Loop quantum gravity and black hole singularity

0.61 MB·English

by  Leonardo Modesto

Loop quantum gravity and black hole singularity
Leonardo Modesto
Department of Physics, Bologna University V.Irnerio 46, I-40126 Bologna & INFNBologna, EU
Abstract
Inthispaperwesummarizeloop quantum gravity(LQG)andweshowhowideasdevelopedin
LQGcan solve the black hole singularity problem when applied to a minisuperspace model.
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0
0
Introduction
2

n
Quantumgravityisthetheorybywhichwetrytoreconcilegeneralrelativityandquantummechanics.
a
Becauseingeneralrelativitythe space-timeisdynamical,itisnotpossibletostudyotherinteractions
J
on a fixed background. The theory called “loop quantum gravity” (LQG) [1] is the most widespread
5
2 nowadays and it is one of the non perturbative and background independent approaches to quantum
gravity (another non perturbative approach to quantum gravity is called asymptotic safety quantum

1 gravity [2]). LQG is a quantum geometricalfundamental theory that reconciles generalrelativity and
v
quantum mechanics at the Planck scale. The main problem nowadaysis to connect this fundamental
9
theory with standard model of particle physics and in particular with the effective quantum field
3
2 theory. In the last two years great progresses has been done to connect LQG with the low energy
1 physics by the general boundary approach [3], [4]. Using this formalism it has been possible to
0 calculate the graviton propagator in four [5], [6] and three dimensions [7]. In three dimensions it has
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been showed that a noncommuative field theory can be obtained from spinfoam models [8]. Similar
0
efforts in four dimension are going in progress [9]. Algebraic quantum gravity, a theory inspired by
/
h LQG, contains quantum field theory on curved space-time as low energy limit [10]. About an unified
t
- theoryofparticlephysicsandgravityauthorsinarecentpaper[11]haveshowedthatspinfoammodels
p
[1], including loop quantum gravity, are also unified theories, in which matter degrees of freedom are
e
automatically included and a complete classification of the standard model spectrum is realized.
h
: Early universe and black holes are other interesting places for testing the validity of LQG. In the
v
past years applications of LQG ideas to minisuperspace models lead to some interesting results in
i
X thosefields. Inparticularithasbeenshowedincosmology[12],[13]andrecentlyinblackholephysics
r [14], [15], [16], [17] that it is possible to solve the cosmologicalsingularity problemand the black hole
a
singularity problem by using tools and ideas developed in full LQG. Other recent results concern a
semiclassical analysis of the black hole interior [18] and the evaporation process [19].
We can summarize the “loop quantum gravity program” in the following researchlines
the first one dedicated to obtain quantum field theory from the fundamental theory rigorously
•
defined;
the second one dedicated to apply LQG to cosmology and black holes where extreme energy
•
conditions need to know a quantum gravity theory.
The paper is organized as follow. In the first section we briefly recall loop quantum gravity at
kinematical and dynamical level. In the second section we recall a simplified model [14] showing how
1
quantum gravity solves the black hole singularity problem. In the third section we summarize “loop
quantumblackhole”(LQBH)[17]. ThisisaminisuperspacemodelinspiredbyLQGwherewequantize
theKantowski-Sachsspace-timewithoutapproximations. Thismodelisusefultounderstandtheblack
holephysicsnearthesingularitybecausethespace-timeinsidetheeventhorizonisofKantowski-Sachs
type.
1 Loop quantum gravity in a nutshell
In this section we recallthe structure of the theory introducing the Ashtekar’s formulation of general
relativity [20], the kinematical Hilbert space,quantum geometry and quantum dynamics.
1.1 Canonical gravity in Ashtekar variables
The classical starting point of LQG [1] is the Hamiltonian formulation of general relativity. In ADM
Hamiltonian formulationofthe Einsteintheory, the fundamental variablesare the three-metric q of
ab
the spatial section Σ of a foliation of the four-dimensional manifold = R Σ, and the extrinsic
M ∼ ×
curvature K . In LQG the fundamental variables are the Ashtekar variables: they consist on an
ab
SU(2) connection Ai and the electric field Ea, where a,b,c, = 1,2,3 are tensorial indices on the
a i ···
spatial section and i,j,k, =1,2,3 are indices in the su(2) algebra. The density weighted triad Ea
··· i
is related to the triad ei by the relation Ea = 1ǫabcǫ ejek. The metric is related to the triad by
a i 2 ijk b c
q =ei ejδ . Equivalently,
ab a b ij
det(q)qab =EaEbδij. (1)
i j
The rest of the relation between the vapriables (Ai,Ea) and the ADM variables (q ,K ) is given by
a i ab ab
Ai =Γi +γK Ebδij (2)
a a ab j
where γ is the Immirzi parameter and Γi is the spin connection of the triad, namely the solution of
a
Cartan’s equation: ∂ ei +ǫi Γj ek =0.
