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\begin{document}
\hspace*{\fill} LMU--TPW 96--18 \\[2ex]
\section*{}
\begin{center}
\large\bf
Complex Structures in Quantum Field Theory
\end{center}
%\vspace{-6ex}
\section*{}
\normalsize \rm
\begin{center}
{\bf Rainer Dick}\\[1ex]
{\small \it Sektion Physik der Universit\"at M\"unchen\\
Theresienstr.\ 37, 80333 M\"unchen, Germany}
\end{center}
\vspace*{\fill}
{\footnotesize Invited contribution to the II.\ International Workshop on
Classical and Quantum Integrable Systems, Dubna (Russia),
8--12 July 1996.}
%\newpage
%\noindent
%{}
\newpage
\noindent
\begin{center}
\large\bf
Complex Structures in Quantum Field Theory
\end{center}
%\vspace{-6ex}
%\section*{}
\normalsize \rm
\begin{center}
{\bf Rainer Dick}\\[1ex]
{\small \it Sektion Physik der Universit\"at M\"unchen\\
Theresienstr.\ 37, 80333 M\"unchen, Germany}
\end{center}
%\newpage
\noindent
{\bf 1.\ Introduction}\\[0.5ex]
Covariance under symmetry transformations is a powerful
constraint and provides a very useful tool
for the investigation of physical systems.
A particular example is provided by the infinite--dimensional
symmetry of two--dimensional field theories under the Virasoro group
\cite{BPZ}. This symmetry is partially inherent to
all generally covariant two--dimensional models,
since any Riemannian two--manifold admits a conformal gauge,
whence two copies of a subalgebra of the Virasoro algebra generate locally
the residual symmetry group remaining from the diffeomorphism group
after gauge fixing.
It is a remarkable fact, however, that many physically interesting
models in two dimensions carry representations of the full Virasoro
algebra.
The statement that any Riemannian two--manifold admits a
conformal gauge is equivalent to the statement that the manifold
carries a complex structure, and it is compatibility of
quantum field theory on two--manifolds
with the underlying complex structure which permits
us to draw
far reaching conclusions both on operator product expansions
and on the field content of these models \cite{FQS}.
Complex structures show up in other places in quantum field theory:
In twistor theory a complexification of space--time is employed to set
up the twistor correspondence between complexified Minkowski space
and twistor space \cite{P}. In string theory it is known that preservation
of
supersymmetry under compactification from ten to four dimensions
as a particular implication requires existence of a complex structure
on the internal six--manifold \cite{CHSW}. Furthermore, holomorphy constraints
on the field content of superpotentials play an important role in the
investigation of low energy effective actions in supersymmetric theories \cite{ADS},
while holomorphy of K\"ahler potentials is a powerful tool in the
investigation of extended supersymmetry \cite{SW}.
In this talk I will not review these important subjects but
concentrate on a few topics: After a brief repetition of a few
facts in two--dimensional conformal field theory in section 2, section 3 is
devoted to Ward identities and reproducing kernels and section 4 comments
on a higher--dimensional Witt algebra. Section 5 comprises
a digression into the physics
of chiral symmetry breaking, while in section 6 I discuss a map between primary
fields of half--integer conformal weight on spheres in momentum space
and Weyl spinors in 3+1 dimensions and an application of this
map to correlation functions
on the cone $p^2=0$.\\[1ex]
{\bf 2.\ Covariant Field Theory in Two Dimensions}\\[0.5ex]
There are a few facts in two--dimensional field theory which I would like
to recall, since they proved useful in constructing the isomorphy between
Weyl spinors and half--differentials in Minkowski space.
A remarkable fact of two--dimensional field theory is the isomorphy between
tensors and spinors on a two--manifold with the set of primary fields $\Phi$,
which in a conformal gauge transform according to\footnote{The
general factorized
transformation
behaviour of primary fields without conformal gauge fixing is described
in Ref.\ \cite{rd2}.}
\[
\Phi(z',\overline{z}')=
\Phi(z,\overline{z})\bigg(\frac{\partial z'}{\partial z}\bigg)^{-\lambda}
\bigg(\frac{\partial \overline{z}'}{\partial \overline{z}}\bigg)^{-\overline{\lambda}}
\]
More precisely, the set of tensors and spinors is isomorphic to the set
of primary fields with spins $\sigma=\lambda-\overline{\lambda}$
satisfying $2\sigma\in \mathbb{Z}$ \cite{rd2,hn}.
