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\begin{titlepage}

\noindent Nuclear Physics B \hspace*{\fill} hep-ph/0101087 \newline
                            \hspace*{\fill} KA-TP-24-2000
\vspace{1cm}
\begin{center}
{\Large \bf Causality and CPT violation from an Abelian \\[0.25ex]
            Chern--Simons-like term
}
\\[1cm]
C. Adam\footnote{E-mail address: adam@particle.uni-karlsruhe.de},
F.R. Klinkhamer\footnote{E-mail address:
frans.klinkhamer@physik.uni-karlsruhe.de}
\\[0.5ex]
{\it Institut f\"ur Theoretische Physik, Universit\"at Karlsruhe, 76128
Karlsruhe, Germany}

\vfill
{\bf Abstract} \\
\end{center}
We study a class of
generalized Abelian gauge field theories where CPT symmetry is violated by
a Chern--Simons-like term which selects a preferred direction in spacetime.
Such Chern--Simons-like terms may either emerge as part of the
low-energy effective action of a more fundamental theory or be
produced by chiral anomalies over a nonsimply connected spacetime manifold.
Specifically, we investigate the issues of unitarity and causality.
We find that the behaviour of these gauge field theories depends on
whether the preferred direction is spacelike or timelike.
For a purely spacelike preferred direction,  a well-behaved Feynman
propagator exists and microcausality holds,
which indicates the possibility of a consistent
quantization of the theory. For timelike preferred directions,
unitarity or causality is violated and a consistent quantization does not
seem to be possible.

\vspace{1\baselineskip}
\begin{tabbing}
PACS \hspace{1.25em} \= : \hspace{0.25em} \=
                          11.15.-q; 11.15.Bt; 11.30.Cp; 11.30.Er
                    \\[0.5ex]
Keywords       \> : \> Gauge invariance; Causality;
                       Lorentz noninvariance; CPT violation
\\
\end{tabbing}
\vfill
\end{titlepage}


\section{Introduction}

Lorentz and CPT invariance are two of the cornerstones of modern
quantum field theory. Both invariances are respected by the Standard Model of
known elementary particles (quarks and leptons) and their interactions.
Possible signals of Lorentz and CPT violation could, therefore, be indicative of
new physics, e.g. quantum gravity \cite{Wal1,Haw1} or superstrings
\cite{KoPo1}.
But even within local quantum field theory an anomalous
breaking of Lorentz and CPT symmetry might occur, at least for
a nontrivial global spacetime structure \cite{Kli1,Kli2}.

Consequently,
a considerable amount of attention has been devoted  over the last years to
the possible occurrence of Lorentz and CPT noninvariance.
Phenomenological consequences of breaking Lorentz and CPT symmetry in
electromagnetism were studied in Ref. \cite{CFJ}. It was shown that the
symmetry breaking would result in, for example, optical activity (birefringence)
of the vacuum, that is, a
direction-dependent rotation of the linear polarization of an
electromagnetic plane wave.
Reference \cite{CoKo1}, in turn,
investigated CPT- and Lorentz-noninvariant extensions of the Standard Model
(interpreted as low-energy limits of more fundamental theories).
Furthermore, there have been extensive discussions in the literature
on the possibility of
CPT- and Lorentz-symmetry breaking in the gauge field sector induced by
radiative corrections of an explicitly symmetry-breaking matter sector,
see Refs. \cite{CoKo1}--\cite{CCGF} and references therein.

At this point, the question arises whether or not a quantum field theory with
Lorentz- and CPT-violating terms can be consistent at all,
cf. Refs. \cite{Jost}--\cite{C67}. Also,
in each of the papers quoted in the previous paragraph, the CPT- and
Lorentz-noninvariant terms in the
gauge field sector were of the Chern--Simons type \cite{CS74}.
In this paper, therefore, we intend to study possible implications of a
Chern--Simons-like term for the quantization of Abelian gauge
fields, focusing on the issues of unitarity and causality.

We start from the following Lagrangian density:
\beq \label{eq:lagrangian}
{\cal L}(x)=- \textstyle{\frac{1}{4}}\, F_{\mu\nu}(x)\,F^{\mu\nu}(x)
         -\textstyle{\frac{1}{2}}\,\xi^{-1} \left( n_\mu A^\mu(x)  \right)^2
         + \mathcal{L}_{\rm \,CS-like}(x) \; ,
\eeq
with the \CS-like term
\beq \label{eq:CS-like}
\mathcal{L}_{\rm \,CS-like}(x) =
{\textstyle \frac{1}{4}}\,m\, k_\mu \,
\epsilon^{\mu\nu\rho\sigma}A_\nu(x) \, F_{\rho\sigma}(x)\; ,
\eeq
in terms of the Abelian gauge potential $A_\mu(x)$ and field strength tensor
$F_{\mu\nu}(x)$ $\equiv$ $\partial_\mu A_\nu(x)$ $-$ $\partial_\nu A_\mu(x)$.
The spacetime metric is taken to have Lorentzian signature
$(-$, $+$, $+$, $+)$ and $\epsilon_{\mu\nu\rho\sigma}$
is the  completely antisymmetric Levi-Civita symbol, normalized to
$\epsilon_{0123}=+1$. (Our conventions, with $\hbar$ $=$ $c$ $=$ $1$,
will be given in more detail later on.)

The Abelian \CS-like term (\ref{eq:CS-like}) is characterized
by a real mass parameter $m$ and a real symmetry-breaking ``vector''
$k_\mu$ of unit length, which may be spacelike ($k^2 =+1$) or timelike ($k^2 =-1$)
but is fixed once and for all
(hence, the quotation marks around the word vector).
Strictly speaking, $k_\mu$ can also be ``lightlike'' ($k^2 =0$), but the
present paper considers only the extreme cases, spacelike or timelike $k_\mu$.
As long as $k_\mu$ and $m\ne 0$ are fixed external parameters (coupling constants),
both Lorentz and CPT invariance are broken, but translation invariance still
holds. Note that the Lagrangian term (\ref{eq:CS-like}) is called \CS-like,
because a genuine topological \CS~term exists only in an odd number of
dimensions \cite{CS74}.

For later convenience, we have added a gauge-fixing term to the
Lagrangian (\ref{eq:lagrangian}), where
$n_\mu$ determines the axial gauge condition and $\xi$ is a gauge parameter.
Choosing an axial gauge,
which selects a particular direction
$n_\mu$, seems natural because the ``vector'' $k_\mu$ already selects a
preferred direction. In other words, there is no compelling reason to prefer
Lorentz-covariant gauge choices over noncovariant ones, cf. Ref. \cite{Leib}.

The Lagrangian (\ref{eq:lagrangian})
is Abelian and, therefore, describes a photon-like gauge
field. But Eq. (\ref{eq:lagrangian})
may as well be interpreted as one component of the
quadratic part of a non-Abelian Lagrangian. The discussion
that follows is, in principle, also relevant for Lorentz- and
CPT-symmetry breaking in a non-Abelian context. Still, the issue of locality may
be more subtle for the non-Abelian case due to gauge invariance, as
discussed in Section 4 of Ref. \cite{Kli1}.

Let us now give in more detail the reasons for studying the \MCS~the\-o\-ry
(\ref{eq:lagrangian}), with broken Lorentz and CPT symmetry.
First, a nonzero mass scale $m$ may be
introduced by hand as a symmetry-breaking parameter. Possible
physical consequences and experimental bounds on the value of $m$
may be studied, as  was done in Ref. \cite{CFJ}, under the
assumption that the Lagrangian (\ref{eq:lagrangian}) describes the photon.
Recently, it has also been claimed \cite{NR1}
that certain astronomical observations  indicate a nonzero value of $m$
for the case of a spacelike \CS~parameter $k_\mu$,
but this claim is apparently not substantiated by more accurate data
(see Ref. \cite{WPC} and references therein).

Second, the symmetry-breaking term in the
Lagrangian (\ref{eq:lagrangian}) may be thought
of as being part of the effective action which results from integrating out
the fermionic matter fields. Here, the source of the symmetry breaking
might be an explicit symmetry-breaking term in the fermionic matter
sector \cite{CoKo1}--\cite{CCGF}. Alternatively, the symmetry-breaking
term in the effective action might be traced to a quantum anomaly
which occurs when Weyl fermions in suitable representations are quantized
on a nonsimply connected spacetime manifold (e.g.  $\mathbb{R}^3$ $\times$ $S^1$).
This CPT anomaly was
discovered and described in Ref. \cite{Kli1}, where the precise conditions for
its occurrence can be found. In this case, the experimentally
required smallness of $m$ for photons is naturally accounted for,
because the mass scale $m$ is
inversely proportional to the linear extension ($L$) of the universe in the
compact direction,
\beq \label{eq:mCPT}
m_{\rm CPT\; anomaly} \sim \, \alpha\, \hbar \, (L\,c)^{-1} \;,
\eeq
with $\alpha$ the fine-structure constant and the dependence on $\hbar$ and $c$
made explicit. For $L$ $\sim$ $1.5\;10^{10}\,{\rm lightyears}$, this mass scale
corresponds to $10^{-35}\, {\rm eV}$, which might be within reach of future
astronomical observations (this point will be discussed further in Section 6).

Third, the Lagrangian (\ref{eq:lagrangian}) may be interpreted as the quadratic
gauge field part of a low-energy effective action of a truly fundamental theory,
which could, for example, replace point-particles by superstrings,
cf. Ref. \cite{KoPo1}.

Our paper is organized as follows. In Section 2, we focus on the
classical aspects of the \MCS~theory (\ref{eq:lagrangian}) and
discuss the resulting dispersion relations and causality behaviour.
This turns out
to be rather different for spacelike and timelike \CS~parameter $k_\mu$.
In Section 3, the Feynman propagator for the Lagrangian
(\ref{eq:lagrangian}) is
calculated both for Minkowskian and Euclidean spacetime. Again, the cases of
``spacelike'' and ``timelike'' $k_\mu$ have to be discussed separately.
In Section 4, we address the related issue of reflection positivity
for the Euclidean theory corresponding to Eq. (\ref{eq:lagrangian}), which also
depends on the type of parameter $k_\mu$.
In Section 5, we determine the field commutators of the quantum field theory based
on Eq. (\ref{eq:lagrangian}), first for a purely spacelike $k_\mu$.
We find that the usual microcausality holds for this case, which is perhaps
the most important result of this paper.
(Some details of our calculation are relegated to Appendix A.)
On the other hand, unitarity and microcausality cannot be
maintained simultaneously for a timelike $k_\mu$. In Section 6, finally, we
summarize our results and briefly discuss possible applications and open
questions. The present paper is, by necessity, quite technical and the
general reader may wish to concentrate on Sects. 2 and 6.


\section{Dispersion relations}

As a first step we discuss the dispersion relations which result
from the Lagrangian (\ref{eq:lagrangian}) without the gauge fixing term and
investigate the implications for the causal behaviour of the classical
theory. Throughout this section, we take the spacetime manifold
$M$ $=$ $\mathbb{R}^4$ and Minkowskian spacetime metric
$g_{\mu\nu}$ $=$ $\mathrm{diag}$ $(-1,1,1,1)$, with indices
running over 0, 1, 2, 3.

The Lagrangian (\ref{eq:lagrangian}) then leads to the following dispersion
relation for the gauge fields \cite{CFJ}:
\beq \label{eq:disprel}
p^4 +m^2 \left(k^2 p^2 -(k\cdot p)^2\right) =0 \; ,
\eeq
for momentum $p_\mu = (p_0 ,p_1 ,p_2 ,p_3 )$ and $c=1$.\footnote{For the moment,
the velocity parameter $c$ is only used to define the Minkowski
spacetime coordinates $(x^0 \equiv c t,x^1,x^2,x^3)$. As will become clear
later, $c$ corresponds to the front velocity of light propagation
\emph{in} \emph{vacuo} for the electromagnetic
theory based on the Lagrangian (\ref{eq:lagrangian}) with a spacelike
parameter $k_\mu$.}
Due to the breaking of Lorentz invariance, there exist preferred
coordinate systems. A particular preferred coordinate system for spacelike
\CS~parameter $k_\mu$ is one in which $k_\mu$ is purely spacelike
($k_0 =0$), which we shall choose in the sequel.

Let us discuss this last point in somewhat more detail
(see also Ref. \cite{CoKo1}).
As mentioned in the Introduction, the \CS~parameters $k_{\mu}$ are
considered to be fixed coupling constants (four real numbers)
belonging to a particular coordinate system. For localized gauge
fields (that is, $A_\mu(x)$ $=$ $0$ for $|x|$ $\geq$ $R$),
one can nevertheless make a Lorentz transformation
$x^\mu$ $\rightarrow$ $x^{\prime \,\mu}$ $=$
$\Lambda^\mu_{\phantom{\mu}\lambda}\, x^\lambda+ a^\mu$,
so that the \CS-like term (\ref{eq:CS-like}) changes into
\beq \label{eq:kmu-prime}
\textstyle{\frac{1}{4}}\, m\,k_\mu \,\Lambda^\mu_{\phantom{\mu}\lambda}\,
\epsilon^{\lambda\nu\rho\sigma}A_\nu(x^\prime) F_{\rho\sigma}(x^\prime) \equiv
\textstyle{\frac{1}{4}}\, m\,k^\prime_\lambda \,
\epsilon^{\lambda\nu\rho\sigma}A_\nu(x^\prime) F_{\rho\sigma}(x^\prime) \; .
\eeq
The new reference frame (with coordinates $x^{\prime \,\mu}$) thus has its own
\CS~parameters $k^\prime_\mu$, determined by the old $k_{\mu}$ and the Lorentz
parameters for the change of frame. It is, however, not at all obvious
that this change of $k_{\mu}$ parameters is unitarily
implementable for the quantum theory.
The quantization of the \MCS~theory (\ref{eq:lagrangian}) is, therefore,
considered rather explicitly in the following sections.
For now, we continue our discussion of the classical dispersion relation.

