\documentstyle[12pt,eqnnumch]{article}
\begin{document}
\title{PERTURBATIVE QUANTUM FIELD\\ THEORY AT POSITIVE TEMPERATURES:\\
AN AXIOMATIC APPROACH}
\author{O. Steinmann\\
Universit\"at Bielefeld\\
Fakult\"at f\"ur Physik\\
D-33501 Bielefeld\\
Germany}
\date{}
\maketitle
\begin{abstract}
It is shown that the perturbative expansions of the correlation functions
of a relativistic quantum field theory at finite temperature are uniquely
determined by the equations of motion and standard axiomatic requirements,
including the KMS condition. An explicit expression as a sum over generalized
Feynman graphs is derived. The canonical formalism is not used, and the
derivation proceeds from the beginning in the thermodynamic limit. No
doubling of fields is invoked. An unsolved problem concerning existence of
these perturbative expressions is pointed out.
\end{abstract}
\hfill\eject
\section{INTRODUCTION}
The traditional way of describing thermal equilibrium states of an infinitely
extended quantum system, in particular of a quantum field theory, begins by
restricting the system to a finite volume $V$, defining the canonical or grand
canonical equilibrium by means of the familiar density matrices, and then
going to the limit $V \to \infty$ (the "thermodynamic limit") for the quantities
for which this limit can be expected to exist [1,2]. This applies especially
to the correlation functions of the fields and closely related objects like
the expectation values of time ordered field products. Up to now most actual
calculations of such functions have been based on this approach, using a
Hamiltonian or Lagrangian formalism at finite $V$.
Another description of equilibria and their local disturbances, which can be
used directly in the thermodynamic limit, has been developed in the framework
of the algebras of local observables [3]. In this approach equilibrium states
are characterized through an analyticity requirement for correlation functions,
the so-called KMS condition. In the present paper we intend to show that this
axiomatic method, suitably adapted to a field theoretical context, is
perfectly capable of handling dynamical problems. More exactly, it will be
shown that perturbative expansions for the correlation functions of a
relativisitic field theory, and related functions, can be derived directly
in the thermodynamic limit, not making use of the canonical formalism, but using
as only inputs the equations of motion and the axiomatic requirements that
the correlation functions must satisfy. The result is represented as a sum
over generalized Feynman graphs. For the special case of time ordered
functions it agrees with the well-known result of the canonical approach.
Dispensing with the canonical formalism is also a major difference between
our approach and thermo field dynamics [2,4], a Fock space method developed
by H.~Umezawa and coworkers. We differ also from thermo field dynamics by
not invoking a doubling of fields, and by not assigning a basic role to
particles, including quasi-particles. We hold particles to be secondary
objects of the theory, of great phenomenological importance, but little
fundamental significance. In this respect we differ also from the views put
forward by Landsman in ref. [5].
We consider only the $\Phi^4_4$-model, i.e.~the theory of a scalar hermitian
field $\Phi(x)$ satisfying the equation of motion
\begin{equation}
(\Box + m^2) \Phi = - \frac{g}{6} N (\Phi^3) ,
\end{equation}
where $N$ stands for a normal-product prescription taking care of
renormalization. The restriction to $\Phi^4_4$ is merely a matter of
convenience. The generalization of the method to other models, including
gauge theories in local gauges, is straightforward.
We are interested in the correlation functions
\begin{equation}
W(x_1,\ldots, x_n) = \langle\Phi(x_1) \ldots \Phi(x_n) \rangle ,
\end{equation}
where $\langle\cdot\rangle$ denotes the expectation value in a thermal
equilibrium state
with temperature $T\ge 0$. These correlation functions describe the full
physical content of the theory: all observable quantities can in principle
be derived from them (for examples see e.g.~[1,2]). This is so because
knowledge of the $W$ allows the reconstruction of the full representation
of the field algebra by means of the GNS construction [2,3], which yields
a Hilbert space representation of the field algebra with a cyclic vector
$|\rangle$, such that $\langle A\rangle = \langle |R(A)|\rangle$ for any
sufficiently regular function
$A(\Phi)$, with $R(A)$ its representative. All observables of the theory
are supposed to be of this form, and local disturbances of the equilibrium
are created by applying suitable functions $F(\Phi)$ to $|\rangle$.
More generally, we consider the set of functions (or rather, distributions)
\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 $X_\alpha$ are non-overlapping sets of 4-vectors $x_i$, the
$s_\alpha$ are signs, and $T^\pm (X)$ denotes respectively
the time-ordered or
anti-time-ordered product of the fields $\Phi (x_i), x_i \in X$.
