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% K-H REHREN
% COMMENTS ON A RECENT SOLUTION TO WIGHTMAN'S AXIOMS (July 1995)
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\noindent DESY 95-149 \hfill ISSN 0418-9833 \newline
July 1995
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\def\CR{1} \def\SW{2} \def\GF{3} \def\BH{4} \def\B{5} \def\L{6}
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\centerline{\hbf Comments on a recent solution} \vskip2mm
\centerline{\hbf to Wightman's axioms}
\mgap
\centerline{\caps K.-H.\ Rehren} \sgap
\centerline{II.\ Institut f\"ur Theoretische Physik, Universit\"at
Hamburg ({\caps Germany})}
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{\narrower {\sbf Abstract:} \newline
{\srm A class of exact Wightman functionals satisfying all fundamental
physical requirements in an arbitrary number of space-time dimensions,
which bear the appearance of describing interacting fields, was recently
constructed by C. Read \cite\CR. It is shown here, that the construction can
be considerably generalized, and that even the enlarged class belongs to the
Borchers class of a system of generalized free fields. } \par }
\baselineskip12.5pt
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\newsec{Introduction}
%
Ever since Wightman's formulation of the axioms \cite\SW\ to be satisfied by
the collection of $n$-point functions of local quantum fields, there has been
a discomforting lack of models. Apart from models with polynomial interaction
in two and three space-time dimensions, there are essentially only
constructions based upon free fields and generalized free fields \cite\GF\
available. These constructions involve Wick polynomials of derivatives of a
given field, as well as so-called $p$- and $s$-products \cite\BH\ (i.e.,
pointwise products resp.\ sums of independent fields in different Hilbert
spaces). Although one can easily produce non-vanishing truncated Wightman
functionals, such models do not describe interacting particles.
We recall the well-known list of axioms for a hermitean scalar field,
referring to the standard literature \cite\SW\ for the precise formulation:
Positivity and Hermiticity permit to reconstruct a Hilbert space containing
the cyclic vacuum vector, and an (in general unbounded) hermitean field
$\phi(f)$ on this Hilbert space whose vacuum correlation functions are given
by the Wightman functional. Poincar\'e Invariance of the Wightman functional
ensures the invariance of a vacuum vector along with the Poincar\'e covariance
of the reconstructed field. The Spectrum Condition and Cluster Property
ensure the positivity of the energy spectrum and the uniqueness of the vacuum.
Finally, Locality expresses the commutativity of field operators $\phi(f)$
smeared in causally disconnected regions of space-time.
In a recent paper \cite\CR, a new class of solutions was presented which
satisfy all Wightman axioms in any number of space-time dimensions. The models
are based on Feynman-like rules without at the same time being perturbative;
instead, every $n$-point function is obtained as a sum over finitely many
graphs. Some of the free input parameters of the models play a similar role
as coupling constants in ordinary Feynman rules (although it will become
clear in the course of this communication that they rather determine the
structure of the field as a Wick polynomial), while the remaining parameters
serve as appropriate cutoffs to ensure convergence of all sums and integrals
involved. Unlike regulators in perturbative approaches, these cutoffs need
not be removed in the end. They comprise a smooth space-time cutoff
function, a mass distribution in a finite mass interval, and a numerical
limitation of the number of vertices.
In the present contribution, we intend to shed new light onto these models.
It is found that the class of models \cite\CR\ can be considerably extended
by a generalization which essentially makes the ``interaction polynomial''
obsolete. We shall then reduce the entire construction to Wick products of
generalized free fields, and discuss aspects of the sharp mass limit
of the extended class of models in comparison with the original class.
\newsec{A quick review of the original construction}
%
The structure of the $n$-point functions
$$ \ww_n(f_1,\dots,f_n) \equiv \vac{\phi(f_1) \cdots \phi(f_n)} \eqno(1) $$
given in ref.\ \cite\CR\ is the following (in somewhat schematic notation).
$$ \eqalign{\ww_n&(f_1,\dots,f_n) := \sum_{{\rm banded} \atop {\rm graphs}}
\frac 1 {G\inter!} \int \biggl( \prod_{e \in E\inter} \!\!
dg\,dp\; \fmeas gp \biggr) \times
\cr & \times \prod_{i=1}^n \biggl[\hatf_i(P_i[p]) \cdot
\frac 1 {G_{i,{\rm dom}}!} \int \biggl( \prod_{e \in E_{i,{\rm dom}}} \!\!\!
dq\; \fmeas {}q \biggr) \prod_{v \in V_i}
\bigl(r! a_r \del_v(q,gp)\bigr) \biggr], \cr} \eqno(2) $$
where the sum extends over a class of ``banded graphs''. A banded graph is a
(possibly disconnected) graph which
contains one connected subgraph (``band graph'' or simply ``band'') for every
field entry $\phi(f_i)$ in (1) such that the sets $V_i$ of vertices of the
$n$ bands are disjoint and exhaust all vertices of the full graph; every band
has a distinguished ``external'' vertex of degree 1. The internal vertices
are of degree $2 \leq r \leq R$ for some finite number $R$. In each band the
number $s_i$ of vertices is limited by some finite number $S$, and the vertices
are labelled $1,\dots s_i$. Inequivalent labellings of the
vertices of the same abstract graph are considered as different banded graphs
to be summed over. The banded graph has no external lines and no edges
connecting a vertex with itself.