For a purely spacelike \CS~parameter $k_\mu$ $=$ $(0, \vec k)$ with
$|\vec k|^2$ $=$ $1$, Eq. (\ref{eq:disprel}) is a quadratic equation in $p_0^2$,
with the following
solutions:
\beq \label{eq:disprel-space}
p_0^2 = |\vec p|^2 + \textstyle{\frac{1}{2}}\, m^2\,
\pm \textstyle{\frac{1}{2}}\, m\,
\sqrt{m^2 + 4 \,(\vec p\cdot \vec k\,)^2}\; .
\eeq
Apparently, there are two very different
degrees of freedom, especially
towards the infrared ($|\vec p|$ $\lsim$ $m$).
(The identification of these two degrees of freedom with circular
polarization modes depends on the sign of $\vec p\cdot \vec k$,
cf. Eq. (26) of Ref. \cite{CFJ}.)
For directions $\vec p$ per\-pen\-di\-cu\-lar to $\vec k$,
Eq. (\ref{eq:disprel-space}) effectively
describes one massive degree of freedom with mass $m$, corresponding to
the plus sign,  and one massless degree
of freedom, corresponding to the minus sign.
In the $\vec p$ direction parallel to $\vec k$,
both the massive and the massless dispersion relations get
distorted. However, both degrees of freedom may still be separated into
positive and negative frequency parts, as is obvious from
Eq. (\ref{eq:disprel-space}).
In Fig. 1 we plot the dispersion relation (\ref{eq:disprel-space})
restricted to the $(p_0,p_3)$ plane, where $ k_\mu $ is assumed to point
into the $x^3$ direction as well.
The separation into positive and negative
frequency parts is clearly seen in Fig. 1.

Without loss of generality we now assume
that $k_\mu $ points into the
$x^3$ direction, i.e. $k_\mu =(0,0,0,1)$. The dispersion relations
for the two degrees of freedom then read
\beq \label{eq:frequency-space}
p_0^2 = \omega_\pm^2 \equiv p_1^2 + p_2^2 +\wt\omega_\pm^2 \; ,
\eeq
with
\beq \label{eq:omega-tilde}
  \wt\omega_\pm \equiv
  {\textstyle \frac{1}{2}}\,\left(\sqrt{4 p_3^2 + m^2} \pm m \right)\; .
\eeq
Consider, for simplicity, the case of $p_1 = p_2 =0$ and $p_3$ $\geq$ $0$.
Then, the dispersion relations $p_0^2$ $=$ $\wt\omega_\pm^2$
lead to the phase velocities
\beq
v_{\rm ph}^\pm = \frac{\wt\omega_\pm}{p_3}=\frac{\sqrt{4 p_3^2
    +m^2} \pm m}{2p_3}
\eeq
and group velocity
\beq \label{eq:group-velo}
v_{\rm g} = \frac{\rmd \wt\omega_\pm}{\rmd p_3}=
\frac{2p_3}{\sqrt{4 p_3^2 +m^2}} \le 1 \; .
\eeq
For the case considered and $m \neq 0$, both velocities
approach 1 in the limit $p_3\to \infty$. More generally, the front
velocity  $v_{\rm f}\equiv \lim_{|\vec p| \to\infty} |\vec v_{\rm ph}|$,
which is relevant for signal propagation \cite{B60}, has the same
value 1 in all directions (recall that $c=1$ in our units).

This classical reasoning already indicates that the causal structure of
the theory remains unaffected by the additional CPT-violating term in Eq.
(\ref{eq:lagrangian}), at least for the case $k_\mu =(0,0,0,1)$.
In Section 5.1, we shall find further evidence for this statement by
calculating the commutators of the quantized fields.

Before closing this section, we want to contrast the discussion
above with that for the case of a timelike \CS~parameter $k_\mu$,
which has already been studied in detail
by the authors of Ref. \cite{CFJ}. Here, a particular preferred coordinate
system is one where $k_\mu$ is purely timelike, $k_\mu
=(1,0,0,0)$, which we assume in the
following. Again, Eq. (\ref{eq:disprel}) leads to a
quadratic equation in $p_0^2$, with the solutions
\beq \label{eq:frequency-time}
p_0^2 = \omega_\pm^2 \equiv |\vec p|^2 \pm m |\vec p| \; .
\eeq
(These two degrees of freedom correspond to circular
polarization modes, cf. Eq. (26) of Ref. \cite{CFJ}.)
The dispersion relation (\ref{eq:frequency-time}) is plotted in Fig. 2.
It is obvious that
there is no separation
into positive and negative frequency parts.\footnote{This fundamental
difference of the \MCS~theory (\ref{eq:lagrangian}) for the case of,
say, $k_\mu =(1,0,0,0)$
and $k_\mu =(0,0,0,1)$ traces back to our fixed choice of the time
coordinate, namely $x^0 \equiv c t$.}
Worse, the energy becomes imaginary at low momenta $|\vec p|<m$ for the minus
sign in Eq. (\ref{eq:frequency-time}). In addition,
the group velocities of both degrees of freedom may become arbitrarily large.
For the minus sign in Eq. (\ref{eq:frequency-time}), one has, for example,
\beq
\frac{\rmd \omega_-}{\rmd |\vec p|}=
\frac{2|\vec p|-m}{2\sqrt{|\vec p|^2 -m|\vec p|}} \; ,
\eeq
which is singular at $|\vec p|=m$. These results indicate that
the case of timelike \CS~parameter $k_\mu$ is rather different from the case of
spacelike $k_\mu$ and does not allow for quantization.
In the next section, we
shall find further evidence for this statement by investigating the Feynman
propagator.


\section{Feynman propagator in a general axial gauge}

We now consider the Feynman propagator which may be formally
derived from the Lagrangian (\ref{eq:lagrangian}),
and investigate what may be inferred for the
possible quantization of the theory.

The action corresponding to Eq. (\ref{eq:lagrangian}) can be
re-expressed as follows:
\beq
S={\textstyle \frac{1}{2}}\,\int_{\mathbb{R}^4} \rmd ^4 x \,
A_\mu \left(g^{\mu\nu}\Box -\partial^\mu \partial^\nu
-(1/ \xi)\,n^\mu n^\nu -m\, \epsilon^{\mu\nu\rho\sigma}k_\rho
\partial_\sigma \right) A_\nu \; ,
\eeq
so that the inverse propagator in momentum space becomes
\beq
\left(G^{-1}\right)^{\mu\nu}(p)=
-p^2g^{\mu\nu} +p^\mu p^\nu -(1/\xi)\, n^\mu n^\nu
-i\, m\, \epsilon^{\mu\nu\rho\sigma}k_\rho p_\sigma \; .
\eeq
The corresponding propagator, which obeys
$(G^{-1})^{\mu\nu}G_{\nu\lambda}
= -\delta^\mu_\lambda$, reads
\beqa \label{eq:propagator}
G_{\nu\lambda}(p) &=& \left( g_{\nu\lambda} + \frac{\xi \, p^2 + n^2 +
  (m^2 /p^2)(k\cdot n)^2 + \xi \,m^2 k^2 -\xi \,  (m^2/p^2)(k
\cdot p)^2}{(p\cdot n)^2}\, p_\nu p_\lambda \right.
\no \\
&&
-\frac{1}{(p\cdot n)} \, (p_\nu n_\lambda + n_\nu p_\lambda )
+\frac{m^2}{p^2} \, k_\nu k_\lambda - \frac{m^2 (k\cdot n)}{p^2 (p\cdot n)}
\, (p_\nu k_\lambda + k_\nu p_\lambda )
\no \\
&&
+\left. i\,m\,\epsilon_{\nu\lambda\alpha\beta}
\left(\frac{ (p\cdot k)}{p^2 (p\cdot n)} \, p^\alpha n^\beta
-\frac{1}{(p\cdot n)} \, k^\alpha n^\beta
 \right)
\right) K\; ,
\eeqa
with
\beq \label{eq:feyn-poles}
K\equiv \frac{p^2}{p^4 + m^2 \left(k^2 p^2 - (k\cdot p)^2\right)} \; .
\eeq
(Note that the equivalent propagator in a covariant gauge has
already been computed in Ref. \cite{ASS1}.)

Up till now, the calculation of the propagator was formal and purely
algebraic. We did not discuss the pole structure, nor even define whether we
are in Minkowski or Euclidean spacetime.\footnote{The classical Euclidean theory
is derived from Eq. (\ref{eq:lagrangian}) by rotating both $x^0$ and $k^0$ to
the imaginary axis.
(See Ref.  \cite{MM1} for a general discussion of Euclidean field theories.)
An alternative Euclidean theory could perhaps be defined
by keeping $k^0$ absolutely fixed,
but this is not what has been discussed in, for example, Refs.
\cite{Kli1,CoKo1}.} A systematic treatment can
be given for the spurious singularities related to the axial gauge vector
$n_\mu$ (see, for example, Ref. \cite{Leib}), and we
ignore these singularities in the following. Instead, we focus on the pole
structure of the propagator function $K$ as given by Eq. (\ref{eq:feyn-poles}).
For clarity, we relabel our previous \CS~parameter $k_\mu$ in Minkowski
spacetime as $k^{\mathrm M}_\mu$ and use $k^{\mathrm E}_\mu$ in Euclidean
space.

First, let us discuss the case of a purely spacelike $k^{\mathrm M}_\mu
=(0,0,0,1)$, with index $\mu$ running over 0, 1, 2, 3. For Minkowski spacetime
with metric signature $(-$, $+$, $+$, $+)$, we get
\bdi
K = \frac{-p_0^2 +|\vec p|^2}
{\left( p_0^2 -|\vec p|^2 -\frac{m^2}{2} -
\frac{m}{2}\sqrt{m^2 +4 p_3^2}+i\epsilon \right)
\left( p_0^2 -|\vec p|^2 -\frac{m^2}{2} +
\frac{m}{2}\sqrt{m^2 +4 p_3^2}+i\epsilon \right)} \; ,
\edi
\beq \label{eq:prop-mink-space}
\eeq
where both poles are displaced with the help of the usual Feynman
$i\epsilon $ prescription ($\epsilon$ $=$ $0^+$),
cf. Refs.  \cite{V94,W96}. For Euclidean space
with metric signature $(+$, $+$, $+$, $+)$ and indices running over
4, 1, 2, 3, we find instead
\beq \label{eq:prop-eucl-space}
K=\frac{p_4^2 +|\vec p|^2}
{\left(p_4^2 +|\vec p|^2 +\frac{m^2}{2} +
\frac{m}{2}\sqrt{m^2 +4 p_3^2}\right)
 \left(p_4^2 +|\vec p|^2 +\frac{m^2}{2} -
\frac{m}{2}\sqrt{m^2 +4 p_3^2}\right)} \; .
\eeq
The poles from both factors in the denominator are placed on the positive
and negative imaginary axis of the complex $p_4$ plane.
A Wick rotation \cite{V94,W96} to Minkowski
spacetime can be performed and leads to the $i\epsilon$
prescription (\ref{eq:prop-mink-space}) for the Feynman propagator
(\ref{eq:propagator}) of both degrees of freedom.
Hence, the propagator is well-behaved, at least for the case of
a purely spacelike \CS~parameter.

For a purely timelike $k^{\mathrm M}_\mu =(1,0,0,0)$ in Minkowski spacetime,
with index $\mu$ running over 0, 1, 2, 3,  we obtain
\beqa \label{eq:prop-mink-time}
K &=& -\frac{p_0^2 -|\vec p|^2}{(p_0^2 -|\vec p|^2 + m|\vec p|)\,
(p_0^2 -|\vec p|^2 - m|\vec p|)}
\no \\
&=& -\frac{1}{2(p_0^2 -|\vec p|^2 +m|\vec p|)}-\frac{1}{2(p_0^2 -
|\vec p|^2 -m|\vec p|)}\; .
\eeqa
For low momenta $|\vec p|<m$, the poles in the first term are
placed on the imaginary $p_0$ axis, which means that the energy becomes imaginary.
This, in turn, implies that unitarity is violated already at tree
level, i.e. for the free theory (\ref{eq:lagrangian}).
The region $|\vec p|<m$ has, therefore, to be excluded for this degree of freedom.
(The situation is similar to the
case of a tachyon field with dispersion relation
$p_0^2 -|\vec p|^2 +m^2 =0$, where
the region $|\vec p|<m$ has to be excluded
in order to maintain unitarity of the quantum field theory
at tree level. See, for example, the discussion in Refs.  \cite{DhSu1,JTW}.)
But we shall find in Section 5.2 that exclusion of the region $|\vec p|<m$ leads to a
violation of microcausality.

If we now assume a purely ``timelike'' $k^{\mathrm E}_\mu =(1,0,0,0)$ in Euclidean
space, with index $\mu$ running over 4, 1, 2, 3, the function $K$ becomes
\beqa \label{eq:prop-eucl-time}
K &=& \frac{p_4^2 +|\vec p|^2}{(p_4^2 +|\vec p|^2 + im|\vec p|)\,
(p_4^2 +|\vec p|^2 - im|\vec p|)}
\no \\
&=& \frac{1}{2(p_4^2 +|\vec p|^2 +im|\vec p|)}+\frac{1}{2(p_4^2 +
|\vec p|^2 -im|\vec p|)}\; .
\eeqa
Here, the poles of the first  (second) term are placed in
the  second and fourth (first and third) quadrants of the complex
$p_4$ plane. In order to determine the behaviour of the propagator
(\ref{eq:prop-eucl-time}) under a Wick rotation to Minkowski space,
we have to remember that according to our prescription we have to rotate
$k_4$ as well. This makes that the poles of Eq.
(\ref{eq:prop-eucl-time}) move under Wick rotation. For
sufficiently small $|\vec p|$, two poles will, in fact, move to the real axis and,
therefore, cross the Wick-rotated $p_4$-axis. In short, the analytic
behaviour of the propagator is problematic for the case of a purely timelike
\CS~parameter.

For  spacelike $k_\mu$ in Minkowski spacetime, we have used up till now a
special coordinate system in which
$k_\mu$ is purely spacelike, that is, $k_0 =0$ exactly.
Let us, finally,  relax this condition and
investigate what happens if we allow for $k_0 \ne 0$. In general, the
four roots of the denominator of Eq. (\ref{eq:feyn-poles}) are rather complicated.
We shall, therefore, make some simplifying assumptions. By choosing $k_1 = k_2 =0$,
we can restrict ourselves to the plane $p_1 = p_2 =0$. Also, we choose units of
energy and momentum such that $\tilde k_\mu \equiv mk_\mu =(\tilde k_0 ,0
,0,1)$. With these assumptions, we still find four real roots $p_0
=r_i$, $i=1 \ldots 4$, as long as $| \tilde k_0 | <1$.

