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An Introduction to Loop GravitySpinfoam Models and Toy Models

Probing the Geometry through CorrelationsFluctuating Geometry Models

Quantum Gravity and Spinfoams:

From Evolving Graphs to Smooth Geometries?

Quantum Fields, Geometry and Information

Etera LivineLaboratoire de Physique - ENS Lyon (France)

April 2013 @

Quantum Gravity and Spinfoams: From Evolving Graphs to Smooth



Why Quantum Gravity?

Goal: Describe Gravity at all scales of energy and length






There exist otherregimes based ondifferent limits forG , c and ~. . .






Describe the Structure of Space-Time atsmallest distances

Unify General Relativity (large scale) &Quantum Mechanics (small scale)

Test their domains of validity at High Energy

Understand how they are Low Energy Approximations to aNew Theory

Solve Singularities of GR? Understand Emergence of QM?

Discover the Physical Principles underlying this new theory ofQuantum Gravity




Quantifying the Space-Time Geometry?

Going beyond General Relativity

Geometry is dynamical and thus acquires quantum fluctuations.

Applying Quantum Mechanics to General Relativity:

At first: Precise measurements would create black holes. . .◮ Running of G , c , ~? Loss of notion of “points”?◮ Emergence of geometry from more fundamental d.o.f.,Phase transition

Metric becomes Quantum Operator ... Discrete Spectra ofGeometrical Observables?

Standard QFT Techniques lead to Divergent Fluctuations . . .

How to Localize in a Quantum Dynamical Space-Time? background independence, problem of time, . . .

New Concept of Observer? New Relativity Principle?




The Path to Quantum Gravity. . . is not straight: QG Approaches

Spinfoams & Loop quantum gravity :Non-perturbative quantization of Gen-eral Relativity with quantum states ofgeometry and path integral

String theory : Particlesare tiny vibrating strings.Unifies all interactions andmuch more

Asymptotic Safety Scenarios and Non-trivial Fixed Point for(Exact) Renormalization Group

Non-Commutative Geometry (Spectral Triple, DSR,..)

Dynamical Triangulations, Emergent Gravity, CausalSets/Histories, Topos . . .




Quantum Gravity: From Evolving Graphs to Smooth Geometries?

1 An Introduction to Loop Gravity and Spinfoams

2 Spinfoam Models and Loop Dynamics

3 Probing the Geometry through Correlations?

4 Fluctuating Geometry Models and Research in Progress




Loop Quantum GravityQuantization and Spin Network StatesThe Geometry of Loop GravityTowards Coarse-Graining Discrete Geometries










Loop Gravity: Reformulating General Relativity

There exists various classical variables to describe geometry. . .⇒ Leads to different degrees of freedom at quantum level!Instead of using Metric, we can use a couple of new variables:

tetrad: defines local coordinate system.local reference frame

connection: defines parallel transport.how to relate frames of diff observers (Lorentz transf forspace-time, 3d rotations for space)

Possible to re-formulate all of GeneralRelativity in these variables⇒ a more relational formulation





One-slide Loop Quantum Gravity

LQG = a Non-Perturbative Quantization of General Relativity

Writes General Relativity as a Diffeomorphism-InvariantGauge Field Theory◮ Basic variable is Connection relating reference framesbetween different observers

Defines Quantum Dynamics of ψ(3-Metric)

Derives a Discrete Spectra of Area & Volumes

Describes Fundamental Space-Time Structures at PlanckScale

A Background-Independent Framework◮ No background geometry/topology is assumed, space-timeis constructed from scratch





Two Dual Perspectives: Canonical vs Covariant

Two equivalent point of views and formalisms:

Canonical Framework

Quantum State of 3-GeometryEvolving in Time

Hamiltonian (constraint) de-fined by GR quantum dynam-ics?

Covariant Framework

Transition Amplitude definedby Path Integral of GR Actionas a Sum over 4-Geometries

Physical boundary states?





What do we have?