Edges which connect vertices of the same
band will be called ``domestic''; they are oriented from the vertex with
lower ordinal number to the vertex with higher ordinal number; if a domestic
edge connects to the external vertex of the band, then it may carry both
orientations. [This restriction on the orientations of domestic edges in
\cite\CR\ seems not really to be necessary. We view it as another model input
parameter.] Edges which connect vertices of different bands will be called
``interband''; they are oriented from the band with lower ordinal number to
the band with higher ordinal number. The integration rules are the following.
($i$) Associated with every domestic edge $e \in E\dom$ is a momentum variable
$q \in \Mink$, to be integrated over with the measure $dq\, \fmeas {}q$,
where $\theta$ is the Heaviside step function for the energy, and
$\meas k = \mu(k^2) \cdot \vert\hat\Theta(k)\vert^2$ consists of a smooth
mass distribution $\mu$ with support in the mass interval
$m_0^2 \leq k^2 \leq m_1^2$ where $0 < m_0 < m_1$, and the square modulus of
the Fourier transform of a real space-time cutoff function $\Theta(x)$ in the
Schwartz space $\Sch$ (i.e., smooth and all derivatives decaying faster than
any power of the arguments).
($ii$) Associated with each interband edge $e \in E\inter$ are a momentum
variable $p \in \Mink$ and a group variable $g \in L$ which runs over the
four connected components of the full Lorentz group. These variables are
integrated with the measure $d\mu(g,p) = dg\,dp\, \fmeas gp$ involving the
right invariant Haar measure $dg$ on $L$. We notice that the variables $g$
enter only in the combination $gp$; thus effectively, the group integrations
extend only over the Lorentz boosts and the time inversion in the rest frames
of the momenta $p$. One may normalize to unity the redundant integral over the
compact stabilizer group $O(d)$ of $p$, and therefore ignore it.
($iii$) The combinatorial weights $G\inter!$ and $G_{i,{\rm dom}}!$ are given
by $\prod_{v,v'}m(v,v')!$ where $m(v,v')$ is the number of edges connecting
the vertices $v$ and $v'$, and the product extends over all pairs of vertices
in different bands, and within the band $i$, respectively.
($iv$) The integrand contains a ``coupling constant'' $r!a_r$ for every
internal vertex of degree $r$ along with a momentum conservation delta
function for the momentum flow at that vertex involving the domestic momenta
$q$ and the Lorentz transformed interband momenta $gp$.
($v$) Finally, the momentum transfer at each band is given by $P = P[p] =
\sum\aus p - \sum\ein p$, where the sums refer to all interband momenta flowing
out of the band, resp.\ into the band. There contributes to the integrand a
factor $\hatf_i(P_i)$ for every band, where $\hatf_i$ are the Fourier
transforms of the real test functions $f_i \in \Sch$. It is worth noting that
the momentum transfer of the field operators (involving the interband momenta
$p$) is decoupled from the momentum flow within a graph (involving the
transformed momenta $gp$ and the domestic momenta $q$).
The rapid decay of the integrand along with the momentum conservation delta
functions guarantees that every integral in (2) converges absolutely. Due to
the limitations $S$ of the number of vertices per band and $R$ of the degree
of vertices, there are only finitely many different banded graphs, so the
functional (2) is well defined. It is evident that it is translation invariant
since the arguments of the test functions sum up to zero, $\sum_i P_i = 0$.
It is invariant under the orthochronous Lorentz group $\Loro$ since
for $\g \in \Loro$ the change of integration variables $p \mapsto \g p$,
$g \mapsto g \g\inv$ and $q \mapsto q$ takes the functional (2) into its
Lorentz transform.
We note here that the domestic variables $q$, being coupled by momentum
conservation to the Lorentz invariant variables $gp$, may also
be considered as Lorentz invariant. This explains why the momentum cutoff
does not violate Lorentz invariance: it affects only Lorentz invariant
variables.
The spectrum condition is satisfied since for every $j \leq n$, the sum of
momentum transfers $\sum_{i\leq j} P_i$ is a sum of
interband momenta $p$ restricted to the forward light-cone. The uniqueness
of the vacuum will become apparent later when we identify the Hilbert space.
The axiom with the most unprecedented realization in the new models is
Locality. We shall limit ourselves here to the simple core of the exact but
tedious argument given in \cite\CR. Namely, we shall discuss the
commutativity of $\phi(f)$ and $\phi(f')$ when $f$ and $f'$ are delta
functions at space-like separated points $x$ and $x' \in \Mink$. Although
these are not {\it a priori} admitted as test functions (the convergence
argument given in \cite\CR\ will fail for such functions), we may argue as
follows for the validity of the simplified argument: Given the spectrum
condition and Poincar\'e invariance, it is a standard result \cite\SW\ that at
the Jost points where all coordinates are space-like separated, the Wightman
distributions are in fact functions. By the Reeh-Schlieder theorem cite\SW,
it is
then sufficient to test the commutativity of $\phi(x)$ and $\phi(x')$ within
Wightman functions at Jost points.