[a b] jk [a b]
The action is
1
S = dt d3x 2Tr(EaA˙ ) N Na Ni , (3)
a a i
κγ − − H− H − G
Z ZΣ h i
where Na is the shift vector, N is the lapse function and Ni is the Lagrangemultiplier for the Gauss
constraint . WehaveintroducedalsothenotationE =Ea∂ =Eaτi∂ andA=A dxa =Aiτidxa.
Gi a i a a a
The functions , and arerespectivelytheHamiltonian, diffeomorphismandGaussconstraints,
a i
H H G
given by
(Ea,Ai)= 4e 1Tr F EaEb 2e 1(1+γ2)EaEbKi Kj
H i a − − ab − − i j [a b]
(Ea,Ai)=EaFj (cid:0)(1+γ2)K(cid:1)iG
Hb i a j ab− a i
(Ea,Ai)=∂ Ea+ǫk AjEa, (4)
Gi i a a i ij a k
wherethecurvaturefieldstrengthisF =∂ A ∂ A +[A ,A ]ande=det(ei). Theconstraints(4)
ab a b− b a a b a
arerespectivelygeneratorsforthefoliationreparametrization,fortheΣsurfacereparametrizationand
for the gaugetransformations. The symplectic structure for the Ashtekar Hamiltonian formulationof
general relativity is
Ea(x),Ai(y) =κγδaδiδ(x,y), Ea(x),Eb(y) = Aj(x),Ai(y) =0. (5)
j b b j j i a b
General rel(cid:8)ativity in met(cid:9)ric formulation is defin(cid:8)ed by the Ein(cid:9)stein(cid:8)equations G(cid:9) = 8πG T . The
µν N µν
Ashtekar Hamiltonian formulation of general relativity is instead defined by the constraints H = 0,
H =0, G =0 and by the Hamilton equations of motion: A˙i = Ai, and E˙a = Ea, .
a i a { a H} i { i H}
2
1.2 The Dirac program for quantum gravity
The general strategy to quantize a system with constraints was introduced by Dirac. The program
consist on :
1. find a representation of the phase space variables of the theory as operators in an auxiliary
kinematical Hilbert space H satisfying the standard commutation relations, i.e., ,
kin
i/~[ , ]; { } →
−
2. promote the constraints to (self-adjoint) operators in H . For gravity we must quantize a set
kin
ofsevenconstraints (A,E), (A,E),and (A,E)andwemustsolvethequantumEinstein’s
i a
G H H
equations (for γ =i)
ˆ ψ = 4e 1Tr F EaEb + ψ =0,
− ab M
H| i − H | i
ˆ ψ =(cid:2) EaFj +(cid:0) ψ (cid:1)=0, (cid:3)
Hb| i j ab HMb | i
ˆ ψ = h∂ Ea+ǫk AjEia+ ψ =0 (6)
Gi| i a i ij a k GMi | i
We will consider pure gravity then(cid:2)the matter constraints a(cid:3)re identically zero.
3. introduce an inner product defining the physical Hilbert space H .
phys
1.3 Kinematical Hilbert space
The fundamental ingredient of LQG is the holonomy of the Ashtekar connection along a path e,
h [A] = P exp A SU(2). Given two oriented paths e and e such that the end point of e
e − e ∈ 1 2 1
coincides with the starting point of e so that we can define e = e e we have the composition rule
2 1 2
R
h [A]=h [A]h [A]. By a gauge transformation the holonomy transforms as
e e1 e2
h′e[A]=g(x(0)) he[A] g−1(x(1)), (7)
and by a Diffeomorfism of the three dimensional variety φ Diff(Σ) we have
∈
he[φ∗A]=hφ−1(e)[A], (8)
where φ A denotes the action of φ on the connection. In other words, transforming the connection
∗
with a diffeomorphism is equivalent to simply moving the path with φ 1.
