%%%%%%%%%%%%%%%%%%%%%%%%%% mic44.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%==============================================================================
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% Adapted from Knuth's \answer macro in the TeXbook.
% Requires Plain TeX.  Maybe other flavors will work too?
% Jamie Stephens, jamies@math.utexas.edu, 28 Nov 94

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\centerline{\bf Microcausality and Energy-Positivity in all frames} 
\centerline{\bf imply Lorentz Invariance of dispersion laws} 


\vskip 0.2cm 
\centerline { by }

\vskip 0.2cm 
\centerline { {\bf Jacques Bros$^1$} and {\bf Henri Epstein$^2$}} 

\vskip 0.2cm 
\centerline {${ }^1$ \bf Service de Physique Th\'eorique, CEA-Saclay,  
F-91191 Gif-sur-Yvette, France}

\centerline {${ }^2$ \bf Institut des Hautes Etudes Scientifiques,  
F-91440 Bures-sur-Yvette, France}

\vskip 0.5cm 
\noindent  
{\bf Abstract.}\  
 A new presentation of the Borchers-Buchholz result of the Lorentz-invariance 
of the 
energy-momentum spectrum in theories with broken Lorentz symmetry is given 
in terms of properties of the Green's functions of microcausal Bose and
Fermi-fields. 
Strong constraints based on complex geometry phenomenons are shown to 
result from the interplay of the basic 
principles of causality and stability in Quantum Field Theory:  
if microcausality and energy-positivity in all Lorentz frames 
are satisfied, then 
it is unavoidable that all stable particles of the theory   
be governed by Lorentz-invariant dispersion laws: in all the field sectors, 
discrete parts outside the continuum as well as the thresholds 
of the continuous parts of the energy-momentum spectrum, with possible holes
inside it,  are necessarily 
represented by mass-shell hyperboloids (or the light-cone). No violation of this
geometrical fact can be produced by   
spontaneous breaking of the Lorentz symmetry.  

\vskip 0.5cm

\centerline{\bf 1- Introduction}

\vskip 0.5cm

In a recent work [1], it has been advocated that 
the occurrence of spontaneous Lorentz and CPT violations in Quantum Field 
Theories governed by suitable {\sl non-local} Lagrangians can very well 
generate {\sl non-Lorentz-invariant dispersion laws} 
\footnote{${ }^{(1)}$}{We prefer keeping here
the terminology of 
``dispersion law'' (used traditionally e.g. in Thermal Quantum Field Theory) 
rather than adopting the new usage of ``dispersion relation'', which is of
course confusing 
in a domain where (Cauchy-type) dispersion relations relating the absorptive and
dispersive parts of Feynman-type amplitudes 
remain a basic tool of frequent use.}
{\sl which avoid the problems with stability and causality}.  
Such Lorentz violation effects produced at Planck scale might then in principle
be observed at 
lower energies in particle physics. In support of their claim, the authors of
[1]  
have produced examples of possible ``non-local models'' in which the quadratic
part of the Lagrangian is 
supposed to yield a dispersion law 
$p_0 = \omega(\vec p)$ enjoying the following properties:

a) The hypersurface ${\cal M}$ with equation  
$p_0 = \omega (\vec p)$ 
differs from a Lorentz-invariant mass shell hyperboloid, 

b) ${\cal M}$ is contained in the positive energy-momentum cone $\overline V^+ \
(p_0 \geq |\vec p|),$ 

c) For every momentum $\vec p$, the ``group velocity condition'' $|\partial
\omega (\vec p)| \leq 1$ holds, 
which means that ${\cal M}$ admits a space-like (or light-like) tangent
hyperplane at each of its points.  

\vskip 0.3cm

While condition b) expresses energy-positivity in all Lorentz frames, condition c)
ensures that all 
wave-packets satisfying the dispersion law $p_0 = \omega (\vec p)$ propagate
``essentially'' 
with a {\sl subluminal (or luminal) velocity};  
essentially means ``up to the quantum spreading of wave-packets, of the order of
the Planck constant'',  
as it is the case for the solutions of the Klein-Gordon and Dirac equations.  

However, we wish to stress that the latter condition c) should {\sl by
no means} 
be taken as a criterion of {\sl microcausality } for the underlying Quantum
Field Theory. 
Microcausality states that the commutator (resp. anticommutator)
$[\Phi(x),\Phi(x')]_{\mp}$ of 
a boson (resp.fermion) field $\Phi(x)$ should vanish in the whole region of 
{\sl relativistic spacelike separation} $\{(x,x');\  (x-x')^2 <0 \}.$ As we
shall see below,
the ``group velocity condition'' c) only appears as a necessary consequence of
microcausality,
but the converse is not true. This is why, in the various examples presented in
[1], checking 
the validity of condition c) does {\sl not} constitute a check of the validity
of microcausality. 
On the contrary, 
the requirement of microcausality represents such a strong constraint that,
when combined with energy positivity in all frames, it definitely implies the
following properties:  
 
 i) any dispersion law describing particles generated by the field 
is Lorentz invariant, namely the corresponding hypersurface ${\cal M}$ is a
sheet of 
hyperboloid with equation of the form $p_0 = \sqrt {{\vec p}^2 + m^2}$ (or the
light cone  
$\partial V^+$ if $m=0$).  

 ii) In all the sectors (or collision channels) of the 
space of states of the (interacting) field theory considered,   
the hypersurfaces which border the continuous part of the 
energy-momentum spectrum, including possible holes 
in the latter,
are also Lorentz-invariant, namely sheets of hyperboloid 
of the form $p_0 =\sqrt{{\vec p}^2 + M_i^2} $ (or the light-cone).  

\vskip 0.1cm
It is the purpose of the present paper to give a hopefully elementary
presentation of the
latter facts, which have been established long ago in a general, although
slightly different, 
framework by Borchers and Buchholz. 
As a matter of fact, the interest for the possible occurrence of
Lorentz-symmetry 
breaking is not new and it has already been a subject of deep investigation in
the 
framework of the basic principles of Quantum Field Theory (QFT): the latter two
properties
of Lorentz-invariance of the energy-momentum spectrum have indeed been proven   
in a paper by H.J. Borchers and D. Buchholz entitled 
``The Energy-Momentum Spectrum in Local Field Theories with Broken 
Lorentz-Symmetry''[2] completed by a paper by H.J. Borchers 
entitled ``Locality and covariance of the spectrum''[3] in the general framework
of 
Algebraic QFT (or ``Local Quantum Physics'') [4].  
In this deep analysis, generalizing similar results already obtained in [5] 
(see also [6] for a complete survey of the question), it was proven that 
the interplay of a weak form of microcausality, namely the commutativity of 
local observables attached to pairs of mutually space-like regions, together
with 
energy-positivity in all Lorentz frames was sufficient to produce a 
Lorentz-invariant shape of the energy-momentum spectrum, even if the
Lorentz-symmetry
was broken in the considered physical representation of the field observables. 
In view of the always vivid interest of the community for the possible
occurrence of 
some form of Lorentz-symmetry violation 
emerging from the spontaneous breaking at Planck scale 
of  a ``fundamental field or string theory''  
(see [1] and references therein), 
but also of its apparent unawareness of the results of [2,3],  
we think it useful to give a revival to these results in a way which 
we hope to be accessible to the current field-theorist reader.   
In fact, we wish 
to give here a new presentation of these unexpected properties of geometrical
nature
in energy-momentum space in terms of  
propagators and Green's functions of microcausal Bose and Fermi-fields
of usual type. We shall thus avoid  using the more abstract formulation of
Algebraic QFT,  
and will focus on the contrary on the phenomenons of complex geometry which play
a basic role in this
matter.  

As in [2,3], the proof of properties i) and ii) which we give below is 
of general nature, i.e. non-perturbative 
and even independent of any Lagrangian formulation of the field model. 
We wish to stress that the somewhat surprising phenomenon of geometrical
Lorentz-invariance
produced in the present problem has to do with peculiar properties of  
complex geometry in several complex variables; such properties, which are also
closely related 
to the Jost-Lehmann-Dyson (JLD) formula [7], have been thoroughly exploited in
[2,3] precisely 
in the spirit of [7].  
Here we propose to  
give a completely clear-cut and self-contained account of the previously stated
properties, 
by exploiting the simplest\break geometrical situation, which is provided by
propagators 
(as commented below, these are in fact the typical objects considered in [1])
and by indicating subsequently  
how and why the same phenomenons still occur for the spectral properties of
four-point (and 
general $n-$point) functions, which provide a complete framework for interacting
fields.  
Under this respect, our presentation is in the spirit of the   
{\sl analyticity properties of Green's functions in general QFT} (see [8,9,17]
and references therein) and 
therefore differs from 
that of [2,3] which always deals with the properties of 
expectation values of 
commutators 
in general states (with appropriate energy-momentum spectrum) in the JLD-way.   



In the models considered in [1], the dispersion laws of particles are always  
associated with given quadratic parts of field Lagrangians incorporating
explicit Lorentz-symmetry 
breaking coefficients of appropriate type. Such dispersion laws  
therefore correspond to particles which are ``elementary'' with respect to the
field 
introduced in the Lagrangian, namely they appear as associated with poles  
of the propagator of this field in energy-momentum space. Another case of   
dispersion laws should also be considered, namely those which  
correspond to ``composite'' particles  
of the field: the latter appear as associated with poles of the four-point (or
higher n-point) 
functions of the field in energy-momentum space; for example, this is the case
for the 
hadronic particles if the fundamental fields are those of the standard model.

Here we shall show in detail the previously announced geometrical properties for
the 
poles of propagators (corresponding to the case considered in [1]) 
and we shall also indicate the 
derivation of the corresponding equally valid results for the poles of 
four-point (or n-point) functions. 
The essential point is that we are only concerned here with stable particles,
corresponding to 
discrete parts of the spectrum, not embedded in the continuum. The case of
unstable particles corresponding to possible complex poles of the Green's
functions in 
unphysical sheets is excluded from our study. 


\vskip 0.1cm
In our section 2, we shall recall the basic analyticity properties of retarded
and advanced 
two-point functions which express microcausality in the complexified
energy-momentum space, 
and the procedure through which information on the energy-momentum spectrum 
is encoded in this framework. 
We then formulate 
three basic results of complex geometry, called  Properties A, B and C, whose
physical 
consequences in terms of {\sl admissible dispersion laws} 
are derived in a straightforward way:
Property A explains why the velocity group condition c) of dispersion laws is
implied 
by microcausality under a weak requirement of energy-positivity.  
Properties B and C provide a proof of the previous statements of
Lorentz-invariance for  
the dispersion laws of elementary particles and for the thresholds
(and possible holes) of the continuous spectrum,  
under the joint requirement of microcausality and energy-positivity in all
frames.
A complete proof of Properties A, B and C is given in this section. 
In section 3, it is shown that similar consequences of  
microcausality and (weak or strong) energy-positivity 
requirements can be formulated in terms of    
momentum-space analyticity properties  
of four-point (resp.  more generally $2n-$point)  
Green's functions established in [8,9] (resp. [17d),e)]).   
The exact counterparts of Properties A,B,C, called respectively 
A',B',C', are then described and these phenomenons of complex geometry are shown
to imply
the corresponding constraints for the dispersion laws of composite particles and
for the 
thresholds (and possible holes) of the continuous spectrum in the channel
considered. 
Section 4 gives concluding remarks.  


