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\begin{document}
\title{\Large \bf De Sitter Invariant Vacuum States, Vertex Operators,
and Conformal Field Theory Correlators  }

\author{Aharon Casher$^{a,b}$, Pawel O. Mazur$^a$ and Andrzej J. Staruszkiewicz$^c$}
%\preprint{} \preprint{}
%\preprint{hep-th//yymmdd}
\address{$^a$ Department of Physics and Astronomy \\
University of South Carolina \\
\sl Columbia, S.C. 29208, U.S.A. \\
%{\tt mazur@mail.psc.sc.edu}
}
\address{$^b$ School of Physics and Astronomy \\ Tel Aviv University \\ Tel Aviv 69978, Israel
\\
%{\tt ronyc@post.tau.ac.il}
}
\address{$^c$ Marian Smoluchowski Institute of Physics \\
Jagellonian University \\
Reymonta 4, 30-059 Krak\'{o}w, Poland \\
%{\tt astar@th.if.uj.edu.pl}
}
%\date{\today}
\abstract {\small{\noindent We show that there is only one
physically acceptable vacuum state for quantum fields in de Sitter
space-time which is left invariant under the action of the de
Sitter-Lorentz group $SO(1,d)$ and supply its physical
interpretation in terms of the Poincare invariant quantum field
theory (QFT) on one dimension higher Minkowski spacetime. We
compute correlation functions of the generalized vertex operator
$:e^{i\hat{S}(x)}:$, where $\hat{S}(x)$ is a massless scalar
field, on the $d$-dimensional de Sitter space and demonstrate that
their limiting values at timelike infinities on de Sitter space
reproduce correlation functions in $(d-1)$-dimensional Euclidean
conformal field theory (CFT) on $S^{d-1}$ for scalar operators
with arbitrary real conformal dimensions. We also compute
correlation functions for a vertex operator $e^{i\hat{S}(u)}$ on
the \L obaczewski space and find that they also reproduce
correlation functions of the same CFT. The massless field
$\hat{S}(u)$ is the nonlocal transform of the massless field
$\hat{S}(x)$ on de Sitter space introduced by one of us. }}

\pacs{04.62.+v, 11.10.Kk, 11.25.Hf, 11.30.Cp, 11.90.+t \qquad
\qquad } \maketitle2 \narrowtext

{\bf Introduction.} The maximally symmetric and homogeneous
solution to Einstein's equations with the positive vacuum energy
density is the de Sitter space-time. It is therefore important to
understand the behavior of quantum fields on this space. Recently
this subject has been taken up with renewed interest after the new
%and spectacular
evidence for the positive cosmological constant
%and the accelerating expansion of the Universe
has turned up in recent cosmological measurements. Although
quantum field theory (QFT) on de Sitter spacetime is a rather well
studied subject
\cite{as89,as90,as92,cherntag,tag,mott,wyroz,lokas,amm,mazmott,strom1,strom2}
there are some aspects of it which attracted considerable interest
most recently \cite{amm,mazmott,strom1,strom2}. One of them is the
question of uniqueness of the de Sitter invariant vacuum state
\cite{as89,as90,as92,cherntag,tag,mott,wyroz,lokas,strom2} while
the second one is the fact first established in
\cite{as89,amm,mazmott} that the limiting values of correlation
functions at timelike infinities on de Sitter space reproduce
correlation functions of Euclidean CFT in one dimension lower.
This property is now referred to as the de Sitter/CFT
correspondence in the recent literature \cite{strom1,strom2}. In
some early papers on the subject of QFT on de Sitter space an
analysis of solutions to the wave equation for a massive scalar
field led to the conclusion that there exists a one-parameter
family of de Sitter invariant vacuum states for a massive scalar
field $\hat{\Phi}(x)$ \cite{cherntag,tag,mott}. In the current
literature the origin of this one-parameter family of vacuum
states is regarded as mysterious \cite{cherntag,tag,mott,strom2}.
Similarly, the proposal of de Sitter/CFT correspondence based on
the analysis of the propagator for massive fields on de Sitter
space leads to some difficulties because the scaling dimensions
$\Delta_{\pm} = \frac{d - 1}{2} {\pm} \sqrt{(\frac{d-1 }{2})^2 -
m^2}$ become complex for $m^2 > (\frac{d-1}{2})^2$
\cite{amm,strom1}.