With the same simplifications,
we find in Euclidean space the following four roots $p_4 =
q_i$, $i=1 \ldots \, 4$, for the
denominator of Eq. (\ref{eq:feyn-poles}):
\bdi
q_1 = \frac{i}{2} \left( 1 - \sqrt{1 +4i\tilde k_4 p_3 +4 p_3^2}\right)
\, ,\quad q_2 = q_1^*  \; ,
\edi
\beq \label{eq:poles}
q_3 = \frac{i}{2} \left( 1 + \sqrt{1 +4i\tilde k_4 p_3 +4 p_3^2}\right)
\, ,\quad q_4 = q_3^*  \; .
\eeq
For $\tilde k_4 \ne 0$, $q_1$ and $q_3$ have nonzero real parts of opposite
sign (${\rm Re}\, q_1 =-{\rm Re}\, q_3$) and the four poles of
Eq. (\ref{eq:feyn-poles}) are
placed in all four quadrants of the complex $p_4$ plane.
Under a Wick rotation, with $k_4$ rotated as well, all four poles
(\ref{eq:poles}) move towards the imaginary axis together with $p_4$
(as long as $\tilde k_\mu$ is ``spacelike,'' $ |\tilde k_4| <1$),
and a Wick rotation may be performed without crossing poles in the
complex $p_4$ plane.  Hence, the propagator is well-behaved, provided
the \CS~parameter is spacelike.

This completes our elementary discussion of the Feynman propagator for the
\MCS~theory (\ref{eq:lagrangian}).
In the next section, we will study the Euclidean propagator in somewhat
more detail.


\section{Reflection positivity}

An important condition for the quantization of a field theory in the
Euclidean formulation is reflection positivity \cite{OS1,MM1}.
This condition is essential for establishing the existence of a positive
semi-definite self-adjoint Hamiltonian $H$ in Minkowski spacetime,
with the corresponding unitary time evolution operator
$\exp (-i H t)$.

The reflection positivity condition for an Euclidean two-point function is simply
\beq
\langle \Theta \left(\phi (x^4, \vec x)\right)\phi (x^4, \vec x) \rangle \ge 0 \; ,
\eeq
where $x^4$ is the Euclidean time coordinate, $\phi(x^4, \vec x)$
a scalar field of the theory,
and $\Theta :\phi (x^4, \vec x)$ $\ra$ $\phi^\dagger (-x^4, \vec x)$
the reflection operation.
Reflection positivity then gives the following inequality
for the scalar Euclidean propagator function $G(p_4 ,\vec p)$:
\beq  \label{eq:reflposG}
\int \rmd^3 p\int_{-\infty}^{\infty}\rmd p_4\; e^{-i p_4 x^4} \,G(p_4 ,\vec p)
\equiv
\int \rmd^3 p \; G(x^4 ,\vec p) \ge 0\; ,
\eeq
for arbitrary values of $x^4$. By choosing suitable smearing functions,
it is even possible to derive the stronger condition $G(x^4 ,\vec p)$ $\ge$ $0$,
but for our purpose the condition (\ref{eq:reflposG})  suffices.

For the gauge-invariant degrees of freedom of the \MCS~theory (\ref{eq:lagrangian}),
it turns out to be sufficient
to check the issue of reflection positivity for the Euclidean propagator
function $K$ (as introduced in Eq. (\ref{eq:propagator}) above), thereby
effectively reducing the
problem to the investigation of a scalar two-point function.
Concretely, we then have to verify whether or not the inequality
(\ref{eq:reflposG}) holds for our propagator function $K(p_4 ,\vec p)$.

For the case of purely ``spacelike'' $k^{\mathrm E}_\mu=(0,0,0,1)$,
with index $\mu$ running over 4, 1, 2, 3, the function
$K(p_4 ,\vec p)$ is given by Eq. (\ref{eq:prop-eucl-space}) and we get
 \beqa \label{eq:refl-space}
K(x^4 ,\vec p) &=& \int_{-\infty}^{+\infty}\rmd p_4 \; e^{-ip_4 x^4}\frac{p_4^2
  +|\vec p|^2}{(p_4^2 +\omega_+^2)\,(p_4^2 +\omega_-^2)}
\no \\
&=& \pi \,  \frac{\omega_+ e^{-\omega_+ |x^4|} - \omega_- e^{-\omega_- |x^4|}}{
  \omega_+^2 - \omega_-^2}
+\pi  \, \frac{|\vec p|^2}{\omega_+\omega_-} \,
\frac{\omega_+ e^{-\omega_- |x^4|} - \omega_- e^{-\omega_+ |x^4|}}{
  \omega_+^2 - \omega_-^2}
\no \\
& =& \pi \,  \frac{(|\vec p|^2 -\omega_-^2)\,\omega_+ e^{-\omega_- |x^4|} +
(\omega_+^2 -|\vec p|^2)\,\omega_- e^{-\omega_+ |x^4|}}{\omega_+ \omega_-
(\omega_+^2 - \omega_-^2)} \; ,
\eeqa
where Eqs. (3.728.1) and (3.728.3) of Ref. \cite{GrRy1}
have been used to evaluate the integral and
the frequencies $\omega_\pm$ are defined in
Eq. (\ref{eq:frequency-space}). This expression
is manifestly positive semi-definite, since $\omega_- \le |\vec p|\le
\omega_+$,  and reflection positivity (\ref{eq:reflposG}) holds.

For the case of purely ``timelike'' $k^{\mathrm E}_\mu=(1,0,0,0)$, the function
$K(p_4 ,\vec p)$ is given by
Eq. (\ref{eq:prop-eucl-time}) and we get
\beqa
K(x^4 ,\vec p) &=& \int_{-\infty}^{+\infty}\rmd p_4 \; e^{-i p_4 x^4} \,
\frac{p_4^2 + |\vec p|^2}{p_4^4 +2p_4^2 |\vec p|^2 +|\vec p|^4 +m^2 |\vec p|^2}
\no \\
&=&  \int_{0}^{+\infty}\rmd p_4  \;2 \cos\left(p_4 x^4\right) \, \frac{p_4^2 +
|\vec p|^2}{p_4^4 +2p_4^2 b^2 \cos 2a +b^4} \; ,
\eeqa
with
\bdi
\cos 2a \equiv\ |\vec p|^2/b^2\, ,\quad
b^2 \equiv |\vec p |\sqrt{|\vec p|^2 +m^2} \; .
\edi
The integration over $p_4$ can be performed explicitly and we obtain,
using Eqs. (3.733.1) and (3.733.3) of Ref. \cite{GrRy1},
\beqa \label{eq:refl-time}
K(x^4 ,\vec p) &=& \pi \exp(-|x^4|\, b\cos a)
\left(\frac{\sin[a-|x^4|\,b\sin a]}{b\sin 2a} +
\frac{|\vec p|^2}{b^2} \frac{\sin[a+|x^4|\,b\sin a]}{b\sin  2a}\right)
\no \\
&=& \pi  \exp(-|x^4|\, b\cos a)\, \cos[a+|x^4|\,b\sin a]/b\; .
\eeqa
Clearly, this expression is not positive semi-definite (as long as $m\ne 0$)
and numerical integration over $\vec p$ shows
reflection positivity (\ref{eq:reflposG}) to be violated for large enough
values of $m |x^4|$.
The different behaviour of, respectively,  Eqs. (\ref{eq:refl-space}) and
(\ref{eq:refl-time}) is caused by the different pole structure in Eqs.
(\ref{eq:prop-eucl-space}) and (\ref{eq:prop-eucl-time}), which was also
the crux of the previous section.


\section{Microcausality}

Having dealt with unitarity, we continue
our investigation of the hypothetical quantum field theory
based on the Lagrangian (\ref{eq:lagrangian})
and focus on the issue of causality.
Minkowskian conventions, with metric signature $(-,+,+,+)$ and
indices running over 0, 1, 2, 3,
are assumed throughout this section and the units are such that
$c$ $=$ $\hbar$ $=$ $1$.

\subsection{Purely spacelike \CS~parameter}

Let us, again, start with the case of
a purely spacelike ``vector'' $k_\mu =(0,\vec k)$.
We prefer to use a physical gauge condition, in order to
avoid the problem of constructing the subspace of physical
states. Furthermore, we will try to connect to the well-known results of
Quantum Electrodynamics, i.e. the Lagrangian
(\ref{eq:lagrangian}) for $m=0$. We, therefore,
switch from the general axial gauge to the Coulomb gauge,
$\vec \partial \cdot \vec A =0$, cf. Refs. \cite{W96, Heitler}.

The resulting commutator for the gauge field $\vec A(x^0,\vec x)$
is then given by
\beq \label{eq:comm}
[A_i (x), A_j (0)] = i\,
T_{ij}(-i\partial_0 ,-i\vec \partial \,)\,D (x) \;,
\eeq
with the commutator function
\beq \label{eq:comm-func-space}
D(x)=(2\pi)^{-4}\,\oint_C \rmd p_0 \int \rmd^3 p \; \frac{e^{ip_0 x^0
+i\vec p \cdot \vec x}}{(p^2)^2 +m^2
 \left(p^2 |\vec k|^2 -(\vec p\cdot \vec k)^2\right)  }\;,
\eeq
for an integration contour $C$ that encircles all four poles of the
integrand in the counterclockwise direction, cf. Appendix A1 of Ref. \cite{JR}.
The denominator of Eq. (\ref{eq:comm-func-space})
is given by the dispersion relation (\ref{eq:disprel})
for purely spacelike \CS~parameter $k_\mu$. The ``tensor'' $T_{ij}$
on the \rhs~of the commutation relation (\ref{eq:comm}) is found to be given
by
\beq \label{eq:Tij}
T_{ij}(p_0,\vec p )=(p_0^2 -|\vec p|^2)\,\pi_{ij} -m^2 \,s_{ij} +imp_0 \,a_{ij}\;,
\eeq
with
\beq
\pi_{ij}\equiv \delta_{ij} -\frac{p_i p_j}{|\vec p|^2}  \;,
\eeq
\beq
s_{ij}\equiv\left( k_i -\frac{\vec p\cdot \vec k}{|\vec p|^2}\, p_i \right)
\left(k_j -\frac{\vec p\cdot \vec k}{|\vec p|^2}\, p_j \right)  \;,
\eeq
\beq
a_{ij}\equiv\epsilon_{ija}k_a +\frac{p_i}{|\vec p|^2}\,\epsilon_{jab}p_a k_b
-\frac{p_j}{|\vec p|^2}\,\epsilon_{iab}p_a k_b = \frac{\vec p\cdot \vec
  k}{|\vec p|^2}\,\epsilon_{ijl}p_l \; .
\eeq
One immediately verifies
that the commutator (\ref{eq:comm}) respects the
Coulomb gauge, since $p_i\,T_{ij}=0$ and $p_j\,T_{ij}=0$. Further
details on the derivation of this commutation relation can be found
in Appendix A.

Microcausality (i.e. the commutativity of local observables with spacelike
separations, cf. Refs. \cite{SW64,V94,JR}) holds, provided that:
\begin{enumerate}
\item the commutator function $D(x)$ vanishes for spacelike separations $|x^0|
  <|\vec x|$,  and
\item the poles of the type $|\vec p|^{-2}$ which occur in the ``tensor''
$T_{ij}$ are
absent in the commutators of physical, gauge-invariant fields (i.e. the
electric and magnetic fields).
\end{enumerate}
In our case, the commutators of the electric field $\vec E \equiv \partial_0 \vec A
-\vec \partial A_0$ and magnetic field $\vec B \equiv \vec \partial \times
\vec A$ are found to be the following (see Appendix A for details):
\beqa \label{eq:comm-e}
[E_i (x) ,E_j (0)]&=&\Big( (\partial_0^2 -\vec \partial^2 )(
\delta_{ij}\partial_0^2 -\partial_i \partial_j ) + m^2 \partial_0^2 k_i k_j
\no \\
&&- m\partial_0^3 \epsilon_{ijl} k_l -m \partial_0 (\partial_i
\epsilon_{jab} \partial_a k_b - \partial_j \epsilon_{iab} \partial_a k_b)
\Big)\, i \, D(x)  \;,
\eeqa
\beqa  \label{eq:comm-eb}
[E_i (x),B_j (0)] &=&\Bigl(  (\partial_0^2 -\vec \partial^2
)\epsilon_{ijl} \partial_l\partial_0 +m^2 \partial_0 k_i \epsilon_{jab} \partial_a
k_b
\no \\
&&+ m\partial_0^2 (\vec k\cdot \vec \partial ) \delta_{ij} -m
(\vec k\cdot \vec \partial ) \partial_i \partial_j -m
(\partial_0^2 -\vec \partial^2 ) \partial_i k_j \Bigr)\, i \,  D(x)\; ,
\eeqa
\beqa \label{eq:comm-b}
[B_i (x), B_j (0)] &=&\left( (\partial_0^2 -\vec \partial^2 )(\delta_{ij}
\vec \partial^2 -\partial_i \partial_j )+m^2 \{\delta_{ij}(\vec \partial^2
|\vec k|^2 - (\vec k\cdot \vec \partial )^2 ) -|\vec k|^2 \partial_i
\partial_j \right.
\no \\
&& \left. -k_i k_j \vec \partial^2 +(\partial_i k_j + k_i \partial_j
)(\vec k \cdot \vec \partial )\} -m \partial_0 (\vec k\cdot \vec \partial
) \epsilon_{ijl} \partial_l\right)\, i \, D(x) \;.
\eeqa
Remark that poles of the type $|\vec p|^{-2}$, which could spoil causality,
are indeed absent in these commutators of physical field operators. In
addition, we recover the Jordan--Pauli commutators
of Quantum Electrodynamics \cite{Heitler} in the limit $m\to 0$
(remember that our $D(x)$ for $m\to 0$ obeys the massless
Klein--Gordon equation squared, $\Box\Box D=0$).

We still have to discuss the commutator function
(\ref{eq:comm-func-space}). Henceforth, we assume that $\vec k$
points into the $3$ direction, $\vec k =(0,0,1)$. We first observe that
(\ref{eq:comm-func-space})
vanishes for equal times ($x^0 =0$), because of the
symmetry in $p_0$ of the integrand (\ref{eq:comm-func-space}),
which results in a cancellation of the residues (compare Eq.
(\ref{eq:Dcontour}) for $\wt x^0 =0$ below).
The commutator function is also zero for
$(x^0)^2 < (x^1)^2 + (x^2)^2 $, because the integrand can be made to be
symmetric
in a new variable $p_0'$, which is related to $p_0$ by a
conventional Lorentz boost involving only $p_0$, $p_1$, and $p_2$.
We will now show that the commutator function $D(x)$ vanishes, in fact,
over the \emph{whole} spacelike region $|x^0|$ $<$ $|\vec x|$.
The reader who is not interested in the
details may skip the rest of this subsection.

