deficit angle and curvature at the hinge h X δ(h) ns · ns 0 −1 R(h) ∝ δ(h) = 2π − cos |ns ||ns 0 | ACh (h) 0 gij (s) =
s6=s ⊃h
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Lattice Path Integral 1 2 δgij (s) = δl0j + δl0i2 − δlij2 2 R R 2 Substitution Dg → Dl Z ∞Y Z Y 2 Vdσ (s) dlij2 Θ(lij2 ) Dl →
s
ij
Pure Gravity Action S(l 2 ) = λ
X s
Vd (s) − k
X
Vd (h)R(h)
h
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Outline
1
Introduction
2
Lattice Quantum Gravity
3
Numerics
4
Applications
5
Conclusions
An introduction to Lattice Quantum Gravity M.Beria
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 simplicial gluing 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.
An introduction to Lattice Quantum Gravity M.Beria
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 simplicial gluing 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.
An introduction to Lattice Quantum Gravity M.Beria
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 simplicial gluing 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.
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Random Surface Theory R
2
Dl 2 e −S(l ) → Partition Function e.g. 3D − 4D the discretized action turns to be the linear combination S3D = k3 N3 − k0 N0
,
S4D = k4 N4 − k2 N2
Sum over simplicial complexes made of Ni i-simplicies and Nj j-simplicies Z (ki , kj ) =
X
W (Ni , Nj )e −S
Ni ,Nj
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Random Surface Theory R
2
Dl 2 e −S(l ) → Partition Function e.g. 3D − 4D the discretized action turns to be the linear combination S3D = k3 N3 − k0 N0
,
S4D = k4 N4 − k2 N2
Sum over simplicial complexes made of Ni i-simplicies and Nj j-simplicies Z (ki , kj ) =
X
W (Ni , Nj )e −S
Ni ,Nj
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Causal Moves Alexander Moves Select a subcomplex change with another complex that glues with the rest of the manifold. define a causal time direction restrict moves to causal one
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Causal Moves Data Structure (id, type, t, v1 , v2 , . . . , id1 , id2 , . . .)
2 ↔ 6, 4 ↔ 4, 2 ↔ 3. An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Outline
1
Introduction
2
Lattice Quantum Gravity
3
Numerics
4
Applications
5
Conclusions
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Large distance behavior S = −2m2
X
Rt (At (l 2 ), δt (l 2 )) + λ
t
GV (d) = h GR (d) = h
X
Vs
s 0 ⊃vd
s⊃v0
X
X
Rs
s⊃v0
X
s 0 ⊃vd
X
Vs (l 2 )
,σ = 0
s
Vs 0 ic Rs 0 ic
Expected large distance functional form GR,V (d) ∼
e −md d a∼2
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Large distance behavior (a) Conventional Regge gravity:
GV
0.006 0.005 0.004 0.003 0.002 0.001 0 -0.001
! . . . m2P = −0.0775, 0.05
" ! "
" " !
! " "
! " "
" " !
2
3
4
" ! "
! " "
0.005 GR
-0.1 " " !
0
-0.005
-0.2
-0.01 1
2
d (b) Group theoretical approach:
" ! "
-0.05
-0.15
".
0.01
0 GR
1
0.006 0.005 0.004 0.003
" . . . m2P = −0.0785,
3
4
d ! . . . m2P = −0.055,
" . . . m2P = −0.0555,
0.5 0
" ! "
" " !
! " "
An introduction to Lattice Quantum Gravity
-0.5
"
0.2 0.15 0.1 0.05 M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Phase transition S = ki Ni − kj Nj , ki = Kcrit (kj )
dH =description 5, 4, 2 A pictorial of the smooth (left) and rough (right) phases of four-dimension
gravity. An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Phase transition
dH = 5, 4, 2 An introduction to Lattice Quantum Gravity M.Beria
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 rest of the manifold by narrow necks.
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Baby Universes and Fractality Self-Generating Universe Fractal decomposition of the Universe in Baby Universes.
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Outline
1
Introduction
2
Lattice Quantum Gravity
3
Numerics
4
Applications
5
Conclusions
An introduction to Lattice Quantum Gravity M.Beria
Introduction
Lattice Quantum Gravity
Numerics
Applications
Conclusions
Conclusions Lattice Quantum Gravity: approximate quantum with fluctuating simplicial lattice. Random Surface Theory. Large distance behavior, phase transition, Baby Universes and fractality.
An introduction to Lattice Quantum Gravity M.Beria
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 universes in quantum gravity Hagura, Tsuda, Yukawa: Fractal Structures of the 3d simplicial gravity Beril, Hauke, Homolka, Markum, Riedler: Correlation function in lattice formulation of quantum gravity Rosen: Self-Generating Universe and Many Worlds
An introduction to Lattice Quantum Gravity M.Beria
