duality of two causally separated double cones is violated by a charge-
anticharge pair. In constrat to gauge bridges which have no material content
(since they can be changed by gauge transformation implemented by indefinite
metric gauge charges) string-bridges consist of quantum matter. Hence the
existence of string bridges provides a local method to distinguish an
interacting net from a free one.

Another somewhat more metaphoric way of saying the same is that SLFT results
from gauge theory by applying Occam’s razor to indefinite metric- and ghost-
degrees of freedom. Gauge theory is its best placeholder within the setting of
Lagrangian quantization.

Interestingly the same string bridges which save Haag duality of observables
also allow to view interpolating fields as resulting from space- or light-like
limits in which the anticharge component is disposed of at infinity but leaves
a trail of quantum matter behind.

Whereas all sl fields in the absence of interactions can be obtained as
semiinfinite line integrals from pl fields 282828Note that the interaction
density and the S-matrix only uses such free fields., this breaks down in the
presence of interactions. In that case the necessarily sl localized
interpolating $s<1$ fields receive their sl localization metaphorically
speaking through being ”infected” by their contact with higher order $s\geq
1~{}$sl potentials with which they share the interaction density. This will be
exemplified in a number of models in the next section.

For anybody who has grown up with Haag’s way of looking at QFT it is deeply
satisfying that the model-dependent division between gauge invariant
observables and gauge dependent interpolating fields corresponds to the SLFT
localization dichotomy between pl observables and sl interpolating fields.

The fault line, which unfortunately still separates the ST community from
those who are working on the successful but still largely unfinished project
of QFT, runs precisely alongside the issue of causality and its refinements.
The recent progress on entanglement entropy [32] [33] requires a profound
understanding of causality in the context of operator algebras. The fact that
these ideas are presently also leading to a revision of perturbative QFT
raises hopes that this schism will be overcome.

##  5 Renormalization in the presence of massive sl vector mesons

###  5.1 Remarks on scalar QED; induced interactions and counterterms

As mentioned in the introduction SLFT differs both in its concepts as well as
in its calculational techniques from Lagrangian- (or Euclidean action)-based
quantization theories. In section 3 these differences played a role in the
solution of the Velo-Zwanziger causality conundrum and in the present section
they will be exemplified in a full QFT.

The simplest nontrivial model for illustrating these differences (for reasons
which will become clear in the sequel) is scalar QED.

Being an S-matrix-based QFT, the starting point of SLFT is an S-matrix which,
following Bogoliubov, is formally written as a time ordered product of an
interaction density $L(x,e)~{}$as in (45). The construction of interacting sl
fields uses Bogoliubov’s map (46) which converts free fields into their
interacting counterparts whose large-time asymptotic behavior reproduces the
scattering amplitudes associated to $S.$ In this section we will be
exclusively interested in the S-matrix; localization properties of interacting
fields will be mentioned in section 6.

The simplest nontrivial model for which the preservation of positivity
requires the use of sl localized fields is scalar QED. Here ”nontrivial”
refers to the appearance of a $2^{nd}$ order induced term $A\cdot
A\varphi^{\ast}\varphi$ and a $4^{th}$ order counterterm
$(\varphi^{\ast}\varphi)^{2}$ which has no counterpart in spinorial QED.

The independence of the large-time LSZ limits of causally separable fields on
their original localization is the basis of the SLFT perturbation theory. The
following is in part a recollection of arguments presented in section 3.

In lowest order we may start with the pl interaction density
$L^{P}=A_{\mu}^{P}j^{\mu},~{}j^{\mu}=i\varphi^{\ast}\overleftrightarrow{\partial^{\mu}}\varphi$
and use the linear relation with its short distance improved sl counterpart
and its escort $A_{\mu}^{P}=A_{\mu}-\partial_{\mu}\phi~{}$to write

$\displaystyle L^{P}$
$\displaystyle=L-\partial^{\mu}V_{\mu},~{}with~{}V_{\mu}=j_{\mu}\phi$ (66)
$\displaystyle S^{(1)}$ $\displaystyle=\int L^{P}(x)d^{4}x=\int L(x,e)d^{4}x$
(67)

This solves two problems in one stroke, the highest short distance
contribution to $L^{P}$ has been encoded into a divergence which drops out in
the adiabatic limit (second line) so that S is string-independent (the left
hand side) as well as renormalizable (the right hand side).