−
We introduce now the space of cylindrical functions (Cyl ) where γ denotes a general graph.
γ
A graph γ is defined as a collection of paths e Σ (e stands for edge) meeting at most at their
⊂
endpoints. If N is the number of paths or edges of the graph and e , for i= 1, N , are the edges
e i e
of the corresponding graph γ a cylindrical function is an application f :SU(2)Ne··· C, defined by
→
ψ [A]:=f(h [A],h [A], h [A]). (9)
γ,f e1 e2 ··· eNe
Twoparticularexamplesofcylindricalfunctions arethe holonomyarounda loop,W [A]:=Tr[h [A]]
γ γ
and the three edges function Θ1,1/2,1/2[A] = 1D(h [A])ij1/2D(h [A]) 1/2D(h [A]) fABCD,
e1 e2 e3 e1 e2 AB e3 CD ij
wherejD(h )istheSU(2)represe∪nta∪tionfortheholonomyalongthepathe andfABCD arecomplex
ei i ij
coefficients. The algebra of generalized connections is given by Cyl= Cyl .
∪γ γ
We introduce now the space of spin networks states. We label the set of edges e γ with spins
⊂
j . Toeachnode n γ oneassignsaninvarianttensor,calledintertwiner,ι ,inthetensorproduct
e n
{ } ⊂
ofrepresentationslabellingtheedgesconvergingatthecorrespondingnode. Thespinnetworkfunction
is defined
s [A]= ι jeD(h [A]), (10)
γ,{je},{ιn} n e
n γ e γ
O⊂ O⊂
3
wheretheindicesofrepresentationmatricesandinvarianttensorsareimplicittosimplifythenotation.
An example of spin network state is
Θ1,1/2,1/2[A]= 1D(h [A])ij1/2D(h [A]) 1/2D(h [A]) σACσBD, (11)
e1 e2 e3 e1 e2 AB e3 CD i j
∪ ∪
where for the particular representations converging to the two three-valent nodes of the graph the
intertwiner tensor is the Pauli matrix. The spin network states are gauge invariant because for any
node of the graphwe haveinvarianttensors(intertwiners),then onthe spinnetworkstates the Gauss
constraint is solved as asked from the Dirac program of the previous subsection.
To complete the Hilbert space definition we must introduce an inner product. The scalar product
is defined by the Ashtekar-Lewandowskimeasure
<ψ ,ψ >=µ (ψ ψ )= dh f(h ,...h )g(h ,...h ), (12)
γ,f γ′,g AL γ,f γ′,g e e1 eNe e1 eNe
Z e Γ
⊂Yγγ′
where we use Dirac notation and f(h ,...h ), g(h ,...h ) are cylindrical functions; Γ is
e1 eNe e1 eNe γγ′
any graph such that both γ Γ and γ Γ ; dh is the Haar measure of SU(2). The scalar
γγ ′ γγ e
⊂ ′ ⊂ ′
product (12) is non zero only if the two cylindrical functions have support on the same graph. The
kinematical Hilbert space H is the Cauchy completion of the space of cylindrical functions Cyl in
kin
the Ashtekar-Lewandowski measure. In other words, in addition to cylindrical functions we add to
H the limits of all the Cauchy convergent sequences in the norm defined by the inner product :
kin
ψ = ∞n=1anψn, ||ψ||2 = ∞n=1|an|2||ψn||2.
We complete the construction of the theory at kinematical level solving the diffeomorphism con-
P P
straint. Given ψ Cyl the finite action of a Diff. transformation is implemented by an unitary
γ,f
∈
operator such that
UD
UD[φ]ψγ,f[A]=ψφ−1γ,f[A]. (13)
The states invariant under Diff. transformations satisfy [φ]ψ = ψ and are distributional states in
the dual space of H , ψ Cyl⋆ UD
kin
∈
([ψ ] = <ψ [φ]= <ψ , (14)
γ,f γ,f φγ,f
| |UD |
φ∈DXiff(Σ) φ∈DXiff(Σ)
wherethesumisoveralldiffeomorphismswhichmodifiedthespinnetwork. Thebracketsin([ψ ] de-
γ,f
|
note thatthe distributionalstate depends onlyonthe equivalenceclass[ψ ]underdiffeomorphisms.
