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\begin{document}
\title{Holographic dimensional reduction from entanglement in Minkowski space}
\author{Ram Brustein, Amos Yarom}
\address{ Department of Physics, Ben-Gurion University,
Beer-Sheva 84105, Israel \\
{\rm E-mail:} {\tt ramyb@bgumail.bgu.ac.il,
yarom@bgumail.bgu.ac.il } }


\begin{abstract}

We present evidence that dimensional reduction in Minkowksi space is  
induced by quantum entanglement. 
First, we show that correlation functions of a class of
operators restricted to a sub-volume of D-dimensional Minkowski 
space scale as its surface area. 
A simple example of such area scaling is provided by the
energy fluctuations of a free massless quantum field in its vacuum
state. This is reminiscent of area scaling of entropy of
entanglement but applies to quantum expectation values in a pure
state, rather than to statistical averages over a mixed state.
We then show, in specific cases,  that the 
bulk theory in the sub-volume has a holographic 
representation in terms of a boundary 
theory at high temperature. 
\end{abstract}

\maketitle


%\section{Introduction}

Physics up to energy scales of about one TeV is very well
described in terms of quantum field theory, which uses, roughly,
one quantum mechanical degree of freedom (DOF) for each point in
space. This seems to imply that the entropy $S(V)$, or the
dimensionality of phase space, of a quantum system in a sub-volume
$V$, is proportional to $V$. The Bekenstein entropy bound
\cite{BEB}, that can be applied to any sub-volume $V$ in which
gravity is not dominant implies, roughly, that $S(V)$ has to be
less than or equal to the boundary area $A$ of $V$ in Planck units
. An interpretation of this, relying on arguments involving
physics of black holes, was proposed by 't Hooft and by Susskind
\cite{HOL} who suggested that the number of independent quantum
DOF contained in a given spatial sub-volume $V$ is bounded by the
surface area of the region measured in Planck units. The
holographic principle (see \cite{bousso} for a recent review)
postulates an extreme reduction in the complexity of physical
systems.  It is widely believed that quantum gravity has to be
formulated as a holographic theory, and it is implicitly assumed
that gravity is somehow responsible for this massive reduction of
the number of DOF. This point of view has received strong support
from the AdS/CFT duality (see \cite{adscft} for a review), which
defines quantum gravity non-perturbatively in a certain class of
space-times and explicitly exposes only the physical variables
admitted by holography.

Another route leading to area dependent entropy originated in
evidence of dimensional reduction due to quantum entanglement,
mainly from the calculation of entanglement entropy of
sub-systems. Srednicki \cite{srednicki} (and previously Bombelli
et. al. \cite{bombal}) considered the Von-Neumann entropy of
quantum fields in a state defined by taking the vacuum and tracing
over the DOF external to a spherical sub-volume of Minkowski
space. They discovered that this ``entropy of entanglement" was
proportional to the boundary area of the sub-volume.  It is
possible to show that the entropy obtained by tracing over the DOF
outside of a sub-volume of any shape is equal to the one obtained
by tracing over the DOF inside this sub-volume \cite{srednicki}
but an explicit numerical calculation was needed to show that it
is linear in the area for a spherical sub-volume. Following this
line of thought, some other geometries were considered, in
particular, half of Minkowski space in various dimensionalities
(see for example, \cite{Holzhey:1994we,callwil,sussugl,alwis1,alwis2}).
Dimensional reduction due to entanglement seems to be quite
different from that in AdS, for example, and its relationship to
dimensional reduction due to gravity was never clarified (see
however \cite{maldacena,kleban,shenker}).