\vskip 0.5cm
\centerline{\bf 2 Shape of the energy-momentum spectral supports for the
two-point functions} 


\vskip 0.5cm
Let $F^+(p)$ and $F^-(p)$ (with $p=(p_0,\vec p)$) be respectively the Fourier
transforms of 
the vacuum expectation values of the retarded and advanced (anti-)commutators of
a general
(fermion or boson) quantum field $\Phi(x)$, which we write formally
\footnote{${ }^{(2)}$}{The distribution character of the integrand at $x=0$ is
treated rigorously by a
standard mathematical procedure.}
$$ F^+(p) = \int {\rm e}^{ip\cdot x} \ \theta(x_0)\ <[\Phi({x\over 2}),
\Phi(-{x\over 2})]_{\pm}>\ dx_0
d\vec x,\ \ \ \ \eqno(1)$$ 
$$ F^-(p) = -\int {\rm e}^{ip\cdot x} \ \theta(-x_0)\ <[\Phi({x\over 2}),
\Phi(-{x\over 2})]_{\pm}>\ dx_0
d\vec x.\ \ \ \ \eqno(2)$$ 

For writing the latter, we have assumed as usual that the space of states in
which the field is acting 
carries a representation of the group of spacetime translations and that the
field is 
invariant under this representation; energy and momentum operators are the
corresponding generators of
this group.  
It is of current use to exploit the analyticity properties of 
$F^+(p)$ and $F^-(p)$ respectively in the upper and lower half-planes of the
complexified energy
variable $p_0$. However, the postulate of microcausality for the field $\Phi(x)$
implies much more. 
In fact, it requires that the retarded and advanced propagators occurring under
the integrals at the 
r.h.s. of Eqs (1) and (2) have respectively their supports contained in the
closed forward   
and backward cones $\overline V^+ \ (x_0 \geq |\vec x|)$ and $\overline V^-\
(x_0 \leq -|\vec x|)$.  
It then follows that the integrals (1) and (2) remain convergent and define
analytic functions 
of the complex energy-momentum vector $k=p+iq$, still denoted by $F^+(k)$ and
$F^-(k)$,  
in the respective domains $T^+ \ ( p\  {\rm arbitrary} , \ q \in V^+)$ and  
$T^-\ ( p\  {\rm arbitrary},\ q \in V^-)$;\  $V^+ = -V^-$ is the open forward
cone: $q_0 > |\vec q|$.      
\ $T^+$ and $T^-$ are called the ``forward and backward tubes''; they  contain
respectively the upper and 
lower half-planes in all their one-dimensional sections by (complexified)
time-like straight 
lines, interpreted as energy variables in all possible Lorentz frames.
$F^{\pm}(k)$ are
the ``Fourier-Laplace transforms'' 
of the retarded and advanced propagators in complex energy-momentum space;  
their boundary values $F^{\pm}(p)$ on the reals from (respectively) 
$T^{\pm}$ are the Fourier transforms themselves of these propagators.  

So {\sl in the sector generated by ``one-field vector-states'' of the form $\int
\varphi(x)\Phi(x) dx >$} 
(with $\varphi$ arbitrary in the Schwartz space of smooth and rapidly decreasing
functions), 
microcausality is fully expressed by the analyticity of the pair of functions
$(F^+,F^-)$   
in the corresponding domains $T^+, T^-$. 
Now any usable information on the 
support of the energy-momentum spectrum 
of the theory in this sector  
amounts to specifying an open subset ${\cal R}$ of the (real) 
energy-momentum space
in which the distributions 
\footnote{${ }^{(3)}$}{This ``bracket notation'' in terms of operator products 
and of (anti-)commutators is used purely for  
its suggestive content; no (infinite!) energy-momentum conservation
$\delta-$function 
is involved in it.}  
$<\tilde \Phi(p)\tilde \Phi(-p)>$
and $ <\tilde \Phi(-p),\tilde \Phi(p)>$ vanish simultaneously.  
In fact, such a support property implies the
{\sl coincidence relation} 
$F^+_{|{\cal R}} = F^-_{|{\cal R}}$, since  
the expression${ }^{(3)}$ 
$$ F^+(p) - F^-(p) = <[\tilde \Phi(p),\tilde \Phi(-p)]_{\pm}> \ \ \ \ \eqno(3)$$
vanishes in $\cal R.$
It then follows from a standard theorem of complex analysis, 
called the ``edge-of-the-wedge theorem'' (see [10] and references therein),
that $F^+(k)$ and $F^-(k)$ then admit a {\sl common analytic continuation}
$F(k)$ which is 
analytic in the union of $T^+$, $T^-$ and of a complex neighborhood of the real
set ${\cal R}$;
in other words, $F^+$ and $F^-$ ``communicate analytically'' through ${\cal R}$,
as functions of the
set of complex variables $k= (k_0, \vec k)$. . 


It is one of the basic phenomenons of Analysis and Geometry in several complex
variables 
that arbitrary (connected) subsets of complex space ${\bf C}^n$ are not in
general ``natural'' for the 
class of holomorphic functions: this means that for such a general subset
$\Sigma$, all the  
functions holomorphic in $\Sigma$ admit an analytic continuation in a common
larger  
domain $\hat \Sigma$, called the holomorphy envelope of $\Sigma$. This
phenomenon,
which does not exist in the single-variable case, involves exclusively   
geometrical properties of the set $\Sigma$ and the extension from $\Sigma$ to
$\hat \Sigma$
can always be done in principle by an appropriate use of the Cauchy integral
formula; 
this {\sl analytic completion procedure} presents a strong analogy with the
procedure of taking the 
convex hull $\check S$ of a subset $S$ in the ordinary real space ${\bf R}^n$,
the 
notion of a ``natural 
holomorphy domain'' in ${\bf C}^n$  being a certain generalisation of the
notion of ``convex domain'' in ${\bf R}^n$ (see e.g. [11,12] and references
therein). As a matter of fact, 
the most standard and useful result 
in this connection is the so-called ``tube theorem'' (see e.g. [12]) which we
shall apply below: 
{\sl Any domain $D$ in ${\bf C}^n$ which is ``tube-shaped'', i.e. of the form 
${\bf R}^n + i B$ 
admits a holomorphy envelope which is the tube 
$\hat D = {\bf R}^n + i \check B$, where $\check B$ is the convex hull of $B$ in
${\bf R}^n$.}   

It turns out that sets of the form $\Sigma_{\cal R} = T^+ \cup T^- \cup {\cal
R}$  
are not natural and that,
for various choices of ${\cal R}$ of physical interest,  
the corresponding holomorphy envelope $\hat \Sigma$ or parts of it can be
computed
and unexpectedly strong results then follow.
Cases when ${\cal R}$ itself can be extended to a larger real region $\hat {\cal
R}$ 
(namely $\hat{\cal R} =\hat \Sigma \cap {\bf R}^n  \ \supset \ {\cal R}$) 
are specially interesting, since they correspond   
to enlarging the region on which the ``spectral function'' 
$ <[\tilde \Phi(p),\tilde \Phi(-p)]_{\pm}>$ is proven to vanish, and therefore
to refining our 
information on the support of the distribution 
$<\tilde \Phi(p)\tilde \Phi(-p)>$, called {\sl ``spectral support''}.  
Properties A and C given below are precisely of this
type. Property B is a basic example of holomorphy envelope for a domain 
${\Sigma}_{\cal R}$ which exactly corresponds to 
the case when energy-positivity is satisfied in all frames.  

\vskip 0.2cm
{\bf 2.1 Microcausality implies dispersion laws with subluminal velocities}  

\vskip 0.4cm
If energy-positivity is required to hold {\sl only in privileged frames, 
such as the laboratory frame and a set of frames which have  
small velocities with respect to the latter} \footnote{${ }^{(4)}$}{This refers 
to the notion of ``concordant
frames'' introduced in [1]}, 
there exists a maximal region $\hat{\cal R}$ of the form $-\omega(\vec p) < p_0
< \omega(\vec p)$ 
(with $\omega(\vec p) \geq \gamma |\vec p|$ for some positive constant $\gamma$)
in which the (anti-)commutator function   
$<[\tilde \Phi(p),\tilde \Phi(-p)]_{\pm}>$ vanishes. We claim that, {\sl due to
microcausality, 
the hypersurface with equation 
$p_0 = \omega(\vec p)$ is not arbitrary: it has to be a space-like
hypersurface.} 
In fact, the geometry of the relativistic light-cone is deeply involved in the
implications of  
microcausality, as it results from the following  


{\sl Property A (``Double-cone theorem''):  

\noindent
Let ${\cal R}_{a,b}$ be a neighborhood (in real $p-$space) of a given 
time-like segment $]a,b[$ with end-points $a$ and $b$ 
($b$ in the future of $a$). Then any function $F(k)$ holomorphic in  
$\Sigma_{{\cal R}_{a,b}}$ admits an analytic continuation in a (complex) domain
which contains  
the real region 
$\diamond_a^b$, where   
$ \diamond_a^b$ is the ``double-cone'' defined as the set of all points $p$ such
that 
$p$ is in the future of $a$ and in the past of $b$.} 

{\sl Interpretation of Property A:}\  


Let ${\cal M}$ 
($p_0 = \omega (\vec p)$)  
be the hypersurface bordering the vanishing region $\hat{\cal R}$  
of the (anti-)commutator function 
of a certain field theory satisfying microcausality and energy-positivity in
privileged frames.  
Then for each point 
$b=(\omega (\vec p), \vec p)$ in ${\cal M}$,  
there exists some 
interval of the form $ \omega (\vec p) -\epsilon < p_0 < \omega (\vec p)$ and
some open neighborhood 
${\cal R}_{a,b}$ of the time-like segment $]a,b[$ defined by this interval 
(i.e. $a\equiv (\omega (\vec p) - \epsilon, \vec p)$)  
which lies in $\hat{\cal R}$.  
It then follows from Property A that the propagator $F(k) $ of this theory has
to be analytic
in the full double-cone $\diamond_a^b$, and therefore that the corresponding
(anti-)commutator function must vanish in this double-cone: therefore,
$\diamond_a^b$ belongs to
$\hat {\cal R}$, 
and this argument holds for every point $b$ of ${\cal M}$,   
which shows that ${\cal M}$ has to be a spacelike hypersurface.  

Similarly, assume that the vanishing region $\hat {\cal R}$ is accompanied by
another pair of  
maximal vanishing regions $\hat {\cal R}_1^{\pm} $
of the form $\omega (\vec p) < |p_0| < \omega_1(\vec p)$
of the  (anti-)commutator function.  
Then $p_0 = \omega (\vec p)$ appears as the dispersion law of a particle
corresponding to a pole 
$Z(\vec p) \over 
k_0 - \omega (\vec p)$ of the propagator $F(k)$. 
So the previous argument shows in this case that both the hypersurface ${\cal
M}$ describing the 
dispersion law of the particle and the hypersurface ${\cal M}_1\ (p_0
=\omega_1(\vec p))$ 
bordering the region 
$\hat {\cal R}_1^+$ have to be spacelike.  
The argument extends of course to the case of any (ordered) set of dispersion
laws corresponding to 
several particles. 
{\sl Therefore, for every particle appearing with an energy gap in 
the propagator of the field considered, microcausality alone  
implies that condition c) (subluminal or luminal velocities) 
is satisfied by such a particle.}  


\vskip 0.2cm
{\sl Proof of Property A:}

This theorem, which can be seen as a generalisation of a similar property 
(corollary of the ``mean value Asgeirsson theorem'') for the solutions of 
the wave-equation [13], has been proved by Vladimirov [14] and by Borchers [15].
The main geometrical idea is displayed by treating a typical case in  
two-dimensional energy-momentum space with coordinates $(p_0,p_1)$.  
We take for $]a,b[$ the segment $\delta =]-1,+1[$ 
of the time axis and for ${\cal R}_{a,b}$ a thin rectangle $\delta_{\epsilon}$
of the form: 
$|p_0| < 1,\ |p_1| <\epsilon$. 
The tubes $T^+,T^-$ in the complexified space with coordinates  
$(k_0 = p_0 +i q_0,\ k_1= p_1 +i q_1)$ are defined respectively by the
conditions 
$q_0 +q_1>0,\ q_0-q_1 >0$ 
and 
$q_0 +q_1<0,\ q_0-q_1<0,$ 
and we shall show that the real region obtained by analytic 
completion of $T^+ \cup T^- \cup \delta_{\epsilon}$ contains the ``double-cone''
$\diamond$ (a square in this case!) defined by the inequalities: $|p_0 - p_1|
<1,\  |p_0 + p_1| <1.$   
One introduces the family of complex curves $h_{\lambda}$ with equation
$[k_0^2 -(k_1-1)^2]=  
\lambda [k_0^2 -(k_1+1)^2]$, 
where the parameter $\lambda$ varies in a 
complex neighborhood ${\cal V}$ of the real interval $]0,+\infty[$. 
Except for $h_1$ which is the (complexified) $p_0-$axis,  
all these curves are hyperbolae, and $\diamond $ is generated by the (real) arcs
$\breve h_{\lambda}$ of $h_{\lambda}$ parametrized by $-1 <p_0 < 1$ (with $|p_1|
<1$)  
when $\lambda $ varies from $0$ to $+\infty$; in a subinterval  
of the form $|\lambda -1| < \eta$ (for some $\eta$ 
determined by $\epsilon$),  
$\breve h_{\lambda}$ remains inside  
the rectangle $\delta_{\epsilon}$ (see fig 1). 

