23 research outputs found
The Quantized Sigma Model Has No Continuum Limit in Four Dimensions. I. Theoretical Framework
The nonlinear sigma model for which the field takes its values in the coset
space 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 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 model which preserves
these properties does not exist.Comment: 25 pages, no figure
Privately Subsidized Recycling Schemes and their Potential Harm to the Environment of Developing Countries: Does International Trade Law Have a Solution?
The Quantized Sigma Model Has No Continuum Limit in Four Dimensions. II. Lattice Simulation
A lattice formulation of the 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 vanishes for some value of the bare scale
constant~. The geodesic action has a special form that allows direct
access to the small- 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 -independent action which is noteworthy in being
unbounded from below yet yielding integrable averages. Both the exact action
and the -independent action are used to obtain from Monte
Carlo computations of field-field averages (2-point functions) and
current-current averages. Many consistency cross-checks are performed. It is
found that there is no value of for which 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
model has no continuum limit.Comment: 32 pages, 7 postscript figures, UTREL 92-0