Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals - - PowerPoint PPT Presentation

▶
loop quantum gravity from spin networks to spinor path
SMART_READER_LITE
LIVE PREVIEW

Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals - - PowerPoint PPT Presentation

Feb 15, 2024 433 likes •753 views

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals Etera

slide-1
SLIDE 1

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

Etera Livine

ENS Lyon - Laboratoire de Physique

July 2014 in Marseille @

FFP ’14 Frontiers of Fundamental Physics

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-2
SLIDE 2

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

A Large Project: Probe the Geometry of Loop Quantum Gravity Goal of the programme: Understand the space-time geometry of the LQG quantum states and parameterize their deformations (quantum diffeomorphisms?) in order to describe the dynamics of this QG theory

LQG spin network states define quantum geometry, but. . . Interpretation as an effective discrete classical geometry? Emergence of smooth classical geometry at large scale? Symmetries encode the whole physics of GR and maps between observers. We assume the same will be true for quantum gravity, but. . . How to go from “discrete diffeomorphisms” at Planck scale to classical diffeomorphisms of smooth manifolds?

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-3
SLIDE 3

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

1 Loop Quantum Gravity in a Nutshell 2 Twisted Geometries and Coherent States 3 Spinfoam Amplitudes as Spinor Integrals 4 The Hamiltonian Constraints as Differential Operators Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-4
SLIDE 4

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

A 1-slide Loop Quantization Programme for General Relativity Loop Quantum Gravity = Canonical Quantization of GR

1 GR as a gauge theory with gauge group SU(2) in terms of

Ashtekar-Barbero connection A and conjugate triad field E. Metric reconstructed a posteriori from A and E.

2 Canonical quantization with quantum states of geometry

evolving in time, the spin network states.

3 Length, area, volume, holonomy, raised to quantum operators.

Spin networks describe quantum excitations of geometry.

4 Fully constrained theory with Hamiltonian constraints:

Gauss constraints imposing SU(2) gauge symmetry Scalar & vector constraints imposing diffeomorphism invariance

5 Transition amplitudes given by spinfoam path integral over

random discrete histories of spin networks

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-5
SLIDE 5

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Whats’s to be done in Loop Quantum Gravity? The hard tasks:

1 Implement the Hamiltonian constraints as quantum operators

(without anomaly)

2 Develop a mathematically consistent perturbative scheme to

identify physical states solving the constraints and compute their scalar product

3 Identify suitable physical states corresponding to actual

physical configurations (spherical symmetry, two-body problem, cosmology, ..)

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-6
SLIDE 6

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Loop Quantum Gravity

  • Concretely. . .

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-7
SLIDE 7

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Spin Networks and Geometric Operators Choice of wave-functions: cylindrical functionals of connection A

1

Choose oriented graph Γ.

2

Build holonomies Ue of connection A along edges e.

3

Consider gauge-invariant functions under SU(2) transformations at vertices v. ψ(A) = ψΓ({Ue[A]}e∈Γ) = ψΓ({hs(e)Ueh−1

t(e)}e∈Γ),

∀hv ∈ SU(2)

Spin Network Basis of L2(SU(2)E/SU(2)V ) on fixed graph Γ

1

SU(2) irreps on edges : spin je ∈ N/2

2

Intertwiners (singlet states) at vertices, Iv ∈ InvSU(2)V je1 ⊗ .. ⊗ V jen

3

Spin network ψje,Iv

Γ

(Ue) = Tr

e Dje(Ue) v Iv

Gluing of SU(2) elements by Clebsh-Gordan coefficients

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-8
SLIDE 8

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Spin Networks and Geometric Operators Geometric Operators built out of triad E raised to operator ˆ E

1

Area ˆ A of S intersecting edge carrying spin je: AS ∼ jel2

Planck in Planck units 2

Volume excitations at spin network vertices: ˆ V diagonalized by spin network basis and value depend on intertwiners Iv inside region

LQG Hilbert Space allowing superposition of different graphs

֒ → Have to describe how graphs share parts and can be refined ֒ → Done cleanly through projective limit by Ashtekar & Lewandowski ֒ → Compatible with diffeomorphism invariance

A discrete picture of geometry. . . ?

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-9
SLIDE 9

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Spinfoam Amplitudes and Evolving States of Geometry Spinfoams as histories of evolving spin networks: Graphs evolving into 2-complexes Spinfoam vertices are Space-time events Models define probability amplitude for each spin- foam vertex. Spinfoam models define transition amplitudes between spin network states, as product of vertex ampli- tudes Framework allow more general non- trivial space-time topologies.