Consider therefore the change in the sum (2) when the field entries $\phi(x)$
at position $i$ and $\phi(x')$ at position $i+1$ are interchanged. Along with
every banded graph contributing to $\ww_n = \ww_n(\dots x,x' \dots)$ there
corresponds a banded graph contributing to $\ww'_n = \ww_n(\dots x',x \dots)$
which differs only by the numbering of bands and therefore by the orientation
of the interband edges extending from band $i$ to band $i+1$. Let there be
$\sig$ such edges in a given graph. The associated integration variables $g$
and $p$, here and later on collectively indicated by $(g,p)_\sig$, enter the
arguments of the test functions $\hatf_i(P) = \exp-iPx$ and $\hatf_{i+1}(P)
= \exp-iPx'$ through $P = P[p]$, the measure factors $\fmeas gp$, and the
momentum conservation
delta functions. Due to Poincar\'e invariance, we are free to choose the
Lorentz frame such that $x^0 = x^{\prime 0}$. In this frame, each integral
contributing to $\ww_n$ is of the form
$$ \int \dots \times \int \prod_1^\sig \biggl( dg\,dp\,\fmeas gp
\exp(i\vec p(\vec x - \vec x') \biggr) \prod_v \del_v( \dots, (gp)_\sig)
\times \dots \eqno(3a) $$
while the corresponding integral contributing to $\ww'_n$ is of the form
$$ \int \dots \times \int \prod_1^\sig \biggl( dg\,dp\,\fmeas gp
\exp(i\vec p(\vec x' - \vec x) \biggr) \prod_v \del_v( \dots, (-gp)_\sig)
\times \dots \eqno(3b)$$
where $\dots$ stands for further factors and dependences on other variables
common to both contributions to $\ww_n$ resp.\ $\ww'_n$, and independent of
$(g,p)_\sig$. By the change of integration variables $(g,p)_\sig \mapsto
(gT,Pp)_\sig$ where $T$ resp.\ $P$ are the time resp.\ space inversion in the
given Lorentz frame, the integrands are transformed into each other (note
that the cutoff function $\Theta$ is real, hence $\ovl{\hat\Theta(k)} =
\hat\Theta(-k)$, so $\meas {gp}$ is invariant under $gp \mapsto -gp$).
This establishes Locality, graph by graph.
Finally, Positivity of the Wightman functional (2) becomes manifest if one
views every graph integral as an operator product of integration kernels
(= the square brackets in (2)) in the Fock space $\ff(H)$ over the
underlying Hilbert space $H = L^2(L \times \Mink, d\mu)$ with the measure
$d\mu(g,p) = dg\,dp\; \fmeas gp$.
More precisely, every band subgraph with $\nu\aus$ resp.\ $\nu\ein$
interband edges flowing out of resp.\ into it, and $\mu$ ``bypassing''
interband edges flowing from some lower band to some higher band, corresponds
to a kernel interpolating from the $(m = \nu\aus + \mu)$-``particle'' subspace
$\ff_m \equiv H^{\otimes m}$ to the $(n = \nu\ein + \mu)$-``particle''
subspace $\ff_n \equiv H^{\otimes n}$. As a kernel, it is a function of $m$
pairs of variables $(g,p)$ to be integrated (with the measure $d\mu$) with the
variables of a wave function in $\ff_m$ (``annihilation part''), and $n$ free
pairs of variables $(g,p)$ (``creation part''). It is given by the expression
in square brackets in (2) which is a function of the integration
variables $(g,p)\aus$ and $(g,p)\ein$ associated with the out- and ingoing
interband edges, while the bypassing edges contribute as delta function
kernels $\del(g,g')\del(p-p')/\meas{gp}$. (Note that every application of an
integral kernel operator in $H$ or $\ff(H)$ involves the integral measure
$d\mu(g,p) = dg\,dp\; \fmeas gp$, so one has not to worry about denominators.)
A careful analysis of the combinatorics reveals that actually every field
operator in (1) arises as a finite sum of integral kernels,
sandwiched between the completely symmetrizing projection operator
$\Pi = \bigoplus_n \Pi_n$ onto the symmetric Fock space $\Fock = \Pi\ff(H)$.
Namely, let $(k)\ein$ resp.\ $(k)\aus$ denote two collections of $\nu\ein$
ingoing resp.\ $\nu\aus$ outgoing momentum variables, and let the functions
$K^{\nu\ein}_{\nu\aus}((k)\ein;(k)\aus)$ be given by the domestic integrals
summed over all band graphs with these variables assigned to the given number
of in- and outgoing external lines:
$$ K^{\nu\ein}_{\nu\aus}((k)\ein;(k)\aus) := \sum_{\rm band \atop \rm graphs}
\! \frac 1 {G\dom!} \int \! \biggl( \prod_{e \in E\dom} \!\! dq\;\fmeas {}q
\biggr) \prod_{v} \bigl(r! a_r \del_v(q,k)\bigr) \eqno(4) $$
as read off eq.\ (2). Since there are no according band graphs,
$K^{\nu\ein}_{\nu\aus}$ vanish for $\nu\ein+\nu\aus>(R-1)^S$. Let furthermore
$$ \eqalign{ \PPhi^{\nu\ein,\mu}_{\nu\aus,\mu}((g,p)\ein,&(g,p)_\mu \, ; \,
(g,p)\aus,(g',p')_\mu) := \cr & := \hatf(P[p]) \;
K^{\nu\ein}_{\nu\aus}((gp)\ein;(gp)\aus) \, \otimes \,
{(\rm delta\;kernels)}^{\otimes \mu} \cr} \eqno(5) $$
be integral kernels interpolating between $H^{\otimes \nu\aus+\mu}$ and
$H^{\otimes \nu\ein+\mu}$ where $\hatf$ resp.\ $K^{\nu\ein}_{\nu\aus}$ are
functions of the indicated in- and outgoing interband variables
$P[p] = \sum\aus p -\sum\ein p$ resp.\ $(gp)\ein$ and $(gp)\aus$ only, and
(delta kernels)$^{\otimes \mu}$ stands for the delta functions in $\mu$
pairs of bypassing interband variables $(g,p)_\mu$ as explained before.
