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\begin{document}

\begin{titlepage}
\begin{flushleft}
       \hfill                      {\tt hep-th/0304241}\\
       \hfill                      HIP-2003-28/TH \\
       \hfill                      CERN-TH/2003-097 \\
       \hfill                      April 28, 2003\\
\end{flushleft}
\vspace*{3mm}
\begin{center}
{\Large {\bf The Taming of Closed Time-like Curves}}
\end{center}
\begin{center}
\vspace*{12mm}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}
{\large Rahul Biswas${}^{3}$\footnote{E-mail: rbiswas@students.uiuc.edu}, Esko
Keski-Vakkuri${}^{1}$\footnote{E-mail:
esko.keski-vakkuri@helsinki.fi}, Robert G.
Leigh${}^{2,3}$\footnote{E-mail: rgleigh@uiuc.edu}, \\  Sean
Nowling${}^{3}$\footnote{E-mail: nowling@students.uiuc.edu}
and Eric Sharpe${}^{3}$\footnote{E-mail: ersharpe@uiuc.edu}}
\vspace*{4mm}

{\em ${}^{1}$Helsinki Institute of Physics \\
P.O. Box 9,
FIN-00014  University of Helsinki,
Finland \\}
\vspace*{5mm}
{\em ${}^{2}$CERN-Theory Division\\
CH-1211, Geneva 23, Switzerland\\}
\vskip .5cm
{\em ${}^{3}$Department of Physics,
University of Illinois\\
1110 W. Green Street, Urbana, IL 61801, U.S.A.}
\vspace*{10mm}
\end{center}

\begin{abstract}
We formulate QFT on a $\orbid$ orbifold, in a manner which is
invariant under the $\Zmath_2$ time and space reversal. This is a
background with closed time-like curves. It is also relevant for
the elliptic interpretation of de Sitter space. We calculate the
one-loop vacuum expectation value of the stress tensor in the
invariant QFT, and show that it does not diverge at the boundary
of the region of closed time-like curves. Rather, the only
divergence is at the initial time slice of the orbifold, analogous
to a spacelike Big-Bang singularity. We then calculate the
one-loop graviton tadpole in bosonic string theory, and show that
the answer is the same as if the target space would be just the
Minkowski space $\Rmath^{1,d}$, suggesting that the tadpole
vanishes for the superstring. Finally, we argue that it is
possible to define local S-matrices, even if the spacetime is
globally time-nonorientable.
\end{abstract}

\end{titlepage}

\baselineskip16pt

\section{Introduction and Summary}
A technical obstacle in exploring string theory in time-dependent
space-times is to find suitable backgrounds where string
quantization is tractable. Early work includes
\cite{Horowitz:1991ap,
Nappi:1992kv,Kounnas:1992wc,Kiritsis:1993av,Martinec:1995xj,Lawrence:1996ct}.
More recently, interest has been revitalized, motivated in part
by novel string-based cosmological scenarios (see for example
\cite{Khoury:2001bz,Seiberg:2002hr,Veneziano:2000pz}).
An obvious path to follow was
to construct such backgrounds as time-dependent orbifolds of
Minkowski space
\cite{vijayetal,Nekrasov:2002kf,Cornalba:2002fi,Simon:2002ma,Liu:2002ft,
Liu:2002kb,Horowitz:2002mw,Lawrence:2002aj,Fabinger:2002kr}
or anti-de Sitter space \cite{Elitzur:2002rt,Craps:2002ii,Simon:2002cf}.
Further related work includes
\cite{Aharony:2002cx,Buchel:2002kj,Alishahiha:2002bk,
Dudas:2002dg,Berkooz:2002je,Tolley:2002cv,Gordon:2002jw,
Figueroa-O'Farrill:2002tb,Fabinger:2003jn,Hashimoto:2002nr,Cornalba:2003ze}.
However, depending on
how the orbifold identifications are defined, potentially dangerous issues
may arise. The resulting time-dependent orbifolds can have regions
with closed time-like curves (CTCs) or closed null curves (CNCs),
or may not even be globally time-orientable. Therefore,
one could choose to first make a list of desirable features for the
orbifolds and then try to limit the study only to those backgrounds
that possess those features.  This sensible strategy was laid out and
pursued by Liu, Moore and Seiberg \cite{Liu:2002ft,Liu:2002kb}.
For orbifolds of type $\Rmath^{1,3}/\Gamma $ where $\Gamma$
is a discrete subgroup of the Poincar\'e group, the list turned
out to be very short containing only the null branes with $R>0$.
However, the null brane construction involves identifications by arbitrarily large
boosts. This turns out to be another
potential reason for instabilities, and it was argued by Horowitz
and Polchinski \cite{Horowitz:2002mw} that such backgrounds become
unstable after just a single particle is added, because on the
covering space the particle can approach its infinitely many
images with increasingly high momenta and produce a black hole.
Additional discussion of potential problems can be found in \cite{Lawrence:2002aj,
Liu:2002kb,Berkooz:2002je}.

Aside from constructing and studying time-dependent backgrounds
by alternative methods, one might speculate if the
list of desirable features for
suitable orbifold backgrounds was too prohibitive and reconsider the
reasons for including each item on the list. In any case, it is important to understand
if and/or why string theory actually has problems with these features. The reason for
demanding that there be no regions containing closed time-like
curves appears obvious. CTCs violate causality, and it has been
conjectured by Hawking \cite{Hawking:1992nk} that the laws of
physics prevent CTCs from appearing. The arguments in support of
this chronology protection conjecture (CPC) are usually based on
general relativity plus matter at the classical or semiclassical
level. A recent summary can be found in \cite{Visser:2002ua}.
Essential features are that perturbations can keep propagating
around a CTC so that backreaction accumulates, or quantum effects
can lead the matter stress tensor to diverge at the boundary of
the CTC region, leading to infinite backreaction. However, the
trouble with CTCs and CNCs seems to arise from propagation along
them, rather than merely from their existence. It is not clear if
the two are equivalent. For example, the model studied in
\cite{vijayetal} involves CTCs and CNCs, but it was argued that
they do not necessarily pose a problem in quantum mechanics if one
projects to a subspace of states which do not time evolve along
the CTCs and CNCs. Another desirable feature on the list was
time-orientability. This was included to avoid problems in
defining an S-matrix, and problems associated with the existence of
spinors \cite{Kay,Chamblin:1995jz,Chamblin:1995nq}.
However, the consequences of a
lack of time-orientability have not yet been subject to extensive
investigation and are thus less well understood. From the point of view of
local physics, one might wonder if the whole Universe could be
globally time-nonorientable, but in such a way that the global
feature could only be detected by meta-observers and never be
revealed by local experiments. The orbifold studied in
\cite{vijayetal} is an example of a spacetime which is globally
time-nonorientable. In any case, its structure appears to allow for a definition of an S-matrix for local experiments.

To summarize, there are many reasons to investigate the chronology
protection conjecture and time-nonorientability. We also note that
recently the former topic has been investigated from other points
of view in the context of string theory and holography
\cite{Boyda:2002ba,Herdeiro:2002ft,Harmark:2003ud,Gimon:2003ms,
Dyson:2003zn}.

The $\orbid$ orbifold, obtained by identifying points $X$ with
reflected points $-X$, provides a simple model which incorporates
both issues. Some comments were made in passing in
\cite{vijayetal}. In this paper we perform a more detailed
investigation. First, we formulate quantum field theory in a
manner which appropriately incorporates the $\Zmath_2$
identification under time reversal and space reflection. This
construction is also relevant for the elliptic interpretation of
de Sitter space ($dS$)
\cite{Schrodinger1,Gibbons:1986dd,Folacci:1987gr,Parikh:2002py}. A
$d$-dimensional de Sitter space is a time-like hyperboloid embedded
in $\Rmath^{1,d}$. The $\Zmath_2$ reflection on $\Rmath^{1,d}$
induces an antipodal reflection on the $dS$ spacetime. The elliptic de
Sitter space $dS/\Zmath_2$ is then defined by identifying the
reflected antipodal points. Previous studies of the elliptic $dS$
spacetime have discussed problems in defining a global Fock space in
the global patch; however, it was possible to construct QFT and a
Fock space by restricting to the static patches of observers at
the (identified) north and south poles. Similar problems are
encountered in trying to formulate QFT on $\orbid$. We circumvent
these problems by first doubling the field degrees of freedom,
with the copy fields propagating towards the reversed time
direction, and then identifying the degrees of freedom under the
$\Zmath_2$ reflection. The doubling of fields is motivated by (the
zero temperature limit of) the real-time formulation of finite
temperature QFT. The doubling of degrees of freedom helps to
overcome problems with causality when the light-cones of the
identified points $X$ and $-X$ intersect, as we will assume that
the two copies of the fields (at $X$ and $-X$) are dynamically
decoupled. Note that in the limit in which the cosmological
constant approaches zero, the $dS$ spacetime becomes locally Minkowski
spacetime. Correspondingly, it has been argued that in this limit the
elliptic $dS$ spacetime goes to two copies of Minkowski spacetime, related
by the $\Zmath_2$ reflection \cite{Parikh:2002py}. In the present
work, we would instead propose that QFT on the elliptic $dS$ spacetime goes to
QFT on $\orbid$, with two copies of {\em fields}, identified under
the $\Zmath_2$ reflection.


After introducing the $\Zmath_2$ invariant QFT, we show that the
reformulation of QFT {\em removes} the infinite backreaction
normally associated with the boundary of the CTC region. This is
not necessarily a contradiction with the CPC, since the
reformulation can also forbid propagation {\em around} a CTC. We
then study the  backreaction at one-loop level in string theory.
We calculate the one-loop graviton tadpole in the $\orbid$
background, and show that the answer is the same as if the
background were just $\Rmath^{1,d}$! While the answer by
itself would at first appear puzzling, it appears very natural in
relation to the $\Zmath_2$ invariant formulation of QFT. Indeed,
the low-energy limit of string theory should be the $\Zmath_2$
invariant QFT. Finally, we argue that it is possible to define
S-matrices in a manner that makes sense locally.
The definition only breaks down at the point which can be regarded
as the initial ``Big Bang singularity'' of the orbifold, and at
that point we also find that even in the invariant reformulation
of the QFT, the stress tensor diverges. However, it is also
possible that stringy effects lead to a smooth blow-up of the
orbifold singularity. Then the QFT would need to be reconsidered
in this smooth background.

We have organized the paper as follows.
In Section \ref{sec:oldorbid}, we review some features of the
time-dependent orbifold background, and focus on some novel features
of these orbifolds. In particular, we point out that a choice of time
orientation must be made. In Section \ref{sec:tradcalc}, we review
the (na\"ive) analysis of the gravitational back reaction in this
geometry. In Section \ref{sec:qftnew}, we present a proper formulation
of quantum field theory on the $\orbid$ background, and show that the
only divergence of the stress tensor, beyond the familiar short-distance
Minkowski spacetime divergence, can be interpreted as a ``cosmological
initial condition.'' In Section \ref{sec:stringcalc}, we present some
details of similar calculations in string theory which demonstrate the
same results (complementary calculations in a different formalism are shown
in the Appendix.) Finally, in Section \ref{sec:smat}, we discuss further
features of the interacting QFT, including a discussion of the S-matrix.

\section{Overview of $\orbid$}
\label{sec:oldorbid}

Let us first review some features of the $\orbid$ orbifold
\cite{vijayetal}.
We begin with the covering space $\Rmath^{1,d}$ and identify the
time and space coordinates under the reflection
\begin{equation}
\label{eq:orbidaction}
(t,x^a) \sim (-t,-x^a) \ .
\end{equation}
The resulting orbifold is a space-time cone, depicted in
Figure \ref{fig:cone.eps} for $d=1$.
Points in the opposite quadrants are identified. The
orbifold contains closed time-like curves, an example of which is depicted
in the figure.
\myfig{cone.eps}{12}{The orbifold $\Rmath^{1,1}/\Zmath_2$.}
Orbifolds that act purely spatially are familiar and are certainly well
understood. New problems arise when the identification involves
the time direction; for example it is not guaranteed that the string
spectrum will be free from tachyons and ghosts.
Ref. \cite{vijayetal} investigated bosonic and type II
superstrings on $\orbid \times \Rmath^{n}$, with $n$ additional
spacelike directions added to bring the total spacetime dimension
to 26 or 10. It was shown, using a Euclidean continuation, that although the background is
time-dependent and quantization had to be done in the covariant
gauge, the physical spectrum did not contain any negative norm
states (ghosts). The superstring spectrum did not contain any
tachyons. Also, the one-loop partition function vanishes for the
superstring.