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-10
SLIDE 10

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

First List of Questions What discrete (classical) geometry behind spin networks? related to Regge triangulations? Can we build coherent spin network states of geometry? Can spinfoam transition amplitudes be written as discrete path integrals of 4d geometry? Do they admit some semi-classical interpretation as transition amplitudes for quantum general relativity, e.g. at least in the framework of cosmology? What are the symmetries and invariance of the spinfoam amplitudes? and what are their relation to deformations and diffeomorphisms of the geometry? A second list of questions will come later. . .

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-11
SLIDE 11

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

1 Loop Quantum Gravity in a Nutshell 2 Twisted Geometries and Coherent States 3 Spinfoam Amplitudes as Spinor Integrals 4 The Hamiltonian Constraints as Differential Operators Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-12
SLIDE 12

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

The Holonomy-Flux Poisson algebra on a fixed graph Phase space of loop gravity in terms of standard holonomy-flux variables on a fixed oriented graph: Holonomy along edge e ge ∈ SU(2) Flux for edge e around vertex v X v

e ∈ su(2) ∼ R3

These are discretization of connection-triad fields A and E ge = P exp(i

  • e A)

X v

S =

  • p∈S gv→p E g−1

v→p

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-13
SLIDE 13

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

The Holonomy-Flux Poisson algebra on a fixed graph Phase space of loop gravity in terms of standard holonomy-flux variables on a fixed oriented graph: Holonomy along edge e ge ∈ SU(2) Flux for edge e around vertex v X v

e ∈ su(2) ∼ R3

T ∗SU(2) Poisson bracket : {ge, ge′} = 0 {X a

e , X b e′} = iδee′ǫabcX c e

{ Xe, ge} = σge With constraints: Parallel transport of vectors by 3d rotations along edges: X t(e)

e

= −ge ⊲ X s(e)

e

Closure constraint around vertices:

e∋v X v e = 0

generating local SU(2)-inv

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-14
SLIDE 14

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Minkowski theorem for Polyhedra and Twisted Geometries Interpretation as Discrete Geometries Closure constraint ⇒ convex polyhedra dual to each vertex

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

Polyhedra glued along the edges by area matching ֒ → ∃ge ∈ SO(3), X t

e = −ge ⊲

X s

e ⇔ |

X t

e | = |

X s

e |

We have polyhedra, representing chunks of volume, glued together by their faces, with area matching but no shape matching = ⇒ Twisted Geometries

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-15
SLIDE 15

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

The twisting angle, extrinsic curvature and torsion Twisted geometries are not triangulations, they generalize Regge geometries . . . to torsion Where does the torsion come from?

֒ → connection A = Γ[E] + γK mixes intrinsic and extrinsic data and naturally leads to torsion dAE ∝ γA ∧ K = 0

So how to separate intrinsic and extrinsic geometry data?

Shape mismatch encodes extrinsic curvature and torsion. . . On an edge, the vectors X s,t

e

do not entirely determine the holonomy ge: ge = nt(e) ei ξe

σ3 2 ǫ n−1

s(e)

Twisting angle ξe ∼ γK encodes extrinsic curvature at 1st order

Twisted Geometry = Chunks of flat space (living in tangent space) glued together with curvature and torsion

Can impose gluing constraints to restrict to Regge geometries. . . . . . but that’s more limited than LQG!

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-16
SLIDE 16

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Loop Gravity Phase Space: Spinorial Parametrization Let us re-visit this phase space replacing vectors X v

e ∈ R3 by

spinors |zv

e ∈ C2 . . .

We describe twisted geometries as Spinor Networks

Can reconstruct both g’s and X’s from spinors : ◮ X = z| σ|z with Pauli matrices, area as norm | X| = z|z ⇒ spinor |z carries vector X plus a U(1) phase ◮ Unique SU(2) element ge mapping zs

e to zt e since we are in the

fundamental representation of SU(2) ge = |zt

e]zs e| − |zt e[zs e|

  • zs

e|zs ezt e|zt e

, |z] = ǫ|¯ z = −¯ z1 ¯ z0

  • Darboux coordinates with canonical bracket {za, ¯

zb} = −iδab

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-17
SLIDE 17

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

The Spinor Network phase space We keep area-matching constraints and closure constraints : around vertices:

e∋v |zv e zv e | ∝ I

⇒ Generates SU(2) transformations on spinors along edges: zs

e|zs e = zt e|zt e

⇒ Generates inverse U(1) phase transformations on spinors New feature: U(1) phase of spinors ֒ → Vectors X invariant under U(1) transformations ֒ → Holonomies ge inv under joint U(1) on both spinors ֒ → Relative phase along edge gives twisting angle ξe, conjugate to area

Better understanding of link between twisting angles and embedding data (extrinsic curvature) through twistor formalism, allowing embedding of SU(2) action in full Lorentz group

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-18
SLIDE 18

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Quantizing: Building Coherent Spin Network States We have canonical Poisson brackets, so quantization is direct . . .

