\input harvmac.tex
\def\CD {{\cal D}}
\def\CF {{\cal F}}
\def\CP {{\cal P }}
\def\CL {{\cal L}}
\def\CV {{\cal V}}
\def\CO {{\cal O}}
\def\p {\partial}
\def\CS {{\cal S}}
\def\hb {\hbar}
\def\inbar{\,\vrule height1.5ex width.4pt depth0pt}
\def\IB{\relax{\rm I\kern.18em B}}
\def\IC{\relax\hbox{$\inbar\kern.3em{\rm C}$}}
\def\IP{\relax{\rm I\kern.18em P}}
\def\IR{\relax{\rm I\kern.18em R}}
\Title{\vbox{\baselineskip12pt\hbox{YCTPP891}\hbox{RU9112}}}
{\vbox{\centerline{DoubleScaled Field Theory at $c=1$}}}
\centerline{Gregory Moore}
\centerline{Department of Physics and Astronomy}
\centerline{Rutgers University}
\centerline{Piscataway, NJ 088550849}
\centerline{and}
\centerline{Department of Physics\footnote{*}{Permanent address}}
\centerline{Yale University}
\centerline{New Haven, CT 065118167}
\bigskip
\noindent
We investigate the doublescaled free fermion theory of the
$c=1$ matrix model. We compute correlation functions of the
eigenvalue density field and compare with the predictions of
a relativistic boson theory. The $c=1$ theory behaves as a
relativistic theory at long distances, but has softer behavior
at short distances. The soft short distance behavior is closely
related to the breakdown of the topological expansion
at high energies. We also compute macroscopic loop amplitudes
at $c=1$, finding an integral representation for $n$
loop amplitudes to all orders of perturbation theory. We evaluate
the integrals explicitly for two, three, and four macroscopic
loops. The small loop length asymptotic expansion then
gives correlation functions of local operators in the theory.
The twomacroscopicloop formula gives information on the spectrum
and wavefunctions in the theory. The three and four loop
amplitudes give scattering amplitudes for tachyon operators
to all orders of
perturbation theory. Again, the topological expansion
breaks down at high energies. We compare our amplitudes with
predictions from the liouville theory.
\Date{Feb. 20, 1991}
\newsec{Introduction}
The discovery of the double scaling limit of matrix models
has opened the way for a study of nonperturbative effects in
subcritical string theory
\ref\brkz{E.Br\'ezin and V. Kazakov, Phys. Lett. {\bf 236B}(1990)144.}
\ref\dgsh{M.R. Douglas and S. Shenker, Nucl. Phys. {\bf B335}(1990)
635.}
\ref\grmgdl{D. Gross and A. Migdal, Phys. Rev. Lett. {\bf 64}(1990)127}\ .
The exact solvability of the
the $c<1$ models is closely connected to the equivalence of
the relevant matrix models to theories of free fermions
\ref\bpiz{E. Br\'ezin, C. Itzkyson, G. Parisi, and J.B. Zuber,
Comm. Math. Phys. {\bf 59}(1978)35.}
\ref\bdss{T. Banks, M.R. Douglas, N. Seiberg, and S. Shenker,
Phys. Lett. {\bf 238B}(1990)279} .
In the $c<1$ models the
free fermions provide an efficient calculational tool
but their relevance goes much
deeper. First, the connection of the $c<1$ models to integrable systems
established in \brkz\dgsh\grmgdl\bdss\
\ref\mike{M.R. Douglas, Phys. Lett. {\bf 238B}(1990)176}
may be understood very naturally in terms of the theory of
the doublescaled free fermion field
\ref\geom{G. Moore, Comm. Math. Phys. {\bf 133}(1990)261;
``Matrix Models of 2D
Gravity and Isomonodromic Deformation,'' to appear in the proceedings
of the 1990 Yukawa International Seminar, and in the proceedings
of the Carg\'ese Workshop on Random Surfaces}\ . Second,
there are many indications that the doublescaled
free fermion field theory will play a role in formulating a natural
spacetime interpretation of the $c<1$ models.
The
fermionic field theory for the $c=1$ matrix model has been explored to some
extent in
\ref\senwad{A.M. Sengupta and S.R. Wadia, ``Excitations and interactions
in d=1 string theory,'' Tata preprint 9033, July 1990. To
appear Int. Jour. of Mod. Phys. A.}
\ref\gk{D.J. Gross and I.R. Klebanov, ``Fermionic String Field
Theory of c=1 TwoDimensional Quantum Gravity,'' PUPT1198}\ ,
but these treatments have made important approximations
limiting their applicability to genus zero physics.
Consequently, the full potential
of the fermionic formulation has not yet been exploited in
the study of $c=1$ string physics.
In this paper we continue the study of the $c=1$
free fermions in the doublescaling limit.
In section two we derive the
theory and comment on the relation of the fermi field
to a relativistic fermi field. In section three we compute
correlation functions of the eigenvalue density field, and
compare with the results of the DasJevicki formulation
\ref\dsjv{S.R. Das and A. Jevicki, ``String field theory and physical
interpretation of D=1 strings,'' Brown preprint BROWNHET750}\
of $c=1$. In section four we compute macroscopic loop amplitudes
and use these to define scaling operators. From explicit
expressions we obtain information about
the spectrum of the theory, the
wavefunctions for operators in the theory, and some
scattering amplitudes.
For other approaches to $c=1$ scattering amplitudes see
\nref\kostov{I. Kostov, Phys. Lett. {\bf B215}(1988)499}
\nref\bmh{S. BenMenahem, ``Two and threepoint functions in
the $d=1$ matrix model,'' SLACPUB5262}
\nref\grkltwpt{D.J. Gross, I.R. Klebanov, and M.J. Newman,
``The twopoint correlation function of the onedimensional
matrix model,'' PUPT1192}
\nref\JOE{J. Polchinski, ``Critical Behavior of Random Surfaces in
One Dimension,'' UTTG1590}\
\nref\kutasov{D. Kutasov and P. Di Francesco, ``Correlation Functions
in 2D String Theory,'' PUPT1237, to appear. We thank D. Kutasov for
communicating the results of this paper prior to publication.}
\refs{\kostov{}\kutasov}\ .
The conclusion summarizes some of the
physical lessons which may be learned from our computations.
\newsec{The doublescaled field theory at $c=1$.}
\subsec{Derivation}
We begin with the standard matrix model integral at finite
$N$
\ref\kazrv{For a recent review see V. Kazakov, ``Bosonic
strings and string field theories in one dimensional target
space,'' to appear in the proceedings of the Carg\'ese workshop
on random surfaces and quantum gravity.}\ .
For definiteness we
choose a potential $V(\lambda)=\half(A^2\lambda^2)$ with
infinite walls at $\lambda=\pm A$. Dependence on $A$
reflects dependence on a nonuniversal cutoff. We will see
below that all dependence on $A$ disappears in the
double scaling limit. The schr\"odinger equation
is
\eqn\scrho{
\biggl({1\over 2\beta^2}{d^2\over d\lambda^2}V(\lambda)+
\varepsilon\biggr)\psi=0}
with boundary conditions $\psi(\varepsilon,\pm A)=0$,
leading to a discrete spectrum $\varepsilon_i(\beta,A)$,
$i=1,\dots$.