Although string theory passed the first tests, questions
associated with the time identification on the orbifold remained.
In the orbifold (\ref{eq:orbidaction}), there is actually
extra data that must be specified. To see this, we note that to specify a
Lorentzian metric on an orientable space $M$ ($w_1(M)=0$), we must
specify a {\it time orientation}. Mathematically, this implies a real
rank 1 subbundle $L\subset TM$, the time orientation bundle. ($M$ is said to be
globally time-orientable when $w_1(L)=0$.) A time-like Killing vector defining time's arrow, if available, would be  a
global section of this line bundle.

In the case of the orbifold (\ref{eq:orbidaction}), we must ask how various
quantities descend from the covering space to the orbifold. In particular,
$\partial/\partial t$ is manifestly not invariant under
the group action, and so does not define a time's arrow, or
time-like Killing vector, in the quotient.  Thus, this orbifold
leaves ambiguous the direction on which time flows in the quotient
 -- we must manually make a choice of direction of time-flow.

Furthermore, the natural time orientation bundle on the covering space does
descend to the quotient space, but (omitting the singularity at the origin)
the class $w_1(L)$ is non-trivial. Thus the image of $L$ on the quotient is
not time-orientable. Although locally we can choose a perfectly sensible
notion of time orientation, this is not possible globally.

In fact, there are essentially three choices.
\myfig{pic1.eps}{12}{Three possible time-arrows
on the quotient $\Rmath^{1,1}/\Zmath_2$.}
To illustrate, let us consider the case of $\Rmath^{(1,1)}/\Zmath_2$.
The obvious choice of time's arrow on the
covering space $\Rmath^{1,1}$,
namely $\partial/ \partial t$, is not invariant under the group action,
a property which manifests itself in the observation that by picking different
fundamental domains for the group action on the cover, the time's arrow
in those fundamental domains restricts to a different time's arrow on the
quotient.

In Fig. \ref{fig:pic1.eps} we have shown three possible time-arrows that one
can construct on $\Rmath^{1,1}/\Zmath_2$.
The left-most case corresponds to taking the fundamental domain
to be regions II and IV, the middle case corresponds to taking the
fundamental domain to be regions I and II, and the right-most case
corresponds to taking the fundamental domain to be one
side of a wall of the lightcone.
In this case, omitting the origin, the time-orientation line bundle
on the quotient is not orientable ($w_1(L) \neq 0$),
hence each choice of time's arrow depicted in figure~\ref{fig:pic1.eps} has
zeroes -- in case (a), along the left vertical crease,
and in case (b), along the bottom horizontal crease. Note that in
each case it would also be possible to choose a reverse time
orientation (reversed arrows). Then {\em e.g.} Fig.
\ref{fig:pic1.eps}(b) would depict a "big crunch".

In Figure \ref{fig:figgg3.ps}, we have drawn the quotient space
corresponding to Fig. \ref{fig:pic1.eps}(a). In this case, there are
asymptotic regions for both $t\to\pm\infty$.
\myfig{figgg3.ps}{4}{A view of the quotient spacetime (for 1+1 dimensions).
Note the absence of the $x=0$ axis for $t<0$.}
Another choice for the quotient space, corresponding to
Fig. \ref{fig:pic1.eps}(b),  is shown in Fig. \ref{fig:figg4.ps}. In this case, there is no is no asymptotic region corresponding to $t\to -\infty$.
\myfig{figg4.ps}{5}{Another view of the quotient spacetime (for 1+1 dimensions).
Note the absence of the $t=0$ axis for $x<0$. The $t=0$ axis represents
a ``big bang'' singularity--the beginning of the spacetime.}
Instead, we have a ``big bang'' singularity at $t=0$. It is interesting to
contemplate the properties of quantum field theory on such a spacetime.
It is of even more interest to ponder the role of string theory.
We will return to a more thorough discussion of these issues in a later section.


\section{Backreaction in Quantum Field Theory}
\label{sec:tradcalc}

Here we give a short review of the standard QFT calculation
for the vacuum expectation value (vev) of the stress tensor, showing
that it diverges at
the boundary of the CTC region. Later, we will contrast this with a calculation
in string theory.

The standard argument for the instability of a spacetime (here, specifically,
a spacetime
orbifold) with closed time-like curves proceeds as follows. The gravitational
backreaction from the renormalized stress energy of a quantum field may be
evaluated semi-classically
\begin{equation}
  G_{\mu\nu} = -8\pi G_N\bra T_{\mu\nu} \ket_{ren}.
\end{equation}
Here the subscript refers to the fact that we subtract off the usual
vacuum energy contribution ---  the curvature is well-defined if there are
no divergences other than the usual flat space short distance singularities.
When one evaluates the right hand side, one expects to find it
to diverge at the boundary of the CTC region, signaling infinite
backreaction and instability of the spacetime.

The $\orbid$ orbifold
has a region of closed time-like curves. On the covering
space, it is the region bounded by the future and
past light-cones emanating from the origin. Then one would expect
that the vacuum expectation value of the stress tensor diverges on
the light-cones, signaling potential instability. Indeed, a quick
calculation yields just such a divergence. To illustrate the point, consider
the  $\Rmath^{1,1}/\Zmath_2$ orbifold and
a free massless scalar field. The field decomposes into left- and
rightmovers. Let us focus on the right-movers only. The right-moving
component of the stress tensor is
\begin{equation}
T_{uu}(u) = :\pat_u \phi (u) \pat_u \phi (u): \ ,
\end{equation}
where $u=t-x$. In Minkowski space, this is obtained by subtracting the divergence
that is
uncovered through point-splitting
\begin{equation}
T_{uu}(u) = \left.\lim_{u'\rightarrow u} \pat_u \phi (u)
\pat_{u'} \phi (u')\right|_{ren}
\end{equation}
making use of the two-point function
\begin{equation}
G(u,u') = \bra 0 |\phi (u)\phi (u')|0\ket \sim -\ln (u-u') \ .
\end{equation}

On the orbifold, we require the field operator
and the stress tensor to be invariant under the $(t,x)\rightarrow (-t,-x)$
$\Zmath_2$ reflection. For the right-moving component, let us impose the
invariance by considering
\begin{equation}
  \tilde{\phi}(u) = \frac{1}{\sqrt{2}} (\phi (u)+\phi (-u)) \ .
\label{phitilde}
\end{equation}
The renormalized expectation value of the $\Zmath_2$ invariant
stress tensor is then\myfig{figgg5.ps}{4}{Correlator of point-split
composite operator. The 'short' contractions, between $x$ and $x'$ are the usual
short-distance ones, and should be subtracted. The 'long' contractions
give rise to the Casimir energy.}
\bea
 \bra \tilde{T}_{uu} (u) \ket_{ren} &=&
 \lim_{u'\rightarrow u} \half~\pat_u \pat_{u'}~
 \bra 0| \{\phi (u)+\phi(-u)\}\{\phi (u')+\phi(-u')\}\ket_{ren}
 \nonumber \\
 \mbox{} &=& \lim_{u'\rightarrow u} \pat_u\pat_{u'}
 \{ -\ln (u-u') -\ln (u+u')\}_{ren} \nonumber \\
 \mbox{} &=& \lim_{u'\rightarrow u}~\frac{1}{(u+u')^2} =
 \frac{1}{4u^2} \ .
\label{comm}
\eea
This diverges on the null line $u=0$, which is a part of the
boundary of the CTC region. A similar calculation for the
left-movers yields a divergence at the other part of the boundary, $v=0$.
Hence one concludes that the orbifold is potentially unstable.
Similar calculations can be done in higher dimensions.

However, upon closer inspection the above argument has some
puzzling features. The most cumbersome one is that the $\Zmath_2$
invariant field operator (\ref{phitilde}) has the mode expansion
\begin{equation}
 \tilde{\phi}(u) = \sqrt{2}
 \int d\omega~(a_\omega + a^\dagger_\omega)\cos(\omega u)
\end{equation}
so it is not clear what exactly is meant by the naive notion
of particles and vacuum. The problem of constructing a global
Fock space is also well known from investigations of
elliptic de Sitter space $dS/\Zmath_2$
\cite{Schrodinger1,Gibbons:1986dd,Folacci:1987gr,Parikh:2002py}.
In the above, the problem has been lifted onto $\orbid$, where
the $dS/\Zmath_2$ can be embedded.

Actually, we will argue that the orbifold
identification requires identifying a particle with positive energy
at $(t,x)$ with a particle with negative energy at $(-t,-x)$. Particles
of the latter kind cannot be created with $a^\dagger_\omega$. A quick
look at the mode expansion of $\phi (-u)$ might give a false impression
that this would happen, but really $\phi (-u)$ is just the field operator
$\phi$ evaluated at point $-u$ rather than a new operator with the
creation and annihilation operators acting in a different way.
Another problem is that the usual
prescription calls us to evaluate commutators of field operators
at equal time. On the orbifold covering space this becomes
problematic, since "equal time" now corresponds to times $t$
and $-t$. For these reasons we would like to take a step
back and reconsider the formulation of field theory on the
$\Rmath^{1,d}/\Zmath_2$ orbifold.


\section{Quantum Field Theory on $\orbid$ Revisited}
\label{sec:qftnew}

Consider a point particle on the fundamental domain of $\orbid$.
On the covering space, it corresponds to two particles: one with
positive energy (propagating forward in time), and its image
with negative energy (propagating backward in time) with opposite
momentum (Fig. \ref{fig:partic.eps}). In other words, for each particle with a
momentum $(k^0,\vek)$ we must include its image with momentum
$(-k^0,-\vek)$.

\myfig{partic.eps}{7}{A point particle on the orbifold.
a) depicts a single point particle on the fundamental domain,
while b) depicts the point particle and its image on the
covering space, moving towards the opposite
time and space directions.}

The situation is similar for strings on $\orbid$, analyzed in
\cite{vijayetal}. The
states in the untwisted sector which survive the $\Zmath_2$
projection are of the type
\begin{equation}
  |\psi \ket_S = \left( \alpha^{\mu_1}_{-n_1}\cdots
  \tilde{\alpha}^{\nu_1}_{-m_1}\cdots \right)_{S,A}~(|0,k\ket
  \pm |0,-k\ket ) \ ,
\end{equation}
{\em i.e.} symmetrized combinations of string states with
opposite pairs of center-of-mass momentum $k$.
We conclude that in order to have a good description of quantum
mechanics on the orbifold, we must start with pairs of states
with opposite energy and momentum.

In quantum field theory in Minkowski space,
1-particle states in the Fock space are associated with
positive energy,
\begin{equation}
   |\omega_k , \vek \ket = a^{\dagger}_{\vek} | 0 \ket \ \ {\rm
   with} \ k^0 = \omega_k >0.
\end{equation}
However, in order to formulate
a quantum field theory on the orbifold covering space,
we must also be able to include states with
$k^0 =-\omega_k <0$. In other words, what we need is a Fock space
$\Hcal$ which involves sectors with both sign choices for the
energy:
\bea
  \Hcal^+ &=& \prod_{\vek} \Hcal^+_{\vek} \ \ {\rm with} \ k^0
  =\omega_k > 0 \nonumber \\
  \Hcal^- &=& \prod_{\vek} \Hcal^-_{\vek} \ \ {\rm with} \ k^0
  =-\omega_k < 0 \ .
\eea
The full Fock space is then the direct sum
\begin{equation}
  \Hcal = \Hcal^+ \oplus \Hcal^- \ .
\end{equation}
In order to construct an invariant Fock space on the orbifold, we
need to first implement the
$\Zmath_2$ action as an isomorphism $\Hcal^\pm \rightarrow
\Hcal^\mp$ which acts by flipping the sign of energy
and momentum in the orbifolded directions. In particular, the
usual vacuum $|0\ket\in \Hcal^+$ must map to a state in $\Hcal^-$; we
will call it $|\tilde{0}\ket$. We will later define it and other
states in $\Hcal^-$ more
precisely. The invariant Fock space is then
\begin{equation}
    \Hcal_{inv} = \Hcal / \Zmath_2 \ .
\end{equation}
Given this orbifold identification, it should be noted that there is no particular problem with the stability of the theory related to a negative energy sea.