\vskip 0.25 truecm 
%\input micf1.tex
\newdimen\fixhoffset
\fixhoffset=\hsize \relax
\advance\fixhoffset by -16.000\varcm
\divide \fixhoffset by 2
\hbox{\kern\fixhoffset\vbox to 12.00 \varcm{\offinterlineskip
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\raise #2 \varcm \hbox{#3}}}
\def\spot{{\kern -0.2em\lower.55ex\hbox{$\bullet$}}}
\vfill\special{psfile=nmicf1.ps}
\smash{\hbox to \hsize{%
\point 14.10 6.00 {$p_1$}
\point 8.20 11.50 {$p_0$}
\point 3.50 5.60 {$-1$}
\point 8.10 5.60 {0}
\point 12.10 5.60 {$1$}
\point 8.10 1.50 {$-1$}
\point 8.10 10.10 {$1$}
\point 9.50 10.85 {$\delta_\varepsilon$}
\point 9.82 9.50 {$\check h_\lambda\ \hbox{for}\ |\lambda-1| < \eta$}
\point 11.05 4.00 {$\check h_1$}
\point 5.89 2.30 {$\check h_\lambda$}
\point 3.60 4.10 {$\diamond$}
\hfill}}}\hfill}


\vskip 0.25 truecm
\centerline{Fig.~1. The ``double-cone'' $\diamond$ and the curves 
$\check h_\lambda$}
\vskip 1 truecm
 

One then checks that for any function $F(k_0,k_1) $ holomorphic 
in $T^+ \cup T^- \cup {\delta_{\epsilon}}$  
the change of complex 
variables $(k_0,k_1) \to (k_0, \lambda)$ is admissible and allows one to define 
$\underline F(k_0,\lambda) = F(k_0, k_1(k_0,\lambda)) $ as an analytic function 
in the domain where   
$\lambda $ varies in ${\cal V}$  and $k_0$ varies in the unit disk  
$ |k_0| < 1$ {\sl deprived from a neighborhood of a real interval of the form } 
$-1 +\alpha (\lambda) \leq p_0 \leq 1- \alpha(\lambda)$ (fig 2a). 
This comes from the fact that for $0 < \lambda < +\infty$, the full upper (resp.
lower) 
half-plane in the variable $k_0$ represents a set of 
points $(k_0,k_1)$ of $h_{\lambda}$ in $T^+$ (resp. $T^-$) 
\footnote{${ }^{(5)}$}{To see this, one can e.g. rewrite the equation of
$h_{\lambda}$ as follows:
${U-1 \over U+1}=\lambda\  {V-1\over V+1}$ with $U= k_0 +k_1, \ V= k_0 -k_1,$ 
which entails (for 
$\lambda >0$)  
the condition $\Im m\, U \times \Im m\, V >0$, and therefore the fact that 
all complex points $(k_0,k_1) \equiv (U,V)$ in $h_{\lambda}$ belong either to 
$T^+$ or to $T^-$ according to whether $\Im m\, k_0 \equiv {1\over 2}(\Im m\, U
+ \Im m\, V)$ is 
positive or negative.}
and that these two half-planes are connected by small real intervals       
$]-1, -1+ \alpha[$, $]1- \alpha, 1[$ which represent points in
$\delta_{\epsilon}$.  
Moreover, for   
$1-\eta < \lambda <1+ \eta$ {\sl the full unit disk} $|k_0| <1$ is in the
analyticity domain 
of $\underline F$ (fig 2b) since the corresponding arcs  
$\breve h_{\lambda}$ are all contained in   
$\delta_{\epsilon}$. 
\vskip 0.25 truecm 
%\input micf2.tex
\newdimen\fixhoffset
\fixhoffset=\hsize \relax
\advance\fixhoffset by -16.000\varcm
\divide \fixhoffset by 2
\hbox{\kern\fixhoffset\vbox to 10.00 \varcm{\offinterlineskip
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\point 6.10 4.50 {$1$}
\point 3.50 4.50 {0}
\point 1.30 4.50 {$\scriptstyle -1+\alpha(\lambda)$}
\point 4.70 4.50 {$\scriptstyle 1-\alpha(\lambda)$}
\point 1.50 1.00 {$a)\ \lambda\ \hbox{arbitrary in}\ {\cal V}$}
\point 12.50 5.00 {\spot}
\point 9.50 4.50 {$-1$}
\point 15.10 4.50 {$1$}
\point 12.50 4.50 {0}
\point 10.50 1.00 {$b)\ 1-\eta < \lambda\ < 1+\eta$}
\hfill}}}\hfill}


\vskip 0.25 truecm
\centerline{Fig.~2. Initial analyticity domains of 
$\underline F(k_0,\ \lambda)$}
\centerline{in the $k_0$-plane}
\vskip 0.5 truecm

\noindent
Now consider the Cauchy integral 
$$ I(k_0, \lambda) = {1\over 2i\pi} \oint_{|k'_0| =1-{\alpha(\lambda) \over 2}} 
{{\underline F}(k'_0,\lambda) \over k'_0 - k_0} dk'_0,$$ which is a holomorphic
function of $k_0$ and 
$\lambda$ for $k_0$ varying in the unit disk and $\lambda$ varying in ${\cal
V}$; 
in view of the 
latter analyticity property of $\underline F$, one has  
$ I(k_0, \lambda) = \underline F(k_0, \lambda)$  
for $1-\eta < \lambda < 1+\eta$ and therefore  
$ I(k_0, \lambda) $ provides an analytic continuation of  
$ \underline F(k_0, \lambda)$ itself inside the full unit disk $|k_0| <1$ {\sl
and therefore on 
the real interval $]-1,+1[$ which represents the arc  
$\breve h_{\lambda}$ for all $\lambda $ in the interval $]o,+\infty[$.}   
By coming back to the original variables $(k_0,k_1)$, this shows that $F$ admits
an 
analytic continuation in the full region $\diamond$.
In the most general version of the theorem in two dimensions,  
the neighborhood ${\cal R}_{a,b}$ of the given time-like segment $]a,b[$ is
considered as a
union of rectangles of the previous $\delta_{\epsilon}-$type, whose thickness
$\epsilon$ 
tends to zero while they tend to $]a,b[$: the double-cone (or square) 
${\diamond_a^b}_{|d=2}$ is then clearly 
obtained as the union of the corresponding squares ${\diamond}$ obtained in the
previous 
procedure of analytic completion . Finally   
the proof of the theorem in the $d-$dimensional case is obtained by 
applying the two-dimensional result in all the planar sections passing
by $a$ and $b$, since i) the two-dimensional sections  of the tubes $T^{\pm}$
are 
the corresponding tubes of the (complexified) planar sections, and   
ii) $\diamond_a^b$ is generated by the union of all double-cones of 
the previous type ${\diamond_a^b}_{|d=2}$ in these planar sections.    


\vskip 0.5cm
{\bf 2.2 Microcausality and energy-positivity in all frames imply Lorentz
invariant   
spectral supports} 
 
A basic implication of microcausality together with energy-positivity in all
frames is the 
fact that propagators $F(k)$ of the underlying fields have to be {\sl
holomorphic in a  
domain which is invariant under all complex Lorentz transformations}, even if
these 
propagators are not Lorentz invariant functions due to the fact that the Lorentz
symmetry is broken in the  
representation of the fields under consideration. The key property which is at
the origin 
of this peculiarity is the 
following

\vskip 0.2cm
{\sl Property B (``K\"allen-Lehmann domain''): 

\noindent
Let ${\cal R}= {\cal R}_0$ be the set of all 
space-like energy-momentum vectors $p= (p_0, \vec p):\   
|p_0| < |\vec p|$. Then  
any function $F(k)$ holomorphic in  
$\Sigma_{{\cal R}_0} =T^+ \cup T^-\cup {\cal R}_0$ admits an analytic
continuation in 
the domain $\hat \Sigma_{{\cal R}_0} $ which is the set of all 
{\sl complex} vectors $k= (k_0, \vec k)$ 
such that $k^2 \equiv k_0^2 - {\vec k}^2 $ is different from any positive number
and from zero.}  

\vskip 0.1cm
{\sl Interpretation of Property B:}\ \  

Energy-positivity in all Lorentz frames implies that the distribution 
$<\tilde \Phi(p)\tilde \Phi(-p)>$ vanishes in the complement of ${\overline
V}^+$   
and therefore, in view of (3),     
that the coincidence relation   
$F^+_{|{\cal R}_0} = F^-_{|{\cal R}_0}$ is satisfied .   
Property B then implies the analyticity of the propagator $F(k)$ in the full
``cut-domain'' 
$\hat \Sigma_{{\cal R}_0} $.
Our denomination of  
``K\"allen-Lehmann domain'' for the latter     
is motivated by the fact that 
in the usual case when Lorentz invariance (or covariance) of the field is
postulated, 
the analyticity domain 
$\hat \Sigma_{{\cal R}_0} $ is directly  
obtained as a byproduct of the 
K\"allen-Lehmann integral representation of the propagator 
$$ F(k) \equiv {\underline F}(k^2)  =
{1\over 2i\pi} \int_0^{\infty} {\rho (\sigma)\over k^2- \sigma} d\sigma,$$
since the image of 
$\hat \Sigma_0 $
in the variable $k^2$ is   
the usual cut-plane domain ${\bf C} \setminus {\bf R}^+.$ 
Here, however, this 
Lorentz-invariant domain (considered in the full complex $k-$space) 
is obtained {\sl without any assumption
of Lorentz covariance  and of boundedness of the functions}, 
but purely on the basis of microcausality and energy-positivity.  

\vskip 0.1cm
Moreover, one will  show that any further information on the 
spectral support which is superimposed to the conditions of Property B 
implies the Lorentz-invariant
shape of all the components of the spectral support together with 
the invariance under complex Lorentz transformations
of the corresponding analyticity domain of the propagator. 
This is the purpose of the following property, whose statement 
in the present form is valid for any
spacetime dimension $d \geq 3$; we postpone to the proof the 
corresponding statement for the two-dimensional case, which requires  
a little more care in view of the decomposition of the light-cone 
into two straight-lines (the so-called ``left and right-movers'').  

