]. 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].,
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.,
56,
57,
58,
59,
156,
92,
94,
131,
89].,
146,
20], Maxwell [21], linearized gravity [22], and, much later, 2d Yang-Mills theory [19].]. These states, denoted ``weaves'', have a ``polymer'' like
structure at short scale, and can be viewed as a formalization
of Wheeler's ``spacetime foam''.
algebraic framework
], 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.] 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.,
78].,
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].
measure and the scalar product
,
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].], 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].] 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].,
159,
94,
79].] 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).], Roberto DePietri proved the equivalence of the two
formalisms, using ideas from Thiemann [205] and Lewandowski [139].,
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.
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.], on the basis of the ideas of Kirill Krasnov [134,
135]. Recently,
Ashtekar, Baez, Corichi and Krasnov
have announced an alternative derivation [11].,
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.,
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].
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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 |