# 3 History of Loop Quantum Gravity, Main Steps

The following chronology does not exhaust the literature on loop quantum gravity. It only indicates the key steps in the construction of the theory, and the first derivation of the main results. For more complete references, see the following sections. (Due to the attempt to group similar results, some things may appear a bit out of the chronological order.)
1986
Connection formulation of classical general relativity
Sen, Ashtekar.

Loop quantum gravity is based on the formulation of classical general relativity, which goes under the name of ``new variables'', or ``Ashtekar variables'', or ``connectio-dynamics'' (in contrast to Wheeler's ``geometro-dynamics''). In this formulation, the field variable is a self-dual connection, instead of the metric, and the canonical constraints are simpler than in the old metric formulation. The idea of using a self-dual connection as field variable and the simple constraints it yields were discovered by Amitaba Sen [190]. Abhay Ashtekar realized that in the SU (2) extended phase space a self-dual connection and a densitized triad field form a canonical pair [7, 8] and set up the canonical formalism based on such pair, which is the Ashtekar formalism. Recent works on the loop representation are not based on the original Sen-Ashtekar connection, but on a real variant of it, whose use has been introduced into Lorentzian general relativity by Barbero [39, 40, 41, 42].
1986
Wilson loop solutions of the hamiltonian constraint
Jacobson, Smolin
.
Soon after the introduction of the classical Ashtekar variables, Ted Jacobson and Lee Smolin realized in [128] 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-selfintersecting 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.
1987
The Loop Representation
Rovelli, Smolin
.
The discovery of the Jacobson-Smolin Wilson loop solutions prompted Carlo Rovelli and Lee Smolin [182, 163, 183, 184] to ``change basis in the Hilbert space of the theory'', choosing the Wilson loops as the new basis states for quantum gravity. 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 [211], and its application to continuous Yang-Mills theory had been explored by Gambini and Trias [95, 96], who developed a continuous ``loop representation'' much before the Rovelli-Smolin one. The difficulties of the loop representation in the context of Yang-Mills theory are cured by the diffeomorphism invariance of GR (see section 6.8 for details). The loop representation was introduced by Rovelli and Smolin as a representation of a classical Poisson algebra of ``loop observables''. The relation to the connection representation was originally derived in the form of an integral transform (an infinite dimensional analog of a Fourier transform) from functionals of the connection to loop functionals. Several years later, this loop transform was shown to be mathematically rigorously defined [12]. The immediate results of the loop representation were two: The diffeomorphism constraint was 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 [195] concrete; and (suitable [184, 196] extensions of) the knot states with support on non-selfintersecting loops were proven to be solutions of all quantum constraints, namely exact physical states of quantum gravity.
1988 -
Exact states of quantum gravity
Husain, Brügmann, Pullin, Gambini, Kodama
.
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 and has continued since [112, 56, 57, 58, 59, 156, 92, 94, 131, 89].
1989 -
Model theories
Ashtekar, Husain, Loll, Marolf, Rovelli, Samuel, Smolin, Lewandowski, Marolf, Thiemann
.
The years immediately following the discovery of the loop formalism were mostly dedicated to understanding the loop representation by studying it in simpler contexts, such as 2+1 general relativity [13, 146, 20], Maxwell [21], linearized gravity [22], and, much later, 2d Yang-Mills theory [19].
1992
Classical limit: weaves
Ashtekar, Rovelli, Smolin
.
The first indication that the theory predicts that Planck scale discreteness came from studying the states that approximate geometries flat on large scale [23]. These states, denoted ``weaves'', have a ``polymer'' like structure at short scale, and can be viewed as a formalization of Wheeler's ``spacetime foam''.
1992
algebraic framework
Ashtekar, Isham
.
In [12], Abhay Ashtekar and Chris Isham showed that the loop transform introduced in gravity by Rovelli and Smolin could be given a rigorous mathematical foundation, and set the basis for a mathematical systematization of the loop ideas, based on algebra ideas.
1993
Gravitons as embroideries over the weave
Iwasaki, Rovelli
.
In [125] Junichi Iwasaki and Rovelli studied the representation of gravitons in loop quantum gravity. These appear as topological modifications of the fabric of the spacetime weave.
1993
Alternative versions
Di Bartolo, Gambini, Griego, Pullin
.
Some versions of the loop quantum gravity alternative to the ``orthodox'' version have been developed. In particular, the authors above have developed the so called ``extended'' loop representation. See [80, 78].
1994
Fermions,
Morales-Tecotl, Rovelli
.
Matter coupling was beginning to be explored in [150, 151]. Later, matter's kinematics was studied by Baez and Krasnov [133, 35], while Thiemann extended his results on the dynamics to the coupled Einstein Yang-Mills system in [200].
1994
The measure and the scalar product
Ashtekar, Lewandowski, Baez
.
In [14, 15, 16] Ashtekar and Lewandowski set the basis of the differential formulation of loop quantum gravity by constructing its two key ingredients: a diffeomorphism invariant measure on the space of (generalized) connections, and the projective family of Hilbert spaces associated to graphs. Using these techniques, they were able to give a mathematically rigorous construction of the state space of the theory, solving long standing problems deriving from the lack of a basis (the insufficient control on the algebraic identities between loop states). Using this, they defined a consistent scalar product and proved that the quantum operators in the theory were consistent with all identities. John Baez showed how the measure can be used in the context of conventional connections, extended it to the non-gauge invariant states (allowing the E operator to be defined) and developed the use of the graph techniques [24, 28, 27]. Important contributions to the understanding of the measure were also given by Marolf and Mourão [149].
