Canonical approach to quantum gravity

Dirac's canonical programme for the quantisation of gravity is discussed. The order of operators in the 'kinetic' part of the generator of normal deformations, H_{perpendicular to} , is chosen to give the Laplace-Beltrami form. This follows from the requirement of invariance of the quantum theory under arbitrary contact transformations of the canonical coordinates, g_ij(x). The remaining constraint operators H_i are read off from the commutation relations for H_{perpendicular to} . A regularisation is introduced in order to give meaning to the (divergent) formal expressions of these operators. In this regularisation, the operators H_mu are Hermitian (symmetric) and the algebra they generate is isomorphic to the classical one. The structure functions of the algebra are independent of the number of spatial dimensions and the infinite renormalisation constant (ultraviolet cut-off). If there is a cosmological constant, the infinities can be absorbed by an appropriate renormalisation of the physical parameters.