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authorKarl Berry <karl@freefriends.org>2009-05-24 00:31:38 +0000
committerKarl Berry <karl@freefriends.org>2009-05-24 00:31:38 +0000
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treec76f77b61118c210746b2b9a513369908249ae7e /Master/texmf-dist/doc/otherformats/texsis/base/Example.tex
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+%% 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 <<<