Main developments in LQG:

Well-defined Hilbert space as wave-functions over generalizedconnections, discrete spectrum for areas and volumes

Discretization and regularization of constraints, but. . .

EPRL-FK spinfoam models defining transition amplitudesbetween spin network states

Relation with Regge calculus and discretized GR

Loop quantum cosmology





The BIG Issues to Solve in Loop Gravity

Open problems (or items in progress):

Symmetries, discrete diffeomorphisms Algebra of deformations of spin networks, Topologicalinvariance and recursion relations for spinfoam amplitudes,. . .

Spin network Dynamics from Spinfoams EPRL model, extracting effective Hamiltonian for (loop)cosmology,. . .

Large scale behavior, coarse-graining, QG corrections to GR Radiative corrections for spinfoams, Renormalization ofgroup field theory, large N limit of (colored) tensor models,. . .

Deriving solvable reduced models Developing loop cosmology taking into accountinhomogeneities, . . .





Triad-Connections Variables: the Loop Gravity Phase Space

Loop Gravity based on formulation of GR as SU(2) Gauge FieldTheory with canonical variables the connection A and triad E :

{Aai (x), E

jb(y)} ∝ δji δ

abδ

(3)(x , y) {A,A} = {E , E} = 0

Physical meaning:

Triad E defines 3d local coordinate systems - ref framehab = E i

aEia

Connection A = Γ(E) + γK defines the change of ref framebetween these coordinates systems

◮ Constraints: dAE = 0 and space-time diffeos Hµ = 0





Wave-functions and Quantum States of Geometry

LQG quantizes these variables, A = A and E = −i ∂∂A

◮ Introduce wave-functions ψΓ(ge) of the holonomies ge = Pe∫eA

along the edges e of a graph Γ◮ Hilbert space as a sum over graphs Γ (refinement processthrough projective limit)

Require SU(2) Gauge-Invariance generated by dAE = 0

ϕ({ge}e∈Γ) = ϕ({hs(e)geh−1t(e)}e∈Γ), ∀hv ∈ SU(2)V

LQG does NOT describe gravity variations around flat metric BUTaround the no-metric state: basic excitations of geometry areelementary volume blocks “Space Points” and Connection betweenthem⇒ Discrete Quantum Geometry as a Network of Relations





Decomposing onto SU(2) Representations

Fourier decomposition for L2(SU(2)) given by Peter-Weyldecomposition:f (g) =

∑

j ,a,b fjabD

jab(g), δ(g) =

∑

j(2j + 1)χj (g)

Representations of SU(2) labeled by spin j ∈ N/2◮ SU(2) acts on each space V j = {|j ,m〉, m = −j ..+ j}

A state in Hilbert space V j represents a “quantum vector” oflength j . . . the quantum version of 3-vector ~X

Intertwiner is a state in the tensor product V j1 ⊗ ..⊗ V jn which isinvariant under SU(2) (usually given by Clebsh-Gordan coeffs)◮ SU(2)-inv is Quantization of Closure Constraint

Intertwiner is “quantum polyhedra”





Spin Network states diagonalizing Geometrical Operators

A basis of states given by spin networks diagonalizing geom ops

spins or SU(2) irreps on edgesje ∈ N/2 area

intertwiners or singlets orinvariant tensors on verticesiv ∈ je1 ⊗ ..⊗ jen volume

f Γ{je ,iv}(ge) =∑

ms,te

∏

e

〈je ,mse |ge |je ,m

te〉

∏

v

〈⊗t(e)=v je ,mte |iv |⊗s(e)=v je ,m

se〉





Spin Networks

Simplifying Spin networks: edge on/off between two vertices◮ Graphity models: fluctuating random graph encoded inadjacency matrix◮ then dress graph with spins: the higher the spin, the mostpowerful the relationship along the link

Important feature: there is no background graph◮ No a priori closest neighbour. . . Need to reconstruct the notionof locality from graph combinatorics◮ Non-trivial probability of creating non-local links. . . derivation of quantum mechanics from fluctuating graphs(stochastic interpretation) [Markopoulou&Smolin 03]