γ,f
Clearly we have ([ψ ] [α]=([ψ ] α Diff(Σ).
γ,f γ,f
|UD | ∀ ∈
We conclude that the Dirac’s programat kinematical level is realized by the Gelfand triple
SU(2) Diff.
Cyl⊂Hkin ⊂Cyl∗ → Cyl0 ⊂Hkin ⊂Cyl∗0 → HDiff ⊂Cyl∗, (15)
where Cyl is the subspace of cylindrical functions invariant under SU(2).
0
At quantum level the phase space variables operators are represented on the spin network space
by the holonomy operator hˆ [A] that acts multiplicatively on the states and by the smearing of the
e
triad Ea on a two dimensional surface S Σ
i ∈
∂xa ∂xb ∂xa ∂xb δ
Eˆ[S,α]= dσ1dσ2 αiEˆaǫ = i~κγ dσ1dσ2 αi ǫ , (16)
∂σ1∂σ2 i abc − ∂σ1∂σ2 δAi abc
Z Z c
S S
where αi is the smearing function with values on the Lie algebra of SU(2). The action of Eˆ[S,α] on
the spin network states can be calculated using
Eˆ[S,α]h (A)= il2γαih (A)τ h (A), e=e e ,
e − P e1 i e2 1∪ 2
δ
Eˆa(x)h (A)= il2 h (A)= dse˙a(s)δ3(e(s),x)h (A)τ h (A), (17)
i e − PδAi e e1 i e2
a Z
4
Figure1: Inthispictureweshowthespinnetworkphysicalmeaning. Thegraphontheleftrepresentsa
particularspinnetwork. Inthecenterwerepresentthesamespinnetworkandthedualdecomposition
ofthespacesectioninchunkofspace. Inthelastpictureontherightweconsideranotherspinnetwork
and a particular dual volume region. The yellow region is a chunk of space with volume eigenvalues
related to the red intertwiners and area eigenvalues given by the SU(2) representations associated to
the green edges.
where e˙a(s) is tangent to the curve e(s) in the graph γ. The pair (hˆ [A],Eˆ[S,α]) realizes the first
e
point of the Dirac’s program.
1.4 Quantum geometry and dynamics
WearegoingtogiveaphysicalinterpretationoftheHilbertspacepreviouslyintroduced. Weconsider
the spatialsectionΣ of the space-time and we study the spectrum of the area S and volume R in the
section Σ [21]. We define the area of a surface S as the limit of a Riemann sum
N
A = lim AN , AN = E (S )Ei(S ) (18)
S S S i I I
N
→∞ I=1
Xp
where N is the number of cells. The quantum area operator is A = lim AN. Using (17) we
S N S
calculate the action E (S )Ei(S )jD(h [A]) = (l2γ)2(j(j +1))jD(h [A])→∞. The area spectrum
i I I e mn P e mn
for spin network without edges and nodes on the surface is b b
b b
Aˆ γ,j ,ι =l2γ j (j +1)γ,j ,ι , (19)
S| e ni P p p | e ni
pX∩Sq
where j are the representations on the edges that cross the surface S. Now we consider a region R
p
with a number n of nodes inside. The spectrum of the volume operator for the region R is
VˆR|γ,je,ιni=lP3γ3/2 w(ιn)(jn)|γ,je,ιni. (20)
nX⊂Rq
We have all the ingredients to give a physical interpretation of the Hilbert space. The states have
support on graphs that are a collection of nodes and edges converging in the nodes. The dual of a
spin network corresponds to a discretization of the three dimensional surface Σ. The dual of a set of
edges is the 2-dimensional surface crossed by the links and the dual of a set of nodes is the volume
chunk contained nodes (see Fig.1).