\vskip .1cm \noindent The purpose of this Letter is to address
these two aspects of QFT on de Sitter space. In particular, we
shall demonstrate the existence of the one-parameter family of de
Sitter invariant vacuum states $\mid \lambda>$ by solving
equations following from the conditions that they are annihilated
by generators of the de Sitter group, $\hat{M}_{_{ab}}\mid
\lambda> = 0$ and we will connect their origin to the same
property of the Lorentz covariant QFT on Minkowski space. It is
well known that the spatial infinity of $d+1$-dimensional
Minkowski space-time is the $d$-dimensional de Sitter space
\cite{pgb}. Quantum field theory of massless and massive fields on
de Sitter space inherits its vacuum structure from the Lorentz
invariant QFT of massless fields in Minkowski spacetime. There
exists a one-parameter family of Lorentz invariant vacuum states
for QFT on Minkowski spacetime. This implies the presence of
$\lambda$-vacua for QFT on de Sitter space. It turns out that
these states are physically unacceptable for $\lambda \neq 0$ as
they introduce correlations between the antipodal points on de
Sitter spacetime. We shall show that there exists only one
physically acceptable and unique de Sitter invariant vacuum state
for massive and massless fields on de Sitter space, corresponding
to $\lambda = 0$, and supply the physical interpretation of this
state in terms of the Poincare invariant QFT of a massless
particle in $d+1$-dimensional Minkowski space-time
\cite{as89,as90,as92,wyroz,lokas}. The positive frequency modes
$e^{-ik\cdot x }$ of a massless particle in Minkowski spacetime
properly projected on de Sitter space lead to the unique positive
frequency modes on de Sitter space
\cite{as89,as90,as92,wyroz,lokas} and to a unique physically
acceptable de Sitter invariant vacuum state.

We shall also address the now well known fact
\cite{as89,amm,mazmott,strom1} that behavior of two-point
correlation functions at timelike infinities on $d$ dimensional de
Sitter space (and at spatial infinity on \L obaczewski space) is
characteristic of the $d-1$ dimensional Euclidean conformal field
theory (CFT). In order to make the de Sitter/CFT correspondence as
transparent as possible and at the same time satisfy the
requirement that conformal dimensions $\Delta$ of scalar operators
in CFT be real and positive we shall introduce a family of
generalized vertex operators $e^{i\hat{S}(x)}$ for a massless
scalar field $\hat{S}(x)$ and compute their two-point correlation
functions \cite{as89,as90,as92}. We demonstrate that every
correlation function for a scalar operator of conformal dimension
$\Delta$ is reproduced by our vertex operators on de Sitter
spacetime. \noindent {\bf De Sitter invariant vacuum states.} De
Sitter space is a single-sheeted hyperboloid in the $d+1$
dimensional Minkowski space: $x^a\eta_{ab}x^b = {x {\cdot} x} =
-1$. This representation of de Sitter space makes it transparent
that the de Sitter isometry group is $SO(1,d)$. The generators of
this symmetry corresponding to Killing vectors on de Sitter space
are $M_{ab} = i(x_a\partial_b - x_b\partial_a)$. Using the
standard global parametrization of the hyperboloid $x^0 =
sinh\tau$, $x^i = n^i cosh\tau$, $\textbf{n}^2 = 1$ one obtains
the induced metric on de Sitter space: $ds^2 =
g_{\mu\nu}dx^{\mu}dx^{\nu} = d\tau^2 - cosh^2\tau
d\Omega^2_{d-1}$. The d'Alambertian on de Sitter space is
${\Box}_g =
|det(g)|^{-\frac{1}{2}}\partial_{\mu}(|det(g)|^{\frac{1}{2}}g^{\mu\nu}\partial_{\nu})
= (cosh\tau)^{1-d}\partial_{\tau}((cosh\tau)^{d-1}\partial_{\tau})
+ cosh^{-2}\tau \textbf{L}^2$, where $\textbf{L}^2$ is the
Laplacian on the unit $S^{d-1}$ sphere, and it is the same as the
Casimir operator ${\cal{C}}=\frac{1}{2}M_{ab}M^{ab}$ of the de
Sitter group. The Casimir operator ${\cal{C}}$ enters a very
useful identity relating the d'Alambertian on Minkowski space
${\Box}_{d+1} = \eta^{ab}\partial_{a}\partial_{b}$ to the one on
de Sitter space

