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\title{AXIOMATIC APPROACH TO PERTURBATIVE QUANTUM FIELD
THEORY\thanks{Talk delivered at the ``Colloquium on New Problems in the
General Theory of Fields and Particles", Paris, July 1994.}}
\author{O. Steinmann\\
Universit\"at Bielefeld\\
Fakult\"at f\"ur Physik\\
D-33501 Bielefeld\\
Germany}
%\date{Revised version}
\maketitle
\begin{abstract}
A derivation with axiomatic methods of a perturbative expansion for the
Wightman functions of a relativistic field theory is described. The method
gives also the correlation functions of the fields in KMS states. Using these
results, a scattering formalism for QED is introduced, which does not
involve any infrared divergent quantities.
\end{abstract}
\newpage
\section{INTRODUCTION}
I will report on a method of developing the perturbation theory of
relativistic quantum fields within the context of axiomatic field theory
[1-3]. The central objects of the approach are the Wightman functions
\begin{equation}
W(x_1, \ldots, x_n) = \langle \Phi (x_1) \ldots \Phi(x_n)\rangle\ ,
\end{equation}
where $\langle \cdot\cdot \rangle$ denotes the vacuum expectation value,
and $\Phi(x_i)$ is any of the basic fields of the theory under consideration,
taken at the point $x_i$ of Minkowski space. Perturbative expansions, in the
form of sums over generalized Feynman graphs, are derived, starting from
equations of motion as dynamical input and using the Wightman properties [1]
as subsidiary conditions for the unambiguous solution of these equations.
The $W$-functions are known to determine the theory completely.
More generally, the method yields the following functions:
\begin{equation}
{\cal W} (X_1, s_1 | \ldots | X_n, s_n) = \langle T^{S_1} (X_1) \ldots
T^{S_n} (X_n) \rangle\ .
\end{equation}
Here the $X_i$ are finite sets of $4$-vectors $x_{i_1} , \ldots,
x_{i_{\alpha_i}}$, the $s_i$ are signs, and $T^+(X), T^-(X)$ is a
time-ordered or anti-time-ordered product respectively of fields
$\Phi(x_1), \ldots, \Phi(x_\alpha)$,
\linebreak
$X = \{ x_1, \ldots, x_\alpha\}$.
These functions include as special cases the Wightman functions (all
$\alpha_i = 1$), the completely time ordered Green's functions
$\tau (x_1, \ldots, x_\alpha)$ (for $n=1$), and the functions
\begin{equation}
\langle T^\ast (x_1, \ldots, x_n) T(y_1, \ldots, y_m)\rangle
\end{equation}
which occur in unitarity relations and play an important part in the
description of particle scattering (see section 4).
The method can also be applied to thermal field theories, in which case
the symbol $\langle \ldots \rangle$ stands for the expectation value in
a thermal equilibrium state with temperature $T \ge 0$, these states being
characterized by the KMS condition [4]. The two cases (vacuum and thermal)
will be discussed in parallel.
The interest of being able to calculate the correlation functions $W$ in
thermal field theories is obvious. In the vacuum case they are held to be
somewhat remoter from the quantities of direct physical interest: $S$-matrix
elements are easier to calculate from $\tau$ than from $W$. But the $W$ are
more suitable than the $\tau$ for studying some fundamental problems of field
theory, because their properties are more directly related to the
fundamental assumptions of the theory like locality, spectral properties,
and the like. An example of such a problem is finding the exact relation
between the physical state space of a gauge theory and its state space in a
non-physical gauge, especially a Gupta-Bleuler gauge. This problem has not
yet been satisfactorily solved, especially in the Yang-Mills case, where it
is connected with the confinement problem. Also, for setting up a convincing
scattering formalism for the particles of a gauge theory, for which the
customary asymptotic conditions do not hold, the functions (3) turn out to
be useful, as will be indicated for QED in section 4. It is therefore an
advantage of the present method that it produces directly expressions for the
general ${\cal W}$-functions.
Of course, these expressions can in principle also be derived from the
conventional Feynman rules for the $\tau$-functions [5]. But our method
is also of a more fundamental significance, since it avoids some of the
doubtful ingredients of the conventional canonical formalism. Using $W$
instead of $\tau$ as its basic objects, it avoids the notorious ambiguities
of time-ordering. It does not assume asymptotic conditions, which are not
satisfied for the charged fields of gauge theories. And it does not need
the canonical commutation relations, which have a doubtful status in
relativistic field theory, because, according to the available evidence,
interacting relativistic fields cannot in general be restricted to sharp
times.