It follows that the wavefunctions are expressed in
terms of parabolic cylinder functions (see appendix A for
our conventions) as
\eqn\qmwvf{\eqalign{
\psi^+_i&=N^+\biggl(W(\nu_i,\tilde\lambda)+W(\nu_i,\tilde\lambda)\biggr)\cr
\psi^_i&=N^\biggl(W(\nu_i,\tilde\lambda)W(\nu_i,\tilde\lambda)\biggr)\cr}
}
where $\nu_i=\beta(\half A^2\varepsilon_i)$
and $\tilde\lambda=\sqrt{2\beta}
\lambda$. In the quasiclassical limit $\beta\to\infty$ the
normalization factors may be evaluated using the asymptotic
expansions of $W$ to give:
\eqn\normfc{
(N^+)^2=(N^)^2={1\over 2\sqrt{2(1+e^{2\pi \nu})}\beta^{1/2}
log\sqrt{2\beta} A+\CO(\beta^{1/2})}
}
From WKB estimates we find that the fermi level for $N$ fermions is
\eqn\fermlevel{
\pi{N\over \beta}=\half A\sqrt{2\varepsilon_N}
\half(A^22\varepsilon_N)
log\biggl((A+\sqrt{2\varepsilon_N})\sqrt{A^22\varepsilon_N}\biggr)}
while the density of states at the fermi level is
$\rho(\varepsilon)={1\over \pi}\beta log\sqrt{2\beta}A$.
It follows that the $c=1$ doublescaling limit of
\ref\brkzzm{E. Br\'ezin, V. A. Kazakov, Al. B. Zamalodchikov,
Nucl. Phys. {\bf B333}(1990)673}
\ref\ginsparg{P. Ginsparg and J. ZinnJustin, Phys. Lett. {\bf 240B}
(1990)333}
\ref\grmil{D.J. Gross and N. Miljkovi\'c, Phys. Lett. {\bf B238}(1990)217}
\ref\parisi{G. Parisi, Phys. Lett. {\bf B238}(1990)209}
\ref\grkli{D.J. Gross and I.R. Klebanov, Nucl. Phys. {\bf B344}(1990)475}\
is obtained by taking $N,\beta\to \infty$,
with $N/\beta\to A^2/(2\pi)$ such that the cosmological constant
\eqn\dscl{
\mu\equiv\beta\bigl(\half A^2\varepsilon_N(\beta,A)\bigr)
}
is held fixed.
Correlation functions in the model at finite $N$ may be computed
in a free fermion formalism much as in
\bdss .
The fermion field is simply
\eqn\finfer{
\hat \Psi(\lambda,x)=\sum_{i=1}^\infty a_\epsilon(\varepsilon_i)
\psi^\epsilon(\varepsilon_i,\lambda)e^{i\varepsilon_i x}
}
Here $\epsilon=\pm$ refers to the parity of the wavefunction;
we always sum over parity states when the $\epsilon$ index
is repeated. The $a$'s anticommute and satisfy
$\{a_\epsilon(\varepsilon_i),a_{\epsilon'}^\dagger(\varepsilon_j)\}
=\delta_{ij}\delta_{\epsilon,\epsilon'}$.
We may take the double scaling limit by defining
\eqn\sclfer{
\hat\Psi_\beta(\lambda,x)\equiv {1\over (2\beta)^{1/4}}
\hat\Psi
\bigl({\lambda\over\sqrt{2\beta}},\beta x\bigr)e^{\half i A^2 \beta x}
}
In the doublescaling limit
%
%$\hat\Psi_\beta(\lambda,x)\to
%\hat\psi(\lambda,x)$,
%
the discrete energy levels become continuous so the
sum in \finfer\ becomes an integral $\int d\nu \rho(\nu) \cdots$
with $\rho(\nu)\sim {1\over \pi}log\sqrt{2\beta}A$ near the
fermi level.
The oscillators $a_\epsilon(\varepsilon_i)$ are related to
continuum oscillators satisfying
$\{ a_{\epsilon}^\dagger(\nu),a_{\epsilon'}(\nu')\}=\delta_{\epsilon,
\epsilon'}\delta(\nu\nu')$ by the rescaling:
\eqn\contosc{
a_\epsilon(\varepsilon_i)\rightarrow {a_\epsilon(\nu)\over
\bigl({1\over \pi}log\sqrt{2\beta}A\bigr)^{1/2} }
\qquad .}
Thus, taking into account \normfc\ the {\it operator}
$\hat\Psi_\beta(\lambda,x)$ has a limit as $\beta\to\infty$
for fixed $\lambda,x$
\eqn\field{
\hat\Psi_\beta(\lambda,x)\to \hat\psi(\lambda,x)=\int d\nu e^{i\nu x}
a_{\epsilon}(\nu)\psi^\epsilon(\nu,\lambda) }
where $\psi^\epsilon(\nu,x)$ are normalized as in appendix A.
Thus we may compute quantities in the double scaling limit
directly from the fermionic field theory defined by the
action
\eqn\action{S=\int_{\infty}^\infty dx d\lambda
\hat\psi^\dagger\biggl( i{d\over dx}+{d^2\over d\lambda^2}
+{\lambda^2\over 4}\biggr)\hat\psi}
The infinitely negative energies of the upsidedown oscillator are
filled by the Fermi sea, i.e., the vacuum is defined by
\eqn\vac{\eqalign{
a_{\epsilon}(\nu)\mu\rangle&=0\qquad \nu<\mu\cr
a_{\epsilon}^\dagger(\nu)\mu\rangle&=0\qquad \nu>\mu\cr}
}
and illustrated in \fig\fermsea{The fermi sea for
the upside down oscillator}\ ,
where $\mu$ is the cosmological constant.
When we compute physical quantities below we will often
be interested in their topological expansion to obtain
the results which should be reproduced by string perturbation
theory. We can reintroduce the loop counting parameter $\kappa^2$
by the substitutions $\lambda\to\kappa^{1/2}\lambda$
and $\mu\to\kappa^{1}\mu$. It is worth emphasizing that the
above theory provides a nonperturbative definition of the
$c=1$ model.
\foot{This has also recently been emphasized in \kazrv\ .}
Indeed, we will discuss some nonperturbative effects
in sections three and four.
We will see below that this field theory reproduces many of the
known results for $c=1$. For example, one thing which follows immediately
from the identity (A.12) of appendix A is that we can compute
the derivative of the density of states as
\eqn\dos{\eqalign{
Tr\delta'(H\mu)&\equiv Im{1\over\pi}\int d\lambda {\p
\over \p\mu} R(\mui\epsilon;\lambda,\lambda)\cr
&={1\over 2\pi}Im\int_0^{\infty}ds e^{i\mu s}{s\over sh~s/2}\cr
&={\p\over\p\mu}\biggl[{1\over 2\pi }Re~\Psi(\half+i\mu)\biggr]\cr}
}
reproducing the wellknown answer for the partition function.