Revisiting the Loop Gravity Phase Space

LQG based on choice of discretized observables built on finiteoriented graphs Γ:

Holonomy along edge e ∈ Γ: ge ∈ SU(2)

Flux through surface transversal to edge e:

Xe = Xs(e)e ∈ su(2) ∼ R

3 at source vertex v = s(e) and

Xt(e)e = −ge ⊲ X

s(e)e at target vertex

{ge , ge′} = 0 {X ae ,X

be′} = iδee′ǫ

abcX ce {~Xe , ge} = ~σge

3d Rotations ge along edgesVectors ~X v

e around vertices

Xt(e)e = ge ⊲ X

s(e)e on edges

∑

e∋v Xve = 0 at vertices





Triad-Connections Variables: Geometrical Meaning

Vectors ~X ve describe discrete geometries :

Closure constraint∑

e∋v Xve = 0 defines closed polyhedra dual

to each vertex

Each edge dual to a face with~Xe as the normal vector and itsnorm |~Xe | giving the area of theface

Polyhedra glued along the edges but only area matchingconditions (no shape matching!)

◮ ∃ge ∈ SO(3), ~X te = −ge ⊲ ~X s

e ⇔ |~X te | = |~X s

e |

We get Twisted Geometries◮ shape mismatch encodes extrinsic curvature





Twisted Geometries

Twisted geometries are not triangulations and do not give theusual Regge geometries

Loop gravity phase is more flexible and can encode moreinformation about the geometry! Can impose gluing constraint to restrict to Regge geometries





Using Spinor Variables

Can describe twisted geometries asSpinor Networks Replace vectors ~X v

e ∈ R3 by

spinors zve ∈ C2

Can reconstruct both g ’s and X ’s from spinors:◮ ~X = 〈z |~σ|z〉◮ Unique SU(2) element ge mapping z se to z te

ge =|z te ]〈z

se | − |z te 〉[z

se |

√

〈z se |zse 〉〈z

te |z

te 〉

Darboux coordinates with canonical Poisson bracket{za, zb} = −iδab





Using Spinor Networks

A fruitful formalism:

Gluing polyhedra through phase matching of spinors








Easy to build SU(2)-inv observables and explore dynamics(reduced models and algebra of constraints)









Straightforward to quantize HOs and build coherent spinnetwork states peaked on classical spinor networks gives coherent boundary states for spinfoams










U(N) deformations for polyhedra and intertwiners for fixedtotal area entropy counting at kinematical level, effective boundaryHamiltonian at dynamical level










U(N) deformations for polyhedra and intertwiners for fixedtotal area entropy counting at kinematical level, effective boundaryHamiltonian at dynamical level

Generalizable to twistors for SL(2,C) networks, in order towrite spinfoam amplitudes and get exact evaluations





Coarse-graining the spin network geometry

Spin(or) networks describe the geometry at the Planck scale. . .. . .We want to understand how to coarse-grain them and whattype of geometries we get at larger scales

smooth geometries? curvature? metric?





Why coarse-graining?

Need to study the coarse-graining of geometry states, in order tounderstand the transition for graph-like excitations to smooth(semi-)classical geometries on large scales

Understand the relevant degrees of freedom whencoarse-graining

Understand how to reconstruct the geometry, distances andcurvature

Understand how the dynamics gets coarse-grained. . . do weget phase transitions?

Applications to cosmological settings: how to imposehomogeneity? isotropy? How to blow up or contract quantumgeometries (action of dilatations)?





