j. fernando barbero g. instituto de estructura de la...

Loop quantum gravity: A brief introduction.

J. Fernando Barbero G.Instituto de Estructura de la Materia, CSIC.

Graduate Days Graz, March 30-31, 2017

LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, 2017 1/51

Summary1 Motivation (Why canonical quantum gravity?).2 A non-metric Hamiltonian approach: The Ashtekar variables.3 Quantization.4 Applications:

Black hole entropy.Loop Quantum Cosmology.

5 The rest of LQGThe covariant approach: spin foams.Group field theory.The continuum limit.

6 Where do we stand today?

Disclaimer

Not exhaustive.

I will talk about some selected items.

I will try to give enough details about them.


Motivation

Should we quantize gravity?

The interplay between the quantum world and the geometry of space-time must be understood.

Quantum gravityDifferent approaches to the quantization of gravity:

Perturbative (including strings and asymptotic safety).Canonical.Causal Dynamical Triangulations.Causal sets, shape dynamics,...

The LQG choice

Canonical quantization


Particle physics QGPerturbative quantum gravity: Try to use the same approach that workedso well for the other interactions.

The goal is to obtain S matrix elements (containing the physics ofparticle interactions, giving predictions for “accelerator experiments”,and providing necessary information for cosmology).

The use of perturbation theory for this means that one wants toobtain physical predictions as power series in GN .For this to be at all possible there are some “consistency” require-ments (that can be met for the other interactions) such as renor-malizability (the ability to absorb the infinities appearing in physicalamplitudes by a redefinition of the coupling constants).

Does it work?

Maybe: Naive perturbative QG is non-renormalizable but the asymp-totic safety scenario offers some hope and string theories containgravity in a arguably consistent way.


Relativist’s QG

What kind of phenomena would a relativist love to understand in aquantum theory of gravity?

Singularities: In a sense QFT solves some problems associated withsingularities of the classical EM field, Coulomb potential, infinite en-ergy of a single charge. It does it without introducing extended ob-jects but rather by changing the description of the interactions.The high curvature regime of general relativity, in particular,black holes and the Big-Bang.The origin of black hole entropy and its detailed microscopicaldescription?Black hole evaporation, in particular solve the problems related tounitarity and information loss.


Relativist’s QG (goals)

Define a mathematically rigorous quantum theory of the gravita-tional field (general relativity or something very close). To this endwe need to understand how to quantize background free fieldtheories (related to diff-invariance).

Find useful quantum gravitational observables (hopefully leadingto verifiable experimental or observational predictions).

Solve the problem of time and, in general, the problem of spacetimecovariance (time does not exist as an external object in GR).

As the gravitational field is a manifestation of space-time ge-ometry the quantization of general relativity will require us to under-stand the fate of geometry after quantization or, in other words,the meaning of quantized geometry.

Get a detailed picture of emergence of classical geometry at largescales from the purely quantum gravity theory (the semiclassicalregime and the continuum limit).


A possible choice: “canonical” quantization

Use Dirac’s method (quantization of constrained systems)

The starting point is a Hamiltonian formulation for gravity. Thiswas first obtained in the early sixties by Arnowitt, Deser, and Misner(the ADM formalism) after some pioneering work by Dirac.

Obtain a quantum version of the constraints. In metric variablesthis leads to the famous Wheeler-DeWitt equation, apparently re-ferred to by one of his authors as that damned equation!.

Work on this issue has provided lots of interesting insights but hasnot produced a completely satisfactory theory of QG.

An important issue: learn how to deal with diff-invariance andunderstand quantization in the absence of a metric background.


Dirac’s method in a nutshell

Some important physical systems (such as EM) are singular (there is aproblem to go from a Lagrangian to a Hamiltonian formulation).

The fiber derivative (a.k.a. definition of conjugate momenta) is not adiffeomorphism between TQ and T ∗Q.Hamiltonian dynamics can only be defined in a consistent way on asubmanifold of the phase space T ∗Q. This is usually described as thevanishing of some phase space functions (constraints).

