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- % file: Example.tex TeXsis version 2.15
- % $Revision: 15.3 $ : $Date: 92/06/30 14:42:28 $ : $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
-
-
- \texsis % this tells everyone that it's a TeXsis manuscript file
-
-
- % 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
- %\twinout\tenpoint\def\Tbf{\twelvebf}\def\tbf{\tenbf}
- \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 <<<
-