Moreover, $\check{\mbox{C}}$ech cohomology tells us that models
with a particular primary field content can be formulated
on any Riemannian two--manifold if and only if all spins in the model
satisfy $2\sigma\in \mathbb{Z}$.
Of special interest are chiral primary fields of weights $(\lambda,0)$,
since all
anomalous terms in their operator product expansions
with conserved currents arise
from extensions of the Lie algebra
generated by the Noether currents and from extensions of the
modules ${\cal M}_{1-\lambda}$:
\[
L_m\circ a_n=[(\lambda -1)m-n]a_{m+n}
\]
and
have to be determined from the first cohomology groups
of these modules\footnote{This yields the second cohomology group
of the Witt algebra ${\cal A}_1$ for $\lambda=2$.
See \cite{rd2} for a definition of the
complex involved in the construction of
$H^1({\cal A}_1,{\cal M}_{1-\lambda})$.}. One finds \cite{rd1}
\[
\mbox{dim}H^1({\cal A}_1,{\cal M}_{1-\lambda})=\delta_{\lambda,2}+
\delta_{\lambda,1}+2\delta_{\lambda,0}
\]
The corresponding short distance expansions of operator
products of chiral primary fields
with the holomorphic components
of the stress tensor $T$ or a Noether current $J$
follow from the Cauchy theorem
and are well
known\footnote{The cohomology of modules of the Virasoro algebra tells
us that there may appear one further anomaly in (\ref{ope1})
for $\lambda=0$. However, this anomaly corresponds to a
background charge and breaks
translational invariance of the operator product.}:
\begin{equation}\label{ope1}
T(z)\phi(\zeta)=\frac{c_\lambda}{(z-\zeta)^{\lambda +2}}
\delta_{\lambda^3-3\lambda^2+2\lambda,0}+\frac{\lambda}{(z-\zeta)^2}
\phi(z)+\frac{1}{z-\zeta}\partial_z\phi(z)
\end{equation}
\begin{equation}\label{ope2}
J(z)\phi(\zeta)=\frac{d_2}{(z-\zeta)^3}
\delta_{\lambda,2}+\frac{d_1}{(z-\zeta)^2}
\delta_{\lambda,1}+\frac{1}{z-\zeta}\delta\phi(z)
\end{equation}
where the Noether current is supposed to correspond to
a symmetry $\phi\to\phi+\delta\phi$.\\[1ex]
{\bf 3.\ Ward Identities and Reproducing Kernels}\\[0.5ex]
The derivation of operator products from conformal Ward identities
in two--dimensional field theory relies heavily on the Cauchy kernel.
Therefore, it should be possible to turn the argument around and infer from
Ward identities in a symmetric theory the existence of reproducing
kernels on the set of classical trajectories of the theory.
Ward identities between correlation functions in quantum
field theory
can be derived from the generating functional
\begin{equation}\label{wi1}
Z[J]=\int D\phi\exp\bigg(iS[\phi]+i\int d^4 x J(x)\phi(x)\bigg)
\end{equation}
since invariance of the correlation functions under
symmetry transformations
of the fields
$\phi_a(x)\to\phi_a(x)+\delta\phi_a(x)$ implies
\begin{equation}\label{wi2}
\langle i\delta\phi_a(x)\rangle=
\langle \delta S[\phi]\phi_a(x)+\phi_a(x)\int d^4 x'
J^b(x')\delta\phi_b(x')\rangle
\end{equation}
This equation holds for any correlation functions involving
also products of fields. However, for the derivation of the
short distance parts of the operator product it is enough
to consider only the expectation values of single
operators $\langle\phi(x)\rangle$. Writing down the Ward
identity for correlations of operator products tells us
that the Noether currents
satisfy a Leibnitz property, i.e.\ the operator
product is associative in the sense of Lie products.
In the sequel we allow for transformations which are
restricted to domains $\cal D$, but do not need to be continuous
on boundaries $\partial\cal D$.