The physical Hilbert space is obtained by quantizing these constraintsand finding their kernels (well, one tries to do that...)

There is a very concrete method due to Dirac (Dirac’s algorithm inthe following) that consistently gives the constraints starting from anyLagrangian.A warning: one must follow it to the letter.

There are other ways to arrive at the same results (GNH method) thathave some conceptual advantages and are somehow simpler to use.


Metric Hamiltonian gravity: ADM formulation

Is there a Hamiltonian formulation for GR?

YESDerive it from the Einstein-Hilbert action S =

1

2κ

∫M

e√σgR ; κ =

8πGN

c3

by following Dirac’s approach

M is a 4-dim manifold M = R× Σ (global hyperbolicity [Geroch]).Σ ia a smooth, orientable, closed (i.e. compact and without bound-ary) 3-manifold, e is a fiducial volume form and σ is the space-timesignature (σ = −1 Lorentzian, σ = +1 Riemannian).

Introduce a “time function” t defining a foliation ofM by smooth 3-dim hypersurfaces Σt diffeomorphic to Σ and a “time flow directionta” (a globally defined smooth vector field such that ta∇at = 1).Alternatively one can introduce a congruence of spacetime filling curves.



Given a metric of signature (−+++) it defines a unit time-like normalna on the points of each Σt .

Notation: I use Penrose notation (no coordinate charts!) (ta ∈X(M), ta ∈ X∗(M), tab ∈ X(M)⊗ X(M),...)

Let us define:The induced metric hab = gab + nanb (on vectors X a tangent to eachΣt we have habX

b = gabXb := Xb, and habn

a = 0).

The lapse N := −gabtanb = (na∇at)−1, (∇a denotes a torsion-freeconnection on M).

The shift Na := habtb = ta − Nna.

Let us define also:The unique, torsion-free, derivative operator Da on each Σt compatiblewith hab.

This is given by DaTa1,...ak

b1...b`:= ha1

d1· · · h e`

b`h fa ∇f T

d1,...dke1...e` .

The extrinsic curvature Kab := h ca ∇cnb = 1

2N (Lthab − 2D(aNb)).



The information contained in gab is the same as the one present in(N,Na, hab) but these are “3-dimensional objects”.

Let us rewrite the E-H action in terms of these (The fiducial, non-dynamical volume form e := eabcd is chosen so that it satisfiesLteabcd = 0).

Remember that Kab can be written as a “time derivative” of hab andDaNb.

3 + 1 decomposition

S =

∫M

e√hN[σ(3)R + KabK

ab − K 2] :=

∫R

∫Σt

(3)eLG :=

∫RLG

Here (3)eabc := tdedabc .



Canonical conjugate momenta

pab :=∂LG∂hab

=√h(K ab − Khab)

The momenta associated with the lapse N and the shift Na are zero.

These become first class constraints implying that N and Na arearbitrary (a strange thing for Lagrange multipliers to do!).

Hamiltonian density

S =

∫R

∫Σt

(3)eHG :=

∫RHG

HG = pabhab − LG = N

[σ√h

(3)R +

1√h

(pabpab −1

2p2)

]−2NbDa(h−1/2pab)



N and Na are Lagrange multipliers enforcing the secondary con-straints:

σ√h (3)R +

1√h

(pabpab −1

2p2) = 0 (Scalar constraint)

Da(h−1/2pab) = 0 (Vector constraint)

They generate gauge transformations. Time evolution (!!!) and3-dimensional diffeomorphisms respectively. For instance:

V (Λ) :=

∫ΛaDap

ab , V (Λ), qab = LΛqab , V (Λ), pab = LΛpab .

