Loop quantum gravity - a short review

Abstract

In this article we review the foundations and the present status of loop
quantum gravity. It is short and relatively non-technical, the emphasis is on
the ideas, and the flavor of the techniques. In particular, we describe the
kinematical quantization and the implementation of the Hamilton constraint, as
well as the quantum theory of black hole horizons, semiclassical states, and
matter propagation. Spin foam models and loop quantum cosmology are mentioned
only in passing, as these will be covered in separate reviews to be published
alongside this one.

Full text

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Loop quantum gravity – a short review∗
Hanno Sahlmann
Institute for Theoretical Physics, Karlsruhe University
Karlsruhe Institute for Technology
Preprint KA-TP-19-2009
Abstract
In this article we review the foundations and the present status of loop quantum
gravity. It is short and relatively non-technical, the emphasis is on the ideas, and
the flavor of the techniques. In particular, we describe the kinematical quantization
and the implementation of the Hamilton constraint, as well as the quantum theory
of black hole horizons, semiclassical states, and matter propagation. Spin foam
models and loop quantum cosmology are mentioned only in passing, as these will
be covered in separate reviews to be published alongside this one.
1Introduction
Loopquantumgravityisanon-perturbativeapproachtothequantumtheoryofgravity,
in which no classical background metric is used. In particular, its starting point is not
a linearized theory of gravity. As a consequence, while it still operates according to
the rules of quantum field theory, the details are quite different from those of field
theories that operate on a fixed classical background space-time. It has considerable
successes to its credit, perhaps most notably a quantum theory of spatial geometry in
which quantities such as area and volume are quantized in units of the Planck length,
and a calculation of black hole entropy for static and rotating, charged and neutral
black holes. But there are also open questions, many of them surrounding the dynamics
(“quantum Einstein equations”) of the theory.
In contrast to other approaches such as string theory, loop quantum gravity is rather
modest in its aims. It is not attempting a grand unification, and hence is not based on an
overarching symmetry principle, or some deep reformulation of the rules of quantum
field theory. Rather, the goal is to quantize Einstein gravity in four dimensions. While,
as we will explain, a certain amount of unification of the description of matter and
gravity is achieved, in fact, the question of whether matter fields must have special
properties to be consistently coupled to gravity in the framework of loop quantum
gravity is an important open question.
Loop quantum gravity is, in its original version, a canonical approach to quantum grav-
ity. Nowadays, a covariant formulation of the theory exists in the so called spin foam
models. One of the canonical variables in loop quantum gravity is a connection, and
many distinct technical features (such as the ‘loops’ in its name) are directly related to
∗Talk delivered at the workshop “Foundations of Space and Time – Reflections on Quantum Gravity”
in honor of George Ellis, STIAS, Stellenbosch, South Africa, 10-14 August 2009.
1
arXiv:1001.4188v1 [gr-qc] 23 Jan 2010

