hamiltonian and diffeomorphism generators of a quantum field in r x s^2 spacetime

Within the adm formalism, spacetime is foliated by spacelike hypersurfaces, dividing spacetime into space and time. in this formalism, the evolution of fields can be separated into evolution on these hypersurfaces and evolution from one hypersurface to the next. the commutators of these generators form an algebra known as the dirac algebra or hypersurface deformation algebra. in this paper, we expand the matter field on the 3-dimensional r x s^2 spacetime using the basis of spherical harmonics on s^2. inspired by the approach of second quantization, we promote the coefficients of this expansion to creation and annihilation operators. we demonstrate that the algebra of the evolution generators, determines the hamiltonian and diffeomorphism generators on s^2. the tensor character of the matter field is classified based on its lie derivative with respect to its diffeomorphism generators. this quantization method determines the ordering of creation and annihilation operators in the evolution generators.