For $(x^0)^2 \geq (x^1)^2 + (x^2)^2$, it is still useful to perform a
Lorentz transformation involving only $p_0$, $p_1$ and $p_2$, because
there exists a transformation which allows us to rewrite
Eq. (\ref{eq:comm-func-space}) as
\beqa \label{eq:comm-func-space-2}
D(\wt x^0 ,x^3) &=&
(2\pi)^{-4}\,\oint_C \rmd p_0 \int \rmd^3 p \;\frac{e^{ip_0
\wt x^0 +i p_3 x^3}}{(p^2)^2 +m^2 \left(p^2 |\vec k|^2 -(\vec p\cdot \vec
k)^2\right)  }\;,
\no\\
&=& \label{eq:Dcontour}
(2\pi)^{-4}\, \oint_C \rmd p_0 \int \rmd^3 p \;\frac{e^{ip_0
\wt x^0 +ip_3 x^3}}{(p_0 -\omega_+)(p_0 +\omega_+)(p_0 -\omega_-)(p_0 +
\omega_-) }\; ,
\eeqa
with
\beq \label{eq:x0tilde}
\widetilde x^0 \equiv
               \sqrt{\,\frac{(x^0)^2 -(x^1)^2 -(x^2)^2}{(x^0)^2}}\;\, x^0 \; .
\eeq
(For $(x^0)^2 < (x^1)^2 + (x^2)^2 $, we can effectively set $\widetilde x^0 =0$.)
The contour integral is readily performed,
\beq \label{eq:D}
D(\wt x^0 ,x^3)=(2\pi)^{-3} \int \rmd^3 p \; e^{ip_3 x^3}\left( \frac{\sin \omega_-
\wt x^0 }{\omega_- (\omega_+^2 - \omega_-^2)} - \frac{\sin \omega_+
\wt x^0}{\omega_+ (\omega_+^2 - \omega_-^2)}\right)\;,
\eeq
with the roots $\omega_\pm$ explicitly given by Eq. (\ref{eq:frequency-space}).
The integral (\ref{eq:D}) obviously vanishes for $\wt x^0 =0$
(as long as $x^3 \neq 0$), and we are
interested in determining its behavior for other values of $\wt x^0$.
We will start by demonstrating that $D(\wt x^0 ,x^3)$ at the time slice
$\wt x^0$ $=$ $0$ is ultra-local in $x^3$,
i.e. $\partial^n_{\wt x^0} D(\wt x^0 ,x^3)|_{\wt x^0 =0}$ is a sum of
derivatives of the delta function $\delta (x^3)$.

If $n$ is even, then $\partial^n_{\wt x^0} D(\wt x^0 ,x^3)|_{\wt x^0 =0}$
is obviously zero. If $n=2l+1$ is odd, then one has
\beq \label{eq:Dderivatives}
\left. \partial^{2l+1}_{\wt x^0} D(\wt x^0 ,x^3)\right|_{\wt x^0 =0} = (-1)^{l+1}
\,(2\pi)^{-3} \int \rmd^3 p \;e^{ip_3 x^3}\, \frac{(\omega_+^2)^l
-(\omega_-^2)^l}{\omega_+^2 - \omega_-^2} \;.
\eeq
Two remarks are in order. First, the fraction
in the integrand is, in fact, a polynomial in $\omega_+^2$ and
$\omega_-^2$, namely
\beq \label{eq:poly}
\frac{(\omega_+^2)^l -(\omega_-^2)^l}{\omega_+^2 - \omega_-^2}
=(\omega_+^2)^{l-1} + (\omega_+^2)^{l-2}\omega_- + \ldots +\omega_+^2
(\omega_-^2)^{l-2} + (\omega_-^2)^{l-1} \equiv P_{2l-2} \;.
\eeq
Second, if we temporarily re-express $\omega^2_\pm$ as
\beq
\omega^2_\pm = a\pm b \; ,
\eeq
where $a$ is a polynomial in the momenta $\vec p$ and $b$ is the
square-root of a polynomial, then the above polynomial (\ref{eq:poly})
only depends on even powers of $b$, $P_{2l-2}=P_{2l-2}(a,b^2)$.
The last observation follows from
the simple fact that $P_{2l-2}$ is invariant under the interchange $\omega_+^2
\leftrightarrow \omega_-^2$.
These two remarks make clear that $P_{2l-2}$ is a polynomial in the
momenta  $p_1$,  $p_2$, and, especially, $p_3$, which implies
ultra-locality.

The finite domain of vanishing $D$ may be determined by a direct
evaluation of the integral (\ref{eq:D}). The end result of a straightforward
calculation is that
\beq \label{eq:Dspacelike} D(\wt x^0 ,x^3)=0\,, \quad
{\rm for}\;\; |\widetilde x^0 |<|x^3| \; ,
\eeq
which corresponds to the
usual spacelike region $(x^0)^2$ $<$ $(x^1)^2$ $+$ $(x^2)^2$ $+$ $(x^3)^2$ ,
see Eq. (\ref{eq:x0tilde}) above. The calculation proceeds in three steps.

First, one notes that the factors $(\omega_+^2 - \omega_-^2)$
in the denominators of (\ref{eq:D}) are independent of $p_1$ and
$p_2$, so that these integrals can be readily performed for
$\wt x^0$ $\neq$ $0$,
\beq \label{eq:Dp3integral}
D(\wt x^0 ,x^3)= -\frac{1}{4\pi^2 \,\wt x^0}\int_{-\infty}^\infty \rmd p_3\;
e^{ip_3 x^3} \,
\frac{\cos \wt \omega_+ \wt x^0 -\cos \wt \omega_- \wt  x^0}
     {\wt \omega_+^2 - \wt \omega_-^2} \; ,
\eeq
where the
$\wt \omega_\pm \equiv \left. \omega_\pm \right|_{p_1^2+p_2^2 =0}$ are
given by Eq. (\ref{eq:omega-tilde}).
(To arrive at Eq. (\ref{eq:Dp3integral}) we have dropped the contribution at
$p_1^2 + p_2^2 $ $=$ $\infty$, which corresponds to a rapidly
oscillating function of $\wt x^0$ that vanishes upon integration.)
Note that the Taylor expansion of the integrand of (\ref{eq:Dp3integral})
in powers of $\wt x^0$ has
precisely the polynomials (\ref{eq:poly}) as coefficients, but now in terms
of $\wt \omega_\pm $.

Second, we replace the variable $p_3$ by $\phi$, which is defined as follows
\beq \label{eq:phi}
p_3 \equiv {\textstyle \frac{1}{2}}\, m \sinh \phi \; , \quad
\sqrt{p_3^2 +{\textstyle \frac{1}{4}}\,m^2} \equiv
{\textstyle \frac{1}{2}}\, m \cosh \phi \; .
\eeq
This change of variables eliminates the denominator
$(\tilde{\omega}_+^2 - \tilde{\omega}_-^2)$ in (\ref{eq:Dp3integral}),
so that only exponentials remain in the integrand.
(The same procedure is followed in Section 15.1 of Ref. \cite{BS} for
the standard commutator function of massive scalars.) The result is then
\beq \label{eq:Dphiintegral}
D(\wt x^0 ,x^3)= -\frac{1}{8 \pi^2 m \,\wt x^0}\int_{-\infty}^\infty \rmd
                 \phi \, e^{i p_3 x^3} \,
(\cos \wt \omega_+ \wt x^0 - \cos \wt \omega_- \wt x^0 )  \; ,
\eeq
with $p_3$ and $\tilde{\omega}_\pm$ defined in terms of $\phi$.

Third, the integral over $\phi$ can be evaluated, taking care of the relative
signs and magnitudes of $\wt x^0$ and $x_3$.
For the case of $0$  $<$ $\wt x^0$ $<$ $x^3$, we write
\beq \label{eq:phi0}
\wt x^0 \equiv \sqrt{ x^2}\sinh \phi_0 \, ,\quad x^3
\equiv \sqrt{ x^2}\cosh \phi_0  \; .
\eeq
Defining $\mu$ $\equiv$ $\frac{1}{2}m \sqrt{x^2}$,
the $\phi$ integral in (\ref{eq:Dphiintegral}) becomes after a simple manipulation
\beq \label{eq:phiintegral}
\int_{-\infty}^\infty \rmd \phi  \;
\left( i \sin \left[ \mu \sinh \phi_0 \right]\right)
\left( e^{i\mu \sinh (\phi +\phi_0)}
- e^{i\mu \sinh (\phi - \phi_0)}\right)  \; .
\eeq
The first factor in brackets is a constant which can be taken out of
the integral.
But the remaining integral of the second factor in brackets vanishes
trivially (in the second term shift $\phi$ $\rightarrow$ $\phi + 2\phi_0$).
Since the $\phi$
integral (\ref{eq:Dphiintegral}) is even in both $\wt x^0$ and $x^3$,
and the original commutator function $D$ as given by Eq. (\ref{eq:D})
manifestly vanishes for $\wt x^0$ $=$ $0$,
this establishes the result (\ref{eq:Dspacelike}) announced above.

It may also be instructive to see what
happens for the case of, say, $0< x^3 <\wt x^0$. Defining
\beq \label{eq:newphi0}
\wt x^0 \equiv \sqrt{-x^2}\cosh \phi_0 \, ,\quad x^3
\equiv \sqrt{-x^2}\sinh \phi_0  \; ,
\eeq
one now gets for the $\phi$ integral in Eq. (\ref{eq:Dphiintegral})
\beq \label{eq:newphiintegral1}
\int_{-\infty}^\infty \rmd \phi  \;
\left( i \sin \left[ \mu \cosh \phi_0 \right]\right)
 \left( e^{i\mu \cosh (\phi +\phi_0)}
- e^{-i\mu \cosh (\phi - \phi_0)}\right)  \; ,
\eeq
with $\mu$ $\equiv$ $\frac{1}{2}m  \sqrt{-x^2}$.
Taking out the constant factor and making a change of variables
$\phi$ $\rightarrow$ $\phi + 2\phi_0$ for the second term, one obtains
\beq \label{eq:newphiintegral2}
\int_{-\infty}^\infty \rmd \phi \, \left( e^{i\mu \cosh (\phi +\phi_0)}
- e^{-i\mu \cosh (\phi + \phi_0)}\right)  \; ,
\eeq
which need not vanish. Using Eq. (8.421) of Ref. \cite{GrRy1},
the integral (\ref{eq:newphiintegral2}) gives $2 \pi i J_0\left( \mu\right)$,
where the Bessel function $J_0\left( \mu\right)$ is, in general, nonzero.
All together, one has
\beq \label{eq:Dtimelike}
D(\wt x^0 ,x^3)= \frac{1}{8 \pi} \, \epsilon(\widetilde x^0)\;
                 \frac{\sin\left( \frac{1}{2}m \widetilde x^0\right)}
                      { \frac{1}{2}m \widetilde x^0}\;
                 J_0\left( {\textstyle \frac{1}{2}} m \sqrt{-x^2}\right)\,,
                 \quad {\rm for}\;\; |\widetilde x^0 |>|x^3| \; ,
\eeq
where the antisymmetry in $\widetilde x^0$ has been made
explicit.\footnote{The structure of our commutator function (\ref{eq:Dtimelike})
closely
resembles the one obtained for a CPT-violating massive Dirac fermion,
as given in Appendix E of the first paper in Ref.  \cite{CoKo1}. Note, however, that
in our case $m$ sets the scale of the CPT violation for the photons.}
For fixed timelike separation $x^\mu$ and \CS~mass scale $m \rightarrow 0$, the
commutator function approaches a constant value $\pm (8\pi)^{-1}$.
(Remark that derivatives operating
on $D$ will result in further singularities on the  null-cone $x^2=0$.)
This completes our discussion of the gauge field commutator (\ref{eq:comm})
for the case of a purely spacelike \CS~parameter $k_\mu =(0,0,0,1)$, with
microcausality established.

\subsection{Purely timelike \CS~parameter}

Let us, briefly, discuss the commutator function for the case of a purely
timelike ``vector'' $k_\mu $. In this case there is no invariant separation
of the dispersion relation into positive and negative frequency
parts (see Fig. 2).
However, as Lorentz invariance is broken anyway, we may simply choose to
quantize in the particular coordinate frame where $k_\mu =(1,0,0,0)$.
Specifically,
we want to study the degree of freedom with dispersion relation
$p_0^2$ $=$ $|\vec p|^2  - m|\vec p|$. For this degree of freedom,
the region $|\vec p|<m$
has to be excluded in order to maintain unitarity, as was mentioned
a few lines below Eq. (\ref{eq:prop-mink-time}).
The relevant commutator \cite{BS} is then
\beq \label{eq:phicommutator}
[\Phi (0),\Phi (x)]= i \, \tilde{D}(x) \; ,
\eeq
with $\Phi(x)$ the quantum field corresponding to this particular
degree of freedom of the gauge field (recall that $\hbar =c=1$)
and the commutator function
\beqa  \label{eq:comm-timelikeCS}
\tilde{D}(x) &=& \frac{1}{i (2\pi)^3}\int \rmd ^4 p
\; \delta \left(p_0^2 - |\vec p|^2  + m|\vec p|\right) \,
\theta \left(|\vec p|-m\right) \, \epsilon (p_0) \,  e^{ip\cdot x}
\no \\
&=& \frac{1}{(2\pi)^3}\int \rmd ^3 p \;  \frac{\theta (|\vec p|-m)}{\sqrt{
|\vec p|^2  - m|\vec p|}} \;  \sin \left( \sqrt{|\vec p|^2  - m|\vec p|}\; x^0
\right) \,  e^{i\vec x\cdot \vec p} \; .
\eeqa
Here, $\epsilon(x)$ $\equiv$ $x / |x|$ and  $\theta$ is
the usual step function, $\theta(x)$ $=$ $0$ for $x$ $<$ $0$ and
$\theta(x)$ $=$ $1$ for $x$ $>$ $0$.
We will now demonstrate that microcausality is violated for this
commutator function, i.e. $\tilde{D}(x) \ne 0$ somewhere in the
spacelike region $|x^0| <|\vec x|$.