Gauge-fixing Spin(or) Networks

Here, will only discuss coarse-graining at kinematical level.Consider a spin(or) network and a bounded region:







1 Identify gauge-invariant data describing the state of the region gauge-fixing procedure







1 Identify gauge-invariant data describing the state of the region gauge-fixing procedure

2 Decide what d.o.f. interact with exterior and what data tokeep coarse-graining step






We can use the gauge-invariance at every vertex of the graph tocollapse the entire bulk to a single vertex (without loss ofinformation for an external observer)

Choose a maximal tree T going through any bulk vertex andset all the holonomies on tree edges to ge∈T = I by usinggauge-inv ψ(ge ) = ψ(hs(e)geh

−1t(e)) with

hv =−→∏

e∈v0→vge

We keep the information about all the internal loops(representing excitations of the connection/curvature) buterase the detail of the graph not accessible to the exterior

⇒ Dynamics with exterior through boundary interaction andaction on internal loops





A Fock Space of Loopy Spin Networks

We can shift the perspective:Fix a background graph, representing the network of space pointspostulated by observer, then account for possible internal structureof each point through local loopy excitations of curvature

Now can model LQG dynamics on this background provided withFock space structure. . . Action of holonomy operators can be either on large loops or onlittle internal loops





Coarse-Grained Spin Networks and Tag. . . and Mass?

We can go further and coarse-grain on loopy structure and onlyretain tag at each vertex accounting the non-trivial recoupling ofboundary spins tag ∼ global “mass” of little loops

Physical meaning of tag?◮ Non-closure of polyhedron. . .







Physical meaning of tag?◮ Non-closure of polyhedron. . . indicates curvature!







Physical meaning of tag?◮ Non-closure of polyhedron. . . indicates curvature!

Independent of choice of tree

Tag K bounded by total boundary area

Equivalent in the continuum:∫

V [A, E ]

⇒ Mass/energy?Quantum Gravity and Spinfoams: From Evolving Graphs to Smooth




Modeling the Interaction with the Environment/Exterior

Master equation? Decoherence? Research in progress. . .

Model interaction region ↔ exterior Use operators that increase/decrease spins on boundaryedges to define effective Hamiltonian for the region’s dynamics

Fixed or changing boundary area? What’s an “isolated” region? At equilibrium?

Dynamics that change the number of boundary edges?

Any interesting generic phenomenon due to interaction withexterior treated as environment?




The Spinfoam FrameworkKitaev’s model and Z2 BF theoryPonzano-Regge model and 3d Quantum GravityWhat’s Next for Spinfoams?










And spinfoams. . .

Let us discuss the dynamics of spin networks!





What is a Spinfoam Model?

We have these spin networks evolve (in time) and createspace-time. . .





What is a Spinfoam Model?

Spinfoam is 2-complex describing evolution of spin(or) networks:

graph geometry spin network → spinfoam

vertex polyhedron intertwiner → edgelink normal vector spin → face

And all the dynamics is implemented at the spinfoam vertices,when graph and spin network state changes

One intertwiner/polyhedron splitting intofour intertwiners/polyhedra at a spinfoamvertex: the graph changes

Spinfoam model = Transition Amplitudes for each possibleHistory of Spin Network





Dynamics: Evolving Networks to Create Space-Time

Spinfoam Vertex = Space-Time Event

Now Spin Network represent discrete quantum space geometry . . .And Spinfoam represent quantum space-time geometry .We construct spinfoam 2-complexes from space-time triangulationsby taking the dual 2-skeleton. Let’s consider 4-valent graphs onboundary:

graph ↔ 3d triangulation → spinfoam ↔ 4d triangulation

link triangle → face triangle (2d)node tetrahedron → edge tetrahedron (3d)

vertex 4-simplex

Allows for a direct Comparison Between Spinfoam Amplitudes andDiscretized General Relativity (Regge calculus)





Spinfoams merges perspectives on QG

Many ingredients to the spinfoam framework:





The Recipee for a Spinfoam Model

The recipe for spinfoam models for 4d quantum gravity:

Start with formulation of GR as a BF gauge field theory withconstraints◮ Gravity as an “almost-topological” field theory





From BF Theory to Gravity (Technical Slide)

Write general relativity as a BF gauge theory with potential

S [B ,A] =

∫

MǫIJKLB

IJ ∧ FKL[A] + λφ[B ]

with simplicity constraints φ[B ] ∼ B∧B on the 2-form B valued insl(2,C) ensuring that B comes a tetrad field e:

φ[B ] = 0 ⇒ B = ⋆(e∧e)

We recover the Palatini action for GR:

S [e, ω] =

∫

MǫIJKLe

I∧eJ∧FKL[ω]

Equivalent to Standard Action for GR in term of Metric:

S [g ] =

∫

Md4x

√

− det g R [g ]








Spinfoam quantize topological BF theory (our “free theory”)◮ An exact discretization of BF path integral!









