## 6.9 Dynamics

Finally, the definition of the theory is completed by giving the hamiltonian constraint. A number of approaches to the definition of a hamiltonian constraint have been attempted in the past, with various degrees of success. Recently, however, Thiemann has succeeded in providing a regularization of the hamiltonian constraint that yields a well defined, finite operator. Thiemann's construction [206, 201, 202] is based on several clever ideas. I will not describe it here. Rather, I will sketch below the final form of the constraint (for the Lapse=1 case), following [175].

I begin with the Euclidean hamiltonian constraint. We have

Here i labels the nodes of the s-knot s ; (IJ) labels couples of (distinct) links emerging from i . are the colors of the links emerging from i . is the operator that acts on an -knot by: (i) creating two additional nodes, one along each of the two links I and J ; (ii) creating a novel link, colored 1, joining these two nodes, (iii) assigning the coloring and, respectively, to the links that join the new formed nodes with the node i . This is illustrated in Figure 2.

Figure 2: Action of .

The coefficients , which are finite, can be expressed explicitly (but in a rather laborious way) in terms of products of linear combinations of 6- j symbols of SU (2), following the techniques developed in detail in [77]. Some of these coefficients have been explicitly computed [51]. The Lorentzian hamiltonian constraint is given by a similar expression, but quadratic in the operators.

The operator defined above is obtained by introducing a regularized expression for the classical hamiltonian constraint, written in terms of elementary loop observables, turning these observables into the corresponding operators and taking the limit. The construction works rather magically, relying on the fact, first noticed in [188], that certain operator limits turn out to be finite on diff invariant states, thanks to the fact that, for and , sufficiently small, and are diffeomorphic equivalent. Thus, here diff invariance plays again a crucial role in the theory.

For a discussion of the problems raised by the Thiemann operator and of the variant proposed, see section 8 .

 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