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  and references therein. See also [113,
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 , 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 , 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 . 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 have been added to the theory [150,
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 . In , Thiemann shows that the Yang-Mills term in the quantum
hamiltonian constraint can be defined in a rigorous manner,
extending the methods of [206,
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,
20] (on 2+1 quantum gravity, in the loop and in other
representations, see ) and others .
Lattice and simplicial models.
A number of interesting discretized versions of the theory are
being studied. See in particular [141,