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 $N$-dimensional quasi-Euclidean space-time; the vacuum is defined in accordance with the Hamiltonian diagonalization method. We extend the $n$-wave procedure to the many-dimensional case. We find all the counterterms in the case $N=5$ and 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-dimensional case and the first three subtractions in the many-dimensional case correspond to the renormalization of the parameters in the bare gravitational Lagrangian. We discuss the geometric structure of the other counterterms in the many-dimensional case and the problem of eliminating the conformal anomaly in the four-dimensional case.