finds that their scattering amplitude has divergences which cannot be absorbed into the redefinition of the Newton constant. Canonical quantum gravity theorists do not accept this argument; however they have not so far provided an alternative calculation of the graviton scattering amplitude which could be used to understand what happens with the terms found non-renormalizable in the perturbative treatment. A long-held expectation is that in a theory of quantum geometry such as canonical quantum gravity that geometric quantities such as area and volume become quantum observables and take non-zero discrete values, providing a natural regulator which eliminates infinities from the theory including those coming from matter contributions. This `quantization' of geometric observables is in fact realized in loop quantum gravity (LQG). Canonical quantization in metric variables The quantization is based on decomposing the metric tensor as follows, where the summation over repeated indices is implied, the index 0 denotes time , Greek indices run over all values 0, . . . ,3 and Latin indices run over spatial values 1, . . ., 3. The function is called the lapse function and the functions are called the shift functions. The spatial indices are raised and lowered using the spatial metric and its inverse : and , , where is the Kronecker delta. Under this decomposition the Einstein–Hilbert Lagrangian becomes, up to total derivatives, where is the spatial scalar curvature computed with respect to the Riemannian metric and is the extrinsic curvature, where denotes Lie-differentiation, is the unit normal to surfaces of constant and denotes covariant differentiation with respect to the metric . Note that . DeWitt writes that the Lagrangian "has the classic form 'kinetic energy minus potential energy,' with the extrinsic curvature playing the role of kinetic energy and the negative of the intrinsic curvature that of potential energy." While this form of the Lagrangian is manifestly invariant under redefinition of the spatial coordinates, it makes general covariance opaque. Since the lapse function and shift functions may be eliminated by a gauge transformation, they do not represent physical degrees of freedom. This is indicated in moving to the Hamiltonian formalism by the fact that their conjugate momenta, respectively and , vanish identically (on shell and off shell). These are called primary constraints by Dirac. A popular choice of gauge, called synchronous gauge, is and , although they can, in principle, be chosen to be any function of the coordinates.