Then $\phi(f)$ is represented by the integral kernels
$$ \phi(f) = \sum_{\nu\ein,\nu\aus,\mu} \PPi_{\nu\ein+\mu} \; \left(
{\textstyle
\frac{\sqrt{(\nu\ein+\mu)!(\nu\aus+\mu)!}}{\mu!\,\nu\ein!\,\nu\aus!} }
\; \PPhi^{\nu\ein,\mu}_{\nu\aus,\mu} \; \right) \;
\PPi_{\nu\aus+\mu}. \eqno(6) $$
We shall refer to the functions (4) as the ``reduced kernels''. They are
symmetric in both their in- and outgoing sets of variables.
They encode the entire dependence of the model on the choice of the constants
$a_r$ ($r \leq R$) and the number $S$ (limiting the number of graphs),
as well as the previously mentioned restriction on the orientations of
domestic edges. Since the remaining input parameters determine the integral
measure of the Hilbert space $H$, the model is now specified by the reduced
kernels $K^{\nu\ein}_{\nu\aus}$ and the measure $d\mu$.
The assertion that the prescriptions (2) and (4--6) produce the same
Wightman functional is the central claim of this section. Since it is
essential for the rest of the paper, we formulate it as a Lemma.
\sgap {\narrower
{\bf Lemma.} {\sl Let the fields $\phi(f)$ be represented as integral
kernels (4--6) on the symmetrized Fock space $\Fock$. Let the vacuum vector
$\Omega$ be represented by the number $1 \in \CC \equiv \ff_0 \subset \Fock$.
Then the vacuum correlations of $\phi(f)$ coincide with eq.\ (2).} \par}
\sgap
{\it Proof:} A vacuum correlation of $n$ operators $\phi(f_i)$ is a sum over
single band graphs
contributing to every reduced kernel (4) and therefore to (5), and over
single permutations contributing to the symmetrizing projections in (6).
Every such contribution clearly corresponds to a banded graph contributing
to the sum (2), and vice versa. However, the correspondence is not always
one to one. What remains to be checked is that multiple counting and
numerical coefficients according to eqs.\ (4--6)
together produce the correct combinatorial weights
$(G\inter!\prod_i G_{i,{\rm dom}}!)\inv$ as in (2). The domestic factors
are explicitly present in (4) and need not be considered any longer.
At this point, in order to get the global factor $1/G\inter!$, it is
crucial that the sum (2) extends over all inequivalent labellings of the
vertices of the banded graphs, while the sum (6) extends over all inequivalent
labellings of the vertices {\rm and} external lines of the band graphs.
We start with a two-point function and consider a banded graph $G$
contributing to (2) with its two band subgraphs $G_1$ and $G_2$. Let the
vertices of $G_1$ be labelled as $v_i$, those of $G_2$ as $w_j$ (for this
matter not distinguishing the external vertex of each band graph from its
internal vertices), and let there $m_{ij}$ edges connect $v_i$ with $w_j$.
Thus $\nu_i = \sum_j m_{ij}$ interband edges connect to $v_i$, and $\kappa_j =
\sum_i m_{ij}$ interband edges connect to $w_j$. Let finally $\nu = \sum_i
\nu_i = \sum_j \kappa_j$ denote the total number of interband edges of $G$.
Apart from the domestic combinatorial weights (which are common to
eq.\ (2) and eq.\ (4)), the banded graph enters eq.\ (2) with the weight
$1/G\inter! = \prod_{ij}1/m_{ij}!$. Assume for the moment that the symmetry
groups of the vertices of $G_1$ and $G_2$ considered as abstract subgraphs of
$G$, i.e., ignoring the labelling of vertices and orientation of edges,
are trivial. This assumption is equivalent to the assumption that each
labelling of the vertices gives rise to a different labelled band graph.
Then there are $\nu!/\prod_i\nu_i!$ inequivalent assignments of $\nu$
distinguished momenta $(k) = (gp)$ to the external lines of $G_1$, and
similarly $\nu!/\prod_j \kappa_j!$ assignments of momenta to the external
lines of $G_2$, each giving rise to one term in the sums (4). Furthermore,
each set of $\nu_i$ lines extending from $v_i$ can be partitioned in
$\nu_i!/\prod_j m_{ij}!$ ways to join the vertices $w_j$ with multiplicities
$m_{ij}$, and there is a similar number of partitions for the vertices $w_j$.
Finally, each of the $m_{ij}!$ contractions of $m_{ij}$ lines between
$v_i$ and $w_j$ is counted separately according to the prescription (4--6).
Thus the total number of contractions of band graphs contributing to (4)
which give rise to the same banded graph $G$ equals
$$ \frac{\nu!}{\prod_i \nu_i!} \frac{\nu!}{\prod_j \kappa_j!}
\prod_i \frac{\nu_i!}{\prod_j m_{ij}!}
\prod_j \frac{\kappa_j!}{\prod_i m_{ij}!}
\prod_{ij} m_{ij}! = \frac{(\nu!)^2}{G\inter!}. $$
Since the symmetrizing projection operator between the two kernels contributes
a weight $1/\nu!$ for each permutation, and the explicit numerical
coefficients in (6) contribute another factor $(\sqrt{\nu!}/\nu!)^2$ $ =
1/\nu!$ to the two-point function, the weight of each term is $1/(\nu!)^2$ and
the combinatorial weight as in (2) is reproduced.