If $\tilde{D}(x)$ were to vanish for $|x^0| <|\vec x|$, this would imply that
$\mathcal{D}(x^0,x^3)$ $\equiv$ $\int \rmd x^1 \rmd x^2 \tilde{D}(x)$
$=$ $0$ for $|x^0| <| x^3|$.
So, let us show that the latter relation is violated.
If $\mathcal{D}(x^0,x^3)$, in turn, were to vanish for
$|x^0| <| x^3|$, this would imply that
$\partial_0 \mathcal{D}(x^0, x^3)|_{x^0 =0}$ had to be an ultra-local expression
in $x^3$, but it can be easily checked that this is not the case. Indeed,
we calculate
\bdi
\left. \partial_0 \, \mathcal{D}(x^0,x^3)\right|_{x^0 =0}
\edi
\bdi
= \left. \partial_0
\int_{-\infty}^{\infty} \frac{\rmd p_3}{2\pi}  \,
\frac{\theta (| p_3|-m)}{\sqrt{
|p_3|^2 - m| p_3|}}  \; \sin \left( \sqrt{|p_3|^2 - m| p_3|}\; x^0 \right)
e^{i x^3 p_3}\right|_{x^0 =0}
\edi
\beq
=\int_{-\infty}^{\infty} \frac{\rmd p_3}{2\pi}  \, \theta (| p_3|-m) \,
e^{i x^3 p_3}
= \delta (x^3)-\frac{\sin mx^3}{\pi x^3} \; ,
\eeq
which can be nonzero for finite $x^3$, as long as $m\ne 0$.
(Recall that for purely spacelike $k_\mu =(0,0,0,1)$
we have shown the ultra-locality of
Eq. (\ref{eq:Dderivatives}) above.)
For the case of a purely timelike \CS~parameter $k_\mu $,
the commutator (\ref{eq:phicommutator}) thus violates microcausality.


\section{Summary and discussion}

Lorentz- and CPT-violating field theories might emerge as low-energy
effective theories of a more fundamental theory,
where Lorentz and CPT symmetry are broken
spontaneously or dynamically \cite{CoKo1}.
Alternatively, these symmetry-breaking theories might result
from a quantum anomaly within the realm of quantum field theory itself
\cite{Kli1}. In both
instances, the question arises whether or not these Lorentz- and
CPT-violating theories are valid quantum field theories, that is,
whether or not a consistent quantization is possible.
For theories which contain an Abelian Chern--Simons-like term
(\ref{eq:CS-like}), issues like power-counting
renormalizability and conservation of energy-momentum have already
been discussed in Ref. \cite{CoKo1}, where it was
demonstrated that these features continue to hold.

In this paper, we have focused on the issues of unitarity and causality for
theories containing an Abelian Chern--Simons-like term (\ref{eq:CS-like}).
The results found strongly depend on
whether the preferred direction $k_\mu$ of the Lagrangian term
(\ref{eq:CS-like}) is spacelike or timelike.

For a purely spacelike \CS~parameter $k_\mu$ $=$ $(0, \vec k)$, our results
are certainly encouraging for the issue of quantization.
By investigating the dispersion relations we have found in Section 2
that a universal,
direction-in\-de\-pen\-dent signal propagation speed $c$ can still be
defined (in this paper, we have chosen units such that $c=1$). In addition,
the group velocity is less or equal to $c$. This suggests that the CPT-violating
term in Eq. (\ref{eq:lagrangian}) does not change the causal structure of
spacetime, but rather acts like a medium with a direction-dependent dispersion
for the field excitations (e.g. the photons).

In fact, the anisotropic propagation of the circular polarization modes makes
clear that the Abelian \CS-like term for purely spacelike $k_\mu$ $=$ $(0,
\vec k)$ is T-odd (and P- and C-even). According to the dispersion relation
(\ref{eq:frequency-space}) for \CS~parameter $\vec k$ $=$ $(0,0,1)$,
left-handed wave packets propagating in different directions along, say, the
$x^2$-axis (with an infinitesimal $p_3$-component added) have unequal group
velocities, $|\,\vec v^{\,L}_{\rm g}(0,p_2,\delta p_3)|$ $\neq$
$|\,\vec v^{\,L}_{\rm g}(0,-p_2,-\delta p_3)|$, as long as $m \neq 0$.
The physics is thus
non-invariant under time reversal, which flips the momentum, $\vec p$
$\rightarrow$ $-\vec p$, and preserves the helicity, ${L}$ $\rightarrow$
${L}$, cf. Ref. \cite{W96}.

We have also found from the dispersion relations that a separation into
positive and negative frequency modes is still possible for purely
spacelike $k_\mu$.
Therefore, particles and antiparticles may be defined and the
field may be quantized in the usual fashion \cite{Heitler}--\cite{BS}.
As shown in Section 5.1, the resulting commutators of the electric and
magnetic fields vanish for spacelike separations, which demonstrates
that microcausality holds for the potential quantum field theory based on the
Lagrangian (\ref{eq:lagrangian}). This result is not quite trivial.
Recall, for example, that the CPT-violating theories of Ref. \cite{C67},
with self-conjugate bosons of
odd-half-integer isospin, fail precisely in this respect, because certain
fields do not commute outside the light-cone.
The well-known result \cite{Jost} that
microcausality, Lorentz invariance
and the existence of a unique vacuum state imply CPT invariance
does not contradict our result, because  in  the Lagrangian (\ref{eq:lagrangian})
Lorentz invariance is broken as well ($k_\mu$ is fixed once and for all).

In addition, we have demonstrated in Section 3 that a Feynman
propagator for the relevant Lagrangian (\ref{eq:lagrangian}) can be
defined in Minkowski spacetime with the usual $i\epsilon$ prescription
and that this propagator can be Wick-rotated to Euclidean space.
As shown in Section 4, reflection
positivity  holds for the Euclidean Feynman propagator.
Again, this requires the choice of a (purely) spacelike $k_\mu$.

For both a classical and a quantum treatment,
the causal structure of the \MCS~theory (\ref{eq:lagrangian}) thus
remains unaltered by the inclusion of the CPT-violating term, provided
that the \CS~parameter $k_\mu$ in Eq. (\ref{eq:CS-like}) is
spacelike. This suggests that a CPT- and Lorentz-symmetry-violating
theory like (\ref{eq:lagrangian}) may lead to a consistent local quantum
field theory. If so,  the particular chiral gauge field theories discussed in
Ref. \cite{Kli1}, which display the remarkable phenomenon of a CPT anomaly,
could perhaps be realized in nature (see also the remarks below).

On the other hand, a consistent quantization for
a timelike \CS~parameter $k_\mu$ does not seem to be possible.
As noted in Section 3, the presence of imaginary energies at low
momenta requires the exclusion of these momenta if unitarity is to be
maintained.\footnote{Moreover, tachyon pair production would destabilize
the perturbative vacuum state, see Section 4 of Ref. \cite{ASS1}.}
But we have seen in Section 5.2 that this exclusion leads to a violation of
microcausality. It is, therefore, not possible to maintain both unitarity and
causality. In fact, these results are just the quantum
analogs of the results of Ref. \cite{CFJ}.
The authors of that paper have pointed out that the Green's functions of
the classical equations of motion resulting from the Lagrangian
(\ref{eq:lagrangian}) either are causal but with exponential growth in
time, or without exponential growth but noncausal. For timelike $k_\mu$,
there are no Green's functions that are both
causal (i.e. propagating signals only into the future) and without exponential
growth. As shown in Section 4 of this paper,
reflection positivity in the Euclidean formulation
is indeed violated for the Feynman propagator with \CS~parameter
$k_\mu$ $=$ $(1,0,0,0)$.

Hence, the Abelian \CS-like term (\ref{eq:CS-like}) for timelike
$k_\mu$, which contains a P-odd
(and C- and T-even) part, does not allow for a consistent quantization.
It does appear, however,
that a \CS-like term could play a role for T-violation (leaving C- and
P-invariance intact), provided the parameter $k_\mu$ is (purely) spacelike.
Such a \CS-like term would, in fact,  provide a ``fundamental arrow-of-time,''
cf. Ref. \cite{Wal1}. (Recall the Gedankenexperiment with the left-handed wave
packet presented at the beginning of this section.)
This problem is currently under investigation.

As briefly mentioned in Ref. \cite{Kli1}, the birefringence of a
photonic \CS-like term (\ref{eq:CS-like})  with purely spacelike
parameter $k_\mu$ could also affect the polarization of the \cmb.
The expected polarization pattern \cite{CCT}
around temperature hot- and cold-spots would be modified, due to the
action of the \CS-like term on the photons traveling between the last-scattering
surface (redshift $z$ $\sim$ $10^3$) and the detector ($z$ $=$ $0$).
Future satellite experiments such as NASA's Microwave Anisotropy Probe and
ESA's Planck Surveyor could look for this effect,
see Ref. \cite{L98} for further details.

Let us end this section with three somewhat more theoretical issues.
The first issue concerns stability, which has been discussed recently
for a theory of a massive Dirac fermion with (spontaneous) Lorentz and CPT
breaking \cite{KL00}. For the \MCS~theory (\ref{eq:lagrangian}),
the photon is, of course, stable. This holds, in particular, for the case of
purely spacelike \CS~parameter $k_\mu$.
But even in the context of the
CPT anomaly \cite{Kli1}, with all chiral fermions integrated out, the effective
action can be expected to have additional quartic (and higher-order)
interaction terms for the photons. A rough estimate suggests an extremely small,
but nonzero, effect for the decay of the photon, cf. Eq. (\ref{eq:mCPT}).
(The photons of the \cmb~would not be affected significantly.) Whether or not
photon decay is physically acceptable remains an open question, though.

The second issue concerns the case of a ``lightlike'' \CS~parameter
$k_\mu$, which has not been discussed so far. No imaginary energies appear
for ``lightlike'' $k_\mu$. There is, therefore, no obvious obstacle against
quantization, as was the case for timelike  \CS~parameters.
But explicit calculations of the type performed in Sects. 4 and 5 are
hampered by the complicated pole structure for the ``lightlike'' case.
For the CPT anomaly, there may still be a problem with the
``lightlike'' theory, as will become clear shortly.

The third, and final, issue concerns the possible implications of our
microcausality results for the case of an anomalous origin of the Lagrangian
(\ref{eq:lagrangian}) considered here.
As mentioned several times by now, it has been shown in Ref. \cite{Kli1} that
certain chiral gauge field theories defined over the Euclidean spacetime
manifold $M$ $=$ $\mathbb{R}^3$ $\times$ $S^1$, for example,
could give rise to a CPT anomaly of
the form of a Chern--Simons-like term (\ref{eq:CS-like}), with an additional
factor $i$ for the Euclidean signature of the metric.
(In the context of the CPT anomaly, the exponential of the
integrated Euclidean \CS-like term appears as the phase factor of the
chiral determinant, whereas the absolute value of the chiral determinant is CPT
invariant.) In that case, the specific parameters of Eq. (\ref{eq:CS-like})
would be $m$ $=$ $(2\,n+1)\,\alpha /L$ and $k_\mu$ $=$ $(0,0,0,1)$,
with the integer $n$ defining the theory in the ultraviolet and
the nonzero entry of $k_\mu$ corresponding to the single compact
dimension of length $L$.

The microcausality results of the present paper then imply that this compact
dimension should correspond to a spatial direction after the Wick rotation
to the Lorentzian signature of the metric ($x^3 \in S^1$ becoming a spatial
coordinate and $x^4 \in \mathbb{R}$, say, the time coordinate).
If, on the other
hand, the compact dimension were to correspond to the time direction
($x^4 \in S^1$), our results would lead us to expect
problems with unitarity or causality. Indeed, we would then have started from a
spacetime manifold with closed timelike curves, which has
a built-in violation of what might be called ``macrocausality,''
cf. Section 8.2 of Ref. \cite{Wal2}.\footnote{Another spacetime manifold with
closed timelike curves would be, for example,
$M$ $=$ $S^1$ $\times$ $\mathbb{R}^2$ $\times$ $S^1$, with periodic
coordinates $x^4$ $\in$ $[0,L_4]$ and  $x^3$ $\in$ $[0,L_3]$
and noncompact coordinates $x^1,x^2$ $\in$ $\mathbb{R}$.
The corresponding effective action could then contain  a \CS-like term
(\ref{eq:CS-like}) with parameter $k_\mu$ $\propto$ $(L_4^{-1},0,0,L_3^{-1})$.
For $L_4$ $=$ $L_3$ exactly, the  \CS~parameter $k_\mu$ would be
``lightlike.''
Note also that the transformation of $k_\mu$ as given by Eq. (\ref{eq:kmu-prime})
may not be applicable in general for this manifold (compare with Eq. (13) of
Ref. \cite{Kli2}).} Still, the proper fundamental theory
(gauge fields and chiral fermions over a spacetime manifold with a
separable compact dimension that is spacelike) and
the corresponding effective theory (gauge fields with a \CS-like term for purely
spacelike parameter $k_\mu$) appear to be consistent as far as causality is
concerned.

\section*{Acknowledgements}
It is a pleasure to thank the anonymous referee for useful comments.


\begin{appendix}

\section{Commutators for the Coulomb gauge}
% new type of equation numbers for the Appendix A
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}

In this appendix, we first discuss the equations of motion of the
Lagrangian (\ref{eq:lagrangian}) for purely spacelike
\CS~parameter $k_\mu$ $=$ $(0, \vec k)$, but with the general axial gauge
replaced by the Coulomb gauge $\vec \partial \cdot \vec A =0$. From these
equations of motion, we then determine the
``tensor'' structure $T_{ij}$ of the gauge field commutator
(\ref{eq:comm}). Finally, we calculate the commutators of the electric and
magnetic fields.