Following a calculation in \cite{befo}, we have discovered that in
the vacuum state, energy fluctuations of a free massless scalar
field in a sub-volume of Minkowski space are proportional to the
boundary area of this sub-volume.  Energy fluctuations of quantum
fields in their vacuum state in the whole of Minkowski space
vanish, of course, because the vacuum is an eigenstate of the
hamiltonian, but energy fluctuations in a sub-volume of Minkowski
space do not \footnote{
    As the sub-volume in question is of the order of the volume of
    Minkowski space, the energy fluctuations rapidly decrease to
    zero.}.
These are energy fluctuations due to quantum entanglement. They
are, in some sense, similar to entanglement entropy discussed
above but do not involve tracing over DOF, and are calculated as
quantum expectation values in a pure state, rather than as
statistical averages over a mixed state. 
Area scaling is not specific to energy fluctuations of free
massless fields. We show that it is valid for 
correlation functions of a class of integral operators and
persists for interacting field theories. In specific cases we
show that the area scaling of correlation functions results from a
boundary field theory at high temperature. Our results
were obtained for systems 
which do not involve gravity explicitly, 
so they seem to suggest that it is the nature of quantum entanglement 
which allows for a boundary description of
bulk theories.


Let us start with a simple example: energy fluctuations in a
sub-volume $V$ of Minkowski space of a free massless quantum field
theory in its vacuum state. We define the (normal-ordered) energy
operator for this sub-volume $ E^V=\int_V
:\!\mathcal{H}(\vec{x})\!: d^dx $, where $:\mathcal{H}(\vec{x}):$
is the normal-ordered zero-zero component of the energy-stress
tensor, i.e., the hamiltonian density. For a free massless field
theory in $d$ spatial dimensions it is possible to express $E^V$
in terms of canonical creation and annihilation operators
$a^\dagger_{\vec{p}}$ and $a_{\vec{p}}$: $
    E^V = \frac{1}{4}\frac{1}{(2\pi)^{2d}}
        \int_V d^dx\int
        \frac{d^dp}{\sqrt{p}}\,\frac{d^dq}{\sqrt{q}}
        e^{\imath(\vec{p}+\vec{q})\cdot \vec{x}}$
       $ \left\{2\left(p q -\vec{p} \cdot \vec{q} \right)
            a^\dagger_{\vec{p}}a_{\vec{q}} -
            \left(\vec{p} \cdot \vec{q} + p q \right)
            \left(a_{\vec{p}}a_{\vec{q}} +
                a^\dagger_{\vec{p}}a^\dagger_{\vec{q}} \right)
        \right\}.
$ The vacuum expectation value of $E^V$ vanishes: $\langle 0| E^V
|0\rangle=0$. However, the energy fluctuations do not:
\begin{equation}
\label{E:HV2}
    \langle 0|(E^V)^2|0\rangle = \frac{1}{8} \frac{1}{(2\pi)^{2d}}
               \int_V d^dy_1\int_V d^dy_2 \int d^dp \int d^dq
                e^{\imath(\vec{p}+\vec{q})\cdot
               (\vec{y}_1-\vec{y}_2)}
               {\left(\frac{\vec{p} \cdot \vec{q}}
                    {\sqrt{ p  q}} +
                    \sqrt{p q}\right)}^2.
\end{equation}
After performing the integration over the momenta, the integrand
is a function of the variable $|\vec{y}_1\!-\!\vec{y}_2|$.
Rewriting eq. ({\ref{E:HV2}) as $\langle(E^V)^2\rangle= \int
\limits_0^\infty F(y) \D_V(y) dy, $ where $
    F(y)=\frac{1}{8} \frac{1}{(2\pi)^{2d}}
           \int
            \left( pq+
                2\vec{p} \cdot \vec{q}+
                \frac{(\vec{p} \cdot {\vec{q}})^2}{pq} \right)
            e^{-\imath(\vec{p}+\vec{q})\cdot \vec{y}}
                d^dp\,d^dq,
$
and
\begin{equation}
\label{hol5}
    \D_V(y)=\int_V d^dy_1\int_V d^dy_2 \,
            \delta^{(d)}(y-|\vec{y}_1-\vec{y}_2|),
\end{equation}
the integral is separated into a geometric factor, $\D_V(y)$,
which depends only on the shape of the sub-volume $V$, and a
factor $F(y)$ which is specific to the operator that is being
evaluated.