Actually one may forget the $L^{P}$ and formulate the SLFT construction solely
in terms of a $L,V_{\mu}$ pair fulfilling the $L-\partial V=0$ pair condition.
It turns out that the $L,V_{\mu}$ pair corresponding to vector mesons
interacting with lower spin particles and possibly among themselves is
uniquely determined: the interaction is completely determined in terms of its
particle content! In other words the $L^{P}$ can be defined in terms a SLFT
pair which is in turn defined of the particle content.

For the formulation of the higher order pair property it is convenient to use
the differential form of the pair property (as in section 3) and write
$d_{e}(L-\partial^{\mu}V_{\mu})=0.$ Further simplification is obtained by
using the ”$Q$-formalism” with

$\displaystyle Q_{\mu}$ $\displaystyle=d_{e}V_{\mu}=j_{\mu}u,~{}u=d_{e}\phi$
(68) $\displaystyle d_{e}L-\partial^{\mu}Q_{\mu}=0$

The $Q_{\mu}$ turns out to have a better $m\rightarrow 0$ behavior; it is only
logarithmically divergent and $\partial^{\mu}Q_{\mu}$ remains finite. The
logarithmic infrared divergence is a perturbative spacetime indication that
the massless limit of vacuum expectation values cannot be described in terms
of Wigner-Fock space which is simply a tensor product of a helicity space of
photons with that of charge-carrying Wigner particles [50]. In other words the
S-matrix based perturbative SLFT formalism indicates that its massless limits
needs a (presently unknown) extended formulation of scattering theory.

A notational simplification for the higher order pair conditions is obtained
by using lightlike strings (not possible for $m=0$). In this case the massive
sl fields are functions in $e$ rather than distributions and hence all $e$ may
be set equal. The second order pair condition reads

$d_{e}TLL^{\prime}-\partial^{\mu}TQ_{\mu}L^{\prime}-\partial^{\prime\mu}TLQ_{\mu}^{\prime}=0$
(69)

and the extension to higher than second order is straightforward. They have no
counterpart in the standard pl setting and account for the strength of SLFT as
compared to the standard pl perturbation theory.

Violations of these relations are referred to as obstructions. The Bogoliubov
S-matrix formalism is preserved by encoding these obstructions into a
redefinition of the interaction density $L\rightarrow L_{tot}=L+L_{2}+..$just
as it was done for external potential interactions in section 3.

The time-ordering $T$ which fulfills (69) is not necessarily the ”kinematic”
time ordering $T_{0}$ which is defined by attaching a
$-i2\pi(p^{2}-m^{2}-i\varepsilon)$ denominator to the momentum space 2-ptfct.
The scaling rule of renormalization requires that $T$ and $T_{0}$ share the
same scaling degree which in the presence of two derivatives leaves a
normalization freedom

$\left\langle
T\partial_{\mu}\varphi^{\ast}\partial_{\nu}^{\prime}\varphi^{\prime}\right\rangle=\left\langle
T_{0}\partial_{\mu}\varphi^{\ast}\partial_{\nu}^{\prime}\varphi^{\prime}\right\rangle+icg_{\mu\nu}\delta(x-x^{\prime})$
(70)

which leads to

$\partial^{\mu}\left\langle
T\partial_{\mu}\varphi^{\ast}\partial_{\nu}^{\prime}\varphi^{\prime}\right\rangle-\left\langle
T\partial^{\mu}\partial_{\mu}\varphi^{\ast}\partial_{\nu}^{\prime}\varphi^{\prime}\right\rangle=i(1+c)\partial_{\nu}^{\prime}\delta(x-x^{\prime})$
(71)

with an initially undetermined $c.$

The fulfillment of the second order pair requirement (69) in the tree
approximation fixes $c=-1.$ The action of $S$ on one-particles states as the
identity operator $S\left|p\right\rangle=\left|p\right\rangle~{}$takes care of
the contribution from 2 contractions. The change of $T_{0}~{}$to $T$ in
$Tj_{\mu}j_{\nu}^{\prime}=T_{0}j_{\mu}j_{\nu}^{\prime}-g_{\mu\nu}\delta$ in
all $Tj_{\mu}j_{\nu}~{}$accounts for the occurrence of the second order
induced $A_{\mu}A^{\mu}\left|\varphi\right|^{2}$ term

$\displaystyle TLL^{\prime}$
$\displaystyle=T_{0}LL^{\prime}-i\delta(x-x^{\prime})L_{2}$ (72)
$\displaystyle L_{2}$ $\displaystyle=gA_{\mu}A^{\mu}\left|\varphi\right|^{2}$
(73)

which is usually attributed to the implementation of gauge symmetry, but here
it follows from the causality and positivity principle of interpolating fields
which guaranties the $e$-independence of $S$.