We must now implement the Hamiltonian constraint on the Hilbert space. The Euclidean part of
the constraint SE(N)= d3xN(x) (Ea,Ai),
Σ H i a
R EaEb
SE(N)= d3x N(x) i j ǫij Fk. (21)
det(E) k ab
Z
Σ
p
5
Using the Thiemann’s trick [1] we can express the inverse of det(E) by
EbEc 4 p
i j ǫijkǫ = Ak,V , (22)
det(E) abc κγ a
(cid:8) (cid:9)
and the Hamiltonian constraintpis
4
SE(N)= dx3 N(x) ǫabcδ Fi Aj,V . (23)
κγ ij ab c
Z
Σ (cid:8) (cid:9)
Now we define the Hamiltonianconstraintin terms ofholonomies. Givenaninfinitesimal loopα on
ab
the ab-plane in the surface Σ with coordinate area ǫ2, we can define Fi in terms of holonomies by
ab
h [A] h 1[A]=ǫ2Fi τ + (ǫ4) and h 1[A] h [A],V =ǫ Ai,V + (ǫ2) (e is a path along
αab − −αab ab i O −ea { ea } a O a
the a-coordinateofcoordinatelength ǫ). With these ingredientsthe quantumconstraintcanformally
(cid:8) (cid:9)
be written
4
SE(N)= κγ ǫlim0 NI ǫabcTr (hαIab[A]−h−αIa1b[A])h−eIc1[A] heIc[A],V , (24)
→ XI h h ii
b b b b b
where we have replaced the integral by a Riemann sum over cells of coordinate volume ǫ3. It is easy
toseethattheregularizedquantumscalarconstraintactsonlyonspinnetworknodes,becausein(21)
F and EaEb are respectively antisymmetric and symmetric in indexes on spin network states. In
ab
fact EaEbψ e˙ae˙bψ (this is a consequence of (17)). The action of (24) on spin network states
γ,f γ ,f
∼ ′ ′
is S (N)ψ = N Sn ψ , where Sn acts only on the node n γ and N is the value of the
ǫ γ,f nγ n ǫ γ,f ǫ ⊂ n
lapse N(x) at the node. The scalar constraint modifies spin networks by creating new exceptional
P
linkbs around nodes. ThebEuclidean constbraint action on 4-valent nodes is [1]
k k k k
Sǫn j l = Sjklm,opq j p q l + Sjlmk,opq j p q l + Sjmkl,opq j poq l .
(cid:12)(cid:12) m + Xop (cid:12)(cid:12) o m + Xop (cid:12)(cid:12) o m + Xop (cid:12)(cid:12) m +
b (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (25)
(cid:12) (cid:12) (cid:12) (cid:12)
The only dependence on the regularization parameter ǫ is in the position of the extra edges in the
resulting spin network states, then the limit ǫ 0 can be defined on diffeomorphism invariant states
→
in H . The key property is that in the diffeomorphism invariant context the position of the new
Diff
linkisirrelevant. Therefore,givenadiffeomorphisminvariantstate([φ] H Cyl⋆,thequantity
Diff
|∈ ⊂
(φS (N)ψ is well defined and independent of ǫ.
ǫ
| | i
The operator (24) defines the dynamics and an unitary implementation of the constraint SE(N)
givebs the evolution amplitude from a spin network s to a new spin network s′. Introducing the
projector operator P = D[N] exp(i N(x)SE(x)) we can characterizes the solutions of quantum
Einstein equations by P s , s H . The matrix elements of P define the physical scalar product,
kin
|Ri ∀| i∈ R
W(s,s) := ss := sP s . The amplitude W(s,s) solves the Hamiltonian constraint in the
′ ′ phys ′ ′
h | i h | | i
following sense. If Ψ := P s we have that SE(N)Ψ = 0, but sΨ := Ψ (s) =
phys ′ phys phys phys
| i | i | i h | i
W(s,s), then we obtain
′
sSE(N)Ψ = sSE(N)s s Ψ =
phys ′′ ′′ phys
h | | i h | | ih | i
Xs′′
= SE(N) Ψ (s )= SE(N) W(s ,s)=0. (26)
ss phys ′′ ss ′′ ′
′′ ′′
s s
X′′ X′′
Relation (26) shows the amplitude W(s,s) is in the Hamiltonian constraint kernel. W(s,s) realizes
′ ′
the Dirac’s programbecause corresponds to the finite implementation of the scalar constraint on the
kinematical states and defines a class of models called spinfoam models [1].