\subsec{Propagators}
Correlation functions in the matrix model are given by the
analytic continuation of correlators in the fermi field theory
from minkowski space to euclidean space (in $x$). It is useful to
have an operator formalism for computation directly in euclidean
space. Thus after computing a correlation function from wick's
theorem, using the minkowski space propagator, which is a
sum of ``hole'' and ``particle'' propagators:
\eqn\propmink{\eqalign{
\langle\mu
T\bigl(\hat\psi^\dagger(x_1,\lambda_1)\hat\psi(x_2,\lambda_2)
\bigr)\mu\rangle
&=\theta(\Delta x)S_h(1,2)\theta(\Delta x)S_p(2,1)\cr
=\int d\nu [\theta(\nu\mu)\theta(\Delta x)&
\theta(\mu\nu)\theta(\Delta x)] e^{i\nu \Delta x}
\psi^\epsilon(\nu,\lambda_1)\psi^\epsilon(\nu,\lambda_2)\cr}
}
where $\Delta x\equiv x_1x_2$,
we analytically continue $\Delta x\to i \Delta x$, so we can
equivalently work with ``Euclidean time ordered'' correlation
functions with propagator:
\eqn\propeuc{\eqalign{
S^E(x_1,\lambda_1;x_2,\lambda_2)=&
\int d\nu [\theta(\nu\mu)\theta(\Delta x)
\theta(\mu\nu)\theta(\Delta x)]e^{\nu(\Delta x)}
\psi^\epsilon(\nu,\lambda_1)\psi^\epsilon(\nu,\lambda_2)\cr
&=e^{\mu\Delta x}\int d\nu\int {d~p\over 2\pi}
e^{ i p\Delta x}{i\over p+i(\nu\mu)}
\psi^\epsilon(\nu,\lambda_1)\psi^\epsilon(\nu,\lambda_2)\cr
=i e^{\mu\Delta x} \int {d~p\over 2\pi}
e^{ i p\Delta x}&
\int_0^{sgn(p)\cdot\infty} ds
{e^{sp+i\mu s}\over (4\pi i sh s)^{1/2}}
exp\bigl({i\over 4}({\lambda_1^2+\lambda_2^2\over
th~s}2{\lambda_1\lambda_2\over sh~s})\bigr)
\cr}
}
where we have again used eq. A.12.
\subsec{Simple variant theories}
The above theory can be modified simply by generalizing to
the case in which the infinite walls are at $\lambda=B,A$.
Unless $B$ or $A$ is in the scaling region near $\lambda=0$
choosing $B\not=A$ has no effect on the universal physics.
On the other hand, if $B=\CO(\beta^{1/2})$, the physics
can be modified. The simplest example of this is to take
$B=0$. In this case we have wavefunctions $\psi_i^$ and
the doublescaled propagator is modified to
\eqn\newprop{
S(\lambda_1,\lambda_2;\Delta x)\to\half\biggl[
S(\lambda_1,\lambda_2;\Delta x)
S(\lambda_1,\lambda_2;\Delta x)\biggr]
}
Most of the calculations below can be redone for this theory.
\subsec{Relation to relativistic fermions}
One of our goals in this paper is to investigate
correlation functions in this theory, and the extent to which they
are related to correlators in a relativistic field theory.
Let us begin by considering the anticommutator in minkowski space.
We have
\eqn\anti{
\{\hat\psi(x_1,\lambda_1),\hat\psi^\dagger(x_2,\lambda_2)\}=
{1\over (4\pi~ i sh(\Delta x))^{1/2}}exp\biggl[{i\over
4}\bigl({\lambda_1^2+\lambda_2^2\over th(\Delta
x)}2{\lambda_1\lambda_2\over sh(\Delta x)}\bigr)\biggr]
}
where $\Delta x\equiv x_1x_2$. For $\Delta x\to 0$ this
approaches $\delta(\lambda_1\lambda_2)$ as it should. Note, however
that for $\lambda_1=\lambda_2$ the short distance singularity is
$\sim 1/(\Delta x)^{1/2}$ and differs from relativistic field theory.
Similarly the minkowski propagator has the following
asymptotics:
\eqn\asympprop{\eqalign{
S^M&\sim \CO(1)\qquad\qquad\qquad \qquad\qquad\qquad\qquad \Delta x\to 0^+\cr
&\sim {1\over 4\pi \Delta x^{1/2}}\qquad\qquad\qquad\qquad\qquad\quad
\lambda_1=\lambda_2,\Delta x\to 0^\cr
&\sim \CO(1)\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\lambda_1\not=\lambda_2,\Delta x\to 0^\cr
&\sim \pm i {1\over \Delta x} e^{i\mu \Delta x}\psi(\mu,\lambda_1)
\psi(\mu,\lambda_2)\qquad\qquad \Delta x\to \pm\infty\cr}
}
The different singularity structure from relativistic field
theory might come as a surprise to some readers. Naively one
might expect the following.
Important contributions to correlation functions come from
the neighborhood of the fermi level $\nu\cong \mu$.
Using the ``planewave'' linear combinations of appendix A, which
have the property that
\eqn\appxi{
\chi^\pm(\nu,\lambda)\cong \chi^\pm(\mu,\lambda)e^{\pm i (\nu\mu)
\tau(\lambda,\mu)}}
where
\eqn\timflt{
\tau(\lambda,\mu)\equiv \int_{2\sqrt{\mu}}^\lambda {dy\over
\sqrt{y^24\mu}}=
log\biggl({\lambda+\sqrt{\lambda^24\mu}\over
2\sqrt{\mu} }\biggr)
}
is the ``timeofflight'' coordinate we may be tempted to rewrite
the fermi field \field\ as
\eqn\nwfield{
\hat\psi(\lambda,x)=e^{i\mu x}\chi^+(\mu,\lambda)\int d\nu e^{i(\nu\mu)
(\tau+x)}\alpha^+(\nu)
+
e^{i\mu x}\chi^(\mu,\lambda)\int d\nu e^{i(\nu\mu)
(\taux)}\alpha^(\nu)}
where $\alpha$'s are linear combinations of the $a's$.
It thus appears that we should have a relativistic field theory
perhaps with small corrections.
It is important to realize
that \appxi\ is only a valid approximation for $\lambda^2\gg \mu$,
$\lambda^2\gg\nu$, and $\mu\gg \nu\mu$.
On the other hand, short distance
singularities always come from an integration in the region
$\nu\gg \lambda$, that is, from energies well above
the top of the parabola in fig. 1, where the parabolic cylinder functions
again behave like planewaves, but of a different sort:
\eqn\appxii{
\psi^+\pm i\psi^
\sim {1\over (4\pi)^{1/2}\nu^{1/4}} e^{\pm i\sqrt{\nu}\lambda}
}
Since we have taken the double scaling limit, all nonuniversal
quantities have dropped out so this $\nu\to \infty$
behavior is physical. Equivalently, energies with $\nu$ large and negative are
still infinitesimally close to the fermi level in unscaled variables
and thus can affect the universal physics.
It is, of course, possible to find an expansion around the relativistic
theory which allows us to calculate corrections to lowmomentum/
longdistance physics. We may do this by using the WKB expansion of
parabolic cylinder functions to calculate the corrections to \appxi\
as a power series in $(\mu\nu)^n/\lambda^m$. These correction
terms in the wavefunction may be traded in for correction
terms to a relativistic lagrangian. The region of
validity of this expansion defines the region of validity
of the effective theories of
\senwad\gk\dsjv\ .
\newsec{Correlation Functions of the Eigenvalue Density}
\subsec{$n$Point function}
We derive
\foot{I would like to thank T. Banks for some useful
conversations relevant to these calculations.}
an integral representation for
the $n$point function of $\hat\psi^\dagger\hat\psi$.