\be \label{identity} {\cal{C}} = \frac{1}{2}M_{ab}M^{ab} = - x^2
{\Box}_{d+1} + {\cal E}({\cal E} + d - 1)  \ , \ee

\noindent where ${\cal E} = x^a\partial_a$ is the Euler operator
(generator of dilatation subgroup of the conformal group). This
identity allows us to describe massive (and massless) scalar
fields on de Sitter space in terms of a massless scalar field on
Minkowski space. A massless scalar field $\Phi_{_\Delta}(x)$ in
Minkowski space with the scaling dimension $\Delta$,
$\Phi_{_\Delta}(\lambda x) = \lambda^{-\Delta} \Phi_{_\Delta}(x)$,
${\cal{E}}\Phi_{_\Delta}(x) = - \Delta\Phi_{_\Delta}(x)$,
corresponds to a massive scalar field on de Sitter space with a
mass $m$ such that $m^2 = \Delta(d - 1 - \Delta)$. For a given
mass there are two scaling dimensions $\Delta_{\pm} = \frac{d -
1}{2} {\pm} \sqrt{(\frac{d-1 }{2})^2 - m^2}$. In the following we
choose $\Delta = \Delta_{-}$ for the scaling dimension of a
massless scalar field on Minkowski space corresponding to a
massive scalar on de Sitter space. The dynamics of this massive
scalar field $\Phi(x)$ is defined by the following Lagrangian
density: ${\cal {L}} =
\frac{1}{2}(g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi -
m^2\Phi^2)$. The Poincare invariant QFT of a massless scalar field
on Minkowski space induces a de Sitter invariant QFT of a massive
(or massless) scalar field on de Sitter space with a unique vacuum
state. This follows immediately from (\ref{identity}) and the
observation that the sign of the Klein-Gordon norm for a massless
scalar field on Minkowski space is preserved upon projection on
the de Sitter hyperboloid \cite{as89,as90,as92}. This has two
immediately clear implications. First, a positive frequency
solution for a massless scalar on Minkowski space \be
\label{posfreq} \Phi^{(+)}_{_\Delta}(x) = \int
\frac{(dk)_{d}}{k_0} e^{-i{k{\cdot}x}} a_{_\Delta}(k) \ , \ee
\noindent where $a_{_\Delta}(\lambda k) =
\lambda^{-\tilde{\Delta}}a_{_\Delta}(k)$, $\tilde{\Delta} = d -1 -
\Delta$, and $k$ is a null momentum vector $k^2 = 0$, when
projected on the de Sitter hyperboloid ${x{\cdot}x} = -1$ becomes
a positive frequency solution for the Klein-Gordon wave equation
on de Sitter space. Choosing $a_{_\Delta}(k) = {\sum_{nM}} N_n
{k_0}^{-\tilde{\Delta}}Y_{nM}(\hat{k}) a_{nM}$ with
$\tilde{\Delta} = \Delta_{+}$, $\hat{k} = \frac{k}{k_0}$, and
$Y_{_{nM}}(\hat{k})$ the spherical harmonics on $S^{d-1}$, one
obtains the positive frequency modes of the massive scalar field
on de Sitter space $f_n(\tau)Y_{nM}(\textbf{n})$, \be \label{mode}
f_n(\tau)Y_{nM}(\textbf{n}) = N_n \int \frac{(dk)_{d}}{k_0}
e^{-i{k{\cdot}x}} {k_0}^{-\tilde{\Delta}}Y_{nM}(\hat{k})  \ . \ee
\noindent The integral (\ref{mode}) can be easily evaluated
\cite{lokas} and it is given below in (\ref{modes}) with \be
\label{coeffs} A^{(0)}_n = \frac{1}{2}\sqrt{\frac{\Gamma(\frac{n +
{\Delta}_{+}}{2})\Gamma(\frac{n +
{\Delta}_{-}}{2})}{\Gamma(\frac{n + 1 +
{\Delta}_{+}}{2})\Gamma(\frac{n + 1 + {\Delta}_{-}}{2})}} \,\,\,\
, \,\,\ B^{(0)}_n = - \frac{i}{2A^{(0)}_n}  \,\ . \ee \noindent
The spherical harmonics $Y_{nM}(\hat{k})$ and $Y_{nM}(\textbf{n})$
are eigenfunctions of the Laplacian on $S^{d-1}$ corresponding to
the eigenvalue $n(n+d-2)$ and the degeneracy $d_n(d) = \frac{(2n +
d-2)}{(d-2)!}(n + 1)...(n + d -3)$. Second, by promoting the
amplitudes $a_{nM}$ to the annihilation operators $\hat{a}_{nM}$
one also obtains the positive frequency part of the massive scalar
field operator $\hat{\Phi}^{(+)}(x)$ on de Sitter space. The
unique de Sitter invariant vacuum state $\mid 0>$ is defined by
the condition $\hat{\Phi}^{(+)}(x)\mid 0> = 0$.

