Karim Noui- Loop Quantum Gravity : a review

8/3/2019 Karim Noui- Loop Quantum Gravity : a review

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Loop Quantum Gravity : a review

Karim NOUI

Laboratoire de Mathematiques et de Physique Theorique, TOURSFederation Denis Poisson, ORLEANS-TOURS, FRANCE

LQG : - Southampton - october 2008 Karim NOUI Loop Quantum Gravity : a review 1/14
http://find/http://goback/


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Overview

Introduction : LQG in a nutshell

1. The starting point: Ashtekar variables

Why standard quantisation schemes fail ? Ashtekar variables : similarities with Yang-Mills Quantisation strategy in LQG

2. The Quantum Geometry: space is discrete

Quantum states : loops and spin-networks Geometrical operators have a discrete spectrum

Discussion : pros and cons Successes : black holes and cosmology Open issues : dynamics and graviton propagator



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What is Loop Quantum Gravity ?In a nutshell...

The essential characteristics : LQG is supposed to be Hamiltonian quantisation of pure 4D Gravity : M = R Non-perturbative quantisation : question of renormalisation avoided Background independent quantisation : no background metric needed

The main results : LQG is supposed to provide

Structure of Quantum geometry at Planck scale : discretness Microscopic explanation of Black holes thermodynamics : S = A/4 Resolution of cosmological singularity : LQ Cosmology

The remaining important issues :

Performing the quantisation completely : scalar constraint Free parameters to be understood : Barbero-Immirzi ambiguity Consistent coupling to other interactions

LQG offers a very nice, still under construction but fascinating framework

to Quantum Gravity : worth being studied...LQG : - Southampton - october 2008 Karim NOUI Loop Quantum Gravity : a review 3/14


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The starting point : Ashtekar variables

Why standard quantisation schemes fail ?

Lagrangian formulation : M a 4D manifold Einstein-Hilbert action : functional of the metric g

SEH[g] =

d4x

|g|R

Hamiltonian formulation :M =

R

(61) ADM variables : ds2 = N2dt2 (Nadt + habdxb)(Nadt + hacdx

c) ADM action : (h, ) canonical variables

SADM[h, ; N,Na] =

dt

d3x(h + NaHa[h, ] + NH[h, ])

Constraints H = 0 = Ha generate the diffeomorphisms

What about the quantisation ? Path integral : no renormalisability

Canonical : too complicated constraints !LQG : - Southampton - october 2008 Karim NOUI Loop Quantum Gravity : a review 4/14


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Similarities with SU(2) Yang-Mills theory I : Lagrangian

First order Lagrangian : variables are Cartan data A tetrad eI = eIdx such that g = e

Ie

JIJ

eI is a 4 4 matrix : local flat frame

a so(3, 1) spin-connection = iRi + 0iBi ; F() its curvature

I is related to Levi-Civitta coefficients

Einstein-Palatini-Holst action : depends on the free parameter = 0

SP[e, ] =

eI eJ (FIJ()

1

FIJ())

, e.o.m. are equivalent to EH if e invertible

It looks like a topological field theory Plebanski action (79) in terms of and BIJ = BIJdx

dx

SPl[B, ] =

BIJ FIJ() + (B)

where enforces B to be simple : B = (1 or )e eLQG : - Southampton - october 2008 Karim NOUI Loop Quantum Gravity : a review 5/14


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Similarities with SU(2) Yang-Mills theory II : Hamiltonian

Canonical analysis of Palatini action Quite technical but reproduces exactly ADM resultsThe Ashtekar variables (86)

New variables : Eai =12ijk

abcejbe

kc and A

ia =

ia +

0ia

Pair of canonical variables :

{Aia(x),Ebj (y)} = (8G)

ba

ij

3(x, y)

It is a canonical transformation Historically, Ashtekar considered = i : complex gravity

Canonical analysis with the time gauge choice e0 = 0 The constraints are almost polynomial

Gi = DaEai , Ha = F

iabE

bi , H = (F

ijab + (

2 + 1)Ki[aKjb])E

ai E

bj

The constraints generate symmetries : Gauss and diffeomorphismsLQG : - Southampton - october 2008 Karim NOUI Loop Quantum Gravity : a review 6/14