If $\delta\phi$ perturbes
a solution of the equations of motion
\[
-J(x)=\frac{\delta S[\phi]}{\delta\phi(x)}
\]
by an internal symmetry,
Eq.\ (\ref{wi2}) implies
\begin{equation}\label{wi3}
\langle i\delta\phi_a(x)\rangle=
\langle\oint_{\partial\cal D} d^3\sigma'_\mu\,\phi_a(x)
\frac{\partial\cal L}{\partial(\partial_\mu\phi_b(x'))}\delta\phi_b(x')\rangle
=-\langle\oint_{\partial\cal D} d^3\sigma'_\mu\,\phi_a(x)
j^\mu(x')\rangle
\end{equation}
where $d^3\sigma'$ denotes the surface element on $\partial\cal D$,
and $j(x)$ is the corresponding Noether current.
Therefore,
\[
K_a{}^{b\mu}(x,x')=-i\phi_a(x)
\frac{\partial\cal L}{\partial(\partial_\mu\phi_b(x'))}
\]
provide the components of a reproducing kernel on the set of classical
trajectories of $S[\phi]$.
This clearly suggests far reaching generalizations
of the notions of
analyticity
and reproducing kernels we are aware of, in the sense
that the space of
classical solutions
of a quantum field theory with local symmetries allows
for a notion
of reproducing kernels and a generalized Cauchy theorem.
On the other hand, in cases where notions of analyticity
are well
under mathematical control, we may exploit established
knowledge
on reproducing kernels to infer results on operator product
expansions from (\ref{wi3}), as was done for euclidean quantum
field theory on two--manifolds.
For another example, consider a spinor field $\psi$ satisfying
the Dirac equation $(i\gamma^\mu
\partial_\mu -m)\psi(x)=0$ (which could be considered as a
a generalized monogenicity condition from a mathematical
point of view). In this case we would infer the operator product
\[
\psi(x)j^\mu(x')=S(x-x')\gamma^\mu\delta\psi(x')
\]
with $S(x-x')$ denoting a Dirac propagator.\\[1ex]
{\bf 4.\ A Witt Algebra in Higher Dimensions}\\[0.5ex]
In this section I would like to make a cautionary remark
about extensions of two--dimensional conformal field theory
to higher dimensions.
An immediate way to generalize two--dimensional conformal field theory
to higher dimensions relies on higher--dimensional complex manifolds,
which locally carry a higher--dimensional analog of the Witt algebra.
This can be inferred from a
local basis of meromorphic vector fields on ${\mathbb C}^n$
\[
L_{j\{m\}}=-\prod_{i=1}^n {z_i}^{m_i}\cdot z_j\frac{\partial}{\partial z_j}
\]
where $\{m\}=\{m_1,\ldots m_n\}$ is a vector of integers.
This yields an algebra ${\cal A}_n$:
\[
[L_{i\{k\}},L_{j\{m\}}]=k_j L_{i\{k+m\}}-m_i L_{j\{k+m\}}
\]
To determine the anomalous extensions of the product of
two stress tensors, we have to classify the central extensions
\[
[L_{i\{k\}},L_{j\{m\}}]=k_j L_{i\{k+m\}}-m_i L_{j\{k+m\}}
+C_{ij\{k\}\{m\}}
\]
satisfying a cocycle condition due to
preservation of the Jacobi identity:
\begin{equation}\label{cocon}
k_b C_{ac\{k+l\}\{m\}}-k_c C_{ab\{k+m\}\{l\}}
-l_c C_{ab\{k\}\{l+m\}}-l_a C_{bc\{k+l\}\{m\}}
\end{equation}
\[
-m_a C_{bc\{l\}\{k+m\}}+m_b C_{ac\{k\}\{l+m\}}
=0
\]
The possibility to shift the generators by central elements
implies the trivial solutions or coboundaries
\begin{equation}\label{cobon}
C_{ab\{k\}\{m\}}^{\mbox{\footnotesize{trivial}}}=m_a f_{b\{k+m\}}-k_b f_{a\{k+m\}}
\end{equation}
Unfortunately, in solving this cohomology problem we find the somewhat
disappointing result
\[
\mbox{dim}H^2({\cal A}_n)=\delta_{n,1}
\]
Therefore, when we really want to imitate the successes of two--dimensional
conformal field theory in higher dimensions, we seem to
end up with three canonical
choices: We may rely on the finite--dimensional conformal group, or we may
rely on the twistor correspondence employing a complexification of Minkowski
space, or we may employ some other variant of quaternionic analyticity.
