Dimensional regularization and the n-wave procedure for scalar fields in many-dimensional quasi-Euclidean spaces

We obtain the vacuum expectation values of the energy momentum tensor for a scalar field arbitrarily coupled to a curvature in the case of an N-dimens ional quasi-Euclidean space-time; the vacuum is defined in accordance with the Hamiltonian diagonalization method. We extend the n-wave procedure to t he many-dimensional case. We find all the counterterms in the case N = 5 an d the counterterms for the conformal scalar field in the cases N = 6, 7. We determine the geometric structure of the first three counterterms in the N -dimensional case. We show that all the subtractions in the four-dimensiona l case and the first three subtractions in the many-dimensional case corres pond to the renormalization of the parameters in the bare gravitational Lag rangian. We discuss the geometric structure of the other counterterms in th e many-dimensional case and the problem of eliminating the conformal anomal y in the four-dimensional case.