The full evolution of hab and Kab is given by the Hamilton equations(HG denotes the Hamiltonian)

hab =δHG

δpab= HG , hab, pab = −δHG

δhab= HG , p

ab


ADM formulation: The final description

Phase space: Γ(hab, pab)

Symplectic structure: Ω =

∫Σ

(3)eδhab ∧ δpab. Alternatively the

Poisson brackets are [these should be understood as brackets betweenweighted versions of the configuration and momentum variables]

hab(x), hcd(y) = 0

pab(x), hcd(y) = δa(cδbd)δ

3(x , y)

pab(x), pcd(y) = 0

Constraints (first class):

σ√h (3)R +

1√h

(pabpab −1

2p2) = 0 (Scalar constraint)

Da(h−1/2pab) = 0 (Vector constraint)

Two local d.o.f. (the two polarizations of the gravitational field!)


A new point of view (as of the mid eighties...)

A surprising development [Ashtekar 1986]: The best way to approachthe canonical quantization of gravity is to describe it as a theory ofSU(2) connections (or SO(3) in some simple situations).

This approach is the starting point of Loop Quantum Gravity (LQGin short).

It is a canonical approach.

The label non-perturbative (which has become a trade mark forLQG) refers to the fact that no splitting of a metric is used.

Notice, however, that some approximation scheme (perturbation the-ory) may well have to be developed to obtain sensible and testable(at least in principle) physical predictions.


The SO(3)− ADM formulation

Let us perform now a simple change of variables.

Introduce a triad i.e. three 1-forms e ia, i = 1, 2, 3 defining a frame ateach point of Σ (det e 6= 0)

Write the metric as hab = e iaejbδij

Introduce E ai = (det e)eai with eai eaj = δij (densitized inverse triad).

Define K ia =

1

det eKabE

bj δ

ij .

The new variables are canonical.

We can rewrite the constraints in terms of these variables. Beforewe do that it is important to realize that we have now local SO(3)rotations of e ia and K i

a that do not change neither hab nor Kab sothere must be extra constraints to generate them.

These can easily be found from the condition K[ab] = 0 (the 2nd

fundamental form is symmetric).


The SO(3)− ADM formulation

Phase space: Γ(K ia, E

ai )

Symplectic structure (Poisson brackets):

K ia(x),K j

b(y) = E ai (x), Eb

j (y) = 0

E ai (x),K j

b(y) = δij δabδ

3(x , y)

Constraints (first class): R is the scalar curvature of hab := e iaebi .

εijkKjaE

ak = 0

Da

[E akK

kb − δabE c

kKkc

]= 0

−σ√hR +

2√hE

[ck E

d ]l K k

c Kld = 0

Two local degrees of freedom (of course!)


GR in Yang-Mills phase space: Ashtekar vars.

Consider the transformation

γEai = −1

γE ai

γAia = Γi

a + γK ia

Γia is a SO(3) connection that defines a covariant derivative compatible

with the triad.

∂[aeib] + εi jkΓj

[aekb] = 0 can be inverted to get Γi

a (a SO(3) connection).

γ ∈ C is known as the Immirzi parameter.

The Poisson brackets between the new variables γAia and γE

ai are

γAia(x), γA

jb(y) = γE a

i (x), γEbj (y) = 0

γAia(x), γE

bj (y) = δij δ

ba δ

3(x , y)

So this is a canonical transformation!


GR in Ashtekar variables.

Phase space: Γ(γAia,γ E

ai ) [smooth SO(3) connections and triads on

Σ, i.e. a Yang-Mills phase space]

Symplectic structure (Poisson brackets):The variables γAai (x) and γE

bj (y) are canonical;

Constraints (first class):

DaEai = 0 Gauss

F iabE

bi = 0 Vector

E[ai E

b]j

[εijkF

kab +

2(σ − γ2)

γ2(Ai

a − Γia)(Ai

a − Γjb)

]= 0, Scalar

Where DaEai = ∂aE

ai + εijkA

jaE

ak , and F iab = 2∂[aA

ib] + εijkAajAbk is

the SO(3) curvature.