If each $X_\alpha$ contains only one variable, then $\cal W$ is the correlation
function $W(x_1, \ldots, x_N)$ irrespective of the choice of the signs
$s_\alpha$. For $N=1$ we obtain the usual time-ordered and anti-time-ordered
functions (Green's functions) of the theory. In the sequel the signs
$s_\alpha$ will be frequently suppressed when they are not essential to
understanding.
In a previous work [6], henceforth quoted as $V$, we have derived perturbative
expressions for the functions $\cal W$ in terms of generalized Feynman graphs
for
the case $T=0$, in which case $\langle \cdot \rangle$ denotes the vacuum expectation value.
This derivation uses neither the Hamiltonian nor the Lagrangian formalism,
but is instead relying on the Wightman axioms as an essential input. We
propose to generalize this method to the case $T>0$. Our way of proceeding
is closely modeled on that taken in V. The ideas and results of that paper
will be freely used. Equation (n.m.) of V will be referred to as eq.
(V.n.m.). The method conists in solving the differential equations for $W$ which
follow from the field equation (1.1), by a power series expansion in the
coupling constant $g$, using the axiomatic properties of $W$ as subsidiary
conditions.
The paper is organized as follows. The assumptions on which the formalism
is based will be stated and discussed in section 2. An explicit formal
expression for the $\cal W$'s as sums over generalized Feynman graphs will be
stated in section 3 and shown to possess the required properties. In section 4
the ultraviolet (UV) divergences of these graphs will be removed by
renormalization. It will be pointed out, however, that renormalization does
not guarantee the existence of the resulting finite-order expressions, on
account of the local singularities of the integrands. This problem remains
unsolved. Finally we will show in section 5 that the expressions of sections
3 and 4, provided they exist, are the only ones satisfying the assumptions
stated in section 2.
\section{ASSUMPTIONS}
In this section the assumptions on which our derivations are based,
in addition to the field equation (1.1), will be enumerated.
The following conditions for $W$ and $\cal W$ are taken over unchanged from
V.
a) The $W$ and $\cal W$ are {\it invariant} under space-time
translations and under space rotations. Invariance under Lorentz boosts
cannot be demanded.
b) {\it Locality} holds, i.e. $W(X)$ is invariant under the
exchange of two neighbouring variables $x_i, x_{i+1},$ if $x_i - x_{i+1}$
is space-like.
c) The {\it reality condition}
\begin{equation}
{\cal W} (X_1, s_1| \ldots | X_N, s_N)^\ast = {\cal W} (X_N, -s_N|\ldots |
X_1, -s_1)
\end{equation}
holds.
d) The {\it cluster property} (V.1.7) holds: the truncated
correlation function $W^T(X)$ vanishes if two complementary non-empty
subsets of $X$ are removed from each other to infinity in a space-like
direction. $W^T$ is recursively defined through the cluster expansion
\begin{equation}
W(X) = \sum W^T (X_1) \ldots W^T(X_N) , \qquad W^T(x_1, x_2) = W(x_1, x_2) ,
\end{equation}
the sum running over all partitions of $X = \{ x_1, \ldots , x_n\}$ into
an arbitrary number of complementary subsets with an even number of elements.
The cluster condition implies that the state $\langle\cdot\rangle$
describes a pure phase
[3]. Perturbation theory is not the proper vehicle for studying phase
transitions.
e) The functions $\cal W$ are {\it permutation invariant}
within each sector $X_\alpha$, and they satisfy the {\it splitting
property}
\begin{equation}
{\cal W} (\ldots |X_1 \cup X_2, +|\ldots ) = {\cal W} (\ldots |X_1 ,+|X_2 ,+|
\ldots)
\end{equation}
if $x_i^0 > x_j^0$ for all $x_i \in X_1, x_j \in X_2$. This property shall
hold in every Lorentz frame, not only in the rest frame of the infinite system
under consideration. To that extent we retain Lorentz invariance. These
conditions are not merely a definition of time ordering. That they can be
satisfied is intimately connected with locality.
The spectral condition of the vacuum representation does not hold at
positive temperatures. It is replaced by the {\it KMS condition}, which
we use in its p-space form (see [3], lemma 1.1.1 of chapter V). Define the
Fourier transform $\tilde{\Phi} (p)$ of $\Phi$ by
\begin{equation}
\tilde{\Phi} (p) = (2\pi)^{-5/2} \int dx e^{ipx} \Phi (x),
\end{equation}
and the Fourier transform $\tilde{\cal W} (P_1 |\ldots |P_N)$ of
${\cal W} (X_1 |\ldots |X_N)$ accordingly. Let $P_1, \ldots , P_N, Q_{N+1},
\ldots , Q_M , M > N,$ be finite, non-empty sets of 4-momenta. Define
\begin{equation}
P^0 = \sum_{p_i \in \cup P_\alpha} p^0_i .