\vskip 0.2cm
{\sl Property C (Lorentz-invariance of the borders of the spectral supports);
case $d\geq 3$: 

\noindent
If $\cal R$ is any real open set, not necessarily connected, containing ${\cal
R}_0$ then 
every function $F(k)$ holomorphic in 
$\Sigma_{\cal R} = T^+ \cup T^- \cup {\cal R}$ admits an  
analytic continuation in the (Lorentz-invariant) 
set $\hat {\cal R}$ of all real vectors $p$ 
whose Minkowskian norm $p^2$ has a value already taken at some vector in  ${\cal
R}$. 
Moreover the domain
$\hat \Sigma_{\cal R}$ in which every  
such function $F(k)$  can be analytically continued 
is the set of vectors $k$ such that $k^2$ takes all possible complex
values and all real values taken by $p^2$ when $p$ varies in ${\cal R}$.}  

\vskip 0.2cm
{\sl Interpretation of Property C:}\ \  

It is easy to see that Property $C$ (in its first part) implies that if
microcausality and
energy-positivity are satisfied, then the most general type of 
set $\hat {\cal R}$ where the (anti-)\break commutator function 
(3) has to vanish is a set composed of one distinguished region ${\cal R}_{M_0}$
of the  
form $-\infty < p^2 < {M_0}^2$, with ${M_0} \geq 0$ and of zero, one   
or several disjoint Lorentz-invariant regions of the form 
${M'}_i^2 < p^2 < {M}_{i}^2,$ where 
${M'}_1 \geq M_0 $ and 
${M'}_{i+1} \geq {M}_i,\ \ i= 1,\ldots, l-1,\ $
${M}_l \leq \infty$.
This implies in turn that the support of  
$<\tilde \Phi(p)\tilde \Phi(-p)>$ is exactly the union of all the ``thick (or
thin) 
hyperbolic shells'' defined by  
${M}_i^2 \leq p^2 \leq {M'}_{i+1}^2,\ p_0 \geq 0,$ ($i=0,1,\ldots,l-1$), $p^2
\geq M_l^2$   
(and of the origin if    
$<\tilde \Phi(p))> \not = 0$). 
The equality case  $M_i = {M'}_{i+1}$ corresponds to some ``thin shell'' $p^2=
M_i^2$. 
This thin shell situation occurs precisely    
when the distribution 
$<\tilde \Phi(p)\tilde \Phi(-p)>$ 
describes a particle with dispersion law $p_0 = \sqrt{ {\vec p}^2 + M_i^2 }.$   
No possibility is left for a Lorentz-symmetry breaking dispersion law. 
(Note that in this argument, 
the positivity of the Hilbert-space norm, 
implying the fact that the previous distribution is 
a positive measure factoring out 
a $\delta (p^2 - M_i^2)$, has not been used).  


\vskip 0.2cm
The proofs of Properties B and C given below are based on purely geometrical
arguments.   
Both of them  rely on a standard analytic completion procedure of geometrical
type, namely  
the ``tube theorem'' (stated at the beginning of this section); apart from the  
recourse to this piece of knowledge in complex geometry, 
these proofs are completely self-contained.    
The analytic completion procedure is actually at work in the two-dimensional
case, which we treat 
at first, while the general $d-$ dimensional case will be reducible to the
latter.  


\vskip 0.2cm
For the two-dimensional case, Property C must be properly restated as follows:

\vskip 0.2cm
{\sl Property C (Lorentz-invariance of the borders of the spectral supports);
case $d=2$: 

\noindent
If $\cal R$ is any real open set, not necessarily connected, containing ${\cal
R}_0$ then 
every function $F(k)$ holomorphic in 
$\Sigma_{\cal R} = T^+ \cup T^- \cup {\cal R}$ admits an  
analytic continuation in the (Lorentz-invariant) set 
$\hat \Sigma_{\cal R}$ obtained by adding to 
$\hat \Sigma_{{\cal R}_0}$  
the set of all (real or complex) vectors $k$ obtained by the action of 
real or complex Lorentz transformations  
on all vectors in ${\cal R}$. }   

\vskip 0.1cm
We note that in the $d-$dimensional case, the latter version of Property C is
equivalent to the 
former. In fact, the set of all vectors $k$ obtained 
from a given vector $p =\underline p \neq 0$ in ${\cal R}$ by real or 
complex Lorentz transformations is the full complexified hyperboloid $k^2 =
\underline p^2$ if 
$\underline p^2 \not= 0$ or the full complexified light cone $k^2=0$ if
$\underline p^2=0$. 
However in the two-dimensional case, the latter
statement differs from the former if ${\cal R}$ contains vectors $\underline p$ 
such that $\underline p^2=0$. 
In that case, the set of vectors $k$ obtained from such a vector $\underline p$
by the action of  
real or complex Lorentz transformations {\sl is not the full light cone} but
only the complexified line 
of left or right-movers 
which the given vector $\underline p$ itself belongs to. In other words, one of 
these two lines may very well be a singular set of the propagator, and therefore
contribute 
to the spectral support, although the other line doesn't; 
in such a case the parity symmetry of the 
spectral support is then  
broken but its Lorentz invariance is still preserved.   

\vskip 2 truecm 
%\input micf3.tex
\newdimen\fixhoffset
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\point 6.42 10.43 {$b_{-1/2}$}
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\point 12.43 4.42 {$b_{1}$}
\point 14.43 2.42 {$b_{3/2}$}
\point 2.50 12.50 {$B^{-}_{-1}$}
\point 4.50 10.50 {$B^{+}_{-1}$}
\point 6.50 8.50 {$B^{-}_{0}$}
\point 8.50 6.50 {$B^{+}_{0}$}
\point 10.50 4.50 {$B^{-}_{1}$}
\point 12.50 2.50 {$B^{+}_{1}$}
\point 14.50 0.50 {$B^{-}_{2}$}
\point 7.50 5.50 {0}
\point 15.00 5.50 {$\Im u$}
\point 8.20 13.50 {$\Im v$}
\hfill}}}\hfill}


\vskip 0.25 truecm
\centerline{Fig.~3. The set $B$ (dark gray)}
\centerline{and its convex hull $\check B$ (light gray)}


\vfill\eject
{\sl Proof of Properties B and C in the two-dimensional case:}

We here consider the case when $k= (k_0,k_1)$ varies in ${\bf C}^2$,  
corresponding to two-dimensional field-theory. 
In the complex variables $(U= k_0 + k_1,\ V= k_0- k_1),$ the domains $T^{\pm}$
are described as 
$T^+:\ \Im m\, U >0, \ \Im m\, V >0,$ \  
$T^-:\ \Im m\, U <0, \ \Im m\, V <0,$  and ${\cal R}_0$ is the real set: $p^2 =
UV <0.$ 
Let us then pass to the logarithmic variables $u = \log U,\ v = \log V$ and use
the 
fact that any function $F(k)\equiv F(U,V) = F({\rm e}^u, {\rm e}^v)\equiv
f(u,v)$  
is holomorphic and 
$2\pi-$periodic with respect to the variables $u$ and $v$ in the image of 
$T^+ \cup T^- \cup {\cal R}_0$ in the space of these variables.   
One easily sees that 
the domain $T^+$ is one-to-one mapped (periodically) onto each one of the
following (tube-shaped) 
domains $\Theta^+_l = {\bf R}^2 + i B^+_l$ ($l$ integer) where $B^+_l$ is the
square $\ 0<\Im m\, u -2l\pi< \pi,\ 0 <\Im m\, v +2l\pi<\pi$ and similarly 
for $T^-$ onto each one of the  
domains $\Theta^-_l = {\bf R}^2 + i B^-_l$ ($l$ integer) where $B^-_l$ is the
square $\ -\pi<\Im m\, u -2l\pi< 0,\ \pi <\Im m\, v +2l\pi<2\pi$.   
As seen on fig 3, the set of all squares $B^+_l$ and $B^-_l$ form a connected
set {\sl if one adds to them
the common boundary vertices} represented by all the points 
$b_{l\over 2} =(\Im m\, u = l\pi,\ \Im m\, v= (-l+1)\pi) $, with $l$ integer.
But  
as one easily checks, the sets $\theta_{l\over 2} = {\bf R}^2 +i b_{l\over 2}$ 
belong precisely to the (periodic) image 
of the set ${\cal R}_0$ ($UV= {\rm e}^{u+v}<0;\ \ {\rm e}^u,\ {\rm e}^v $\
real).  
The function $f(u,v)$ is therefore holomorphic  
in the union of all the tube-shaped sets $\Theta^+_l,\ \Theta^-_l$ and
$\theta_{l\over 2}$ and 
even (in view of the invariance of this edge-of -the-wedge configuration  
by all real translations in ${\bf R}^2$) in a {\sl connected open tube} $\Theta
={\bf R}^2 +i B$ such that 
$B$ is the union of all 
sets $B^+_l,\ B^-_l$ together with {\sl open neighborhoods} of all the points
$b_{l\over 2}$.   
Then in view of the tube theorem, $f(u,v)$ admits a ($2\pi-$periodic) analytic
continuation in 
the tube $\hat \Theta = {\bf R}^2 +i \check B$, where  
$\check B$, namely the convex hull of $B$, is (as shown by fig 3) the 
domain $\check B: \ 0< \Im m\, u+ \Im m\, v < 2\pi$. 
$F(k)$ therefore admits an analytic continuation in the inverse image of    
the tube $\hat \Theta$  
in the original variables, which is the set of all $k\equiv (U,V)$ such that
$0< \arg U + \arg V \equiv \arg k^2 <2\pi, $  namely  
the domain $\hat \Sigma_{{\cal R}_0}$ described in Property B. 

The domain $\hat \Sigma_{{\cal R}_0}$ can also be seen as the union of all
complex hyperbolae $h_{\zeta}$ 
in ${\bf C}^2$ with equation $k^2 = UV =\zeta$ such that $\zeta$ belongs to the
cut-plane   
${\bf C} \setminus [0,\infty[.$   
Let us now assume that in addition to   
${\cal R}_0$, the set ${\cal R}$ contains 
a given point $\underline p= (\underline U,\underline V)$ with 
$\underline p^2 = \underline \zeta \geq 0$. To be specific, consider the case
when one has: $\underline U >0$ and $\underline V \geq 0$ and put  
$\underline U= {\rm e}^{\underline t}\ >0,
\underline V= \underline \zeta {\rm e}^{-\underline t}\ \geq 0$, with $
\underline t$ real;   
the remaining cases would be treated similarly by i) exchanging 
the roles of $U$ and $V$ and ii)  changing $(U,V)$ into $(-U,-V)$ in the
following.  
We now use the fact that any 
function $F(k) \equiv F(U,V)$ analytic in $\Sigma_{\cal R} =T^+ \cup T^- \cup
{\cal R}$   
is analytic in a complex neighborhood of $\underline p$ 
and therefore in particular in a set of the form  
${\cal N}(\underline p) =\{ k=(U,V);\ U={\rm e}^t,\ V=\zeta {\rm e}^{-t};\
(\zeta,t) \in S_1 \}$, 
where $S_1 = \{(\zeta,t);  
\underline \zeta -\epsilon <\zeta <\underline \zeta + \epsilon,\   
|t-\underline t| < \rho \}. $   
It also follows from Property B that the image $G$ of such a function $F(U,V)$
in the space of
complex variables $(\zeta, t)$, namely $G(\zeta,t) \equiv F({\rm e}^t, \zeta
{\rm e}^{-t})$, is  
analytic in the set $S_2 =  \{ (\zeta,t);\ |\zeta -\underline \zeta| < \epsilon,
\  
\Im m\, \zeta \not=0;\ t \in {\bf C}\}$ (with periodicity with respect to the
translations 
$t \to t +2il\pi$). Putting these two facts together, namely the analyticity of
$G(\zeta,t)$ in the union of the sets $S_1$ and $S_2$,  
and making the new change of variables 
$$\alpha= \log {\zeta-\underline \zeta +\epsilon \over \underline \zeta
+\epsilon -\zeta},
\ \  \beta =i \log (t-\underline t),$$ 
one checks that the function $g(\alpha, \beta) \equiv G(\underline \zeta + 
\epsilon {{\rm e}^{\alpha}-1 \over 
{\rm e}^{\alpha}+1},\ \underline t + {\rm e}^{-i\beta})$ is holomorphic in the
following
tube-shaped domain ${\cal T} = {\bf R}^2 +i {\cal B}$, where 
${\cal B}$ is the union of the (disconnected) open set $\{(\Im m\, \alpha, \Im
m\, \beta);\ 
0 < |\Im m\, \alpha| < {\pi\over 2};\ \Im m\, \beta \ {\rm arbitrary}\}$ with   
the ``connection interval''  
$\{(\Im m\, \alpha, \Im m\, \beta);\ 
\Im m\, \alpha =0 ;\ \Im m\, \beta  <\log \rho\}$ (see fig 4).  Now 
since the convex hull of ${\cal B}$ is obviously the domain  
$\check {\cal B} =\{(\Im m\, \alpha, \Im m\, \beta);\ 
{-\pi\over 2} < \Im m\, \alpha < {\pi\over 2};\ \Im m\, \beta \ {\rm
arbitrary}\}$,       
the tube theorem implies that $g(\alpha,\beta)$ admits an analytic continuation 
in ${\bf R}^2 +i \check {\cal B}$, and therefore that $G(\zeta,t)$ admits an
analytic continuation 
in the set 
$ \{ (\zeta,t);\ |\zeta -\underline \zeta| < \epsilon, \  
t \in {\bf C}\}$ (with periodicity with respect to the translations 
$t \to t +2il\pi$). 
Coming back to $F(U,V)$, this shows that $F$ admits an analytic continuation in 
a set which is the union of all complex curves parametrized by  
$U={\rm e}^t,\ V=\zeta {\rm e}^{-t};\ t\in {\bf C}$, for $\zeta$ varying in the 
disk $|\zeta -\underline \zeta| < \epsilon $. These curves are complex
hyperbolae 
except for the one corresponding to the value $\zeta =0$, which is the 
straight-line $V=0$, namely the (complexified) 
``right-mover'' component of the light-cone.  
All these curves can be seen as generated  by the action of all real or complex
Lorentz
transformations (parametrized by $t$) on the set ${\cal N}(\underline p)$   
and Property C is therefore established for the two-dimensional case.