There are of course well known reasons to involve
both positive and negative energy sectors in the formulation of
QFT. One of them is QFT in curved spacetime, or other cases
where we compare observers who are not related by proper orthochronous
%Poincar\'{e}
Lorentz
transformations.  The mode expansions of the field operator
relevant for such observers are related by mixing of positive
and negative energies. In the present case, the $\orbid$ spacetime
is locally flat, but we want to identify (as opposed to compare) observers
related by the time (and space) reflection, and identify
the corresponding degrees of freedom.

Actually, a closely related starting point is
QFT in flat spacetime but
at finite temperature (FTQFT). The real-time formulation of FTQFT
also leads to mixing between positive and negative energies. For
inspiration, we shall review it briefly.
The starting point in the path integral formulation of real-time FTQFT
is the generating functional
\begin{equation}
  Z[J] = \int \Dcal \phi~\exp \left\{ i \int_C d^4x~[\Lcal (\phi (x))
  + J(x)\phi (x)] \right\}
\label{ZFT}
\end{equation}
where the time integral has been promoted to a contour
integral along a complex time path $C$, starting from some initial
time $t_i$ and ending at a complex final time $t_i-i\beta$, where
the imaginary part is given by the inverse temperature $\beta
=T^{-1}$ \cite{Niemi:1984nf}. The functional integral is taken over all field
configurations which satisfy the periodic boundary condition
\begin{equation}
  \phi (t_i-i\beta ,\vex ) = \phi (t_i,\vex ) \ .
\end{equation}
One convenient choice of the complex time path consists of three
segments, $C=C_1\cup C_2\cup C_3$, where $C_1$ runs along
the real axis from $t_i$ to some $t_f\gg t_i$, $C_2$ runs
backwards along the time axis from $t_f$ to $t_i$, and finally $C_3$
runs parallel to the imaginary axis from $t_i$ to $t_i-i\beta$
(Fig. 7).
\begin{figure}
  \begin{center}
 \epsfysize=2in
   %\mbox{\epsfbox{contour.eps}}
    \mbox{\epsfbox{tpath2.eps}}
\end{center}
\caption{The time contour for FTQFT.}
\label{pocket}
\end{figure}

From the generating functional, one can calculate the thermal
Green's function
\begin{equation}
    iD_C(x-x')= \bra T_C \phi (x)\phi (x') \ket \ ,
\end{equation}
where time ordering has been promoted to path ordering $T_C$ along
the complex time path $C$.
Equivalently, one can rewrite the thermal Green's function in
terms of a $3\times 3$ matrix $(D^{rs})_{r,s=1,2,3}$ with components
\begin{equation}
   D^{rs}(t-t') = D_C (t_r - t_s)
\end{equation}
where $t=t_r,\ t'=t_s$ if $t\in C_r$ and $t'\in C_s$ along $C=C_1\cup C_2\cup C_3$.
Furthermore, it can be shown that if one takes $t_i,t_f\rightarrow
-\infty, +\infty$ in an appropriate manner, the contributions
involving the segment $C_3$ decouple from the rest. The matrix $(D^{rs})$
then reduces to a $2\times 2$ matrix, but the temperature
dependence remains, as the components depend on the distribution
function of the thermal background.
The contour $C$ reduces
to the Schwinger-Keldysh contour \cite{Schwinger:1961qe,Keldysh:1964ud}
$C_1\cup C_2$. The propagator is reproduced by
breaking up the field $\phi$ and the source $J$ into two-component
vectors,
\bea
  \phi &=& (\phi_1,\phi_2) \ \ {\rm with}\
  \phi_r(x)=\phi(t_r,\vex),\ t_r\in C_r \nonumber \\
  J &=& (J_1,J_2) \ \ {\rm with }\ J_r(x)=J(t_r,\vex),\ t_r\in C_r
  \ .
\eea
The generating functional (\ref{ZFT}) then reduces to a form
\begin{equation}
  Z[J_1,J_2] = \int \Dcal \phi_1\Dcal \phi_2~\exp
  \left\{ i\int^{\infty}_{-\infty} d^4x~[\phi_r (D^{-1})^{rs} \phi_s
  + \Lcal_{int} (\phi_1)-\Lcal_{int} (\phi_2) + J_r\phi_r] \right\} \
\label{ZFT2}
\end{equation}
where $\Lcal_{int}$ is the interaction part of the Lagrangian.
In particular, the diagonal components of the
$2\times 2$ propagator $D^{rs}$ have the momentum space
representation
\bea
iD^{11}(k) &=& \frac{i}{k^2-m^2+i\epsilon}+2\pi \delta (k^2-m^2) n_T(k_0)
\nonumber \\
iD^{22}(k) &=& \frac{-i}{k^2-m^2-i\epsilon}+2\pi \delta (k^2-m^2) n_T(k_0)
%\nonumber \\
%iD^{12}(k) &=& 2\pi {\rm sign}(k_0) \delta (k^2-m^2) n(|\omega|)
%\nonumber \\
%iD^{21}(k) &=& 2\pi {\rm sign}(k_0) \delta (k^2-m^2) e^\omega
%n(|\omega|) \ ,
\eea
where $n_T(k_0)$ is essentially the thermal distribution function.
It is then evident that in addition to the physical field
$\phi_1$, the theory contains another degree of freedom $\phi_2$,
called the thermal ghost, which propagates backwards in time. The
two fields $\phi_{1,2}$ are coupled together only by the off-diagonal
elements $D^{12,21}$ of the propagator. One is interested in correlation functions
of $\phi_1$ only. Furthermore, it can be shown that in
the zero temperature limit $\beta \rightarrow \infty$ the
off-diagonal elements of the propagator vanish, $D^{12,21}\rightarrow
0$, so that the thermal ghost decouples from the physical degree
of freedom. Hence at zero temperature one can ignore the thermal
ghost and the theory reduces back to the usual form involving only the
physical degree of freedom. But at any finite $T$, both fields make
physical contributions.



\paragraph{The orbifold case.} In the above example, the Fock
spaces associated with the physical field and thermal ghost are
the positive and negative energy sectors $\Hcal^+$ and $\Hcal^-$.
At zero temperature, before removing the thermal ghost, the
generating functional (\ref{ZFT2}) is symmetric under $\Zmath_2$
reflection which reverses the direction of time. This is precisely
what we need as a starting point for QFT on the covering space of
$\orbid$. We will also start with a path integral involving the
Schwinger-Keldysh contour as the complex time path. Then, as
before, we break up the field $\phi$ as a two-component vector
$\phi =(\phi_+,\phi_-)$ where $\phi_+$ and $\phi_-$ involve times
at the forward and backward running segments of the Schwinger-Keldysh contour.
The path integral can then be rewritten as
\begin{equation}
  Z = \int \Dcal \phi_+ \Dcal \phi_-~\exp \left\{
  i \int^{\infty}_{-\infty}dt \int d^d\vex~[\Lcal (\phi_+)-\Lcal (\phi_-)]
  \right\} \ ,
\label{Zorbi}
\end{equation}
where $\Lcal$ is the (for example) scalar field Lagrangian
\begin{equation}
 \Lcal (\phi) = \half (\pat \phi )^2 -\half m^2 \phi^2
 -V_{int}(\phi) \ .
\end{equation}
We have made physical input here by the choice of
propagator for $\{\phi_+,\phi_-\}$. On the covering space, our picture
is that the field $\phi_+$
propagates forward and its copy field $\phi_-$ propagates backward in time,
decoupled from each other for $t\neq 0$. Hence the propagator is diagonal
in $\phi_+,\phi_-$. We then choose the $t=0$ hypersurface as the time
slice where we define initial conditions\footnote{Hence we are choosing
the fundamental domain to be that of Figures \ref{fig:pic1.eps} b) and
\ref{fig:figg4.ps}. While this is the most convenient choice
for our QFT construction, other choices would also be possible.}.
More precisely, we could
consider the time evolution of $\phi_+$ from $t<0$ up to a specified
profile at $t=0$ and then forward to $t>0$, and the reverse for
$\phi_-$. The orbifold identification
then calls us to identify the fields and time evolutions (elaborated
further below). However, note first a subtlety in defining the initial condition.
At $t=0$ the orbifold identification is $(0,x)\sim (0,-x)$, hence
the profiles of $\phi_+$ and $\phi_-$ must become symmetric at $t=0$.
The most natural initial  condition is to set the profiles to be equal
at $t=0$. Thus
our initial condition is
\begin{equation}
x>0:\ \ \ \phi_+(0,x)=\phi_+(0,-x)=\phi_-(0,x)=\phi_-(0,-x)=\phi_0 (x)
\end{equation}
where $\phi_0(x)$ is the specified initial profile on $x>0$. This can be satisfied
as follows. Decompose the fields $\phi_\pm$ into symmetric and
antisymmetric parts under $x\mapsto -x$:
\bea
&& \phi_\pm(t,x)=\phi_{\pm,S}(t,x) + \phi_{\pm,A}(t,x) \ , \nonumber \\
&& \phi_{\pm,S}(t,x)=\half ( \phi_\pm(t,x)+\phi_\pm(t,-x)) \nonumber \\
&& \phi_{\pm,A}(t,x)=\half ( \phi_\pm(t,x)-\phi_\pm(t,-x)) \ .
\eea
The initial condition can be satisfied if the antisymmetric parts
$\phi_{\pm,A}$ decay to strictly zero sufficiently
rapidly as $t\rightarrow 0$ (and the symmetric parts become equal).
There is a subtlety here in what exactly should be
meant by ``sufficiently rapid,'' and we will
comment on it further below.

While the above serves as a starting point for our construction of the
theory on the covering space, we must also take into
account the identification which is part of the orbifold
construction. We already discussed this in the context of Fock space
states, and can now do it more explicitly. Let us leave
the path integral formalism and return
back to the canonical quantization prescription. In the remainder
of this section, we focus only on the free field part of the
Lagrangian. This is sufficient for the construction of the
invariant Fock space, and for the improved invariant version of the
back-reaction calculation which will replace that of Section \ref{sec:tradcalc}.
We will present a
tentative discussion of interacting theory and the S-matrix in
Section \ref{sec:smat}.