And, finally, these considerations show that the methods of axiomatic
field theory {\it can} be used to handle dynamical problems.
For the sake of simplicity I will only discuss the $\phi^4_4$-model.
But the method can be applied to any relativistic, local, field theory,
in particular to gauge theories in covariant, local, gauges. The assumptions
underlying our formalism will be stated and briefly discussed in section 2.
The results will be described in section 3. In the time available it will
be impossible to give proofs, even in a sketchy form. For the proofs I must
refer to the original publications [6,7]. Finally, in section 4, an
application of these ideas to a proper description of scattering events
in QED will be discussed.
\section{ASSUMPTIONS}
The $\phi^4_4$ model is the theory of a scalar, hermitian, Wightman field
$\Phi(x)$ on 4-dimensional Minkowski space, satisfying the equation of
motion
\begin{equation}
(\Box + m^2) \Phi(x) = - \frac{g}{6} N(\Phi^3(x))\ .
\end{equation}
In the vacuum case, $m$ is the physical mass of a stable particle with
$\Phi$ as interpolating field, $g$ is a coupling constant, and $N$,
standing for ``normal product", denotes the renormalization prescription
needed to make sense out of the a priori undefined third power of the
distribution-valued field $\Phi(x)$. Renormalization, leading to the
disappearence of all ultraviolet divergences, can be handled by conventional
methods and will not be discussed further.
In thermal field theory we demand that the field equation (4) is exactly the
same independently of the temperature $T$. In other words: the KMS states
with any value of $T$ should all generate representations of the same
abstract field algebra. This means that the parameters $m$ and $g$, and
the subtraction prescription $N$ are independent of $T$. In particular,
$m$ denotes the mass of the $\Phi$-particle at $T=0$. It is {\it not}
the mass of any corresponding quasi-particle at positive temperatures,
which physical mass is temperature-dependent. Such an independence
prescription is necessary to make the temperature dependence of
physical quantities like specific heats, transport coefficients, and others
[8,9] unambiguous.
The equation of motion (4) implies the following infinite system of
partial differential equations for the Wightman functions:
\begin{equation}
(\Box_i + m^2) \langle \Phi(x_1) \ldots \Phi(x_i) \ldots \Phi(x_n)\rangle
= - \frac{g}{6} \langle \ldots N(\Phi^3 (x_i)) \ldots \rangle\ ,
\quad i = 1, \ldots, n\ .
\end{equation}
The right-hand side can be expressed in terms of $W$-functions, once the
normalization prescription $N$ has been fixed.
These equations must be solved, using the Wightman properties of the $W$
as subsidiary conditions. In our perturbative approach we will not need all
of these properties. Needed in an essential way are: translation invariance,
locality, the cluster property in a weak formulation (see below), and for
$T=0$ the spectral property. In the thermal case the latter is replaced
by the KMS condition [4], which we use in its $p$-space form: let
${\tilde W}$ be the Fourier transform of $W$, and let $\{ p_1, \ldots, p_n\},
\{ q_1, \ldots, q_m\},$ be two sets of 4-momenta. Then
\begin{equation}
{\tilde W} (p_1, \ldots, p_n, q_1, \ldots, q_m) = e^{\beta P_0} {\tilde W}
(q_1, \ldots, q_m, p_1, \ldots, p_n)\ ,
\end{equation}
where $\beta = (kT)^{-1}$ is the inverse temperature, and $P_0$ is defined
as
\begin{equation}
P_0 = \sum_i p_{i,0}\ .
\end{equation}
For $\beta \to \infty$ the equation (6) becomes the spectral condition in the
vacuum
\begin{equation}
{\tilde W} (p_1, \ldots, q_m) = 0 \quad \mbox{if} \quad P_0 < 0\ .
\end{equation}
Lorentz invariance is only used in a marginal manner, for fixing the
$N$-prescription in such a way that the field equation (4) transforms
covariantly. This condition is implemented at $t=0$ and fixes then $N$ also
for $T > 0$ because of its required $T$-independence.