\vskip 1 truecm 
%\input micf4.tex
\newdimen\fixhoffset
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\advance\fixhoffset by -16.000\varcm
\divide \fixhoffset by 2
\hbox{\kern\fixhoffset\vbox to 10.00 \varcm{\offinterlineskip
\def\point#1 #2 #3 {\rlap{\kern #1 \varcm
\raise #2 \varcm \hbox{#3}}}
\def\spot{{\kern -0.2em\lower.55ex\hbox{$\bullet$}}}
\vfill\special{psfile=nmicf4.ps}
\smash{\hbox to \hsize{%
\point 13.20 5.00 {$\Im \alpha$}
\point 8.20 9.50 {$\Im \beta$}
\point 8.20 6.40 {$\log \rho$}
\point 8.20 4.50 {0}
\point 4.20 4.50 {$-{\pi \over 2}$}
\point 11.20 4.50 {${\pi \over 2}$}
\hfill}}}\hfill}


\vskip 0.25 truecm
\centerline{Fig.~4. The set ${\cal B}$ (gray)}
\vskip 1 truecm




\vskip 0.1cm
As a by-product of the latter, we stress the following result which is used
below:

\vskip 0.1cm
{\sl Property B with masses:

\noindent
Let ${\cal R}= {\cal R}_{\mu}$ be the set of all 
real energy-momentum vectors $p$ such that $p^2 < {\mu}^2$.  
Then any function $F(k)$ holomorphic in  
$\Sigma_{{\cal R}_{\mu}} =T^+ \cup T^-\cup {\cal R}_{\mu}$ admits an analytic
continuation in 
the domain $\hat \Sigma_{{\cal R}_{\mu}} $ which is the set of all 
{\sl complex} vectors $k$ 
such that $k^2$ belongs to the cut-plane 
${\bf C} \setminus [\mu^2, +\infty[.$}   

{\sl Remark}\ \  It is sufficient that ${\cal R}$ is known to contain
(neighborhoods of) 
one point $\underline p$ on the line $V=0$ and one point 
$\underline p'$ on the line $U=0$ (besides ${\cal R}_0$) 
in order to obtain  
an analyticity domain 
$\hat \Sigma_{\cal R}$ 
of the previous type  
$\hat \Sigma_{{\cal R}_{\mu}} $: in fact, Property C implies that both complex 
lines $U=0$ and $V=0$ are contained in the domain, except maybe for the 
point $U=V=0$ which is not obtained by the previous analytic completion
procedure.
However, this point must also belong to the domain since 
an analytic function of two complex variables {\sl cannot be singular at an
isolated point} surrounded by its domain of analyticity (see e.g. [11]): it
admits an  
analytic continuation at this isolated point defined by an appropriate Cauchy
integral.   

\vskip 0.2cm
{\sl Proof of Properties B and C in the d-dimensional case:}

The general case when $k=(k_0,\vec k)$ varies in ${\bf C}^d$ (e.g. $d=4$ for
field theory in the 
physical Minkowskian space) will be treated by appropriately using the previous 
two-dimensional results  
in sections of ${\bf C}^d$ by (complexified) planes 
containing a time-direction. 

Let $\underline k=\underline p+i\underline q$ be any vector in 
${\bf C}^d$ such that ${\underline k}^2 \in {\bf C} \setminus [0,+\infty[.$ 
In the affine Minkowskian space ${\bf R}^d$ 
consider the point $P$ such that $[OP] \equiv \underline p$ and the 
time-like plane $\Pi$ passing by $P$ and generated by $\underline q$ 
and the unit vector $e_0$ of the time-axis (or choose one of these planes 
and call it $\Pi$ in the degenerate case 
when $\underline q$ is along $e_0$ or is the null vector).  
There is a unique decomposition $\underline p= \underline p' + p_{\perp}$ such
that 
$\underline p'$ is parallel to $\Pi$ and $p_{\perp}$ is orthogonal
to $\Pi$ and therefore spacelike, if not the null vector: 
$p_{\perp}^2 = - \rho^2 \leq 0$. Introducing the 
complexified space $\Pi^{(c)}$ of $\Pi$ and the  
two-dimensional vector variable $k' = p' +iq$ such that every point $k=p+iq$ in 
$\Pi^{(c)}$ can be uniquely written as $ k= k' + p_{\perp}$
with $k'$ orthogonal to $p_{\perp}$, one has:  
${k'}^2= k^2 + \rho^2$.  
In $\Pi^{(c)}$ the section of the domain 
$\Sigma_{{\cal R}_0} =T^+ \cup T^-\cup {\cal R}_0$ is represented in the
vector-variable $k'$ as 
the union of the two-dimensional tubes ${T'}^+$ and ${T'}^-$ defined by ${\Im
m\, k'}^2 >0$ 
and respectively ${\Im m\, k'_0} >0, $ 
${\Im m\, k'_0} <0 $,  and of the real region defined by ${p'}^2 = p^2
-p_{\perp}^2= p^2+ \rho^2
< \rho^2.$ Therefore  
since the given vector 
$\underline k=\underline p' + p_{\perp} +i\underline q \equiv \underline k' +
p_{\perp} $   
is such that ${\underline k'}^2 ={\underline k}^2 + \rho^2 \in {\bf C} \setminus
[\rho^2,+\infty[,$ 
it follows from the two-dimensional {\sl Property B with masses}, applied in  
$k'-$space to the restriction $F'(k') = F_{|\Pi^{(c)}}(k)$ of any function
$F(k)$ analytic in
$\Sigma_{{\cal R}_0}$, that $F'$ admits an analytic continuation at $\underline
k'$ and 
therefore that $F$ itself can be analytically 
continued at the given vector $\underline k$. This shows that Property B 
holds in the $d-$dimensional case. 

Proof of Property C: let us assume that in addition to   
${\cal R}_0$, the set ${\cal R}$ contains 
a given vector $\underline p =[OP]$ with $\underline p^2 \geq 0$. 
Considering at first the case $\underline p^2 >0$, we know that the 
two-sheeted hyperboloid $H(P)$  
with equation $p^2 = \underline p^2$ can be seen as the union of all the
hyperbolae 
$h_{\alpha}(P)$ passing by $P$ 
which are the sections of $H(P)$ by all the two-dimensional planes
$\Pi_{\alpha}$ containing the
parallel to the time axis passing by $P$. In the complexified space of each 
(Minkowskian-type) plane $\Pi_{\alpha}$,
the domain $\Sigma_{\cal R} $ 
admits a restriction represented by a two-dimensional domain of the form
$\Sigma_{{\cal R}_{\alpha}}$, where ${\cal R}_{\alpha}$ contains $P$ 
in addition to a region of the form $p_{\alpha}^2 < \rho_{\alpha}^2$,
corresponding to the intersection of
${\cal R}_0$ by $\Pi_{\alpha}$. Therefore, in view of Property C for the
two-dimensional case 
the whole hyperbola  
$h_{\alpha}(P)$ (and even its complexified) belongs to the holomorphy envelope  
$\hat \Sigma_{{\cal R}_{\alpha}}$ of   
$\Sigma_{{\cal R}_{\alpha}}$.    
Since this is true for all hyperbolae
$h_{\alpha}(P)$, the full hyperboloid $H(P)$ itself belongs to the holomorphy
envelope  
$\hat \Sigma_{\cal R} $ of  
$\Sigma_{\cal R} $.  In the case $\underline p^2 =0$ (with $ P\neq 0)$), $H(P) $
is the light-cone 
and the previous argument 
of analytic completion in the union of all hyperbolic sections by the planes
$\Pi_{\alpha}$ yields 
the whole light-cone {\sl deprived from} the ``light-ray'' distinct from $[OP]$
and  
contained in the (unique) plane $\Pi_{{\alpha}_0}$ passing by the origin.  
However, this exceptional light-ray can be recovered by replacing $P$ by a
neighbouring point $P'$ 
also such that $[OP']^2 =0$: this is always possible since ${\cal R}$ is an open
set.  
(We also note that for the same reason one thus obtains in that case an {\sl
open set} 
$\hat {\cal R}$ of the form $p^2 < \epsilon^2$, the isolated  
point $p=0$ being also obtained 
according to the remark given at the end of the two-dimensional case).   
We have thus 
established the first part of Property C, namely the analytic completion at all
{\sl real} vectors 
$p$ such that the value $p^2$ is taken by some vector $\underline p= [OP] $ with
$P$ in ${\cal R}$.  

In order to establish the second part, we can now assume that ${\cal R}$ is the
union of  
${\cal R}_0$ together with a set of 
hypersurfaces $H_{\mu}$ of the form $p^2 = \mu^2,$ with $\mu \geq 0$; 
then there 
remains to prove that all the points of the corresponding {\sl complex} 
hypersurfaces $H_{\mu}^{(c)}$ can be 
reached by the previous analytic completion procedure. Here again, one can
proceed as in the proof of
Property B, namely taking any given vector   
$\underline k=\underline p+i\underline q$ in $H_{\mu}^{(c)}$, one considers the
complex two-dimensional
configuration in the corresponding plane $\Pi^{(c)}$ (specified above in the
proof of Property B).   
Now the section of $H_{\mu}$ by the plane $\Pi$ is a hyperbola contained in the
region ${\cal R}_{\Pi}$ 
of the corresponding section, so that as a result of Property C in the
two-dimensional case, 
the holomorphy envelope contains all the points of the corresponding {\sl
complex} hyperbola,
which includes 
by construction the given point $\underline k$. For 
the case $\mu =0$, the same method still works, including the treatment of the
vectors  
$\underline k=\underline p+i\underline q$ such that $p^2=q^2=0$, which belong to
complexified 
light-rays: the latter are again obtained by the two-dimensional version of
Property C    
in the special case of the right and left movers (no complex light-ray can be
excluded since  
each light-ray has all its real points in the analyticity domain). 
This ends the proof of Property C in the general case.    




\vskip 0.5cm
\centerline{\bf 3 Shape of the energy-momentum spectral supports for the
$N-$point functions} 

\vskip 0.3cm
\noindent
We shall now show that the previous study can be repeated  
{\sl for the sector generated by ``two-field vector-states'' of the form 
$\int \varphi(x,x') \Phi(x)\Phi(x')> dxdx'$}. It   
is in fact possible  to perform a similar treatment in complex momentum space, 
in which propagators of the 
fields are now replaced by four-point functions of the latter: the corresponding
results
on the form of dispersion laws will then apply to composite particles appearing
as 
``two-field bound-states''. Subsequently, we shall indicate 
the existence of a similar treatment for the sectors of ``$n-$field 
vector states'' in terms of $2n-$point Green's functions with applications to
dispersion 
laws of composite particles appearing as ``$n-$field bound states'', with $n\geq
3$.  
The validity of such a general study relies in an essential way 
on the general formalism of 
the analytic Green's functions of interacting fields in complex momentum space
[17]. 