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thechoiceofthesevariables. Anotherdistinctfeatureofloopquantumgravityisthatno
fixed classical geometric structures are used in the construction. New techniques had to
be developed for this, and the resulting Hilbert spaces look very different than those in
standard quantum field theory, with excitations of the fields one- or two-dimensional.
But it has also simplified the theory, since it can be shown that some choices made in
the quantum theory are actually uniquely fixed by the requirement of background in-
dependence. Furthermore, the requirement of background independence seems to lead
to a theory which is built around a very quantum mechanical gravitational “vacuum”,
a state with degenerate and highly fluctuating geometry. This is exciting, because it
means that when working in loop quantum gravity, the deep quantum regime of grav-
ity is ‘at one’s fingertips’. However, it also means that making contact with low energy
physics is a complicated endeavor. The latter problem has attracted a considerable
amount of work, but is still not completely solved. Another (related) challenge is to
fully understand the implementation of the dynamics. In loop quantum gravity the
question of finding quantum states that satisfy ‘quantum Einstein equations’ is refor-
mulated as finding states that are annihilated by the quantum Hamilton constraint. The
choices that go into the definition of this constraint are poorly understood in physical
terms. Moreover the constraint should be implemented in an anomaly-free way, but
what this entails in practice, and whether existing proposals fulfill this requirement are
still under debate. This is partially due to the lack of physical observables with man-
ageable quantum counterpart, to test the physical implication of the theory.
While these challenges remain, remarkable progress has happened over the last couple
of years: The master constraint program has brought new ideas to bear on the imple-
mentation of the dynamics [1]. Progress has been made in identifying observables for
general relativity that can be used in the canonical quantization [2, 3, 4]. A revision of
the vertex amplitudes used in spin foam models has brought them in much more direct
contact to loop quantum gravity [5, 6]. And, last not least, in loop quantum cosmology,
theapplicationofthequantizationstrategyofloopquantumgravitytomini-superspace
models has become a beautiful and productive laboratory for the ideas of the full the-
ory, in which the quantization program of loop quantum gravity can be tested, and, in
many cases, brought to completion [7, 8, 9, 10]. The present review will not cover these
developments in any detail, partially because they are ongoing, and partially because
there will be separate reviews on group field theory and loop quantum cosmology pub-
lished alongside the present text. But we hope that it makes for good preparatory read-
ing. In fact, the basic connection between loop quantum gravity and spin foam models
is explained in section 3.3, the master constraint program is briefly described in section
3.2, and there are some references to loop quantum cosmology in section 4. Certainly
the present review can also not replace the much more complete and detailed reviews
that are available. We refer the interested reader in particular to [11, 12, 13].
The structure of the review is as follows: In section 2 we explain the classical theory
and kinematical quantization underlying loop quantum gravity. Section 3 covers the
implementation of the Hamilton constraint. In section 4 we consider some physical
aspects of the theory: quantized black hole horizons, semiclassical states, and matter
propagation. We close with an outlook on open problems and new ideas in section 5.
2Kinematical setup
Loop quantum gravity is a canonical quantization-approach to general relativity, thus
it is based on a splitting of space-time into time and space, and on a choice of canonical
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variables. Implicit in the splitting is the assumption that the space-time is globally hy-
perbolic. Whethertopologychangecanneverthelessbedescribedintheresultingquan-
tum theory is a matter of debate. The choice of canonical variables is characteristic to
loop quantum gravity: One of the variables is a connection, and hence the phase space
(before implementation of the dynamics) has the same form as that of Yang-Mills the-
ory. As with any canonical formulation of general relativity, the theory has constraints
that have to be handled properly both in the classical and in the quantum theory.
The quantization strategy applied in loop quantum gravity is that of Dirac, for the
case of first class constraints: First, a kinematical representation of the basic fields by
operators on a Hilbert space Hkinis constructed. In this representation, operators cor-
responding to the constraints are defined. Then, quantum solutions to the constraints
are sought. Such solutions, also called physical states, are quantum states that are in
the kernel of all the constraints. They form the physical Hilbert space Hphys. Finally,
observable quantities are quantized. The corresponding operators should form an al-
gebra A, and commute with the quantum constraints. Thus A leaves Hphysinvariant.
The pair (A,Hphys) then constitutes the quantum theory of the constrained system in
question. Technical aspects of this procedure have to be refined in loop quantum grav-
ity. For example, if the zero eigenvalue in the continuous part of the spectrum of one of
the constraints, the resulting physical space is not part of the Hilbert space but part of
its dual. But there are also some fundamental questions about this procedure, such as
what guides the choice of the kinematical Hilbert space, and how the quantization and
implementation of the constraints is checked. Also, it is notoriously difficult to write
down explicit examples of observables for general relativity in the canonical setting,
even in the classical theory.
While some of the above questions are not yet answered for loop quantum gravity, the
quantum theory is successful in many respects: It includes a fully quantized spatial
geometry, and an implementation of the constraints that is anomaly-free at least in a
certain sense. In the following, we will give a short, and mostly non-technical introduc-
tion to the kinematical aspects of the quantization. The quantization of the Hamilton
constraint will be discussed in section 3.
2.1Connection formulation of general relativity
Loop quantum gravity rests on a reformulation of ADM canonical gravity in terms
