Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics
In this article we explore the origin of black hole thermodynamics using semiclassical states in loop quantum gravity. We re-examine the case of entropy using a density matrix for a coherent state and describe correlations across the horizon due to SU(2) intertwiners. We further show that Hawking ra...
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irk-123456789-1492172019-02-20T01:29:09Z Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics Dasgupta, A. In this article we explore the origin of black hole thermodynamics using semiclassical states in loop quantum gravity. We re-examine the case of entropy using a density matrix for a coherent state and describe correlations across the horizon due to SU(2) intertwiners. We further show that Hawking radiation is a consequence of a non-Hermitian term in the evolution operator, which is necessary for entropy production or depletion at the horizon. This non-unitary evolution is also rooted in formulations of irreversible physics. 2013 Article Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics / A. Dasgupta // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 43 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 83C57; 81Q35; 81T20 DOI: http://dx.doi.org/10.3842/SIGMA.2013.013 http://dspace.nbuv.gov.ua/handle/123456789/149217 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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In this article we explore the origin of black hole thermodynamics using semiclassical states in loop quantum gravity. We re-examine the case of entropy using a density matrix for a coherent state and describe correlations across the horizon due to SU(2) intertwiners. We further show that Hawking radiation is a consequence of a non-Hermitian term in the evolution operator, which is necessary for entropy production or depletion at the horizon. This non-unitary evolution is also rooted in formulations of irreversible physics. |
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Dasgupta, A. Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics Symmetry, Integrability and Geometry: Methods and Applications |
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Dasgupta, A. |
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Dasgupta, A. |
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Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics |
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Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics |
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Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics |
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Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics |
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Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics |
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semiclassical loop quantum gravity and black hole thermodynamics |
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Інститут математики НАН України |
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2013 |
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Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics / A. Dasgupta // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 43 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT dasguptaa semiclassicalloopquantumgravityandblackholethermodynamics |
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2025-07-12T21:06:28Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 013, 25 pages
Semiclassical Loop Quantum Gravity
and Black Hole Thermodynamics?
Arundhati DASGUPTA
University of Lethbridge, 4401 University Drive, Lethbridge T1K 7R8, Canada
E-mail: arundhati.dasgupta@uleth.ca
Received March 22, 2012, in final form February 05, 2013; Published online February 16, 2013
http://dx.doi.org/10.3842/SIGMA.2013.013
Abstract. In this article we explore the origin of black hole thermodynamics using semi-
classical states in loop quantum gravity. We re-examine the case of entropy using a density
matrix for a coherent state and describe correlations across the horizon due to SU(2) in-
tertwiners. We further show that Hawking radiation is a consequence of a non-Hermitian
term in the evolution operator, which is necessary for entropy production or depletion at the
horizon. This non-unitary evolution is also rooted in formulations of irreversible physics.
Key words: black holes; loop quantum gravity; coherent states; entanglement entropy
2010 Mathematics Subject Classification: 83C57; 81Q35; 81T20
1 Introduction
With new results in lattice gravity, loop quantum gravity (LQG) and various other approaches to
quantum gravity, there is hope that we might find the quantum of space-time [2,4,21,22,39,41].
Though unifying gravity with other forces of nature will remain a puzzle. General theory of
relativity is a theory of gravity and quantum field theory is the theory of particle dynamics
and both are experimentally verified. Yet when we describe quantum fields in gravitational
backgrounds, new phenomena arise which seem to signify the existence of new physics. Black
hole thermodynamics and Hawking radiation are particularly interesting phenomena. The laws
of black hole mechanics derived in [7] were the first indication that the black hole might be
a system similar to a thermodynamic ensemble. The area increasing theorem [26] led to the
conjecture that the entropy of the thermodynamic system is proportional to the area of the
horizon [9]. The area of the horizon divided by four Planck length squared today is known as
the Bekenstein–Hawking entropy. The use of quantum fields near the horizon then showed that
black holes emit radiation in thermal spectrum at a temperature given by the surface gravity
of the black hole [27], and the Planck’s constant naturally got included in the quantification of
entropy, temperature etc. This confirmed that there is a ‘quantisation’ of the space-time and
the microscopic structure will explain the origin of thermodynamic quantities. There are several
explanations of the nature of the microscopic structure of the black hole entropy [4, 39], and
the origin of thermodynamics, some of which will be reported in this volume. In this article
I shall discuss the use of ‘coherent states’ in LQG to derive the physics of the system [25, 42].
The use of coherent states is well known in quantum electrodynamics in particular for lasers,
and we will use similar coherent states for gravity. Once the coherent states are identified for
the black hole, the derivation of thermodynamics follows the standard entropy formulation of
correlated systems, part of which has been traced away [18]. A reduced density matrix is derived
for a region outside the horizon and the wavefunction inside the horizon is traced over. This
?This paper is a contribution to the Special Issue “Loop Quantum Gravity and Cosmology”. The full collection
is available at http://www.emis.de/journals/SIGMA/LQGC.html
mailto:arundhati.dasgupta@uleth.ca
http://dx.doi.org/10.3842/SIGMA.2013.013
http://www.emis.de/journals/SIGMA/LQGC.html
2 A. Dasgupta
reduced density matrix gives an entropy proportional to the area of the horizon. There are
semiclassical corrections to the Bekenstein–Hawking term, and the nature of the corrections is
a function of the discretisation or graph embeddings used to describe the system [19]. In this
article we describe the derivation of entropy using coherent states, and discuss briefly the nature
of correlations which arise due to the gauge invariant coherent states using SU(2) intertwiners.
These calculations are a preliminary indication of what a ‘physical coherent state’ might carry
as correlations at the horizon [15]. The correlations make the state inside the horizon ‘entangled’
with the state outside the horizon.
As the coherent state is defined in the canonical formalism where space-time is foliated with
constant time slices, the coherent state in discussion is defined in one time slice, which we
take as the initial time slice. We then use a semi-classical Hamiltonian to evolve the coherent
state in time. The derivation of time evolution is trivial if we restrict the evolution operator to
be unitary. Non-trivial evolution occurs if the classically forbidden regions behind the horizon
get exposed using a non-Hermitian term added to the Hamiltonian which facilitates entropy
production [17, 20]. The change in entropy of the black hole is carried away as Hawking flux
which escapes to infinity. This time evolution is irreversible in nature and is very similar to such
time evolutions in complex systems [34].
In the next section we briefly describe coherent states for gravity. We also review the concept
of entanglement entropy in this section. The third section describes the origin of entropy using
a density matrix. It includes a description of the tracing mechanism when the coherent state
is defined using intertwiners at vertices. In the fourth section we describe time evolution and
the relation of this entropy production process to physics of complex systems. The fifth section
discusses the results and the open problems yet to be solved using this formalism.
2 Coherent states in loop quantum gravity
The coherent states are useful semiclassical states in a quantum theory. In case of the simple
harmonic oscillator or in quantum electrodynamics, these can be formulated as the eigenstate
of the annihilation operator. Explicitly
â|ψ〉 = z|ψ〉,
where â is the annihilation operator for the positive energy modes of the theory and z is the
eigenvalue. These are usually Gaussians, e.g. for the harmonic oscillator
ψ =
1√
2π~
exp
(
−(x− z)2
2~
)
,
where z = x0 + ip0 is a point in the classical complexified phase space, ~ = h/2π, and h is
Planck’s constant. Thus the coherent state is a Gaussian ‘peaked’ on the classical phase space
point. In these coherent states the expectation values of operators are obtained as their classical
values and trajectories also evolve along classical paths, preserving the coherence.
The coherent states appear as the kernel of a transformation from the Hilbert space L2(R)
to the Segal–Bergmann representation of the wave functions or H(C) ∩ L2(C) [25]. Using the
definition of the coherent state as a kernel Hall [25] generalised the coherent states to obtain
those defined for a SU(2) Hilbert space. These appear as kernels in the transformation from
the SU(2) Hilbert space to the intersection of the Hibert space defined in the complexified
SU(2) phase space which incidentally is SL(2,C) space. These are therefore often named as
‘complexifier coherent states’. In a particular definition of the kernel, it is obtained as
K(h, g) = e
t
2
∆δ(h′, h)h′→g, (2.1)
Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics 3
where h is a SU(2) element, and g is a SL(2,C) element, ∆ is the Laplacian on the group
manifold, t is a parameter. The Hall coherent states can be directly applied to gravity as
written in terms of LQG variables gravity is a SU(2) theory [42].
We begin by writing gravity in terms of LQG variables. The phase space of loop quantum
gravity is obtained using a canonical formulation of gravity. The manifold is taken diffeomorphic
to Σ×R. The Σ slices foliate the space-time, and using the ADM metric, the dynamical variables
are identified as the intrinsic metric qab of the slice and the extrinsic curvature Kab with which
the slices are embedded in the space-time. The intrinsic metric can be written in terms of the
tangent space triads or soldering forms eIa. The phase space for LQG is again a redefinition of
these and described thus [21,22,41]
AIa = ΓIa − βKabe
Ib, EaI =
1
β
(det e)eaI , (2.2)
where eIa are the usual triads the ‘square root’ of the metric qab = eIae
I
b , Kab is the extrinsic
curvature, ΓIa are the associated spin connections to the triads, β is the one parameter ambiguity,
which is known as the Immirzi parameter. The quantisation of the Poisson algebra of these
variables is done by smearing the connection along one dimensional edges e of length δe of
a graph Γ to get holonomies he(A). The triads are smeared in a set of 2-surface decomposition
of the three dimensional spatial slice to get the corresponding momentum P Ie . (The momenta are
also labeled by the edges as each edge intersects the corresponding 2 surface once by construction,
or every edge has a corresponding unique two surface and vice versa.) The regularised LQG
variables are
he(A) = P exp
(∫
e
A
)
, P Ie =
∫
S
∗EI . (2.3)
The algebra is then represented in a kinematic ‘Hilbert space’, in which the physical con-
straints have been ‘formally’ realised. Once the phase space variables have been identified, one
can write a coherent state for these [25], i.e. minimum uncertainty states peaked at classical
values of he, P
I
e for one edge of the graph. The tensor product of these coherent states for
each edge of the graph gives the coherent state for the entire space-time. Using the Peter–Weyl
theorem for the delta function δ(h, h′) =
∑
j(2j + 1)χj
(
h′h−1
)
, where χj is the character of the
jth representation of SU(2), the complexifier coherent state (2.1) can be written as
ψ̃t(ge) =
∑
j
(2j + 1)e−tj(j+1)/2χj
(
geh
−1
e
)
. (2.4)
These are coherent states peaked at the classical holonomy with a width controlled by the
parameter t. A ‘momentum representation’ of this is obtained by defining a Fourier transform
which is such that the ‘momentum eigenstate’ |jmn〉 gives
〈h|jmn〉 = πj(h)mn, P̂e|jmn〉 = j(j + 1)~|jmn〉.
