The quantized 0(1,2)/0(2) X Z 2 sigma model has no continuum limit in four dimensions.
11.A lattice formulation of the 0( 1,2 )/0( 2) X Z 2 sigma model is developed, based on the continuum theory presented in the preceding paper. Special attention is given to choosing a lattice action (the "geodesic" action) that is appropriate for fields having noncompact curved configuration spaces. A consistent continuum limit of the model exists only if the renormalized scale constant PR vanishes for some value of the bare scale constant p. The geodesic action has a special form that allows direct access to the small-P limit. In this limit half of the degrees of freedom can be integrated out exactly. The remaining degrees of freedom are those of a compact model having a P-independent action which is noteworthy in being unbounded from below yet yielding integrable averages. Both the exact action and the Pindependent action are used to obtain PR from Monte Carlo computations of field-field averages (twopoint functions) and current-current averages. Many consistency cross-checks are performed. It is found that there is no value of for which PR vanishes. This means that as the lattice cutoff is removed the theory becomes that of a pair of massless free fields. Because these fields have neither the geometry nor the symmetries of the original model we conclude that the 0( 1,2)/0(2)XZ2 model has no continuum limit.PACS number(s): 1 l.lO.Lm, 04.60. + n, ll.lO.Gh, 12.25. +e
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.
The nonuniversal behavior of two noncompact nonlinear sigma models is described. When these theories are defined on a lattice, the behavior of the order parameter (magnetization) near the critical point is sensitive to the details of the lattice definition. This is counter to experience and to expectations based on the ideas of universality.
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