The Quantized $O(1,2)/O(2)\times Z_2$ Sigma Model Has No Continuum Limit in Four Dimensions. I. Theoretical Framework

TL;DR

This paper develops a theoretical framework for a nonlinear sigma model with noncompact target space, showing that it does not have a continuum limit, which has implications for quantum gravity models.

Contribution

It provides a detailed theoretical analysis of the $O(1,2)/O(2)\times Z_2$ sigma model, including its geometry, symmetries, and renormalization, and connects these to lattice simulations.

Findings

Lattice simulations show the model approaches free massless fields as cutoff is removed.

The continuum limit preserving the model's properties does not exist.

Theoretical framework clarifies the model's nonperturbative structure.

Abstract

The nonlinear sigma model for which the field takes its values in the coset space $O(1,2)/O(2)\times Z_2$ is similar to quantum gravity in being perturbatively nonrenormalizable and having a noncompact curved configuration space. It is therefore a good model for testing nonperturbative methods that may be useful in quantum gravity, especially methods based on lattice field theory. In this paper we develop the theoretical framework necessary for recognizing and studying a consistent nonperturbative quantum field theory of the $O(1,2)/O(2)\times Z_2$ model. We describe the action, the geometry of the configuration space, the conserved Noether currents, and the current algebra, and we construct a version of the Ward-Slavnov identity that makes it easy to switch from a given field to a nonlinearly related one. Renormalization of the model is defined via the effective action and via current algebra. The two definitions are shown to be equivalent. In a companion paper we develop a lattice formulation of the theory that is particularly well suited to the sigma model, and we report the results of Monte Carlo simulations of this lattice model. These simulations indicate that as the lattice cutoff is removed the theory becomes that of a pair of massless free fields. Because the geometry and symmetries of these fields differ from those of the original model we conclude that a continuum limit of the $O(1,2)/O(2)\times Z_2$ model which preserves these properties does not exist.