Evaluating $\langle 0|(E^V)^2|0\rangle$ we find, remarkably, that
it is proportional to the boundary area of $V$ for various
geometries. It is possible to compute $\langle 0|(E^V)^2|0\rangle$
analytically with an exponential momentum cutoff for a spherical
sub-volume of radius $R$ in $d$ odd spatial dimensions. The
results, in the limit of a macroscopic sphere $\Lambda R \gg 1$
($\Lambda$ is the high momentum cutoff), can be summarized in an
`empirical' formula $\langle 0|(E^V)^2|0\rangle=
\frac{K_d}{2^{d+1} \pi^2} \Lambda^{d+1}R^{d-1}$, where $K_d$ is a
fraction of two integers. (for example, $K_3=8/15$,
$K_{11}=11072/189189$,  and
$K_{27}=140737488355328/189060384200625$.) We have also evaluated
energy fluctuations in half of Minkowski space with an exponential
momentum cutoff $\langle 0|(E^V)^2|0\rangle= V_{\bot}
\Lambda^{d+1}
\frac{\left(\Gamma(\frac{d}{2}+\frac{1}{2})\right)^2}{2^d
d~\pi^{d/2+1} \Gamma(\frac{d}{2})}$ (See \cite{bruyar} for
details). Here $V_{\bot}$ is the $d-1$ dimensional sub-volume
(area) of the boundary separating the two halves of Minkowski
space. $\langle 0|(E^V)^2|0\rangle$ can also be evaluated
numerically using different cutoff schemes. We find that the area
dependence is a robust result, while the numerical coefficients
depend on the details of the cutoff procedure.

The surprising result that energy fluctuations in a sub-volume of
Minkowski space are proportional to the boundary area of the
sub-volume is not accidental. Rather, it seems to be a very general
characteristic property of entangled systems. We wish to
understand this fact, and examine how general it is, which we do
in the rest of the paper.


%\section{Operators in a finite sub-volume and their correlation functions  }
As a generalization of the above, we wish to examine expectation
values of operators of the form $\langle O^V_i O^V_j \rangle$
defined as follows. Let us consider a  finite $d$ dimensional
space-like domain of volume $V$ and linear size $R$, which we
think  of as being part of a $d=D-1$ dimensional Minkowski space.
We denote the complement of the sub-volume $V$ by $\widehat{V}$.  A
quantum field theory is defined in the whole space. Generically,
it is an interacting field theory, which comes equipped with a
high momentum (UV) cutoff $\Lambda$, and some regularization
procedure which makes all correlation functions well defined. A
low momentum (IR) cutoff is implicitly assumed, but we will not
discuss any of its detailed properties. The sub-volume is
``macroscopic" in the sense that $R\Lambda\gg 1$. The boundary of
$V$ is an ``imaginary" boundary, since we do not impose any
boundary conditions, or restrictions on the fields on it.


We will be interested in a set of  (possibly composite) operators
$O_i$ which can be expressed as an integral over a density ${\cal
O}_i$: $ O_i=\int_{V+\hat{V}} d^d x {\cal O}_i(\vec{x})$. For this
class we may define the operators $ O_i^V=\int_V d^d x {\cal
O}_i(\vec{x}) $ and $O_i^{\widehat{V}}=\int_{\widehat{V}} d^d x
{\cal O}_i(\vec{x})$. Since the whole space is the union of $V$
and $\widehat{V}$, we find that $ O_i= O_i^V+O_i^{\widehat{V}}$.
We further restrict our attention to a given quantum state of the
large space. We will discuss for concreteness the case in which
the large space is in the vacuum state $|0\rangle$. The operators
which we consider are operators $O_j$ for which the vacuum is an
eigenstate, $O_j|0\rangle=0$, for example the hamiltonian,
momentum, angular momentum, charge etc. However, the vacuum is not
necessarily and eigenstate of $O_j^V$ or $O_j^{\widehat{V}}$, so
$|0\rangle$ is an entangled state with respect to $V$ and its
complement. Our results can be easily generalized to states other
than the vacuum