However, we may also opt for another possibility: Making use of low--dimensional
complex structures in real Minkowski space.
Before discussing a specific example, where this option can be employed,
I will digress from mathematics to physics for a while and discuss some aspects
of chiral symmetry breaking in gauge theories.\\[1ex]
{\bf 5.\ Remarks on Chiral Symmetry Breaking}\\[0.5ex]
Chiral symmetry breaking is a characteristic
feature of low energy QCD which remains puzzling from a theoretical
point of view.
Spontaneous breaking of chiral $SU(N_f)$ symmetry
is expected to arise as
a consequence of confinement or as an instanton
effect \cite{CDG,evs}, but it is not clear which mechanism
drives chiral symmetry breaking.
Chiral symmetry breaking effects of confining forces have been
discussed in \cite{BC,corn}, and it has been pointed out that
in dual QCD
monopole condensation not only yields confinement but also
a chiral condensate through a gap equation \cite{BBZ}.
In QCD we would like to understand
how monopoles break chiral symmetry in spite of their
chiral coupling, or whether
instantons or other non--perturbative effects break chiral
symmetry before confinement.
This problem is also relevant for the nature
of the phase transition, since the absence of an
order parameter for confinement
in the presence of light flavors
excludes a second order phase transition, if
chiral symmetry breaking is causally connected to
confinement.
Pisarski and Wilczek pointed out that an $\epsilon$--expansion
for the corresponding $\sigma$--model indicates a first order
transition
for more than two light flavors, but that a second order transition
in the universality class of the $O(4)$ vector model
is likely to appear in case of two light flavors \cite{PW,fw}.
In the next section I report on recent results for
the the fermion correlation $\langle q(p_1)\overline{q}(p_2)\rangle$
on the orbit $p^2=0$, in an attempt to shed new light on the problem
from an unexpected angle \cite{rd3}.
Dynamical breaking of chiral flavor symmetry in gauge theories is a puzzle,
because it is very different from spontaneous magnetization:
In a ferromagnet the interaction tends to align the dipoles, while
thermal fluctuations restore disorder if the system has enough energy. Gluon
exchange, on the other hand, does not necessarily
align left-- and right--handed
fermions.
Stated differently, the massless Dyson--Schwinger equation for
the trace part of the fermion propagator always admits a
trivial solution, if a quark condensate is not inserted
{\it ab initio}\footnote{In the latter case the corresponding Dyson--Schwinger
equation yields a gap equation for the condensate. The puzzle then concerns
the identification of the "phonons" in QCD.}.
Therefore, chiral symmetry breaking has to be implemented
in unusual ways, if we want to recover it from
gauge dynamics: By requiring a
condensate as part of initial conditions,
in double scaling limits, through chiral symmetry breaking
boundary conditions, in chiral symmetry breaking regularizations, etc.
While this does not invalidate standard approaches to the problem, it serves
to remind the reader that some poorly understood mechanism leaves its
footprint on the long distance properties of the QCD vacuum, and
motivates the group theoretical
construction of Lorentz covariant correlation functions
given below.
The main ingredient of the work reported below is a mapping
between massless spinors in 3+1 dimensions and primary fields,
which relates the order parameter
to automorphic functions
under the Lorentz group. From a mathematical point of view,
the novel feature of the automorphic functions under
investigation is that they provide correlations between primary fields
on spheres of different radii, thus providing true representations of
the Lorentz group and extending the determination of correlation functions
in 2D conformal field theory. The nontrivial behavior of radii under the
boost sector of the Lorentz group allows for chiral symmetry preserving
terms in the correlation
functions which could not appear in a two--dimensional framework, while the
chiral symmetry breaking terms in turn appear
closely related to 2D fermionic correlation
functions.
There exists wide agreement that chiral symmetry
breaking in QCD arises both dynamically, as a genuine QCD phenomenon, and
through electroweak symmetry breaking, which in a standard scenario accounts
for the quark current masses\footnote{The electroweak sector
also contributes to breaking of chiral $SU(N_f)$ through the axial anomaly
since the charge operator $Q^2$ is not flavor symmetric.}.
