%% file: Example.tex (TeXsis version 2.17) % $Revision: 17.0 $ : $Date: 1994/11/30 21:53:40 $ : $Author: myers $ %======================================================================* % This is a sample paper typeset with TeXsis, to give you a quick idea % of how it's done. Note: this is just hacked together from an old % conference proceedings, so it's not a real paper. -EAM % This lets your manuscript run under either mTeXsis (from % mtexsis.tex) or the full TeXsis distribution \ifx\texsis\undefined \input mtexsis.tex \fi \texsis % this turns on TeXsis % Saying \draft puts a time-stamp, page number, etc. on the page, % but you don't want it for the final version of the paper. %%\draft % Document Format: uncomment one of these lines to select the style % in which the paper is printed: % %\preprint % Preprint style %\nuclproc % Nuclear Physics Proceedings style %\PhysRev % Physical Review style % --- % some macros used in this paper: \def\Kb{{\bar K^2 \over \beta_R}} % --- % BEGIN: \titlepage % begin title page material \title Noncompact nonlinear sigma models and numerical quantum gravity \endtitle \author Eric Myers, Bryce DeWitt, Rob Harrington, and Arie Kapulkin Center for Relativity, Department of Physics\\ University of Texas, Austin, Texas 78705 USA \endauthor \abstract Studying the $O(2,1)$ nonlinear sigma model is a useful step toward determining whether or not a consistent quantum theory of gravity (based on the Einstein-Hilbert action) exists. Like gravity, the sigma model is not perturbatively renormalizable, and corresponding Feynman graphs in the two theories have the same na\"{\i}ve degrees of divergence. Both theories also have a single overall dimensionful coupling constant, and both have a configuration space which is noncompact and curved. The sigma model allows one to study the renormalizability properties of such theories without the added complications of local symmetries. \endabstract \bigskip \endtitlepage % will start \doublecolumns for \nuclproc % --- % Now start in on the text of the paper... % --- Quantum Field Theory and the theory of General Relativity are, separately, probably the two most successful physical theories of this century. This notwithstanding, nobody has yet been able to bring the two together into one complete and consistent quantum theory of gravity. One major impediment to such a theory is that, unlike gauge field theories, gravity with the Einstein-Hilbert action $$ S = {1 \over 16\pi G_N} \int d^4x\, \sqrt{g} R \EQN 1$$ is not renormalizable, at least not by the usual methods of perturbation theory. This has lead a number of physicists to adopt the position that General Relativity is only the low energy limit of some other quantum theory, such as superstring theory. An alternative view which one can adopt, however, is that the failure of perturbation theory in the case of gravity is not an indication that the theory is inconsistent, but only that the mathematical tools one has used are inadequate. To pursue a quantum theory of gravity in this direction one needs a nonperturbative method of calculation: the methods of lattice field theory, which have already been applied to gauge theories, are immediately suggested. One also needs a simple model with which to test the ideas of nonperturbative renormalizability without the complicated structure of the full theory of General Relativity. This paper describes our work with such a model, the $O(2,1)$ noncompact nonlinear $\sigma$-model.\reference{DeWitt, 1989} B.S.~DeWitt, ``Nonlinear sigma models in 4 dimensions: a lattice definition,'' lectures given at the International School of Cosmology and Gravitation, ``Ettore Majorana'' Centre for Scientific Culture, Erice, Sicily, May 1989 \endreference\relax The model we consider consists of three scalar fields $\varphi_a$ described by the action $$ S = \half \mu^2 \int d^4 x \, \eta^{ab} \del_\mu \varphi_a \del^\mu \varphi_b \,, \EQN 2$$ with $\eta_{ab}={\rm diag}(-1,+1,+1)$ and with the fields obeying the constraint $$ -\varphi_0^2 + \varphi_1^2 + \varphi_2^2 = -1 \qquad (\varphi_0 > 0) \,. \EQN 3$$ The manifold of constraint is the two dimensional surface of constant negative curvature represented schematically in \Fig{1}. It is the coset space $O(2,1)/O(2)\times Z_2$, but for simplicity we refer to \Eqs{2} and \Ep{3} as the $O(2,1)$ nonlinear $\sigma$-model. There are several reasons this model is of interest: \item{1)} For dimensionless fields $\varphi_a$ the coupling constant $\mu^2$ has units of $(length)^2$, the same as $1/G_N$ in the Einstein-Hilbert