We now construct the generators ${\hat{M}_{ab}}$ of the de Sitter
group acting on the Fock space of a massive scalar field
$\hat{\Phi}(x)$. To claim de Sitter invariance of a vacuum state
one needs to impose the conditions ${\hat{M}_{ab}}\mid 0> = 0$,
where conserved ${\hat{M}_{ab}} = \int (cosh\tau)^{d-1}
d\Omega_{d-1} :\hat{T}^0_{\mu}(x):\xi^{\mu}_{ab}$ are the
generators of the de Sitter group evaluated at $\tau = 0$,
$\xi^{\mu}_{ab}$ are the de Sitter space Killing vectors, and
$:\hat{T}_{\mu\nu}(x): =
:\partial_\mu\hat{\Phi}(x)\partial_{\nu}\hat{\Phi}(x): -
\frac{1}{2} g_{\mu\nu}:{\cal{L}}(x):$ is the stress-energy
momentum tensor of a massive scalar field on de Sitter space. The
field operator of a massive scalar field on the de Sitter space is
$\hat{\Phi}(x) = {\sum_{nM}} (f_n(y)Y_{nM}(\textbf{n})
\hat{a}_{nM} + h.c.)$, where

\bea f_n(y) &=& (1 - y)^{s_{-}}[A_n F(s_{-}+ \frac{n}{2},s_{-} +
\frac{2 -d -n}{2};\frac{1}{2};y) + \,\nonumber\\
&& + B_n y^{\frac{1}{2}}F(s_{-}+ \frac{n +1}{2},s_{-}+ \frac{3 - d
- n}{2};\frac{3}{2};y)] \,, \label{modes}\eea \noindent $s_{-} =
\frac{1}{2}\Delta_{-}$, $y = tanh^2\tau$, and $F$ is the
hypergeometric function. The coefficients $A_n$ and $B_n$ satisfy
the condition $i(\overline{A}_nB_n - A_n\overline{B}_n) = 1$
following from the positivity of the Klein-Gordon norm for the
positive frequency modes $f_n(y)Y_{nM}(\textbf{n})$. To find the
de Sitter invariant vacuum state it is sufficient to compute the
de Sitter group generators ${\hat{M}_{0i}}$ and demand that they
annihilate this vacuum state. The Killing vectors
$\xi^{\mu}_{ab}$, $a=0$, $b=i$, corresponding to these generators
are expressed in terms of the $n =1$ spherical harmonics, $\xi_{0}
= Y_{1M_0}(\textbf{n})$, $\xi_{\alpha} = - sinh{\tau}cosh{\tau}
\partial_{\alpha}Y_{1M_0}(\textbf{n})$.
We obtain the following formula for the only
nonvanishing terms in the normal ordered expression for the
generators ${\hat{M}_{0i}}$ acting on the vacuum state $\mid 0>$

\be {\frac{1}{2}}{\sum_{nM}} {\sum_{n'M'}} C_{nn'} {\int
d\Omega_{d-1}} Y_{1M_0} \overline{Y}_{nM}
\overline{Y}_{n'M'}{\hat{a}^{\dagger}}_{nM}{\hat{a}^{\dagger}}_{n'M'}
+ ...
 \ , \label{generators}\ee