The equations of motion in momentum space are (with the conventions
$g_{\mu\nu}$ $=$ $\mathrm{diag}$ $(-1,1,1,1)$,
$\epsilon_{0123}=\epsilon_{123}=1$, and using $\vec p
\cdot \vec A =0$)
\beq
(p_0^2 -|\vec p|^2 )g^{\mu\nu}A_\nu -p^\mu p_0 A_0 +im\epsilon^{\mu\nu\rho
  l}A_\nu p_\rho k_l =0 \; .
\eeq
This leads to a nondynamical equation for $A_0$,
\beq \label{eq:eom-A0}
A_0 =i\,\frac{m}{|\vec p|^2}\,\epsilon_{abc}A_a p_b k_c \;,
\eeq
and dynamical equations for $A_i$,
\beqa  \label{eq:eom-Ai}
M_{ij}A_j &\equiv& \left( \left[p_0^2 -|\vec p|^2 -m^2 \,\frac{|\vec p|^2 |\vec k|^2
  -(\vec p \cdot \vec k)^2}{|\vec p|^2}\right]\delta_{ij} \right.
\no \\
&& \left. -imp_0
\left[\frac{p_i}{|\vec p|^2}\,\epsilon_{jab}p_a k_b + \epsilon_{ija}k_a \right] -m^2
\,\frac{\vec p \cdot \vec k}{|\vec p|^2}\, p_i k_j +m^2 k_i k_j\right) A_j =0 \; .
\eeqa

The gauge field commutator (\ref{eq:comm}) will obey the equations of motion
(\ref{eq:eom-Ai}) provided that
\beq \label{eq:eom-tensor}
M_{ij}T_{jk}(p) = \pi_{ik}\,D^{-1}(p)  \; ,
\eeq
with
\beq
D^{-1}(p)=(p^2)^2 +m^2 \left(p^2 |\vec k|^2 -(\vec p\cdot \vec k)^2\right) \; .
\eeq
(The tensor structure on the right-hand side of Eq. (\ref{eq:eom-tensor})
could, in principle,
be more complicated, but turns out to be just $\pi_{ik}$.) With some effort,
it can be verified that (\ref{eq:eom-tensor}) holds for the $T_{ij}$ as given by Eq.
(\ref{eq:Tij}).

The magnetic fields then have  the following commutator:
\beqa
[B_i ,B_j](p) &=&i^2 \epsilon_{iab}p_a \epsilon_{jcd}(-p_c) T_{bd}\, i \,D(p)
=\Big( (p_0^2 -|\vec p|^2 )(\delta_{ij} |\vec p|^2 -p_i p_j )
\no \\
&&
+imp_0 (\vec p\cdot \vec k)\epsilon_{ijl}p_l
-m^2
\{\delta_{ij} (|\vec p|^2 |\vec k|^2 -(\vec p\cdot \vec k)^2)
\no \\
&&\left. -p_i p_j |\vec k|^2
-k_i k_j |\vec p|^2 +(p_i k_j + k_i p_j)(\vec p\cdot \vec k)\}\right)
\, i \, D(p) \; .
\eeqa
The electric fields are more involved,
\beq
\vec E(x)=
\partial_0 \vec A (x) - m\int d^3 z \,\Delta^{-1}(\vec x -\vec z)\; \vec
\partial^z \epsilon_{abc}\, k_a\, \partial_b^z \,A_c (x^0, \vec z) \; ,
\eeq
and one finds after some algebra the following commutators:
\beqa
[E_i ,B_j ](p)&=&\Bigl( (p_0^2 -|\vec p|^2 ) \epsilon_{ijl} p_l p_0 -m^2 p_0
k_i \epsilon_{jab} p_a k_b -imp_0^2 (\vec p\cdot \vec k) \delta_{ij}
\no \\
&&
+ im(\vec p\cdot \vec k) p_i
p_j +im (p_0^2 -|\vec p|^2 )p_i k_j \Bigr) \, i \, D(p) \;,
\eeqa
\beqa
[E_i ,E_j ](p)&=&\Big( (p_0^2 -|\vec p|^2 )\delta_{ij} p_0^2 -(p_0^2 -|\vec p|^2)
p_i p_j
\no \\
&&
-m^2 p_0^2 k_i k_j + imp_0^3 \epsilon_{ijl}k_l +imp_0
(p_i\epsilon_{jab}p_a k_b - p_j \epsilon_{iab} p_a k_b)
\no \\
&&
- \frac{p_i p_j}{|\vec p|^2}\, \{(|\vec p|^2 - p_0^2 )^2 +m^2 ((|\vec p|^2 -p_0^2
)|\vec k|^2 -(\vec p \cdot \vec k)^2)\}\Big)\, i \, D(p) \; .
\eeqa
Apparently, the electric field commutator contains a term with a
$|\vec p|^{-2}$ pole. But this term is multiplied by precisely the function
$D^{-1}(p)$, which cancels the over-all factor $D(p)$. Therefore, this term
does not contribute to the contour integral (\ref{eq:comm-func-space}),
and we reproduce the commutators (\ref{eq:comm-e})
-- (\ref{eq:comm-b}) as given in the main text.
\end{appendix}


\begin{thebibliography}{99}
\bibitem{Wal1}
R.M. Wald, Phys. Rev. D 21 (1980) 2742.
\bibitem{Haw1}
S.W. Hawking, Phys. Rev. D 32 (1985) 2489.
\bibitem{KoPo1}
V.A. Kosteleck\'{y},  R. Potting, Nucl. Phys. B 359 (1991) 545.
\bibitem{Kli1}
F.R. Klinkhamer, Nucl. Phys. B 578 (2000) 277.
\bibitem{Kli2}
F.R. Klinkhamer,  J. Nishimura, Phys. Rev. D 63 (2001) 097701.
\bibitem{CFJ}
S.M. Carroll, G.B. Field,  R. Jackiw, Phys. Rev. D 41 (1990) 1231.
\bibitem{CoKo1}
D. Colladay,  V.A. Kosteleck\'{y}, Phys. Rev. D 55 (1997) 6760;
D 58 (1998) 116002.
\bibitem{CoGl1}
S. Coleman,  S.L. Glashow, Phys. Rev. D 59 (1999) 116008.
\bibitem{JaKo1}
R. Jackiw,  V.A. Kosteleck\'{y}, Phys. Rev. Lett. 82 (1999) 3572.
\bibitem{CCGF}
M. Chaichian, W.F. Chen,  R. Gonzales Felipe, Phys. Lett. B 503 (2001) 215.
\bibitem{Jost}
R. Jost, Helv. Phys. Acta 30 (1957) 409.
\bibitem{SW64}
R. Streater, A. Wightman, \emph{PCT, Spin and Statistics, and All That},
Benjamin, New York, 1964.
\bibitem{C67}
P. Carruthers, Phys. Rev. Lett. 18 (1967) 353; Phys. Lett. B 26 (1968) 158.
\bibitem{CS74}
S. Chern,  J. Simons, Ann. Math. 99 (1974) 48.
\bibitem{Leib}
G. Leibbrandt, \emph{Noncovariant gauges}, World Scientific, Singapore,
1994.
\bibitem{NR1}
B. Nodland,  J.P. Ralston, Phys. Rev. Lett. 78 (1997) 3043.
\bibitem{WPC}
J.F.L. Wardle, R.A. Perley,  M.H. Cohen, Phys. Rev. Lett. 79 (1997) 1801.
\bibitem{B60}
L. Brillouin, \emph{Wave Propagation and Group Velocity},
Academic, New York, 1960.
\bibitem{ASS1}
A.A. Andrianov, R. Soldati,  L. Sorbo, Phys. Rev. D 59 (1999) 025002.
\bibitem{MM1}
I. Montvay,  G. Munster, \emph{Quantum Fields on a Lattice}, Cambridge
U.P., Cambridge, 1994.
\bibitem{V94}
M. Veltman, \emph{Diagrammatica--The Path to Feynman Rules},
Cambridge U.P., Cambridge, 1994.
\bibitem{W96}
S. Weinberg, \emph{The Quantum Theory of Fields I},
Cambridge U.P., Cambridge, 1996.
\bibitem{DhSu1}
J. Dhar,  E. Sudarshan, Phys. Rev. 174 (1968) 1808.
\bibitem{JTW}
T. Jacobson, N.C. Tsamis,  R.P. Woodard, Phys. Rev. D 38 (1988) 1823.
\bibitem{OS1}
K. Osterwalder,  R. Schrader, Comm. Math. Phys. 31 (1973) 83;
42 (1975) 281.
\bibitem{GrRy1}
I.S. Gradshteyn,  I.M. Ryzhik, \emph{Table of Integrals, Series and
Products}, Academic, New York, 1980.
\bibitem{Heitler}
W. Heitler, \emph{The Quantum Theory of Radiation}, 3rd ed.,
Oxford U.P., London, 1954.
\bibitem{JR}
J.M. Jauch,  F. Rohrlich, \emph{The Theory of Photons and Electrons},
2nd ed., Springer, New York, 1976.
\bibitem{BS}
N.N. Bogoliubov,  D.V. Shirkov, \emph{Introduction to the Theory of Quantized
Fields}, Wiley, New York, 1959.
\bibitem{CCT}
D. Coulson, R. Crittenden, N. Turok, Phys. Rev. Lett. 73 (1994) 2390.
\bibitem{L98}
N.F. Lepora, arXiv:gr-qc/9812077;
M. Kamionkowski, A. Kosowsky, Ann. Rev. Nucl. Part. Sci. 49 (1999) 77.
\bibitem{KL00}
V.A. Kosteleck\'{y}, R. Lehnert, Phys. Rev. D 63 (2001) 065008.
\bibitem{Wal2}
R.M. Wald, \emph{General Relativity}, Chicago U.P., Chicago, 1984.
\end{thebibliography}

\newpage
\input psbox.tex
\begin{figure}
$$ \psboxscaled{1000}{causalfig1.ps} $$
\caption{The dispersion relation (\ref{eq:disprel-space})
in the $(p_0,p_3)$ plane for a purely spacelike \CS~parameter $k_\mu
=(0,0,0,1)$ and mass scale $m$, with broken (solid) curves corresponding
to the plus (minus) sign in Eq. (\ref{eq:disprel-space}).}
\end{figure}

\begin{figure}
$$ \psboxscaled{1000}{causalfig2.ps} $$
\caption{The dispersion relation (\ref{eq:frequency-time})
in the $(\mathrm{Re}\, p_0,|\vec p|)$ halfplane for a purely timelike
\CS~parameter $k_\mu = (1,0,0,0)$ and mass scale $m$, with broken (solid)
curves corresponding to the plus (minus) sign in Eq. (\ref{eq:frequency-time}).
For the minus sign in Eq. (\ref{eq:frequency-time}), the energy $p_0$
becomes imaginary at low momenta $|\vec p|<m$.}
\end{figure}