6
2 Avoidance black hole singularity in quantum gravity
In this section we study the black hole system inside the event horizon in ADM variables considering
a simplified minisuperspace model [14] and in particular using the fundamental ideas suggested by
full LQG and introduced in the first section. The simplification consist on a semiclassical condition
which reduce the phase space from four to two dimensions. This is an approximate model but it is
useful to understand the ideas before to quantize the Kantowski-Sachssystem in Ashtekar varables.
2.1 Classical theory
Consider the Schwarzschildsolution inside the event horizon; the metric is homogeneous and it reads
dT2 2MG
ds2 = + N 1 dr2+T2(sin2θdφ2+dθ2). (27)
− 2MGN 1 T −
T − (cid:18) (cid:19)
This metric is a particul(cid:0)ar represen(cid:1)tative of the Kantowski-Sachs class [25]
ds2 = dt2+a(t)2dr2+b(t)2(sin2θdφ2+dθ2). (28)
−
Introducing (28) in the Einstein-Hilber action we obtain S = R dt ab˙2 +2a˙b˙b a (where
−2GN −
R = dr is a cut-off on the radial cell) [14]. We introduce in theR achtion the classicalirelation
a2 =2MG /b(t) 1, obtaining
N
R −
S = R dt √b 1 b −12 b˙2+ √2MGN 1 b 21 . (29)
2GN Z "√2MGN(cid:16) − 2MGN(cid:17) √b (cid:16) − 2MGN(cid:17) #
1/2
The momentum conjugates to the variable b(t) is p= R√b 1 b − b˙, and the Hamil-
GN√2MGN − 2MGN
tonian constraint, by Legendre transform, is (cid:16) (cid:17)
G p2 R √2MG b 1/2
H =pb˙ L= N N 1 . (30)
− (cid:18) 2R − 2GN(cid:19)" √b (cid:18) − 2MGN(cid:19) #
Weintroduceafurtherapproximation. Inquantumtheory,wewillbeinterestintheregionofthescale
the Planck length l around the singularity. We assume that the Schwarzschild radius r = 2MG
P s N
is much larger than this scale.In this approximation we can write 1 b/2MG 1 and H becomes
N
− ∼
G p2 R √2MG
N N
H = .
(cid:18) 2R − 2GN(cid:19) √b
In the same approximation the volume of the space section is V =4πR√2MG b3/2 :=l b3/2.
N o
The canonical pair is given by b x and p, with Poisson brackets x,p = 1. We now assume
that x R. This choice it not corre≡ct classically, because for x = 0 we{hav}e the singularity, but it
∈
allows us to open the possibility that the situation be different in the quantum theory. We introduce
an algebra of classical observables and we write the quantities of physical interest in terms of those
variables. We are motivated by loop quantum gravity to use the fundamental variables x and
8πG δ
N
U (p) exp ip (31)
δ
≡ L
(cid:18) (cid:19)
where δ is a real parameter (see next paragraph for a rigorous mathematical definition of δ) and L
fixes the unit of length. The operator(31) canbe seen as the analogof the holonomyvariable ofloop
quantum gravity.
7
A straightforwardcalculation gives
iδ
x,U (p) =8πG U (p),
δ N δ
{ } L
Uδ−1{Vn,Uδ}=l0nUδ−1{|x|32n,Uδ}=i8πGNl0n Lδ 32nsgn(x)|x|32n−1. (32)
These formulas allow us to express inverse powers of x in terms of a Poissonbracketbetween U and
δ
the volume V, following Thiemann’s trick [26]. For n=1/3 (32) gives
sgn(xx) =−(8πG2L)il1/3δ Uδ−1{V 13,Uδ}. (33)
| | N 0
We will use this relationin qupantum mechanics to define the physicaloperators. We are interestedto
the quantity 1/x because classically this quantity diverge for x 0 and produce the singularity.
| | | | →
WearealsointerestedtotheHamiltonianconstraintandthedynamicsandwewilluse(33)forwriting
the Hamiltonian.
2.2 Polymer quantization
In this section we recall the polymer representation[22] of the Weyl-Heisenberg algebra and we com-
parethisrepresentationwithfullLQG.ThepolymerrepresentationoftheWeyl-Heisenbergalgebrais
unitarilyinequivalent totheSchroedi-ngerrepresentation. NowweconstructtheHilbertspaceH .