\bea C_{nn'} &=&
\overline{B}_{n}\overline{B}_{n'} + (m^2+{\frac{1}{2}}n(n+d-2)+ \,\nonumber\\
&& + {\frac{1}{2}}n'(n'+d-2)-{\frac{1}{2}}(d-1))\overline{A}_{n}
\overline{A}_{n'} \,. \label{coefs}\eea

\noindent The generators $\hat{M}_{0i}$ annihilate the vacuum
state $\mid 0>$ only when the coefficients in front of two
creation operators in (\ref{generators}) vanish. The
Clebsch-Gordan coupling coefficients for $SO(d)$ do not vanish for
$\mid n - n'\mid = 1$. This implies that $C_{nn'}$ for $\mid n -
n'\mid = 1$ must vanish. We obtain the following simple equation
for $\alpha_n = i\frac{B_n}{A_n}$  \be \label{secondcontraint}
\alpha_n \alpha_{n+1} = m^2 + n(n+d-1) \ , \ee

\noindent which together with the K-G norm condition leads to the
following result: for $n$ even, $A_n = e^{\lambda}A^{(0)}_n$, $B_n
= e^{-\lambda}B^{(0)}_n$ and for $n$ odd, $A_n =
e^{-\lambda}A^{(0)}_n$, $B_n = e^{\lambda}B^{(0)}_n$, with
$\lambda$ real. $A^{(0)}_n$ and $B^{(0)}_n$ are the coefficients
obtained before (\ref{coeffs}) from the evaluation of the integral
(\ref{mode}). The closure of the Lorentz-de Sitter Lie algebra
implies that the remaining generators will also annihilate the
vacuum state. There exists a simple canonical transformation
between the field operator $\hat{\Phi}_{\lambda}(x)$ corresponding
to the modes (\ref{modes}) and the field operator $\hat{\Phi}(x)$
corresponding to the unique vacuum state (\ref{mode}):
$\hat{\Phi}_{\lambda}(x) = cosh{\lambda}\hat{\Phi}(x) +
sinh{\lambda}\hat{\Phi}(-x)$, which makes the correlations between
antipodal points transparent. The modes (\ref{modes}) correspond
to the following Lorentz, but not Poincare, invariant choice for
positive frequency modes for a QFT on Minkowski spacetime:
$f_{k}(x) = cosh{\lambda}e^{-ik{\cdot}x} +
sinh{\lambda}e^{ik{\cdot}x}$. This concludes our demonstration of
the existence of a one-parameter family of de Sitter invariant
vacuum states $\mid \lambda>$ and elucidates their physical
meaning.

\noindent {\bf Vertex operators on de Sitter spacetime and their
correlation functions.} The special case of a massless scalar
field which is the phase $\hat{S}(x)$ canonically conjugate to the
electric charge operator $\hat{Q}$ has been studied extensively in
the physically most interesting case $d=3$ in
\cite{as89,as90,as92}. It turns out that a special care is needed
in the treatment of the zero mode $n=0$ of the Laplacian on the
unit sphere. Indeed, our formula for $A^{(0)}_n$ has a simple pole
at $n=0$ for a massless field and as such is meaningless. We need
to solve the d'Alambert equation for the $n=0$ part of a massless
scalar field which from now on we denote $\hat{S}(x)$. The
dynamics of $\hat{S}(x)$ is derived from the somewhat more
conveniently normalized Lagrangian density ${\cal{L}} =
(2e^2{\Omega}_{d-1})^{-1}g^{\mu\nu}\partial_{\mu}\hat{S}\partial_{\nu}\hat{S}$,
where ${\Omega}_{d-1}$ is the volume of $S^{d-1}$ and $e$ is the
coupling constant whose meaning is that of a unit of electric
charge in the $d = 3$ case. We find the following expression for
the massless field operator $\hat{S}(x)$ \be \label{massless}
\hat{S}(x) = \hat{S}_0 + e\hat{Q}f_0(\tau) +
{\sum_{nM}}(f_n(\tau)Y_{nM}(\textbf{n})\hat{a}_{nM} + h.c.) \, \ee
\noindent where the sum is over $n\neq 0$ and the zero mode
$f_0(\tau)$ satisfies the following equation:
$(cosh{\tau})^{d-1}\partial_{\tau}f_0(\tau) = 1$. The operators
$\hat{S}_0$ and $\hat{Q}$ are canonically conjugate:
$[\hat{S}_0,\hat{Q}] = ie$. The annihilation and creation
operators satisfy the following commutation relations:
$[\hat{a}_{nM},\hat{a}^{\dag}_{n'M'}] =
e^2\Omega_{d-1}\delta_{nn'}\delta_{MM'}$. It is easy to show that
the de Sitter group generators evaluated for a massless scalar
$\hat{S}(x)$ do not depend on the constant phase $\hat{S}_0$ but
do depend on the charge operator $\hat{Q}$. The de Sitter
invariant vacuum state is then characterized by the following
conditions: $\hat{Q}\mid{0}> = 0$, $\hat{a}_{nM}\mid{0}> = 0$. In
the representation in which the operator $e^{i\hat{S}_0}$ is
diagonal and eigenstates of $\hat{Q}$ are normalizable $S_0$ is a
periodic variable with the usual period of $2{\pi}$. It is a phase
conjugate to a charge $\hat{Q}$.