We have already considered energy fluctuations in a given
sub-volume $V$. In general, in addition to fluctuations of
integral operators, namely, two point functions of the same
operator, we may look at two point functions of different
operators. Such correlation functions have the surprising property
\begin{equation}
 \label{hol2}
 \langle 0| O_i^V O_j^V |0\rangle = \langle 0|
  O_i^{\widehat{V}}O_j^{\widehat{V}} |0\rangle,
\end{equation}
which holds irrespective of the linear size $R$ of the inner
domain, or the scaling dimensions of the operators. To prove this
look at
\begin{eqnarray}
\label{hol3} 0&=&\langle 0|\left(O_i^V- O_i^{\widehat
V}\right)\left( O_j^{V}+ O_j^{\widehat V}\right)|0\rangle
\nonumber \\
&=& \langle 0|O_i^{V}O_j^{V} - O_i^{\widehat V}O_j^{\widehat V} +
O_i^{V}O_j^{\widehat V}-O_i^{\widehat V}O_j^{V} |0\rangle,
\end{eqnarray}
where the first equality relies on the fact that $O_j|0\rangle=0$.
The correlation function  $\langle 0|{\cal O}_i(\vec{y}_1){\cal
O}_j(\vec{y}_2)|0\rangle=
f_{ij}\left(|\vec{y}_1-\vec{y}_2|\right), $  depends only on
$|\vec{y}_1-\vec{y}_2|$ if the whole system is translation and
rotation invariant. Now,
\begin{equation}
\label{hol4}
    \langle 0| O_i^{V}O_j^{\widehat V}-O_i^{\widehat V}O_j^{V} |0\rangle =
    \int\limits_V\!\! d^d y_1 \int\limits_{\widehat V}\!\!  d^d y_2
    \left(
        f_{ij}\left(|\vec{y}_1-\vec{y}_2|\right)-
        f_{ij}\left(|\vec{y}_2-\vec{y}_1|\right)
    \right)=0,
\end{equation}
which gives us eq (\ref{hol2}). This is the
first hint of non-extensivity in such systems. Equation
(\ref{hol2}) shows that the correlation functions $\langle
0|O_i^{V}O_j^{V}|0\rangle$ must depend only on properties of the
boundary of the sub-volume $V$, and suggests that there might be a
holographic representation of them on the boundary. We will show
later that the correlation functions depend linearly on the
boundary area of the sub-volume $V$, in analogy with the entropy of
entanglement in such situations.

We now evaluate the correlation function $\langle
0|O_i^{V}O_j^{V}|0\rangle= \int\limits_V d^d y_1 \int\limits_{V}
d^d y_2~ f_{ij}\left(|\vec{y}_1-\vec{y}_2|\right)$. Following
\cite{befo}, as in the energy fluctuations case, we evaluate all
the integrals except for the $y=|\vec{y}_1-\vec{y}_2|$ integral,
$\int\limits_V d^d y_1 \int\limits_{V} d^d y_2~
f_{ij}\left(|\vec{y}_1-\vec{y}_2|\right)=\int\limits_0^{\infty}
dy~ \D_V(y)~ f_{ij}(y)$.