Now let the vertices of the abstract band graphs $G_i$ possess some symmetry
groups $S_i$ and let $S \subset S_1 \times S_2$ be the symmetry group of the
vertices of $G$. Then one may sum in (2) over all labellings of the internal
vertices of $G_1$ and over all labellings of the internal vertices of $G_2$
independently, if one includes a correction factor $1/\vert S \vert$ for
overcounting of banded graphs. Similarly, one may sum in (4) over all
labellings of internal vertices and over all labellings of external lines
independently, if one includes a correction factor $\vert S_i \vert$ for each
of the two kernels. On the other hand, the counting of inequivalent
contractions of two band graphs with labelled vertices and external lines
which give rise to the same banded graph with unlabelled interband edges
provides an additional multiplicity factor $\vert S_1 \times S_2/S \vert$
which cancels the correction factors for overcounting. We conclude that for
two-point functions, the prescriptions (2) and (4--6) produce the same
combinatorial weights.
Turning now to higher $n$-point functions, we repeat the previous reasonings
with the obvious generalization. The total number of contractions of band
graphs $G_i$ ($i = 1,\dots n$) contributing to (4) which give rise to the same
banded graph $G$ is found to exceed the expected weight $1/G\inter!$ by the
factor $\prod_i \nu\ein^i!\nu\aus^i!$ where $\nu\ein^i$ and $\nu\aus^i$ are
the number of in- and outgoing interband edges of $G_i$. This factor is
cancelled by the corresponding factors in the denominators of the numerical
coefficients in (6). The square root numerators of the latter (which arise
twice each) are compensated by the weights $1/(\nu+\mu)!$ of each permutation
within the projections $\Pi_{\nu+\mu}$ between two kernels, while the remaining
factors $1/\mu!$ in the denominators of (6) are cancelled by the number of
permutations of the sets of bypassing variables at each kernel. The discussion
of symmetries of the vertices of band graphs also parallels the two-point case.
This completes the proof of the Lemma. \qed
Due to the Lemma, Positivity of the Wightman functional (2) is manifest.
Let us now complete the list of arguments that eq.\ (2) fulfils all Wightman
axioms.
The orthochronous Lorentz group $\Loro$ is represented on $H$, and therefore
on the Fock space by second quantization, by the tensor product of the natural
action (on $p \in \Mink$) and the right regular action (on $g \in L$). Since
the measure is invariant under this action, the representation is
unitary. For the
same reason, functions of $k = gp$ represent Lorentz invariant elements of
$H$ or $\Fock$, and the reduced kernels (4) commute with the action of
$\Loro$. Consequently, the full kernels (5) and finally the fields (6)
transform like scalar fields.
The invariant Wightman domain of the field operators $\phi(f)$ is $\dd_\phi =
{\rm span} \, (\prod \phi(f_i)) \Omega$ $\subset \Fock$. It is clear that
$\Omega$ is the only translation invariant vector in the Hilbert space
$\ovl{\dd_\phi}$. The property of the reduced kernels
$$ \ovl{K_{\nu\aus}^{\nu\ein}((k)\ein;(k)\aus)} =
K_{\nu\ein}^{\nu\aus}((k)\aus;(k)\ein) \eqno(7) $$
ensures that $\langle\Phi,\phi(f)\Psi\rangle = \langle\phi(f)\Phi,\Psi\rangle$
for $\Phi,\Psi \in \dd_\phi$, i.e., $\phi$ is a hermitean field.
\sgap {\narrower
{\bf Corollary \cite\CR:} {\sl The $n$-point distributions given by eq.\
(2) define a hermitean scalar local Wightman field.} \par}
\newsec{An enlarged class of local Wightman fields}
%
We make the following crucial observation. As was remarked before, the fields
$\phi(f)$ are completely specified as operators on
$\Fock$ by the reduced kernels (and the measure) while it is irrelevant how
these kernels were produced by domestic integrals over band graphs. One may
indeed {\it choose} a sequence of reduced kernels as the primary model input.
None of the arguments for Finiteness, Positivity, Translation Invariance,
Lorentz Invariance and Spectrum Condition along with the Cluster Property is
affected if one replaces the reduced integral kernels
$K^{\nu\ein}_{\nu\aus}((k)\ein;(k)\aus)$ given by (4) by arbitrary smooth
polynomially bounded functions of the respective in- and outgoing variables
$k = gp$, and defines $\phi(f)$ by (5) and (6). Since the kernels only act
between symmetrizing projections, these functions may be chosen symmetric in
both of their two sets of variables.
Furthermore, the argument for Hermiticity of $\phi(f)$ is unaffected provided
the reduced kernels satisfy the condition (7) above. These assertions are
obvious except, maybe, the one concerning Finiteness, for which we refer to
the estimate (11) in the Lemma below.
Finally, the above simplified argument for Locality remains unaffected when
in (3) the delta functions due to each band (integrated over the domestic
variables) are replaced by the respective reduced kernels, provided
$$ K_{\mu}^{\nu+\sig}((k)_\nu \cup (k)_\sig;(k)_\mu) =
K_{\sig+\mu}^\nu((k)_\nu;(-k)_\sig \cup (k)_\mu) \; , \eqno(8) $$
where $\cup$ indicates the union of the respective sets of variables.
(The reduced kernels (4) have this symmetry.)