Poly
First of all we define a graph γ as a finite number of points x on the real line R. We denote by
i
Cyl the vector space of function f(k) (f :R C) of the type{ }
γ
→
f(k)= fje−ixjk (34)
j
X
where k R, x R and f C and j runs over a finite number of integer (labelling the points of
j j
∈ ∈ ∈
the graph). We will call cylindrical with respect to the graph γ the function f(k) in Cyl . The real
γ
number k is the analog of the connections in loop quantum gravityand the plane wave e−ikxj can be
thought as the holonomy of the connection k along the graph x .
j
{ }
Now we consider all the possible graphs (the points and their number can vary from a graph to
another)andwe denote Cylthe infinite dimensionalvectorspaceoffunctions cylindricalwith respect
to some graph: Cyl= γCylγ. A basis in Cyl is givenby e−ikxj with he−ikxi|e−ikxji=δxi,xj. HPoly
is the Cauchy completion of Cyl or more succinctly H = L (R¯ ,dµ ), where R¯ is the
S Poly 2 Bohr 0 Bohr
Bohr-compactification of R and dµ is the Haar measure on R¯ .
0 Bohr
The Weyl-Heisenberg algebra is represented on H by the two unitary operators
Poly
Vˆ(λ)f(k)=f(k λ), Uˆ(δ)f(k)=eiδkf(k), (35)
−
where λ,δ R. In terms of eigenkets of Vˆ(λ) (we associate a ket xj with the basis elements e−ikxj)
∈ | i
we obtain
Vˆ(λ)x =eiλxj x , Uˆ(δ)x = x δ . (36)
j j j j
| i | i | i | − i
It is easy to verify that Vˆ(λ) is weakly continuous in λ, whence exists a self-adjoint operator xˆ such
that xˆx =x x [22], [24].
j j j
| i | i
The operator analogy between loop quantum gravity and polymer representationis the following:
the basic operator of loop quantum gravity, holonomies and electric field fluxes, are respectively
analogous to the operators Uˆ(δ) and xˆ with commutator [xˆ,Uˆ(δ)] = δUˆ(δ). The commutator is
−
paralleltothe commutatorbetweenelectricfields andholonomies. As,inthe polymerrepresentation,
8
the unitary operator Uˆ(δ) is well defined but the operator pˆdoesn’t exist, in loop quantum gravity
the holonomies operators are unitary represented self-adjoint operators but the connection operator
doesn’t exist. As xˆ, the electric flux operators are unbounded self-adjoint operators with discrete
eigenvalues.
AfterthisshortreviewonpolymerrepresentationoftheWeyl-Heisenbergalgebrawereturntoour
system.
2.3 Polymer black hole quantization
FollowingtheprevioussectionwequantizetheHamiltonianconstraintandtheinversevolumeoperator
in the Polymer representation of the Weyl-Heisenberg algebra. The operators are xˆ, acting on the
basis states according to
xˆµ =Lµµ , µν =δ (37)
µν
| i | i h | i
(we haveredefinedthe continuumeigenvaluesofthe positionoperatorofthe previoussectionx µ)
i
→
and the operator corresponding to the classical momentum function Uδ =ei8πGNδp/L. We define the
action of Uˆ on the basis states using the definition (37) and using a quantum analog of the Poisson
δ
bracket between x and U
δ
Uˆ µ = µ δ , [xˆ,Uˆ ]= δLUˆ . (38)
δ δ δ
| i | − i −
Using the standardquantizationprocedure[, ] i~ , ,the Poissonbracket(32)and(38)weobtain
the value of the length scale L=√8πG ~=l→. { }
N P
2.3.1 Avoidance black hole singularity and regular dynamics
We recall that the dynamics is all in the function b(t), which is equal to the the radial Schwarzschild
coordinate inside the horizon. The important point is that b(t=0)=0 and this is the Schwarzschild
singularity. We now show that the spectrum of the operator 1 does not diverge in quantum me-
b(t)
chanics and therefore there is no singularity in the quantum theory.