Add constraints and re-introduce local degrees of freedom◮ BF theory gives flat connection Add curvature!










Compute discrete path integral and Derive spinfoamamplitudes◮ Obtain a quantized version of discrete GR











Define semi-classical boundary state peaked on suitable 3dgeometries and analyze behavior of 4d geometry in the bulk











Define semi-classical boundary state peaked on suitable 3dgeometries and analyze behavior of 4d geometry in the bulk

Extract Physically-relevant Correlation Functions??





The Spinfoam Ansatz

The data: Consider the gauge group G, then dress up thetriangulation:

Triangle → representation (spin) jt of G

Tetrahedron → intertwiner IT between its 4 triangles

4-Simplex → evaluation of the boundary spin network= contraction of its 5 tetrahedra

A local ansatz:

A[∆] =∑

{jt}

∏

t

At(jt)∏

T

AT (jt ,IT )

︸︷︷︸

Statistical weights

∏

σ

Aσ(jt ,IT )

︸︷︷︸

Dynamics

A model: Choice of irreps and intertwiners, Modify evaluation





Actual Spinfoam Models

Ponzano-Regge and Turaev-Viro models for 3d qg◮ gauge groups SU(2) and SU(1, 1)◮ q-deformation for cosmological constant◮ vertex amplitude for tetrahedra: {6j}-symbol

Crane-Yetter model for 4d BF theory◮ {15j}-symbol

Barrett-Crane model for 4d qg◮ based on geometric quantization of 4-simplex◮ {10j}-symbol

EPRL-FK models for 4d qg → see Ricardo’s talk◮ Implements the simplicity constraints through coherentstate techniques

Also spinfoam models for lattice QCD or Kitaev’s model for Z2 BFtheory. . .





First Toy-Model: Kitaev’s Topological Model and Toric Code Hamiltonian

Let’s consider a spin system on an arbitrary graph Γ embedded in2d surface S:Spin | ± 1〉 on each edge with Hamiltonian

H = −J∑

v

Av − K∑

f

Bf

Av =∏

e∈∂v

σxe

Bf =∏

e∈f

σze

[Av ,Bf ] = [Av ,Aw ] = [Bf ,Bg ] = 0





Kitaev’s model as Z2 BF Theory

Ground states satisfy Av |ψ〉 = |ψ〉 = Bf |ψ〉⇒ Physical states for BF theory in 2+1d with gauge groupZ2 = {0, 1}

Av |ψ〉 = |ψ〉 imposing Z2 gauge invariance at vertices(invariant under flips of all states around v)Bf |ψ〉 = |ψ〉 imposing Z2 flatness around faces (product ofspins is 1)

From the quantum gravity point of view:

Ground states do not depend on details of graph Γ, only thetopological properties of surface SCan be seen as action of holonomy operators around all(elementary) loops (projecting on flat states) on arbitrarystate string condensate. . .Can build 3d spinfoam model projecting on physical states ofBF theory Quantum Gravity and Spinfoams: From Evolving Graphs to Smooth




Interplay: Kitaev Model ↔ Quantum Gravity

Some insight for Quantum Gravity?

Data/Matter/Qubits as Topological Defects (excited states)(Av |ψ〉 = −|ψ〉 or Bf |ψ〉 = −|ψ〉) Particles as topology/geometry Defects in QG (matterencoded in the geometry)

Toric code as effective Hamiltonian in phase of other spinmodels (e.g. honeycomb) Emergent gravity from condensed matter systems





Interplay: Kitaev Model ↔ Quantum Gravity

Analogy can be pushed/studied further:

Kitaev’s toric code generalizable to discrete groups and tonon-abelian phases◮ Also spinfoam and spinnet models for discrete groups

Interplay between Spinfoam QG (TQFT) and QuantumComputing◮ Propagation and braiding of topological defects used forQuantum Computing

Relaxing flatness constraint in (more complex) (spin) systems?