Dynamical chiral symmetry breaking is then expected
to account partially for the difference
between current and constituent masses \cite{pol}. From this point of view,
the large discrepancy between current and constituent
masses, and the fact that there is not even an approximate parity degeneracy
in the hadron spectrum provides strong evidence for dynamical breaking of
chiral
symmetry.
Another argument in favor of dynamical breaking
of chiral symmetry comes from the Gell-Mann--Oakes--Renner relation:
\begin{equation}
2m_q\langle \overline{q}q\rangle = -f_{\pi}^2 m_{\pi}^2
\end{equation}
where $m_q$ stands for a mean value of current quark masses.
This relation is expected to hold in the sense of a leading approximation
in $m_q$, and works phenomenologically
the better the smaller the value of $m_q$
is \cite{leut}.
While this does not strictly imply $\lim_{m_q\to 0}\langle \overline{q}q\rangle
\neq 0$, it implies at least that the condensate vanishes weaker than first
order
in $m_q$.
The necessity of including non--vanishing condensates in QCD sum rules
provides further strong indication for spontaneous breaking of
chiral symmetry\footnote{This becomes particularly evident in
heavy--light systems, where the
condensate of the light quark is expected
to contribute to the meson propagator even
in the limit of vanishing current mass.},
while
yet another hint for
chiral symmetry breaking is provided by
`t Hooft's result that decoupling of heavy fermions does not comply
with local chiral flavor symmetry \cite{tH}.
Last not least, chiral symmetry breaking can also be addressed
in lattice simulations of QCD. In this framework
non--vanishing condensates have been
reported e.g.\ in \cite{born,GuBh}.
This review comprises a short summary of some
compelling arguments in favor of dynamical chiral symmetry breaking.
There is strong evidence that
chiral symmetry breaking in QCD is not solely of electroweak
origin.\\[1ex]
{\bf 6.\ Half--Differentials and Fermion Propagators}\\[0.5ex]
Chiral spinors in 3+1 dimensions can be described as
primary fields of conformal weight $\frac{1}{2}$
on spheres in momentum space \cite{rd3}.
To exploit this observation, we work in
the Weyl representation of Dirac matrices,
and parametrize the unit sphere in momentum space in terms of
stereographic coordinates:
\begin{equation}\label{zdef1}
z=\frac{p_1+ip_2}{|{\bf p}|-p_3}\qquad\qquad\tilde{z}=
-\frac{p_1-ip_2}{|{\bf p}|+p_3}
\end{equation}
Proper orthochronous Lorentz transformations act on these coordinates
according to
\begin{equation}\label{zlor1}
z^{\prime}=z({\bf p}^{\prime})=\overline{U}\circ z({\bf p})=
\frac{\bar{a}z+\bar{b}}{\bar{c}z+\bar{d}}
\end{equation}
if $E=|{\bf p}|$, and
\begin{equation}\label{zlor2}
z^{\prime}=U^{-1T}\circ z({\bf p})=
\frac{dz-c}{a-bz}
\end{equation}
if $E=-|{\bf p}|$.\\
$U$ denotes the positive chirality spin $\frac{1}{2}$
representation of the Lorentz
group:
\[
U(\omega)=\exp(\frac{1}{2}\omega^{\mu\nu}\sigma_{\mu\nu}^{})
=\left(\begin{array}{cc} a & b\\ c & d\end{array}\right)\in
SL(2,{\mathbb C})
\]
We identify local functions written in co-ordinates
$(z,\bar{z},|\bf p|)$ and
$(\tilde{z},\bar{\tilde{z}},|{\bf p}|)$ via
\begin{equation}\label{weyl}
\psi(\tilde{z},\bar{\tilde{z}},|{\bf p}|)= -z\psi(z,\bar{z},|{\bf p}|)
\end{equation}
\begin{equation}\label{antiweyl}
\phi(\tilde{z},\bar{\tilde{z}},|{\bf p}|)= \bar{z}\phi(z,\bar{z},|{\bf p}|)
\end{equation}
and these overlap conditions can be rephrased as Weyl equations:
\[
(|{\bf p}|+{\bf p}\cdot{\bf\sigma})
\left(\begin{array}{c}\psi(z,\bar{z},|{\bf p}|)\\
\psi(\tilde{z},\bar{\tilde{z}},|{\bf p}|)\end{array}\right)=0
\]
\[
(|{\bf p}|-{\bf p}\cdot{\bf\sigma})\left(\begin{array}{c}
\phi(\tilde{z},\bar{\tilde{z}},|{\bf p}|)\\ \phi(z,\bar{z},|{\bf p}|)
\end{array}\right)=0
\]
Under (\ref{zlor1}) $\phi$ and $\psi$ transform according to
\begin{equation}\label{traphi}
\phi^{\prime}(z^{\prime},\bar{z}^{\prime},|{\bf p}^{\prime}|)=
(c\bar{z}+d)\phi(z,\bar{z},|{\bf p}|)
\end{equation}
\begin{equation}\label{trapsi}
\psi^{\prime}(z^{\prime},\bar{z}^{\prime},|{\bf p}^{\prime}|)=
(\bar{c}z+\bar{d})\psi(z,\bar{z},|{\bf p}|)
\end{equation}
if $E=|{\bf p}|$.