Two local degrees of freedom X


The self-dual and Holst actions

The new formulation can be derived from an action principle

The Holst action

S =

∫M

e I ∧ eJ ∧ (εIJKLΩKL − 2

γΩIJ)

e I takes values in a 4-dimensional R-vector space (I = 0, 1, 2, 3).

ωIJ takes values in the Lie algebra of SO(1, 3) (Lorentzian signature) or

SO(4) (Riemannian signatures).

εIJKL is the alternationg tensor in V .

ΩIJ = dωI

J + ωIK ∧ ωK

J is the curvature 2-form of ωIJ .

If γ = i (Lor. case) or γ = 1 (Riem. case) the action can be written interms of the self-dual curvature of a self-dual connection ωI+

J .

This action is invariant under diffeomorphisms of M and “internal”gauge transformations (SO(1, 3) or SO(4) depending on the signature).



Comments

GR in these new variables is a background independent relative ofSO(3) [or SU(2)] Yang-Mills theory.

The fact that the configuration variable is a connection is a cornerstoneof the formalism.

What happens with γ?

If σ = +1 (Riemannian signature) we can cancel the last term bychoosing γ = ±1. In this case the variables Ai

a and E ai are real and

the scalar constraint takes a very simple form.

If σ = −1 (Lorentzian signatures, i.e. the real thing) we face twochoices:

If we want to remove the ugly term we have to take γ = ±i .If we want to have real variables we have to live with complicatedconstraints.



This parameter shows up in the definition of the area and volumeobservables that are an essential ingredient of the formalism.It is not an unobservable ambiguity but, rather, has physical conse-quences.It plays a significant role in the description of black holes and LQC.The fact that the “internal” symmetry group is compact is very im-portant in the construction of the Hilbert spaces used to quantize thetheory (a good reason to use real variables).

Once it was understood that even the complicated form of the con-straints could be handled (more or less...) the emphasis was placedon the geometric meaning of the new variables.


(Dirac) quantization

Several steps:

1 Find a representation of the basic classical variables in a suitablekinematic Hilbert space Hkin.

2 Represent the constraints in this Hilbert Space as self adjoint op-erators.

3 Find their kernels (solutions to the quantum constraints) to definephysical states and a suitable scalar product in Hphys .

4 In practice the process is much subtler, in particular with regards tothe implementation of the vector constraint (the generator of 3-dimdiffeos). Group averaging methods.

5 Find a complete set of (gauge invariant) observables and phrasethe relevant physical questions in terms of them. This is highly non-trivial for background independent field theories with local degrees offreedom.


Quantization

A formal approach to quantization would consist in copying thestandard “quantization rules” used in the familiar finite-dimensionalquantum mechanical models. [Configuration variables Multiplica-tion operators XΨ(x) = xΨ(x)]. Momentum variables Derivativeoperators [PxΨ(x) = −i~∂xΨ(x)].

Choosing here the Ashtekar connection A as configuration variableand the triad as momentum one would be led to consider wave func-tionals Ψ[A] and represent the connection itself as a multiplicationoperator and the triad as a “functional derivative” −iδ/δA.

In order to properly define the kinematical Hilbert space in a rigor-ous way one introduces a different set of variables –holonomies andfluxes– defined in a space of wave functionals Ψ[A] which are squareintegrable with respect to a diff and SU(2) invariant measure knownas the Ashtekar-Lewandowski measure dµAL.


Quantization

dµAL defines a kinematical inner product〈Ψ,Φ〉=∫

Ψ[A]Φ[A]dµAL

Holonomies

Given a path e in the spatial manifold Σ we define

he [A] := P exp

(−∫eA

)∈ SU(2) ,

where P de notes the path ordered exponential.

1 he [A] is parametrization independent.2 he [A] defines a representation of the group of paths:

he [A] = he1 [A]he2 [A] if e = e1 e2.3 Under SU(2) gauge transformations the holonomies change as

h′e = g(x(0)he [A]g−1(x(1))).4 Under a diffeo φ it transforms as he [φ∗A] = hφ−1(e)[A].