The basic fact is that there exists an analog of formula (3) for 
the four-point function, which can be written as follows (see again ${ }^{(3)}$
for our 
use of the bracket notation):
$$ F^+(p; p_1,p_2) - F^-(p; p_1,p_2) = 
<[\tilde R(p_1, p-p_1),  
\tilde R(p_2, -p-p_2)]_{\pm}>\ \ \ \ \eqno(4)$$ 
where $\tilde R$ denotes the Fourier transform of a retarded two-point field
operator 
carrying the total energy-momentum $p$: 
$$\tilde R(p',p-p') = 
\int {\rm e}^{ip\cdot x} 
\ {\rm e}^{ip'\cdot (x'-x)} 
\ \theta(x'_0 -x_0)\ [\Phi(x'), \Phi(x)]_{\pm}\ dx dx' \ \ \ \ \ \eqno(5)$$ 
and where 
$ F^+(p; p_1,p_2)$ and $ F^-(p; p_1,p_2)$ are distributions affiliated with the
{\sl ``generalized 
retarded four-point functions''} (see [8,9]).

\vskip 0.3cm
Here again, 
any usable information on the 
support of the energy-momentum spectrum 
of the theory in the corresponding two-field sector  
will amount to specifying an open subset ${\cal R}$ in the space of 
energy-momentum vectors $(p,p_1,p_2)$ {\sl whose boundary only depend on the
total  
energy-momentum vector} $p$   
in which the distributions  
$<\tilde R(p_1, p-p_1)  
\tilde R(p_2, -p-p_2)>$ 
and 
$<\tilde R(p_2, -p-p_2) 
\tilde R(p_1, p-p_1)>$  
vanish simultaneously.  
In view of (4), such a support property (corresponding to the 
knowledge of the ``intermediate states in the latter matrix elements'') then
implies the
{\sl coincidence relation} 
$F^+_{|{\cal R}} = F^-_{|{\cal R}}$.   

\vskip 0.3cm
Moreover, as in the case of propagators, 
the {postulate of microcausality} for the field $\Phi(x)$  
implies {\sl properties of analytic continuation} of the previous  
objects in {\sl complex energy-momentum space}, which play a crucial role.  
Even if the description of these properties is more complicated, due to the
occurrence of  
{\sl three complex energy-momenta} 
$k=p+iq,$ $ k_1= p_1 +iq_1,\ k_2=p_2 +iq_2$,  
the situation reproduces 
the case of propagators {\sl as far as the total energy-momentum $p$ is 
concerned}. In fact, $F^+$ and $F^-$ are boundary values of holomorphic
functions from 
tubes ${\cal T}^+,{\cal T}^-$  whose projections onto the space of {\sl complex}
total energy-momentum $k=p+ iq$ are 
respectively $T^+:\ q\in V^+$ and    
$T^-:\ q\in V^-$, so that formula (4) still appears (like (3)) as a 
discontinuity formula: it indicates that 
the discontinuity between the two holomorphic functions $F^+(k;k_1,k_2)$ and 
$F^-(k;k_1,k_2)$
is known to vanish on the set ${\cal R}$.        

\vskip 0.3cm
However we must describe more carefully the situation concerning the analyticity
properties of these functions in the 
{\sl ``internal momenta''}
$k_1$ and $k_2$.   
First, it is clear from formula (5) that in view of the support 
property of the retarded product ($x'-x $ contained in $\overline V^+$), 
$\tilde R$ is the boundary value of an (operator-valued) analytic function  
$\tilde R(k', p-k')$ from the tube $k'=p'+iq':\ q' \in V^+$ for all real $p$.   
Therefore the r.h.s. of Eq.(4) is the 
boundary value of a holomorphic function $\Delta F(p; k_1,k_2)$  of $(k_1,k_2)$ 
in the tube $\Theta$ defined by the conditions $q_1 \in V^+,\ q_2 \in V^+$ for
all real $p$.  

Now it is also shown [8,9] that the domains of analyticity of $F^+,F^-$ implied 
by microcausality are the tubes
${\cal T}^+$ and ${\cal T}^-$ defined by the following conditions:
$${\cal T}^+:\ \ q \in V^+,\  q_1\in V^+,\  q_2 \in V^+\ \ \ \ \ \eqno(6)$$ 
$${\cal T}^-:\ \ -q \in V^+,\  q+q_1\in V^+,\  q+q_2 \in V^+\ \ \ \ \ \eqno(7)$$
and one easily checks that these two tubes admit precisely {\sl as their 
common boundary} (at $q=0$) the tube $\Theta$ for all real $p$. On the latter,
there holds the
following discontinuity formula for the boundary values of $F^+$ and $F^-$:  
$$ \Delta F(p; k_1,k_2) = F^+(p; k_1,k_2) - F^-(p;k_1,k_2).\ \ \eqno(8)$$ 

The main geometrical difference with respect to the case of propagators is that 
the tubes ${\cal T}^+$ and 
${\cal T}^-$ in the big complex $(k,k_1,k_2)-$space {\sl are not opposite} as 
it is the case for $T^+$ and $T^-$ in $k-$space. As a matter of fact, in view of
(6) and (7), the union of  
the tubes ${\cal T}^+$ and 
${\cal T}^+$ admits a convex hull $\check{\cal T}$ which is contained in the 
tube defined by the conditions 
$q_1\in V^+,\  q_2 \in V^+,\  
q+q_1\in V^+,\  q+q_2 \in V^+.$ 
Now in such a situation, and provided  
the {\sl coincidence relation} 
$F^+_{|{\cal R}} = F^-_{|{\cal R}}$ holds true,    
there exists a generalized version of the 
edge-of-the-wedge theorem [10], which states that  
$F^+(k;k_1,k_2)$ and $F^-(k;k_1,k_2)$ still admit a {\sl common analytic
continuation} 
$F(k;k_1,k_2)$. The latter is  
analytic in the union of ${\cal T}^+$, ${\cal T}^-$ and of a complex set ${\cal
N({\cal R})}$ of the
following form: ${\cal N({\cal R})}$ is the intersection of a complex
neighborhood of ${\cal R}$ 
with the convex hull
$\check{\cal T}$ of  
${\cal T}^+ \cup {\cal T}^-$;  
in other words, $F^+$ and $F^-$ ``communicate analytically'' through  
the complex set ${\cal N({\cal R})}$  
which is bordered by ${\cal R}$, although not being analytic anymore in ${\cal
R}$ itself.

\vskip 0.2cm
In the present situation, the open set ${\cal R}$ is always of the following
``cylindric'' form: 
$p_1$ and $p_2$ are arbitrary and $p$ varies in an open set $\underline {\cal
R}$ (namely the
projection of ${\cal R}$ onto $p-$space). Then the equivalence of the following
two statements 
(proved in [8,9]) 
deserves to be stressed:  

a) the boundary values of 
$F^+(k;k_1,k_2)$ and $F^-(k;k_1,k_2)$ 
coincide on ${\cal R}$, 

b) $\Delta F(p;k_1,k_2)$ vanishes {\sl as an analytic function of $(k_1,k_2)$ in
$\Theta$} 
for all $p$ in $\underline {\cal R}$. 

Property b) means that the ``bridge'' in which $F^+$ and $F^-$ have a common
analytic continuation 
contains not only the ``small''set ${\cal N}({\cal R})$ but the ``large common
face'' 
defined by the conditions  $(k_1,k_2)$ in $\Theta$ 
for all $p$ in $\underline {\cal R}$. 

\vskip 0.3cm
As in Sec. 2, one is then led to make use of   
an analytic completion procedure in order to enlarge the 
primitive (``non-natural'') set  
$\Sigma_{\cal R} = {\cal T}^+ \cup {\cal T}^- \cup {\cal N({\cal R})},$  
in which $F(k;k_1,k_2)$ is 
known to be analytic.
It turns out that one can obtain results very similar to those of Sec 2, 
which reproduce the corresponding physical interpretations. 
In fact, the Properties A', B' and C' listed below can be seen as exact
counterparts of the   
respective Properties A, B and C, since they involve identical  
regions (now called) $\underline {\cal R}$ and $\underline {\hat {\cal R}}$ 
in the space of the total energy-momentum $p$, while the
additional analyticity properties with respect to the internal energy-momenta
$k_1$ and $k_2$ 
are a remnant of microcausality in these variables.    

\vskip 0.4cm
i) {\sl Dispersion laws with subluminal velocities}  

\vskip 0.2cm
{\sl Under the} weak {\sl assumption that energy-positivity only holds in} 
privileged {\sl Lorentz frames (see Sec 2-1
and ${ }^{(4)}$),  
microcausality implies that all the hypersurfaces ${\cal M}_i$ and ${\cal M}$
representing  
respectively dispersion laws $p_0=\omega_i(\vec p)$ of one-particle states and
the border of the
continuous energy-momentum spectrum of ``intermediate states in the  
matrix elements'' 
$<\tilde R(p_1, p-p_1)  
\tilde R(p_2, -p-p_2)>$ 
have to be space-like hypersurfaces.} This follows from  


\vskip 0.2cm
{\sl Property A':  

\noindent
Let ${\cal R}_{a,b}$ be the set of all points $(p,p_1,p_2)$
such that $p$ belongs to a neighborhood of a given  
time-like segment $]a,b[$  
with end-points $p=a$ and $p=b$ 
($b$ in the future of $a$). Then any function $F(k;k_1,k_2)$ holomorphic in  
$\Sigma_{{\cal R}_{a,b}} = {\cal T}^+ \cup {\cal T}^- \cup {{\cal N}({\cal
R}_{a,b})}$  
admits an analytic continuation in a (complex) domain which contains  
the set of all points $(p,k_1,k_2)$ such that $p$ belongs to the double-cone  
$\diamond_a^b$ and $(k_1,k_2)$ varies arbitrarily in the tube $\Theta$.}   


\vskip 0.2cm
The argument of Sec 2-1, based on the consideration of time-like segments
$]a,b[$ with $b$ 
contained in ${\cal M}$ or
${\cal M}_1$, then shows again the necessity of the 
space-like character of these hypersurfaces. 
In fact, for all such choices of $]a,b[$, 
the conclusion of Property A' implies that the discontinuity 
$\Delta F(p;k_1,k_2)$ of $F$ vanishes 
for all $p$ in $\diamond_a^b$ and $(k_1,k_2) $ in $\Theta$ and therefore that  
the distribution 
$<[\tilde R(p_1, p-p_1),  
\tilde R(p_2, -p-p_2)]_{\pm}>$ 
vanishes   
for all $p$ in $\diamond_a^b$ and $(p_1,p_2) $ arbitrary.  