\end{equation}
Then the KMS condition states that
\begin{equation}
\tilde{\cal W} (P_1 | \ldots |P_N |Q_{N+1} | \ldots | Q_M) = e^{\beta P^0}
\tilde{\cal W} (Q_{N+1}|\ldots |Q_M | P_1 | \ldots | P_N ) ,
\end{equation}
where $\beta = \frac{1}{kT}$ is the inverse temperature.
The normalization conditions stipulated in V are taken over unchanged
for the case $T=0$. They are the standard conditions demanding that $m$ be the
physical mass of the particles associated with the field $\Phi$ at $T=0$,
and defining the coupling constant $g$ and the field normalization in terms
of the Green's functions. In addition we demand that $\Phi$, considered as
an element of an abstract, representation independent, field algebra, is
$T$-independent. This implies that the field equation is $T$-independent,
meaning that the parameters $m$ and $g$ and the subtraction prescription
$N$ do not depend on the temperature. In perturbation theory the
$T$-independence of $N$ means that the UV subtractions and the subsequent
finite renormalizations have the values used at $T=0$ for all $T$. Note that
in the BPHZ method [7], which we will be using, the subtractions do not
involve prior regularizations or any formal juggling with divergent
quantities. The procedure is therefore well defined. For more details we
refer to section 4. A nonperturbative definition of $N$ can possibly be given
with the help of a Wilson expansion, as explained in section IV.2. of ref.
[7], i.e.~by a point-splitting method with suitable singular coefficient
functions. $T$-independence of $N$ could then be defined as $T$-independence
of these coefficients. We will not explore this possibility any further.
Note that the parameter $m$ is for $T>0$ not the physical mass of a particle.
Quasi-particle masses are determined by the positions of the singularities
of the clothed propagator, which are $T$-dependent.
\section{UNRENORMALIZED SOLUTION}
The coefficient $W_\sigma (X)$ of order $\sigma$ in the power series
expansion
\begin{equation}
W(X) = \sum^\infty_{\sigma =0} g^\sigma W_\sigma (X)
\end{equation}
is determined as a solution of the system of differential equations
\begin{equation}
(\Box_i + m^2) \langle \ldots \Phi (x_i) \ldots \rangle_\sigma =
- \frac{1}{6} \langle\ldots N(\Phi(x_i)^3)\ldots \rangle_{\sigma-1}
\end{equation}
satisfying the subsidiary conditions described in section 2. For $\sigma = 0$
the right-hand side is zero, for $\sigma > 0$ it can be calculated if the
problem has already been solved in order $\sigma -1$.
We first state a formal, unrenormalized, graphical representation for\linebreak
${\cal W}_\sigma (X_1 | \ldots | X_N)$ and then show that it does indeed
satisfy all the requirements.
For the free propagators we use the following notations. We define first
the p-space expressions
\begin{equation}
\tilde{\Delta}_+ (p) = \frac{1}{(2\pi)^3} \delta_+ (p) , \quad
\tilde{\Delta}_F (p) = \frac{i}{(2\pi)^4} \frac{1}{p^2 - m^2+i\epsilon} ,
\end{equation}
which are the usual vacuum propagators, and their thermal extensions
\begin{equation}
\tilde{D}_+ (p) = \tilde{\Delta}_+ (p) + \tilde{C} (p) , \qquad
\tilde{D}_F (p) = \tilde{\Delta}_F (p) + \tilde{C} (p)
\end{equation}
with the $T$-dependent correction term
\begin{equation}
\tilde{C} (p) = \frac{1}{(2\pi)^3} \frac{1}{e^{\beta\omega}-1} \delta
(p^2 - m^2) .
\end{equation}
Here $\delta_+(p) = \theta (p_0) \delta(p^2 - m^2)$ is the $\delta$-function
of the positive mass shell, and $\omega = ({\bf p}^2 + m^2)^{1/2}$. Note that the
additional term $\tilde{C}$ is the same for $\tilde{D}_+$ and $\tilde{D}_F$.
The $x$-space versions of these propagators are defined by
\begin{equation}
\Delta \ldotp (x) = i \int d^4 p \tilde{\Delta} \ldotp (p) e^{-ipx} , \qquad
D \ldotp (x) = i \int d^4 p \tilde{D} \ldotp (p) e^{-ipx} .
\end{equation}
The thermal propagators (3.4-5) need not be taken over from the traditional
formalism. They can be derived in our framework by the methods that will
be used in section 5 to establish uniqueness of our solution. But in the
present section we rely on an a posteriori justification, by showing that these
forms give rise to expressions with all the required properties.
${\cal W}_\sigma$ is represented as a sum over generalized Feynman graphs which
are defined as follows. Draw first an ordinary Feynman graph of the $\Phi^4$
theory with $|X| = \sum_\alpha |X_\alpha|$ external and $\sigma$ internal
vertices. Here $|X_\alpha |$ is the number of points in the set $X_\alpha$.