The geometric factor $\D_V(y)$ can be evaluated explicitly in some
cases, but we will only need to use $ \D_V(y)\sim y^{d-1}$  for
$y\sim 0$: considering $\vec{y}_1$ and $\vec{y}_2$ in eq.
(\ref{hol5}), the limit $y\sim 0$ is approached when both vectors
are almost equal. Fixing one of them, say $\vec{y}_1$, at some
arbitrary value makes the integral over $\vec{y}_2$ an
unrestricted $d$-dimensional integral, which in spherical
coordinates yields the above result. We may now define ${\cal
D}_V(y) = y^{d-1}R^d\widetilde \D_V\left(\frac{y}{R}\right)$,
which captures its behavior near $y\sim 0$, and its
dimensionality. We will also need to know that for generic
sub-volumes $ {\widetilde \D_V}^{\prime}(0) \ne 0$.

The short distance/large momentum  behavior of  correlation
functions is determined by the full mass dimensionalities
$\delta_i$, $\delta_j$ (including anomalous dimensions induced by interactions)  of the
operator densities ${\cal O}_i$ and ${\cal O}_j$.  At short distances
$f_{ij}\left(|\vec{x}|\right)\sim
\frac{1}{|\vec{x}|^{\delta_i+\delta_j} }$, equivalently for large
momentum, $ \widetilde f_{ij}\left(|\vec{q}|\right) \sim
|\vec{q}|^{\delta_i+\delta_j-d}$,
where we have defined
$\widetilde f_{ij}\left(|\vec{q}|\right)= \int ~ d^dx
 ~ e^{i\vec{q}\cdot\vec{x}}
f_{ij}\left(|\vec{x}_1-\vec{x}_2|\right)$.
Note that this estimate is strictly valid for large momentum. In
general, $\widetilde f_{ij}\left(|\vec{q}|\right)$, will be a
function of several powers of $q$,
which may be fractional. We will consider operators
whose correlation functions are sharply peaked at short distances,
and decay at large distances. The short distance/large momentum
behavior of correlation functions is in many cases singular, and
needs to be regularized. This is done by
introducing an ultraviolet momentum cutoff.

So after all is said and done, we need to evaluate the following
integral,
\begin{equation}
 \label{corr3}
   \int\limits_0^{2R} dy~ {\cal D}_V(y)~ f_{ij}(y)    =
   R^d\int\limits_0^{2R} dy~ y^{d-1}
  ~\widetilde{\cal D}_V\left(\frac{y}{R}\right)
    \int \hbox{\large$\frac{d^dq}{(2\pi)^d}$} ~ e^{-iqy \cos\theta_q}
        \widetilde{f}_{ij}(|\vec{q}|),
\end{equation}
where $\theta_q$ is the angle of the vector $\vec q$ with respect
to some fixed axis. The last momentum integral needs to be
regularized, since it is divergent in the large momentum region
whenever $\delta_i+\delta_j>0$.

Integrals of the form appearing in eq.(\ref{corr3}) can be
evaluated analytically for integer powers $(q^2)^n$ with a
gaussian momentum cutoff, and for integer and fractional powers
$q^\alpha$ with an exponential momentum cutoff. We will discuss in
some detail the integer case, and quote the results for the other
case.