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The quantisation strategy in Loop Quantum Gravity

Choice of polarisation The variables (A,E) become non-commutative operators Representation of the operators algebra on (A)

(Aia(x))(A) = Aia(x)(A), (

Eai (x)

8G)(A) = i

(A)

Aia(x)

Imposing the constraints successively K : Quantum states are functions of holonomies U(A) SU(2)

KGi

K0Ha

KdiffH

H

State of the art : K0 and Kdiff are known but not H

Construct physical observables for eventual predictions Observables are fundamentally non-local functions



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The quantum geometry : space is discrete

Quantum states I : kinematical states

The space K of kinematical states Constructed by analogy with gauge theories defined from a graph with L links and f C(SU(2)L)

:i are oriented links

:ni are nodes

1

2

3n1 n2

,f(A) = f(U1

(A), , UL

(A))

Scalar product : ,f|,f = ,

d(U) f(U)f(U)with d(U) the SU(2) Haar measure



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Quantum states II : loops, spin-networks, knots

The space K0 of spin-networks Imposing the Gauss constraint : Gi = 0 A basis of the space K0 in terms of spin-networks |S = |,ji, n

The links i are colored with SU(2) representations : spins jiThe nodes are colored with SU(2) intertwiners : n

States |S form an orthonormal basis of K0

Diffeomorphism invariance : the space Kdiff

identify states related by a diffeomorphism States of Kdiff are labelled by knots



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Geometrical operators have discrete spectrum (95)

Area operator A(S) acting on K0

Classical area of a surface S : A(S) =S

naE

ai nbE

bi d

2

Quantum area operator : S = Nn Sn

A(S) = limN

nEi(Sn)Ei(Sn) with Ei(Sn) =

Sn

Ei

Spectrum and Quanta of area

S

A(S)|S = 8Gc3 PSjP(jP + 1)|S

Volume operator V(R) acting on K0

Classical volume on a domain R : V(R) =

Rd3x

|abcijkEaiEbjEck|

3!

It acts on the nodes of |S : discrete spectrumLQG : - Southampton - october 2008 Karim NOUI Loop Quantum Gravity : a review 10/14
http://goforward/http://find/http://goback/


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How looks space at the Planck scale ?

Space is fundamentaly discrete...

... and might be non-commutative !



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Discussion

Successes of LQG : Black holes and cosmology

Uniqueness theorem for the quantisation (LOST)

Structure of space at Planck scale : discrete

Black holes thermodynamics Very simple arguments lead to S = A/4 Fixing Barbero-Immirzi parameter Robust counting that works for isolated BH Relation to quasi-normal modes By Ashtekar, Baez, Krasnov, Rovelli ...

Loop Quantum Cosmology No more singularity Apply to BH : no more information lost paradox... By mostly Bojowald and team around Ashtekar



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Discussion

Open issues : the dynamics and graviton propagator ?

Solving the Hamiltonian constraint is fundamental To find physical states : construct P s.t. |Sphys = P|S To find physical observables and make robust physical predictions

The different approaches to solve the problem Thiemann proposes a very tricky regularisation of H

Spin-Foam models : based on a covariant quantisation

A = S|Sphys

The graviton propagator : failure for the moment !



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References

Politic and polemic science

C. Rovelli, a dialog on QG [hep-th/0310077] H. Nicolai & co, LQG : an outside view [hep-th/0501114] T. Thiemann, LQG : an inside view [hep-th/0608210] A. Ashtekar, LQG : 4 recent advances and a 12 FAQs

[gr-qc/0702030]

Books and reviews A. Ashtekar and J. Lewandowski [gr-qc/0404018] C. Rovelli, Quantum Gravity, Camb.Univ.Press T. Thiemann, Camb.Univ.Press

3D quantum gravity K.N. and A. Perez [gr-qc/0402111][gr-qc/0402112] K.N. [gr-qc/0612144][gr-qc/0612145] E. Joung, J. Mourad and K.N. [0608.4121]


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Karim Noui- Loop Quantum Gravity : a review