Namely, when $K$ resp.\ $K'$ refer to the reduced kernels due to the field
entries $\phi(x)$ resp.\ $\phi(x')$, then in a typical contribution to
$\ww_n$ products of reduced kernels
$$ K^\lam_{\sig+\kappa}(\dots;(gp)_\sig \cup \dots)
K_\mu^{\prime\nu+\sig}(\dots \cup (gp)_\sig; \dots) $$
replace the delta functions in (3$a$), while in the corresponding contribution
to $\ww'_n$,
$$ K^{\prime\nu}_{\sig+\mu}(\dots;(gp)_\sig \cup \dots)
K_\kappa^{\lam+\sig}(\dots \cup (gp)_\sig; \dots) $$
replace the delta functions in (3$b$). As before in Sect.\ 2, the dots
indicate dependences on other variables which are common to both
contributions. With the same change of the integration variables $(g,p)_\sig$
as before, the crossing symmetry (8) ensures $\ww_n = \ww'_n$.
In view of the two conditions (7) and (8), it is sufficient to specify a
terminating (in order to guarantee temperedness of the ensueing distribution)
sequence $F \equiv (F_\nu(k_1,\dots,k_\nu))_{\nu \in \NN}$ of smooth
polynomially bounded symmetric functions satisfying
$$ \ovl{F_\nu(k_1,\dots,k_\nu)} = F_\nu(-k_1,\dots,-k_\nu) \eqno(9) $$
(i.e., the Fourier transforms of real functions). We shall call functions
satisfying (9) ``hermitean''. Then the reduced kernels
$$ K^\nu_\mu((k)_\nu;(k)_\mu) := F_{\nu+\mu}((k)_\nu \cup (-k)_\mu)
\eqno(10) $$
satisfy the conditions (7) and (8).
The reduced kernels (10) inserted into (5) and (6) define a manifestly finite
hermitean
scalar local Wightman field $\phi_F$. Due to the Lemma above, this class
of fields extends the class constructed in \cite\CR.
The preceding arguments apply also without substantial change to mixed
Wightman functionals with field entries $\phi_{F^{(i)}}(f_i)$ specified by
different sequences $F^{(i)}$ of reduced kernels. The following conclusion
is immediate.
\sgap {\narrower
{\bf Corollary:} {\sl Every terminating sequence $F =
(F_\nu(k_1,\dots,k_\nu))_{\nu \in \NN}$
of (smooth polynomially bounded symmetric) hermitean functions defines,
upon insertion of the reduced kernels (10) into (5) and (6), a hermitean
scalar local Wightman field $\phi_F$ on the symmetric Fock space $\Fock$.
The fields $\phi_F$ associated with different sequences $F$ are defined (as
operator-valued tempered distributions) on the joint Wightman domain
$\dd = {\rm span} \; (\prod \phi_{F^{(i)}}(f_i))\Omega \subset \Fock$.
They are relatively local with respect to each other.} \par}
\sgap
We shall call a field $\phi_F$ ``of order $\nu$'' if $F_\nu \neq 0$
and all other $F_\mu$ vanish. In the general case, $\phi_F$ is a finite sum
over its components of order $\nu$.
The assignment $F \mapsto \phi_F$ is clearly real linear (in the obvious sense
for each component of order $\nu$). It is continuous in the following sense.
\sgap {\narrower
{\bf Lemma.} {\sl The correlations of fields $\phi_{F^{(i)}}$ (of fixed order
$\nu_i$) are bounded by
$$ \vert \vac{\phi_{F^{(1)}}(f_1)\cdots\phi_{F^{(n)}}(f_n)} \vert
\leq C_{(\nu_i)} \prod_i \vert\vert F^{(i)} \vert\vert \eqno(11)$$
where the constants $C_{(\nu_i)} < \infty$ depend on the test functions
$f_i \in \Sch$, and
$$ \vert\vert F_\nu \vert\vert^2 := \sup_{m_0^2 \leq m_i^2 = p_i^2 \leq m_1^2}
\int \biggl( \prod_i dg_i\, \meas {g_ip_i} \biggr) \vert
F_\nu(g_1p_1,\ldots,g_\nu p_\nu)\vert^2 \; . \eqno(12) $$
Thus, for each test function, the assignment $F \mapsto \phi_F(f)$ is weakly
continuous on the joint Wightman domain $\dd$.} \par}
\sgap
{\it Proof:} Every correlation of field operators in the vacuum state is a
finite sum of integrals of the form
$$ \int \biggl(\prod dp\,\theta(p^0)\biggr) \prod_{i=1}^n \hatf_i(P_i[p])
\int \biggl(\prod dg\,\meas{gp} \biggr) \prod_{i=1}^n
K^{(i)}((gp)_{\nu_i-\mu_i};(gp)_{\mu_i}) $$
such that each momentum $gp$ enters precisely one of the reduced kernels
$K^{(i)} \equiv K^{(i)\nu_i-\mu_i}_{\mu_i}$ as an ingoing (creation) variable,
and another one as an outgoing
(annihilation) variable. The $g$-integrals over products of reduced kernels
with the measure $dg\,\meas{gp}$ can be estimated, by repeated use of the
Cauchy-Schwarz inequality, by the product of the corresponding finite
$L^2$-norms of the reduced kernels. The latter are functions of the involved
$p^2$ only, and can in turn be estimated by the supremum over the mass
interval $[m_0,m_1]$, i.e., the norms given by (12). After these crude
estimates, which are common to all terms in the sum, the remaining
$p$-integrals of the form
$\int \bigl( \prod dp\,\theta(p^0) \bigr) \prod_i \hatf_i(P_i[p])$ still
converge absolutely due to the decay of the test functions \cite\CR. Summing
all these integrals yields the constants $C_{(\nu_i)}$.