Let
\eqn\gfns{\eqalign{
G(x_1,\lambda_1,\dots , x_n,\lambda_n)&\equiv
\langle \mu \hat\psi^\dagger\hat\psi(x_1,\lambda_1)\cdots
\hat\psi^\dagger\hat\psi(x_n,\lambda_n)\mu\rangle_c\cr
G(q_1,\lambda_1,\dots q_n,\lambda_n)&\equiv \int \prod_i dx_i
e^{i q_i x_i}
G(x_1,\lambda_1,\dots x_n,\lambda_n)\cr}
}
Applying wick's theorem \gfns\ can be expressed in terms of
a sum of ring diagrams.
These are easily evaluated using
\propeuc\ and after fourier transforming and taking a
derivative with respect to $\mu$ we have
\eqn\nptev{\eqalign{
{\p\over\p\mu}G(q_1,\lambda_1,\dots,q_n,\lambda_n)= i^{n+1}&
\delta(\sum q_i)
\sum_{\sigma\in\Sigma_n}\int_{\infty}^{\infty} d~\xi e^{i\mu\xi}
\int_0^{\epsilon_1\infty} ds_1\cdots\int_0^{\epsilon_{n1} \infty}
ds_{n1}\cr
e^{s_1 Q^\sigma_1\cdotss_{n1}Q^\sigma_{n1}}&
\langle\lambda_{\sigma(1)}e^{2 i s_1 H}\lambda_{\sigma(2)}\rangle
\cdots
\langle\lambda_{\sigma(n)}e^{2 i \bigl(\xi\sum_1^{n1}s_i\bigr)H}
\lambda_{\sigma(1)}\rangle\cr}
}
where $Q_k^\sigma\equiv q_{\sigma(1)}+\cdots q_{\sigma(k)}$,
$\epsilon_k=sgn[Q^\sigma_k]$ and $H=\half p^2
{1\over 4}\lambda^2$ so that
\eqn\ker{
\langle\lambda_1e^{2 i t H}\lambda_2\rangle
={1\over (4\pi i~sh~t)^{1/2}}
e^{{i\over 4}({\lambda_1^2+\lambda_2^2\over
th~t}2{\lambda_1\lambda_2\over sh~t})}
}
We can recover the result eq. 6.10 of
\gk\
by taking the limit of \nptev\
as $q_i\to 0$ with $q_i>0$ for $i=1,\dots, n1$. All terms for which
$Q_k^\sigma$ change sign as $k$ runs from $1$ to $n1$
may be seen to cancel, and the remaining terms give:
\eqn\zeromo{
{2\over\ n}Im~i^n
\sum_{\sigma\in\Sigma_n}
\int_0^{\infty} ds_1\cdots\int_0^{\infty}
ds_{n} e^{i\mu\bigl(\sum s_i\bigr)}
\langle\lambda_{\sigma(1)}e^{2 i s_1 H}\lambda_{\sigma(2)}\rangle
\cdots
\langle\lambda_{\sigma(n)}e^{2 i s_n H}
\lambda_{\sigma(1)}\rangle }
for the zeromomentum eigenvaluedensity correlator.
\subsec{One point function}
The onepoint function is simply the eigenvalue density:
\eqn\evdens{
\langle \rho(\lambda)\rangle=
\langle\hat\psi^\dagger\hat\psi(\lambda,x)\rangle=
\int_\mu^\infty d\nu ~ \psi^\epsilon(\nu,\lambda)^2}
In the region $\lambda^2\gg 4\mu\gg 1$ we have the
asymptotics:
\eqn\derevi{
{\p\over\p\mu}\langle
\rho(\lambda)\rangle=\psi^\epsilon(\mu,\lambda)^2
\sim {2\over
\pi\sqrt{\lambda^24\mu}}\biggl[1sin\bigl(
\lambda\sqrt{\lambda^24\mu}2\mu\tau(\lambda,\mu)\bigr)\biggr]
}
We may use a similar wkb approximation in the integrand of \evdens\
for an energy range $1\ll C\ll\lambda^24\nu\le\lambda^24\mu$ for any
$C$ to conclude that the eigenvalue distribution has a smooth
envelope
\eqn\envelope{\rho^{cl}={1\over 2\pi}\sqrt{\lambda^24\mu} }
(away from $\lambda^2=4\mu$)
on which are superposed rapid oscillations of small amplitude.
Higher order
corrections to \envelope\ are easily obtained from the asymptotic
expansions of parabolic cylinder functions.
We thus see that in the planar limit the ``Wigner
distribution'' (in the sense of the rate of vanishing of the
distribution) holds also for $c=1$, where there are two
cuts with support $(\infty,2\mu^{1/2}]$ and
$[2\mu^{1/2},\infty)$.
\foot{It also follows that $c=1$ is a good place to look for
a physical interpretation of socalled 2cut models. Indeed
the present model bears many fascinating similarities with a
recently discovered ``topological phase'' of the 2cut
models
\ref\crdglmr{\v C. Crnkovic, M. Douglas, and G. Moore, ``Loop
equations and the topological phase of multicut matrix models,''
to appear.}}
The region $\mu\gg\lambda^2$ is therefore
far outside the cut. Here we can estimate the integral \evdens\
to be (for $\lambda>0$):
\eqn\outcut{\langle\rho\rangle\sim {e^{\pi\mu+2\sqrt{\mu}\lambda}
\over 2\pi(\pi\mu)^{1/4}}
}
Across the turning point $\lambda=\pm 2\mu^{1/2}$ the
eigenvalue density smoothly matches the behavior
\outcut\ to \envelope\ as in
\fig\evdens{A sketch of the eigenvalue density at $c=1$.}.
\subsec{Twopoint function}
Since the eigenvalue density $\hat\psi^\dagger\hat\psi$ plays the
role of the bosonic field $\rho$ in \dsjv
we would like to investigate the singularity structure of
the two point function and compare with the relativistic
field theory described in
\dsjv\senwad
\gk\ . The singularity structure is
most directly understood working in position space:
\eqn\tptsp{\eqalign{
\langle\rho(1)\rho(2)\rangle_c=
&\biggl(\int_\mu^\infty e^{\nu \Delta x}\psi^\epsilon
(\nu,\lambda_1)\psi^\epsilon(\nu,\lambda_2)\biggr)
\biggl(\int_{\infty}^\mu e^{\nu \Delta x}\psi^\epsilon
(\nu,\lambda_1)\psi^\epsilon(\nu,\lambda_2)\biggr)\cr
&={e^{{1\over 4}({\lambda_1^2+\lambda_2^2\over
th~\Delta x}2{\lambda_1\lambda_2\over sh~\Delta x})}
\over (4\pi sin\Delta x)^{1/2}}
\int_\mu^\infty e^{\nu \Delta x}\psi^\epsilon(\nu,\lambda_1)
\psi^\epsilon(\nu,\lambda_2)\cr
&\biggl(\int_\mu^\infty e^{\nu \Delta x}\psi^\epsilon
(\nu,\lambda_1)\psi^\epsilon(\nu,\lambda_2)\biggr)
\biggl(\int_\mu^\infty e^{\nu \Delta x}\psi^\epsilon
(\nu,\lambda_1)\psi^\epsilon(\nu,\lambda_2)\biggr)\cr}
}
The integrals in the second equality make sense for
$\Delta x<\pi$.