For a Gaussian cutoff eq. (\ref{corr3}) with $f_{ij}(q)\sim q^{2n}$, reduces to
\begin{equation}
 \label{eyeen1}
I_n=R^d\int\limits_0^{2R} dy~ y^{d-1}~\widetilde{\cal
D}_V\left(\frac{y}{R}\right) \int d^dk ~ e^{ i k y
\cos\theta_k-\frac{1}{2} k^2/\Lambda^2}
(k^2)^n.
\end{equation}
Naively, this integral scales extensively, as $R^d\Lambda^{2 n}$,
as the prefactor might suggest.
The $k^2$ term may be rewritten as a d-dimensional laplacian in
spherical coordinates: $\frac{1}{y^{d-1}}\partial_y y^{d-1}
\partial_y$. If we now integrate by parts once, the surface term will vanish due
to the properties of $\cal D_V$, and the integrand will include a
term $\widetilde{\cal D_V}$. For $R\Lambda\gg 1$, due to the
gaussian term, we may approximate $y\sim 0$, and we are left with
\begin{equation}
\label{eyeen2}
I_n=(-1)^n~a_V~C_d~R^{d-1} \Lambda^{2n-1},
\end{equation}
where $C_d\sim\int\limits_0^{\infty} dx
 x^{d-2} \left[ \left(\frac{1}{x^{d-1}}\partial_x x^{d-1}
\partial_x\right)^{n-1}\hbox{\large $e$}^{ \hbox{${-\frac{1}{2}
x^2}$}}\right]$ is a finite $d$- and $n$-dependent remnant of the
momentum integral, and $a_V= (2\pi)^{d/2}~ \widetilde{\cal
D}'_V(0)$. Thus we have shown that $I_n$ scales as
$R^{d-1}\Lambda^{2n-1}$. Similarly $ {\cal I}_\alpha=R^{d}
\int\limits_0^{2R} dy~y^{d-1} ~\widetilde{\cal
D}_V\left(\frac{y}{R}\right) \int d^dk ~ e^{
i\vec{k}\cdot\vec{y}-k/\Lambda} k^\alpha\sim \Lambda^{\alpha-1}
R^{d-1} $, where $\alpha$ is not necessarily integer.

It is possible to show that the area scaling law
$\langle O_{i}^VO_{j}^V \rangle \propto
 \Lambda^{\delta_i+\delta_j-1}R^{d-1}$ is robust to changes in the
cutoff procedure. Any cutoff function $C(k/\Lambda)$ will give
similar results provided that it and all of its derivatives vanish
at $k\rightarrow\infty$, and that its integral $\int d^d k
C(k/\Lambda)$ is finite.


%\section{Holographic representation of correlation functions}
The fact that correlation functions of operators scale as the
boundary area of the sub-volume rather than as the volume as
expected, suggests that it might be
possible to find a holographic representation of them on the
boundary $\partial V$ of $V$ (and of $\widehat {V}$).  To show
this we shall express $\langle 0|O_i^{V}O_j^{V}|0\rangle$ as a
double derivative and use Gauss' law. Since $\widetilde f_{ij}(0)=0$, this is
possible in general. Approximating these correlation functions by their
small distance behavior, we have
\begin{align}
 \label{boundary2.5}
    \langle 0|O_i^{V}O_j^{V}|0\rangle &\sim
        \int\limits_V d^d y_1
        \int\limits_{V} d^d y_2~
        \frac{c_{ij}^d}{|\vec{y}_1-\vec{y}_2|^{\delta_i+\delta_j}} \\
\notag
    &= {\alpha^d_{ij}}\int\limits_V d^d y_1
        \int\limits_{V} d^d y_2~
            \vec{\nabla}_{1}\cdot \vec{\nabla}_{2}
            \frac{c_{ij}^{d-1}}{|\vec{y}_1-\vec{y}_2|^{\delta_i+\delta_j-2}}\\
\notag
    & = {\alpha^d_{ij}}\oint\limits_{\partial V} d^{d-1} z_1
        \oint\limits_{\partial V} d^{d-1} z_2~
        \frac{c_{ij}^{d-1} \hat n_1 \cdot\hat n_2}{|\vec{z}_1-\vec{z}_2|^{\delta_i+\delta_j-2}},
\end{align}
where $\hat n$ is a boundary unit normal, and
$\alpha^d_{ij}=\frac{c_{ij}^d}{(\delta_i+\delta_j-2)(d-\delta_i-\delta_j)c_{ij}^{d-1}}$.
In the special case $\delta_i+\delta_j=2$ the power becomes a
logarithm, and for $\delta_i+\delta_j=d$ it becomes a delta
function, so they need to be handled with special care. Note that
the term $|\vec{z}_1-\vec{z}_2|$, even though evaluated at
$\vec{z}_1$ and $\vec{z}_2$ on the boundary, generally expresses
distances in the bulk. In the case where the boundary is a plane,
$|\vec{z}_1-\vec{z}_2|$ indeed expresses distances on the
boundary.