\vskip 0.4cm
ii) {\sl Lorentz invariance of dispersion laws }  

\vskip 0.2cm
{\sl Under the (usual)} strong {\sl assumption that energy-positivity holds in} 
all {\sl Lorentz frames, 
microcausality implies 
(as in  Sec 2-2)  
that all the hypersurfaces ${\cal M}_i$ and ${\cal M}$ representing  
respectively dispersion laws $p_0=\omega_i(\vec p)$ of one-particle states 
and the border of the continuous energy-momentum spectrum of ``intermediate
states in the  
matrix elements'' 
$<\tilde R(p_1, p-p_1)  
\tilde R(p_2, -p-p_2)>$ 
have to be hyperboloid-shells with equations of the form $p_0= \sqrt {{\vec p}^2
+ m_i^2}$,
$p_0= \sqrt {{\vec p}^2 + M^2}$.}  
This follows from the applicability of  

\vskip 0.2cm
{\sl Property B':   

\noindent
Let ${\cal R}= {\cal R}_0$ be the set of all (real) configurations $(p,p_1,p_2)$
such that the total energy-momentum vector $p= (p_0, \vec p)$ belongs to the
following region       
$\underline {{\cal R}_0}:\  |p_0| < |\vec p|$. Then  
any function $F(k;k_1,k_2)$ holomorphic in  
$\Sigma_{{\cal R}_0} ={\cal T}^+ \cup {\cal T}^-\cup {\cal R}_0$ admits an
analytic continuation in 
the domain $\hat \Sigma_{{\cal R}_0} $ which is the set of all 
{\sl complex} configurations $(k,k_1,k_2)$ belonging  
to the convex hull
$\check{\cal T}$ of  
${\cal T}^+ \cup {\cal T}^-$ and   
such that $k^2 \equiv k_0^2 - {\vec k}^2 $ is different from any positive number
and from zero,}   

\vskip 0.2cm
supplemented by

%\eject  
\vskip 0.2cm
{\sl Property C' (Lorentz-invariance of the borders of the spectral supports):  

\noindent 
If $\cal R$ is any real open set, not necessarily connected, containing ${\cal
R}_0$ 
and of ``cylindric form'' $p\in 
\underline {\cal R}$, with  
$\underline {\cal R} \supset   
\underline {{\cal R}_0}$, $p_1,p_2$ arbitrary,  
then 
every function $F(k; k_1,k_2)$ holomorphic in 
$\Sigma_{{\cal R}} ={\cal T}^+ \cup {\cal T}^-\cup {\cal R}$ admits an analytic
continuation in 
the set of all configurations $(p,k_1,k_2)$ such that $(k_1,k_2)$ belongs to the
tube $\Theta$ 
and $p$ varies in an open set 
$\hat {\underline  {\cal R}}$ defined as in Property C: 
it is (for $d\geq 3$) the set of all real vectors $p$ 
whose Minkowskian norm $p^2$ has a value already taken at some vector in 
$\underline {\cal R}$. 
Equivalently (but then including the case $d=2$), it is the set 
of all vectors $p$ obtained from vectors in  
$\underline {\cal R}$ by the action of a (real) Lorentz transformation. } 


\vskip 0.2cm
The conclusion of Property C' implies that the discontinuity 
$\Delta F(p;k_1,k_2)$ of the holomorphic function $F(k;k_1,k_2)$ vanishes 
for all $p$ in 
$\hat {\underline  {\cal R}}$  
and $(k_1,k_2) $ in $\Theta$ and therefore that  
the distribution 
$<[\tilde R(p_1, p-p_1),  
\tilde R(p_2, -p-p_2)]_{\pm}>$ 
vanishes   
for all $p$ in  
$\hat {\underline  {\cal R}}$  
and $(p_1,p_2) $ arbitrary.  
It thus expresses the property of Lorentz invariance of the borders of the 
energy-momentum spectrum and therefore (according to the same analysis as in Sec
2-2) 
the results announced above follow.  

\vskip 0.2cm
A derivation of Properties A',B' and C' 
can be given along the same line as the proofs of Properties A,B and C 
presented above in Sec. 2.  
Let us only mention here that Property A' corresponds to a specific case of the 
double-cone theorem for tubes ${\cal T}^+$, ${\cal T}^-$ in general (i.e.
non-opposite) 
situations (see [16])  and that Property B' is exactly the statement given in
Theorem 1 of [8] 
for the case of $n=3$ vector variables, with $m=0$. 

\vskip 0.3cm
{\sl Remark:}\ \  In the statements previously given under i) and ii), the
constraints which 
were obtained concern the shape of the energy-momentum spectrum as it appears in
the subspace of 
two-field states generated by retarded products of the following form  
$R [\varphi]> = 
\int \varphi(x,x') 
\ \theta(x'_0 -x_0)\ [\Phi(x'), \Phi(x)]_{\pm}> \ dx dx'. $ 
\footnote{${ }^{(6)}$}{Rigorously speaking, the passage from support properties
of 
the ``scalar''  distribution\break  
$<\tilde R(p_1, p-p_1)  
\tilde R(p_2, -p-p_2)>$ 
to corresponding support properties of the vector-valued 
distribution $\varphi \to R[\varphi]>$  
relies on a Hilbert-space-norm argument.}     
However, it is clear  
that the same treatment and results are valid as well for two-field 
states generated by the corresponding advanced products, and therefore for 
the subspace generated by all states of the form  
$C [\varphi]> = 
\int \varphi(x,x') 
\ [\Phi(x'), \Phi(x)]_{\pm}> \ dx dx' $ (for all admissible test-functions
$\varphi$).  

\vskip 0.3cm
\noindent
{\sl The general case:}

\vskip 0.2cm
We shall now end this section by explaining  why the previous treatment of 
spectral properties of the space of ``two-field states'' can be generalized to
the 
spaces of ``$n-$field states'' for all $n\geq 3$. Although it is not here the 
right place for presenting this general treatment with all its technical
details, it  
is still possible to indicate briefly how it works.  

The formalism of {\sl generalized retarded operators (g.r.o.)} [17]  
allows one to introduce {\sl generalized absorptive parts}: these are  
expectation values of (anti-) commutators of the following form 
$<[\tilde R_{\alpha}(\{p_i;\ i\in I\}),  
\tilde R_{\alpha'}(\{p'_{i'};\ i'\in I'\})]_{\pm}>$,   
where the operators 
$\tilde R_{\alpha}(\{p_i;\ i\in I\})$ and 
$\tilde R_{\alpha'}(\{p'_{i'};\ i'\in I'\})$   
denote the Fourier transforms of $n-$point g.r.o. 
$R_{\alpha}$, $R_{\alpha'}$ with supports  
contained in relevant corresponding salient cones 
${\cal C}_{\alpha}$
and ${\cal C}_{\alpha'}$ in the space of differences $x_j-x_k$ 
(resp. $x'_{j'} - x'_{k'}$) of space-time vectors:  
these cones are (non-trivial) analogs of the supports of the usual
retarded and advanced operators of the case $n=2$ (i.e. $x_1 -x_2 \in {\bar
V}^{\pm}$).  
In our notation, $I$ and $I'$ represent disjoint subsets of $n$ elements
($|I|=|I'|=n$)
of the set $\{1,2,,\cdots, 2n\}$ and the corresponding energy-momenta $p_i,
p'_{i'}$ are 
linked by the energy-momentum conservation law 
$p=\sum_{i\in I} p_i
=-\sum_{i'\in I'} p'_{i'}$, $p$ being the  
total energy-momentum of the corresponding channel $(I,I')$ of the 
$2n-$point function of the fields considered; as previously (see ${ }^{(3)}$), 
it is understood that the distribution  
$\delta(\{\sum_{i\in I} p_i\} + \{\sum_{{i'}\in I'} p'_{i'}\})$    
has been factored out 
in the brackets $<\  >$.  

\vskip 0.3cm 
We then claim that for each $n$ and each $(I.I')$ 
there exists a complete set of g.r.o.  
$R_{\alpha}$, $R_{\alpha'}$ whose Fourier transforms   
satisfy a discontinuity formula analogous to (4) of the following form  
$$ F^+_{\alpha,\alpha'}(\{p_i;\ i\in I\}; \ \{p'_{i'};\ i'\in I'\})  
-F^-_{\alpha,\alpha'}(\{p_i;\ i\in I\}; \ \{p'_{i'};\ i'\in I'\}) = $$ 
$$<[\tilde R_{\alpha}(\{p_i;\ i\in I\}),  
\tilde R_{\alpha'}(\{p'_{i'};\ i'\in I'\})]_{\pm}>;\ \ \ \ \ \eqno(9)$$ in the
latter,   
$ F^{\pm}_{\alpha,\alpha'}(\{p_i;\ i\in I\}; \ \{p'_{i'};\ i'\in I'\}) $   
are distributions affiliated with the {\sl ``generalized 
retarded $2n-$point functions''}   
which are boundary values of analytic functions (still denoted by)  
$ F^{\pm}_{\alpha,\alpha'}(\{k_i;\ i\in I\}; \ \{k'_{i'};\ i'\in I'\}) $   
from respective tubes
${\cal T}^+_{\alpha,\alpha'},$  
${\cal T}^-_{\alpha,\alpha'},$ 
in the space of complex vectors $k_i= p_i+ iq_i, \ k'_{i'}= p'_{i'} +i q'_{i'},$
such that
$k =p+iq=\sum_{i\in I} k_i
=-\sum_{i'\in I'} k'_{i'}.$ 
These pairs of tubes 
play the same role as the pair 
(${\cal T}^+,$  
${\cal T}^-$) of the case of two-field states: all points in   
${\cal T}^+_{\alpha,\alpha'}$ 
(resp. ${\cal T}^-_{\alpha,\alpha'},$)  
satisfy the condition $q= \Im m k \in V^+$ (resp. $V^-$).  
Microcausality is a basic ingredient in the proof of the previous statement,
which relies on 
the results of [17 d), e)].  

\vskip 0.3cm
We are again led to express  
the energy-momentum spectral assumptions   
of the theory in the corresponding $n-$field sector  
by specifying an open subset ${\cal R}$ in the space of 
energy-momentum vectors $p_i,\ p'_{i'} $ whose boundary {\sl only depend on the
total  
energy-momentum vector} $p =\sum _{i\in I} p_i,$   
in which the distributions  
$<\tilde R_{\alpha}(\{p_i;\ i\in I\})  
\tilde R_{\alpha'}(\{p'_{i'};\ i'\in I'\})>$ and   
$<\tilde R_{\alpha'}(\{p'_{i'};\ i'\in I'\})  
\tilde R_{\alpha}(\{p_{i};\ i\in I\})>$   
vanish simultaneously. 
Here again, the edge-of-the-wedge theorem [10] implies that  
$ F^+_{\alpha,\alpha'}$  
and $ F^-_{\alpha,\alpha'}$ have a common analytic continuation 
$ F_{\alpha,\alpha'}$  
in a set of the form 
$\Sigma_{\cal R} = {\cal T}^+_{\alpha,\alpha'}   
\cup {\cal T}^-_{\alpha,\alpha'} \cup {\cal N}({\cal R}).$  


\vskip 0.3cm
One could then present the ``$n-$field-state version'' of Properties A', B' and
C' 
in a way which closely parallels the two-field state case. For brevity , we
shall not repeat 
the full statements and the corresponding physical interpretations which are
identical to 
those listed above in paragraphs i) and ii) under the respective ``weak'' and
``strong''
forms of the energy-positivity condition. To exhibit the parallelism of the
geometry of the   
$n-$field case with the one of the two-field case, it is sufficient to make a
little more precise 
the description of the situation in the sets of energy-momentum vectors $k_i$
and $k'_{i'}$
and the characterization of the domains  
${\cal T}^+_{\alpha,\alpha'},$  
${\cal T}^-_{\alpha,\alpha'},$ and of their common face in the subspace $k=p$
real.  

For $p$ real, we introduce 
the sets of complex vectors 
$K_I= \{\underline k_i =k_i- {p\over n};\ i\in I\}$ and  
$K'_{I'}= \{\underline k'_{i'} =k'_{i'} +{p\over n};\ i'\in I'\}$   
linked by the relations 
$\sum_{i\in I}\underline k_i =  
\sum_{i'\in I'} \underline k'_{i'} = 0$; correspondingly 
$Q_I= \Im m K_I$ 
(resp.$Q'_{I'}= \Im m K'_{I'}$) is the set of all $q_i$ (resp. $q'_{i'}$) such
that 
$\sum_{i\in I} q_i =0$  
(resp. $\sum_{i'\in I'} q'_{i'} = 0$). 
Each of the sets of vectors $K_I$, $K'_{I'}$ (resp.  
$Q_I$, $Q'_{I'}$) varies in a space of $(n-1)$ independent complex (resp. real) 
energy-momentum vectors. 