The graph need not be connected, but must not contain any components without
external points. This graph is called the "scaffolding" of the generalized
graph. Next, it is partitioned into non-overlapping subgraphs, called "sectors",
such that the external points of a set $X_\alpha$ belong all to the same
sector, but variables in different $X_\alpha$ to different sectors. There may
also exist "internal sectors" not containing external points. The sectors
are either of type $T^+$ or $T^-$. For external sectors this sign is given by
$s_\alpha$. To each sector $S$ we affix its number $\nu (S)$ according to
the following rules.
\begin{itemize}
\item[{i)}] $\nu (S) = \alpha$ for the external $X_\alpha$-sector.
\item[{ii)}] If $s_\alpha = s_{\alpha +1}$ there may be an internal sector
with number $\alpha + \frac{1}{2}$. Its type is the reverse of the adjacent
external sectors: $s_{\alpha + \frac{1}{2}} = - s_\alpha$. If
$s_\alpha = - s_{\alpha +1}$ no such intermediate internal sector exists.
\item[{iii)}] If $s_1 = s_N$ there may be an internal sector with number
$N + \frac{1}{2}$ and $s_{N + \frac{1}{2}} = - s_N$. Equivalently we could
give this additional sector the number $\frac{1}{2}$, but we must choose
one of these possibilities and use it consistently.
\end{itemize}
With a partitioned graph we associate a Feynman integrand, which we state
first in $x$-space. To the external points correspond the external variables
$x_i$. To each internal vertex we assign an integration variable $u_j,\quad
j = 1, \ldots, \sigma$. $z_i$ denotes a variable which may be either external
or internal. Within a $T^+$ sector the usual Feynman rules hold: each internal
vertex carries a factor $-ig$, a line connecting the points $z_i$ and $z_j$
carries the propagator $-iD_F (z_i - z_j)$. In a $T^-$ sector the
complex-conjugates of these rules apply. A line connecting points $z_i, z_j$
in different sectors, with $z_i$ lying in the lower-numbered sector, carries
the propagator $-iD_+(z_i - z_j)$. The graph must be divided by the usual
symmetry number if it is invariant under certain permutations of points and
lines.
In p-space, integration variables are assigned to the lines. The vertex
factors in $T^\pm$ factors are $\mp(2\pi)^4g$ and the propagators are
$\tilde{D}_F(p)$ or $(\tilde{D}_F(p))^\ast$ respectively. Lines connecting
different sectors carry propagators $\tilde{D}_+(p)$, the momentum $p$
flowing from the lower to the higher sector. For each external point there
is a factor $(2\pi)^{3/2}$. Momentum is conserved at each internal vertex.
For $T=0$ the rule ii) governing internal sectors seems to differ from the
corresponding rule in V, where instead of one (possibly empty) intermediate
sector of different type we had chains of intermediate sectors of the same
type as the bracketing external sectors. However, it can be shown with the
help of Lemma V.3.1 that the two formulations are equivalent. Graphs
containing non-empty internal sectors according to iii) vanish for $T=0$,
wherefore they do not occur in V.
For the Green's functions $(N=1)$ our rules agree with those of the conventional
real-time formalism. Our graphs are equal to those of the Keldysh formulation
[8].
\bigskip
The first point to be checked is that the above prescription gives an
unambiguous result for the correlation functions $W$. The problem is that in
the definition $W(x_1, \ldots , x_n) = {\cal W}(x_1, s_1 | \ldots | x_n , s_n )$ the
signs $s_\alpha$ can be chosen arbitrarily. But it can be shown that the
sum over all graphs with the same scaffolding does not depend on the choice
of these signs. The proof is modeled closely on the corresponding proof in
V and will only be briefly indicated. Consider a given scaffolding.
Let $S$ denote an internal $T^+$ sector, $S_x$ an external $T^+$ sector with
external variable $x, \bar{S}$ and $\bar{S}_x$\, $T^-$ sectors. A product
$S_xS$ or the like denotes the sum over all partitions of a given subgraph
into two adjacent sectors $S_x, S$. The propagators connecting these two
sectors are included. We will first show that the choice of $s_N$ is irrelevant.
The graphs belonging to $s_N = +$ or $s_N = -$ respectively differ only in the
sectors with number $\nu (S) > N-1$. In the case $s_{N-1} = +, s_1 = -$
these variable sectors are (we put $x_N = x$) $S_x + \bar{SS}_x$ for
$s_N = +$, and $\bar{S}_x + \bar{S}_x S$ for $s_N = -$. But
\[
S_x + \bar{SS}_x = \bar{S}_x + \bar{S}_x S
\]
is a special case of Lemma V.3.1, which remains valid for our new propagators.