First, we quantize the field operators $\phi_\pm$.
While $\phi_+$ has the standard free field mode expansion
\begin{equation}
  \phi_+ (t,\vex ) = \int \frac{d^d\vek}{(2\pi )^d}~\frac{1}{\sqrt{2\omega_k}}
  \left\{ a_{\vek}~e^{-i\omega_{\vek} t + i\vek \cdot \vex }
  +  a^{\dagger}_{\vek}~e^{+i\omega_{\vek} t - i\vek \cdot \vex }
  \right\} \ ,
\end{equation}
for the operator $\phi_-$ we write the mode expansion as
\begin{equation}
 \phi_- (t,\vex ) = \int \frac{d^d\vek}{(2\pi )^d}~\frac{1}{\sqrt{2\omega_k}}
  \left\{ \atilde_{\vek}~e^{+i\omega_{\vek} t - i\vek \cdot \vex }
  +  \atilde^{\dagger}_{\vek}~e^{-i\omega_{\vek} t + i\vek \cdot \vex }
  \right\} \ .
\end{equation}
The initial condition at $t=0$ and the required rapid decay of the
antisymmetric parts of $\phi_\pm$ create subtleties, but the above
mode expansions are valid sufficiently far from the $t=0$ slice.
Since the field $\phi_-$ is decoupled from $\phi_+$, we
introduced a new set of annihilation and creation operators
$\atilde_{\vek},\atilde^\dagger_{\vek}$ which commute
with $a_{\vek},a^\dagger_{\vek}$:
\begin{equation}
   [a_{\vek},\atilde_{\vek'}]=[a_{\vek},\atilde^\dagger_{\vek'}]
   = [a^{\dagger}_{\vek},\atilde_{\vek'}]
   =[a^{\dagger}_{\vek},\atilde^\dagger_{\vek'}]=0 \ .
\end{equation}
However, the new operators $\atilde$ satisfy the standard
commutation relation
\begin{equation}
  [\atilde_{\vek},\atilde^{\dagger}_{\vek'}] =
  (2\pi)^d\delta^{(d)}(\vek -\vek') \ .
\end{equation}
Thus, whereas $a_{\vek}$ destroys a particle with wavefunction
$e^{-i\omega_{\vek} t+i\vek
\cdot \vex}$ which is positive energy and momentum with respect to the Killing
vectors $E=i\pat_t$ and $P=-i\nabla$, the new annihilation operator
$\atilde_{\vek}$ destroys a particle with wavefunction $e^{i\omega_{\vek}t
-i\vek \cdot \vex}$ which is negative energy and opposite momentum
with respect to $E$ and $P$. Let us then define a new vacuum $|\tilde{0}\ket$
and 1-particle states:
\bea
 |\tilde{0}\ket\ &:& \ \atilde_{\vek}|\tilde{0}\ket =0 \nonumber
 \\
 |-\omega_{\vek},-\vek\ket \ &:& \
|-\omega_{\vek},-\vek\ket = \atilde^{\dagger}_{\vek}|\tilde{0}\ket
\ .
\eea
We use the notation $|-\omega_{\vek},-\vek\ket$ to
emphasize that these particles carry negative energy and opposite
momentum. The above states are the images of the usual vacuum and
1-particle states of $\Hcal^+$ under the $\Zmath_2$ isomorphism
$\Hcal^+\rightarrow \Hcal^-$, that we discussed earlier.

Next we need to take into account the identification and define
$\Zmath_2$ invariant states on the fundamental domain; these are
states in the invariant Fock space $\Hcal_{inv}=(\Hcal^+\oplus
\Hcal^-)/\Zmath_2$. {\em E.g.} for 1-particle states we define
\begin{equation}
     |\omega_{\vek},\vek\ket_{inv}   = \frac{1}{\sqrt{2}}
      (|+\omega_{\vek},+\vek\ket \otimes |\tilde{0}\ket
      + |0\ket\otimes  |-\omegak ,-\vek\ket )
      = \frac{1}{\sqrt{2}}
     \left( \begin{array}{l} |+\omega_{\vek},+\vek\ket \\
     |-\omegak ,-\vek\ket \end{array} \right) \ .
\end{equation} Invariant multiparticle states are constructed in an analogous
fashion. Next we define a number operator on the fundamental
domain,
\begin{equation}
   N^{inv}_{\vek} = a^\dagger_{\vek}a_{\vek} \oplus \atilde^\dagger_{\vek}
   \atilde_{\vek}
   \mbox{} = \left( \begin{array}{cc} a^\dagger_{\vek}a_{\vek} &
   \mbox{} \\ \mbox{} & \atilde^\dagger_{\vek}
   \atilde_{\vek} \end{array} \right) \ .
\end{equation}
Clearly,
\begin{equation}
  N^{inv}_{\vek}~|\omegak ,\vek\ket_{inv} = 1\cdot |\omegak
  ,\vek\ket_{inv} \ .
\end{equation}
We propose that the natural energy operator on the orbifold is
\begin{equation}
 H_{inv} = \sum_{\vek}N_{\vek}\omega_{\vek}
 \mbox{} = \left( \begin{array}{cc}
 \sum_{\vek} \omega_{\vek} a^\dagger_{\vek}a_{\vek} & \mbox{} \\
 \mbox{} & \sum_{\vek} \omega_{\vek}\atilde^\dagger_{\vek}
 \atilde_{\vek}  \end{array} \right) \ .
\label{enop}
 \end{equation}
Let us compare this with the stress tensor. The action (\ref{Zorbi})
\begin{equation}
 S = \int^{\infty}_{-\infty}dt\int d^{d}x~[\Lcal (\phi_+)
 -\Lcal (\phi_-)]
 \end{equation}
yields a stress tensor (for brevity, we consider a massless scalar field)
 \bea
  && T_{\mu\nu} = \frac{\delta S}{\delta \eta^{\mu\nu}}
  = \pat_\mu \phi_+ \pat_\nu \phi_+
  -\pat_\mu \phi_- \pat_\nu \phi_- \nonumber \\
  && = T^+_{\mu\nu}-T^-_{\mu\nu} \ .
 \eea
From $T_{00}$ we derive the associated Hamiltonian
\begin{equation}
  H = \half \sum_{\vek}
   \omegak [a^\dagger_{\vek}a_{\vek}+a_{\vek}a^{\dagger}_{\vek}]
   -\half \sum_{\vek}
   \omegak [\atilde^\dagger_{\vek}\atilde_{\vek}
   +\atilde_{\vek}\atilde^{\dagger}_{\vek}] \ ,
\end{equation}
before normal ordering. For
$\atilde^\dagger,\atilde$, the normal ordering prescription is
\begin{equation}
    :\atilde_{\vek}\atilde^\dagger_{\vek}: = \atilde^\dagger_{\vek}
    \atilde_{\vek} \ ,
\end{equation}
since we imposed the usual canonical commutation relations.
Hence the normal ordered Hamiltonian is
\begin{equation}
 H = \sum_{\vek}  \omegak a^\dagger_{\vek}a_{\vek}
   +\half \sum_{\vek}\omegak
   - \sum_{\vek} \omegak \atilde^\dagger_{\vek}\atilde_{\vek}
   -\half \sum_{\vek}\omegak \equiv H^+ - H^- \ .
\end{equation}
However, the reason why this is not a good Hamiltonian on the
orbifold is that it generates time translations in $t$. It is
associated with a preferred time direction on the covering space,
as it is the energy operator associated with the Killing
vector $-i\pat_t$. The individual pieces $H^\pm$ generate time
translations in $\pm t$ directions. The orbifold identifies the
two, hence on the covering space neither direction is preferred. So
the theory on the covering space must start with a symmetric
combination of the two Hamiltonians $H^\pm$. Indeed, the energy
operator (\ref{enop}) which we defined above is
\begin{equation}
H_{inv} = \left( \begin{array}{cc} H^+ & \mbox{} \\
\mbox{} & H^- \end{array} \right)
\label{enop2}
\end{equation}
with zero point energies included. Note that the $\Zmath_2$ invariance
also extends to the zero energy contributions. We can then evaluate
the vacuum energy on the orbifold,
\bea
  {}_{inv}\bra 0 | H_{inv} |0\ket_{inv} &=& \half \bra 0 | H^+ |0\ket
  + \half \bra \tilde{0} | H^- | \tilde{0} \ket
  \nonumber \\
  \mbox{} &=& \half \bra 0 | \half \sum_{\vek}\omegak
  |0\ket + \half \bra \tilde{0} | \half \sum_{\vek}\omegak
  | \tilde{0} \ket = \half \sum_{\vek}\omegak \ .
\eea
This is the usual vacuum divergence. The above calculation should
be contrasted with the naive calculation
\begin{equation}
  \bra 0 | H | 0 \ket = \bra 0 | (H ^+-H^-)| 0 \ket
  =\half \sum_{\vek} \omegak
   -\half \sum_{\vek} \omegak =0 \ ,
\end{equation}
which is not $\Zmath_2$ symmetric. Note also that the invariant
Hamiltonian $H_{inv}$ is bounded from below, since the notion
of ``boundedness" depends on the direction of time and both
directions are identified.

The invariant Hamiltonian operator
(\ref{enop2}) must derive from an invariant stress tensor. Thinking
of the latter as an operator, it will also reduce to components
which act on the subspaces $\Hcal^\pm$. In our notation, the
invariant stress tensor should be written
\begin{equation}
  T^{inv}_{\mu\nu}(t,\vex) = \left( \begin{array}{cc}
  T^+_{\mu\nu} (t,\vex) & \mbox{} \\ \mbox{} & T^-_{\mu\nu} (-t,-\vex)
  \end{array} \right) % \nonumber \\
\mbox{} = \left( \begin{array}{cc}
  \pat_\mu \phi_+ \pat_\nu \phi_+ & \mbox{} \\ \mbox{} &
  \pat_\mu \phi_- \pat_\nu \phi_-
  \end{array} \right) \ .
\end{equation}
We can now present an improved (completely
$\Zmath_2$ invariant) version of the calculation of
the vacuum expectation value of the stress tensor.
Recall that the initial condition
creates subtleties near $t=0$, so we first assume $|t|>0$ so that
we can trust the mode expansions. Then, simply
\bea
 {}_{inv}\bra 0| T^{inv}_{\mu\nu}(x)|0\ket_{inv}
 &=& \half \left( \bra 0| , \bra \tilde{0}| \right)
\left( \begin{array}{cc}
  T^+_{\mu\nu} (x) & \mbox{} \\ \mbox{} & T^-_{\mu\nu} (-x)
  \end{array} \right)
  \left( \begin{array}{c} | 0\ket \\
     |\tilde{0}\ket \end{array} \right) \nonumber \\
     \mbox{} &=& \lim_{x'\rightarrow x}
     \half \left\{ \bra 0 |\pat_\mu \phi_+ (x)
     \pat_\nu \phi_+ (x') |0\ket +
\bra \tilde{0} |\pat_\mu \phi_- (-x)
     \pat_\nu \phi_- (-x') |\tilde{0}\ket \right\} \nonumber \\
\mbox{} &=& \lim_{x'\rightarrow x}  \half\left\{
\frac{1}{(x-x')^2} +\frac{1}{(x-x')^2}\right\}
= \lim_{x'\rightarrow x} \frac{1}{(x-x')^2} \ .
\eea
This is again just the usual vacuum divergence. The renormalized
expectation value of $T_{inv}$ would then be equal to zero.
There are two  main
differences with the previous calculation of Section 2:
i) Everything is $\Zmath_2$ invariant, including the vacuum state.
ii) Essentially, $\phi(-u)$ is now replaced by $\phi_-$. But
$\phi_\pm$ are decoupled, so there are no $"\bra 0 |\phi_+\phi_-|0\ket"$
cross contractions, as were depicted in Fig. \ref{fig:figgg5.ps}.
Near the initial slice $t=0$ the situation is more subtle. As noted
previously, the antisymmetric
part of the fields must die off sufficiently rapidly. Such
behavior will alter the mode expansion of the fields. If
we insist on trusting the mode expansion everywhere at $t\neq0$, then
we must switch off the antisymmetric parts abruptly with step functions:
\begin{equation}
  \phi_{+,A} (t,\vex ) = (1-\theta(t))~f_+(\vex ) \ ; \
  \phi_{-,A} (-t,-\vex ) = (1-\theta(-t))~f_-(-\vex)  \ ,
\end{equation}
separating out the time dependence.
But then the $tt$ component of the invariant stress tensor will have
a $\delta^2 (t)$ singularity and the $tx^i$ components a $\delta (t)$
singularity at the initial slice $t=0$. However, if we interpret
the $\orbid$ orbifold as a toy model of cosmology, then $t=0$ slice
plays the role of the initial singularity.
Having a divergent stress tensor
at the $t=0$ slice is then natural in such a cosmological
interpretation --- it could represent the necessity for appropriate
boundary conditions.

Moreover, we are really interested in
strings on $\orbid$, and one should recall that there is a twisted sector
which is localized
at the singularity. The initial divergence of the stress
tensor is related to the intricate story of whether the orbifold singularity
can be blown up and whether the singular geometry really is the actual
geometry on which to consider the QFT.