It is convenient, following \cite{as89,as90,as92}, to introduce
the generalized vertex operator on de Sitter space $\hat{V}(x) =
:e^{i\hat{S}(x)}:$ and compute its two-point correlation function
\be \label{correlator} <0\mid
:e^{i\hat{S}(x)}::e^{-i\hat{S}(y)}:\mid 0> = e^{F(x,y)} \ , \ee
\noindent where $F(x,y) = e^2(<0\mid
\hat{S}^{+}(x)\hat{S}^{-}(y)\mid 0> - \frac{i e^2}{2}(f_0(x) -
f_0(y)))$, with $\hat{S}^{+}(x)$ and $\hat{S}^{-}(x)$ the positive
and negative frequency parts of $\hat{S}(x)$. The master function
$F(x,y)$ can be computed exactly but its exact form is of no
concern for us here. The remarkable property of this function is
its universal asymptotic behavior when $\tau(x) = \tau(y) =
{\pm}\infty$. One finds in this limit \be \label{conf}
F(\hat{x},\hat{y}) = - C(d)e^2 \left( \frac{1}{(d-3)!}
ln(\frac{1-cos\theta}{2}) + C'(d)\right) \ , \ee \noindent where
$cos\theta = \textbf{n}(x)\cdot\textbf{n}(y)$ and $C(d) =
{{2^{d-3}\Gamma^2({{d-1}\over 2})}\over {(d-2)\pi}}$, $C'(d) = (1
+ (-1)^d)ln2 + (-1)^d{\sum_{n=1}^{d-2}}(-1)^n \frac{1}{n}$. This
implies that the two-point correlation function of the vertex
operator on timelike infinity $S^{d-1}$ is equal to the two-point
correlation function of the scalar operator with the conformal
dimension $\Delta = C(d)e^2$ in the Euclidean CFT on $S^{d-1}$.
The underlying reason for this relation is the isomorphism of the
conformal group of $R^{d-1}$ (or $S^{d-1}$) and the de Sitter
group $SO(1,d)$ on $d$-dimensional de Sitter space which is the
basic kinematical fact with many different dynamical realizations
\cite{amm,mazmott}. Our basic formula (\ref{correlator})
generalizes the one obtained in the special case of $d=3$
\cite{as89}. Unlike the case of the two-point correlation function
of a massive scalar field first discussed in \cite{amm} there are
no problems with the vanishing with $\tau\rightarrow\infty$ of an
overall amplitude of the $d-1$-dimensional CFT two-point function
and the complex values of the scaling dimensions $\Delta_{{\pm}}$
for $m^2
> (\frac{d-1}{2})^2$. Using our vertex operators we can reproduce
all multi-point correlation functions of scalar primary scaling
operators of any scaling dimension in the Euclidean
$d-1$-dimensional CFT. \noindent {\bf Correlation functions on the
\L obaczewski space.} The operator $\hat{S}_0$ can be extracted
from $\hat{S}(x)$ by averaging over the sphere $S^{d-1}$, $\tau =
0$, which is an intersection of $x^0 = 0$ with the de Sitter
hyperboloid $x{\cdot}x = -1$. The proper Lorentz invariant
generalization of $\hat{S}_0$ is the following nonlocal transform
from de Sitter space to the \L obaczewski space of velocities $u$
\cite{as89,as90,as92} \be \label{StarTransform} \hat{S}(u) =
\frac{2}{\Omega_{d-1}} \int (dx)_{d+1}\delta(x{\cdot}x + 1)
\delta(u{\cdot}x) \hat{S}(x) \ , \ee \noindent where $u{\cdot}u =
1$. The induced metric on the \L obaczewski space is $ds^2 =
d{\psi}^2 + sinh^2{\psi}d\Omega^2_{d-1}$. One can show that
$\hat{S}(u)$ satisfies the Laplace equation on \L obaczewski
space: $\Delta \hat{S}(u) = 0$. A quick inspection of
(\ref{StarTransform}) reveals that only the even part of
$\hat{S}(x)$, $\hat{S}_e(-x) = \hat{S}_e(x)$ enters the formula
for $\hat{S}(u)$. This means that for $n$ even only $A_n$ and for
$n$ odd only $B_n$ part of the positive (negative) frequency modes
enters the formula for $\hat{S}(u)$. We obtain the following
expression for $\hat{S}(u)$ \be \label{phaseLob} \hat{S}(u) =
\hat{S}_0 + {\sum_{nM}}(f_n(y)Y_{nM}(\textbf{n})\hat{a}_{nM} +
h.c.) \ , \ee \noindent where the sum is over $n\neq 0$. The
$f_n(y)$ modes on \L obaczewski space are the same functions of
$y$ as before but here we have $y = coth^2{\psi}$. The fact that
only even modes in $\hat{S}(x)$ appear in $\hat{S}(u)$ implies
that $\hat{Q}$ does not appear in it. The two-point correlation
function of the vertex operator $\hat{U}(u) = e^{i\hat{S}(u)}$ is
a finite function which does not require that the normal ordering
prescription be applied in marking contrast to the de Sitter case
\be \label{correlLob} <0\mid e^{i\hat{S}(u)}e^{-i\hat{S}(v)}\mid
0> = e^{H(u,v)} \ , \ee