Similarly, if $s_1 = s_{N-1} = +$ we must show that
\[
S_x + S_x \bar{S} + \bar{SS}_x + \bar{SS}_x \bar{S} = \bar{S}_x ,
\]
or
\[
(S_x + S_x \bar{S} - \bar{S}_x - \bar{SS}_x) + \bar{S} (S_x + S_x\bar{S} -
\bar{S}_x - \bar{SS}_x) + (S+\bar{S}+\bar{SS}) \bar{S}_x = 0 ,
\]
which is correct because all three brackets vanish as a result of Lemma
V.3.1. The proof of independence on the other $s_\alpha$ follows similar
lines. The case $1 < \alpha < N$ is considerably simplified compared to V
by the new formulation of the rules concerning internal sectors.
Of the conditions stated in section 2 invariance under translations and
rotations, and the symmetry of $\cal W$ within $T^\pm$ factors are trivially
satisfied. That the equation of motion (3.2) is satisfied, is shown
exactly as in V. At the present formal level the product $N(\Phi^3)$ denotes
Wick ordering $:\Phi^3(x):$ with respect to the free vacuum field.
The proof of the reality condition (2.1) and Ostendorf's proof [9] of the
splitting relation (2.3) can also been taken over unchanged from V. Since the
non-invariant part $C(x-y)$ is the same in $D_+$ and in $D_F$, the crucial
relation $D_F(x-y) = D_+ (x-y)$ if $x^0 > y^0$ holds in every orthochronous
Lorentz frame. Hence the same is true for the splitting relation. From this
and the symmetry within sectors we can prove locality. Let $(x-y)^2 < 0$.
Then there exist two Lorentz frames such that in one $x^0 > y^0$, in the
other $x^0 < y^0$. Hence
\begin{eqnarray*}
& W_\sigma(\ldots, x, y, \ldots ) = {\cal W}_\sigma (\ldots | x,y,+|
\ldots ) = \nonumber \\
&= {\cal W}_\sigma (\ldots |y, x, + | \ldots ) = W_\sigma
(\ldots , y, x, \ldots ).
\end{eqnarray*}
Finally, the decay properties of $D \ldotp (\xi )$ for $\xi \to \infty$ in
space-like directions are the same as those of $\Delta \ldotp (\xi )$, so that
the cluster property is also unaffected by the change of propagators.
Yet to be proved remains the KMS condition (2.6). It is easy to see that a
graph contributing to the left-hand side of (2.6) becomes a graph
contributing to the right-hand side, if the directions of all lines connecting
sectors with $\nu (S) < N+1$ to sectors with $\nu (S) \ge N+1$ are reversed,
and vice versa. This means that the corresponding propagators $\tilde{D}_+ (k)$
are replaced by $\tilde{D}_+ (-k)$. From the definitions (3.3-5) we find
$\tilde{D}_+ (k) = e^{\beta k_0} \tilde{D}_+ (-k)$. Hence the two variants
of the considered graph differ by the factor $\exp (\beta \sum_\alpha
k^\alpha_0)$, the sum extending over the momenta of the lines in question.
But $K = \sum k^\alpha$ is the total momentum flowing from the $P$-part of
the graph to the $Q$-part, so that $K^0 = P^0$ as defined in (2.5). As a
side remark we note that a similar argument can be used to show that it is
immaterial whether extremal internal sectors are assigned the number
$\frac{1}{2}$ or $N + \frac{1}{2}$.
\section{RENORMALIZATION AND THE
EXIS\-TENCE PROBLEM}
As yet, the expressions derived in the preceding section have only a formal
meaning. There remains the question of the existence of the integrals
symbolized by the graphs.
The UV problem is concerned with the behaviour of the integrands at infinity
in momentum space. It can be handled exactly as was done in V for the vacuum
case. We note first that loop integrals over loops extending over more than
one sector are finite because of the strong decrease of $\tilde{D}_+ (k)$
for $k_0 \to - \infty$ and momentum conservation: primitively divergent
subgraphs exist only within sectors, where they can be treated with
conventional methods. We choose BPHZ renormalization (see [7]), which
introduces suitable subtractions in the integrand, before integrating,
thereby avoiding the need for regularization. The subtractions are found by
expanding the integrands of potentially dangerous subgraphs, the
"renormalization parts", into power series of sufficiently high degree in their
external momenta. The temperature dependent part $\tilde{C}$ of the
propagator $\tilde{D}_F$ decreases exponentially fast at large momenta and is
innocent of any UV problems. We can therefore define the mentioned subtraction
terms using $\tilde{\Delta}_F$ instead of $\tilde{D}_F$ inside the
renormalization parts, without destroying the UV convergence achieved by the
subtraction. We also define the finite renormalizations of the BPHZ prescription
to be $T$-independent, giving them the values needed to satisfy the normalization
conditions at $T=0$ stated in V. These rules are the expression of the
required $T$-independence of the renormalization prescription $N$ in the
field equation (1.1). An important effect of this limited subtraction is the
emergence for $T>0$, of lines connecting a vertex to itself. But they carry
the propagator $\tilde{C} (k)$, hence the loop integral over $k$ exists,
and is the same in $T^+$ and in $T^-$ sectors.