The main point that we would like to stress here is that the
stress tensor does not diverge in the boundary of the CTC region. That
kind of a singularity would have been a signal
of a more serious instability. The
reason why it does not happen, without conflict of the chronology protection
conjecture, is that in the reformulation of QFT it is apparent that
nothing actually propagates along a CTC. Instead of a single quantum
propagating around and around in a CTC, there is a quantum and its copy
which propagate in opposite directions.

\paragraph{Summary.} We have presented a $\Zmath_2$ formulation of QFT,
needed as a starting point to define QFT on the $\orbid$ orbifold.
In particular, we presented a new, invariant derivation of the
vacuum expectation value
$\bra T_{\mu\nu}\ket$. It turns out to have the same form
as if there was no orbifold, except at the initial slice $t=0$ which
can be interpreted as the ``big bang'', thinking of the orbifold
as a toy cosmological model.  This initial divergence is the only
pathology from the time-nonorientability of the orbifold. Everywhere
away from the $t=0$ slice the local physics is well defined.

In the following sections, we will calculate the backreaction
in string theory, using standard orbifold techniques. The result turns
out to be the same: the vev has the same form as if the target
space were Minkowski space instead of the orbifold. If one
were to have performed the string calculation first (as we in fact did),
the result would seem puzzling. However, in the light of the
invariant formulation of QFT on the orbifold, it appears natural.
Further, at low energies string theory should reduce to QFT: since the
former must respect the symmetries of the orbifold, so must the
latter. The low energy limit of string theory on $\orbid$ must
be a symmetric QFT; the latter is precisely what we have formulated
in this section.

We will now give some details of the complementary string theory calculation,
and later return to  a discussion of other aspects of the theory.

\section{The Stress Tensor and the Graviton Tadpole}
\label{sec:stringcalc}

Our next goal is to calculate the backreaction on the orbifold
at one-loop level in string theory. In practice, this is done by
calculating the one-loop graviton tadpole.

If we write the metric tensor as
$g_{\mu\nu}(x)= \eta_{\mu\nu}+2\kappa h_{\mu\nu}(x)$, the
vev of the stress tensor may be written \cite{Birrell:1982ix}
\begin{equation}
\bra T_{\mu\nu}\ket = -i\frac{\delta}{\delta g^{\mu\nu}}\ln
Z^{2nd}_{EFT}|_{h^{\mu\nu}=0} = -\frac{i}{2\kappa}\frac{\delta
Z_{1st}}{\delta h^{\mu\nu}}|_{h^{\mu\nu}=0} \ . \label{T1}
\end{equation}
In the above, we used the relation between the vacuum amplitudes
in the second quantized and first quantized formalism, $Z_{2nd} =
e^{Z_{1st}}$, to replace the effective field theory action $\ln
Z^{2nd}_{EFT}$ by the point particle partition function $Z_{1st}$.


Now we replace point particles by strings. At one-loop level
\cite{bigbook1}%p. 217, p.104
\begin{equation}
  Z^{ST}_{1-loop}[g] = \int \frac{d\tau d\bar{\tau}}{4\tau_2}~Z(\tau)
  = \int \frac{d\tau d\bar{\tau}}{4\tau_2} \int_{T^2}
  \Dcal X~e^{i\frac{T}{2}\int d^2w~g_{\mu\nu}(X)\pat X^\nu \patb
  X^\nu} \ .
\end{equation}
This is then inserted\footnote{This is somewhat reminiscent
of a recent calculation in \cite{Larsen:2002wc}.} in (\ref{T1}).
Suppressing the integral over $\tau$, we
have
\begin{equation}
  Z^{ST}_{1-loop} = \int
  \Dcal X~e^{i\frac{T}{2}\int d^2w~\eta_{\mu\nu}\pat X^\mu \patb X^\nu}
  \left\{ 1+i\frac{g_{str}}{\alpha'}\int d^2w~h_{\mu\nu}(X)\pat X^\mu \patb
  X^\nu +\cdots \right\} \ .
\end{equation}


Now Fourier expand the perturbation,
\begin{equation}
 h_{\mu\nu}(X) = \int \frac{d^{D+1}k}{(2\pi)^{D+1}}~e_{\mu\nu}(k)e^{ik\cdot X}
\end{equation}
and introduce
\begin{equation}
  V_{\mu\nu}(k) = \pat X^\mu \patb X^\nu e^{ik\cdot X} \ ,
\end{equation}
then
\begin{equation}
  Z^{ST}_{1-loop} [g] =
   Z^{ST}_{1-loop} [\eta ] +
  i\frac{g_{str}}{\alpha'} \int \frac{d^{D+1}k}{(2\pi)^{D+1}}\int d^2w~e_{\mu\nu}(k)
  \bra V^{\mu\nu}(k;w) \ket +\cdots \ .
\label{Z}
\end{equation}
We then get
\begin{equation}
\bra T^{\mu\nu}(x) \ket = \frac{1}{4\pi \alpha'}
\int \frac{d^{D+1}k}{(2\pi)^{D+1}}\int d^2w
\bra V^{\mu\nu}(k;w) \ket e^{-ik\cdot x}
\end{equation}
the relation between the Fourier transformed tadpole and the
stress tensor.

Note that in Minkowski space, one obtains
\begin{equation}
 \bra V^{\mu\nu}(k)\ket = -\left( \frac{g_{str}}{4\pi\tau_2} \right)
 \frac{\delta^{(D+1)}(\sqrt{\alpha'}k)}{V_{D+1}}
 \eta^{\mu\nu} Z_{1-loop} \ ,
\end{equation}
so that
\begin{equation}
  \bra T_{\mu\nu}\ket \sim \frac{1}{{\alpha'}^{13}V_{26}} Z_{1-loop} \times
  \eta_{\mu\nu}
\end{equation}
which is of the right form for the stress tensor of a cosmological
constant $\Lambda \sim Z_{1-loop}$.

On $\Zmath_2$ orbifolds, the story is essentially the same.
What is different in the string graviton tadpole calculation is
that the relevant vertex operator must be $\Zmath_2$ invariant: it
is the sum of vertex operators carrying $k$ and $-k$ in the
directions of the orbifold. The Fourier
transform of the tadpole will then be the sum
\begin{equation}
   \bra T_{\mu\nu}(X)\ket + \bra T_{\mu\nu}(-X)\ket ,
\label{Tsum}
\end{equation}
where $X$ are the coordinates along the orbifold directions.
This is obtained from the effective action by including the functional
differentiation $\delta /\delta h_{\mu\nu}(-X)$. By comparing with
the discussion in Section \ref{sec:tradcalc}, it can be seen that (\ref{Tsum})
corresponds to what we called
\begin{equation}
  \bra T^+_{\mu\nu}\ket + \bra T^-_{\mu\nu} \ket \ .
\end{equation}

\subsection{One-loop Graviton Tadpole}

Now we proceed to give some of the details of the calculation of the
one-loop graviton tadpole in string theory described above.
Our calculations are based on the functional method.
We begin with a brief review of the latter, following \cite{bigbook1}.
As it turns out, an immediate difference with tadpole calculations
on Euclidean orbifolds is in kinematics and in appropriate choice of polarization
of vertex operators.
We have also performed
the same calculations in the oscillator formalism. It also turns out
that there are some interesting subtleties and differences with the
standard discussion; detailed notes may be found in
the appendix.

We should note that in the string computations, one usually
performs a Wick rotation in both spacetime and worldsheet,
necessary for formal convergence.
If the target space is time-dependent,
the standard techniques of analytic continuation may not be applicable.\footnote{See {\em e.g.} \cite{Mathur:1993tp} for a proposal to modify the
standard approach.}
In the context of the $\orbid$ orbifold, the
issue was already noted in \cite{vijayetal}. In the present paper,
we simply adopt the same strategy as in \cite{vijayetal}, namely
we formally continue to Euclidean signature in the calculations to obtain
an expression for the tadpole, which we then formally continue
back to Lorentzian signature. The result turns out to be well
defined and compatible with the low-energy limit, the invariant field theory
calculation of Section \ref{sec:qftnew}. In that section, propagation on the
orbifold was essentially shown to be an identification of forward and
backward propagation on the covering space $\Rmath^{1,d}$. This
may also explain why the formal analytic continuation prescription
continues to work in the calculations of this section.

\subsection{The generating functional on $\Rmath^{1,d-1}$}
%\comment{This subsection follows Polchinski Section 6.2.
%Unfortunately, Joe's stuff doesn't make dimensional sense. Using the
%convention that worldsheet coordinates do not have dimension and $X$ has
%dimension one, it is simplest to take the $x_I^\mu$ to have no dimension
%by explicitly putting a power of $4\pi^2\alpha'$ in front of (\ref{eq:modes}).
%The currents also have dimension $1/L$, which I absorb by putting in a factor
%of string tension. The result is then as follows.}

Following \cite{bigbook1}, the generating functional is
\begin{equation}
  Z[J] = \bra \exp \{ i\int d^2w J_\mu (w,\wbar ) X^{\mu}(w,\wbar )
  \} \ket \ .
\label{genfun}
\end{equation}
In order to perform the functional integrals, we introduce a
complete set eigenmodes $X_I$ of the Laplacian $\nabla^2$ on the
toroidal worldsheet,
\bea
  && \nabla_w^2 X_I (w,\wbar ) = -\omega^2_I X_I (w,\wbar )\ \ \ \  , \nonumber \\
 &&  \int d^2w~X_I(w,\wbar)X_J(w,\wbar )= \delta_{IJ} \
\eea
and expand the string embedding coordinates in the eigenmodes,
\begin{equation}\label{eq:modes}
  X^\mu (w,\wbar ) = \sqrt{4\pi^2\alpha'}\sum_I x^\mu_I X_I(w,\wbar )\ .
\end{equation}
We also denote
\begin{equation}
      J_{\mu, I}=  \sqrt{4\pi^2\alpha'}\int d^2 w J_\mu (w,\wbar ) X_I(w,\wbar ) \ .
\end{equation}
We then integrate out the expansion coefficients $x^\mu_I$ by completing
the squares in the generating
functional and performing the resulting Gaussian integrals. In particular,
the integrals will include zero mode contributions from
$x^\mu_0$. The result in $d$ target space dimensions is
%\begin{equation}
%  Z[J]= N[J_0]\left[ \det \left( \frac{-\nabla^2}{4\pi^2
%  \alpha'}\right) \right]^{-d/2} \exp \left\{ -\half \int d^2 w
%  \int d^2 w' J(w)\cdot J(w')~G'(w,w') \right\} \ ,
%\end{equation}
\begin{equation}
  Z[J]= N[J_0]\left[ {\det}'  \left( -\nabla_w^2\right) \right]^{-d/2}
  \exp \left\{ -\half \int d^2 w
  \int d^2 w' J(w)\cdot G'(w,w')\cdot J(w') \right\} \ ,
\end{equation}
where $N[J_0]$ is the zero mode contribution
\begin{equation}
  N[J_0]= i(2\pi)^d \delta^{(d)}(J_0) \ ,
\end{equation}
(with $i$ coming from the Wick rotation $x^0_I\equiv ix^d_I$), the
determinant factor is
\begin{equation}
  {\det}'   \left( -\nabla_w^2\right) \equiv \prod_{I\neq 0} \omega^2_I\ ,
\label{det}
\end{equation}
and $G'(w,w')$ is the Green's function
%\begin{equation}
%   G'(w,w') = \sum_{i\neq 0} \frac{2\pi
%   \alpha'}{\omega^2_I}X_I(w)X_I(w') \ .
%\end{equation}
\begin{equation}
   G'(w,w') = \sum_{I\neq 0} \frac{2\pi\alpha'}{\omega^2_I}X_I(w)X_I(w') \ .
\end{equation}
The latter satisfies the differential equation
%\begin{equation}
%   -\frac{\nabla^2_w}{2\pi \alpha'} G'(w,w') = g^{-1/2}\delta
%   (w-w')-X^2_0 \ ,
%\label{gfeqn}
%\end{equation}
\begin{equation}
   -\frac{1}{2\pi\alpha'}\nabla^2_w G'(w,w') = g^{-1/2}\delta^{(2)}(w-w')-X^2_0 \ ,
\label{gfeqn}
\end{equation}
where $X_0$ is the zero mode of the Laplacian on the torus. The
functional determinant (\ref{det}) gives the torus partition function,
\begin{equation}
   Z_{T^2} [0] = V_d\left[ \alpha' X_0^2{\det}'
   \left(-\nabla_w^2\right) \right]^{-d/2}
\end{equation}