\end{document}





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%       %               %       %     %  %     %   %   %
%       %         %%%%%%        %%%%%%    %%%%%   %     %
%
%       By Jean Orloff
%       Comments & suggestions by e-mail: ORLOFF@surya11.cern.ch
%       No modification of this file allowed if not e-sent to me.
%
% WHAT IS IT:
% psbox is a set of machine-independent TeX macros to
% 1) allow (Encapsulated) PostScript figure inclusion in all versions
%    of TeX (Plain, LaTeX) on all machines using a PostScript printer
% 2) facilitate the communication (e-mail, ftp, ...) of all the files
%    (text, macros, figs) needed to reproduce a TeX document by grouping
%    them together into a single, TeXable file.
% For more info, get the file pub/TeX/psbox/PSBOXALL.TEX by anonymous
%     ftp from cs.nyu.edu(=128.122.140.24)
%
% History:
%  1.34  \readfilename=final fix for all filename scans; try \psforptips
%  1.33: corrects \psnewinput for LaTeX (still fails if fname=a{b}c)
%  1.32: corrects \psfordvialw and adds .TEX to PSBOXALL(!)
%  1.31: adds \psfordvialw(?)
%  1.30: adds \splitfile & \joinfiles for multi-file management
%  1.24: fix error handling & add \psonlyboxes
%  1.22: makes \drawingBox \global for use in Phyzzx
%  1.21: accepts %%BoundingBox: (atend)
%  1.20: tries to add \psfordvitps for the TeXPS package.
%  1.10: adds \psforoztex, error handling...
%2345678 1 2345678 2 2345678 3 2345678 4 2345678 5 2345678 6 2345678 7 23456789
%
% Checking version no to avoid multiple loadings
\def\temp{1.34}%
\let\tempp=\relax
\expandafter\ifx\csname psboxversion\endcsname\relax
  \message{PSBOX(\temp) loading}%
\else
    \ifdim\temp cm>\psboxversion cm
      \message{PSBOX(\temp) loading}%
    \else
      \message{PSBOX(\psboxversion) is already loaded: I won't load
        PSBOX(\temp)!}%
      \let\temp=\psboxversion
      \let\tempp=\endinput
    \fi
\fi
\tempp
\let\psboxversion=\temp
\catcode`\@=11
% Every macro likes a little privacy...
%
%Trying to tame the variety of \special commands for Postscript: the
%  universal internal command \PSspeci@l##1##2 takes ##1 to be the
%  filename and ##2 to be the integer scale factor*1000 (as for usual
%   TeX \scale commands)
%
\def\psfortextures{%     For TeXtures on the Macintosh
%-----------------
\def\PSspeci@l##1##2{%
\special{illustration ##1\space scaled ##2}%
}}%
\def\psfordvitops{%      For the DVItoPS converter on IBM mainframes
%----------------
\def\PSspeci@l##1##2{%
\special{dvitops: import ##1\space \the\drawingwd \the\drawinght}%
}}%
\def\psfordvips{%      For DVIPS converter on VAX, UNIX and PC's
%--------------
\def\PSspeci@l##1##2{%
%    \special{/@scaleunit 1000 def}% never read dox without trying!
\d@my=0.1bp \d@mx=\drawingwd \divide\d@mx by\d@my% BUG! for large \drawingwd
\special{PSfile=##1\space llx=\psllx\space lly=\pslly\space%
urx=\psurx\space ury=\psury\space rwi=\number\d@mx
}}}%
\def\psforoztex{%        For the OzTeX shareware on the Macintosh
%--------------
\def\PSspeci@l##1##2{%
\special{##1 \space
      ##2 1000 div dup scale
      \number-\psllx\space \number-\pslly\space translate
}}}%
\def\psfordvitps{%       From the UNIX TeXPS package, vers.>3.12
%---------------
% Convert a dimension into the number \psn@sp (in scaled points)
\def\psdimt@n@sp##1{\d@mx=##1\relax\edef\psn@sp{\number\d@mx}}
\def\PSspeci@l##1##2{%
% psfig.psr contains the def of "startTexFig": if you can locate it
% and include the correct pathname, it should work
\special{dvitps: Include0 "psfig.psr"}% contains def of "startTexFig"
\psdimt@n@sp{\drawingwd}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\drawinght}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psllx bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\pslly bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psurx bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psury bp}
\special{dvitps: Literal "\psn@sp\space startTexFig\space"}
\special{dvitps: Include1 "##1"}
\special{dvitps: Literal "endTexFig\space"}
}}%
\def\psfordvialw{%   Try for dvialw, a UNIX public domain
%---------------
\def\PSspeci@l##1##2{
\special{language "PostScript",
position = "bottom left",
literal "  \psllx\space \pslly\space translate
  ##2 1000 div dup scale
  -\psllx\space -\pslly\space translate",
include "##1"}
}}%
\def\psforptips{%   For MS-DOS; LUOMA@brandeis.bitnet
%---------------
\def\PSspeci@l##1##2{{
\d@mx=\psurx bp
\advance \d@mx by -\psllx bp
\divide \d@mx by 1000\multiply\d@mx by \xscale
\incm{\d@mx}
\let\tmpx\dimincm
\d@my=\psury bp
\advance \d@my by -\pslly bp
\divide \d@my by 1000\multiply\d@my by \xscale
\incm{\d@my}
\let\tmpy\dimincm
\d@mx=-\psllx bp
\divide \d@mx by 1000\multiply\d@mx by \xscale
\d@my=-\pslly bp
\divide \d@my by 1000\multiply\d@my by \xscale
\at(\d@mx;\d@my){\special{ps:##1 x=\tmpx, y=\tmpy}}
}}}%
\def\psonlyboxes{%     Draft-like behaviour if none of the others works
%---------------
\def\PSspeci@l##1##2{%
\at(0cm;0cm){\boxit{\vbox to\drawinght
  {\vss\hbox to\drawingwd{\at(0cm;0cm){\hbox{({\tt##1})}}\hss}}}}
}}%
\def\psloc@lerr#1{%
\let\savedPSspeci@l=\PSspeci@l%
\def\PSspeci@l##1##2{%
\at(0cm;0cm){\boxit{\vbox to\drawinght
  {\vss\hbox to\drawingwd{\at(0cm;0cm){\hbox{({\tt##1}) #1}}\hss}}}}
\let\PSspeci@l=\savedPSspeci@l% restore normal output for other figs!
}}%
%\def\psfor...  add your own!
%
% Some common defs
%
\newread\pst@mpin
\newdimen\drawinght\newdimen\drawingwd
\newdimen\psxoffset\newdimen\psyoffset
\newbox\drawingBox
\newcount\xscale \newcount\yscale \newdimen\pscm\pscm=1cm
\newdimen\d@mx \newdimen\d@my
\newdimen\pswdincr \newdimen\pshtincr
\let\ps@nnotation=\relax
{\catcode`\|=0 |catcode`|\=12 |catcode`|%=12 |catcode`~=12
|catcode`#=12 |catcode`*=14
|xdef|backslashother{\}*
|xdef|percentother{%}*
|xdef|tildeother{~}*
|xdef|sharpother{#}*
}%
% useful to display special chars in \tt; fails for \,#,%
\def\R@moveMeaningHeader#1:->{}%
\def\uncatcode#1{%
\edef#1{\expandafter\R@moveMeaningHeader\meaning#1}}%
%
\def\execute#1{#1}% NOT stupid: cs in #1 are then identified BEFORE execution
\def\psm@keother#1{\catcode`#112\relax}% borrowed from latex
\def\executeinspecs#1{%
\execute{\begingroup\let\do\psm@keother\dospecials\catcode`\^^M=9#1\endgroup}}%
\def\@mpty{}%
% \if\matchin#1#2<=> \iftrue if #1 contains #2, <=>\iffalse otherwise:
% \if\matchexpin: idem, but #1 & #2 are first fully expanded (no \if
% inside!)
% \tmpa & \tmpb contain what's before and after the occurence of #2
\def\matchexpin#1#2{
  \fi%
%\message{(#1>#2)}
  \edef\tmpb{{#2}}%
  \expandafter\makem@tchtmp\tmpb%
  \edef\tmpa{#1}\edef\tmpb{#2}%
  \expandafter\expandafter\expandafter\m@tchtmp\expandafter\tmpa\tmpb\endm@tch%
  \if\match%
}%
\def\matchin#1#2{%
  \fi%
  \makem@tchtmp{#2}%
  \m@tchtmp#1#2\endm@tch%
  \if\match%
}%
\def\makem@tchtmp#1{\def\m@tchtmp##1#1##2\endm@tch{%
  \def\tmpa{##1}\def\tmpb{##2}\let\m@tchtmp=\relax%
  \ifx\tmpb\@mpty\def\match{YN}%
  \else\def\match{YY}\fi%
}}%
% converts any dimen in cm, with 1E-4 cm precision
\def\incm#1{{\psxoffset=1cm\d@my=#1
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \xdef\dimincm{\number\d@mx.}
  \advance\d@my by -\number\d@mx cm
  \multiply\d@my by 100
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \edef\dimincm{\dimincm\number\d@mx}
  \advance\d@my by -\number\d@mx cm
  \multiply\d@my by 100
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \xdef\dimincm{\dimincm\number\d@mx}
}}%
%
%  \ReadPSize{PSfilename} reads the dimensions of a PostScript drawing
%      and stores it in \drawinght(wd)
\newif\ifNotB@undingBox
\newhelp\PShelp{Proceed: you'll have a 5cm square blank box instead of
your graphics (Jean Orloff).}%
\def\s@tsize#1 #2 #3 #4\@ndsize{
  \def\psllx{#1}\def\pslly{#2}%
  \def\psurx{#3}\def\psury{#4}%  needed by a crazyness of dvips!
  \ifx\psurx\@mpty\NotB@undingBoxtrue% this is not a valid one!
  \else
    \drawinght=#4bp\advance\drawinght by-#2bp
    \drawingwd=#3bp\advance\drawingwd by-#1bp
%  !Units related by crazy factors as bp/pt=72.27/72 should be BANNED!
  \fi
  }%
\def\sc@nBBline#1:#2\@ndBBline{\edef\p@rameter{#1}\edef\v@lue{#2}}%
\def\g@bblefirstblank#1#2:{\ifx#1 \else#1\fi#2}%
{\catcode`\%=12
\xdef\B@undingBox{%%BoundingBox}}%
%% is not a true comment in PostScript, even if % is!
\def\ReadPSize#1{
 \readfilename#1\relax
 \let\PSfilename=\lastreadfilename
 \openin\pst@mpin=#1\relax
 \ifeof\pst@mpin \errhelp=\PShelp
   \errmessage{I haven't found your postscript file (\PSfilename)}%
   \psloc@lerr{was not found}%
   \s@tsize 0 0 142 142\@ndsize
   \closein\pst@mpin
 \else
% each entry in \GlobalInputList should be unique
   \if\matchexpin{\GlobalInputList}{, \lastreadfilename}%
   \else\xdef\GlobalInputList{\GlobalInputList, \lastreadfilename}%
     \immediate\write\psbj@inaux{\lastreadfilename,}%
   \fi%
   \loop
     \executeinspecs{\catcode`\ =10\global\read\pst@mpin to\n@xtline}%
     \ifeof\pst@mpin
       \errhelp=\PShelp
       \errmessage{(\PSfilename) is not an Encapsulated PostScript File:
           I could not find any \B@undingBox: line.}%
       \edef\v@lue{0 0 142 142:}%
       \psloc@lerr{is not an EPSFile}%
       \NotB@undingBoxfalse
     \else
       \expandafter\sc@nBBline\n@xtline:\@ndBBline
       \ifx\p@rameter\B@undingBox\NotB@undingBoxfalse
         \edef\t@mp{%
           \expandafter\g@bblefirstblank\v@lue\space\space\space}%
         \expandafter\s@tsize\t@mp\@ndsize
       \else\NotB@undingBoxtrue
       \fi
     \fi
   \ifNotB@undingBox\repeat
   \closein\pst@mpin
 \fi
\message{#1}%
}%
%
% \psboxto(xdim;ydim){psfilename}: you specify the dimensions and
%    TeX uniformly scales to fit the largest one. If xdim=0pt, the
%    scale is fully determined by ydim and vice versa.
%    Notice: psboxes are a real vboxes; couldn't take hbox otherwise all
%    indentation and all cr's would be interpreted as spaces (hugh!).
%
\def\psboxto(#1;#2)#3{\vbox{%
   \ReadPSize{#3}%
   \advance\pswdincr by \drawingwd
   \advance\pshtincr by \drawinght
   \divide\pswdincr by 1000
   \divide\pshtincr by 1000
   \d@mx=#1
   \ifdim\d@mx=0pt\xscale=1000
         \else \xscale=\d@mx \divide \xscale by \pswdincr\fi
   \d@my=#2
   \ifdim\d@my=0pt\yscale=1000
         \else \yscale=\d@my \divide \yscale by \pshtincr\fi
   \ifnum\yscale=1000
         \else\ifnum\xscale=1000\xscale=\yscale
                    \else\ifnum\yscale<\xscale\xscale=\yscale\fi
              \fi
   \fi
   \divide\drawingwd by1000 \multiply\drawingwd by\xscale
   \divide\drawinght by1000 \multiply\drawinght by\xscale
   \divide\psxoffset by1000 \multiply\psxoffset by\xscale
   \divide\psyoffset by1000 \multiply\psyoffset by\xscale
   \global\divide\pscm by 1000
   \global\multiply\pscm by\xscale
   \multiply\pswdincr by\xscale \multiply\pshtincr by\xscale
   \ifdim\d@mx=0pt\d@mx=\pswdincr\fi
   \ifdim\d@my=0pt\d@my=\pshtincr\fi
   \message{scaled \the\xscale}%
 \hbox to\d@mx{\hss\vbox to\d@my{\vss
   \global\setbox\drawingBox=\hbox to 0pt{\kern\psxoffset\vbox to 0pt{%
      \kern-\psyoffset
      \PSspeci@l{\PSfilename}{\the\xscale}%
      \vss}\hss\ps@nnotation}%
   \global\wd\drawingBox=\the\pswdincr
   \global\ht\drawingBox=\the\pshtincr
   \global\drawingwd=\pswdincr
   \global\drawinght=\pshtincr
   \baselineskip=0pt
   \copy\drawingBox
 \vss}\hss}%
  \global\psxoffset=0pt
  \global\psyoffset=0pt
  \global\pswdincr=0pt
  \global\pshtincr=0pt % These are local to one figure
  \global\pscm=1cm %should not be necessary
}}%
%
% \psboxscaled{scalefactor*1000}{PSfilename} allows to bypass the
%   rounding errors of TeX integer divisions for situations where the
%   TeX box should fit the original BoundingBox with a precision
%   better
%   than 1/1000.
%
\def\psboxscaled#1#2{\vbox{%
  \ReadPSize{#2}%
  \xscale=#1
  \message{scaled \the\xscale}%
  \divide\pswdincr by 1000 \multiply\pswdincr by \xscale
  \divide\pshtincr by 1000 \multiply\pshtincr by \xscale
  \divide\psxoffset by1000 \multiply\psxoffset by\xscale
  \divide\psyoffset by1000 \multiply\psyoffset by\xscale
  \divide\drawingwd by1000 \multiply\drawingwd by\xscale
  \divide\drawinght by1000 \multiply\drawinght by\xscale
  \global\divide\pscm by 1000
  \global\multiply\pscm by\xscale
  \global\setbox\drawingBox=\hbox to 0pt{\kern\psxoffset\vbox to 0pt{%
     \kern-\psyoffset
     \PSspeci@l{\PSfilename}{\the\xscale}%
     \vss}\hss\ps@nnotation}%
  \advance\pswdincr by \drawingwd
  \advance\pshtincr by \drawinght
  \global\wd\drawingBox=\the\pswdincr
  \global\ht\drawingBox=\the\pshtincr
  \global\drawingwd=\pswdincr
  \global\drawinght=\pshtincr
  \baselineskip=0pt
  \copy\drawingBox
  \global\psxoffset=0pt
  \global\psyoffset=0pt
  \global\pswdincr=0pt
  \global\pshtincr=0pt % These are local to one figure
  \global\pscm=1cm
}}%
%
%  \psbox{PSfilename} makes a TeX box having the minimal size to
%      enclose the picture
\def\psbox#1{\psboxscaled{1000}{#1}}%
%------------------------------------------------------
%  \joinfiles file1, file2, ...n \into joinedfilename .
%     makes one file out of many
%  \splitfile joinedfilename
%     the opposite
\newif\ifn@teof\n@teoftrue
\newif\ifc@ntrolline
\newif\ifmatch
\newread\j@insplitin
\newwrite\j@insplitout
\newwrite\psbj@inaux
\immediate\openout\psbj@inaux=psbjoin.aux
\immediate\write\psbj@inaux{\string\joinfiles}%
\immediate\write\psbj@inaux{\jobname,}%
%
% INPUT REDEFINITION
%
% works if #1 is a single character
\def\toother#1{\ifcat\relax#1\else\expandafter%
  \toother@ux\meaning#1\endtoother@ux\fi}%
\def\toother@ux#1 #2#3\endtoother@ux{\def\tmp{#3}%
  \ifx\tmp\@mpty\def\tmp{#2}\let\next=\relax%
  \else\def\next{\toother@ux#2#3\endtoother@ux}\fi%
\next}%
%
% \readfilename defs:
%
\let\readfilenamehook=\relax
\def\re@d{\expandafter\re@daux}% spares typing 10 \expandafter's...
\def\re@daux{\futurelet\nextchar\stopre@dtest}%
\def\re@dnext{\xdef\lastreadfilename{\lastreadfilename\nextchar}%
  \afterassignment\re@d\let\nextchar}%
\def\stopre@d{\egroup\readfilenamehook}%
\def\stopre@dtest{%
  \ifcat\nextchar\relax\let\nextread\stopre@d
  \else
    \ifcat\nextchar\space\def\nextread{%
      \afterassignment\stopre@d\chardef\nextchar=`}%
    \else\let\nextread=\re@dnext
      \toother\nextchar
      \edef\nextchar{\tmp}%
    \fi
  \fi\nextread}%
\def\readfilename{\bgroup%
  \let\\=\backslashother \let\%=\percentother \let\~=\tildeother
  \let\#=\sharpother \xdef\lastreadfilename{}%
  \re@d}%
%
% redefines \input using \readfilename
%
\xdef\GlobalInputList{\jobname}%
\def\psnewinput{%
  \def\readfilenamehook{% each entry in \GlobalInputList should be unique
    \if\matchexpin{\GlobalInputList}{, \lastreadfilename}%
    \else\xdef\GlobalInputList{\GlobalInputList, \lastreadfilename}%
      \immediate\write\psbj@inaux{\lastreadfilename,}%
    \fi%
    \ps@ldinput\lastreadfilename\relax%
    \let\readfilenamehook=\relax%
  }\readfilename%
}%
\expandafter\ifx\csname @@input\endcsname\relax    % then Plain
  \immediate\let\ps@ldinput=\input\def\input{\psnewinput}%
\else
  \immediate\let\ps@ldinput=\@@input
  \def\@@input{\psnewinput}%
\fi%
%
\def\nowarnopenout{%
 \def\warnopenout##1##2{%
   \readfilename##2\relax
   \message{\lastreadfilename}%
   \immediate\openout##1=\lastreadfilename\relax}}%
\def\warnopenout#1#2{%
 \readfilename#2\relax
 \def\t@mp{TrashMe,psbjoin.aux,psbjoint.tex,}\uncatcode\t@mp
 \if\matchexpin{\t@mp}{\lastreadfilename,}%
 \else
   \immediate\openin\pst@mpin=\lastreadfilename\relax
   \ifeof\pst@mpin
     \else
     \errhelp{If the content of this file is so precious to you, abort (ie
press x or e) and rename it before retrying.}%
     \errmessage{I'm just about to replace your file named \lastreadfilename}%
   \fi
   \immediate\closein\pst@mpin
 \fi
 \message{\lastreadfilename}%
 \immediate\openout#1=\lastreadfilename\relax}%
% % will have an unusual catcode below; use * instead
%\vbox
{\catcode`\%=12\catcode`\*=14
\gdef\splitfile#1{*
 \readfilename#1\relax
 \immediate\openin\j@insplitin=\lastreadfilename\relax
 \ifeof\j@insplitin
   \message{! I couldn't find and split \lastreadfilename!}*
 \else
   \immediate\openout\j@insplitout=TrashMe
   \message{< Splitting \lastreadfilename\space into}*
   \loop
     \ifeof\j@insplitin
       \immediate\closein\j@insplitin\n@teoffalse
     \else
       \n@teoftrue
       \executeinspecs{\global\read\j@insplitin to\spl@tinline\expandafter
         \ch@ckbeginnewfile\spl@tinline%Beginning-Of-File-Named:%\endcheck}*
       \ifc@ntrolline
       \else
         \toks0=\expandafter{\spl@tinline}*
         \immediate\write\j@insplitout{\the\toks0}*
       \fi
     \fi
   \ifn@teof\repeat
   \immediate\closeout\j@insplitout
 \fi\message{>}*
}*
\gdef\ch@ckbeginnewfile#1%Beginning-Of-File-Named:#2%#3\endcheck{*
 \def\t@mp{#1}*
 \ifx\@mpty\t@mp
   \def\t@mp{#3}*
   \ifx\@mpty\t@mp
     \global\c@ntrollinefalse
   \else
     \immediate\closeout\j@insplitout
     \warnopenout\j@insplitout{#2}*
     \global\c@ntrollinetrue
   \fi
 \else
   \global\c@ntrollinefalse
 \fi}*
\gdef\joinfiles#1\into#2{*
 \message{< Joining following files into}*
 \warnopenout\j@insplitout{#2}*
 \message{:}*
 {*
 \edef\w@##1{\immediate\write\j@insplitout{##1}}*
\w@{% This collection of files was produced with CERN psbox package}*
\w@{% To decompose and tex it:}*
\w@{%-save this with a filename CONTAINING ONLY LETTERS and a .TEX}*
\w@{% extension (say, JOINTFIL.TEX), in some uncrowded directory;}*
\w@{%-make sure you can \string\input\space psbox.tex (version>=1.3);}*
\w@{%  (else ftp cs.nyu.edu(=128.122.140.24):pub/TeX/psbox/, then get}*
\w@{%  and tex the file psboxall.tex; more info in psbREAD.ME)}*
\w@{%-tex JOINTFIL.TEX using Plain, or LaTeX, or whatever is needed by}*
\w@{%  the first file in the joining (after splitting JOINTFIL.TEX into}*
\w@{%  it's constituents, TeX will try to process it as it stands).}*
\w@{\string\input\space psbox.tex}*
\w@{\string\splitfile{\string\jobname}}*
\w@{\string\let\string\autojoin=\string\relax}*
}*
 \expandafter\tre@tfilelist#1, \endtre@t
 \immediate\closeout\j@insplitout
 \message{>}*
}*
\gdef\tre@tfilelist#1, #2\endtre@t{*
 \readfilename#1\relax
 \ifx\@mpty\lastreadfilename
 \else
   \immediate\openin\j@insplitin=\lastreadfilename\relax
   \ifeof\j@insplitin
     \errmessage{I couldn't find file \lastreadfilename}*
   \else
     \message{\lastreadfilename}*
     \immediate\write\j@insplitout{%Beginning-Of-File-Named:\lastreadfilename}*
     \executeinspecs{\global\read\j@insplitin to\oldj@ininline}*
     \loop
       \ifeof\j@insplitin\immediate\closein\j@insplitin\n@teoffalse
       \else\n@teoftrue
         \executeinspecs{\global\read\j@insplitin to\j@ininline}*
         \toks0=\expandafter{\oldj@ininline}*
         \let\oldj@ininline=\j@ininline
         \immediate\write\j@insplitout{\the\toks0}*
       \fi
     \ifn@teof
     \repeat
   \immediate\closein\j@insplitin
   \fi
   \tre@tfilelist#2, \endtre@t
 \fi}*
}%
% To be put at the end of a file, for making a tar-like file containing
%   everything it used.
\def\autojoin{%
 \immediate\write\psbj@inaux{\string\into{psbjoint.tex}}%
 \immediate\closeout\psbj@inaux
 \expandafter\joinfiles\GlobalInputList\into{psbjoint.tex}%
}%
%----------------------------------------------------------------
%  Annotations & Captions etc...
%
%
% \centinsert{anybox} is just a centered \midinsert, but is included as
%    people barely use the original inserts from TeX.
%
\def\centinsert#1{\midinsert\line{\hss#1\hss}\endinsert}%
\def\psannotate#1#2{\vbox{%
  \def\ps@nnotation{#2\global\let\ps@nnotation=\relax}#1}}%
\def\pscaption#1#2{\vbox{%
   \setbox\drawingBox=#1
   \copy\drawingBox
   \vskip\baselineskip
   \vbox{\hsize=\wd\drawingBox\setbox0=\hbox{#2}%
     \ifdim\wd0>\hsize
       \noindent\unhbox0\tolerance=5000
    \else\centerline{\box0}%
    \fi
}}}%
% for compatibility with older versions, but \psfig is a bad name!
%\def\psfig#1#2#3{\pscaption{\psannotate{#1}{#2}}{#3}}
%\def\psfigurebox#1#2#3{\pscaption{\psannotate{\psbox{#1}}{#2}}{#3}}
%
% \at(#1;#2)#3 puts #3 at #1-higher and #2-right of the current
%    position without moving it (to be used in annotations).
\def\at(#1;#2)#3{\setbox0=\hbox{#3}\ht0=0pt\dp0=0pt
  \rlap{\kern#1\vbox to0pt{\kern-#2\box0\vss}}}%
%
% \gridfill(ht;wd) makes a 1cm*1cm grid of ht by wd whose lower-left
%   corner is the current point
\newdimen\gridht \newdimen\gridwd
\def\gridfill(#1;#2){%
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  {\vrule height1\pscm width.4pt\leaders\hrule\hfill}%
  \gridht=#1
  \divide\gridht by \ht0
  \multiply\gridht by \ht0
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  \divide\gridwd by \wd0
  \multiply\gridwd by \wd0
  \advance \gridwd by \wd0
  \vbox to \gridht{\leaders\hbox to\gridwd{\leaders\box0\hfill}\vfill}}%
%
% Useful to measure where to put annotations
\def\fillinggrid{\at(0cm;0cm){\vbox{%
  \gridfill(\drawinght;\drawingwd)}}}%
%
% \textleftof\anybox: Sample text\endtext
%   inserts "Sample text" on the left of \anybox ie \vbox, \psbox.
%   \textrightof is the symmetric (not documented, too uggly)
% Welcome any suggestion about clean wraparound macros from
%   TeXhackers reading this
%
\def\textleftof#1:{%
  \setbox1=#1
  \setbox0=\vbox\bgroup
    \advance\hsize by -\wd1 \advance\hsize by -2em}%
\def\textrightof#1:{%
  \setbox0=#1
  \setbox1=\vbox\bgroup
    \advance\hsize by -\wd0 \advance\hsize by -2em}%
\def\endtext{%
  \egroup
  \hbox to \hsize{\valign{\vfil##\vfil\cr%
\box0\cr%
\noalign{\hss}\box1\cr}}}%
%
% \frameit{\thick}{\skip}{\anybox}
%    draws with thickness \thick a box around \anybox, leaving \skip of
%    blank around it. eg \frameit{0.5pt}{1pt}{\hbox{hello}}
% \boxit{\anybox} is a shortcut.
\def\frameit#1#2#3{\hbox{\vrule width#1\vbox{%
  \hrule height#1\vskip#2\hbox{\hskip#2\vbox{#3}\hskip#2}%
        \vskip#2\hrule height#1}\vrule width#1}}%
\def\boxit#1{\frameit{0.4pt}{0pt}{#1}}%
%
%
\catcode`\@=12 % cs containing @ are unreachable
%
% CUSTOMIZE YOUR DEFAULT DRIVER:
%    Uncomment the line corresponding to your TeX system:
%\psfortextures%     For TeXtures on the Macintosh
%\psforoztex   %     For OzTeX shareware on the Macintosh
%\psfordvitops %     For the DVItoPS converter for TeX on IBM mainframes
 \psfordvips   %     For DVIPS converter on VAX and UNIX
%\psfordvitps  %     For dvitps from TeXPS package under UNIX
%\psfordvialw  %     For dvialw, UNIX public domain
%\psonlyboxes  %     Blank Boxes (when all else fails).