1994
Discreteness of area and volume eigenvalues
Rovelli, Smolin
.
In my opinion, the most significant result of loop quantum gravity is the discovery that certain geometrical quantities, in particular area and volume, are represented by operators that have discrete eigenvalues. This was found by Rovelli and Smolin in [186], where the first set of these eigenvalues was computed. Shortly after, this result was confirmed and extended by a number of authors, using very diverse techniques. In particular, Renate Loll [142, 143] used lattice techniques to analyze the volume operator and corrected a numerical error in [186]. Ashtekar and Lewandowski [138, 17] recovered and completed the computation of the spectrum of the area using the connection representation, and new regularization techniques. Frittelli, Lehner and Rovelli [84] recovered the Ashtekar-Lewandowski terms of the spectrum of the area, using the loop representation. DePietri and Rovelli [77] computed general eigenvalues of the volume. Complete understanding of the precise relation between different versions of the volume operator came from the work of Lewandowski [139].
1995
Spin networks - solution of the overcompleteness problem
Rovelli, Smolin, Baez
.
A long standing problem with the loop basis was its overcompleteness. A technical, but crucial step in understanding the theory has been the discovery of the spin-network basis, which solves this overcompleteness. This step was taken by Rovelli and Smolin in [187] and was motivated by the work of Roger Penrose [153, 152], by analogous bases used in lattice gauge theory and by ideas of Lewandowski [137]. Shortly after, the spin network formalism was cleaned up and clarified by John Baez [31, 32]. After the introduction of the spin network basis, all problems deriving from the incompleteness of the loop basis are trivially solved, and the scalar product could be defined also algebraically [77].
1995
Lattice
Loll, Reisenberger, Gambini, Pullin
.
Various lattice versions of the theory have appeared in [141, 159, 94, 79].
1995
Algebraic formalism / Differential formalism
DePietri, Rovelli / Ashtekar, Lewandowski, Marolf, Mourão, Thiemann
.
The cleaning and definitive setting of the two main versions of the formalisms was completed in [77] for the algebraic formalism (the direct descendent of the old loop representation); and in [18] for the differential formalism (based on the Ashtekar-Isham algebraic construction, on the Ashtekar-Lewandowski measure, on Don Marolf's work on the use of formal group integration for solving the constraints [145, 147, 148], and on several mathematical ideas by José Mourão).
1996
Equivalence of the algebraic and differential formalisms
DePietri
.
In [76], Roberto DePietri proved the equivalence of the two formalisms, using ideas from Thiemann [205] and Lewandowski [139].
1996
Hamiltonian constraint
Thiemann
.
The first version of the loop hamiltonian constraint is in [183, 184]. The definition of the constraint has then been studied and modified repeatedly, in a long sequence of works, by Brügmann, Pullin, Blencowe, Borissov and others [112, 47, 59, 57, 56, 58, 156, 92, 48]. An important step was made by Rovelli and Smolin in [185] with the realization that certain regularized loop operators have finite limits on knot states (see [140]). The search culminated with the work of Thomas Thiemann, who was able to construct a rather well-defined hamiltonian operator whose constraint algebra closes [206, 201, 202]. Variants of this constraint have been suggested in [192, 160] and elsewhere.
1996
Real theory: solution of the reality conditions problem
Barbero, Thiemann
.
As often stressed by Karel Kuchar, implementing the complicated reality condition of the complex connection into the quantum theory was, until 1996, the main open problem in the loop approach. Following the directions advocated by Fernando Barbero [39, 40, 41, 42], namely to use the real connection in the Lorentzian theory, Thiemann found an elegant way to completely bypass the problem.
1996
Black hole entropy
Krasnov, Rovelli
.
A derivation of the Bekenstein-Hawking formula for the entropy of a black hole from loop quantum gravity was obtained in [176], on the basis of the ideas of Kirill Krasnov [134, 135]. Recently, Ashtekar, Baez, Corichi and Krasnov have announced an alternative derivation [11].
1997
Anomalies
Lewandowski, Marolf, Pullin, Gambini
.
These authors have recently completed an extensive analysis of the issue of the closure of the quantum constraint algebra and its departures from the corresponding classical Poisson algebra [140, 90], following earlier pioneering work in this direction by Brügmann, Pullin, Borissov and others [54, 60, 88, 50]. This analysis has raised worries that the classical limit of Thiemann's hamiltonian operator might fail to yield classical general relativity, but the matter is still controversial.
1997
Sum over surfaces
Reisenberger Rovelli
.
A ``sum over histories'' spacetime formulation of loop quantum gravity was derived in [181, 160] from the canonical theory. The resulting covariant theory turns out to be a sum over topologically inequivalent surfaces, realizing earlier suggestions by Baez [26, 27, 31, 25], Reisenberger [159, 158] and Iwasaki [124] that a covariant version of loop gravity should look like a theory of surfaces. Baez has studied the general structure of theories defined in this manner [33]. Smolin and Markoupolou have explored the extension of the construction to the Lorentzian case, and the possibility of altering the spin network evolution rules [144].

 Loop Quantum Gravity Carlo Rovelli http://www.livingreviews.org/lrr-1998-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de