1 Raise spinor components to creation/annihilation operators :

z0 → a0, z1 → a1, ¯ z0 → a0†, ¯ z1 → a1†

2 Gives Schwinger’s representation of su(2) in terms of two

HOs, with spin j being total energy

3 Quantizing matching and closure constraints, we recover the

Hilbert space of spin network states as (holomorphic) wave-functions in the spinors

4 We project HOs’ coherent states and define coherent spin

network peaked on classical spinor networks : ψ{zv

e }(ge) =

  • [dhv] e
  • e[zs(e)

e

|h−1

s(e) ge ht(e)|zt(e) e

  • Semi-classical Twisted Geometries for LQG

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-19
SLIDE 19

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Quantizing: Building Coherent Spin Network States

Using the properties of Perelomov’s SU(2) coherent states, we decompose: ψ{zv

e }(ge)

=

  • [dhv] e
  • e[zs(e)

e

|h−1

s(e) ge ht(e)|zt(e) e

  • =
  • {je}
  • [dhv]
  • e

[je, zs(e)

e

|h−1

s(e) ge ht(e)|je, zt(e) e

  • (2je)!

֒ → Poisson distribution for the spin je peaked on the norm | Xe| = ze|ze ֒ → Group averaging at the vertices leads to coherent intertwiners peaked on classical polyhedra

But also coherent spin networks are generating functionals for spin networks implementing weighted sums over the spins. . . ⇒ Powerful mathematical tool that we use in spinfoams!

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-20
SLIDE 20

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

1 Loop Quantum Gravity in a Nutshell 2 Twisted Geometries and Coherent States 3 Spinfoam Amplitudes as Spinor Integrals 4 The Hamiltonian Constraints as Differential Operators Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-21
SLIDE 21

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

The Spinfoam Amplitude Ansatz Spinfoams as histories of evolving spin networks

A local ansatz: history amplitude given by product of vertex amplitudes Local vertex amplitude defined in terms

  • f

boundary spin network around vertex Flat ansatz: take spin network evalua- tion, given by contraction of intertwin- ers (with trivial holonomies) Aσ ∝ ψje,Iv (I) Typically, vertex amplitudes given by {3nj} symbols of spin recoupling. Framework inspired from state-sum-models as discrete path integrals for topological field theories. Models defined by choice of

1

gauge group

2

admissible representations and intertwiners

3

vertex amplitude

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-22
SLIDE 22

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

The Spinfoam Amplitude Ansatz Spinfoams as histories of evolving spin networks

A local ansatz: history amplitude given by product of vertex amplitudes Local vertex amplitude defined in terms

  • f

boundary spin network around vertex Flat ansatz: take spin network evalua- tion, given by contraction of intertwin- ers (with trivial holonomies) Aσ ∝ ψje,Iv (I) Spinfoam models for quantum gravity: Ponzano-Regge for 3d QG based on {6j}-symbol Turaev-Viro for 3d QG with Λ = 0 based on {6j}q-symbol Crane-Yetter model for 4d BF theory based on {15j}q-symbol Barrett-Crane model for 4d QG based on {10j}-symbol EPRL and FK models for 3+1d LQG based on non-trivial mapping of spins to Lorentz group irreps and coherent packets of {15j}-symbols

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-23
SLIDE 23

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Semi-classical spinfoam amplitudes for coherent boundary states We look at spinfoam amplitudes between coherent spin network states Locally, we switch from the spin basis to spinor variables. The vertex ampli- tudes are now given by the evaluation

  • f coherent spin network.