As $\Delta x\to 0$ we see that
\eqn\shrtpt{\eqalign{
\langle \rho\rho\rangle_c&\sim (S_h(\lambda_1,\lambda_2))^2\qquad
\lambda_1\not=\lambda_2\cr
\langle \rho\rho\rangle_c
&\sim {1\over \Delta x^{1/2}} \langle\rho(\lambda)\rangle \qquad
\lambda_1=\lambda_2\cr}
}
For the longdistance behavior we estimate the integrals by
expanding the parabolic cylinder functions in powers of $(\nu\mu)$
to find
\eqn\esti{\eqalign{
\int_\mu^\infty
e^{\nu \Delta x}
\psi^\epsilon(\nu,\lambda_1)\psi^\epsilon(\nu,\lambda_2)
&={e^{\mu\Delta x}\over \Delta x}\biggl[
\psi^\epsilon(\mu,\lambda_1)\psi^\epsilon(\mu,\lambda_2)+\CO(1/\Delta
x)\biggr]\cr
\int_{\infty}^\mu
e^{\nu \Delta x}
\psi^\epsilon(\nu,\lambda_1)\psi^\epsilon(\nu,\lambda_2)
&={e^{\mu\Delta x}\over \Delta x}\biggl[
\psi^\epsilon(\mu,\lambda_1)\psi^\epsilon(\mu,\lambda_2)+\CO(1/\Delta
x)\biggr]\cr}
}
so that we have the asymptotics for $\Delta x\to \infty$:
\eqn\lgtpt{
\langle \rho\rho\rangle_c\sim{1\over \Delta x^2}\biggl(
\psi^\epsilon(\mu,\lambda_1)\psi^\epsilon(\mu,\lambda_2)
\biggr)^2+\CO(1/\Delta x^4)
}
In \dsjv\senwad\gk\ it was proposed that the genus zero correlators
of the eigenvalue density would be identicle to that of a free boson.
Taking the continuum limit of the expressions in \senwad\gk\ for
example we obtain the twopoint function:
\eqn\relbs{
\langle \rho\rho\rangle_c={\p \tau_1\over\p\lambda}
{\p \tau_2\over\p\lambda}
{1\over 4\pi}\p_{\tau_1}\p_{\tau_2}
log\biggl({(\Delta x)^2 + (\tau_1\tau_2)^2\over (\Delta x)^2+
(\tau_1+\tau_2)^2}\biggr)
}
which behaves like $1/(\Delta x)^2$ at large and small $\Delta x$.
Taking into account the wkb estimate of the parabolic cylinder
functions we see from \lgtpt\ that \relbs\ agrees with the exact formula
in the region $\lambda^2\gg\mu,\Delta x^2\to \infty$.
The deviation of \relbs\ from the exact result in the short
distance region is due, technically, to the origin of the
short distance singularities in the region $\nu\to\infty$
as explained in section 2.4.
The deviation from relativistic behavior has an
important consequence.
Physically, there is a crossover
in the behavior due to a breakdown in the genus expansion
\foot{Conversations with T. Banks and S. Shenker have been very helpful
in clarifying these points.}.
This is seen most clearly from \nptev\ for the case
$n=2$, which may be expressed in terms of the resolvent
$R(\zeta; \lambda_1,\lambda_2)$ of the Schr\"odinger
operator with potential $u=\lambda^2/4$ :
\eqn\twopt{
{\partial\over\partial\mu} G= \biggl(\psi^\epsilon(\mu,\lambda_1)
\psi^\epsilon(\mu,\lambda_2)\biggr)\biggl[
R(\zeta=\mu+ip;\lambda_1,\lambda_2)
+R(\zeta=\muip;\lambda_1,\lambda_2)\biggr]}
On the diagonal we may use the asymptotic expansion of
\ref\gelfdick{I.M. Gelfand and L.A. Dickii, Russian Math Surveys.
{\bf 30}(1975)77.}\
(with potential $u=\lambda^2/4$) to obtain
\eqn\expres{\psi^\epsilon(\mu,\lambda)^2\sum_{n=0}^\infty
\biggl[{R_n[u]\over(\mu+ip)^{n+1/2}}+
{R_n[u]\over(\muip)^{n+1/2}}\biggr]
}
so that the asymptotic expansion in $1/\mu$, i.e., the
genus expansion, breaks down for
large $p$.
\foot{After the this work was completed we learned that
G. Mandal, A. Sengupta and S. Wadia have obtained related results
\ref\wadsem{S. Wadia, Seminar at Rutgers University.}\ .}
The perturbative regime
has a fermi level well below the tip of the potential
$u=\lambda^2/4$. As we
increase $p$ we probe states at higher energies above the
fermi sea. At $p\sim \mu$ there is a qualitative change in the
nature of the wavefunctions, and a qualitative change in the
physics.
\subsec{Comparison with Collective Field Theory}
In some very interesting papers
\dsjv\
\ref\jvrv{A. Jevicki, ``Matrix models and field theory,'' BROWNHET771}
\ref\jvii{A. Jevicki, ``Collective field theory and schwingerdyson
equations in matrix models,'' BROWNHET777}\
an attempt has been made to use the matrix model to derive a
field theory for the eigenvaluedistribution field $\rho(\lambda,t)$.
In particular, the classical solution of the lagrangian of \dsjv\ is
\envelope\ . Comparing with the exact eigenvalue distribution we
see that the field theory of \dsjv\ omits the oscillatory terms
in the eigenvalue distribution. One might be tempted to call such
terms ``nonperturbative'' since they involve an exponential
$\sim e^{i \lambda^2/\kappa}$ but one must be careful about
such terminology, since these terms are also $\CO(1)$. Indeed
in ordinary quantum mechanics such terms in WKB wavefunctions
can contribute to the {\it perturbative} expansion in $\hbar$ in
the computations of physical expectation values.
Similarly, as we
have seen, the twopoint function only agrees with
the genus zero
predictions of the DasJevicki lagrangian at large distances.
It thus appears that the exact field theory of the eigenvalue
distribution  if it can be written as a field theory at all 
is only approximately relativistic at long distances and has
a different (and much {\it softer}) behavior at short
distances. Moreover,
Polchinski's spacetime effective field theory
interpretation of $c=1$ \JOE\
was interpreted \dsjv\ as being
equivalent to the DasJevicki lagrangian after identification
of the tachyon field with the eigenvalue field.
Combining these observations we are lead to ask
if it is a generic feature of nonperturbative
string field theory that the theory only resembles
a relativistic field theory on long distance scales and is a
much softer theory on short distance scales.
\newsec{Macroscopic loop amplitudes at $c=1$}
\subsec{$n$point function}
Formula \nptev\ for the correlation functions of
the eigenvalue densities allows us
to calculate the $n$point function $M(z_i,x_i)$
of the ``macroscopic loop operators'':
\eqn\macropt{\eqalign{
\hat\psi^\dagger e^{iz\hat \lambda}\hat\psi&\equiv
\int_{\infty}^{\infty}d\lambda \hat\psi^\dagger(\lambda,x)
e^{iz\lambda}\hat\psi(\lambda,x)\cr
M(z_i,x_i)&\equiv \langle
\hat\psi^\dagger e^{iz_1\hat\lambda}\hat\psi
\cdots\hat\psi^\dagger e^{iz_n\hat\lambda}\hat\psi\rangle\cr}
}
From $M(z_i,x_i)$ we can obtain the doublescaled correlation
functions of the resolvent $tr\bigl({1\over \zeta\phi}\bigr)$
via a laplace transform.