But the UV problem is not the only existence problem we are faced with.
There is also the problem of the local singularities of the integrand,
again in $p$-space. The integrand is a product of distributions in variables
which are quadratic functions of a complete set of independent external
and internal momenta. The variables in a given subset of propagators may
be dependent (i.e.~their gradients in momentum space may be linearly
dependent) on certain manifolds, in which case the product of propagators
does not necessarily define a distribution. It is no longer clearly integrable
even in the sense of distributions. Contrary to assertions found in the
literature, the problem is not restricted to the case of two propagators
separated by a self energy insertion, and thus both depending on the same
variable. As an example of a more complex situation, consider the integral
$\int dk \tilde{D}_\cdot (k) \tilde{D} \cdot (p-k)$ over a two-line loop,
the dots
standing for either $+$ or $F$. The mass shell $\delta 's$ in the two factors
coalesce at $p=0$, hence the integral diverges at that point, and this
singularity in the external variable $p$ (external to the considered 2-line
subgraph) is not removed by renormalization. Closer inspection shows that the
singularity is of first order. A chain of $n$ two-line bubbles will then
produce a singularity of order $|p|^{-n}$, which is not integrable for $n\ge 4$.
Nor is it defined in another way as a distribution. The remaining integration
over $p$, which may be an internal or external variable of the full graph,
is therefore not defined: individual graphs containing such chains diverge.
These divergences may cancel between graphs with the same scaffolding but
different sector assignments of the vertices in the chain. But we are not
aware of a proof to this effect, even for this still relatively simple
example.
How is this problem solved in the vacuum case? This is easiest for the fully
time ordered function $\tilde{\tau} (P) = \tilde{\cal W} (P, +)$. If $m > 0$
this function is everywhere the boundary value of an analytic function in
complexified variables $p_i$. The same is true for the Feynman propagator
$(k^2 - m^2 + i \epsilon )^{-1}$. For variables $\{ p_i \}$ in the domain of
analyticity of $\tilde{\tau}$ we can deform the integration contours for the
internal variables $k_j$ into the complex in such a way that they never meet
the singularities at $k_j^2 = m^2$. The integrand is then a smooth function,
and there are no problems with the local existence of the integral. But the
presence of a $\delta$-term in $\tilde{D}_F$ destroys analyticity, so that
the method does not work for positive $T$. For $m=0,\quad \tilde{\Delta}_F (k)$
is singular at $k=0$ and is there not a boundary value of an analytic function.
The same holds for $\tilde{\tau}$ at points where a partial sum of $p_i$'s
vanishes. These singularities can lead to infrared divergences, which need
special attention. An $x$-space method for achieving this, and also for
proving existence of the general ${\cal W}_\sigma$, has been described in V.
It relies heavily on the fact that $\Delta_+ (\xi )$ is analytic in $Im \xi^0
< 0$ and decreases at least of second order for $Im \xi^0 \to - \infty$.
But $D_+ (\xi )$ is only analytic in the strip $-\beta < Im \xi^0 < 0$, so
that this method can also not be extended to positive $T$.
At present the problem of existence of ${\cal W}_\sigma$ remains unsolved.
\section{UNIQUENESS}
It will be shown that the expansion described in sections 3 and 4, assuming
its existence in finite orders, is the only expression satisfying all the
requirements.
Assume that uniqueness has been established up to order $\sigma - 1$. Let
$W_\sigma^1, W_\sigma^2 ,$ be two solutions of the equations (3.2) plus
subsidiary conditions, $W^{T1}_\sigma$ and $W^{T2}_\sigma$ their truncated
(= connected) parts defined by eq. (2.2). Then their difference
\begin{equation}
h_\sigma (x_1, \ldots , x_n) = W^{T1}_\sigma (x_1, \ldots) - W^{T2}_\sigma
(x_1, \ldots)
\end{equation}
satisfies the homogeneous equations
\begin{equation}
(\Box_i + m^2) h_\sigma (\ldots , x_i , \ldots ) = 0
\end{equation}
and all the subsidiary conditions.