The reason for using the kinematical time-ordering $T_{0}$ instead of $T$ is
the comparison with GT. In SLFT it is more natural to use $T$ in which case
the second order $L_{2}$ remains encoded in $TLL^{\prime}.$

As already pointed out in section 3, GT has a formally similar structure. This
is most clearly visible in a setting of gauge theory which avoids the standard
Lagrangian quantization of gauge theory (for spins $s\geq 1$ see [4]) in favor
of a perturbative S-matrix formulation as in [64] [65]. The physical
$A_{\mu}(x,e)$ and its escort $\phi(x,e)$ correspond to the gauge potential
$A_{\mu}^{K}$ and the Stückelberg field $\phi^{K}$ acting in a ghost extended
Krein space; the authors show that $A^{K}-\phi^{K}$ has properties expected
from $A_{\mu}^{P}.$

The two S-matrix-based constructions share the same improved short distance
behavior, but they achieve this in a very different way. Whereas in gauge
theory this is the result of enforced compensations between positive and
negative probabilities in intermediate states, the ultraviolet improvement in
SLFT accomplishes this by lessening the tightness of causal localization (but
not abandoning it !) and in this way reducing the strength of vacuum
polarization which is the only physical way to describe particles in terms of
physical (i.e. not gauge) interpolating fields.

Though both SLFT and gauge theory have the same short distance dimensions and
probably even share their Callen-Symanzik equations (and the related
asymptotic freedom property encoded in the beta-function), gauge theory cannot
account for the physics at finite distances let alone at long distances;
infrared properties and the problem of confinement remain outside its physical
range.

Last not least the functional-analytic and operator-algebraic methods used in
deriving nonperturbative theorems from basic principles are not available in
Krein spaces. For this reason gauge theory is shunned in books addressing the
conceptual structure of QFT [1][2]. The perturbative gauge theoretic
construction of a unitary S-matrix reveals this tension between conceptual
clarity and the efficiency of calculations which account for experimental
observations; it is a blessing for the impressive achievements of the Standard
Model but a curse for a on the principles of positivity and causal
localizability formulated QFT of the books.

Considering these conceptual deficiencies the perturbative calculations of a
gauge-invariant S-matrix of the Standard Model is a truly impressive
achievement. The idea that it represents a successful placeholder of an
unknown QFT is quite old and there have been many failed attempts to find the
real thing. The close formal analogy between gauge theory and SLFT suggest
that both may even exist side by side in a Krein extended Wigner-Fock space
containing additional indefinite metric degrees of freedom 292929J. Mund seems
to have discovered such a ”hybrid” formulation (private communication)..

Presently there exist no higher than second order SLFT calculation. Higher
order loop calculations in SLFT are much more laborious than calculations in
gauge theory. The gauge theoretic $4^{th}$ order calculation establishes the
existence of a $c(\varphi^{\ast}\varphi)^{2}~{}$counterterm. In contrast to
the second order $A\cdot A\varphi^{\ast}\varphi$ contribution its strength $c$
is a new parameter which is not determined by electromagnetism of the e.g.
$\pi^{+}~{}$meson303030Using such a model to describe electromagnetic
interactions of charge-carrying pions one usually sets $c=0.$.

Could this counterterm in GT be an induced contribution in SLFT ? This
question is not as crazy as it sounds. The above $L^{P}$ theory is by itself
nonrenormalizable; its short distance dimensions and the number of
counterterms increase with perturbative order. Yet if ”guided” in the above
sense by a $L,V_{\mu}$ pair it shares the finite number of possible free
varying parameters with the SLFT $L$ description.

The still missing answers to such questions are not only owed to the fact that
the number of theoreticians who are presently working on SLFT problems can be
counted on one hand but they also find their explanation in that the necessary
calculations are more involved than those based on pl fields. The sl setting
of QFT is the only known way to uphold the principles of QFT for all fields.

The SLFT approach also touches on an old mathematical problems which arose
from QFT in the late 60’s. The question was whether fields with
$d_{sd}=\infty$ (polynomial unbounded) fields as e.g. Wick-ordered exponential
functions of pl fields as $\exp g\varphi$ have a well-defined mathematical
status. This led Jaffe to extend the notion of Schwartz distributions to a
general class of distributions which still allows smearing with a dense set of
compact localized Schwartz test functions.