{\it Not used} are: positivity, asymptotic conditions, and canonical
commutation relations. These are decided advantages of the method. The
dubious state of CCR's has already been remarked upon in the Introduction,
and positivity and asymptotic conditions are not satisfied in gauge theories
in local gauges.
In addition, we also demand the usual renormalization conditions fixing
$m, g, N,$ and the field normalization. These conditions, with the
exception of the last one, need again be applied at $T=0$ only, and are
then transferred to the thermal case by means of the $T$-independence of
$m,g,N$.
A perturbative solution of the stated problem is constructed as follows.
We insert the perturbative expansion
\begin{equation}
W(x_1, \ldots, x_n) = \sum^\infty_{\sigma=0} g^\sigma W_\sigma (x_1,
\ldots, x_n)
\end{equation}
into the equations (5) and equate the terms of order $g^\sigma$ on both
sides:
\begin{equation}
(\Box_i + m^2) W_\sigma (\ldots, x_i, \ldots) = - \frac{1}{6} \langle
\ldots N (\Phi^3 (x_i)) \ldots \rangle_{\sigma-1}\ .
\end{equation}
For $\sigma =0$ the right-hand side is zero. For higher $\sigma$ we solve
the equations by induction with respect to $\sigma$. Assuming the problem
to have been solved up to order $\sigma-1$, the right-hand side of (10) is
known, and $W_\sigma$ is determined as solution of the system of $n$
linear differential equations (10), using the Wightman properties and the
normalization conditions as subsidiary conditions. All the needed conditions
except the cluster property are linear in $W$, and must therefore be
satisfied separately in each order of perturbation theory. The cluster
property states that
\begin{equation}
\lim_{a\to\infty} W(x_1, \ldots, x_n, y_1 +a, \ldots, y_m +a) = W(x_1,
\ldots, x_n) W(y_1, \ldots, y_m)
\end{equation}
if $a$ tends to infinity in a space-like direction. Since the $a$-limit
need not commute with the derivation of $W$ with respect to $g$, this condition
cannot be easily transformed into a perturbative statement. We will
therefore only postulate a rather weak corollary of the condition. As is well
known, the perturbative expansion of $\tau$, and therefore also of $W$,
can be considered to be an expansion in powers of $\hbar$ instead of in
powers of $g$. And we demand that the equation (11) hold for each
$W(x_1, \ldots, x_n)$ in the lowest non-vanishing order in $\hbar$ contributing
to it. This suffices to guarantee the uniqueness of our solution. In my
formulas the $\hbar$-dependence is, however, suppressed by setting
$\hbar = 1$.
\section{RESULTS}
The problem described in the previous section is solved in two steps
[6,7]. Firstly one proves that the subsidiary conditions single out a
unique solution of the system (10) in every order $\sigma$. Secondly, a
solution of these equations satisfying all subsidiary conditions is
written down explicitly as a sum over generalized Feynman graphs.
These graphs, for the general case of the functions $\cal W$, are defined
as follows. First draw an ordinary Feynman graph, called a ``scaffolding",
of the $\Phi^4$ theory, with $\sigma$ vertices and with external points
corresponding to the arguments of $\cal W$. This graph is then partitioned
into a number of mutually non-overlapping subgraphs, called ``sectors".
To each factor $T^{s_\alpha} (X_\alpha)$ in $\cal W$ corresponds an
``external sector" containing all the external points of the
variables in $X_\alpha$, but no other external points. This sector
carries the number $\alpha$ and is said to be of type $s_\alpha$
(remember $s_\alpha = \pm$). If the adjacent external sectors with numbers
$\alpha$, $\alpha +1$, are of the same type, there may also exist an
``internal sector" not containing any external points, with number
$\alpha +1/2$, and of type $s_{\alpha +1/2} \not= s_\alpha = s_{\alpha +1}$.
In the thermal case there may also be an internal sector with number
$n + 1/2$ if the extremal external sectors 1 and $n$ are of the same type.
In this case we have $s_{n+1/2} \not= s_n = s_1$. Such extremal internal
sectors are not present in the vacuum case.