Usingtherelation(33),andpromotingthePoissonbracketstocommutators,weobtain(forδ =1)
the operator
x1 = 2πG1~l32 Uˆ−1 Vˆ13,Uˆ 2. (39)
|c| N 0 (cid:16) h i(cid:17)
The actionofthis operatoronthe basis states is (the volume operatoris diagonalonthe basis states,
Vˆ µ =l0 x 32 µ =l0 Lµ 23 µ )
| i | | | i | | | i
1 2 1 1 2
µ = µ 2 µ 1 2 µ . (40)
x | i πG ~ | | −| − | | i
|c| r N (cid:16) (cid:17)
We can now see that the spectrum is bounded from below and so we have not singularity in the
quantum theory. In fact the curvature invariant µνρσ =48M2G2 /x(t)6 is finite in quantum
RµνρσR N
mechanics in µ=0. The eigenvalue of the operator1/x for the state 0 correspondsto the classical
| | | i
singularity and in the quantum case it is 4/l2, which is the largest possible eigenvalue. For this
P
particular value the curvature invariant it is not infinity
\
\µνρσ 0 = 48M2G2N 0 = 384M2G2N 0 . (41)
RµνρσR | i x6 | i π3l6 | i
| | P
9
If we consider the ~ 0 limit we obtain the classicalsingularityso the result is a genuinely quantum
→
gravity effect). On the other hand, for µ the eigenvalues go to zero, which is the expected
| | → ∞
behavior of 1/x for large x.
| | | |
Now we study the quantization of the Hamiltonian constraint near the singularity, in the approx-
imation (2.1). There is no operator p in polymer quantum representationthat we have chosen, hence
we choose the following alternative representation for p2. Consider the classical expression
p2 = L2 lim 2−Uδ−Uδ−1 . (42)
(8πGN)2 δ→0(cid:18) δ2 (cid:19)
We havecangiveaphysicalinterpretationto δ asδ =l /L , whereL is the characteristicsize
p phys Phys
of the system. Using (42) we write the Hamiltonian constraint as
Hˆ = l1/3CL1/2 Uˆδ+Uˆδ−1−(2−C′)ˆI sgn(x) Uˆ−1 Vˆ13,Uˆ (43)
0 h i (cid:16) h i(cid:17)
where C = (8πGNL)55//22δG3NRl10/3~ and C′ = 8πlR2P2δ2. The action of Hˆ on the basis states is
Hˆ µ = (µ)[µ δ + µ+δ (2 )µ ],
′
| i CV | − i | i− −C | i
µ δ 1/2 µ1/2 for µ=0
(µ)= − | − | −| | 6 (44)
V δ 1/2 for µ=0
(cid:26) | |(cid:12) (cid:12)
(cid:12) (cid:12)
We now calculate the solutions of the the Hamiltonian constraint. The solutions are in the dual
space of H . A generic element of this space is ψ = ψ(µ) µ. The constraint equation
Poly h | µ h |
Hˆ ψ =0isnowinterpretedasanequationinthedualspace ψ Hˆ ;fromthis equationwecanderive
| i hP| †
a relation for the coefficients ψ(µ)
(µ+δ)ψ(µ+δ)+ (µ δ)ψ(µ δ) (2 ) (µ)ψ(µ)=0. (45)
′
V V − − − −C V
This relation determines the coefficients for the physical dual state. We can interpret this states as
describing the quantum spacetime near the singularity. From the difference equation (45) we obtain
physical states as combinations of a countable number of components of the form ψ(µ+nδ)µ+nδ
| i
(δ l /L 1); any component corresponds to a particular value of volume, so we can interpret
P Phys
∼ ∼
ψ(µ+δ) as the wave function describing the black hole near the singularity at the time µ+δ. A
solution of the Hamiltonian constraint corresponds to a linear combination of black hole states for
particular values of the volume or equivalently particular values of the time.
3 Loop quantum black hole
InthissectionwequantizetheKantowski-Sachsspace-timeinAshtekarvariablesandwithoutapprox-
imations [17].
3.1 Ashtekar variables for Kantowski-Sachs space-time
The Kantowski-Sachsspace-timeisasimplifiedversionofanhomogeneousbutanisotropicspacetime,
written in coordinates (t,r,θ,φ). An homogeneous but anisotropic space-time of spatial section Σ of
topology Σ=R S2 is characterizedby an invariant connection 1-form A of the form [27], [28]
∼ × [1]
A =A (t)τ dr+(A (t)τ +A (t)τ )dθ+(A (t)τ A (t)τ )sinθdφ+τ cosθdφ. (46)
[1] r 3 1 1 2 2 1 2 2 1 3
−
10