of variables similar to those of Yang-Mills theory. Ashtekar discovered a formulation
[14] in terms of a self-dual SL(2,C) connection, and its canonical conjugate, satisfying
suitable reality conditions. Loop quantum gravity came to use a formulation in terms
of an SU(2) connection [15] for technical reasons. Both of these are actually special cases
of a family of formulations depending on several parameters ([16] and literature given
there). We will only consider one of these, the Barbero-Immirzi parameter ι [17]. The
covariant description in this case is the Holst-Action
?
for an SL(2,C) connection ω and a vierbein e. In the limit ι → ∞, this is the well known
topological term, it depends on the geometry. But, in the absence of fermionic matter,
it does not change the equations of motion, as it vanishes identically on shell, due to
S[e,ω] =
?IJKLeI∧ eJ∧ FIJ(ω) +1
ιeI∧ eJ∧ FIJ(ω)
(2.1)
Palatini action of general relativity. The so called Holst term proportional to ι−1is not a
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the Bianchi identity.1In the presence of fermions, there are small effects that could in
principle be used to distinguish the formulation (2.1) from the Palatini formulation [18].
The Holst-term has a profound effect on the canonical formulation of the theory. A Leg-
endre transform of the Palatini action leads (after solving the second-class constraints)
back to the ADM-formulation, with spatial metric and exterior curvature as canonical
variables. The Legendre-transform of (2.1) with finite Barbero-Immirzi parameter leads,
however, to formulations in which one canonical variable is a connection: For ι = ±i
the theory has special symmetries and one obtains the Ashtekar formulation [14] in
terms of a self-dual SL(2,C) connection. For real ι, and after a partial gauge fixing that
gets rid of second class constraints, one obtains a canonical pair consisting of an SU(2)
connection AI
aand a corresponding canonical momentum Eb
J,
{AI
a(x),Eb
J(y)} = 8πGιδb
aδI
Jδ(x,y).
(2.2)
These fields take values on a spatial slice Σ of the manifold that was chosen in the
process of going over to the Hamilton formulation.
There are several constraints on these variables, and the Hamiltonian is a linear com-
bination of constraints. The equations for time evolution are the usual Hamilton equa-
tions, and together with the constraint equations they form a set of equations which
is completely equivalent to Einstein’s equations. The constraints can be written in the
following way:
GI= DaEa
Ca= Eb
I
(2.3)
(2.4)
IFI
ab
H =1
2?IJK
Ea
√detEFK
IEb
J
ab− (1 + ι2)
Ea
√detEKI
IEb
J
[aKJ
b]
(2.5)
where D is the covariant derivative induced by A, F is the curvature of A, and K is the
extrinsic curvature of Σ in space-time. They have a simple geometric interpretation:
GIgenerates gauge transformations on phase space. It is also called Gauss constraint
to highlight that it is completely analogous to the Gauss-law constraint that shows up
in electrodynamics. Cagenerates the transformations induced in phase space under
diffeomorphisms of Σ. It is therefore also called diffeomorphism constraint. Finally, H
generates (when the other constraints hold) the transformations induced in phase space
under deformations of (the embedding of) the hypersurface Σ in a timelike direction in
space-time. It is also called the Hamiltonian constraint, since such deformations can be
interpreted as time evolution.
The canonical momentum E has a direct geometric interpretation: It encodes the spatial
geometry:
|detq|qab= Ea
where qabis the metric induced on Σ by the space-time metric. Thus E is a densitized
triad field for q. The interpretation of A is slightly more involved.
IEb
JδIJ
(2.6)
AI
a= ΓI
a+ ιKI
a
(2.7)
where Γ is the spin connection related to E.
Matter fields can be added to the canonical description given above. This has to be
done with some care, so as to not change the structure of the gravitational sector.For the
fermionic sector this requires working with slightly unusual (“half density”) variables
[19].
1Actually, instead of adding this term, one can also add the Nieh-Yang term, which is topological. The
resulting canonical formulation is the same as that with a real Barbero-Immirzi parameter.
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2.2Kinematic representation
The basic variables for the quantization in loop quantum gravity are chosen in such a
wayastomaketheirtransformationbehaviorunderSU(2)andspatialdiffeomorphisms
as simple and transparent as possible. The obvious reason behind this goal is that one
wants to simplify the solution of the constraints as much as possible. Early ideas about
this go back to Rovelli and Smolin [20]. we follow here [21]. An obvious choice for the
connection A are its holonomies
hα[A] = Pexp
?
α
A,
(2.8)
or more generally, functions of such holonomies,
f[A] ≡ f(hα1[A],hα2[A],...,hαn[A])
(2.9)
for a finite number of paths α1,...,αn. Such functionals are also called cylindrical func-
tions.
For the field E a natural functional is its flux through surfaces S:
E[S,f] =
?
S
∗EIfI
(2.10)
where f is a function taking values in su(2)∗and *E is the two-form Ea?abcdxb∧ dxc.
To quantize cylindrical functions and fluxes, one is seeking a representation of the fol-
lowing algebraic relations on a Hilbert space:
f1· f2[A] = f1[A]f2[A]
[f,ES,r] = 8πιl2
[f,[ES1,r1,ES2,r2]] = (8πιl2
...
(ES,r)∗= ES,r,
PXS,r[f]
P)2[XS1,r1,XS2,r2][f]
(f[A])∗= f[A]
(2.11)
Here, X is a certain derivation on the space of cylindrical functions. As an example,
consider the case of a surface S that is intersected transversally by a path e, splitting it
into a part e1incoming to, and a part e2outgoing from the surface. Then (with a certain
orientation of the surface assumed)
?
The commutators between cylindrical functions and fluxes come from the Poisson re-
lations 2.2. It is somewhat surprising to see that there are also non-trivial commutators
between fluxes. These are required to turn the algebra of fluxes and cylindrical func-
tions into a Lie-algebra, a structure that has representations in terms of operators on
Hilbert-spaces.
XS,rπ(he)j=
i
ri(p)πj(he1τihe2).
(2.12)
Loop quantum gravity employs a specific representation of (2.11) on a Hilbert space
Hkin. A basis for this Hilbert space is given by the so called generalized spin networks.
Such a network is by definition an oriented graph γ embedded in Σ, together with a
labeling of the edges and vertices of that graph: The edges are labeled by irreducible
representations of SU(2). A vertex carries elements of the dual of the tensor product
of all representations on the edges that are incoming to or outgoing from the vertex
5