Quantum geometric operators

5 Given a graph α with edges ei , i = 1, . . . , n we builda Hilbert space Hα of cylindrical functions ψγ,f =f (he1 [A], . . . , he1 [A]) where f : SU(2)n → C is squareintegrable w.r.t. the Haar measure in SU(2).

Fluxes

Associated to smooth surfaces S (α ∈ SU(2) is a smearing field):

E (S , α) :=

∫Sαi Ei .

The Poisson algebra defined by these variables admits a unique quan-tization in a Hilbert space with diff-inv. states (LOST&F th.).

Spin networks: A very convenient orthonormal basis in this spaceis provided by the spin network states. These are labeled by graphsof the type shown above (consisting in edges with spin labels).


Geometric operators

I will start now discussing geometric operators (quantum Rieman-nian geometry), we are finally doing the promised quantum geometry!

The length and angle operatorsThe area operatorThe volume operatorCurvature operator.

General considerations

They can all be rigourously defined in the Hilbert space used above.They have discrete spectra.

The generalized spin network basis introduced before is well adaptedto their description.

The area operator is important in the computation of black hole en-tropy.

The volume operator is a basic ingredient for the quantization of thescalar constraint.The curvature operator provides an alternative quantization of thescalar constraint.


Area operator

Consider a surface embedded in S ⊂ Σ. We will require it to beclosed.

The densitized triad E ai encodes the metric information. Hence we

can write the area of a surface in terms of it.If we choose a normal na to the points of S the area, as a function(al)of E a

i , takes the form

AS [E ai ] =

∫S

(E ai E

bj δ

ijnanb)1/2

We want now to quantize the operator AS [E ai ]. This means that

we have to define its action on the vectors in H. To this end we needto know its action on the elements of the orthonormal basis that wehave introduced above (spin networks).

A reasonable way to approach this problem is trying to express itin terms of the flux operators E [S , f ].


Area operator

The idea is to decompose S in N two dimensional cells SI of “smallcoordinate” size.

Use the three Lie algebra vectors τi (in place of the αi ) and considerthe flux variables E [SI , τ

i ] on each cell.

Let us take AN [S ] := γ

N∑l=1

(E [SI , τi ]E [SI , τj ]ηij)1/2. This is an ap-

proximate expression for the area (“Riemann sum”) in the sense thatif the number of cells goes to infinity in such a way that their co-ordinate size goes uniformly to zero we recover the area in the limitN →∞.

To quantize we take advantage of the fact that in each cellE [SI , τi ]E [SI , τj ]η

ij is a positive self adjoint operator on H (it thenhas a well defined square root).


Area operator

The action of this operator on an element of Hα for a fixed graph isstraightforward to obtain. The idea is to refine the partition sothat every elementary cell has, at most, one transverse intersectionwith the graph. In this case the only terms contributing come fromthe SI that intersect α. Once this point is reacher further refinementsdo nothing.The resulting operator can be written as the following sum over thevertices of α that lie on S

AS,α = 4πγ`2P

∑v

(−4S,v ,α)1/2

where 4S ,v ,α is an operator given by a quadratic combination ofoperators associated with each edge leaving or arriving at the v ’sappearing in the previous sum.


Area operator: Comments

The previous expression is defined on a Hilbert space Hα for a fixedgraph. In order to see if it is defined on the whole Hilbert space H onehas to check some consistency requirements related to the fact thata function may be cylindrical w.r.t. different graphs. It is possible toprove that this is always possible.The previous operator can be extended as a self-adjoint operator tothe full Hilbert space H.

It is SU(2) invariant and diff-covariant.

The eigenvalues of the area operator are given in general by finitesums of the form

4πγ`2P

∑α∩S

[2j (u)(j (u) + 1) + 2j (d)(j (d) + 1)− j (u+d)(j (u+d)+1)]1/2

where the j (u), j (d), and j (u+d) are half integers (eigenvalues of theangular momentum operator for the edges in the expression of thearea operator subject to some inequality constraints).