Note that these integrals
converge and get their main contribution from the region
near the edge of the classical eigenvalue distribution,
since, for large $\lambda$ the oscillatory function
$e^{i z\lambda}$ gives a small contribution to the
integral.
By contrast, correlation functions of the operators
$\hat\lambda^n$, which correspond to correlators of
$tr~\phi^n$ in the matrix model, are infinite. We believe
this is the source of some of the wavefunctionrenormalization
ambiguities which have plagued previous $c=1$ calculations.
By shrinking the macroscopic loop amplitudes we may extract
the local scaling operators of the theory and their correlation
functions, as in the onematrix model
\bdss\ . Namely, if
$M(z_i,q_i)$ is the fourier transform of \macropt\
we expect on physical grounds that the
small $z_i$ asymptotics of $M$ will have the form
\eqn\expasymps{
M(z_i,q_i)\sim \sum_{\Delta_i}\prod z_i^{\Delta_i}
\langle \CO_{\Delta_1}\cdots\CO_{\Delta_n}\rangle
}
where $\CO_{\Delta}$ are the scaling operators of
dimension $\Delta>0$. We will find below that there are
also divergent terms in the $z_i\to 0$ limit. A
nontrivial prediction of the liouville theory is that
these terms will be analytic in $\mu$ since they arise
from surfaces of zero area
\ref\natiliouv{N. Seiberg, ``Notes on Quantum liouville Theory and
Quantum Gravity,'' Rutgers preprint RU9029, to appear in the
proceedings of the 1990 Yukawa International Seminar, Common
Trends in Mathematics and Quantum Field Theories}
\ref\wdwppr{G. Moore, N. Seiberg, M. Staudacher, Rutgers preprint RU9111}\ .
We will verify below that this is indeed the case in all our
explicit formulae.
The calculation of \macropt\ reduces to the
evaluation of a gaussian integral after we use
\nptev\ .
One finds the result
\foot{Some details of the calculation are
provided in appendix C.}
\eqn\macptii{\eqalign{
{\p\over\p\mu}M(z_i,q_i)=\half i^{n+1}\delta(\sum
q_i)\sum_{\sigma\in\Sigma_n}&
\int_{\infty}^{\infty} d\xi {e^{i\mu \xi}\over sh~\xi/2}
\int_0^{\epsilon_1\infty} ds_1\cdots\int_0^{\epsilon_{n1}\infty}
ds_{n1}\cr
exp\bigl(\sum_{k=1}^{n1}s_k Q^\sigma_k\bigr)
exp\bigl({i\over 2}cth(\xi/2)\sum z_i^2 \bigr)&
exp\bigl(i\sum_{1\leq i\mu$.
In this way we see that for large enough momenta, the
topological expansion ceases to be even asymptotic. As we
have discussed this is due to the qualitatively
different nature of the
wavefunctions for energies above the top of the parabolic
maximum. From another point of view,
this phenomenon is probably related to nonrenormalizability
of the effective theories of \JOE\dsjv\senwad\gk\ .
More physically, the string approximation is seen
to be an effective description only for low momentum processes, but at
high momenta we begin to see the fermi particle itself.
Another application of the formulae of this section
\foot{Suggested by S. Shenker}
is to the study of high energy processes.
From \twptiii\ we see that the scattering amplitudes
at fixed genus are (after dividing by a genusindependent
wavefunction renormalization) polynomials in the
momenta. As we will see,
similar results hold for other correlation functions derived below.
This behavior is rather different from the high energy
behavior of perturbative critical string theory
\ref\grmend{D.J. Gross and P.F. Mende, Phys. Lett. {\bf 197B}(1987)
129; Nucl. Phys. {\bf B303}(1988)407}.
If, instead, we study the high energy behavior of the
nonperturbative result \twptii\ we obtain the large $q$ asymptotics:
\eqn\lrgqtw{
%{\sqrt{\pi}\over sin~\piq}e^{\mu\pi/2}\Gamma(q)\biggl[
%sin\bigl(\piq/2 +\mu log(q/\mu)+\mu\bigr)
%
%sin\bigl(3\piq/2 \mu log(q/\mu)\mu\bigr)\biggr]
\eqalign{
{\pi^{3/2}\over qsin^2\piq}e^{qlog~q+q}\sqrt{cosh(\pi\mu)}
\biggl[{2+e^{2\pi\mu}\over 1+e^{2\pi\mu}}&
cos\bigl(\piq/2 \mu logq+2\Phi(\mu)\bigr)\cr
+{e^{2\pi\mu}\over 1+e^{2\pi\mu}}
cos\bigl(3\piq/2 &+\mu logq2\Phi(\mu)\bigr)\biggr]\cr}
}
This also differs from proposed summations of the
results of \grmend\ given in
\ref\menoog{P.F. Mende and H. Ooguri, Nucl. Phys. {\bf B339}(1990)641} .
\subsec{Wavefunctions and the WheelerdeWitt Equation}
In a recent paper \wdwppr\ the role of the wheelerdewitt
equation in 2D gravity and
matrix models has been clarified. In this
section we sketch how many of the results in \wdwppr\ , which
were derived for $c<1$ may be trivially carried over to the case
of $c=1$.
We may use \expbessl\ to obtain a formula for a macroscopic
loop with one tachyon insertion, by letting $\ell_1\to 0$ holding
$\ell_2$ finite.
The genus zero contribution is
particularly interesting because, following the reasoning of
\wdwppr\
we expect it to solve the wheelerdewitt equation in minisuperspace.
The wavefunction of $\sigma_0(\CO_q)$ is easily extracted from
\gnzrtwlp\ with the result:
\eqn\onbglp{
\psi^{h=0}_q(\ell)\equiv\langle\sigma_0(\CO_q)w(\ell)\rangle=
\bigl(2q\Gamma(q)\bigr)
\mu^{q/2}K_{q}\biggl(2\sqrt{\mu \ell^2}\biggr)
}
in beautiful confirmation of the expectations of the liouville theory.
In particular,
$\psi^{h=0}_q$ satisfies the wheelerdewitt equation
in minisuperspace:
\eqn\wdwi{
\biggl((\ell{\p\over\p\ell})^2+4\mu\ell^2+q^2\biggr)\psi^{h=0}_q=0
}
Inspired by the result at genus zero, and the relatively simple
expression \anint\ summarizing the wavefunction $\psi_q$ to
all orders of the topological expansion we ask if some
simple equation summarizes the corrections to the WdW equation
from summing over topologies. The answer is especially simple
at $q=0$ where we find the wavefunction
\foot{This wavefunction has fascinating large $\ell$ behavior:
$$\psi_{q=0}(\ell)\to\sqrt{8\pi\over 1e^{\pi\mu}}e^{\pi\mu}
\ell^{1}cos\bigl[\half\ell^2\mu log\ell^2+2\Phi(\mu)\pi/4\bigr]$$
To all orders of perturbation theory the wavefunction decays
exponentially at $\ell\to\infty$ but nonperturbatively the
the wavefunction oscillates rapidly with a slowlydecaying envelope.}
\eqn\alwvfn{
\psi_{q=0}=Im \biggl[e^{3\pi i/4}\Gamma(\halfi\mu)
\ell^{1}W_{i\mu,0}(i\ell^2)\biggr]
}
using the whittaker differential equation we find that
$\psi_q$ satisfies the modified wheelerdewitt equation:
\eqn\wdwmdf{
\biggl((\ell{\p\over\p\ell})^2+4\mu\ell^2\kappa^2
\ell^4\biggr)\psi_{q=0}=0
}
where we have explicitly introduced the topological coupling
$\kappa$. The extra term is reminiscent of wormhole effects
arising from nonlocal terms in the liouville action
$\sim (\int_{\Sigma} \varphi_L e^{\gamma\varphi_L})^2$.
