The quantized O(1,2)/O(2)×Z2sigma model has no continuum limit in four dimensions. I. Theoretical framework

Abstract

The nonlinear sigma model for which the field takes its values in the coset space O(1, 2)/O(2) × 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) × 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) × Z 2 model which preserves these properties does not exist.

Erratum to: “Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma-models” [Nucl. Phys. B 720 (2005) 235]

“…An important question is of course whether there exists a nontrivial continuum limit. Conventional wisdom would say no, because the models are (perturbatively) asymptotically free in the infrared but not in the ultraviolet ( [55,10]). To gain some feeling whether this is true beyond perturbation theory, it might be worthwhile to study simplified hierarchical versions of the Renormalization Group.…”

Failure of universality in noncompact lattice field theories

“…Instead, the relevant length scale here is µ −1 R which is associated with the interaction. The continuum limit is sought where the characteristic length scale measured in units of the lattice spacing, 1 √ β R , 1 Of course, non-numerical techniques may also be used.…”

“…The magnetization is relevant for the wavefunction renormalization [1], but apparently does not affect the S matrix of the theory (which is trivial).…”

“…Again, there is a striking qualitative difference between the two lattice models. Extensive numerical simulations of the geodesically-differenced SO(2, 1) model indicate clearly that this theory is trivial in the continuum limit [1,2]. Somewhat less thorough, but still convincing numerical studies for the naively-differenced theory [3] suggest the that this theory is also trivial.…”

“…Numerical evidence strongly suggests [1][2][3] that both lattice O(2, 1) models do belong to the same universality class in the sense that they both describe a pair of free uncoupled massless scalar fields in the continuum limit (they are trivial). The situation for the SO(1, 1) models is not as clear.…”