Area operator: Comments

It is possible to obtain simple expressions for the area operator if werestrict ourselves to the (internal) gauge invariant subspace of H.

For example, if the intersections of α and S are just 2-degree verticesthe spectrum of the area operator takes the simple form

8πγ`2P

∑I

(jI (jI + 1))1/2

Notice that the Immirzi parameter γ appears in all these expres-sions. This means that it is not an irrelevant arbitrariness in thedefinition of some canonical transformations but, rather, shows up in(eventually) observable magnitudes such as areas.

A final pictorial interpretation of this is the following. The “quantumexcitations of geometry” are 1-dimensional and carry a flux of area.Any time a graph pierces a surface it endows it with a quantum ofarea. This picture is completely different from the one in Fock spaces(quantum excitations as particles).


Volume operator

The strategy to define it is similar to the one for the area operator.

The volume of a 3-dim region B (a certain open subset of Σ) isclassically given by

VB =

∫B

√h

This can be expressed in terms of the triad as

VB =

∫B

∣∣∣∣ 1

3!εabcε

ijk E ai E

bj E

ck

∣∣∣∣1/2

.

As before we want to rewrite this expression in terms of flux operators.To this end we divide B in cells of a small coordinate volume. Ineach cell we introduce three surfaces such that each of them splitsthe cell in two disjoint pieces. This defines the so called “internalregularization” (others are possible)


Volume operator

An approximate expression for the volume in terms of flux operatorsis then ∑

cells

∣∣∣∣(8πγ`2P)3

6εijkηabcE [Sa, τ i ]E [Sb, τ j ]E [Sc , τk ]

∣∣∣∣1/2

When the coordinate size of the cells goes to zero this gives thevolume of the region B.As in the case of the area operator we define a family of operatorsfor each graph α (satisfying similar consistency conditions).

The resulting operator is given by

VB,α := α∑v

∣∣∣∣∣(8πγ`2P)3

48

∑e1,e2,e3

εijkε(e1, e2, e3)J(v ,e1)i J

(v ,e2)j J

(v ,e3)k

∣∣∣∣∣1/2

where α is an undetermined constant and ε(e1, e2, e3) is the orienta-tion factor of the family of edges (e1, e2, e3).


Volume operator: Comments

The removal of the regulator (i.e. of the auxiliary partition used todefine the volume operator) is non-trivial now because the volumeoperator obtained by “just taking the limit” keeps some memory ofthe details of the partition. Nevertheless there are ways to handle thisissue.

The orientation function is zero if the tangent vectors to the edgese1, e2, e3 are linearly dependent at the point where they meet. Thismeans, in particular, that the volume operator is zero when actingon state vectors defined on planar graphs i.e. graphs such thatat each vertex the tangent vectors are contained in a plane.

The volume operator is SU(2) gauge invariant and diffeomorphismcovariant as the area operator. The total volume operator (i.e. VΣ)is diff-invariant.


Volume operator

The volume operator is zero also when acting on gauge invariantstates if the vertices are at most of degree 3 (tri-valent).

The eigenvalues of the volume operator are real and discrete.They are not known in general but can be computed in many inter-esting cases. In particular when the vertices are four-valent.The volume operator plays a central role in the implementationof the quantum constraints because the quantum version of thescalar constraint can be written by using Poisson brackets of thetotal volume operator and the basic canonical variables.Other regularizations are possible, for example the so called exter-nal regularization obtained by considering the faces of the cells usedin the approximation of the volume operator in the process of writingit in terms of the flux operators. The volume operators built by usingthese different approaches have different properties.