action. Thus $\mu$ plays the role of the Planck mass in the theory. Furthermore this means that Feynman graphs in the $\sigma$-model have the same na\"{\i}ve degree of divergence as similar graphs in gravity, so that the model has the same renormalizability structure (actually the same perturbative non-renormalizability structure) as the theory of gravity. \item{2)} As in gravity, the fields of the $\sigma$-model obey a constraint, and the configuration space defined by the constraint is both {\it curved} and {\it noncompact}. \item{3)} The surface of constraint is invariant under global $O(2,1)$ transformations, but % unlike gravity there is no local symmetry in the model. This is a great simplification which lets us study just the renormalizability properties of the model without the added complications introduced by local symmetries. \item{4)} Unlike gravity, the Euclidean action of the $\sigma$-model is bounded from below. The unboundedness of the gravitational action is a serious problem which must be dealt with at some point, but one which we want to avoid entirely for now. \figure{1} \forceleft \vskip\colwidth % just leave some space to glue in figure %%\epsfbox{o21.ps} % or include with epsf \caption{The constraint surface of the $O(2,1)$ noncompact nonlinear $\sigma$-model.} \endfigure \medskip The transcription of the $\sigma$-model to the lattice is more or less standard with one exception, our definition of the lattice derivative. Rather than using the simple difference between field values at neighboring lattice sites we use the geodesic distance between two points on the constraint surface. The lattice action is thus $$ S = \half \mu^2 \sum_x a^4 \sum_{\hat\mu} [{ \Delta(\varphi(x+\hat\mu a), \varphi(x)) \over a}]^2 \,, \EQN 4$$ where $\Delta(\varphi,\varphi^\prime)$ is the arc length between $\varphi$ and $\varphi^\prime$ on the manifold, $$ \Delta(\varphi,\varphi^\prime) = \cosh^{-1}(-\eta^{ab} \varphi_a\varphi^\prime_b) \,. \EQN 5$$ Our reason for this choice is that it is consistent with the idea that the fields be restricted only to the constraint surface of the $\sigma$-model. In contrast, simply taking the na\"{\i}ve difference between fields, as is usually done for compact $\sigma$-models, produces a difference vector which does not lie in the manifold of constraint. While both methods lead to the same classical continuum limit there is nothing that guarantees that the quantum theories obtained from the two lattice definitions will be the same. It is convenient to factor the dependence on the lattice spacing $a$ out to the front of \Eq{4} and to define the dimensionless coupling constant $\beta = \mu^2 a^2$. If the theory is nonperturbatively renormalizable the Planck mass $\mu$ will be renormalized to $\mu_R$, which results in a renormalized dimensionless coupling constant $$ \beta_R = \mu_R^2 a^2 \,. \EQN 6$$ The renormalized Planck mass defines a characteristic length scale $1/\mu_R$ for the interactions of the theory. The lattice approximation to the continuum theory will be reliable when $a \ll 1/\mu_R \ll L=Na$. Considering each inequality separately, this requires $$ \mu_R a = \sqrt{\beta_R} \ll 1 \qquad \hbox{\rm and} \qquad N \gg {1\over \sqrt{\beta_R}} \EQN 7$$ In the continuum limit $a \to 0$, hence $\beta \to 0$, and for $\mu_R$ to remain finite this requires $\beta_R \to 0$. If this condition is not fulfilled then it would appear to be impossible to define a consistent quantum field theory from the $\sigma$-model, even nonperturbatively. %---------------------------- \figure{2} \forceright % force this to the righthand column \vskip\colwidth % leave this much space %%\epsfbox{beta.ps} % or include with EPSF \caption{The renormalized dimensionless coupling constant $\beta_R$ plotted against the bare coupling constant $\beta$ for an $N=10$ lattice.} \endfigure %---------------------------- In \Fig{2} we show $\beta_R$ plotted as a function of $\beta$ as obtained from Monte Carlo simulations on an $N=10$ lattice. As can clearly be seen, $\beta_R$ vanishes nowhere. We therefore conclude that the $O(2,1)$ nonlinear sigma model does not have an interacting continuum limit. One may view $1/\mu_R$ as the renormalized coupling constant in the theory, in which case our result implies that the model is ``trivial'' (in the technical sense) in that the continuum limit is a free field theory. This work was supported by NSF grants PHY\-8617103 and PHY\-8919177. \smallskip %\nosechead{References} % header for references %\nobreak \ListReferences \bye %>>> EOF Example.tex <<<