To such a partitioned graph we assign an integrand as follows. Each
external point carries a variable $x_i$, each vertex an integration
variable $u_j$. Within a sector of positive type the usual Feynman rules
apply, with vertex factors -ig and propagators
\begin{equation}
D_F (\xi ) = \Delta_F (\xi) + C_T(\xi)\ ,
\end{equation}
where $\Delta_F$ is the familiar Feynman propagator and $C_T$ is the thermal
correction
\begin{equation}
C_T (\xi) = i(2\pi)^{-3} \int d^4 p [e^{\beta|p_0|} -1]^{-1} \delta(p^2 - m^2)
e^{-ip\xi}\ ,
\end{equation}
which is only present for $T > 0$. In sectors of negative type the
complex-conjugate Feynman rules apply. A line connecting two points (external
or internal) in different sectors, with variables $z_i, z_j,$ carries the
propagator $-iD_+ (z_i - z_j)$. Here $z_i$ is the variable in the lower-
numbered sector, and
\begin{equation}
D_+ (\xi) = \Delta_+ (\xi) + C_T (\xi)\ ,
\end{equation}
where again $\Delta_+$ is the familiar invariant function and $C_T$
is only present for $T>0$. The graph is then integrated over the internal
variables $u_j$, and $\cal W$ is obtained as a formal sum over all
partitioned graphs satisfying the above rules.
Primitively $UV$ divergent subgraphs exist only within sectors, and the
corresponding divergences are removed by any of the conventional
renormalization procedures. The individual renormalized graphs give then
finite contributions if $T=0$ and $m>0$. For $m=0$ and $T=0$ the individual
graphs may be infrared divergent, but these divergences cancel in the sum
over all partitioned graphs with the same scaffolding. For $T > 0$ the
existence problem is open even in finite orders of perturbation theory.
\section{ASYMPTOTIC CONDITIONS IN QED}
Experimentalists usually observe particles, not fields. In order to make
contact with experiment, a field theory must therefore be able to describe
particles. Traditionally this is achieved by means of ``asymptotic
conditions" stating that the interacting fields of the theory, or
appropriate local functions of them, converge for large negative or positive
times to free fields, whose connections with a particle picture are well
understood. In axiomatic field theory there are essentially two different
versions of such conditions: the Haag-Ruelle condition [2,3] involving
strong convergence of time dependent states, and the LSZ condition [3]
involving weak convergence of suitably averaged field operators.
Both these conditions can be proved in theories possessing discrete mass
hyperboloids in their momentum spectrum. Unfortunately, gauge theories do not
fall in this class. Indeed, all available evidence shows that neither of the
two conditions is satisfied for fields carrying gauge charges. (An exception
are theories like the electroweak model with spontaneously broken gauge
symmetry). This raises the problem of a proper description of particle
scattering in such theories. And a concomitant problem is that of formulating
asymptotic completeness in such theories, i.e. the statement that the
scattering states span the full state space.
In the following I propose a solution of these problems in the case of
QED. The solution is based on a theorem which can be proved in perturbation
theory with the same kind of methods as used for what was explained in the
previous sections. Again, the results will be stated without proofs. The
proofs can be found in refs.~[10]. In the first reference a particularly
suitable Gupta-Bleuler gauge is used, in the second it is shown how the
results obtained can be transferred to physical gauges like the Coulomb gauge,
and how
to use them for establishing a scattering formalism.
I shall use a rather condensed notation, not explicitly distinguishing between
the various fundamental fields $\psi, {\bar \psi}, A_\mu$, of the theory.
The symbol $\Phi$ will denote any of these fields, as the case may be.
We define, for suitably normalized $\Phi$'s (the normalization condition
is non-trivial):
\begin{eqnarray}
\lefteqn{S(\ldots, {\bf p}_i, \ldots, t)} \nonumber\\
& & = \int \Pi_i dp_{i0} \exp\{ -it(p_{i0} - ({\bf p}^2_i + m^2_i)^{1/2})\}
T(\Pi_i \Phi(p_i))\ .
\end{eqnarray}
Here $T(\ldots)$ is the Fourier transform of the time ordered product of the
fields $\Phi(x_i)$. Let $\alpha$ be the number of $\psi$ in this product,
$\beta$ that of ${\bar \psi}, \gamma + \gamma^\prime$ that of $A$'s.
Let ${\cal A} \subset R^{3\alpha}, {\cal B} \subset R^{3\beta},
{\cal C} \subset
R^{3\gamma}$ be smooth sets and ${\cal C}_s \subset R^3$ a smooth set containing the
origin in its interior. Then the following statements hold to every finite
order of perturbation theory.