Area-Entropy law! and holographic principle

Spin systems as Toy-Models for QG Graphity Models

◮ graph with simplified data:

∣∣∣∣

0 = no-link

1 = link





2nd toy-model: the Ponzano-Regge model for 3d Quantum Gravity

3d Quantum Gravity is much easier than 4d Quantum Gravitybecause it’s exactly a topological BF theory. . .

⇒ Can apply the full spinfoam program!





The Ponzano-Regge Model: 3d Quantum Gravity

Start with a triangulation of the 3d Space-time :Tetrahedra glued together along triangles

Put SU(2) representations or spins je on edges

Associate triangles with 3-valent intertwiners

Glue 4 intertwiners together in a 6j-symbol associated to eachtetrahedron

Ponzano-Regge model for 3d Quantum Gravity:

ZPR =∑

{je}

∏

e

(2je + 1)∏

T

{6j}





Ponzano-Regge Model: Canonical Interpretation

Ponzano-Regge Path Integral defines Transition amplitudesbetween 2d Triangulations, dual to Spin Networks

A clear picture of 3d quantumspace-time

Transition amplitude projects onphysical spin network states for flatconnections, imposing δ(g) around allfaces





Ponzano-Regge Model: Evolution Moves

Evolution Move = Put a Tetrahedron on top of 2d triangulation⇒ Two types of moves:

1 ↔ 3 Moves

Add/Remove Point

2 ↔ 2 Moves

Re-Organize Triangles

That’s enough to generate all 3d triangulations which don’t changethe space topology.





Pachner Moves and Topological Invariance

PR defines a topological state-sum model invariant under the 4-1and 3-2 Pachner moves:

∑

j{6j}{6j}{6j} ={6j}{6j}

∑

j{6j}{6j}{6j}{6j} ∝ {6j}

The discrete path integral Z∆ is a topological invariant!

Can deform, refine, coarse-grain the 3d triangulation withoutaffecting the transition amplitude (as long as the globaltopology is not changed)

Topological Invariance (equivalent to Diffeo Inv for 3d gravity)protects the Renormalization Flow





Ponzano-Regge Model: features

Proper topologically-invariant discrete path integral for 3dquantum gravity

Semi-classical interpretation given by large spin asymptotics◮

{6j} ∼je≫1

e+iSR [je ] + e−iSR [je ],

with Regge action for tetrahedra SR [je ] =∑6

e=1(je +12 )θe [je ]

◮ Adding dihedral angles around an edge e, you get thedeficit angle which measures the curvature.

Include particles/matter as topological defects, creating ashift in the holonomy around the world-line depending on themass and momentum◮ similar to defects in Kitaev’s model





What’s going on with Spinfoams?

Many on-going projects in the spinfoam programme:







Recursion relations, symmetries and Hamiltonian constraints(for 4d QG models)








Matter coupling, Effective non-commutative QFT and QGphenomenology









Group field theory reformulation, renormalization andsearching for phase transitions









Group field theory reformulation, renormalization andsearching for phase transitions

Defining cosmological models and getting predictions forinflation, big bounce, . . .




The Spinfoam Graviton PropagatorEntanglement in BF TheoryBlack Hole Entropy










Localizing Events in a Fluctuating Geometry

How to reconstruct a smooth classical space(-time) from a spinnetwork state? How to localize in a background independent, diffeo invframework?

Use relational approach: probe the geometry throughcorrelations

Closer?