This allows us to define an
equivalent correlation function in $d-1$ dimensions of some
operators $\Theta_i^{\partial V}$ and $\Theta_j^{\partial V}$
\begin{equation}
\label{boundary4}
    \langle 0|O_i^{V}O_j^{V}|0\rangle \sim
    \langle 0|\Theta_i^{\partial V}\Theta_j^{\partial V}|0\rangle_{d-1}= 
    \alpha^d_{ij} \int\limits_{\partial V}\!\! d^{d-1} z_1
    \int\limits_{\partial V}\!\! d^{d-1} z_2~\hat n_1 \cdot\hat n_2
        \langle 0|\hbox{\large$\vartheta$}_i(\vec{z}_1)
        \hbox{\large$\vartheta$}_j(\vec{z}_2)|0\rangle_{d-1}.
\end{equation}
Only the short distance behavior of the correlation function can be
read off from eq.(\ref{boundary2.5}):
$   \langle 0|\vartheta_i(\vec{z}_1)
    \vartheta_j(\vec{z}_2)|0\rangle_{d-1}\sim
    \frac{c_{ij}^{d-1}}{|\vec{z}_1-\vec{z}_2 |^{\delta_i+\delta_j-2} }
$.

It will be very interesting to determine the consistency
conditions under which a holographic representation of correlation
functions can be derived from an effective action, and to
determine how symmetries of the sub-volume and of the theory in the
bulk are reflected in the properties of their boundary holographic
representations. A covariant formulation, starting from a
Lagrangian in $D=d+1$ space-time dimensions would be very useful.
At this point we would like to present evidence that  the boundary
theory is a theory at high-temperature, and leave the answers to
the questions above open for future research.

We consider a massless, free scalar field theory and take $V$ to
be half of Minkowski space (under some assumptions it is
possible to generalize this to other geometries). We first show that the
n-point (single field) functions for a (d+1) dimensional free
scalar field theory in the vacuum state are equal to n-point
functions of a ((d-1)+1) dimensional free scalar field theory at
high-temperature,
\begin{equation}
\label{E:dimredexp}
    \langle 0| e^{-J \int_V \phi (\vec{x}) d^dx}|0 \rangle_{d}
    = \langle e^{-J \sqrt{\alpha^d_{\phi\phi}}\int_{\partial V} \phi (\vec{x})
    d^{d-1}x}\rangle_{d-1}^{\beta \to 0}.
\end{equation}
We shall show this equality term by term,
$
    \int_V d^dx_1\ldots\int_V d^dx_{2n}\langle 0|\phi (\vec{x}_1) \cdots
    \phi (\vec{x}_{2n})|0 \rangle_{d}  =
    ({\alpha^d_{\phi\phi}})^n \int_{\partial V} d^{d-1}x_1 \cdots\int_{\partial V}
    d^{d-1}x_{2n}
        \langle\phi (\vec{x}_1) \cdots \phi (\vec{x}_{2n})
        \rangle_{d-1}^{\beta \to 0}.
$
For an odd number of fields this is trivially satisfied, since
both sides vanish. For an even number of free fields in any
dimension, $
    \langle 0|\phi (\vec{x}_1) \cdots
    \phi (\vec{x}_{2n})|0 \rangle=
     \sum\limits_{\substack{\text{\tiny{all}} \\ \text{\tiny{perm.}}}}
        \langle 0|\phi(\vec{x}_{i_1})\phi(\vec{x}_{i_2})|0 \rangle \cdots
        \langle 0|\phi(\vec{x}_{i_{2n-1}})\phi(\vec{x}_{i_{2n}})|0 \rangle$.
So to prove eq.(\ref{E:dimredexp}) we only need to show
that
 $ \int_V d^dx_1\int_V d^dx_2 \langle 0|\phi (\vec{x}_1) \phi
(\vec{x}_2)|0 \rangle_{d} =
    \alpha^d_{\phi\phi} \int_{\partial V} d^{d-1}x_1\int_{\partial V} d^{d-1}x_2
        \langle \phi (\vec{x}_1) \phi (\vec{x}_2)\rangle_{d-1}^{\beta \to 0}.$
This can be shown using eq.(\ref{boundary2.5}). The mass dimension
of a scalar field in $d+1$ dimensions is $\delta=\frac{d-1}{2}$,
so
\begin{align}
\notag
    \int_V d^dx_1\int_V d^dx_2 \langle
        \phi (\vec{x}_1)
        \phi (\vec{x}_2) \rangle_{d} & = 
    \alpha^d_{\phi\phi} \int_{\partial V} d^{d-1}x_{1_\bot}
    \int_{\partial V}d^{d-1}x_{2_\bot}
        \frac{\hat{n}_1 \cdot
        \hat{n}_2}{|\vec{x}_1-\vec{x}_2|^{d-3}} \\ & =
\label{boundaryp}
      \alpha^d_{\phi\phi}
        \int_{\partial V} d^{d-1}x_{1_\bot}
        \int_{\partial V} d^{d-1}x_{2_\bot}
        \langle \phi(\vec{x}_1) \phi(\vec{x}_2)
        \rangle_{d-1}^{\beta \to 0}.
\end{align}
In eq.(\ref{boundaryp}) we have used the fact that a $D=d+1$
dimensional field theory in the limit of high temperature is
equivalent to a $D-1$ dimensional field theory at zero
temperature, and that $\hat{n}_1\cdot\hat{n}_2=1$ when $V$ is half
of Minkowski space. Our choice of viewing the power of the
integrand as that of a $(d-1)$ dimensional theory is motivated by
the region of integration.