Applications: Black hole entropy

Black holes in LQG are modeled by considering general relativity in thepresence of isolated horizons. These are inner boundaries of the space-time where the induced metric satisfies some conditions:

Non expanding horizons ∆: null, 3-dim submanifolds of (M, gab)such that

1 They have the topology of S2 × R.2 The expansion of any null normal qab∇a`b vanishes.3 The Einstein field equations hold on ∆ with Tab satisfying the (mild)

condition that −T ab`

b is a future directed, causal vector. (This istrue for minimally coupled matter fields).

This definition guarantees that the area of ∆ is constant, there isno matter flux through ∆, and the horizon geometry (qab,D) istime-independent.Isolated horizons: Non-expanding horizons with an –essentiallyunique– null normal ` that is a symmetry of the horizon geome-try (L`qab = 0 , [L`,D] = 0).



4 Isolated horizons capture important features of BH physics (forinstance, 0th and 1st laws of BH thermodynamics).

5 They are appropriate to model black holes in equilibrium withoutrequiring that the exterior geometry be stationary.

6 They can model rotating black holes or black holes with distortedhorizons.

7 A “reduction” of general relativity consisting of spacetimes with iso-lated horizons as inner boundaries admits a Hamiltonian descrip-tion. This is a key first step towards quantization.

8 This idea is similar in spirit to the study of the quantization ofmini and midisuperspace models. A subset of the gravitational fieldconfigurations is selected by imposing restrictions on the metrics.

Mini and midisuperspaces Symmetry requirement on the metrics.

Black holes The allowed metrics must have an isolated horizonthat is also an inner boundary of spacetime.



Quantization is carried out by combining LQG methods with ideasborrowed form Chern-Simons models [H = Hbulk ⊗Hsurface].

The entropy is obtained from a maximally mixed (thermal) densitymatrix (tracing over bulk states).

In all the different proposals the entropy computations can be phrasedas concrete combinatorial problems in terms on the spin labelsassociated with the edges of spin networks piercing the BH horizon.

The Bekenstein-Hawking law (S = A/4) is reproduced in all ofthem, furthermore, logarithmic corrections can be found. A caveat:in some old proposals γ must be chosen to get 1/4 factor!

Approach γ Logarithmic correction

ABCK-DL γDL = 0.237 · · · −1/2 log(a/`2P)

GM γGM = 0.274 · · · −1/2 log(a/`2P)

ENP γENP = γGM −3/2 log(a/`2P)



0 5 10 150

2

4

6

8

10


Applications: LQC

1 Loop quantum cosmology (LQC) is the quantization of cosmologi-cal models (hence, most of the physical degrees of freedom are frozenfrom the start) using LQG-inspired methods, in particular, polymerquantizations where the Hilbert space shares some features withthose of full LQG (non-separability and the existence of an area gapa0 —the difference between the two lowest area eignevalues).

2 Discreteness of geometry plays a fundamental role close to the BigBang. One of the consequences of this is the fact that differenceequations (discrete) play a fundamental role in the description of thequantum dynamics of the system.

3 In FLRW models, these effects can be incorporated in the dynamicsthrough the modified equation for the scale factor(a

a

)2

=8πGN

3ρ

(1− ρ

ρ0

)− k

a2+

Λ

3,with ρc =

(8πGN

3γ2a0

)−1


Applications: LQC

4 When the equation is integrated one sees that the Big-Bang is re-placed by a Big-Bounce in which, after a contracting phase, theuniverse reaches a minimum size and starts expanding again.

5 Quantum effects act as the source of a quantum repulsion not unlikethe one responsible for the stability of compact stelar objects such aswhite dwarfs and neutron stars.

6 Other interesting results that have been found in this setting arerelated to inflation. By studying the effective equations for a it ispossible to see that there is a natural inflationary regime in thesemodels and also that, in the presence of scalar fields, inflation isgeneric (“no inflaton potential engineering” i.e. no fine tuning ofthe initial conditions is required to have an inflationary epoch withthe right duration).

7 In order to have a fully fledged cosmological model one has to dealwith all the gravitational (and matter) degrees of freedom.