⇐⇒ More correlated, more entangledThus study the interplay between geometry and correlations &entanglement of gravitational degrees of freedom





Reconstructing the Geometry from Correlations

Possibility of reconstructing geometry effectively from correlations :2-pt function (graviton propagator) gives distance thus metric

〈δg(x)δg(y)〉 ∝1

d(x , y)2

“Naıve” expectation ‘closer ↔ correlated’ Provides local mapping to flat space(-time) Can use tensorial structure to define directions

Geometry reconstructed effectively in semi-classical regimeNatural renormalization of metric/geometry , will depend onenergy scale emergence of relative locality [Amelino-C,Freidel,Kowalski-G,Smolin]

Need to identify geometrical observables, generalize to fullquantum theory and define precisely the correlations

Correlations ↔ Mutual Information ↔ Motion = Change in correlations= Information flow





Correlations for Geometrical Observables: e.g. the Spinfoam Graviton

Now look at “precise” implementation in spinfoam models:

Choose bulk triangulation ∆

Choose semi-classical state ψ on boundary ∂∆ peaked on flat 3d-geometry andembedding, with typical length scale l0

Compute correlations between geometrical observables on the boundary

〈δAa δAb〉ψ ≡

∑j∆δAa(j∂∆)δAb(j∂∆)ψ∂∆(j∂∆)W∆[j∆ ]

∑ψ∂∆W∆

Look at small/large scale behavior in terms of l0

Computed from a single 4-simplex in the BC model:Power law at large scale: 〈δAa δAb〉 ∼

1l20

⇒ Newton’s law for classical gravity

Regularized at small scale⇒ Emergence of dynamical minimal lengthl0 ∼ lP then “distance” increases with resolution

Generalizable to EPRL model. . . but need to go to more complextriangulations & boundaries





Using the Entanglement

Let’s look at entanglement between regions of spin network states!







Easy answer: entanglement vanishes on a pure spin network (basis)state. . .







Easy answer: entanglement vanishes on a pure spin network (basis)state. . .

Actually complicated calculations for 2 regions and for generic spinnetwork states.For now, look at a single region and interior/exteriorentanglement. . .





Entanglement in Kitaev’s model and BF theory

Entanglement in Kitaev’s model:“area”-entropy law on a 2d gridSR = (nL − 1) ln 2depending on nb of edges crossing boundary

Generalizing Entanglement in Topological BF Theory:Consider completely flat state on a graph and split it in tworegions:

State defined by δ(g) around every plaquette.

Then S = nL δ(I) depending on nb of loops crossing theborder

δ(I) infinite for Lie groups but. . . Put cut-off on spins?Compute ∆S for topological defects? Regularize by quantumgroups?





Entanglement in Kitaev’s model and BF theory

Entanglement for gravity as almost-topological theory?

Generalizable to grid-weave states? Replace δ(g) on every plaquette by other holonomyoperator e.g. χ 1

2(g)

How and why should we get the exact 14 factor of black hole

entropy?

Interplay entanglement↔geometry: holographic entanglementand AdS/CFT correspondence. . .





Black Hole Entropy and Evaporation

Can model black holes in LQG:

Classically as isolated horizon withspecific boundary conditions

Quantized as SU(2)-intertwiners i.e.some fuzzy version of classical polyhedra

Counting of intertwiners leads back toarea-entropy law





Black Hole Entropy and Evaporation

But many points to clarify:

1 Model evaporation from LQG dynamics Have to depart from isolated horizon paradigm Necessary to derive temperature

2 Identify a black hole on a spin network state boundaryconditions at the quantum level (crucially depends ondynamics)

3 Relevance of internal degrees of freedom? Bulk structure ofquantum black holes? Related to question of coarse-grainingspin networks. . .




Loop (Quantum) Gravity on a Fixed GraphFluctuating Graphs for Space Geometry and Graphity ModelsGroup Field Theory and Tensor Models










Dynamics: Graph Changing vs. Fixed Graph

Two different contributions to the microscopic dynamics (althoughrelated by coarse-graining):

Dynamics on fixed graph lattice methods, spinfoams onfixed triangulation,. . .

Graph changing dynamics: CDT, GFT (spinfoams), causalsets, Graphity,. . .