Aσ = ψ{je,Iv}(I) → Aσ = ψ{zv

e }(I)

Using completeness of SU(2) coherent state basis, full spinfoam path integral becomes a integral over spinors:

A =

jf ,Iℓ

  • f Af
  • ℓ Aℓ
  • σ Aσ

↓ A =

  • d4zℓ

f

  • f Af
  • ℓ Aℓ
  • σ Aσ

with Aσ(zv

e ) ∼

  • [dhv]e
  • e[zs(e)

e

|h−1

s(e)ht(e)|zt(e) e

  • Etera Livine ENS Lyon - Laboratoire de Physique

Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-24
SLIDE 24

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Using Spinor Path Integrals 3 ways to use spinor formulation for spinfoam path integral:

1

Exact calculations of vertex amplitude Vertex amplitude Aσ(z) is Gaussian integral

  • ver

SU(2)

  • dh e
  • e[zs(e)

e

|h−1

s(e)ht(e)|zt(e) e

  • ֒

→ proposes modification

  • f vertex amplitude

֒ → Spinfoam amplitude as Gaussian integral over spinors

2

Discrete Lagrangians We have an integral over spinors z, which can be re- expressed as integral over vectors X and holonomies

  • g. . .

֒ → Spinfoam amplitude as path integral of dis- crete Lagrangian for dis- crete geometry. ֒ → Extended to twistors, allowing explicitly for 4d Lorentz-covariant objects

3 Spinor amplitude as

generating functional of spin amplitudes ֒ → Tools to derive asymptotics

  • f

spin amplitudes and {3nj}- symbols ֒ → From recursion rela- tion to differential eqns for spinfoam amplitdues

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-25
SLIDE 25

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

1 Loop Quantum Gravity in a Nutshell 2 Twisted Geometries and Coherent States 3 Spinfoam Amplitudes as Spinor Integrals 4 The Hamiltonian Constraints as Differential Operators Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-26
SLIDE 26

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Differential Equations and Symmetries of Spinfoam Amplitudes A diff eqn satisfied by spinfoam amplitudes expresses an invariance under some deformations of the boundary ⇒ expression of diffeomorphism inv?

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-27
SLIDE 27

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Differential Equations and Symmetries of Spinfoam Amplitudes A diff eqn satisfied by spinfoam amplitudes expresses an invariance under some deformations of the boundary ⇒ expression of diffeomorphism inv? Example of 3d quantum gravity:

Hamiltonian constraints Biedenharn-Elliot Identity on 6j-symbols Recursion relation

  • n 6j-symbols

2nd diff eqn on coherent 6z-symbols, as semi-classical Spinfoam amplitude Quantum Hamiltonian constraints as action

  • f holonomy operator

Numerical simulations

  • f 3d quantum gravity

Ponzano-Regge model

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-28
SLIDE 28

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Going beyond: discrete diffeomorphisms for LQG Looking for closed algebra of deformations, as diff eqns on semi-classical spinfoam amplitudes for coherent spin network boundary states

  • D{z∂} ASF({z∂}) = 0

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-29
SLIDE 29

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Example: Application to Homogeneous FRW Cosmology Approach originally by Rovelli & Vidotto: “Dipole Cosmology”

1

Look at simplest non-trivial graph, with two vertices linked by a few edges

2

Define coherent spin network states

3

Compute corresponding spinfoam amplitude, as evaluation of boundary spin network on I

4

Identify Hamiltonian constraint as diff eqn

5

Take classical limit to translate to effective classical Hamiltonian

6

Compare to cosmology

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-30
SLIDE 30

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Example: Application to Homogeneous FRW Cosmology Approach originally by Rovelli & Vidotto: “Dipole Cosmology”

Amplitude computed as series W(zi, wi) = ψ{zi ,wi }(I) =

J 1 J!(J+1)!

  • det

i |zi[wi|

J in area eigenvalue J =

i ji given by sum of spins

Hamiltonian constraint as 2nd order diff operator in spinors Simplifies in isotropic ansatz with all zi equal to wi up to global phase φ, conjugated to total area λ Use new canonical complex variable z = √ λeiφ Amplitude W(z) ∝ I1(z2)/z2 satisfies 2nd order diff eqn, which translates to classical Hamiltonian constraint of Loop Quantum Cosmology H ∝ λ2 sin2 φ

Can add Λ, curvature and scalar matter!.. Promising but remains first order calculations with a single degree of freedom.

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

slide-31
SLIDE 31

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators

Outlook: 2nd list of questions Applications to more complicated toy models,

֒ → mini/midi-superspaces for cosmology with inhomogeneities

Coarse-graining of LQG dynamics & spinfoam renormalisation:

֒ → Phase transition from discrete quantum geometry to smooth Lorentzian geometries?

Consistent expansion in quantum gravity corrections and relation to standard perturbative scheme for quantum GR

֒ → Link with twistorial integrals for perturbative GR as used in string theory?

Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals