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8/3/2019 Karim Noui- Loop Quantum Gravity : a review
1/14
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/8/3/2019 Karim Noui- Loop Quantum Gravity : a review
2/14
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
LQG : - Southampton - october 2008 Karim NOUI Loop Quantum Gravity : a review 2/14
http://find/http://goback/8/3/2019 Karim Noui- Loop Quantum Gravity : a review
3/14
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
http://find/http://goback/8/3/2019 Karim Noui- Loop Quantum Gravity : a review
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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
http://find/http://goback/8/3/2019 Karim Noui- Loop Quantum Gravity : a review
5/14
The starting point : Ashtekar variables
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
http://find/http://goback/8/3/2019 Karim Noui- Loop Quantum Gravity : a review
6/14
The starting point : Ashtekar variables
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
http://find/http://goback/8/3/2019 Karim Noui- Loop Quantum Gravity : a review
7/14
The starting point : Ashtekar variables
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
LQG : - Southampton - october 2008 Karim NOUI Loop Quantum Gravity : a review 7/14
http://find/http://goback/8/3/2019 Karim Noui- Loop Quantum Gravity : a review
8/14
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
LQG : - Southampton - october 2008 Karim NOUI Loop Quantum Gravity : a review 8/14
http://find/http://goback/8/3/2019 Karim Noui- Loop Quantum Gravity : a review
9/14
The quantum geometry : space is discrete
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
LQG : - Southampton - october 2008 Karim NOUI Loop Quantum Gravity : a review 9/14
http://find/http://goback/8/3/2019 Karim Noui- Loop Quantum Gravity : a review
10/14
The quantum geometry : space is discrete
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
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11/14
The quantum geometry : space is discrete
How looks space at the Planck scale ?
Space is fundamentaly discrete...
... and might be non-commutative !
LQG : - Southampton - october 2008 Karim NOUI Loop Quantum Gravity : a review 11/14
http://goforward/http://find/http://goback/8/3/2019 Karim Noui- Loop Quantum Gravity : a review
12/14
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
LQG : - Southampton - october 2008 Karim NOUI Loop Quantum Gravity : a review 12/14
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13/14
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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