For $n > 2$ we prove exactly as in V that $h_\sigma$ is invariant under
permutations of its arguments, in particular that it satisfies in $p$-space
\begin{equation}
\tilde{h}_\sigma (p_1, \ldots , p_{n-1}, p_n) = \tilde{h}_\sigma (p_n, p_1,
\ldots , p_{n-1}) ,
\end{equation}
and that it is different from zero only for $p_i^2 = m^2$ for all $i$. But
from the KMS condition (2.6) we find that the two sides of (5.3) differ by a
factor $\exp (\beta p^0_n ).$ Hence the support of $\tilde{h}_\sigma
(\ldots , p_n)$ is restricted to $p_n = 0$. But a factor
$\delta^4 (p)$ in $\tilde{h}_\sigma$ violates the cluster property,
so that we must have $\tilde{h}_\sigma = 0$.
The Fourier transform of the 2-point function $h_\sigma (x,y)$ is of the form
\begin{equation}
\tilde{h}_\sigma (p,q) = \delta^4 (p+q) \hat{h}_\sigma (p)
\end{equation}
by translation invariance. The equation (5.3) and rotation invariance imply
\begin{equation}
\hat{h}_\sigma (p) = H_\sigma (p_0) \delta (p^2 - m^2) ,
\end{equation}
with $H_\sigma$ a tempered distribution. Next we use the KMS condition (2.6),
from which we find $H_\sigma (-p_0) = e^{-\beta p_0} H(p_0)$. This relation
is solved by the ansatz
\begin{equation}
H_\sigma (p_0) = \frac{1}{1+e^{-\beta p_0}} G_\sigma (|p_0|) .
\end{equation}
Because of the $\delta$-factor in (5.5) we can replace $|p_0|$ by $\omega =
(|{\bf p}|^2 + m^2)^{\frac{1}{2}}$ as the argument of $G_\sigma$.
Define $K_\sigma (x-y) = h_\sigma (x,y) - h_\sigma (y,x)$. Locality demands
that $K^{ret}_\sigma (\xi) = \theta (\xi_0) K_\sigma (\xi)$ has its support
in the closed forward cone. Its Fourier transform
\begin{equation}
K^{ret}_\sigma (p) = \frac{i}{2\pi} G_\sigma (\omega)
\frac{e^{\beta\omega}-1}{e^{\beta\omega}+1} \frac{1}{(p_0 + i\epsilon)^2
-\omega^2}
\end{equation}
is then analytic in the forward tube $\{ Im(\mbox{p})
\in V_+\}$, and is polynomially
bounded in the slightly smaller tube $\{ (Im(\mbox{p})-a) \in V_+\}$
for any $a\in V_+$ [10]. The last
factor in (5.7) has the correct analyticity. In order for this to be the case
for the full function $K^{ret}_\sigma$, the product $g_\sigma (\omega)$ of the
$\omega$-dependent factors in (5.7) must be an entire function of ${\bf p}$, and
thus of $\omega^2$. Furthermore $g_\sigma$ must be polynomially bounded, hence
it is a polynomial in $\omega^2$.
Combining all the information hitherto obtained we find
\begin{equation}
\hat{h}_\sigma (p) = \gamma_\sigma (\omega^2) [\delta_+ (p) + (e^{\beta\omega}
-1)^{-1} \delta (p^2 - m^2)]
\end{equation}
with $\gamma_\sigma$ a polynomial. We want to show that $\gamma_\sigma = 0$.
To that end we show first that $\gamma_\sigma$ must be a constant, because
otherwise the inductive construction could not proceed to the next higher
order. The trouble comes from the fact that the time-ordered 2-point function
$\gamma_\sigma ( - \partial^2_0) D_F (\xi)$ is not a multiple of a Green's
function of the Klein-Gordon operator, if $\gamma_\sigma$ is not a constant.
To see the problem explicitly we consider the function ${\cal W}_{\sigma +1}
(x,y,+|z_1|z_2)$. It satisfies the equation
\begin{eqnarray}
(\Box_x + m^2) {\cal W}_{\sigma +1} (x,y|\ldots) &=& - \frac{1}{6}
< T(N(x) \Phi(y)) \Phi(z_1) \phi (z_2)>_\sigma \nonumber\\
& & + R_\sigma (x,y,z_1,z_2) = : I_\sigma ,
\end{eqnarray}
where we have abbreviated $N(x) = N(\Phi^3(x))$. $R_\sigma$ is a term with
support at $x=y$, stemming from the ambiguity of the $T$-product at that
point. Let $p,q,k_i$, be the momenta corresponding to $x,y,z_i$. For $p,q,$
off the mass shell, the solution of (5.9) reads in $p$-space:
\begin{equation}
\tilde{\cal W}_{\sigma +1} (p,q|k_1|k_2) = \frac{1}{6} \frac{1}{p^2 - m^2}
\hat{I}_\sigma (p, \ldots, k_2) .
