obtained exponentiating the (Euclidean) hamiltonian constraint
in the proper time gauge (the operator that generates evolution
in proper time) can be expanded in a Feynman sum over paths. In
conventional QFT each term of a Feynman sum corresponds naturally
to a certain Feynman diagram, namely a set of lines in spacetime
meeting at vertices (branching points). A similar natural
structure of the terms appears in quantum gravity, but
surprisingly the diagrams are now given by
*surfaces*
is spacetime that branch at vertices. Thus, one has a
formulation of quantum gravity as a sum over surfaces in
spacetime. Reisenberger [158] and Baez [30] have argued in the past that such a formulation should exist,
and Iwasaki has developed a similar construction in 2+1
dimensions. Intuitively, the time evolution of a spin network in
spacetime is given by a colored surface. The surfaces capture the
gravitational degrees of freedom. The formulation is
``topological'' in the sense that one must sum over topologically
inequivalent surfaces only, and the contribution of each surface
depends on its topology only. This contribution is given by the
product of elementary ``vertices'', namely points where the
surface branches.

The transition amplitude between two s-knot states
and
in a proper time
*T*
is given by summing over all (branching, colored) surfaces
that are bounded by the two s-knots
and

The weight
of the surface
is given by a product over the
*n*
vertices
*v*
of
:

The contribution
of each vertex is given by the matrix elements of the
hamiltonian constraint operator between the two s-knots obtained
by slicing
immediately below and immediately above the vertex. They turn
out to depend only on the colors of the surface components
immediately adjacent the vertex
*v*
. The sum turns out to be finite and explicitly computable order
by order.

As in the usual Feynman diagrams, the vertices describe the elementary interactions of the theory. In particular, here one sees that the complicated structure of the Thiemann hamiltonian, which makes a node split into three nodes, corresponds to a geometrically very simple vertex. Figure 3 is a picture of the elementary vertex. Notice that it represents nothing but the spacetime evolution of the elementary action of the hamiltonian constraint, given in Figure 2.

An example of a surface in the sum is given in Figure 4.

The sum over surfaces version of loop quantum gravity provides a link with certain topological quantum field theories and in particular with the the Crane-Yetter model [71, 72, 73, 74, 75], which admit an extremely similar representation. For a discussion on the precise relation between topological quantum field theory and diffeomorphism invariant quantum field theory, see [160] and [171, 124, 83].

The idea of expressing the theory as a sum over surfaces has been developed by Baez [33], who has studied the general form of generally covariant quantum field theories formulated in this manner, and by Smolin and Markopoulou [144], who have studied how to directly capture the Lorentzian causal structure of general relativity modifying the elementary vertices. They have also explored the idea that the long range correlations of the low energy regime of the theory are related to the existence of a phase transition in the microscopic dynamics, and have found intriguing connections with the theoretical description of percolation.

Loop Quantum Gravity
Carlo Rovelli
http://www.livingreviews.org/lrr-1998-1
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