The |jmn〉 are thus the usual basis spin network states given by
√
2j + 1πj(he)mn (normalised
in the he basis), which is the mn element of the jth irreducible representation of the SU(2) mat-
rix he, and P̂e =
√
P̂ Ie P̂
I
e . Taking the scalar product of the configuration coherent state (2.4)
with the momentum eigenstate gives the momentum ‘coherent state’ as ψte = e−tj(j+1)/2πj(ge)jmn,
where πj(ge)mn is the jth irreducible representation of the SL(2,C) element ge. Thus the cohe-
rent state in the momentum representation is
|ψt(ge)〉 =
∑
jmn
e−tj(j+1)/2πj(ge)mn|jmn〉.
4 A. Dasgupta
In the above ge is a complexified classical phase space element eiT
IP Icle /2hcl
e , where P Icl
e and hcl
e
represent classical momenta and holonomy obtained by embedding the edge in the classical
metric and T I are SU(2) generators. The j is the quantum number of the SU(2) Casimir
operator in that representation, and m, n represent azimuthal quantum numbers which run
from −j, . . . , j. Similarly, (2j + 1)× (2j + 1)-dimensional representations of the 2× 2 matrix ge
are denoted as πj(ge)mn. The coherent state is precisely peaked with maximum probability at
the hcl
e for the variable he as well as the classical momentum P Icl
e for the variable P Ie . The
fluctuations about the classical value are controlled by the parameter t (the semiclassicality
parameter). This parameter is given by l2p/a where lp is Planck’s constant and a is a dimensional
constant which characterises the system. For the Schzwarzschild black hole a can be taken as
proportional to the area of the horizon. The coherent state for an entire slice can be obtained
by taking the tensor product of the coherent state for each edge which form a graph Γ,
ΨΓ =
∏
e
ψte. (2.5)
Thus we are considering a semiclassical state, which is a state such that expectation values
of operators are closest to their classical values. The information of the classical phase space
variables are encoded in the complexified SU(2) elements labeled as ge. The fluctuations over the
classical values are controlled by the semiclassical parameter t.
The density matrix which describes the entire black hole slice is obtained as
ρtotal = |ΨΓ〉〈ΨΓ|,
where |ΨΓ〉 is the coherent state wavefunction for the entire slice, a tensor product of coherent
state for each edge. These coherent states have been studied in various other forms [10, 23]. In
this paper we also discuss the SU(2) invariant coherent state described in [42] using intertwiners
at the vertices. We will describe the detailed derivation of the intertwiners for a particular
graph in the next section using this generalisation. We should mention that previously other
semiclassical states had been considered in the framework LQG known as weave states [5], but
none have been yet used to describe black hole entropy. Coherent states have been used to
describe quantum cosmology and FLRW universes [32]. Relatively recent reviews on approaches
to black hole entropy and corrections to black hole entropy in LQG can be found in [6, 14,33].
A review of black hole entropy in LQG and recent description of two dimensional black hole
evaporation can be found in [3]. Before we begin the discussion on derivation of entropy using
density matrices, we briefly describe the concept of ‘entanglement entropy’ and its use in quan-
tum field theory in curved space-time to describe entropy of black holes. Our derivation will be
totally ‘gravitational in origin’ defined using the LQG phase space variables.
3 Entanglement entropy
In quantum field theory, we use the definition of entropy due to von Neumann. It is defined
thus given a density matrix ρ
S = −Tr(ρ ln ρ).
If the state is pure the entropy is zero, if the state is mixed the entropy is obtained as non-zero.
This could be a system in which part of the wavefunction or the state of the system has been
traced away. The tracing can be done for systems which have a product Hilbert spaces: e.g.
H = HI ⊗HO. The HI is the Hilbert space which defines the ‘internal space’, HO is the outside
Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics 5
Hilbert space. The basis states also thus appear in the product structure |i〉 ⊗ |j〉. The states
are said to be entangled if the coefficients of the basis states do not factorise, e.g.
|ψ〉 =
∑
ij
dij |i〉|j〉
is entangled if
dij 6= didj . (3.1)
A density matrix defined thus
ρ = |ψ〉〈ψ|
can have a partial trace performed on it,
Tr ρ =
∑
d∗ijdji′ |i〉〈i′|,
and the entropy computed using the above. We will now describe the ‘entanglement entropy’ of
quantum fields in a black hole background. This shed light on why the entropy was proportional
to the area of the horizon and not the volume of the black hole. A review on entanglement
entropy of black holes can be found in [37]. Srednicki computed the entanglement entropy of
a quantum field, where the field enclosed in a sphere of radius ‘R’ is traced out. The entropy
of the remaining space-time is proportional to the area of the sphere [38]. This was a very
interesting calculation however doing a similar calculation for a black hole background did not
give finite answers. The reason is that the ‘metric’ in the coordinates of an outside observer
is singular at the horizon. The near horizon volume is infinite, and thus the entropy due to
the quantum fields in this volume is also infinite. ’t Hooft [40] introduced a mass dependent
‘cutoff’ as if a brickwall was steeling off the quantum fields, which satisfied a Dirichlet boundary
condition at the wall. The presence of the brick wall could be used to obtain a finite value for
the entropy. We shall briefly review the procedure here.
3.1 The brickwall model
The scalar field φ is taken in the background of a Schwarzschild black hole whose metric in
spherical coordinates is
ds2 = −
(
1− 2M
r
)
dt2 +
(
1− 2M
r
)−1
dr2 + r2
(
dθ2 + sin2 θdφ2
)
, (3.2)
where M is the mass of the black hole, G = 1, c = 1.
The scalar field’s Klein–Gordon equation in this background is
1√
−g
∂µ
(√
−g gµν∂νφ
)
−m2 = 0.
Using the inverse of the metric (3.2) one finds that the Klein–Gordon equation is(
1− 2M
r
)−1 (
−∂2
t
)
φ+
1
r2
∂r
(
r2
(
1− 2M
r
)
∂r
)
φ+
1
r2
L2φ−m2φ = 0,
where L2 is the angular momentum operator. The above can be solved in the eikonal approxi-
mation using
φ = eiEtei
∫
kdr,
6 A. Dasgupta
where
k2 =
r2
r(r − 2M)
((
1− 2M
r
)−1
E2 − 1
r2
l(l + 1)−m2
)
.
As evident the wavenumber is quite big near the horizon, justifying the eikonal approximation
(the phase dominates). Due to the Dirichlet boundary condition, φ = 0 at the wall, a particular
value of the radius taken as r = 2M+h and an outer boundary L, the wavefunction’s wavenumber
gets quantised as integer multiples of π
πn =
∫ L
2M+h
kdr.
The total number of such eigenmodes with energy E is therefore the sum over angular momentum
(each with 2l + 1) degeneracy∫
(2l + 1)dlπn = g(E).
Given this one can compute
e−βF =
∏
n,l,l3
1
1− e−β̃E
,
where F is the free energy, and the righthand side is the thermal distribution, at a temperatu-
re 1/β̃, such that
πβ̃F =
∫
dg(E) log
(
1− eβ̃E
)
= −
∫ ∞
0
dE
g(E)
eβ̃E − 1
.
Using the value of g(E)
F = − 1
π
∫ ∞
0
dE
∫ L
2M+h
dr
(
1− 2M
r
)−1 ∫ (2l + 1)(
eβ̃E − 1
)dl√E2 −
(
1− 2M
r
)(
l(l + 1)
r2
)
(we use the approximation that the scalar fields are massless). F is then approximated in [40]
as
F = −2π3
45h
(
2M
β̃
)4
− 2
9π
L3
∫ ∞
0
dEE3
eβ̃E − 1
.
Using the definition of entropy
S = β̃(U − F ), U =
∂
∂β̃
(
β̃F
)
one obtains S = 4M2. This is indeed proportional to the area of the horizon, but one had to
put the cutoff radius h = 1
720πM , a very particular value to achieve the result.
There are some observations about this derivation of entropy: (i) There is no entanglement
of modes across the horizon of the QFT modes, what this represents is the entropy of a gas of
scalar particles outside the horizon. (ii) Even though the cutoff is mass dependent the proper-
distance of the brickwall from the horizon is a constant. The question still remains why it is
this constant and not another one.
Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics 7
3.2 Scalar entanglement for a bifurcate horizon
It is very interesting that this ‘entropy of a gas of scalar particles’ using the brick wall method
could be placed in an ‘entanglement entropy’ context. In this formalism one traces over the
‘outside modes’ to obtain a density matrix and the entropy of the resultant partially traced
sector of the system. The details of this derivation can be found in [8] and references therein.
The crux of the computation is hinged on the definition of the wavefunction of the black hole
as a ‘path-integral’ in a space-time with a complex metric. The space-time is the complexified
Kruskal extension of the Schwarzschild black hole. The metric in spherical coordinates for the
Schwarzschild black hole is given in (3.2). One then defines the Kruskal coordinates as
U = −e−(t−r−2M ln(r/2M−1))/4M , V = e(t+r+2M ln(r/2M−1))/4M ,
UV =
(
1− r
2M
)
exp
( r
2M
)
,
such that the Kruskal metric is
ds2 =
−32M3
r
exp−(r/2M) dUdV + r2dΩ2.
As evident the metric is extendible past r = 2M , the horizon, and the horizon is bifurcate, with
V = 0 coinciding with the past horizon and U = 0 coinciding with the future horizon. There
are two asymptotics, and the transformations U → −U and V → −V map the two asymptotics.
One could then define the Lorentzian metric as the real section of a complex metric, by defining
the coordinates U and V w.r.t. a ‘complex time’ z = τ + it. The τ is the Euclidean time, and
it is such that in the Euclidean section −2πM < τ < +2πM . The two asymptotics are then
connected in some sense through this ‘Euclidean’ section. The complexified coordinates are
U = −e(iz−i2πM+r+2M ln(r/2M−1))/4M , V = e(−iz+i2πM+r+2M ln(r/2M−1))/4M .
