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diff --git a/Master/texmf-dist/doc/otherformats/texsis/base/Example.tex b/Master/texmf-dist/doc/otherformats/texsis/base/Example.tex new file mode 100644 index 00000000000..9de019e32c3 --- /dev/null +++ b/Master/texmf-dist/doc/otherformats/texsis/base/Example.tex @@ -0,0 +1,232 @@ +%% 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 <<< |