\subsection{The generating functional on Orbifolds}

Next we generalize this to the case of the orbifold. For
comparison, we will consider two related types of orbifolds:
\begin{description}
\item [A)] The Euclidean orbifold $\mathbb{R}^{1,d}\times \mathbb{R}^{25-d}/
\mathbb{Z}_2$
\item [B)] The Lorentzian orbifold $\mathbb{R}^{1,d}/\Zmath_2 \times
\mathbb{R}^{25-d}$.
\end{description}
We split the coordinates $X$ and the components of the
source $J$ into those parallel ($\parallel$) and transverse ($\perp$)
to the orbifold directions.
The generating functional takes the form
\begin{equation}
  Z[J] = \sum^1_{g=0}\sum^1_{h=0}~\langle \exp \{ i\int J_{\parallel}
  \cdot X_{\parallel}
     + i\int J_\perp \cdot X_\perp \} \rangle_{gh}
\label{genfuno}
\end{equation}
including the sum over the untwisted ($g=1$) and twisted ($g=0$)
sectors, with ($h=0$) and without ($h=1$) the $\Zmath_2$
reflection, for string oscillations in the orbifolded directions.
We then again expand $X^\mu$ in the eigenmodes of $\nabla^2$, but
now the eigenvalues and -modes will be different in the orbifolded
directions for each sector, due to the different (anti)periodic boundary
conditions. After integrating over the eigenmode coefficients, the
functional takes the form
\bea\label{genfunoans}
  Z[J] &=& \frac{N_\perp [J_0]}{N_\perp [0]} Z_\perp[0]~\exp\{-\half
  \int d^2w\int d^2w'
  J_\perp(w)\cdot J_\perp(w')~G'(w,w')\}  \\
  \mbox{} &\times&
  \sum_{gh}~\frac{N_{||,gh}[J_0]}{N_{||,gh}[0]}Z_{\parallel,(g,h)}[0]~\exp
  \{-\half \int d^2w\int d^2w'
  J_{\parallel}(w)\cdot J_{\parallel}(w')~G'_{(g,h)}(w,w')\} .\nonumber
\eea
In the above, $N_\perp[J_0]$, $N_{\parallel,(g,h)}[J_0]$ are the zero mode
contributions.
In the orbifolded directions, there are zero modes only in the untwisted sector
without the $\Zmath_2$ reflection, and none in the other sectors
because $X$ satisfies an antiperiodic boundary condition in at
least one of the toroidal worldsheet directions. Thus, for $J=k\delta^{(2)}
(w-w')$,
\begin{equation}
  \frac{N_{\parallel,(1,1)}[J_0]}{N_{\parallel,(1,1)}[0]}
  =\frac{1}{V_{d+1}}\delta^{(d+1)}(k) \ ;
  \ N_{\parallel,(g,h)}[k] = 1\ {\rm for}\
  (g,h)\neq (1,1) \ .
\end{equation}
The factors $Z_\perp[0]$, $Z_{\parallel,(g,h)}$ are the partition
function contributions
from the directions transverse to and parallel
with the orbifold, including the four untwisted and twisted
$(g,h)$-sectors. Explicitly \cite{vijayetal},
\bea
  Z_{\parallel,(1,1)} &=& \frac{V_{25-d}}{2}~\left|
  \frac{1}{\sqrt{\tau_2}~\eta^2 (\tau)}
  \right|^{25-d} \nonumber \\
  Z_{\parallel,(g,h)} &=& \left| \frac{\eta (\tau )}{\theta_{gh} (\tau )}
  \right|^{25-d} ,\ \ \ \ \ (g,h)\neq (1,1)
\eea
There are four different Green's functions, corresponding to the
different periodicities on the toroidal worldsheet. The doubly
periodic one is \cite{bigbook1}
\begin{equation}
 G'_{(1,1)}(w,w')\equiv G'(w,w')=
  -\frac{\alpha'}{2}\ln \left| \theta_{11}\left(\left.\frac{w-w'}{2\pi}
  \right|\tau \right)
  \right|^2 + \pi\alpha' X_0^2 [{\rm Im}(w-w')]^2\ ,
\end{equation}
and the other ones with at least one antiperiodic direction are
\bea
  G'_{(1,0)}(w,w') &=& -\frac{\alpha'}{2} \ln \left|
  \frac{\theta_{11} (\frac{w-w'}{4\pi }|\tau )\theta_{10}(\frac{w-w'}{4\pi}|\tau )}
  {\theta_{00} (\frac{w-w'}{4\pi }|\tau )\theta_{01}(\frac{w-w'}{4\pi}|\tau )}
  \right|^2
  \nonumber \\
  G'_{(0,1)}(w,w') &=& -\frac{\alpha'}{2} \ln \left|
  \frac{\theta_{11} (\frac{w-w'}{4\pi }|\tau )\theta_{01}(\frac{w-w'}{4\pi}|\tau )}
  {\theta_{10} (\frac{w-w'}{4\pi }|\tau )\theta_{00}(\frac{w-w'}{4\pi}|\tau )}
  \right|^2 \nonumber \\
  G'_{(0,0)}(w,w') &=& -\frac{\alpha'}{2} \ln \left|
  \frac{\theta_{11} (\frac{w-w'}{4\pi }|\tau )\theta_{00}(\frac{w-w'}{4\pi}|\tau )}
  {\theta_{01}(\frac{w-w'}{4\pi}|\tau )\theta_{10} (\frac{w-w'}{4\pi }|\tau )}
  \right|^2 \ .
\eea
In $n$-point amplitudes, one also encounters
self-contractions which require renormalization. A simple
prescription is to subtract the divergent part $-\frac{\alpha'}{2}
\ln|w-w'|^2$ from the Green's functions and define their renormalized
versions. The renormalized version of $G'_{11}$ is \cite{bigbook1}
\begin{equation}
  G'_{(1,1),ren}(w,w)= -\frac{\alpha'}{2}\ln \left| \frac{\theta'_1(0|\tau )}
  {2\pi} \right|^2 \ .
\label{g11ren}
\end{equation}
After some manipulations, the renormalized versions of the other
Green functions also turn out to simplify considerably to the
following simple forms:
\begin{equation}
  G'_{(g,h),ren} =-\frac{\alpha'}{2}\ln |\theta_{gh}(0|\tau)|^4
\end{equation}
for $(g,h)\neq (1,1)$.

\subsection{One-loop graviton tadpole on the orbifold}

Consider then the one-loop graviton tadpole
on the orbifold. The vertex operator for a state which is not
projected out by the $\Zmath_2$ reflection must be symmetric under $X\rightarrow
-X$, hence the relevant massless tadpole on the orbifold is
\bea
&& \bra V_{\mu\nu}(k_{\parallel},k_\perp)
+ V_{\mu\nu}(-k_{\parallel},k_\perp) \ket  \nonumber \\
&& \ \ \ \ \ \ \ \ \ \ \mbox{}=   \frac{2g_{str}}{\alpha'}
\bra \pat X^\mu \patb X^\nu e^{ik_{\parallel}\cdot \Xpar +ik_\perp \cdot \Xperp }
+ \pat X^\mu \patb X^\nu e^{-ik_{\parallel}\cdot \Xpar +ik_\perp \cdot \Xperp }
      \ket
\eea
The momentum must satisfy the on-shell condition
$k^2=-m^2=0$. Now there are some immediate choices to be done where
the Euclidean and Lorentzian
orbifolds {\bf A} and {\bf B} differ. In string theory one often
considers Euclidean orbifolds as a way of compactifying extra
dimensions. Therefore one is usually interested in states which only
propagate and carry polarization in the non-orbifolded noncompact
directions, and
the momentum and the polarization are chosen to be entirely transverse to the
orbifold, with $k^2=k^2_\perp=-m^2$. However, in the Lorentzian
orbifold one must also include components in parallel directions
in order to satisfy the on-shell condition. Further, let us make
the usual simplifying choice for the polarization
and assume it to be polarized only in the first
$d+1$ spacetime directions. That is, the $\pat X^\mu \patb X^\nu$
part of the vertex operator will contribute only when $\mu,\nu =0,\ldots
d$. (Note that the polarization part is then invariant under $X\rightarrow
-X$). Then also the polarization is parallel to the orbifolded
directions.

Let us illustrate this
by calculating the tadpole in both cases {\bf A} and {\bf B}, with
the kinematics
\begin{description}
\item [A)] (Euclidean case) $k=(k_\perp,k_{\parallel})=(k_\perp,0)$ with
$k_\perp^2=0$
\item [B)] (Lorentzian case) $k=(k_{\parallel},k_\perp )$ with
$k^2=k_{\parallel}^2+k^2_\perp=0$.
\end{description}

We evaluate the tadpole by first performing a point splitting and then
functional differentiation of the generating functional,
\bea
  && \bra \pat X^\mu (w,\wbar )\patb X^\nu (w,\wbar ) e^{ikX(w,\wbar)} \ket
  = \nonumber \\
 &&  \ \ \ \ \ \ (-i)^2\lim_{w_1,w_2\rightarrow w} \pat_{w_1}
 \patb_{w_2}~\frac{\delta}{\delta J_\mu (w_1)}
 \frac{\delta}{\delta J_\nu (w_2)}~\bra
 \exp \{i\int d^2 w' J_\lambda (w')X^\lambda (w')\} \ket \ ,
\eea
evaluated at $J (w')=k~\delta^{(2)} (w'-w)$.
Before the functional differentiation, for
the generating functional we substitute the integrated form
(\ref{genfunoans}). We will also substitute the on-shell condition
$k^2=0$.

In the Euclidean case, the functional differentiation and the
on-shell condition leaves
\bea
  && \bra \pat X^\mu (w) \patb X^\nu (w) e^{ikX} \ket = \nonumber
  \\
  && \ \ \frac{N_\perp[k]}{N_\perp[0]}Z_\perp[\tau]~\lim_{w_1,w_2\rightarrow w}
  [~\eta^{\mu\nu}\pat_{w_1}\patb_{w_2}
  G'(w_1,w_2)-k^\mu k^\nu~\pat_{w_1}G'(w_1,w)\patb_{w_2}G'(w_1,w)]
  \nonumber \\
  && \ \ \times  \sum_{g,h} Z_{\parallel,(g,h)}[\tau ]
\label{euc1}
\eea
whereas in the Lorentzian case, the corresponding result is
\bea
 && \bra \pat X^\mu (w) \patb X^\nu (w) e^{ikX} \ket = \nonumber
  \\
  && \ \ \frac{N_\perp[k]}{N_\perp[0]}Z_\perp[\tau]
  \times \sum_{g,h} \frac{N_{\parallel,(g,h)}[k]}{N_{\parallel,(g,h)}[0]}Z_{\parallel,(g,h)}[\tau ] \nonumber \\
 &&\ \ \ \ \ \lim_{w_1,w_2\rightarrow w} [~\eta^{\mu\nu}\pat_{w_1}\patb_{w_2}
  G'_{(g,h)}(w_1,w_2)-k^\mu k^\nu~\pat_{w_1}G'_{(g,h)}(w_1,w)\patb_{w_2}G'_{(g,h)}(w_1,w)]
  \ .
\label{lor1}
\eea
In both cases, the Green's function will need to be replaced by
their renormalized versions.
We can already see that the expressions are quite different. Let
us simplify them further. First, we can use the equation
(\ref{gfeqn}) to simplify the double derivatives of the Green's
functions. First, since $G'(w_1,w_2)=G'(w_1-w_2)$,
\begin{equation}
  \pat_{w_1}\patb_{w_2} G'(w_1,w_2)=
  -\pat_{w_1}\patb_{w_1}G'(w_1,w_2)\ .
\end{equation}
On the other hand, the equation (\ref{gfeqn}) evaluates to
\begin{equation}
 \pat_w\patb_w G'(w,w') =
 -\pi\alpha'\delta^2(w-w')+\frac{\pi\alpha'}{2}X^2_0 \ .
\end{equation}
%Hence we get
%\begin{equation}
% \pat_{w_1}\patb_{w_2} G'(w_1,w_2) = \pi\alpha'\delta^2(w_1-w_2)
% -\frac{\pi}{\alpha'}X^2_0 \ .
%\end{equation}
The first term on the right hand side
originates from the short distance divergence $G'(w_1,w_2)\sim \ln|w_1-w_2|^2$
of the Green's function, which we subtract off when we renormalize
the Green's functions. The latter then satisfy the equation
\begin{equation}
  \pat_{w_1}\patb_{w_2}G'_{ren}(w_1,w_2) = -\frac{\pi\alpha'}{2}X^2_0 \ .
\end{equation}
Similar results hold for the renormalized Green's functions $G'_{(g,h),ren}$
in the Lorentzian orbifold case. Since a zero mode $X_0$ exists
only in the doubly periodic $(g,h)=(1,1)$ sector, the double derivatives
$\pat\patb G'_{gh,ren}$ vanish in all the other three sectors.

