## 6.10 Unfreezing the frozen time formalism: the covariant form of loop quantum gravity

A recent development in the formalism is the translation of loop quantum gravity into spacetime covariant form. This was initiated in [181, 160] by following the steps that Feynman took in defining path integral quantum mechanics starting from the Schrödinger canonical theory. More precisely, it was proven in [160] that the matrix elements of the operator U (T)

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.

Figure 3: The elementary vertex.

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

Figure 4: A term of second order.

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 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de