3 The Development of Loop Gravity
The following chronology does not exhaust the history of loop gravity, it only indicates some steps in the
construction of the theory, in order to provide orientation.
- Connection formulation of general relativity
Loop gravity is based on the “Ashtekar formulation” of classical general relativity. (Abhay
Ashtekar calls it “connectio-dynamics”, in contrast to Wheeler’s “geometro-dynamics”.) [271,
16, 17]. Many recent works in loop gravity are based on a real variant of the original
Ashtekar connection whose utility for Lorentzian general relativity has been emphasized by
Barbero [60, 61, 62, 63].
- Wilson-loop solutions of the Hamiltonian constraint
Soon after the introduction of the Ashtekar variables, Ted Jacobson and Lee Smolin realize
that the Wheeler–DeWitt equation, reformulated in terms of the new variables, admits a simple
class of exact solutions: the traces of the holonomies of the Ashtekar connection around smooth
non-self-intersecting loops . In other words: the Wilson loops of the Ashtekar connection
solve the Wheeler–DeWitt equation if the loops are smooth and non self-intersecting.
- The loop representation
The discovery of the Jacobson–Smolin Wilson-loop solutions suggests that one “change basis in
the Hilbert space of the theory”, choosing the Wilson loops as the new basis states for quantum
gravity [264, 240, 263, 265]. Quantum states can be represented in terms of their expansion
on the loop basis, namely as functions on a space of loops. This idea is well known in the context
of canonical lattice Yang–Mills theory . Its application to continuous Yang–Mills theory
had been explored by Gambini and Trias [132, 133]. The difficulties of the loop representation
in the context of Yang–Mills theory are cured by the diffeomorphism invariance of GR (see
Section 6.4). The immediate results are two: (i) the diffeomorphism constraint is completely
solved by knot states (loop functionals that depend only on the knotting of the loops), making
earlier suggestions by Smolin on the role of knot theory in quantum gravity  concrete;
and (ii) knot states with support on non-self-intersecting loops are proven to be solutions of
all quantum constraints, namely exact physical states of quantum gravity.
- Exact states of quantum gravity
The investigation of exact solutions of the quantum constraint equations, and their relation
to knot theory (in particular to the Jones polynomial and other knot invariants), started soon
after the formulation of the theory [149, 78, 79, 80, 81, 233, 127, 129, 163, 122, 107].
- Model theories
The years immediately following the discovery of the loop formalism are mostly dedicated to
understanding the loop representation by studying it in simpler contexts, such as 2+1 general
relativity [26, 195, 36], Maxwell , linearized gravity , and, later, 2D Yang–Mills
- Discreteness: I. Weaves
The first indication that the theory predicts Planck-scale discreteness derives from studying
the states that approximate flat geometries on large scale . These states, called “weaves”,
can be viewed as a formalization of Wheeler’s “spacetime foam”. Surprisingly, these states turn
out not to require that the average spacing of the loops go to zero.
- algebraic framework
Abhay Ashtekar and Chris Isham show that the loop transform can be given a rigorous
mathematical foundation, and lay the foundation for a mathematical systematization of the
loop ideas, based on algebra ideas .
Fermion coupling is explored in [206, 207]. Later, matter’s kinematics is studied by Baez and
Krasnov [168, 54], while Thiemann extends his results on dynamics to the coupled Einstein
Yang–Mills system in .
- Ashtekar–Lewandowski measure and scalar product
Abhay Ashtekar and Jerzy Lewandowski lay the foundation of the differential formulation of
loop quantum gravity by constructing a diffeomorphism-invariant measure on the space of
(generalized) connections [28, 29, 30]. They give a mathematically-rigorous construction of the
state space of the theory. They define a consistent scalar product and prove that the quantum
operators in the theory are consistently defined. Key contributions to the understanding of
the measure are given by John Baez, Don Marolf and Josè Mourão [43, 44, 42, 198]. Don
Marolf clarifies the use of formal group integration for solving the constraints [194, 196, 197].
The definitive setting of the two versions of the formalism is completed shortly after for
the loop formalism (the actual loop representation)  and for the differential formalism
(the connection representation) . Roberto DePietri proves the equivalence of the two
formalisms , using ideas from Thiemann  and Lewandowski .