Due to (\ref{weyl},\ref{antiweyl}) this is equivalent to
\[
\left(\begin{array}{c}
\phi^{\prime}(\tilde{z}^{\prime},\bar{\tilde{z}}^{\prime},|{\bf p}^{\prime}|)\\
\phi^{\prime}(z^{\prime},\bar{z}^{\prime},|{\bf p}^{\prime}|)\end{array}\right)
=U\cdot
\left(\begin{array}{c}\phi(\tilde{z},\bar{\tilde{z}},|{\bf p}|)\\
\phi(z,\bar{z},|{\bf p}|)\end{array}\right)
\]
\[
\left(\begin{array}{c}
\psi^{\prime}(z^{\prime},\bar{z}^{\prime},|{\bf p}^{\prime}|)\\
\psi^{\prime}(\tilde{z}^{\prime},\bar{\tilde{z}}^{\prime},
|{\bf p}^{\prime}|)\end{array}\right)
=U^{-1\dagger}\cdot
\left(\begin{array}{c}\psi(z,\bar{z},|{\bf p}|)\\
\psi(\tilde{z},\bar{\tilde{z}},|{\bf p}|)\end{array}\right)
\]
The case $E=-|{\bf p}|$ corresponds to
$U\leftrightarrow U^{-1\dagger}$ in the equations above, but we will stick
to positive energy in the sequel. Correlations for $E=-|{\bf p}|$
can easily be
recovered
from the covariance cosiderations for positive energy through
a reflection ${\bf p}\to -{\bf p}$.
The implication of the Weyl equation in the derivation of the
map between massless spinors and half--differentials means,
that we rely
on an interaction picture when we employ the results above
to gain information on Lorentz covariant correlation functions.
To take advantage of this construction, we write a spinor on the
half--cone $E=|{\bf p}|$:
\begin{equation}\label{expand}
\Psi(p)=\left(\begin{array}{c}1\\0
\end{array}\right)
\otimes \left(\begin{array}{c}\bar{z}\\1\end{array}\right)
\phi(z,\bar{z},|{\bf p}|)+
\left(\begin{array}{c}0\\1
\end{array}\right)
\otimes \left(\begin{array}{c}1\\-z\end{array}\right)\psi(z,\bar{z},|{\bf p}|)
\end{equation}
with a corresponding representation of the correlation
function of massless fermions in the Dirac picture
\begin{equation}\label{prop}
\langle\Psi(p)\overline{\Psi}(p^{\prime})\rangle=
\end{equation}
\[
\left(\begin{array}{cc}0&1\\0&0\end{array}\right)\otimes
\left(\begin{array}{cc}\bar{z}z^\prime & \bar{z}\\
z^\prime&1\end{array}\right)\langle\phi({\bf p})\phi^+({\bf
p}^{\prime})\rangle+
\left(\begin{array}{cc}0&0\\1&0\end{array}\right)\otimes
\left(\begin{array}{cc}1 & -\bar{z}^\prime\\
-z& z\bar{z}^\prime
\end{array}\right)\langle\psi({\bf p})\psi^+({\bf p}^{\prime})\rangle
\]
\[
+
\left(\begin{array}{cc}1&0\\0&0\end{array}\right)\otimes
\left(\begin{array}{cc}\bar{z} & -\bar{z}\bar{z}^\prime\\
1&-\bar{z}^\prime
\end{array}\right)\langle\phi({\bf p})\psi^+({\bf p}^{\prime})\rangle +
\left(\begin{array}{cc}0&0\\0&1\end{array}\right)\otimes
\left(\begin{array}{cc}z^\prime & 1\\
-zz^\prime&-z\end{array}\right)\langle\psi({\bf p})\phi^+({\bf p}^{\prime})
\rangle
\]
The 2--point functions on the right hand side
transform under a factorized
representation of the Lorentz group.