Similarly, such dimensional reduction may also be explicitly shown
for other two point functions
$
    \int\limits_ V  d^dx_1 \int\limits_V d^dx_2 \langle 0|
    \nabla_1^m \phi^{n} (\vec{x}_1)
          \nabla_2^{m^{\prime}} \phi^{n^{\prime}}
       (\vec{x}_2) |0 \rangle_{d}
    \cong
    \int\limits_{\partial V} d^{d-1}x_{1_\bot}
            \int\limits_{\partial V} d^{d-1}x_{2_\bot}
            \langle\nabla_{1}^{m+n-1}  \phi^{n} (\vec{x}_1)
            \nabla_2^{m^{\prime}+n-1} \phi^{n^{\prime}}
     (\vec{x}_2) \rangle_{d-1}^{\beta \to 0}
$, where by $\nabla_{1}^{m} \nabla_{2}^{m^{\prime}}$, we mean the scalar operator obtained by
consecutive operations of $\nabla$; in order for this to be a scalar operator we must have that
$m+m^{\prime}$ is even.
This may also be generalized to the case of single field n-point functions with an
arbitrary number of derivatives.

We are able to extend our conclusions to the most general correlation functions
in the theory - products of powers of fields and their derivatives, only in the
case of a large number of fields $N$. To discuss the large $N$ limit we define
$\Phi(x_i)=\hbox{diag}(\phi_1(x_i),\ldots,\phi_N(x_i))$.  In this limit only
correlations functions of the form $\int_V \ldots \int_V
<\hbox{Tr}\Phi(x_1)\ldots\hbox{Tr}\Phi(x_n)> d^dx_1\ldots d^dx_n$ contribute to
leading (zero) order in the $1/N$ expansion. We have already shown that
such correlation functions have a holographic representation in terms of a
free field theory at high temperature, thus we have
shown that, to first order in the large $N$ limit, all bulk correlation
functions have a holographic description.


\acknowledgments

Research supported in part by an Israel Science Foundation grant
174/00-2. A.Y. is supported in part by the Kreitman Foundation. We
thank G. Veneziano for illuminating discussions.

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\end{document}

