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].