\bigskip
\noindent
\underline{Theorem}
\begin{description}
\item{a)} {\it The limit}
\begin{eqnarray}
\Pi_{{\cal A B C}} &=& \lim_{t\to\infty}
\sum^\infty_{\gamma^\prime =0} \frac{1}{\gamma^\prime !}
\int_{{\cal A} \times {\cal B} \times {\cal C} \times {\cal C}^{\times
\gamma^\prime}_s} \prod^{\alpha + \beta + \gamma + \gamma^\prime}_{i=1}
d^3 p_i \times \nonumber\\
& & \times S^\ast (\ldots, {\bf p}_i, \ldots, t) | \rangle K_i ({\bf p}_i) \langle |
S(\ldots, {\bf p}_i, \ldots, t)
\end{eqnarray}
{\it exists and is a projection operator. The $K_i$ are kernels whose exact
form depends on the type of the $i^{th}$ field.}
\item{b)} {\it For} ${\cal A}_\alpha = R^{3\alpha}\ , \ {\cal B}_\beta =
R^{3\beta}\ , \ {\cal C} = \emptyset\ , \ {\cal C}_s = R^3$ {\it we have}
\begin{equation}
\sum_{\alpha,\beta} \frac{1}{\alpha ! \beta !} \Pi_{{\cal A}_\alpha
{\cal B}_\beta \emptyset} = {\bf 1} \ .
\end{equation}
\end{description}
$|\rangle$ is the vacuum ket. The same results hold, of course, also for
$t\to -\infty$.
These results compare as follows with the traditional formulation.
$S^\ast |\rangle $ is the kind of state considered in the Haag-Ruelle condition
and can be proved under favorable conditions to converge strongly to a state
of free particles. Under these conditions the limit (16) without the summation
over $\gamma^\prime$ exists in the strong operator topology, and is a projection.
The expression on the left-hand side of (17) exists then also, if all summations,
including the one over $\gamma^\prime$, are carried out {\it after} taking
the t-limit. In an asymptotically complete theory the result is the identity.
Our result is thus distinguished from the usual formulation by summing
{\it first} over soft photons and taking the t-limit afterwards. A price to be
paid for this change is this: the limit is now not attained in the strong
operator topology, but only in the sense of sesquilinear forms. What
converges are the matrix elements of the expression (16) between smooth
states, i.e. states obtained by applying polynomials in the fields
$\Phi(x)$, averaged over sufficiently smooth test functions, to the
vacuum $|\rangle$.
The method amounts to introducing the time development as a natural
infrared regularization. For finite $t$ the terms in the $\gamma^\prime$-sum
exist individually, but the $t$-limit exists only for the sum, not for the
individual terms. The time $t$ thus takes over the role usually played by ad
hoc regularization parameters like a positive photon mass or a IR momentum
cutoff.
The statement b) in the Theorem is the promised new formulation of asymptotic
completeness.
For establishing a scattering formalism we use part a) of the Theorem for
describing the outgoing state. In terms of particles we can interpret
$\Pi_{{\cal ABC}}$ as the projection onto the set of states with $\alpha$
observed electrons with momenta in ${\cal A}$, $\beta$ observed positrons
in $\cal B$, $\gamma$ observed photons in ${\cal C}$, and any number
$\gamma^\prime$ of non-observed soft photons with momenta in ${\cal C}_s \cdot
{\cal C}_s$ need not be small: no ``small" terms are neglected in (16).
The inclusive cross section for finding such a final state is then given by
\begin{equation}
\sigma_{incl} ({\cal A} \times {\cal B} \times {\cal C}) = (\Psi_{in},
\Pi_{{\cal ABC}} \Psi_{in})\ .