Clearly as the imaginary time τ changes from −2πM to 2πM , one travels from one asymptotics
to another given by z+ = 2πM + it and z− = −2πM + it (−∞ < t <∞) in complex time.
One defines a path-integral through the Euclidean throat which could be labeled as the
propagator connecting the two asymptotics, with φ ≡ φ+ in one and φ ≡ φ− in the other.
Define
Ψ(φ−, φ+) = 〈φ−| exp(−β̃Ĥ/2)|φ+〉. (3.3)
Thus the path-integral is over the Euclidean ‘throat’ connecting the two Lorentzian asymptotics.
Given β̃ = 8πM and Ĥ is a Hamiltonian which in the linearised approximation gives the usual
Hamiltonian of matter fields including scalar fields. The computation in [8] is given for scalar
Hamiltonian using heat kernel techniques. A density matrix is defined thus
ρ =
∫
Dφ+Ψ∗(φ′−, φ+)Ψ(φ−, φ+).
Plugging Ψ from (3.3) and using
∫
Dφ+|φ+〉〈φ+| = 1 this reduces to
ρ = exp(Γ)〈φ−| exp(−8πMĤ)|φ−〉, (3.4)
where the prefactor is introduced to preserve
∫
Dφ−ρ = 1 or the trace condition. The entropy
of this density matrix is then computed using S = −Tr(ρ ln ρ). As the field is in one of the
asymptotic regions, the Hamiltonian is obtained in the one-loop approximation as the due to the
scalar fields solved in the classical black hole background. Due to the definition of the density
matrix as the expectation value of the Hamiltonian at a temperature 1/β (3.4), this calculation
8 A. Dasgupta
can be eventually mapped to the computation of entropy of a gas of the scalar particles near
the horizon and it is obtained as
S =
4M2
45
∫ r0
2M
r3dr
(r − 2M)2
. (3.5)
Though this is proportional to the horizon area some aspects remain as the brickwall model.
The integral in (3.5) is divergent and thus the entropy is infinite. The exact nature of the en-
tanglement is not clear however in the way entangled states are defined in (3.1). ‘Entanglement’
entropy of scalar fields in flat space can be found where the entanglement is obvious as in (3.1)
in the following references [11,38].
In the next section we will compute the ‘entanglement’ entropy of a black hole using ‘non-
perturbative’ coherent states in a ‘quantum gravity’ formalism. The variables are ‘regularised’
gravitational degrees of freedom. The entropy computed is purely gravitational in origin. The
nature of entanglement is also specified clearly exactly similar to the discussion of quantum
mechanical entanglement in equation (3.1). The ‘internal’ Hilbert space is traced out and the
entropy is computed using the density matrix defined in the ‘outside’ Hilbert space. The answer
is finite and no cutoff is introduced.
4 Gravitational entropy
In the definition of the tensor product form of the coherent state (2.5), the total coherent
state is a tensor product of coherent states at each edge. There is a SU(2) Hilbert space at
each edge of the graph Γ. The total Hilbert space is thus H = ⊗He. The question therefore
naturally arises if the coherent state defined in the basis of these Hilbert spaces are entangled
or simply independent? The structure, Ψ =
∏
e ψe is such that the states appears independent
in the tensor product Hilbert spaces, but can be factorised as per equation (3.1). However,
the classical geometry is interwoven from edge to edge due to Einstein’s equations. Thus even
though not written in a manifest way; for two adjacent edges e1 and e2, ψe1 is related to ψe2 due
to the classical solution at which these are constructed to be peaked. We label this entanglement
due to classical geometry as classical correlations. We then compute the entanglement of edges
outside the horizon with those inside the horizon. The coherent states as introduced in (2.5) are
‘gauge covariant’. The gauge SU(2) transformations act at the vertices and they are discussed in
details in Appendix A. Introducing intertwiners at the vertices makes the coherent states ‘gauge
invariant’. They no longer transform due to the SU(2) transformations as the intertwiners map
the vertices to the trivial representation of SU(2). These introduce further ‘entanglement’ of the
coherent states of edges ending/beginning at the same vertex. These correlations arise as the
type of spins assigned to the edges meeting at a vertex are restricted to ensure gauge invariance.
These correlations introduced due to ‘intertwiners’ are labeled as ‘quantum correlations’. We thus
identify two types of correlations. (i) The classical correlations which are due to the relation of
the ge from one edge to the next already discussed in [18]. (ii) Quantum correlations due to the
intertwiners which link the quantum spins of the edges meeting at a vertex. We discuss these
correlations for the first time in this article.
4.1 Classical correlations
The classical correlations can be identified easily in a particular slicing of the classical geometry.
We discuss a particular time slicing of the metric where the intrinsic curvature of the time slices
is flat. One such metric which has the time slices as flat is the Lemaitre metric
ds2 = −dτ2 +
dR2[
3
2rg
(R− τ)
]2/3 +
[
3
2
(R− τ)
]4/3
r2/3
g
(
dθ2 + sin2 θdφ2
)
.
Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics 9
The rg = 2GM (in units of c = 1) and in the τ = const slices one can define the induced metric
in terms of a ‘r’ coordinate defined as dr = dR/ [3/2rg(R− τc)]1/3 (τ = τc is a constant) on the
slice. One gets the metric of the three slice to be
ds2
3 = dr2 + r2
(
dθ2 + sin2 θdφ2
)
. (4.1)
The entire curvature of the space-time metric is contained in the extrinsic curvature or Kµν =
1
2∂τgµν tensor of the τ = const slices. Now if there exists an apparent horizon somewhere in the
above spatial slice, then that is located as a solution to the equation
∇aSa +KabS
aSb −K = 0,
where Sa ((a, b = 1, 2, 3) denote the spatial indices) is the normal to the horizon, Kab is the
extrinsic curvature in the induced coordinates of the slice, and K is the trace of the extrinsic
curvature. If the horizon is chosen to be the 2-sphere, then in the coordinates of (4.1), Sa ≡
(1, 0, 0), the apparent horizon equation as a function of the metric reduces to
Krr
(
1− qrr
)
−Kφφq
φφ − Γφφr −Kθθq
θθ − Γθθr = 0. (4.2)
In this article we provide a very simplified explanation of why this equation provides correlations
across the horizon. Note that the first term of the equation disappears trivially as 1 = qrr for
any point in the spatial slice and not just at the horizon in the classical metric.
If we plug in the exact values of the above in terms of Schwarzschild variables, we find that the
terms cancel groupwise at the horizon due to spherical symmetry. Thus the apparent horizon
equation which respects spherical symmetry can be imposed simply as the equation in the θ
variable or the φ variable
Kφφq
φφ − Γφφr =
1
r
(
1−
√
rg
r
)
= 0, Kθθq
θθ − Γθθr =
1
r
(
1−
√
rg
r
)
= 0. (4.3)
This ‘truncated’ versions of the apparent horizon equation can be implemented on the co-
herent states in the classical limit. Due to spherical symmetry, the other half of the horizon
equation gets automatically satisfied. Note we are not using two different equations to replace
the one equation in (4.2), but observing that due to spherical symmetry, ensuring that the equa-
tion is satisfied for the θ indexed variables ensures that the other half of the equation will also
be zero. In other words the equations are proportional to each other. To regularise the equation
we observe
Γθθr =
1
2
qθθ(qθθ,r) = −1
2
qθθ
(
qθθ,r
)
=
1
2
qθθ
1
δer
(
qθθ(v1)− qθθ(v2)
)
.
In the above we wrote the Christoffel connection as a difference equation, v1 and v2 are the
vertices of the graph and δer is the length of the radial edge connecting the two vertices.
The above then translates to a equation for the truncated apparent horizon equation (4.3) as
qθθ(vout) = qθθ(vin)
[
2δerK
I
θ e
θ
I(vin) + 1
]
, (4.4)
where the vout vertices are outside the horizon and the vin are vertices within the horizon and
Kθθq
θθ = KI
θ e
θ
I (KI
a = Kabe
bI). We then embed a graph with edges along the coordinate
lines r, θ, φ of the spherical coordinate grid, and compute the regularised variables of (2.3). The
details of this can be found in [18]. In terms of the regularised variables (4.4) appears as
V P 2
eθ
(vout) = P 2
eθ
(vin)
[
2 Tr
(
h−1
eθ
(vin)T Iβ
∂heθ(vin)
∂β
)
PeθI(vin) + V
]
, (4.5)
10 A. Dasgupta
where V is the volume operator and P 2
e = P Ie P
I
e . The volume operator appears due to the fact
that it is the densitised operator
√
qeaI which is regularised and
√
q is regularised into the volume
operator. The volume operator in spherically symmetric coordinates is written as PeHδeH and
thus it can be computed for this system. We also use a regularisation introduced in [18] for theKI
a
operator which uses a ‘derivative’ w.r.t. the Immirzi parameter. Using the definition in (2.2),
and the definition of holonomy one gets ∂he/∂β =
( ∫
KI
aT
Idxa
)
he. And thus multiplying
with h−1
e T I and taking trace identifies the appropriate component of the extrinsic curvature
in the continuum limit. This regularisation works for the purposes of the computations of
expectation values as verified in [20]. Note that constants which appear due to the specification
of the graph edges (like δer) are suppressed in the above. And thus it is sufficient to take the
following form of the apparent horizon equation
V P 2
eθ
(vout) = P 2
eθ
(vin)
[
2 Tr
(
h−1
eθ
(vin)T Iβ
∂heθ(vin)
∂β
)
PeθI(vin) + V
]
. (4.6)
We use the apparent horizon equation to solve for Peθ(vout) in terms of Peθ(vin). The apparent
horizon equation is further supplemented with a restriction that the area induced on the horizon
using the radial edges sums up to the total area of the horizon. Which translated into the phase
space operators means∑
eH
PeH = AH . (4.7)
These (4.6), (4.7) therefore specify the horizon information. We then use these to perform
the tracing mechanism on the density matrix and compute entropy. Instead of computing the
entropy with the entire density matrix, we concentrate on a local set of vertices surrounding
the horizon. As we saw, the apparent horizon correlates ‘angular edges’ immediately inside and
outside the horizon. We label this element ρlocal and isolate it from the entire density matrix.