Similar considerations apply to $\psi_q$. The modified
wheelerdewitt equation is most simply written by
taking an inverse laplace transform in $\mu$ to work
at fixed area.
\subsec{Threepoint function}
We now examine the threepoint function. This allows us to
determine the nature of the couplings and fusion rules in
the theory. Moreover, since there is an ambiguity in
the relative normalization of matrixmodel and liouville
operators we cannot obtain unambiguous physical amplitudes
until we compute the threepoint function.
Choosing $q_1,q_2>0$ the rhs of \macptii\ becomes
\eqn\thrpti{\eqalign{
2\delta(\sum q_i)Re\int_0^\infty d\xi{e^{i\mu\xi}\over sh~(\xi/2)}
e^{{i\over 2}cth(\xi/2)\sum z_i^2}\Biggl\{
&\cr
\int_0^\infty ds_1\int_0^\infty ds_2 e^{q_1 s_1+q_3 s_2}
&\biggl[ e^{i(f(s_1)z_1 z_2+f(s_1+s_2)z_1z_3+f(s_2)z_2z_3}
\cr
&+e^{i(f(s_1)z_1 z_2+f(s_1s_2)z_1z_3+f(s_2)z_2z_3}\biggr]\cr
\int_0^\infty ds_1\int_0^\infty ds_2 e^{q_1 s_1q_2s_2}&
\biggl[ e^{i(f(s_1)z_1 z_3+f(s_1s_2)z_1z_2+f(s_2)z_2z_3}
\cr
&+e^{i(f(s_1)z_1 z_3+f(s_1+s_2)z_1z_2+f(s_2)z_2z_3}\biggr]\cr
+\int_0^\infty ds_1\int_0^\infty ds_2 e^{q_2 s_1+q_3 s_2}&
\biggl[ e^{i(f(s_1)z_1 z_2+f(s_1+s_2)z_2z_3+f(s_2)z_1z_3} \cr
&+e^{i(f(s_1)z_1 z_2+f(s_1s_2)z_2z_3+f(s_2)z_1z_3}\biggr]
\Biggr\} \cr}
}
where $f(s)=ch(s\xi/2)/sh(\xi/2)$.
The small $z_i$ behavior of the above integrals is subtle.
For simplicity assume that $q_i<1$.
%
%Denote
%$$\varepsilon_{jk}^{\pm}\equiv {i z_j z_k e^{\pm \xi/2}\over
%2 sh~(\xi/2)}$$
%
Denote the six integrals by $I_j$, $j=1,6$.
These integrals contribute both analytic and nonanalytic terms in
$\mu$. Keeping only the nonanalytic terms in $z_i$ and dropping
terms of order $\CO(z^{2q+n})$ for $n\geq 1$ we find that
$I_1+I_2$ contributes
\eqn\cntrii{\eqalign{
2(i z_2z_3)^{q_3}{\Gamma(q_3)^2\over q_1}F(q_3,q_1,1q_1;z_1/z_2)
&\CF^+(q_3;\mu)\cr
+2(i z_1z_3)^{q_3}{\Gamma(q_1)^2\over q_2}F(q_1,q_2,1+q_2;z_2/z_3)
&\CF^+(q_1;\mu)\cr
+2(i z_1z_3)^{q_1}(iz_2z_3)^{q_2}\Gamma(q_1)\Gamma(q_2)\Gamma(q_3)
&\CF^+(q_3;\mu)\cr}
%
%\Biggl[{\Gamma(\halfi\mu+q_3)\over \Gamma(\halfi\mu)}
%+{\Gamma(\halfi\mu)\over \Gamma(\halfi\muq_3)}\Biggr]\cr
%
}
where $F(\alpha,\beta,\gamma;z)$ is the hypergeometric
function and
we will frequently use the notation
\eqn\dffn{\CF^\pm(a,b;\mu)\equiv
{\Gamma(a+\halfi\mu)\over\Gamma(b+\halfi\mu)}
\pm
{\Gamma(b+\halfi\mu)\over\Gamma(a+\halfi\mu)}
}
We also use the notation $\CF^\pm(a;\mu)=\CF^\pm(a,0;\mu)$.
The contribution of $I_5+I_6$ is obtained by changing
$z_1\leftrightarrow z_2$ and $q_1\leftrightarrow q_2$.
The contribution of $I_3+I_4$ is similar:
\eqn\cntriii{\eqalign{
2(i z_2z_3)^{q_2}{\Gamma(q_2)^2\over q_1}F(q_2,q_1,1+q_1;z_1/z_3)
&\CF^+(q_2;\mu)\cr
%
%\Biggl[{\Gamma(\halfi\mu+q_2)\over \Gamma(\halfi\mu)}
%+{\Gamma(\halfi\mu)\over \Gamma(\halfi\muq_2)}\Biggr]\cr
%
2(i z_1z_3)^{q_1}{\Gamma(q_1)^2\over q_2}F(q_1,q_2,1+q_2;z_2/z_3)
&\CF^+(q_1;\mu)\cr
%
%\Biggl[{\Gamma(\halfi\mu+q_1)\over \Gamma(\halfi\mu)}
%+{\Gamma(\halfi\mu)\over \Gamma(\halfi\muq_1)}\Biggr]\cr
%
2(i z_1z_3)^{q_1}(iz_2z_3)^{q_2}\Gamma(q_1)\Gamma(q_2)\Gamma(q_3)
&\CF^+(q_1,q_2;\mu)\cr}
%
%\Biggl[{\Gamma(\halfi\mu+q_1)\over \Gamma(\halfi\muq_2)}
%+{\Gamma(\halfi\mu+q_2)\over \Gamma(\halfi\muq_1)}\Biggr]\cr}
%
}
Adding these together we find:
\eqn\cntriv{\eqalign{
2(i z_2z_3)^{q_3}{(\Gamma(q_3)^2\over q_1}F(q_3,q_1,1q_1;z_1/z_2)
&\CF^+(q_3;\mu)\cr
%
%\Biggl[{\Gamma(\halfi\mu+q_3)\over \Gamma(\halfi\mu)}
%+{\Gamma(\halfi\mu)\over \Gamma(\halfi\muq_3)}\Biggr]\cr
%
+
2(i z_1z_3)^{q_3}{(\Gamma(q_3)^2\over q_2}F(q_3,q_2,1q_2;z_2/z_1)
&\CF^+(q_3;\mu)\cr
%
%\Biggl[{\Gamma(\halfi\mu+q_3)\over \Gamma(\halfi\mu)}
%+{\Gamma(\halfi\mu)\over \Gamma(\halfi\muq_3)}\Biggr]\cr
%
+
2(i z_1z_3)^{q_1}(iz_2z_3)^{q_2}\Gamma(q_1)\Gamma(q_2)\Gamma(q_3)
&\CF^+(q_3;\mu)\cr
%
%\Biggl[{\Gamma(\halfi\mu+q_3)\over \Gamma(\halfi\mu)}
%+{\Gamma(\halfi\mu)\over \Gamma(\halfi\muq_3)}\Biggr]\cr
+
2(i z_2z_3)^{q_2}(iz_1z_3)^{q_1}\Gamma(q_1)\Gamma(q_2)\Gamma(q_3)
&\CF^+(q_3;\mu)\cr
%
%\Biggl[{\Gamma(\halfi\mu+q_3)\over \Gamma(\halfi\mu)}
%+{\Gamma(\halfi\mu)\over \Gamma(\halfi\muq_3)}\Biggr]\cr
2(i z_1z_3)^{q_1}(iz_2z_3)^{q_2}\Gamma(q_1)\Gamma(q_2)\Gamma(q_3)
&\CF^+(q_1,q_2;\mu)\cr}
%
%\Biggl[{\Gamma(\halfi\mu+q_1)\over \Gamma(\halfi\muq_2)}
%+{\Gamma(\halfi\mu+q_2)\over \Gamma(\halfi\muq_1)}\Biggr]\cr}
}
In isolating nonanalytic powers of $z$ we have to be careful that we can
expand the hypergeometric functions as power series. Thus their
arguments must have absolute value smaller than 1. It follows that
we must apply the inversion formula for $F$ to one of the first
two terms in \cntriv\ . Taking this into account we finally find
\eqn\threptii{\eqalign{