\end{equation}
For the description of the initial state $\Psi_{in}$ we must use a different
method. Let me just briefly indicate this for a two-particle initial state
prepared at time $t=0$, with particles localized at that time in neighbourhoods
of the points ${\bf 0}$ and ${\bf x}$ with a macroscopic distance
$|{\bf x}|$. We define provisionally
\begin{equation}
\Psi_{in} = \int d^4 p \tilde{\varphi}_1 (p) \Phi(-p) \int d^4 q
\tilde{\varphi}_2 (q) e^{-i({\bf q,x})} \Phi (-q)|\rangle\ ,
\end{equation}
where the wave functions $\tilde{\varphi}_i$ are sufficiently smooth test
functions with compact supports in small neighbourhoods of two linearly
independent points $P_1, P_2$, on the respective mass shells, and whose
Fourier transforms $\varphi_i (u)$ in configuration space are negligibly small
(relative to a given experimental accuracy) outside of a small neighbourhood
of the origin. The asymptotic behaviour of the expression (18) for
$|{\bf x}| \to \infty$ can then be determined. It is found that the dominant
contribution $D({\bf x})$ decreases like $|{\bf x}|^{-2}$, provided that
the initial configuration corresponds to a classically possible
scattering event. This means that straight world lines through $u_1 = 0$
and $u_2 = (0,{\bf x})$ in the directions $P_{1,2}$ meet approximately in a
point, the point where the interaction actually takes place, and that a
final configuration in ${\cal A} \times {\cal B} \times {\cal C}$ is
compatible with momentum conservation. For sufficiently large $|{\bf x}|$ this
dominant contribution is a sufficient approximation to (18), and we can define
\begin{equation}
\sigma_{incl} = |{\bf x}|^2 D({\bf x})\ .
\end{equation}
A detailed evaluation leads to an expression which can be considered
as a generalization to inclusive cross sections of the LSZ reduction
formula. No detour through a non-existent S-matrix is needed in this
derivation.
Consider a process with two initial particles with 4-momenta $P_{1,2}$, and
$n$ observed final particles with momenta $Q_1, \ldots, Q_n$. If $P$ is an
electron momentum we define
\begin{equation}
N(P) = - 2\pi i ({{{\mbox P}\!\!\!\! /}} -m) \tau^\prime (P)\ ,
\end{equation}
whre $\tau^\prime$ is the clothed electron propagator, and similarly for
positrons. For photons we set $N(P)=1$. Observed photons must of course
be in physical states (spinor and vector indices have been suppressed
in my formulas). The inclusive cross section for the process in question
is then given by
\begin{eqnarray}
\sigma(P_i \to Q_j) &=& [\Pi N(-P_i)\Pi N (Q_j)]^{-1} (P_1^2 - m_1^2)^2
\ldots (Q_n^2 - m_n^2)^2 \times \nonumber\\
&& \times \langle| T^\ast (Q_1, \ldots, Q_n, - P_1, -P_2) T(Q_1, \ldots,
-P_2)|\rangle^\prime
\end{eqnarray}
where the prime in $\langle |\ldots |\rangle^\prime$ denotes omission of the
$\delta^4$ factor from momentum conservation. All momenta $P_i, Q_j,$ lie
on the respective mass shell. At the mass shell $N(\pm P)$ is IR divergent
in the electron case, and so is the amputated function $\langle | T^\ast
T |\rangle$. The expression (22) must therefore be calculated as a limit
taken from momenta with $P_i^2 < m^2_i, Q^2_j < m^2_j$. The function
$\langle | T^\ast T |\rangle$ is one of the ${\cal W}$ discussed before, and
can be calculated from the rules given in section 3.
\newpage
\noindent
{\large {\bf REFERENCES}}
\begin{description}
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Reading MA: Benjamin/Cummings 1978.
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Am.~Math.~Soc. 1965.
\item{[3]} N.N.~Bogoliubov, A.A.~Logunov, A.I.~Oksak, I.T.~Todorov: General Principles
of Quantum Field Theory. Dordrecht: Kluwer 1990.
\item{[4]} R.~Haag: Local Quantum Physics. Berlin: Springer 1992.
\item{[5]} A.~Ostendorf: Feynman rules for Wightman functions. Ann.~Inst.
H.~Poincar{\'e} 40, 273, 1984.
\item{[6]} O.~Steinmann: Perturbation theory of Wightman functions.
Commun.~Math. Phys.~152, 627, 1993.
\item{[7]} O.~Steinmann: Perturbative quantum field theory at positive
temperatures. To appear in Commun. Math. Phys.
\item{[8]} J.I.~Kapusta: Finite-Temperature Field Theory. Cambridge: Cambridge
University Press 1989.
\item{[9]} N.P.~Landsman, Ch.G.~van Weert: Real- and imaginary-time field
theory at finite temperature and density. Phys.~Reps. 145, 141, 1987.
\item{[10]} O.~Steinmann: Asymptotic completeness in QED. Nucl.~Phys.
B350, 355, 1991; Nucl.~Phys.~B361, 173, 1991.
\end{description}
\end{document}