ρtotal = ρoutsideρlocalρinside,
where ρlocal covers a band of vertices surrounding the horizon one set on a sphere at radius
rg − δer/2 and one set on a sphere at radius rg + δer/2 within the horizon, as described in [18],
and in the Fig. 1 enclosed. This local density matrix and the correlations due to the apparent
horizon equation (4.6) was used to derive entropy [18].
We then further concentrate on two vertices vout outside the horizon, vin inside the hori-
zon which share a radial edge eH and write a density matrix for that ρlocal. At each vertex
vout,in there are two angular edges corresponding to the θ, φ coordinate lines which are in-
going and two angular edges which are outgoing. These angular edges, 4 in number at each
vertex have their classical ge correlated with those meeting at the other vertex. Labeling
ψj(g)mn = 1
〈ψ|ψ〉1/2 e
−t/2j(j+1)πj(ge)mn,
ρlocal =
∑
{jOi}{jH}{jIi},{j′Oi}{j
′
H}{j
′
Ii}
4∏
i=1
ψjOi(ge(vout)[ge(vin)])mOinOiψjH (geH )mHnH
×
4∏
i=1
ψjIi(ge(vin))mIinIi
4∏
i=1
|{jOi}{jH}{jIi}〉〈{j′Oi}{j′H}{j′Ii}|
×
4∏
i=1
ψ̄j′Oi(ge(vout)[ge(vin)])m′
Oin
′
Oi
ψj′H (geH )m′
Hn
′
H
4∏
i=1
ψ̄j′Ii(ge(vin))m′
Iin
′
Ii
.
jOim0in0i label the spins of the edges at the outside vertex. jIimIinIi label the spins at the
inner vertex and the jH , mH , nH label the spins on the radial edge connecting the two vertices.
Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics 11
Figure 1. Graph at the horizon.
We us {j} to label all the three indices jmn. The tracing over the modes inside the horizon
gives in the t→ 0 limit
ρlocal
reduced =
∑
{jOi}{jH}{jIi}{j′Oi}{j
′
H}
4∏
i=1
ψjOi(ge(vout)(ge(vin)))mOinOiψjH (geH )mHnH
×
4∏
i=1
δ(Pei(vin), jIit)|{jOi}{jH}〉〈{j′Oi}{j′H}|ψ̄j′H (geH )m′
Hn
′
H
×
4∏
j′i=1
ψ̄j′Oi(ge(vout)[ge(vin)])m′
Oin
′
Oi
.
In the t → 0 limit, the diagonal terms dominate. The density matrix collapses to a set of
non-zero elements [42]. This can be directly seen from
1
〈ψ|ψ〉
e−tj(j+1)πj(ḡ)mnπj(g)mn
≈ t3/2
4
√
πp
e−jp|m/j−(Rp)3/p|e−jp|n/j−(Lp)3/p|e−[(j+1/2)−p]2/t. (4.8)
In the t → 0 limit, this gets to be peaked like a delta function at the values P = (j + 1/2)t,
mt, nt = LP3,
RP3 are the left and right invariant momentum operators. As in [18] the jOi gets
fixed to be a certain number due to the apparent horizon equation (4.6), and the correspon-
ding mOi, nOi were also taken as fixed numbers using a ‘gauge fixed’ version of the apparent
horizon equation. However this is slightly arbitrary as the classical data is gauge invariant. We
expect that when all the constraints including the Gauss constraints are solved this will emerge
as obvious.
The individual components of L(R)P̂ 3
eOi
, fixes the m0in0i to unique numbers. Thus the outer
classical data jcl
Oi, m
cl
Oi, n
cl
Oi have to be specified completely to specify the ‘outer state’. In other
words we can break the degeneracy of the density matrix with different mOinOi by making the
operator measurements.
Now we come to a subtle point I had not discussed before, and that is the fact that the part
of the radial edge eH (see Fig. 1) is hidden behind the horizon. It thus makes sense to trace over
12 A. Dasgupta
that. To implement that, we therefore divide the horizon edge eH into two further edges eH1
and eH2. We build the basis state for this using the fact that
|jmn〉 =
√
2j + 1πj(he)mn|0〉
=
∑
k
√
2j + 1πj(he1)mkπj(he2)kn|0〉 =
∑
k
1√
2j + 1
|jmk〉|jkn〉,
where |0〉 is the vacuum state. Splitting the horizon edge, and tracing over the internal half
gives
Tr e2|jmn〉〈j′m′n′| =
∑
k′kn
1
2j + 1
|j′m′k〉〈j′k′n′|jkn〉〈j′m′k′|
=
∑
k
1
2j + 1
|jmk〉〈jm′k|δnn′ . (4.9)
Similarly, for the irreducible representation components of the ‘wavefunction’
πj(ge)mnπ̄j(ge)m′n,
which multiplies the basis states in the density matrix, one can perform such a splitting of the
holonomy as
ge = eiT
IP Ihe = eiT
IP Ihe1he2 = ge1he2 ,
and thus∑
kk′
πj(ge1)mkπj(he2)knπ̄j(ge1)m′k′ π̄j(he2)k′n.
The sum over n then gives
πj(ge1)mkπ̄j(ge)m′k = π
(
eιT
IP I
)
mm′ . (4.10)
This and then its complex conjugate gives the delta function in the t → 0 limit peaked at PH .
The contribution from the horizon edge to the density matrix is
1
2jH + 1
∑
mHkH
δ(jHt, PeH )|jHmHkH〉〈jHmHkH |.
This explains the origin of the 1/(2jH + 1) factor anticipated in [18].
Thus the reduced density matrix ‘describing the vertex outside the horizon’ is
Lim
t→0
ρlocal
reduced =
∑
mHnH
1
2jH + 1
×
4∏
i=1
|PeOi ,
Lp3Oi
Rp3Oi〉|PeHmHnH〉〈PeHmHnH |〈PeOi ,
Lp3Oi
Rp3Oi|.
And the tensor product density matrix becomes
ρlocal
reduced =
∏
vout
∑
mHnH
1
2jH + 1
×
4∏
i=1
|PeOi ,
Lp3Oi
Rp3Oi〉|PeHmHnH〉〈PeHmHnH |〈PeOi ,
Lp3Oi
Rp3Oi|,
Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics 13
Figure 2. Spins at the vertices.
the above density matrix is clearly diagonal. If we then compute
−Tr(ρlocal
reduced ln ρlocal
reduced) =
βABH
4l2p
+ corrections,
where β is a prefactor [19]. The exact computation is not obvious, but if we assume that
every jH is the same, then this gives −Tr
(
ρlocal
reduced ln ρlocal
reduced
)
=
∑
vout
ln(2jH+1) = AH ln(2jH+
1)/[l2p(jH + 1/2)] where we used the constraint (4.7) such that the
∑
(jH + 1/2) = AH/l
2
p. For
different distribution of spins one has to use combinatorics [18,19].
Thus to leading order the entropy as computed from the density matrix is indeed Bekenstein–
Hawking modulo the β which is a function of the Immirzi parameter and other details of the
combinatorics of the graph can be set to 1.
4.2 Quantum correlations
We now discuss the correlations across the horizon due to intertwiners placed at the ver-
tices. As evident the coherent states are defined in a SU(2) Hilbert space. For the holonomy
variables, the SU(2) gauge transformations act at the beginning and end point of the edges.
he → g−1(v1)he(v1, v2)g(v2). These changes will change the spin network state. To make the
state gauge invariant one imposes intertwiners at the vertices. These are structures which map
the spin networks at the vertex to the trivial representation. Thus the state does not transform
due to the intertwiners. We concentrate on two vertices (suppress one dimension) and take each
vertex to be four valent at the horizon and observe the role of the intertwiners. From the Fig. 2
two of the edges are ingoing, the other two are outgoing at each vertex. The gauge transforma-
tions act on the holonomies and hence the basis states on each edge at the vertices. Integrating
(group averaging) over the gauge group at the vertices, one ensures gauge invariance [21,22,41].
This solves the Gauss constraint. We find that when done carefully this gives us correlations
between the coherent state at inner and outer vertices which are carried by the horizon edge
which connects the two vertices. This is therefore a ‘quantum correlation’. We describe some
of the details of the calculations here. Let us take the two vertices as vin and vout and the
spin labels jI1, jI2, jI3, jH at the inner vertex and jO1, jO2, jO3, jH at the outer vertex. Let
us assume that at the inner vertex jI1, jH are outgoing at that vertex, or they ‘begin’ at vin
and the jI2, jI3 are ingoing at the vertex or have their end point at that vertex. Similarly at
the outer vertex, jH and jO3 are ingoing or end at vout and jO1, jO2 are outgoing and begin
at the vertex vout. To introduce the intertwiners we observe the basis spin network functions
and their transformations. The basis spin network states |jmn〉 at the inner vertex in the he
representation are given as (
√
2j + 1 factor is not shown but it exists in each basis state for
14 A. Dasgupta
normalisation of the inner product)
πjI1(h)mI1nI1πjI2(h)mI2nI2πjI3(h)mI3nI3πjH (h)mHnH ,
the gauge action U(g) at the vertex is such that
πjI1(gh)mI1nI1πjI2
(
hg−1
)
mI2nI2
πjI3
(
hg−1
)
mI3nI3
πjH (gh)mHnH .
Upon group averaging one obtains a projector onto the gauge invariant Hilbert space which
written using the 3jm symbols is thus (see Appendix A for further details)
∑
{jk}pI1qI2qI3pH
(2jk + 1)
(
jI1 jH jk
mI1 mH mk
)(
jI1 jH jk
pI1 pH nk
)
×
(
jI2 jI3 jk
nI2 nI3 mk
)(
jI2 jI3 jk
qI2 qI3 nk
)
× πjI1(h)pI1nI1πjI2(h)mI2qI2πjI3(h)mI3qI3πjH (h)pHnH . (4.11)
We can then collect the basis vectors and the 3jm symbols contracted with them as a |in〉 basis
=
∑
{jk}
(
jI1 jH jk
mI1 mH mk
)(
jI2 jI3 jk
nI2 nI3 mk
)
|in〉.
If we take ψj(ge)mn = 1
〈ψ|ψ〉e
−t/2j(j+1)πj(ge)mn, the coherent state wavefunction at the inner
vertex is
∑
{j}
3∏
i=1
ψjIi(geIi)mIinIi
(
jI1 jH jk
mI1 mH mk
)(
jI2 jI3 jk
nI2 nI3 mk
)
|in〉.