The SLFT guided construction of the $L^{P}~{}$pl setting requires to identify
pl observables in both settings and suggests to identify the interpolating
state creating fields of the $L^{P}$ theory with Jaffe fields. They correspond
to the well-behaved sl interpolating fields: the two theories share not only
the S-matrix but also their local observables whereas the states in the
$L^{P}$ theory remain singular in the sense of Jaffe. Such singular fields are
not required to have the usual domain properties which one needs to generate
operator algebras from fields so that the algebraic localization of compact
spacetime regions is fully accounted for by observables.

After having exemplified the main difference between gauge theory and SLFT in
the model of scalar QED, the following subsections will present low order
calculations in other models in which vector mesons couple to lower spin
matter fields and among themselves. This includes in particular the Higgs
models for which, different from the standard treatment, the form of the
Mexican hat potential and its spontaneous symmetry breaking is not imposed but
rather induced as a consequence of $e$-independence of $S.$ Even more
surprising is that the division into observables and sl interpolating is very
different from what one naively expects: neither the field strength
$F_{\mu\nu}$ nor the Higgs field is a pl observable.

###  5.2 The perturbative S-matrix in the SLFT setting

The appropriate formalism for the direct perturbative calculation of the on-
shell $S$-matrix is based on the adiabatic limit of Bogoliubov’s operator-
valued time-ordered $S(g)$ functional. Its adjustment to SLFT has been
mentioned in (45) in section 3 and further explored in the previous
subsection.

Time-ordering of quantum fields mathematically represented by operator-valued
distributions is characterized in terms of properties among which the causal
factorization is the physically most important one. The Epstein-Glaser
formalism [82] provides a perturbative computational scheme in which the time-
ordering of $n+1$ pl interaction densities is inductively determined in terms
of the $n^{th}$ ordered time-ordered product. The formulation in the presence
of sl fields is more involved and has not been carried out beyond second
order. Preliminary results reveal that a systematic $n^{th}$ order
construction requires the use of new concepts [96].

The E-G perturbation theory for the S-matrix can be extended to sl fields
(46). The result is a formula which maps a field in the local equivalence
class of Wick-ordered composites of free fields into the equivalence class of
”normal ordered” relative local interacting fields which act in the same
Wigner-Fock Hilbert space but are nonlocal with respect to their free
counterparts. Nowhere does this formalism refer to Lagrangian quantization.
For gauge theory this was first carried out in [61] where it was shown the
time-ordering of the S-matrix passes to that of retarded products in terms of
fields.

The power-counting restriction of renormalizability $d_{sd}(L^{P})\leq 4$ is
violated if one of the spin/helicity of the particles is $\geq 1.$ For
interactions involving particles with highest spin $s=1$ the $d_{sd}(L^{P})=5$
there are two ways to recover renormalizability. Either by converting $L^{P}$
into a ”gauge pair” $L^{K},V_{\mu}^{K}$ which requires the extension of the
Wigner-Fock space by indefinite metric degrees of freedom and possibly BRST
ghosts, or one maintains the physical degrees of freedom (and with it the
positivity of the Wigner-Fock Hilbert space) by converting $L^{P}$ into sl
$L,V_{\mu}$ pair.

For the rest of the paper we will stay with models which are sl renormalizable
$d_{sd}(L)\leq 4.$ This includes all couplings whose particle content consists
of $s=1$ coupled to $s<1~{}$and among themselves. In the case of massless sl
vector potentials the escort $\phi$ diverges as $m^{-1}$ and the large-time
LSZ derivation of the S-matrix breaks down (the on-shell restrictions of
correlations develop logarithmic $m\rightarrow 0$ singularities) and with it
the S-matrix based SLFT construction.

However some remnants of the SLFT construction can be saved; the exact one-
form $d_{e}\phi$ and hence also the $Q_{\mu}=d_{e}V_{\mu}~{}$is only
logarithmically divergent and $\partial^{\mu}Q_{\mu}$ remains convergent.
Hence even in case of breakdown of the S-matrix as a result of infrared
problems the $L,Q_{\mu}$ pair condition

$d_{e}L-\partial_{\mu}Q^{\mu}=0,\text{ }Q_{\mu}=d_{e}V_{\mu}$ (74)

remains a nontrivial condition. In fact it is this weaker formulation of
$e$-independence which corresponds to the BRST invariance of gauge theory.

In the previous subsection it was shown that, although the second order pair
condition in its original form is violated, it is possible to encode the
obstructing contribution $L_{2}$ into a redefinition of the interaction
density. It is helpful to formulate this idea in a model-independent way.