\end{equation}
We develop $I_\sigma$ in a cluster expansion (see (2.2)), treating $N(x)$
as a product of three individual fields. We transform this expansion into
$p$-space and consider it for $p,q,$ not on the mass shell, $p+q\not= 0$.
There the only term affected by the addition of an $h_\sigma$ (remember
that $h_\sigma = 0$ for the 4- and 6-point functions) is
\begin{eqnarray}
\Delta_{\sigma +1} (p, \ldots , k_2) & = & \frac{i}{6(2\pi)^4} \delta^4
(p + \ldots + k_2) \{ \frac{1}{p^2 - m^2} \frac{1}{q^2 - m^2} \nonumber\\
& & \times [\gamma_\sigma (q^2_0) + \gamma_\sigma (k_{10}^2) + \gamma_\sigma (k_{20}^2)]
\tilde{D}_+ (k_1) \tilde{D}_+ (k_2) \nonumber\\
& & + \frac{1}{p^2 - m^2} B_p (q,k_1, k_2)\} ,
\end{eqnarray}
with $B_p$ being a polynomial describing a possible $h$-dependence of
$R_\sigma$. Renormalization terms in $N$ do not contribute to
$\Delta_{\sigma +1}$ at the points considered. By applying to
${\cal W}_{\sigma +1}$ a Klein-Gordon operator in $y$ instead of $x$, we obtain
in the same way
\begin{eqnarray}
\Delta_{\sigma +1} & = & \frac{i}{6(2\pi)^4} \delta^4 (p + \ldots) \{
\frac{1}{p^2 - m^2} \frac{1}{q^2 - m^2} \nonumber\\
& & \times [\gamma_\sigma ((q_0 + k_{10} + k_{20})^2)
+\gamma_\sigma (k_{10}^2 ) + \gamma_\sigma (k_{20}^2)]
\tilde{D}_+ (k_1) \tilde{D}_+ (k_2) \nonumber \\
& & + \frac{1}{q^2 - m^2} B_q (p,k_1, k_2)\} ,
\end{eqnarray}
with $B_q$ again a polynomial in the first variable $p = -q - k_1 - k_2$.
Equality of the expressions (5.11) and (5.12) means
\begin{eqnarray}
& &[\gamma_\sigma (q_0^2) - \gamma_\sigma ((q_0 + k_{10} + k_{20})^2)]
\tilde{D}_+ (k_1) \tilde{D}_+ (k_2) = \nonumber \\
&=& (p^2 - m^2) B_q (p, \ldots) -
(q^2 - m^2) B_p (q, \ldots)
\end{eqnarray}
for $p = -q - k_1 - k_2$. We choose an open neighbourhood $\cal N$ in the
$(q,k_i)$ space, intersecting the mass shell $p^2 = q^2 = k_i^2 = m^2$, in
which the components of $k_i$ and $p^2, q^2, q_0, q_1,$ are independent
variables. Such neighbourhoods clearly exist. We write $B_{p,q}$ as functions
of the new variables. Then we integrate (5.13) over a test function $\varphi
(k_1, k_2)$ with support in $\cal N$, chosen such that the left-hand side
does not give a vanishing result. The left-hand side is then a bounded,
non-vanishing, function of $q_0$ only, independent of $p^2$ and $q^2$.
The integrals $\int dk_1 dk_2 \varphi (k_1, k_2) B_{p,q} (\ldots)$ are
bounded functions of $p^2, q^2, q_0, q_1,$ as long as we remain inside
$\cal N$. Hence the right-hand side of (5.13) converges to zero if we let
$p^2$ and $q^2$ tend towards zero, in contrast to the behaviour of the
left-hand side. Hence (5.13) cannot be satisfied unless $\gamma_\sigma$ is
a constant, in which case the left-hand side vanishes identically.
But a non-vanishing constant $\gamma_\sigma$ leads to an addition to the
2-point function of the same form as $W_0 (x,y)$: it changes the
normalization of the field, hence, if $m, g$ and $N$ remain unchanged, the
new field would no longer satisfy the non-linear equation (1.1).
It is a curious aspect of this argumentation that it does not work in the free
case $g=0$, which should, however, not be confused with the zero-order terms
in our expansion of an interacting theory. For the free theory one could
arrive at the same result faster by postulating the usual canonical commutation
relations. For interacting fields this method is rather less convincing, because
in that case the CCR's have no rigorous meaning, since interacting fields
cannot be restricted to sharp times.
As has been mentioned in section 3, the methods explained here can also be
used to derive the forms of $W_0$ used there.
\bigskip
\noindent
\section{ACKNOWLEDGEMENTS}
I am indebted to R.~Baier and A.K.~Rebhan for enlightening discussions
and comments, and for guidance to the literature on finite-temperature
field theory.
\bigskip
\noindent
\section{REFERENCES}
\noindent
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\end{enumerate}
\end{document}