Effective Lagrangian and the back-reaction problem in a self-interacting $O(N)$ scalar theory in curved spacetime

TL;DR

This paper derives the one-loop effective Lagrangian for a self-interacting $O(N)$ scalar field in curved spacetime, analyzes the back-reaction problem, and demonstrates the quantum-induced existence of a specific spacetime solution.

Contribution

It provides a detailed derivation of the effective Lagrangian using heat-kernel techniques and explores quantum back-reaction effects in a specific curved spacetime setting.

Findings

Existence of a quantum-induced $ eals^2 imes H^2/\Gamma$ spacetime solution.

Effective Lagrangian expressed in curvature and scalar derivatives up to quadratic order.

Leading-log approximation of the renormalization group improved effective action.

Abstract

A derivation of the one-loop effective Lagrangian in the self-interacting $O(N)$ scalar theory, in slowly varying gravitational fields, is presented (using $\zeta$-regularization and heat-kernel techniques). The result is given in terms of the expansion in powers of the curvature tensors (up to quadratic terms) and their derivatives, as well as in derivatives of the background scalar field (up to second derivatives). The renormalization group improved effective Lagrangian is studied, what gives the leading-log approach of the whole perturbation theory. An analysis of the effective equations (back-reaction problem) on the static hyperbolic spacetime $\reals^2 \times H^2/\Gamma$ is carried out for the simplest version of the theory: $m^2=0$ and $N=1$. The existence of the solution $\reals^2 \times H^2/\Gamma$, induced by purely quantum effects, is shown.