8 There are many interpretational issues...LQG: introduction & status J. Fernando Barbero G. (IEM-CSIC) Graz, March 30-31, 2017 43/51

Applications: LQC

Bounce of the wave function of the universe Ψ for a FLRW spacetime coupled toa scalar field Φ. v is the volume of the universe in Planck units.


Applications: LQC


The rest of LQG: Spin foams

jk

l mn

...

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A covariant point of view similar in spirit tothe Feynman path integral approach.Feynman path integrals become discretecombinatorial sums over spin network am-plitudes.They represent transition amplitudes froma spin network state at some time to anotherspin network state at a later time.Faces carry spin labels and edges carry in-tertwiner [invariant SU(2) tensors] labels.

Boundary graphs represent space whereasthe spin network itself represents space-time.The relation with the canonical approachmust be properly understood.


The rest of LQG: Group field theory

A development of tensor models (themselves derived from matrixmodels, which are relevant in string theory).

They are tensor models containing algebraic data encoding quan-tum geometric information (discrete gravity d.o.f.)

They have relations to both LQG and CDT.For instance, spin networks can be used to span the Hilbert spaces ofGFT models.Spacetime becomes similar to a condensed matter system.

GFT’s are QFT’s for spin networks endowed with covariant dynamics


The rest of LQG: The continuum limit

The construction of a suitable continuum limit is necessary to com-ple LQG.Use iterative coarse graining methods to construct physical states.This leads to an understanding of the dynamics of LQG at differentscales.In a sense, this idea replaces the renormalization flow of the stan-dard QFT’s (formulated with the help of a geometric background).

It is necessary to understand the role of diff-invariance for discretesystems and how it reflects on the continumm limit.The starting point is the inductive definition of the Hilbert space ofLQG (the way to get the full Hilbert space H from the Hα).


Conclusions and comments

Where do we stand today?1 The framework provided by the Ashtekar variables provides a tanta-

lizing point of view that brings gravity close to Yang-Mills theories.2 The quantization of the model has provided interesting mathematical

constructions (geometric operators). The discreteness of the areaspectrum gives a glimpse of the microstructure of space at thePlank length.

3 There is a nice interplay between the covariant picture (spin foams)and the canonical one (only partially understood).

4 Despite the fact that both BH and LQC models are not yet completethey provide independent evidence on the soundness of the approach:

The Bekenstein-Hawking law is recovered. Also logarithmic cor-rections of the expected type are found.

The LQC models offer a very appealing picture of the physics ofspace-time close to the Big Bang.


Conclusions and comments

What is left to do?1 The semiclassical and continuum limits: recovering general rel-

ativity.2 The problem with the Hamiltonian constraint: consistent quantum

dynamics. Anomalies and full quantum space-time covariance.3 Can the spin foam models be derived from the canonical approach?4 Extending the results provided by the present models (black holes

and LQC) to full general relativity.5 How is the problem of the non-renormalizability of general relativity

explained away?6 Does LQG have anything to say about the conceptual issues in

quantum gravity?7 Physical predictions.


Bibliography

Introductory texts

R. Gambini and J. Pullin, A first course in Loop Quantum Gravity.Oxford University Press.C. Rovelli and F. Vidotto, Covariant Loop Quantum Gravity: AnElementary Introduction to Quantum Gravity and Spinfoam Theory,Cambridge Monographs on Mathematical Physics.

Intermediate level reviews

A. Ashtekar and J. Lewandowski, Background Independent QuantumGravity: A Status Report, Class. Quant. Grav. 21: R53, 2004.A. Ashtekar and J. Pullin Eds. 100 years of general relativity, Vol.4.Loop quantum gravity, the first 30 years, World Scientific.

Advanced texts

T. Thiemann, Modern Canonical Quantum General Relativity, Cam-bridge Monographs on Mathematical Physics.


j. fernando barbero g. instituto de estructura de la...

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