\qed
\newsec{Reduction to generalized free fields}
%
Let us now study some elementary cases, starting with fields of order 1 which
we denote by $\varphi_F$ with $F_1(k) = F(k)$ a hermitean polynomially bounded
smooth function. These comprise the fields in \cite\CR\ when all
``coupling constants'' vanish, or when $S=0$, hence every band graph has only
its external vertex of degree 1 and $F_1(gp) =1$. One finds that
$$ \vac{\vphi_{F'}(f')\vphi_F(f)} =
\int dp\,\theta(p^0) \biggl( \int dg\,\meas {gp} \ovl{F'(gp)} F(gp) \biggr)
\cdot \hatf'(p) \hatf(-p), \eqno(13) $$
while all truncated higher $n$-point functions for fields of order 1
vanish. Thus, $\vphi_F$ is a generalized free field with mass distribution
$\mu_F$ supported in the mass interval $m_0^2 \leq p^2 \leq m_1^2$
$$ \mu_F(p^2) = \int dg\, \meas {gp} \vert F(gp)\vert^2 \; . \eqno(14) $$
The generalized free fields for different functions $F$ are in general not
independent, i.e., their correlations (13) do not vanish. There are in fact
only countably many independent such fields.
To see this, it is convenient to view a function $F(k)$ in the two-sheeted
region $M = \{k \in \Mink: m_0^2 \leq k^2 \leq m_1^2\}$ as a family (labelled
by the mass) of pairs of functions on velocity space $F_\pm(m;v) := F(k)$
where $k = \pm g_vp_m$ with $p_m = (m,\vec 0)$ a momentum vector in its rest
frame and $g_v \in \Loro$ the Lorentz boost by the velocity $v$. At each mass
$p^2 = m^2$, the value of the mass distribution in the mixed two-point
function (13) is a scalar product in the space
$L^2(\{v \in \RR^d:v^2<1\};d\eta_m) \otimes \CC^2$
$$ \int dg\, \meas {gp} \ovl{F'(gp)}F(gp) = \frac 12 \sum_{\eps = +,-}
\int d\eta_m(v)\, \ovl{F'_\eps(m;v)}F_\eps(m;v) \eqno(15) $$
with the measure $d\eta_m(v) = d^dv\,(1-v^2)^{-(d+1)/2} \meas{g_vp_m}$.
The real linear space of hermitean functions corresponds to the $+1$
eigenspaces of the real linear symmetric involutive operator $(F_+,F_-)
\mapsto (\ovl{F_-},\ovl{F_+})$. These eigenspaces possess countable
orthogonal real bases of the form $(F^\a_+(m;v),F^\a_-(m;v))$ with $F^\a_\pm$
smooth and polynomially bounded in $u \equiv v/\sqrt{1-v^2}$ and $F^\a_- =
\ovl{F^\a_+}$. Since the measure $d\eta_m$ varies smoothly with $m$, the
family of bases can be chosen to vary also smoothly with $m$. The real span of
the functions $(k^2)^n F^\a(k) := m^{2n}F^\a_\pm(m;v)$ at $k = \pm g_vp_m \in
M$ is dense (in the topology (12)) in the space of hermitean polynomially
bounded smooth functions on $M$. Since $\vphi_{k^2F}(f) = \vphi_F(-\Lapl f)$,
it follows from (11) that the countable family of independent generalized free
fields $\vphi^\a = \vphi_{F^\a}$ has a Wightman domain which is dense in the
Hilbert space generated from the vacuum by all order 1 fields $\vphi_F$.
The next case is an order $\nu$ field with $F_\nu$ a constant function.
One finds that
$$ \phi_F(f) = \frac{F_\nu}{\nu!}\, \colon \vphi_1^\nu \colon (f)
\eqno(16) $$
is just a Wick power of the generalized free field $\vphi_{F=1}$ of order 1.
Similarly, if $F_\nu$ is the symmetrized tensor product of single variable
hermitean functions
$$ F_\nu(k_1, \ldots,k_\nu) = \sum_{\pi \in S_\nu}
\prod_i F^{(i)}(k_{\pi(i)}) \; , \eqno(17) $$
then the associated field $\phi_F$ of order $\nu$ is the Wick product
$$ \phi_F(f) = \; \colon \prod_{i=1}^\nu \vphi_i \colon (f) \eqno(18) $$
of the generalized free fields $\vphi_i = \vphi_{F^{(i)}}$ of order 1. It is
not very difficult to prove (16) and (18) by verifying that the combinatorics
of the integral kernels produces precisely the products of two-point functions
required by Wick ordering. Namely, every factor in (17) arising in a creation
kernel will be eventually integrated with another such factor in an
annihilation kernel, yielding a two-point function with mass distribution of
the form (15), while the numerical coefficients in (6) cancel against the
combinatorial factors due to symmetrization.
Now, every symmetric hermitean function $F_\nu$ in $\nu$ variables can be
approximated (in the topology (12)) by real linear combinations of
symmetrized tensor products of $\nu$ hermitean functions in one variable.