\be \label{Hfunct} H(u,v) = - \frac{1}{2}<0\mid (\hat{S}(u) -
\hat{S}(v))^2\mid 0> + \frac{1}{2}[\hat{S}(u),\hat{S}(v)] \ . \ee
\noindent One can show that the commutator function vanishes on \L
obaczewski space. Lorentz invariance implies that $H(u,v)$ is a
function of the scalar product $u{\cdot}v = cosh\lambda = z$,
$H(u,v) = H_d(z)$, which vanishes for $z = 1$ and satisfies an
inhomogeneous Laplace equation on \L obaczewski space: $\Delta_u
H(u,v) = const$. The complete result of integration of the
resulting differential equation on $d$-dimensional \L obaczewski
space is \bea H_{d}(z) &=& - e^2
\frac{2^{d-3}(d-1)}{{\pi}(d-2)!}\Gamma^2(\frac{d-1}{2}) \times
 \,\nonumber\\&& \times{\int_1^z} dx (x^2 - 1)^{-\frac{d}{2}} {\int_1^x}
dy (y^2 - 1)^{\frac{d}{2} - 1} \ . \label{finresult}\eea \noindent
In particular for $d=3$ we find \cite{as89,as90,as92}
$H_3(\lambda) = - {e^2\over \pi}({\lambda}coth{\lambda} - 1)$ and
for $d=4$ we have $H_4(\lambda) = - {e^2\over
2}(lncosh{\lambda\over 2} + {1\over 4 }tanh^2{\lambda\over 2})$.
\noindent This completes our demonstration that the proper
framework for the discussion of the relation between correlation
functions on de Sitter (or \L obaczewski) space and correlation
functions of the Euclidean CFT on a timelike infinity $S^{d-1}$
are the generalized vertex operators and this is the context where
the so-called dS/CFT correspondence \cite{amm,mazmott,strom1} was
first discovered \cite{as89,as90,as92}. \noindent The work of P.
O. M. is partially supported by the NSF grant to the University of
South Carolina. One of us (A. C.) wishes to acknowledge research
support by U.S.C. during his sabbatical leave of absence.

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\end{thebibliography}

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