Note that this way, only the indices which begin or end at the vin are gauge invariant, the
other end of the edges remains ‘free’ and can be attached to another vertex in an invariant way.
Exactly in the same way one can obtain the gauge invariant wavefunction at the outer vertex,
comprising of the 3jm symbols and the corresponding wavefunctions. And therefore one can
define the intertwiners at the outer vertex but notice immediately the horizon edge is shared
by both the outer vertex and the innner vertex, and thus, the correct way to write the coherent
state for the edges meeting at the two vertices is this
|ψ(vout, vin)〉 =
∑
{j}
Ψ({j0i}, {jH}, {jIi})|out〉|H〉|in〉,
where |H〉 is the horizon state for the horizon edge crossing the horizon (see Fig. 1). Clearly the
coefficient of the basis states do not factorise and can be identified as
Ψ({j0i}, {jH}, {jIi}) =
3∏
i=1
ψjOi(geOi)mOinOi
∑
j̃′k
(
j01 jO2 j̃′k
mO1 mO2 m̃′k
)
×
(
j̃k′ jO3 jH
m̃k′ nO3 nk
)
ψjH (geH )mHnH
×
∑
jk
3∏
i=1
ψjIi(geIi)mIinIi
(
jI1 jH jk
mI1 mH mk
)(
jI2 jI3 jk
nI2 nI3 mk
)
.
Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics 15
The density matrix is then
ρinv = |ψ(vout, vin)〉〈ψ(vout, vin)|.
This in a very sure way establishes that correlations exist across the horizon surface, the
question however is that is Bekenstein–Hawking entropy recovered from the tracing mechanism
as previously? Ab initio it is not expected that this will happen, as these gauge invariant
coherent states do not have the appropriate peakedness properties of the gauge non-invariant
coherent states [42]. We simply implement the tracing of the inner edges to find the reduced
density matrix at this stage and demonstrate the entanglement.
The reduced density matrix is easily obtained, using the trace over the inner basis states and
some 3jm symbol identities (see Appendix A). The matrix elements turn out to be∑
{jIi}
Ψ
(
{jOi}, {jH}, {jIi})Ψ̄({j′Ii}, {j′H}, {jOi}
)
= Ψ({j0i}, {jH})Ψ̄({j′H}, {j′Oi})
∑
{jIi},jk
(2jk + 1)δ(jI1jHjk)δ(jkjI2jI3)
(2jH + 1)
×
3∏
i=1
ψjIi(geIi)mIinIi
3∏
i=1
ψ̄jIi(geIi)m′
Iin
′
Ii
δn′
I1nI1
δm′
I2mI2
δm′
I3mI3
×
(
jI1 jH jk
mI1 mH mk
)(
jk jI2 jI3
mk nI2 nI3
)(
jI1 jH jk
m′I1 m′H m′k
)(
jk jI2 jI3
m′k n′I2 n′I3
)
. (4.12)
Whereas it is not obvious what this will yield for the entropy which is SBH = −Tr(ρ ln ρ),
one can use the fact that again in the limit that t→ 0, the product of wavefunctions will assume
the form in equation (4.8)∑
{ji}
Ψ
(
jOi, jH , jIi)Ψ̄(jIi, j
′
H , j
′
Oi
)
= Ψ(j0i, jH)Ψ̄(j′H , j
′
Oi)
∑
jk
(2jk + 1)δ(jI1jHjk)δ(jkjI2jI3)
(2jH + 1)
×
(
jcl
I1 jH jk
mcl
I1 mH mk
)(
jk jcl
I2 jcl
I3
mk ncl
I2 ncl
I3
)(
jcl
I1 jH jk
mcl
I1 m′H m′k
)(
jk jcl
I2 jcl
I3
m′k ncl
I2 ncl
I3
)
.
The above reduced density matrix is precisely of the form expected to yield the entropy, but
the 3jm symbols are non-trivial and one can use some asymptotic form given in [35] for large j
to estimate the form of the density matrix. Details of this calculations and further computations
of the entropy will appear in [15]. This is the absolute proof of the fact that the coherent state
introduced to describe the black hole is entangled across the horizon and does not factorise into
inside and outside coefficients. There has been work in observing entanglement in semiclassical
states independently as in [30].
5 Time evolution
In this section we discuss time evolution of the system. Fortunately as the system is semiclassical,
a natural choice of time is the asymtotic time coordinate. Using the Lemaitre slice and evolving
the state in Lemaitre time would not serve the purpose as that is the frame of an infalling
observer. We use the frame of an observer stationed outside the horizon to evolve the slice in
her time as at the asymptotics this coincides with Minkowski time. Again we concentrate on
the evolution of a local patch surrounding the horizon. Normally the vertices within the horizon
are classically forbidden to the outside observer. However we allow for access to those vertices
in the ‘quantum mechanical’ evolution. The ‘Hamiltonian’ which is used to evolve the horizon
is the Brown and York quasi-local energy.
16 A. Dasgupta
5.1 Semi classical Hamiltonian
Quasi-local Hamiltonian is defined as the energy enclosed in a finite space. This is obtained using
a particular definition [13], as the ‘surface’ integral of the extrinsic curvature with which the
surface is embedded in three space. This generates time translation in the timelines orthogonal
to the spatial slice. In our case, we take the bounding surface to be the horizon and the quasilocal
energy is given by the surface term [13,28]
H̃ =
1
κ
∫
d2x
√
σk,
where k is the extrinsic curvature with which the 2-surface, which in this case is the horizon S2
embedded in the spatial 3-slices, and σ is the determinant of the two metric σµν defined on the
2-surface. This ‘quasilocal energy’ is measured with reference to a background metric. Thus
H = H̃−H0 where H0 is the energy in the background. We concentrate on the physics observed
by an observer stationed at a r = const sphere.
The metric in static r = const observer’s frame is
ds2 = −f2dt2 + r2
(
dθ2 + sin2 θdφ2
)
,
f =
√
1− rg/r, where rg is the Schwarzschild radius. If we take nµ to be the space-like vector,
normal to the 2-surface, then the extrinsic curvature is given by kµν = σαµ∇αnν and the trace is
obviously
k = ∇αnα. (5.1)
In the special slicing of the stationary observer the normal to the horizon 2-surface is given by
(0, f(r), 0, 0). However, we built the density matrix on the Lemaitre slice. The Lemaitre and the
Schwarzschild observer’s coordinates are related by the following coordinate transformations√
r
rg
dr = (dR− dτ), dt =
1
1− f ′
(
dτ − f ′dR
)
, f ′ =
rg
r
.
The r = const cylinder of the Schwarzschild coordinate corresponds to dR = dτ of the
Lemaitre coordinates, and for these dt = dτ . Thus unit translation in the t coordinate coincides
with unit translation in the τ coordinate. Further, the intersection of the r = const cylinder with
a t = const surface coincides with the intersection of r = const and the τ = const surface. Thus
in the initial slice, the QLE Hamiltonian can be written in terms of the Christoffel symbols of
the Lemaitre slice. Even though the generic transformation of the Christoffel symbols from one
coordinate frame to the other is inhomogeneous, in this example, the transformation is trivial.
The trace of the extrinsic curvature (5.1) then written using Christoffel symbols turns out to be
H =
1
2κ
∫
dθdφ
√
gθθgφφ
[
−gθθ ∂gθθ
∂r
− gφφ
∂gφφ
∂r
]
f(r)−H0. (5.2)
The reference frames’ quasilocal energy H0 is a number, it just defines the zero point Hamilto-
nian. Thus, we replace the classical expressions by operators evaluated at the τ = const slice.
In the first approximation we simply take the f(r) as classical
√
1− rg/r =
√
δer/2rg = ε, as
this arises due to the coordinate transformation and the norm of the vector nr in the previous
frame. In the re-writing of (5.2) in regularised LQG variables the Hamiltonian appears rather
complicated.
One can rewrite these in a much simpler form, using the apparent horizon equation. Since
the Hamiltonian is an integral over the horizon, the variables will satisfy the apparent horizon
equation (4.2) up to quantum fluctuations. Thus the Hamiltonian operator is then re-written as
Hhorizon =
ε
κ
∫
dθdφ
√
gθθgφφ
[
KI
θ e
Iθ +KI
φe
Iφ
]
,
Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics 17
where we have used the classical apparent horizon equation (4.2) (with qrr = 1). The regularised
LQG version of this is
Hhorizon =
Ca ε
2κδeθseθ
∑
vout
Tr
[
h−1
eθ
T Iβ
∂
∂β
heθ
]
P Ieθ + h.c. + (θ → φ), (5.3)
where C consists of some dimensionless constants seθ is the 2-dimensional area bit over which EθI
is smeared, a is a dimensionfull constant which appears to get the P Ieθ dimension less. δeθ is
the length for the angular edge eθ over which the gauge connection is integrated to obtain the
holonomy. The sum over vout is the set of vertices immediately outside the horizon. The (5.3)
can then be lifted to an operator. However, this Hermitian operator gives no change to horizon
entropy as the coherent state is evolved from one time slice to the next. Surprisingly a non-
Hermitian term evolves the slices non-trivially. These anti-Hermitian terms can be easily found
in this formalism if we allow for the vertices within the horizon to be exposed to the outside
observer. In this case as the region is within the classical horizon, the norm of the Killing vector
is negative, and nr has components which are imaginary. The ε → ±ιε. One sees why the
inner vertices are relevant in a semiclassical evolution even if they are classically forbidden: In
quantum mechanics tunneling is a well known phenomena, and inclusion of the inner vertices
facilitates the process.
Thus the contribution of ‘inner vertices’ vin is added to the Hamiltonian and it gives the
anti-Hermitian contribution to the Hamiltonian. Thus Hhorizon = 1
2
[∑
vout
Hvout +
∑
vin
Hvin
]
.
The regularised Hamiltonian is not Hermitian, and the evolution equation is
ι~
∂ρ
∂τ
= Hρ− ρH†. (5.4)
Thus one takes the evolved slice to have a density matrix
ρ = ρ0 − ιδτ
~
[
Hρ0 − ρ0H†
]
.
The entropy computed using this density matrix is different from the entropy in the previous
slice and the difference of entropy can be seen as obtained using a simplified model.
