7.1 Technical 

• Solutions of the hamiltonian constraints. One of the most surprising results of the theory is that it has been possible to find exact solutions of the hamiltonian constraint. This follows from the key result that the action of the hamiltonian constraints is non vanishing only over nodes of the s-knots [183, 184]. Therefore s-knots without nodes are physical states that solve the quantum Einstein dynamics. There is an infinite number of independent states of this sort, classified by conventional knot theory. The physical interpretation of these solutions is still rather obscure. Various other solutions have been found. See the recent review [82] and references therein. See also [113, 131, 55, 56, 57, 58, 156, 94, 129]. In particular, Pullin has studied in detail solutions related to the Chern-Simon term in the connection representation and to the Jones polynomial in the loop representation. According to a celebrated result by Witten [212], the two are the loop transform of each other.
• Time evolution. Strong field perturbation expansion. ``Topological Feynman rules''. Trying to describe the temporal evolution of the quantum gravitational field by solving the hamiltonian constraint yields the conceptually well-defined [168], but notoriously non-transparent, frozen-time formalism. An alternative is to study the evolution of the gravitational degrees of freedom with respect to some matter variable, coupled to the theory, which plays the role of a phenomenological ``clock''. This approach has lead to the tentative definition of a physical hamiltonian [188, 48], and to a preliminary investigation of the possibility of transition amplitudes between s-knot states, order by order in a (strong coupling) perturbative expansion [175]. In this context, diffeomorphism invariance, combined with the key result that the hamiltonian constraint acts on nodes only, implies that the ``Feynman rules'' of such an expansion are purely topological and combinatorial.
• Fermions. Fermions have been added to the theory [150, 151, 35, 207]. Remarkably, all the important results of the pure GR case survive in the GR+fermions theory. Not surprisingly, fermions can be described as open ends of ``open spin networks''.
• Maxwell and Yang-Mills. The extension of the theory to the Maxwell field has been studied in [132, 91]. The extension to Yang-Mills theory has been explored recently in [200]. In [200], Thiemann shows that the Yang-Mills term in the quantum hamiltonian constraint can be defined in a rigorous manner, extending the methods of [206, 201, 202]. A remarkable result in this context is that ultraviolet divergences do not seem to appear, strongly supporting the expectation that the natural cut off introduced by quantum gravity might cure the ultraviolet difficulties of conventional quantum field theory.
• Application to other theories. The loop representation has been applied in various other contexts such as 2+1 gravity [13, 146, 20] (on 2+1 quantum gravity, in the loop and in other representations, see [62]) and others [21].
• Lattice and simplicial models. A number of interesting discretized versions of the theory are being studied. See in particular [141, 159, 94, 79].

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