Next, we examine the first derivatives of the renormalized Green's
functions. A calculation shows that in all cases the Green's functions
have a short distance behavior of the type
\begin{equation}
   \pat G'_{(g,h)} (w,w') \approx_{w\rightarrow w'} -\frac{\alpha'}{2}(w-w')^{-1}
   + C_{(g,h)}(\tau) (w-w') + {\cal O}((w-w')^3)
\end{equation}
where $C_{(g,h)}(\tau)$ are rational functions of
derivatives of theta functions at $(0|\tau)$. A similar formula is
found for the antiholomorphic derivative $\patb G'_{(g,h)}$.
There is only one divergent term, due to the self-contraction
of $X$ with $\pat X$. The renormalization prescription again removes the
divergent term, so the renormalized (derivatives of) Green's function
vanish in the limit $w\rightarrow w'$. Hence these terms will not
contribute to the graviton tadpole.

Substituting all the normalization and partition function factors,
the final results are
\begin{equation}
  \bra V^{\mu \nu}(k)+  V^{\mu \nu}(-k) \ket_{1-loop} =
 -\frac{g_{str}}{4\pi\tau_2} \frac{\delta^{(25-d)}(k)}{V_\perp}\times
 Z_\perp[\tau]\sum_{g,h} Z_{\parallel,(g,h)}[\tau ]~\eta^{\mu\nu} \ .
\end{equation}
%\bea
%  && \bra V^{\mu \nu}(k) +  V^{\mu \nu}(-k) \ket_{1-loop}   =
%  \nonumber \\
%  && \ \ \ \ \ \ -\pi g_{str} \delta^{(25-d)}(\sqrt{k_\perp)~\eta^{\mu\nu}
%  ~\left( \frac{1}{\sqrt{\tau_2}|\eta (\tau)|^2}
% \right)^{d-1}  \\
%  && \ \ \ \ \ \ \times  \left(
%  \frac{V_{25-d}}{2}~\left| \frac{1}{\sqrt{\tau_2}~\eta^2 (\tau)}
%  \right|^{25-d} + \left| \frac{\eta (\tau )}{\theta_2 (\tau )}
%  \right|^{25-d} + \left| \frac{\eta (\tau )}{\theta_4 (\tau )}
%  \right|^{25-d} + \left| \frac{\eta (\tau )}{\theta_3 (\tau )}
%  \right|^{25-d} \right) \ , \nonumber
%\eea
for the graviton tadpole in the Euclidean orbifold, and
%\begin{equation}
%  \bra V^{\mu \nu}(k)+  V^{\mu \nu}(-k) \ket_{1-loop}
%  = -g_{str}\frac{\delta^{(d+1)}(\sqrt{\alpha'}k)V_{25-d}}{4\pi\alpha'
%  \tau_2}~\left( \frac{1}{\sqrt{\tau_2}|\eta (\tau)|^2}
% \right)^{24}~\eta^{\mu\nu}
%\end{equation}
\begin{equation}
  \bra V^{\mu \nu}(k)+  V^{\mu \nu}(-k) \ket_{1-loop} =
 -\frac{g_{str}}{4\pi\tau_2}
 \frac{\delta^{(26)}(k)}{V_{26}}~Z_\perp(\tau)
 Z_{\parallel, (1,1)}[\tau]~\eta^{\mu\nu} \ .
\end{equation}
for the tadpole in the Lorentzian orbifold.
%\ack{This is now the correctly normalized result.
%It would look better if we had not set $k_\perp$ to
%zero --- more generally, the result has a
%factor of $ \frac{\delta^{(26)}(k)}{V_{26}}$.
%But not necessary (I put a comment to this effect in the appendix.)}
At first, the final result looks rather surprising, as
it is precisely the
same as for a graviton in the usual $\Rmath^{1,25}$ target space!
However, in the light of the analysis of the Section \ref{sec:qftnew}, this is
precisely what we expect. The low-energy limit of invariant field
theory tells us that the vacuum expectation value of
free field theory on the orbifold has the same
form as in ordinary Minkowski space. The expectation value is just
the vacuum zero-point energy of bosonic string theory. By
analogue, one would then expect the tadpole to vanish for the
superstring.

\section{Time Evolution and Local S-matrix on the Fundamental Domain}
\label{sec:smat}

In all previous discussions, we limited the analysis to what
corresponds to free field theory in the low-energy limit. What
then of the interacting theory? Given that the orbifold is
globally time-nonorientable, is there any way of defining
S-matrices at least locally, for example away from the $t=0$ axis
in the above choice of the fundamental domain?  Let us first
illustrate the problem with a simple figure.
\myfig{apu.eps}{5}{Example of a point particle and its image propagating
on the covering space.}
Figure \ref{fig:apu.eps} depicts a point particle and its image propagating in
opposite time and space directions on the covering space. As
discussed in Section \ref{sec:qftnew}, both trajectories will be identified by
the $\Zmath_2$ reflection, resulting in a single trajectory for a
point particle on the fundamental domain. The details depend on
the choice of the fundamental domain.

Consider first choosing the
right half-space as the fundamental domain, and drawing the
corresponding "pocket", as depicted in Figure \ref{fig:fig9.eps}
(see also Figs. \ref{fig:pic1.eps}(a) and \ref{fig:figgg3.ps}).
\myfig{fig9.eps}{7}{Point particle propagating on the fundamental domain.
The left part of the figure depicts the
fundamental domain, with the dashed negative time axis ($x=0$) identified
with the positive time axis. For example, the points S marked
with a black dot are identified. The identification results in the
pocket shown on the right. The time orientation breaks down on the
time axis. The part of the trajectory drawn with
solid lines depicts propagation on the front fold
of the pocket, while the dashed line depicts propagation on the rear fold. The
point S is on dotted-dashed line, where the time direction
becomes ill-defined.}
Alternatively, we can choose the upper half-space as the
fundamental domain, and identify the negative $x$-axis with the
positive $x$-axis. The result is depicted in Figure \ref{fig:fig10.eps}
(see also Figs. \ref{fig:pic1.eps}(b) and \ref{fig:figg4.ps}).
\myfig{fig10.eps}{7}{Point particle propagating on the fundamental domain.
On the left figure, the points marked with S are identified.
On the right figure, the solid lines depict propagation on the front fold
of the pocket, the dashed line depicts propagation on the rear fold. The
point marked with S is on the dotted-dashed line, where the
time direction becomes ill-defined.}

As discussed in Section \ref{sec:oldorbid}, a possible choice for the
time-arrow on the fundamental domain of Figure \ref{fig:fig9.eps} is to
let it point from
the lower right quadrant to the upper right quadrant, while
becoming ambiguous on the $x=0$ axis. On the pocket, time would
thus flow down the front fold and continue upwards on the rear
fold. Similarly, on the fundamental domain of
Figure \ref{fig:fig10.eps} one can choose the arrow of time to point
upwards on the upper half-space, with the
$t=0$ axis as the origin of time. On the pocket, time would then
flow upwards on both sides of the fold. However, then the
trajectories depicted on the figures would seem to violate causality.
On Figure \ref{fig:fig9.eps}, if we choose a constant time
slice far up on the front fold, the trajectory will cross it
twice. First it crosses the slice on its way down along the front
fold, then it continues to the other side but returns back to the
front slice and crosses the slice again. On Figure \ref{fig:fig10.eps}, the
trajectory would propagate first backwards in time towards $t=0$,
then propagate forward in time on the rear fold and again on the
front fold. Both interpretations are troublesome.

However, we can improve the situation a bit. In Section \ref{sec:qftnew},
we identified forward time evolution of a particle
with backward time evolution of its image on the covering space.
We start the forward evolution from $t=-\infty$ and the backward
evolution from $t=\infty$. The time evolution continues without
problems until we reach the dividing line between the
two half-spaces, depending on the choice of the fundamental
domain. That is, if we choose the right half-space as the
fundamental domain, we can follow the time evolution until the
particle and its image reach $x=0$. If we choose the upper
half-space as the fundamental domain, the time evolution can be
followed up to $t=0$. Similarly, just after crossing the dividing
line, we can again follow the time evolution onwards. For example,
in the latter case we can continue from $t=0+\epsilon$ the
forward time evolution to $t=\infty$ and backward evolution to
$t=-\infty$. The problem is if and how it is possible to continue
the evolution across the dividing line.

If we choose the upper half-space as the fundamental domain, it is
simple to give a more formal definition.
In the Heisenberg picture we define the invariant time
evolution operator from $t_0$ to $t_1>t_0$ on the covering space to be
\bea
   U_{inv}(t_0,t_1) &=& T\left\{\exp \left[ -i\int^{t_1}_{t_0}
   dt~H_{inv}(t)\right] \right\} \nonumber \\
   \mbox{} &=& \left( \begin{array}{cc} T\{\exp [-i\int^{t_1}_{t_0}
   dt~H^+]\} & 0 \\
0 & \tilde{T}\{\exp [-i\int^{-t_1}_{-t_0} dt~H^-]\} \end{array} \right)
\nonumber \\
\mbox{} &=&
\left( \begin{array}{cc} U^+(t_0,t_1) & 0  \\
0 & U^-(-t_0,-t_1) \end{array} \right) \ ,
\eea
where $\tilde{T}$ denotes anti-time ordering. This is unambiguously defined
if both $t_0,t_1 <0$ or both $t_0,t_1>0$. Problems arise
when $t_0<0$ and $t_1>0$.

Let us see what this means for the point particles in the figures.
In Figure \ref{fig:apu.eps}, we launch the particle and its image from $t=-\infty$
and $t=+\infty$, and then propagate them using $U(-\infty ,t)$
towards the $x$-axis, {\em i.e.} up to $t=0-\epsilon$. This gives
the lower half and the upper half of the forward and backward
trajectories of Fig. \ref{fig:apu.eps}. They are identified on the
fundamental domain,
so  this gives the ``downward" trajectory on the left part
in Figure \ref{fig:fig10.eps} all the way to the marked point S, and the downward
trajectory to the point S on the front fold of the pocket in
Figure \ref{fig:fig10.eps}. Similarly, we could propagate forward and backward
from $t=0+\epsilon$, giving the other halves of the trajectories
in Figure \ref{fig:apu.eps}. In Figure \ref{fig:fig10.eps}, the resulting
trajectory is the ``upward"
one on the left diagram, and the upward trajectory starting on the rear
fold and continuing to the front fold of the pocket.