Let the density matrix be defined for a system whose states are given in the tensor product
Hilbert spaces H1 ⊗H2 and given by
|ψ〉 =
∑
ij
dij |i〉|j〉,
where |i〉 is the basis in H1 and |j〉 is the basis in H2 and dij are the non-factorisable coefficients
of the wavefunction in this basis. Let us label the wavefunction at time t = 0 to be given by the
coefficients d0
ij . The density matrix is
ρ0 =
∑
iji′j′
d0∗
i′j′d
0
ij |i〉|j〉〈j′|〈i′|.
The reduced density matrix if one traces over H2 is
Tr2 ρ
0 =
∑
ii′
∑
j
d0∗
i′jd
0
ij |i〉〈i′|.
We now evolve the system using a Hamiltonian which has the matrix elements Hiji′j′ |i〉|j〉〈j′|〈i′|,
we assume that the Hamiltonian does not factorise, that is there exists interaction terms between
the two Hilbert spaces. The evolution equation is
i~
∂ρ
∂τ
= [H, ρ],
18 A. Dasgupta
which in this particular basis gives the density matrix elements at an infinitesimally nearby slice
to be
dδτ∗i′j′ d
δτ
ij = d0∗
i′j′d
0
ij −
i
~
δτ
[∑
kl
(
Hijkld
0
kld
0∗
i′j′ − d0
ijd
0∗
klHkli′j′
)]
.
Thus we evolve the ‘unreduced’ density matrix and then trace over the H2 in the evolved slice.
The reduced density matrix in the evolved slice is
∑
j
dδτ∗i′j d
δτ
ij =
∑
j
d0∗
i′jd
0
ij −
i
~
δτ
∑
klj
(
Hijkld
0
kld
0∗
i′j − d0
ijd
0∗
klHkli′j
) .
This gives
ρδτ = ρ0 − i
~
δτA,
where A represents the commutator. Clearly the entropy in the evolved slice evaluated as
SδτBH = −Tr(ρ ln ρ) can be found as
SδτBH = S0
BH +
i
~
δτ
[
TrA ln ρ0 + Tr ρ0ρ0 −1A
]
.
Given the definition of Aii, one gets
Aii =
∑
jkl
[
ρ0
ijklHklij −Hijklρ
0
klij
]
.
In case both the Hamiltonian and the density operator are Hermitian, one obtains∑
j
Ajj = 2ι Im Tr
(
ρ0H
)
.
This is clearly calculable, and gives the change in entropy ∆SBH. The ln ρ0 term yields
corrections, and we ignore it in the first approximation. However if the Hamiltonian is non-
Hermitian one gets
∆SBH =
ιδτ
~
Tr
[
Hρ0 − ρ0H†
]
(5.5)
instead of the commutator.
In case of the gravitational Hamiltonian, the above can be computed using a U(1) projection
thus. Let us take the U(1) case to make the calculations easier and observe the action of the
Hamiltonian on the evolution of the coherent state. The spin network states are replaced by
|n〉 = eιnζ , 0 < ζ < 2π, n is an integer and the coherent states are
ψt(ge) =
∑
n
e−(tn2)/2ein(χe−ipe)e−ιnζ .
gne = ein(χe−ipe) is the complexified phase space element in the ‘nth’ representation.
The quasi local energy operator (5.3) also takes the simplified form
H
U(1)
horizon = −1
2
C ′ιĥ−1
e β
∂
∂β
ĥep̂e +
1
2
C ′ιp̂eβ
∂ĥ−1
e
∂β
ĥe. (5.6)
Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics 19
The prefactors have been clubbed into C ′. The U(1) coherent states are eigenstates of an
annihilation operator defined thus:
ĝe = et/2ep̂e ĥe, ĝe|ψ〉 = ge|ψ〉.
The holonomy operator can thus be written as
ĥe = e−t/2e−p̂e ĝe.
And the derivative w.r.t. Immirzi parameter of the holonomy which appears in the definition of
the Hamiltonian replaced by
β
∂ĥe
∂β
= e−t/2
[
−β ∂p̂
∂β
e−p̂e ĝe + e−p̂eβ
∂ĝe
∂β
]
= e−t/2
[
p̂ee
−pe ĝe + e−p̂β
∂ĝe
∂β
]
.
The dependence of the operator p on the Immirzi parameter is known (2.2), and thus we
could evaluate the derivative (β∂βpe(β) = β∂β(pe(1)/β) = −pe(β)).
The term
Tr
(
ρ0H
U(1)
horizon
)
(5.7)
is then computable. Let us take the first term of (5.6) and find (5.7). As ρ0 = |ψ〉〈ψ|, (5.7)
gives simply (we drop the ‘e’ label for brevity)
〈ψ|HU(1)
horizon|ψ〉 = 〈ψ| − 1
2
C ′ιĥ−1β
∂
∂β
ĥp̂|ψ〉+ 〈ψ|h.c.|ψ〉
= −1
2
ιC ′〈ψ|ĝ†e−t/2e−p̂e−t/2
[
p̂ e−p̂ĝ + e−p̂β
∂ĝ
∂β
]
p̂|ψ〉+ 〈ψ|h.c.|ψ〉
= −1
2
ιC ′e−tg∗
[
〈ψ|e−p̂p̂e−p̂p̂|ψ〉g + 〈ψ|e−2p̂β
∂ĝ
∂β
p̂|ψ〉
]
+ 〈ψ|h.c.|ψ〉.
We then concentrate on the 2nd term of the above
〈ψ|e−2p̂β
∂ĝ
∂β
p̂|ψ〉 = 〈ψ|e−2p̂β
∂ĝ
∂β
∫
dν(g′)|ψ′〉〈ψ′|p̂|ψ〉 =
∫
dν(g′)β
∂g′
∂β
〈ψ|e−2p̂|ψ′〉〈ψ′|p̂|ψ〉,
where we have used the fact that coherent states resolve unity. It can be shown that the
expectation value of the operators in the t→ 0 collapses the integral to g′ = g point [42]. Thus
one obtains from the above
Tr
(
ρ0H
U(1)
horizon
)
= −1
2
ιC ′e−t/2
[
p+ g∗e−2pβ
∂g
∂β
]
p+ h.c. = C ′β
∂χ
∂β
p, (5.8)
which is real, and thus
∆SBH = 0
(this is actually the classical quasi local energy as it should be from Tr(ρ0Hhorizon)).
This is obvious, as the way the Hamiltonian is defined, this is simply a function of the
Hilbert space outside the horizon, and the matrix elements of this will not yield anything new.
We approximated the horizon sphere by summing over v1 vertices immediately outside the
horizon. We could do the same by summing over v2 vertices immediately within the horizon.
For the Lemaitre slice, the metric is smooth at the horizon, and one can take the ‘quantum
operators’ evaluated at the vertex v2. In this case however, as the region is within the classical
horizon, the norm of the Killing vector is negative, and nr has components which are imaginary.
20 A. Dasgupta
The ε → ±ιε. Thus Hhorizon = 1
2 [
∑
v1
Hv1 +
∑
v2
Hv2 ]. In the evaluation of the quasi local
energy term, the energy would emerge correct in the δer → 0 limit as ε → 0. The regularised
Hamiltonian is not Hermitian, and the evolution equation is (5.4).
The same type of expectation or trace is obtained of the imaginary term (or v2) is obtained
as in (5.8), but this does not cancel in (5.5), and
∆SBH = ∓δτ
~
C ′β
∂χ
∂β
p.
The ‘rate of change’ of entropy is thus
∆̇SBH = ∓ C̃
l2p
β
∂χ
∂β
p
(we extracted the κ from C ′ to get l2p and rewrote the rest of the constants as C̃).
Thus there is a net change in entropy, but, to see if this is Hawking radiation, we have to
couple matter to the system.
In the case of the black hole system, due to spherical symmetry, in the classical limit, from
the regularisation of the extrinsic curvature components KI
θ,φ which appear in the Brown and
York quasi-local energy and its LQG regularisation (5.3), the change of entropy is
∆SBH = ∓ C̃εδτ
l2p
∑
v
β
[
∂χeθ
∂β
Peθ +
∂χeφ
∂β
Peφ
]
.
Due to the nature of the classical metric, the P Ieθ,φ can be gauge fixed to have one non-zero
component in the internal directions [20]. Let that be some fixed index r, then Peθ,φ = δrIP
I
eθ,φ
.
In the computation of the Hamiltonian then it can be shown that if the holonomy is assumed
to be he = eiχ
IT I then, the r component of χr contributes to the Hamiltonian, and this is what
we have set as χeθ,φ in the above. From [16] one finds the χeθ and the derivative of that at the
horizon is a function of
√
rg/r which gives a constant. The Peθ are the regularised densitised
triads at the horizon and thus they give area of the two surfaces comprising the decomposition
of the three sphere in the angular dimensions. As it is evident,
Peθ =
1
a
∫
∗Eθ =
1
a
∫
∗r sin θ.
Thus they will be proportional to rg sin θ at the horizon. Similar analysis is done for the φ sector
∆SBH = ±2C̃ ′εδτ
l2p
∑
v
dAvβrg,
where dAv is the area of the element of the two surface at the vertex sin θ and C̃ ′ has been
redefined to accomodate the regularisation constants which appear in Peθ .
Thus using the fact that TH = 1/4πrg one could define the ‘rate of entropy change’ ∆SBH/δτ
as
∆ṠBH ∝ ±
ε
TH
.
Note that this is change in entropy due to emission of one Hawking quanta of a particular
frequency. It should not be confused with entire black hole decay rate.
Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics 21
5.2 Hawking radiation
The fact that entropy change occurs due to Hamiltonian evolution which includes a non-
Hermitian term is very interesting. Hawking radiation however is a flux of particles emerging
from the horizon, and thus we have to trace the origin of such a flux. In the previous section we
found that as the system evolves in time, the horizon fluctuates and the area decreases. But is
this Hawking radiation? Adding matter to a ‘coherent state’ description of semiclassical gravity
has been discussed [36]. Thus, given a massless scalar field Lagrangian coupled to gravity, whose
Hamiltonian is given by
Hsc =
∫
d3x
[
π2
√
q
+ (∇φ)2
]
,
the ‘gravity’ in the Hamiltonian can be regularised in terms of the he, P
I
e operators in the
coherent state formalism. The integral over the three volume gets converted to a sum over the
vertices dotting the region. Thus
Hv
sc =
∑
v
Hv
(
he, P
I
e , V
)
.