So what type of dynamics dominates which QG regime?





Loop Quantum Gravity on a Fixed Graph: Mini-Superspace Models?

Let us truncate LQG and look at the dynamics of spin(or)networks on a fixed graphFixed graph ∼ Frequency cut-off for gravitational d.o.f. a smooth geometry can be reconstructed by discrete sampling ifit is simple enoughInterpretation: Does not describe discrete geometries butcontinuous 4d geometries which are defined by finite nb ofparameters (e.g. metric with symmetries) ⇒ mini-superspace reconstruct the full metric from data on the graph/triangulationTo go further: Can choose family of graphs with increasing nb ofedges or/and vertices and thus attempt to define limit with infinitenb of d.o.f. (e.g inhomogeneities in cosmology, QFT d.o.f., ...)





Cosmology on the 2-Vertex Graph

α β

e1

e2

e3

e4

eN−3

eN−2

eN−1

eN

2-vertex LQG for isotropic & homogeneousCosmology (FRW):

Discretize LQG Hamiltonian constraint EEF [A]◮ Cgrav =

∑

i ,j TrXiXjgig−1j

Couple to scalar field φα, φβ and de-parameterize

◮ Hα = p2α2 + Cgrav = 0

Impose homogeneity φα = φβ and isotropy◮ Cgrav is U(N)-invariant

Whole dynamics only depends on global boundary area betweenthe 2-vertices and we recover FRW cosmology . . .






α β

e1

e2

e3

e4

eN−3

eN−2

eN−1

eN

2-vertex LQG for isotropic & homogeneousCosmology (FRW):

Discretize LQG Hamiltonian constraint EEF [A]◮ Cgrav =

∑

i ,j TrXiXjgig−1j

Couple to scalar field φα, φβ and de-parameterize

◮ Hα = p2α2 + Cgrav = 0

Impose homogeneity φα = φβ and isotropy◮ Cgrav is U(N)-invariant

Whole dynamics only depends on global boundary area betweenthe 2-vertices and we recover FRW cosmology . . . with corrections:

H2 =8πG

3ρ

(

1−ρ

ρc

)






A very nice and simple model:

Reduction to isotropy done naturally through imposingU(N)-symmetry at the quantum level

Dynamics totally solvable evolution is a boost in SU(1, 1)

Simple derivation of big bounce from holonomy correction inLQG






A very nice and simple model:

Reduction to isotropy done naturally through imposingU(N)-symmetry at the quantum level

Dynamics totally solvable evolution is a boost in SU(1, 1)

Simple derivation of big bounce from holonomy correction inLQG

Nevertheless some “little” problems :

Density at bounce. . .

Curvature k? Cosmological constant Λ?

Inhomogeneities?. . .

Have to move to more complicated graphs !





Graphity Models

Now: no label on graphs, but only fluctuating graphsGraphity models for geometry+matter with Hamiltonian describingevolution in time:

1 Moves for graph changing2 Matter propagating on graph Bose-Hubbard dynamics

3 Moves geometry ↔ matter

The various regimes of the theory not yet understood. . .

But some very interesting promises:

Speed of light from Lieb-Robinson bounds(on correlations)

Ideas for black holes (small worlds)

Gravitational attractionCan entanglement be a measure of locality?





Graphity Models and Social Networks?

Interesting similarities with dynamics of social networks with bignodes attracting most of the links. . .

. . . Can be pushed further?





Generating Space-time Triangulation as Feynman Diagrams

We can define non-perturbative sum over triangulations

Group Field Theories (as generalized matrix models)

Interesting new results on large N limit with hints of phasetransition





Outlook

Some random thoughts:

Continuum regime with semi-classical (space-time) geometries?Which class of geometries in the continuum are we summing over?Role of entanglement in characterizing the geometry?Corrections to GR? Effective QG? Relative Locality and NCgeometry?


quantum gravity and spinfoams: from evolving graphs to ... › uploads › 1 › 1 › 2 › 6 ›...

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