- Discreteness: II. Area and volume eigenvalues
Certain geometrical quantities, in particular area and volume, are represented by operators
that have discrete eigenvalues. The first set of these eigenvalues is obtained in . The
result is confirmed and extended using a number of different techniques. Renate Loll [183, 184]
uses lattice techniques to analyze the volume operator and correct a numerical error in .
Ashtekar and Lewandowski [174, 31] recover and complete the computation of the spectrum
of the area using the connection representation and new regularization techniques. In turn,
the full spectrum of the area is then recovered using the loop representation . The
general eigenvalues of the volume are computed . Lewandowski clarifies the relation between
different versions of the volume operator .
- Spin networks
A long standing problem with the loop basis was its overcompleteness. A technical, but crucial
step in understanding the theory is the discovery of the spin-network basis, which solves this
overcompleteness . The idea derives from the work of Roger Penrose [224, 223], from
analogous bases used in lattice gauge theory, and from ideas by Lewandowski  and Jorge
Pullin on the relevance of graphs and nodes for the theory. The spin network formalism is
cleaned up and clarified by John Baez [47, 48].
- Hamiltonian constraint
The first version of the loop Hamiltonian constraint [263, 265] is studied and repeatedly
modified in a number of works [149, 71, 81, 79, 78, 80, 233, 127, 74]. A key step is the
realization that certain regularized loop operators have finite limits on diffeomorphism-invariant
states . The search culminates with the work of Thomas Thiemann, who is able to
construct a fully well-defined anomaly-free Hamiltonian operator [285, 289, 291].
- Black-hole entropy
The derivation of the Bekenstein–Hawking formula for the entropy of a black hole from loop
quantum gravity is obtained , on the basis of the ideas of Kirill Krasnov [170, 171] and
Lee Smolin . The theory is developed and made rigorous by Ashtekar, Baez, Corichi and
- Spin foams
A “sum over histories” spacetime formulation of loop quantum gravity is derived [257,
236] from the canonical theory. The resulting covariant theory turns out to be a sum
over topologically non-equivalent surfaces, realizing suggestions by Baez [45, 42, 47, 41],
Reisenberger [235, 234] and Iwasaki  that a covariant version of loop gravity should
look like a theory of surfaces. Baez studies the general structure of theories defined in this
manner  and introduces the term “spin foam”.
- The Barrett–Crane vertex
John Barrett and Louis Crane introduce the Barrett–Crane vertex amplitude , which will
become one of the main tools for exploring dynamics in loop gravity and in other approaches.
- Group field theory
The definition of the Barrett–Crane spin-foam model – in its different versions – is completed
in [97, 115], where group-field-theory techniques are also introduced, deriving them from
topological field theories.
- Quantum cosmology
The application of loop quantum gravity to cosmology is started by Martin Bojowald , to
be later extensively developed by Ashtekar, Bojowald and others.
- Spin-foam finiteness
Alejandro Perez gives the first proof of finiteness of a spin-foam model .
- Master constraint
Thomas Thiemann introduces the idea of replacing the full set of quantum constraints with a
single (“master”) constraint .
- Black hole singularity at r = 0
Leonardo Modesto , and, independently, Ashtekar and Bojowald , apply techniques
derived from quantum cosmology to explore the singularity at the center of a black
hole, showing that this is controlled by the quantum theory.
- Loop/spin-foam equivalence in 3D
Karim Noui and Alejandro Perez prove the equivalence of loop quantum gravity and the
spin-foam formalism in three-dimensional quantum gravity .
- The LOST theorem
A key uniqueness theorem for the representation used in loop quantum gravity is proved by
Lewandowski, Okolow, Sahlmann and Thiemann  and, independently and in a slightly
different version, by Christian Fleischhack .
- Noncommutative geometry from loop quantum gravity
Laurent Freidel and Etera Livine show that the low-energy limit of quantum gravity coupled
with matter in three dimensions is equivalent to a field theory on a noncommutative
- Graviton propagator
Beginning of the computation of n-point functions from loop quantum gravity  and first
computation of some components of the graviton propagator .
- The new vertex and the loop/spin-foam relation in 4D
A vertex amplitude correcting some difficulties of the Barrett–Crane model is introduced
in , and gives rise to rapid development [105, 181, 182, 227] leading to the formulation
of a class of spin-foam models that provide a covariant definition of the LQG dynamics [104,