This Hamiltonian is an operator, and one evaluates an expectation value of the Hamiltonian
in the density matrix by computing the trace of the product of the density matrix and the
Hamiltonian
Tr
(
ρτHτ
sc
)
.
This Hamiltonian and the density matrix are then both evolved according to the time-like
observers frame. One gets
i~
∂Hsc
∂τ
= [H,Hsc].
Thus one can compute the Hamiltonian in a slice infinitesimally close to the previous slice τ+δτ .
Using that,
Tr
(
ρτ+δτHτ+δτ
sc
)
− Tr
(
ρτHτ
sc
)
= −(δτ)2 Tr{[H, ρτ ][H,Hτ
sc]}.
It is very clear thus that the order δτ terms are zero for this. However, allowing for the
non-unitary evolution using the non-Hermitian Hamiltonian, the δτ terms survive. In fact the
terms are
Tr
(
ρτ+δτHτ+δτ
sc
)
− Tr
(
ρτHτ
sc
)
= − ιδτ
~
Tr
[
(Hρ− ρH†)Hsc
]
− ιδτ
~
Tr
[
ρ
(
HHsc −HscH
†)]. (5.9)
In the exact Hilbert space of the matter gravity system, the states are not exactly in the tensor
product form, however, in the first approximation we take the states of the matter gravity
system to be
∏
v |ψ(g)〉⊗ |ω〉 where |ω〉 is a matter state with particles of energy ω. At this step
for simplicity we assume that the outer and inner vertex scalar state is at the same energy ω. It
is obvious that the first term in (5.9) gives the change in entropy as per (5.5) times the matter
Hamiltonian’s eigenvalue ∆SBHω. The ‘rate’ of particle creation thus has the form
− εω
TH
,
where TH is the Hawking temperature for the signs +(−)ιε and negative (positive) ω. The ε
can be interpreted as the redshift factor. If a n particle matter state is taken, then the number
will change accordingly. Thus a flux of particles indeed emerges from the horizon. To obtain
a complete finite flux one has to compute the flux production over a finite number of such time
steps. One can see that eventually a exponential will emerge which is the required Boltzmann
factor. Further details can be found in [20].
22 A. Dasgupta
5.3 Irreversible systems
It has been known from the time of Boltzmann that entropic systems usually have irreversible
time evolution. The Boltzmann H-theorem uses a particular collision term in the evolution
equation whose behavior gives rise to this irreversible flow. In complex systems physicists have
been using different techniques to create irreversibility in the dynamics. In [34] this flow of
a complex system is formulated as
ι
∂ρ
∂t
=
(
Φ0 + Φe
)
ρ,
where the usual Liouville operator has been written as broken into two pieces, one Φ0 which is
the reversible part and the other Φe which is irreversible and non-Hermitian. The Φe creates the
entropy production during the flow of the system. Thus it is remarkable that in the derivation
of the time flow in presence of the horizon, precisely such a splitting occurs in the time-evolution
equation. It seems quantum gravity should be formulated such that its dynamics should not
be restricted to unitary operators. A complete re-formulation of quantum gravity using the
language of complex systems and verification of some of the existing results is work in progress.
This will facilitate the description of the system to describe black hole evaporation. Notice that
as the non-Hermitian component of the Hamiltonian contributes from within the horizon, if all
the mass of the black hole is radiated away, the term within the horizon will also disappear, and
the evolution will become unitary again. However this as of yet speculative, a stable Planck size
remnant might also form in the process.
6 Conclusion
Thus in this article we discussed coherent states for non-rotating black holes. We then showed
that these states can be used to explain the origin of entropy due to classical correlations and then
gave a brief introduction to a computation of ‘quantum correlations’ using SU(2) intertwiners at
the vertices. I then derived a time evolution with a non-Hermitian Hamiltonian which generated
entropy and this seems to create a flux of matter which escapes the horizon. This semiclassical
derivation is similar to the process of entropy production observed in complex systems. However,
the non-unitary nature of the evolution equation might be due to the use of semiclassical ‘time’
and an evolution operator which generates evolution in that time. One has to further work
with a ‘quantum Hamiltonian’ to verify these results. Of course this brings us to the problem
of identifying time in quantum gravity, and the evolution in physical time remains a project
of research. There is promise from the reduced ‘spherically symmetric’ solution for the true
Hamiltonian which can be found in the following papers [1, 12, 24]. Other examples of reduced
phase space spherically symmetric quantisation and studies of Hawking radiation can be found
in [29, 31]. For the time being, reformulating quantum gravity in an attempt to answer all the
questions is work in progress.
A Intertwiners
To build the ‘gauge invariant coherent states’ we use the techniques of group averaging, the
details of which can be found in reviews of loop quantum gravity [21,22,41]. For this the graph
and its vertices are specified and the intertwiners at each vertex introduce the gauge invariance.
For us, it is sufficient to take a graph which is comprised of vertices immediately outside the
horizon and those immediately inside the horizon, connected by a radial edge. If we suppress one
angular dimension of the spherical horizon, and open up this graph, it will appear as a ‘ladder’
structure.
Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics 23
If we isolate one rung of the graph and concentrate on that one obtains these two vertices vout
connected to vin with four edges meeting at each vertex (see Fig. 2). From the figure it is clear
that two of the edges are ingoing and two are outgoing at each vertex. To obtain the intertwiners
at each of the vertex, we write the basis states. At vertex vin the basis states are (suppressing
the
√
2j + 1 factor which makes the inner product 1)
πjI1(h)mI1nI1πjI2(h)mI2nI2πjI3(h)mI3nI3πjH (h)mHnH
Implementing a Gauge transformation at vin one obtains
πjI1(gh)mI1nI1πjI2
(
hg−1
)
mI2nI2
πjI3
(
hg−1
)
mI3nI3
πjH (gh)mHnH
= πjI1(g)mI1pI1πjI1(h)pI1nI1πjI2(h)mI2qI2πjI2
(
g−1
)
qI2nI2
× πjI3(h)mI3qI3πjI3
(
g−1
)
qI3nI3
πjH (g)mHpHπjH (h)pHnH .
Integrating over the group element using the measure
∫
D(g) then gives the 3jm symbols. This
then defines the projector onto the gauge invariant space. If we isolate the integrals this is∫
Dg πjI1(g)mI1pI1πjI2
(
g−1
)
qI2nI2
πjI3
(
g−1
)
qI3nI3
πjH (g)mHpH .
Inserting a delta function using Peter–Weyl theorem∑
jkmknk
(2jk + 1)πjk(h)mknkπjk
(
h′−1
)
mknk
= δ(hh′−1).
One obtains∑
jkmknk
(2jk + 1)
∫
Dg πjI1(g)mI1pI1πjH (g)mHpHπjk(g)mknk
×
∫
Dg′πjk
(
g′−1
)
mknk
πjI2
(
g′−1
)
qI2nI2
πjI3(g′−1)qI3nI3 , (A.1)
these integrate to 3jm coefficients as per the definition [43]∫
Dgπj1(g)m1n1πj2(g)m2n2πj3(g)m3n3 =
(
j1 j2 j3
m1 m2 m3
)(
j1 j2 j3
n1 n2 n3
)
.
Writing πj
(
g−1
)
mn
= πj(g
∗)nm = π∗j (g)nm in the second integral of (A.1), one gets the complex
conjugate of the 3jm symbols defined above. Thus the contribution to the inner vertex is as
mentioned in (4.11).
B Tracing
We briefly describe the tracing mechanism at the Inner vertex. To trace over the basis states
of |jIimIinIi〉 ≡
√
2jIi + 1πjIi(h)mIinIi we simply use the inner product of 〈jmn|j′m′n′〉 =
δjj′δmm′δnn′ . Once the Kronecker delta functions have been implemented; the wavefunction at
the inner vertex is of the form∑
{j}
(2jk + 1)
(
jI1 jH jk
mI1 mH mk
)(
jI1 jH jk
pI1 pH nk
)(
jk jI2 jI3
mk nI2 nI3
)(
jk jI2 jI3
nk qI2 qI3
)
×
3∏
i=1
ψjIi(g)mIinIi
3∏
i=1
ψ̄jIi(g)m′
Iin
′
Ii
δn′
I1nI1
δm′
I2mI2
δmI3m′
I3
ψjH (gH)mHnH ψ̄j′H (gH)m′
Hn
′
H
24 A. Dasgupta
×
∑
j′k
(2j′k + 1)
(
jI1 j′H jk
m′I1 m′H mk′
)(
jI1 j′H jk′
pI1 p′H nk′
)
×
(
jk′ jI2 jI3
mk′ n′I2 n′I3
)(
jk′ jI2 jI3
nk′ qI2 qI3
)
.
We then sum the 3jm symbols with common entries, e.g.
∑
qI2qI3
(
jk jI2 jI3
nk qI2 qI3
)(
jk′ jI2 jI3
nk′ qI2 qI3
)
=
δjkj′kδnkn
′
k
(2jk + 1)
{jkjI2jI3},
where {j1j2j3} is the 3j symbol and is such that it is 1 if j1 + j2 + j3 is an integer and |j1− j2| ≤
j3 ≤ j1 + j2, and 0 otherwise. This process then finds equation (4.12).
Acknowledgements
This work was supported by NSERC and University of Lethbridge Research Fund.
References
[1] Álvarez N., Gambini R., Pullin J., Local Hamiltonian for spherically symmetric gravity coupled to a scalar
field, Phys. Rev. Lett. 108 (2012), 051301, 4 pages, arXiv:1111.4962.
[2] Ambjørn J., Jurkiewicz J., Loll R., Lattice quantum gravity – an update, PoS Proc. Sci. (2010),
PoS(LATTICE2010), 014, 14 pages, arXiv:1105.5582.
[3] Ashtekar A., Introduction to loop quantum gravity, arXiv:1201.4598.
[4] Ashtekar A., Baez J., Corichi A., Krasnov K., Quantum geometry and black hole entropy, Phys. Rev. Lett.
80 (1998), 904–907, gr-qc/9710007.
[5] Ashtekar A., Rovelli C., Smolin L., Weaving a classical metric with quantum threads, Phys. Rev. Lett. 69
(1992), 237–240, hep-th/9203079.
[6] Barbero J.F., Lewandowski J., Villasenor E.J., Quantum isolated horizons and black hole entropy,
arXiv:1203.0174.