{\p\over\p\mu}\langle\sigma_0(\CO_{q_1})\sigma_0(\CO_{q_2})
\sigma_0(\CO_{q_3})\rangle&=\cr
\Gamma(q_1)\Gamma(q_2)\Gamma(q_3)
Re\biggl[e^{i\pi q_3/2}&
\biggl(\CF^+(q_3;\mu)\CF^+(q_2,q_1;\mu)\biggr)\biggr]\cr}
}
As a check, one may take the limit $q_1\to 0$ in which case we
obtain the derivative with respect to $\mu$ of the twopoint
function found above. Moreover, we can compute the topological
expansion as before. In particular, the first two terms in
the topological expansion are
\eqn\genzrthr{\eqalign{
\langle\sigma_0(\CO_{q_1})\sigma_0(\CO_{q_2})
\sigma_0(\CO_{q_3})\rangle_{h=0}&=
\prod_i q_i\Gamma(q_i)\mu^{q_31}\qquad\qquad\qquad\qquad
\cr
\biggl[1&{(q_31)(q_32)(q_1^2+q_2^2q_31)\over 24}\mu^{2}
+\cdots\biggr] \cr}
}
Accounting for wavefunction renormalization the leading term
agrees with
the recent calculations of Kutasov and Di Francesco using
the liouville theory \kutasov\ .
\subsec{Fourpoint function}
Next we turn to the fourpoint function. This allows us
to consider issues of factorization and the nature of
intermediate states, a subject fraught with ticklish
issues of principle \natiliouv\ .
The scattering amplitude depends on the kinematic configuration we
are studying. There are, up to inversion and permutation, only
two kinds of configurations:
either two or three momenta have the same sign.
We first consider the kinematic configuration:
\eqn\kincnfi{
q_1>0\qquad q_2>0\qquad q_3>0,\qquad q_4<0
}
i.e., $00\qquad q_2>0\qquad & q_3<0\qquad q_4<0\cr
q_1+q_3&=(q_2+q_4)<0\cr
q_1+q_4&=(q_2+q_3)<0\cr}
}
i.e., $0j}
{ch\bigl(\omega(t_i+\cdots t_{j+1}T/2)\bigr)\over 2\omega sh\omega T/2}
\xi_i\xi_j\biggr]
}
One might object to harmonic oscillators with these
``unphysical'' values of parameters.
Since gaussian integrals are given in terms of the
inverse and determinant of a certain quadratic form,
we may regard the introduction of harmonic
oscillators as a trick to prove a certain {\it algebraic}
identity for the quadratic form arising in \macropt .
\listrefs
\listfigs
\bye
Moreover,
we can express the twopoint functions:
\eqn\twdsc{
\langle \CO^{(k)}_q\CO^{(k)}_{q}\rangle
=
2Im\Biggl[e^{i\piq/2}\CF^(k+q,k;\mu)
\Biggr]
}
A similar discussion can be carried out for the degenerate
fields $\CD^{(n)}_r$. In this case the formulae are more complicated
since the second index of the whittaker function is an integer.
We simply quote the result for the correlator
\eqn\twdg{
\eqalign{
\langle \CD^{(0)}_r\CD^{(0)}_{r}\rangle=\qquad\qquad\qquad\qquad&
\cr
2Im\Biggl[{4(i)^rr\over r^2q^2}({(z_1z_2)^r\over r!^2}
\biggl({\Gamma(r+q+\halfi\mu)\over\Gamma(\halfi\mu)}&
\Psi(r+\halfi\mu)

{\Gamma(\halfi\mu)\over\Gamma(r+\halfi\mu)}
\Psi(\halfi\mu)\biggr)\Biggr]\cr}
}
in fact the above expression is also analytic in $\mu$ for $q$ not
an integer. This is expected since, for momentum $q\not= r$
the operator cannot couple to the macroscopic loop of
momentum $q$. Nevertheless the pole at $q=r$ should be interpreted
as an extra logarithmic factor in $\mu$.
\foot{These last few remarks were pointed out to me by N. Seiberg.}
DISCUSS SPECTRUM
From the expansion of the bessel
function we see that we will obtain terms of the form $z_1^{q+n}
z_2^{q+n}$, where $n\in \IZ_+$, from the bessel function $J_{q}$
and terms of the form $z_1^{r+n}z_2^{r+n}$ from th bessel
functions $J_r$.
Thus, if working at fixed $\xi$ is indeed working at
fixed area, we can already conclude that the scaling operators
of the
theory, $\CO^{(n)}_{q}$, are labelled by
a smoothly varying part, $q$ together with an nonnegative
integer $n$ labelling the tower of ``gravitational descendents.''
In addition there is a discrete spectrum $\CD_m^{(n)}$
arising from the degenerate representations of
the virasoro algebra at $c=1$, together with their gravitational
descendents. NO D's??
We use the double scaled free fermion field theory of
the c=1 matrix model to compute the correlation functions
of eigenvalue densities and macroscopic loops. Integral
representations for these quantities are found. We compare
the eigenvalue density correlators with the predictions of
free relativistic field theory on the spectral curve and
find differences, whose physical origin is explained.
We obtain explicit expressions for the correlation function
of two macroscopic loops to all orders of perturbation
theory. From this one may deduce the spectrum of the model.
We find explicit expressions for the 3 and 4 point scattering
amplitudes of tachyon operators to all orders of perturbation theory.
It is found that for momenta of the order of the cosmological
constant the topological perturbation series ceases to be asymptotic,
explicitly realizing the transition in a string theory from
a theory of strings to a theory of partons.