EFFECTIVE LAGRANGIAN AND THE BACK-REACTION PROBLEM IN A SELF-INTERACTING O(N) SCALAR THEORY IN CURVED SPACETIME

A derivation of the one-loop effective Lagrangian in the self-interact ing O(N) scalar theory, in slowly varying gravitational fields, is pre sented (using zeta-function regularization and heat-kernel techniques) . The result is given in terms of the expansion in powers of the curva ture tensors (up to quadratic terms) and their derivatives, as well as in derivatives of the background scalar field (up to second derivativ es). The renormalization group improved effective Lagrangian is studie d, which gives the leading-log approach of the whole perturbation theo ry. An analysis of the effective equations (back-reaction problem) on the static hyperbolic spacetime R(2) x H-2/Gamma is carried out for th e simplest version of the theory: m(2) = 0 and N = 1. The possibility of the existence of the solution R(2) x H-2/Gamma, induced by purely q uantum effects, is discussed.