An introduction to Lattice Quantum Gravity · The lattice Quantum Gravity assumes that the...

Introduction Lattice Quantum Gravity Numerics Applications Conclusions

An introduction to Lattice Quantum Gravity

Marco Beria

Sissa, Trieste

April 15, 2011

M.Beria



Outline

1 Introduction

2 Lattice Quantum Gravity

3 Numerics

4 Applications

5 Conclusions

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Historical Motivations

Difficulties in traditional quantization of gravity→ non renormalizability

Lattice Quantum Gravity→ Covariant path integral in quantized background

Montecarlo simulation on lattices

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Applicative Motivations

Non perturbative Quantum gravity

AdS/CFT far form the classical limit

Random Surface Theory → Homology, Algebraic topology

Cosmology, Astrophysics → early universe.

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Outline

1 Introduction


3 Numerics

4 Applications

5 Conclusions

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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)

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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).

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Generalities on Simplicial Complexes

A d-Simplex is a generalization of the concept of Triangle ind-dimensions

Point, segment, triangle, tetrahedron...

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Each d-simplex contains (d+1)!(d−k)!(k+1)! k-simplicies.

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Lattice Geometry Simplicial Approximation

Idea: Regge 1960

Triangulation/Simplicial Approximation of Smooth Manifolds

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Lattice Geometry Simplicial Approximation

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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)

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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)

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Outline

1 Introduction


3 Numerics

4 Applications

5 Conclusions

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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.

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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

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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

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Causal Moves

Data Structure (id , type, t, v1, v2, . . . , id1, id2, . . .)

2↔ 6, 4↔ 4, 2↔ 3.

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Outline

1 Introduction


3 Numerics

4 Applications

5 Conclusions

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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

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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).

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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

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Phase transition

dH = 5, 4, 2

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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.

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Baby Universes and Fractality

Self-Generating Universe

Fractal decomposition of the Universe in Baby Universes.

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Outline

1 Introduction


3 Numerics

4 Applications

5 Conclusions

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Conclusions

Lattice Quantum Gravity: approximate quantum with fluctuatingsimplicial lattice.

Random Surface Theory.

Large distance behavior, phase transition, Baby Universes andfractality.

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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

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