By taking into account analogs of formula (5) 
for the operators  $\tilde R_{\alpha}$ 
and $\tilde R_{\alpha'}$ 
together with linear identities between 
them (called ``Steinmann relations'' [17]), one can deduce from the support
properties  
of $R_{\alpha}$ and
$R_{\alpha'}$  
(namely supp $R_{\alpha} \subset {\cal C}_{\alpha}$,  
supp $R_{\alpha'} \subset {\cal C}_{\alpha'}$) the following analyticity
property: 
the r.h.s. of Eq.(9) is for every real $p$ the boundary value of an analytic 
function $\Delta F_{\alpha,\alpha'}(p; K_I,K'_{I'})$ of $(K_I,K'_{I'}),$  
holomorphic in a well-defined tube $\Theta_{\alpha,\alpha'}$ (playing the same
role as $\Theta$ in
the case $n=2$). This tube is specified by 
a set of conditions of the  following type in the space of the 
imaginary parts $(Q_I,Q_{I'})$. There exists 
a set $\Pi_{\alpha}$ of partitions $(J,L)$ of $I$ and  
a set $\Pi'_{\alpha'}$ of partitions $(J',L')$ of $I'$ such that
the defining conditions for 
$\Theta_{\alpha,\alpha'}$ are:  
$q_J =-q_L \in V^+$ and $q'_{J'}= -q'_{L'} \in V^+$   
for all $(J,L)$ in $\Pi_{\alpha}$ and   
all $(J',L')$ in $\Pi'_{\alpha'}$: in the latter the notation 
$q_J$ (resp. $q'_{J'}$) refers to the corresponding partial sum     
$\sum_{i\in J} q_i$ (resp.  
$\sum_{i'\in J'} q'_{i'}$).  The sets  
$\Pi_{\alpha}$ and  
$\Pi'_{\alpha'}$ are not arbitrary but must satisfy 
the so-called ``cell-conditions'' (see [17]) which 
express the fact that no linear subspace with equation $q_{M}=0 $ 
or $q'_{M'}=0$, with $M\subset I$ and  
$M'\subset I'$ intersects the domain     
$\Theta_{\alpha,\alpha'}$.  

\vskip 0.2cm
Now it can be shown that the 
tubes ${\cal T}^+_{\alpha,\alpha'}$  
and ${\cal T}^-_{\alpha,\alpha'}$ in which the functions  
$ F^+_{\alpha,\alpha'}$  
and $ F^-_{\alpha,\alpha'}$ are holomorphic are defined by the 
following conditions:
$${\cal T}^+_{\alpha,\alpha'}:\ \ q\in V^+,\    
-q_L \in V^+\  {\rm and}\  q'_{J'} \in V^+ \ \ \ \eqno(10)$$   
for all $(J,L)$ in $\Pi_{\alpha}$ and   
all $(J',L')$ in $\Pi'_{\alpha'};$
$${\cal T}^-_{\alpha,\alpha'}:\ \ -q\in V^+,\    
q_J =-q_L +q \in V^+\  {\rm and}\  q'_{J'}+q =-q'_{L'} \in V^+\ \ \eqno(11)$$   
for all $(J,L)$ in $\Pi_{\alpha}$ and   
all $(J',L')$ in $\Pi'_{\alpha'}.$ 

\vskip 0.2cm
These two tubes admit as their 
common boundary (at $q=0$) the tube $\Theta_{\alpha,\alpha'}$ for all real $p$. 
On the latter, there holds the
following discontinuity formula for the boundary values of 
$ F^+_{\alpha,\alpha'}$  
and $ F^-_{\alpha,\alpha'}$:  
$$\Delta F_{\alpha,\alpha'}(p; K_I,K'_{I'})= $$  
$$F^+_{\alpha,\alpha'}(\{k_i;\ i\in I\}; \ \{k'_{i'};\ i'\in I'\})_{|q=0}    
- F^-_{\alpha,\alpha'}(\{k_i;\ i\in I\}; \ \{k'_{i'};\ i'\in I'\})_{|q=0}.\ \ \
\eqno(12)$$  

One easily checks that the defining conditions (10), (11) 
of the tubes ${\cal T}^+_{\alpha,\alpha'}$  
and ${\cal T}^-_{\alpha,\alpha'}$   
are completely analogous to 
the defining conditions (6), (7) of ${\cal T}^+$ and  ${\cal T}^-$, up to the 
replacement of the two vector variables $q_1,\ q_2$  by all the vector variables
$-q_L,\ q'_{J'}$  corresponding to the sets of partitions $\Pi_{\alpha}$,
$\Pi_{\alpha'}$.  

As a matter of fact, it is known (see [17]) that it is sufficient to consider a 
subset of g.r.o. called ``Steinmann monomials'' 
$R_{\alpha}$, $R_{\alpha'}$  for which each of the corresponding sets    
$\Pi_{\alpha}$, $\Pi_{\alpha'}$ contains {\sl exactly} $n-1$ partitions (one
also 
says that the corresponding cell-conditions are ``simplicial'');  
in fact, the most general g.r.o. are linear combinations of these Steinmann
monomials.  
It then turns out that in this restricted class of g.r.o. the analog of 
Property B' 
coincides with Theorem 1 of [8] in its general $n-$vector form (with $m=0$): 
this property states that {\sl any function holomorphic in 
$\Sigma_{{\cal R}_0} = {\cal T}^+_{\alpha,\alpha'}   
\cup {\cal T}^-_{\alpha,\alpha'} \cup {\cal N}({{\cal R}_0})$ (with 
${{\cal R}_0} $ now defined by the conditions $|p_0| < |\vec p|$, 
$K_I$ and $K'_{I'}$ real and arbitrary), admits an analytic continuation at  
all the points $(k, K_I, K'_{I'})$ in the convex hull of the tube  
${\cal T}^+_{\alpha,\alpha'}   
\cup {\cal T}^-_{\alpha,\alpha'}$  
such that $k^2\equiv k_0^2 - \vec k^2$ is different from any positive number and
from zero.} 
Property C' then follows from B' as in the case $n=2$, 
while Property A' corresponds again to the double-cone theorem in a  
geometrical situation of general type.  

These considerations can be completed by a remark similar to the one 
given at the end of the case $n=2$ (including footnote ${ }^{(6)}$): 
since the g.r.o. generate (by linear 
combinations of Steinmann monomials) all the multiple (anti-)commutators of 
$n$ field operators, the constraints  on the energy-momentum spectrum 
apply to the subspace generated by all states of the form    
$C[\varphi]= \int \varphi(x_1,\ldots, x_{n-1},x_n) 
[\Phi(x_1),[\ldots,[\Phi(x_{n-1},\Phi(x_n)]..]] > \ dx_1\ldots dx_n. $  (for all
admissible test-functions $\varphi$). 


\vskip 0.3cm

\centerline{\bf 4 Concluding remarks}  
\vskip 0.2cm
In this paper, we have displayed the geometrical constraints on the 
shape of the energy-momentum
spectrum which 
result from microcausality together with (weak or strong)
energy-positivity requirements in any (boson or fermion) interacting field
theory.   
These results apply to field theories involving Lorentz symmetry breaking 
with a rather high degree of generality.
This is due to the purely geometrical character of our method, 
based on analyticity properties in 
several complex variables, which has allowed a strict exploitation of the latter
requirements {\sl in terms 
of Green's functions of the fields}: it is in terms of these objects that 
the spectral constraints are expressed. 
As a matter of fact, the Hilbert space interpretation 
of these constraints can be done separately as for instance in our 
Remark in  Sec. 3 (see our footnote ${ }^{(6)}$). An advantage of the method is 
therefore the fact that the constraints obtained are still proven to hold 
in an indefinite-metric framework, as for example in the usual treatment of the 
QCD-fields with a gauge-fixing preserving the microcausality conditions for the 
Green's functions.  

Another feature of these geometrical results (linked again to the method) is the
fact
that they still remain true if the usual temperateness conditions at infinity 
in energy-momentum space are violated, provided the primitive analyticity
domains
of the Green's functions expressing microcausality in that space are still
valid:
this includes cases when the fields have short-distance singularities which may
be 
wilder than distributions but still allow a generalized form of microcausality
to hold; 
in such cases, the Green's functions may still enjoy a temperate behaviour at
infinity 
in the Euclidean energy-momentum subspace (i.e. at purely imaginary energies)
and therefore admit a corresponding perturbative treatment valid (by analytic
continuation from
the Euclidean subspace) in the usual analyticity domains considered.
\footnote{${ }^{(7)}$}{Note that the method also applies to the (opposite) case
of Green's functions 
enjoying a behaviour at infinite energy-momenta which is very regular at real
energies 
but of exponential increase at purely imaginary energies: this is precisely 
what happens in the case 
of the fields generated (via space-time translations) by local observables in
the  
``local quantum physics'' framework of [4] which were considered in the original
works of 
Borchers and Buchholz [2,3] on the present subject. } 




\vskip 0.6cm
\centerline{\bf References}

\vskip 0.6cm
\noindent
[1] V.A. Kosteleck\'y and R. Lehnert, Phys. Rev. D {\bf 63}, 065008   
(2001).  

\vskip 0.2cm
\noindent
[2] H.J. Borchers and D. Buchholz, Commun. Math. Phys. {\bf 97}, 169-185 (1985).

\vskip 0.2cm
\noindent
[3] H.J. Borchers, Fizika {\bf 17}, 289-304 (1985).


\vskip 0.2cm
\noindent
[4] R. Haag, {\sl Local Quantum Physics},  Springer (1996) 



\vskip 0.2cm
\noindent
[5] H.J. Borchers, Commun. Math. Phys. {\bf 22}, 49-54 (1962).




\vskip 0.2cm
\noindent
[6] H.J. Borchers, {\sl Translation Group and Particle Representations in
Quantum Field\break  
\hskip 1cm
\quad Theory },  
Springer (1996).




\vskip 0.2cm
\noindent
[7] R. Jost and H. Lehmann, Nouvo Cimento {\bf 5}, 1598-1610 (1957); F.J. Dyson,
Phys. Rev. 
\hskip 1cm
{\bf 
\quad 110}, 1460-1464 (1958). 



\vskip 0.2cm
\noindent
[8] J. Bros , H. Epstein and V. Glaser, Nuovo Cimento  {\bf 31 }, 1265-1302
(1964). 


\vskip 0.2cm
\noindent
[9] H. Epstein, {\sl Some analytic properties of scattering amplitudes in
quantum field theory} 
in ``Particle Symmetries and Axiomatic Field Theory'', Brandeis Summer School
1965, Gordon and Breach   
New York. 

\vskip 0.2cm
\noindent
[10] H. Epstein, Journ. Math. Phys. {\bf 1}, 524-531 (1960).   

\vskip 0.2cm
\noindent
[11] S. Bochner and W.T. Martin, {\sl Several complex variables}, Princeton
University Press (1948). 

\vskip 0.2cm
\noindent
[12] A.S. Wightman, {\sl Analytic functions of several complex variables} 
in ``Relations de dispersion et  
particules \'el\'ementaires'', Les Houches Summer School 1960, C. De Witt and R.
Omnes eds, 
Hermann, Paris.  



\vskip 0.2cm
\noindent
[13] L. Asgeirsson, Math. Ann. {\bf 113}, 321 (1936).  



\vskip 0.2cm
\noindent
[14] V.S. Vladimirov, Doklady Akad. Nauk SSSR {\bf 134}, 251 (1960).  



\vskip 0.2cm
\noindent
[15] H.J. Borchers, Nuovo Cimento {\bf 19}, 787-796 (1961).   




\vskip 0.2cm
\noindent
[16] J. Bros , H. Epstein, V. Glaser and R.Stora, in ``Hyperfunctions and 
Theoretical Physics'',  Lecture Notes in Mathematics {\bf 449}, 185 
Springer-Verlag, Berlin 1975.  


\vskip 0.2cm
\noindent
[17]\ -a) D. Ruelle, Nuovo Cimento {\bf 19}, 356 (1961) and Thesis, Zurich 1959;

-b) O. Steinmann, Helv. Phys. Acta {\bf 33}, 257 (1960); {\bf 33}, 347 (1960); 

-c) H.Araki and N. Burgoyne, Nuovo Cimento {\bf 18}, 342 (1960); H. Araki, J.
Math. Phys. 
{\bf 2}, 163 (1961); 

-d) J. Bros, Comptes-rendus RCP no 25, CNRS Strasbourg (1967) and Thesis, Paris
1970; 

-e) H. Epstein, V. Glaser and R. Stora {\sl General Properties of the n-point
Functions in  
Local; Quantum Field Theory} in ``Structural Analysis of Collision Amplitudes'',
Les Houches 
June Institut 1975, R. Balian and D. Iagolnitzer eds, North-Holland, Amsterdam. 




\end