We conclude:
\sgap {\narrower
{\bf Corollary:} {\sl The fields $\phi_F$ of order 1 form a countable system
of generalized free fields $\vphi$. The fields $\phi_F$ of order $\nu$ are
approximated (in the sense of the Lemma of Sect.\ 3) by homogeneous real Wick
polynomials $\colon\! P(\varphi)\colon$ of degree $\nu$ in the latter. The
vacuum vector is cyclic in the Hilbert space $\hh := \ovl\dd \subset \Fock$
with respect to the fields of order 1, which consequently act irreducibly in
$\hh$.} \par}
\newsec{Conclusion and discussion}
%
We can now apply the classical results in \cite\B\ to conclude that the new
fields $\phi_F$, and in particular the fields constructed in \cite\CR\ which
are finite sums of fields $\phi_F$ of order $\nu$, $\nu \leq (R-1)^S$, belong
to the Borchers class of the countable system of generalized free fields
defined by their two-point functions (13). (As a reminder: the Borchers class
of an irreducible field $\phi_0$ consists of all fields on the same Hilbert
space which are relatively local with respect to $\phi_0$, and which are
therefore automatically relatively local with respect to each other.) For
fields with a sharp mass (such that the scattering matrix is defined),
coincidence of the Borchers class implies coincidence of the scattering
matrix \cite\B.
Let us therefore insert a remark concerning a limit of sharp mass
$m_1\!\searrow\!m_0$ which is desirable for a particle interpretation. Note,
however, that scattering aspects of fields without a sharp mass were also
considered in, e.g., \cite\L. In the original class of models \cite\CR, the
naive limit attained by sharpening the bare mass distribution $\mu(m^2)$ is
severely obstructed since due to the momentum flow conservation involving both
domestic and interband momenta there will always occur powers of several
measure factors $\meas q$ at the same argument in the integrands (e.g., in
two-point functions the domestic momenta associated with the edges connected
to the external vertices coincide due to momentum conservation, but are
independently integrated; the problem will be aggravated whenever the
coupling $a_2 \neq 0$). In order to keep the highest of such powers regular in
the sharp mass limit, all the lower powers must become suppressed, so the
limiting Wightman functional will consist of products of two-point functions
only and one ends up with a free field. Apart from this obstruction, every
contribution from the cubic coupling $a_3$ will die out exactly as soon as
$m_1 < 2m_0$, due to momentum conservation at the triple vertex.
On the other hand, no such obstruction prevents us in the enlarged class
of models $\phi_F$ from sharpening the measure $\mu(m^2)$ independently from
the cutoff function $\Theta$ and the reduced kernels $F_\nu$. The singularity
obstruction is absent since there are no domestic integrations, and the
latter effect is absent since there is no momentum conservation at the kernels
for the integration variables $p$ or $gp$. However, as we have seen, in the
limit $\mu(m^2) \rightarrow \del(m^2 - m_0^2)$, the system of generalized
free fields will become a countable family of independent Klein-Gordon fields
$\vphi^\a_{m_0}$, and the limiting fields $\phi_F$ will be Wick polynomials
therein.
To summarize the previous remarks, we have found that although the sharp mass
limit is much more flexible within the enlarged class of models, the limiting
fields will belong to the Borchers class of a countable family of massive free
fields, and hence will not describe scattering \cite\B. Even if it remains to
be clarified in which precise sense the corresponding conclusion is true for
generalized free fields, in view of our results of Sect.\ 4 we do not share
the optimism expressed in ref.\ \cite\CR\ that the new fields might describe
interaction as long as the mass remains smeared. However, the approach of
ref.\ \cite\CR, and in particular the surprising mechanism which restores
locality upon integration over the ``inner degrees of freedom'' associated
with the Lorentz group, might well contribute some new and interesting
stimulations to constructive quantum field theory.
%
\mgap\bigbreak
{\bf Acknowledgments:} The author is very much indebted to C. Read, who
circulated his work prior to publication, and to D. Buchholz and
J. Yngvason for valuable comments and pointing out some of the classical
literature.
\mgap
{\bbf References} \vskip 3mm
\def\ref{\par \noindent \hangafter=1 \hangindent 13pt \cite}
\parskip 4pt
\baselineskip=2.5ex\ninepoint\smallskip
\def\it{\nineit}
\def\bf{\ninebf}
\def\CMP#1{Com\-mun.\ Math.\ Phys.\ {\bf #1}}
\def\AP#1{Ann.\ Phys.\ (N.Y.) {\bf #1}}
\def\NC#1{Nuovo Cim.\ {\bf #1}}
\def\JFA#1{J.\ Funct.\ Anal.\ {\bf #1}}
\def\JMP#1{Journ.\ Math.\ Phys.\ {\bf #1}}
\def\RPP#1{Rep.\ Progr.\ Phys.\ {\bf #1}}
%
\ref\CR\ C. J. Read: {\it Quantum field theories in all dimensions},
Univ.\ Cambridge (UK) preprint (1994), to appear in \CMP{}
%
\ref\SW\ R. F. Streater, A. S. Wightman: PCT, Spin and Statistics, and All
That; Benjamin W. A., New York, Amsterdam (1964). \newline
For a profound guide to more recent achievements of the theory see also:
\newline
R. F. Streater: {\it Outline of axiomatic relativistic quantum field
theory}, \RPP{38}, 771--846 (1975), especially Chapter 3.
%
\ref\GF\ O. W. Greenberg: {\it Generalized free fields and models of local
field theory}, \AP{16}, 158--176 (1961).
%
\ref\BH\ H.-J. Borchers: {\it Algebraic aspects of Wightman field theory},
in: R. N. Sen and C. Weil (eds.), Statistical Mechanics and Field Theory,
Haifa Lectures 1971; Halstedt Press, New York (1972).
%
\ref\B\ H.-J. Borchers: {\it \"Uber die Mannigfaltigkeit der interpolierenden
Felder zu einer kausalen $S$-Matrix}, \NC{15}, 784--794 (1960).
%
\ref\L\ A. L. Licht: {\it A generalized asymptotic condition}, \AP{34},
161--186 (1965).
\bye