This makes this representation very convenient for the investigation
of correlations $\langle\Psi(p)\overline{\Psi}(p^\prime)\rangle$
which comply with Lorentz covariance.
Stated differently, we ask which correlations
of spinors of the form $\Psi(p)$ could be constructed
from a non--trivial vacuum, or more general, between any Lorentz invariant
states.
The investigation in Ref. \cite{rd3} revealed
\begin{equation}\label{f1}
\langle\psi({\bf p}_1)\psi^+({\bf p}_2)\rangle=
\langle\phi({\bf p}_2)\phi^+({\bf p}_1)\rangle=
f_1\!\left(\frac{|{\bf p}_1|}{|{\bf p}_2|}\right)
\frac{1+z_1\bar{z}_2}{\sqrt{|{\bf p}_1||{\bf p}_2|}}\,
\delta_{z\bar{z}}(z_1-z_2)
\end{equation}
\begin{equation}\label{f2}
\langle\psi({\bf p}_1)\phi^+({\bf p}_2)\rangle=
\overline{\langle\phi({\bf p}_2)\psi^+({\bf p}_1)\rangle}=
\frac{1}{z_1-z_2}\,
f_2\!\left(|{\bf p}_1||{\bf p}_2|
\frac{(z_1-z_2)(\bar{z}_1-\bar{z}_2)}
{(1+z_1\bar{z}_1)(1+z_2\bar{z}_2)}\right)
\end{equation}
where Lorentz covariance does not fix $f_1$ and
$f_2$\footnote{The on--shell correlation in the vacuum of the free theory
$
\langle\psi({\bf p})\overline{\psi}({\bf p}^{\prime})\rangle =
-2p\cdot\gamma|{\bf p}|\delta({\bf p}-{\bf p}^{\prime})
$
is recovered from Eqs.\ (\ref{prop},\ref{f1},\ref{f2}) for
$f_1(x)=\delta(x-1)$, $f_2=0$.}.
The orbit $p^2=0$ thus contributes to a condensate
\begin{equation}\label{cond}
tr\langle\Psi(p)\overline{\Psi}(p^{\prime})\rangle=
-2f_2(|{\bf p}||{\bf p}^{\prime}|\sin^2(\frac{\theta}{2}))
\end{equation}
where $\theta$ denotes the angle between ${\bf p}$ and
${\bf p}^{\prime}$.
If the correlation function in configuration space
gives the positive energy contribution to a propagator
of initial conditions (modulo $i\gamma_0$), like in the
perturbative vacuum, then consistency of the result above
is expressed by the fact that
the $f_2$ terms do not
anticommute with $\gamma_5^{}$, while the $f_1$ terms
anticommute with $\gamma_5^{}$ and imply a restriction for external
momenta to be parallel.
On the other hand, since we pretend to deal with a confining theory (albeit disguised
in a non--perturbative vacuum), there is no reason to believe that
the propagator can be reconstructed from data on a single orbit
of the Lorentz group.
The results above presumably make sense only within a confining
theory, if observables are expressed in terms of meson or
baryon correlations, and the 4--point function is on my agenda since
quite a while now.\\[1ex]
{\bf Acknowledgements}:
I would like to thank the organizers of the meeting in Dubna
for their warm hospitality and for the invitation
to give a talk at the workshop. I would also like to thank Yum--Tong Siu and
the staff of the Mathematical Sciences Research Institute in Berkeley
for hospitality and the opportunity
to participate in the program on Several Complex
Variables. Part of this work was performed
during my stay in Berkeley in spring. Research at MSRI is supported in part by
NSF grant DMS--9022140.
This work has also been supported in part by
DFG grants DI 601/1--1 and DI 601/2--1.
{\small
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