To summarize,
on the fundamental domain (the pocket), we can either choose the time
to point downwards, corresponding to Figure \ref{fig:pic1.eps} b) with
the arrows reversed, and consider time evolution up to a ``big
crunch" at $t=0$, or choose the time to point upwards and consider
time evolution forward from a ``big bang". The latter
corresponds to Figure \ref{fig:pic1.eps} b). Either way, the evolution breaks
down at $t=0$. But that is also the point where from Section \ref{sec:qftnew}
we know the stress tensor to diverge\footnote{Similar analysis, based
on the other two choices of the fundamental domain, Figure \ref{fig:pic1.eps}
a) and c),
are also possible. Then the time evolution would break down at $x=0$
(a) or at null cone (c).}.

Consider then an interacting field theory. We could adopt the
interaction picture, and define the time evolution operator as
\bea
 &&  U_{I,inv} (t_0,t_1) = \nonumber \\
 &&
\left( \begin{array}{cc} T\{\exp [+i\int^{t_1}_{t_0}
   dtd^d\vex~{\cal L}_I (\phi_+(t,\vex))]\} & 0 \\
0 & T\{\exp [-i\int^{t_1}_{t_0} dt' d^d\vex~
{\cal L}_I(\phi_-(t',\vex))]\} \end{array} \right)
\eea
where ${\cal L}_I$ is the interaction part of the Lagrangian.
We could consider this as the local S-matrix. Hence, as long
as we stay away from the singular $t=0$ hypersurface, the S-matrix
has the same properties as that of an ordinary field theory on
Minkowski space.

\bigskip


\noindent
{\large {\bf Acknowledgment}}

\bigskip

We would like to thank Vijay Balasubramanian, Fawad Hassan, and
Asad Naqvi for the collaboration which lead us to this
investigation, and many discussions on the interpretational
issues. We also thank Per Kraus for helpful discussions and
useful encouragement. EK-V was in part supported by the Academy
of Finland, and thanks the University of California at Los Angeles
and University of Pennsylvania for hospitality at different stages
of this work. RGL was supported by US DOE under contract DE-FG02-91ER40677
and thanks the Helsinki Institute of Physics for hospitality. ES would like to
thank A. Lawrence, S. Kachru, A. Knutson and R. Bryant for helpful conversations.
EK-V would like to dedicate this work to Anne Keski-Vakkuri, who
was conceived and born during this project.

\bigskip
\begin{appendix}
\section{Appendix}
\newcommand{\RR}{{\Rmath}}
\newcommand{\ZZ}{{\Zmath}}
\newcommand{\pa}{\partial}

In this appendix, we will provide the details of complementary calculations
using oscillator methods. There are several subtleties that are not regularly
seen in the usual backgrounds.
We will use the notation where $\tilde k$ is the image of $k$ under the orbifold.
\bea
\tilde k_o=-k_o\\
\tilde k_u=+k_u
\eea
We want to evaluate
\begin{equation}
T(\tau)=\frac{1}{2}\tr_{U+T} \left[\left( 1+\hat g\right)
\int d^2z V(z,\bar z)\  q^{L_o-a}\bar q^{\tilde L_o-\tilde a}\right]
\end{equation}
It is important to note that in this formalism, obtained by sewing the cylinder
into a torus, there are zero modes in the $U$ sectors of the trace, but not
in the twisted $T$ sectors.
The massless vertex operator is of the form
\begin{equation}
V(z,\bar z)=\frac{2g_{str}}{\alpha'}:\partial X^\mu\bar\partial
X^\nu\frac{1}{2}\left( e^{ik\cdot X(z,\bar z)}
+ e^{i\tilde k\cdot X(z,\bar z)}\right):
\end{equation}

The non-zero mode portion of this expression can be evaluated using
coherent state methods.
For each oscillator $\alpha_n^\mu$ ($n>0$) we introduce a coherent-state
basis $|\rho_{n,\mu})$
and write the trace as a $\rho$-integral. If the $\partial X^\mu$ does not
contribute an oscillator, we find (for each $n>0$ and $\mu$)
\begin{equation}
\int \frac{d^2\rho}{\pi}\ e^{-|\rho|^2} e^{\alpha' k_\mu^2/4n}(\rho|
e^{\sqrt{\alpha'/2}k_\mu \alpha_{-n}^\mu z^n/n}
e^{-\sqrt{\alpha'/2}k_\mu \alpha_{n}^\mu z^{-n}/n}|q^n\rho)
\end{equation}
 for the $1$-insertion, while for the $\hat g$-insertion, we get
\begin{equation}
\int \frac{d^2\rho}{\pi}\ e^{-|\rho|^2} e^{\alpha' k_\mu^2/4n}(\rho|
e^{\sqrt{\alpha'/2}\tilde k_\mu \alpha_{-n}^\mu z^n/n}
e^{-\sqrt{\alpha'/2}\tilde k_\mu \alpha_{n}^\mu z^{-n}/n}|-q^n\rho)
\end{equation}
This is a standard integral whose evaluation can be
found in \cite{Green:1987sp}. The result is
\begin{equation}
\frac{1}{1\mp q^n} e^{\mp \alpha'k_\mu k^\mu \frac{q^n}{2n(1\mp q^n)}}
\end{equation}
again for each $n>0$ and $\mu$. For the $\tilde\alpha$ oscillators, we will
get the same result, with $q$ replaced by $\bar q$. Now by simple re-ordering of
sums and appropriate\footnote{In particular, there is a factor of 2 which must
be absorbed by the (implicit) regulator in the first equation. This can be seen,
for example, as a requirement of modular invariance.} renormalization, we
may compute:
\begin{equation}
\prod_{n\in\ZZ^+}e^{\frac{1}{n}\frac{q^n}{1+q^n}} =
\frac{\theta_2(\tau)}{q^{1/8}},\ \ \ \ \ \
\prod_{n\in\ZZ^+-1/2}e^{\frac{1}{n}\frac{q^n}{\pm 1+q^n}} = \theta_{3,4}(\tau)
\end{equation}

On the other hand, the $\partial X^\mu$ might contribute an oscillator. Then, we
have a new matrix element
\begin{equation}
\sum_{m>0}\int \frac{d^2\rho}{\pi}\ e^{-|\rho|^2} (\rho|
e^{\sqrt{\alpha'/2}k_\mu \alpha_{-n}^\mu z^n/n}
\left[z^{-m-1}\alpha_m^\mu+z^{m-1}\alpha_{-m}^\mu\right]
e^{-\sqrt{\alpha'/2}k_\mu \alpha_{n}^\mu z^{-n}/n}|q^n\rho)
\end{equation}
When $m=n$, we find, recalling that
$[\alpha_m,e^{a\alpha_{-m}}]=mae^{a\alpha_{-m}}$ and
$|\rho_n)=e^{\rho \alpha_{-n}/\sqrt{n}}|0\rangle$
\bea
\frac{\sqrt{m}}{z}\int \frac{d^2\rho}{\pi}\ e^{-|\rho|^2}
\left[z^{-m}q^m\rho+z^{m}\bar\rho\right]
(\rho| e^{k_\mu \alpha_{-n}^\mu z^n/n}
e^{-k_\mu \alpha_{n}^\mu z^{-n}/n}|q^n\rho)\\
=\frac{\sqrt{m}}{z}\int \frac{d^2\rho}{\pi}\ e^{-(1-q^m)|\rho|^2}
\left[z^{-m}q^m\rho+z^{m}\bar\rho\right]
e^{k_\mu(z^m\bar\rho-z^{-m}q^m\rho)/\sqrt{m}}
\eea
It is straightforward to show that this vanishes. Thus only the zero mode
part of the $\partial X^\mu$ factors contribute. As a corollary then, only
the untwisted sector will contribute to the massless tadpole in the Lorentzian
orbifold.

It remains to evaluate the zero modes. These are
\bea
U,1:&&\prod \int \frac{dp}{2\pi}\langle p| \hat P^\mu\hat P^\nu
e^{-\pi\alpha'\tau_2 \hat P^2}e^{(\alpha'/2) k\hat P\ln|z|^2}|p+k\rangle\
|z|^{-\alpha'k_o^2/2}= \frac{\eta^{\mu\nu}}{2\pi\alpha'\tau_2}\prod_o
\frac{\delta(\sqrt{\alpha'}k_o)}{\sqrt{\tau_2}}\nonumber\\
U,\hat g:&&\prod \int \frac{dp}{2\pi}\langle \tilde p| \hat P^\mu\hat P^\nu
e^{-\pi\alpha'\tau_2 \hat P^2}e^{(\alpha'/2) k\hat P\ln|z|^2}|p+k\rangle\
|z|^{-\alpha'k_o^2/2}=\\ &&=\prod_o\frac{e^{-\pi\tau_2\alpha' k_o^2/4}}{2}\times
\left\{\begin{matrix} k_\mu k_\nu/4& \mu,\nu\in o\cr \frac{1}{2\pi\alpha'\tau_2}&
\mu=\nu \in u\end{matrix}\right.\nonumber
\eea
each times a factor $-\left(\frac{\alpha'}{2}\right)^2\frac{1}{|z|^2}
\prod_u\frac{\delta(\sqrt{\alpha'}k_u)}{\sqrt{\tau_2}} $.
In the first case, this is multiplied by
$X_{1,1}= |\eta(\tau)|^{-24}$, while in the second,
we have $X_{1,0}=\prod_n | q^{-1}(1-q^n)^{d-23}(1+q^n)^{-(d+1)}|^2$.
Thus, if we go on-shell, we get
\begin{equation}
T^{\mu\nu}_0=-\left(\frac{g_{str}\alpha'}{2}\right)
\prod_u\frac{\delta(\sqrt{\alpha'}k_u)}{\sqrt{\tau_2}}
\frac{\eta^{\mu\nu}}{ 2\pi\alpha'}
\left( \prod_o \frac{\delta(\sqrt{\alpha'}k_o)}{\sqrt{\tau_2}}
X_{1,1}+2^{-(d+1)}X_{1,0}\right)
\end{equation}
if $\mu,\nu$ are in the unorbifolded directions, while if they are in the
orbifolded directions
\begin{equation}\label{eq:nonmod}
T^{\mu\nu}_0=-\left(\frac{g_{str}\alpha'}{2}\right)^2
\prod_u\frac{\delta(\sqrt{\alpha'}k_u)}{\sqrt{\tau_2}}
\left( \frac{\eta^{\mu\nu}}{ 2\pi\alpha'}
\prod_o \frac{\delta(\sqrt{\alpha'}k_o)}{\sqrt{\tau_2}}
X_{1,1}+\frac{k_\mu k_\nu}{ 4} 2^{-(d+1)} X_{1,0}\right)
\end{equation}
This result is not modular invariant. However there is an ordering
ambiguity\footnote{This ambiguity does not appear for Euclidean orbifolds.}
in zero modes from the $(U,\hat{g})$ sector that we have not taken into account.
To see the problem, suppose we write the vertex operator as
\begin{equation}
V=\alpha \pa X^\mu\bar\pa X^\nu e^{ik.X}+\beta \pa X^\mu e^{ik.X}\bar\pa X^\nu+
e^{ik.X} \pa X^\mu\bar\pa X^\nu
\end{equation}
Then, the $k_\mu k_\nu /4$ in eq. (\ref{eq:nonmod}) is
multiplied by $\alpha+2\beta+\gamma$.
There is a modular invariant choice ($\alpha+\beta+\gamma=1$, $\beta=-1$)
for which the $kk$ terms cancel. (There is no other effect of this ordering issue.)
We then obtain (In the notation of
Section \ref{sec:stringcalc},
$X_{1,1}=\frac{\tau_2^{12}}{V_{26}}Z_{\parallel,(1,1)}Z_\perp$)
\begin{equation}
T^{\mu\nu}_0=-\left(\frac{g_{str}}{4\pi \tau_2}\right)\eta^{\mu\nu}
\frac{ \delta^{(26)}(k)}{V_{26}}Z_{\parallel,(1,1)}[\tau]Z_\perp[\tau]
\end{equation}
(for $\mu,\nu$ in the orbifold directions). This result agrees with the
result in Section \ref{sec:stringcalc} for the case of the Lorentzian orbifold.
\end{appendix}

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\end{thebibliography}\endgroup

\end{document}