[7] Bardeen J.M., Carter B., Hawking S.W., The four laws of black hole mechanics, Comm. Math. Phys. 31
(1973), 161–170.
[8] Barvinsky A.O., Frolov V.P., Zel’nikov A.I., The wave function of a black hole and the dynamical origin of
entropy, Phys. Rev. D 51 (1995), 1741–1763.
[9] Bekenstein J.D., Black holes and entropy, Phys. Rev. D 7 (1973), 2333–2346.
[10] Bianchi E., Magliaro E., Perini C., Coherent spin-networks, Phys. Rev. D 82 (2010), 024012, 7 pages,
arXiv:0912.4054.
[11] Bombelli L., Koul R.K., Lee J., Sorkin R.D., Quantum source of entropy for black holes, Phys. Rev. D 34
(1986), 373–383.
[12] Borja E.F., Garay I., Strobel E., Revisiting the quantum scalar field in spherically symmetric quantum
gravity, Classical Quantum Gravity 29 (2012), 145012, 19 pages, arXiv:1201.4229.
[13] Brown J.D., York Jr. J.W., Quasilocal energy and conserved charges derived from the gravitational action,
Phys. Rev. D 47 (1993), 1407–1419, gr-qc/9209012.
[14] Corichi A., Black holes in loop quantum gravity: recent advances, J. Phys. Conf. Ser. 140 (2008), 012006,
13 pages, arXiv:0901.1302.
[15] Dasgupta A., Entanglement entropy and Bekenstein–Hawking entropy of black holes, in preparation.
[16] Dasgupta A., Coherent states for black holes, J. Cosmol. Astropart. Phys. 2003 (2003), no. 8, 004, 36 pages,
hep-th/0305131.
[17] Dasgupta A., Entropic origin of Hawking radiation, in Proceedings of the Twelfth Marcel Grossmann Mee-
ting on General Relativity (Paris, 2009), Editors T. Damour, R.T. Jantzen, R. Ruffini, World Scientific,
Singapore, 2012, 1132–1134, arXiv:1003.0441.
http://dx.doi.org/10.1103/PhysRevLett.108.051301
http://arxiv.org/abs/1111.4962
http://arxiv.org/abs/1105.5582
http://arxiv.org/abs/1201.4598
http://dx.doi.org/10.1103/PhysRevLett.80.904
http://arxiv.org/abs/gr-qc/9710007
http://dx.doi.org/10.1103/PhysRevLett.69.237
http://arxiv.org/abs/hep-th/9203079
http://arxiv.org/abs/1203.0174
http://dx.doi.org/10.1007/BF01645742
http://dx.doi.org/10.1103/PhysRevD.51.1741
http://dx.doi.org/10.1103/PhysRevD.7.2333
http://dx.doi.org/10.1103/PhysRevD.82.024012
http://arxiv.org/abs/0912.4054
http://dx.doi.org/10.1103/PhysRevD.34.373
http://dx.doi.org/10.1088/0264-9381/29/14/145012
http://arxiv.org/abs/1201.4229
http://dx.doi.org/10.1103/PhysRevD.47.1407
http://arxiv.org/abs/gr-qc/9209012
http://dx.doi.org/10.1088/1742-6596/140/1/012006
http://arxiv.org/abs/0901.1302
http://dx.doi.org/10.1088/1475-7516/2003/08/004
http://arxiv.org/abs/hep-th/0305131
http://arxiv.org/abs/1003.0441
Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics 25
[18] Dasgupta A., Semi-classical quantization of spacetimes with apparent horizons, Classical Quantum Gravity
23 (2006), 635–672, gr-qc/0505017.
[19] Dasgupta A., Semiclassical horizons, Can. J. Phys. 86 (2008), 659–662, arXiv:0711.0714.
[20] Dasgupta A., Time evolution of horizons, J. Modern Phys. 3 (2012), 1289–1297, arXiv:1007.1437.
[21] Dittrich B., Introduction to loop quantum gravity, Lectures given at University of New Brunswick, 2006.
[22] Doná P., Speziale S., Introductory lectures to loop quantum gravity, arXiv:1007.0402.
[23] Freidel L., Livine E.R., U(N) coherent states for loop quantum gravity, J. Math. Phys. 52 (2011), 052502,
21 pages, arXiv:1005.2090.
[24] Gambini R., Pullin J., Spherically symmetric gravity coupled to a scalar field with a local Hamiltonian:
the complete initial-boundary value problem using metric variables, Classical Quantum Gravity 30 (2013),
025012, 7 pages, arXiv:1207.6028.
[25] Hall B.C., The Segal–Bargmann “coherent state” transform for compact Lie groups, J. Funct. Anal. 122
(1994), 103–151.
[26] Hawking S.W., Black holes in general relativity, Comm. Math. Phys. 25 (1972), 152–166.
[27] Hawking S.W., Particle creation by black holes, Comm. Math. Phys. 43 (1975), 199–220.
[28] Hawking S.W., Horowitz G.T., The gravitational Hamiltonian, action, entropy and surface terms, Classical
Quantum Gravity 13 (1996), 1487–1498, gr-qc/9501014.
[29] Husain V., Mann R.B., Thermodynamics and phases in quantum gravity, Classical Quantum Gravity 26
(2009), 075010, 6 pages, arXiv:0812.0399.
[30] Husain V., Terno D., Dynamics and entanglement in spherically symmetric quantum gravity, Phys. Rev. D
81 (2010), 044039, 11 pages, arXiv:0903.1471.
[31] Husain V., Winkler O., Quantum Hamiltonian for gravitational collapse, Phys. Rev. D 73 (2006), 124007,
8 pages, gr-qc/0601082.
[32] Magliaro E., Marcianó A., Perini C., Coherent states for FLRW space-times in loop quantum gravity, Phys.
Rev. D 83 (2011), 044029, 9 pages, arXiv:1011.5676.
[33] Majumdar P., Holography, gauge-gravity connection and black hole entropy, Internat. J. Modern Phys. A
24 (2009), 3414–3425, arXiv:0903.5080.
[34] Prigogine I., From being to becoming: time and complexity in physical sciences, Freeman, San Francisco,
CA, 1980.
[35] Reinsch M.W., Morehead J.J., Asymptotics of Clebsch–Gordan coefficients, J. Math. Phys. 40 (1999),
4782–4806, math-ph/9906007.
[36] Sahlmann H., Thiemann T., Towards the QFT on curved spacetime limit of QGR. I. A general scheme,
Classical Quantum Gravity 23 (2006), 867–908, gr-qc/0207030.
[37] Solodukhin S., Entanglement entropy of black holes, Living Rev. Relativity 14 (2011), 8, 96 pages,
arXiv:1104.3712.
[38] Srednicki M., Entropy and area, Phys. Rev. Lett. 71 (1993), 666–669, hep-th/9303048.
[39] Strominger A., Vafa C., Microscopic origin of the Bekenstein–Hawking entropy, Phys. Lett. B 379 (1996),
99–104, hep-th/9601029.
[40] ’t Hooft G., On the quantum structure of a black hole, Nuclear Phys. B 256 (1985), 727–745.
[41] Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical
Physics, Cambridge University Press, Cambridge, 2007.
[42] Thiemann T., Winkler O., Gauge field theory coherent states (GCS). II. Peakedness properties, Classical
Quantum Gravity 18 (2001), 2561–2636, hep-th/0005237.
[43] Varshalovich D.A., Moskalev A.N., Khersonskii V.K., Quantum theory of angular momentum, World Scien-
tific Publishing Co. Inc., Teaneck, NJ, 1988.
http://dx.doi.org/10.1088/0264-9381/23/3/007
http://arxiv.org/abs/gr-qc/0505017
http://dx.doi.org/10.1139/p07-189
http://arxiv.org/abs/0711.0714
http://dx.doi.org/10.4236/jmp.2012.329166
http://arxiv.org/abs/1007.1437
http://arxiv.org/abs/1007.0402
http://dx.doi.org/10.1063/1.3587121
http://arxiv.org/abs/1005.2090
http://dx.doi.org/10.1088/0264-9381/30/2/025012
http://arxiv.org/abs/1207.6028
http://dx.doi.org/10.1006/jfan.1994.1064
http://dx.doi.org/10.1007/BF01877517
http://dx.doi.org/10.1007/BF02345020
http://dx.doi.org/10.1088/0264-9381/13/6/017
http://dx.doi.org/10.1088/0264-9381/13/6/017
http://arxiv.org/abs/gr-qc/9501014
http://dx.doi.org/10.1088/0264-9381/26/7/075010
http://arxiv.org/abs/0812.0399
http://dx.doi.org/10.1103/PhysRevD.81.044039
http://arxiv.org/abs/0903.1471
http://dx.doi.org/10.1103/PhysRevD.73.124007
http://arxiv.org/abs/gr-qc/0601082
http://dx.doi.org/10.1103/PhysRevD.83.044029
http://dx.doi.org/10.1103/PhysRevD.83.044029
http://arxiv.org/abs/1011.5676
http://dx.doi.org/10.1142/S0217751X09047016
http://arxiv.org/abs/0903.5080
http://dx.doi.org/10.1063/1.533000
http://arxiv.org/abs/math-ph/9906007
http://dx.doi.org/10.1088/0264-9381/23/3/019
http://arxiv.org/abs/gr-qc/0207030
http://arxiv.org/abs/1104.3712
http://dx.doi.org/10.1103/PhysRevLett.71.666
http://arxiv.org/abs/hep-th/9303048
http://dx.doi.org/10.1016/0370-2693(96)00345-0
http://arxiv.org/abs/hep-th/9601029
http://dx.doi.org/10.1016/0550-3213(85)90418-3
http://dx.doi.org/10.1017/CBO9780511755682
http://dx.doi.org/10.1017/CBO9780511755682
http://dx.doi.org/10.1088/0264-9381/18/14/301
http://dx.doi.org/10.1088/0264-9381/18/14/301
http://arxiv.org/abs/hep-th/0005237
1 Introduction
2 Coherent states in loop quantum gravity
3 Entanglement entropy
3.1 The brickwall model
3.2 Scalar entanglement for a bifurcate horizon
4 Gravitational entropy
4.1 Classical correlations
4.2 Quantum correlations
5 Time evolution
5.1 Semi classical Hamiltonian
5.2 Hawking radiation
5.3 Irreversible systems
6 Conclusion
A Intertwiners
B Tracing
References
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