hep-ph0101087.extracted_bib
\begin{thebibliography}{99}
\bibitem{Wal1}
R.M. Wald, Phys. Rev. D 21 (1980) 2742.
\bibitem{Haw1}
S.W. Hawking, Phys. Rev. D 32 (1985) 2489.
\bibitem{KoPo1}
V.A. Kosteleck\'{y},  R. Potting, Nucl. Phys. B 359 (1991) 545.
\bibitem{Kli1}
F.R. Klinkhamer, Nucl. Phys. B 578 (2000) 277.
\bibitem{Kli2}
F.R. Klinkhamer,  J. Nishimura, Phys. Rev. D 63 (2001) 097701.
\bibitem{CFJ}
S.M. Carroll, G.B. Field,  R. Jackiw, Phys. Rev. D 41 (1990) 1231.
\bibitem{CoKo1}
D. Colladay,  V.A. Kosteleck\'{y}, Phys. Rev. D 55 (1997) 6760;
D 58 (1998) 116002.
\bibitem{CoGl1}
S. Coleman,  S.L. Glashow, Phys. Rev. D 59 (1999) 116008.
\bibitem{JaKo1}
R. Jackiw,  V.A. Kosteleck\'{y}, Phys. Rev. Lett. 82 (1999) 3572.
\bibitem{CCGF}
M. Chaichian, W.F. Chen,  R. Gonzales Felipe, Phys. Lett. B 503 (2001) 215.
\bibitem{Jost}
R. Jost, Helv. Phys. Acta 30 (1957) 409.
\bibitem{SW64}
R. Streater, A. Wightman, \emph{PCT, Spin and Statistics, and All That},
Benjamin, New York, 1964.
\bibitem{C67}
P. Carruthers, Phys. Rev. Lett. 18 (1967) 353; Phys. Lett. B 26 (1968) 158.
\bibitem{CS74}
S. Chern,  J. Simons, Ann. Math. 99 (1974) 48.
\bibitem{Leib}
G. Leibbrandt, \emph{Noncovariant gauges}, World Scientific, Singapore,
1994.
\bibitem{NR1}
B. Nodland,  J.P. Ralston, Phys. Rev. Lett. 78 (1997) 3043.
\bibitem{WPC}
J.F.L. Wardle, R.A. Perley,  M.H. Cohen, Phys. Rev. Lett. 79 (1997) 1801.
\bibitem{B60}
L. Brillouin, \emph{Wave Propagation and Group Velocity},
Academic, New York, 1960.
\bibitem{ASS1}
A.A. Andrianov, R. Soldati,  L. Sorbo, Phys. Rev. D 59 (1999) 025002.
\bibitem{MM1}
I. Montvay,  G. Munster, \emph{Quantum Fields on a Lattice}, Cambridge
U.P., Cambridge, 1994.
\bibitem{V94}
M. Veltman, \emph{Diagrammatica--The Path to Feynman Rules},
Cambridge U.P., Cambridge, 1994.
\bibitem{W96}
S. Weinberg, \emph{The Quantum Theory of Fields I},
Cambridge U.P., Cambridge, 1996.
\bibitem{DhSu1}
J. Dhar,  E. Sudarshan, Phys. Rev. 174 (1968) 1808.
\bibitem{JTW}
T. Jacobson, N.C. Tsamis,  R.P. Woodard, Phys. Rev. D 38 (1988) 1823.
\bibitem{OS1}
K. Osterwalder,  R. Schrader, Comm. Math. Phys. 31 (1973) 83;
42 (1975) 281.
\bibitem{GrRy1}
I.S. Gradshteyn,  I.M. Ryzhik, \emph{Table of Integrals, Series and
Products}, Academic, New York, 1980.
\bibitem{Heitler}
W. Heitler, \emph{The Quantum Theory of Radiation}, 3rd ed.,
Oxford U.P., London, 1954.
\bibitem{JR}
J.M. Jauch,  F. Rohrlich, \emph{The Theory of Photons and Electrons},
2nd ed., Springer, New York, 1976.
\bibitem{BS}
N.N. Bogoliubov,  D.V. Shirkov, \emph{Introduction to the Theory of Quantized
Fields}, Wiley, New York, 1959.
\bibitem{CCT}
D. Coulson, R. Crittenden, N. Turok, Phys. Rev. Lett. 73 (1994) 2390.
\bibitem{L98}
N.F. Lepora, arXiv:gr-qc/9812077;
M. Kamionkowski, A. Kosowsky, Ann. Rev. Nucl. Part. Sci. 49 (1999) 77.
\bibitem{KL00}
V.A. Kosteleck\'{y}, R. Lehnert, Phys. Rev. D 63 (2001) 065008.
\bibitem{Wal2}
R.M. Wald, \emph{General Relativity}, Chicago U.P., Chicago, 1984.
\end{thebibliography}