The definition of second order obstruction against the naive form of the
$L,Q_{\mu}~{}$pair property reads (using lightlike $e^{\prime}s$ which can be
identified)

$\displaystyle O^{(2)}$
$\displaystyle:=d_{e}TLL^{\prime}=T\partial^{\mu}Q_{\mu}L^{\prime}-\partial^{\mu}TQ_{\mu}L^{\prime}+TL\partial^{\prime\mu}Q_{\mu}^{\prime}-\partial^{\prime\mu}TLQ_{\mu}^{\prime}$
(75) $\displaystyle O^{(2)}$
$\displaystyle=\delta(x-x^{\prime})d_{e}L_{2}(x,e)$

Encoding them into interaction density one obtains

$L_{tot}:=L+gL_{2},~{}~{}S(g)=T\exp\int ig(x)L_{tot}(x,e)d^{4}x$ (76)

This change of bookkeeping which converts higher order obstruction into
induced contributions $L_{n}~{}$amounts a change of $L\rightarrow L_{tot}$ in
the Bogoliubov $S(g)$ is important. It affects the higher orders; the third
order obstruction is now

$O^{(3)}(g,g,g)=d_{e}\left[TL(g)L_{2}(g^{2})+\frac{i}{3}TL(g)L(g)L(g)\right]$
(77)

In models of interacting $s=1~{}$vector mesons as the Higgs model or scalar
massive QED the third order obstruction vanishes in the adiabatic limit and
the induced contributions account for the Mexican hat potential. As a
consequence the terms in this potential are induced and not postulated for the
purpose of implementing SSB. This will be explicitly verified in the following
subsections.

The $L,Q_{\mu}~{}$pair condition and its higher order extension within the sl
Bogoliubov-Epstein-Glaser setting is also meaningful for $d_{sd}(L)>4.$ The
before mentioned ”minimal” models contain only induced contributions but their
number increases with the perturbative order. By definition of minimal there
are no higher order counterterm parameters so the model depends only on those
parameters which are already present in the interaction density $L.$

The conceptual and mathematical superior aspects of SLFT poses the question
whether it is possible to pass directly from SLFT to pl ultra fields, thus
avoiding the pl counterterm formalism. This problem will come up again in
connection with cubic $h=2$ selfinteractions in the next section (86).

###  5.3 External source models

Consider a vector potential coupled to an conserved classical current
$j_{\mu}$ [11]. The S-matrix and the interacting vector potential are

$\displaystyle L^{P}=A_{\mu}^{P}j^{\mu}=A_{\mu}j^{\mu}-\partial_{\mu}(\phi
j^{\mu}),~{}hence~{}L=A_{\mu}j^{\mu},\ V_{\mu}=\phi j_{\mu}$ (78)
$\displaystyle S_{e}(g)=T\exp i\int g(x)L(x,e)\overset{g(x)\rightarrow
g}{\rightarrow}S=\exp ig\int\int j^{\mu}i\Delta_{F}j_{\mu}^{\prime}:\exp
ig\int A_{\mu}^{P}j^{\mu}:$ $\displaystyle
A_{\mu}^{ret}(x,e,e^{\prime})=S_{e}{}^{-1}(g)\frac{-i\delta}{\delta
f_{\mu}(x,e^{\prime})}S_{e}(g,j\rightarrow j+f)|_{f=0}=A_{\mu}(x,e)+\int
G_{\mu\mu^{\prime}}^{ret}j^{\mu^{\prime}}$ $\displaystyle
G_{\mu\mu^{\prime}}^{ret}(x,e;x^{\prime},e^{\prime})=(-\eta_{\mu\mu^{\prime}}-\partial_{\mu}e_{\mu^{\prime}}I_{e^{\prime}}+\partial_{\mu^{\prime}}e_{\mu}^{\prime}I_{-e}+(ee^{\prime})\partial_{\mu}\partial_{\mu^{\prime}}I_{e}I_{-e^{\prime}})G^{ret}(x-x^{\prime})$

The direct use of $L^{P}$ with $d_{sd}(L^{P})=d_{sd}(A_{\mu}^{P})=2$ leads to
a delta function ambiguity $g_{\mu\nu}c\delta(x-x^{\prime})~{}$in the time-
ordered $A_{\mu}^{P}$ propagator which accounts for a replacement
$i\Delta_{F}\rightarrow i\Delta_{F}+\frac{c}{m^{2}}\delta$ in the second line.
This in the pl formulation undetermined counterterm renormalization parameter
in the S-matrix and in $A_{\mu}^{P,ret}$ is absent in the less singular sl
formulation.

In that case $S$ is independent of $c$ and (by use of current conservation)
the interacting field does not depended on $e^{\prime}.$ As expected the field
strength remains pl. Hence the avoidance of the direct use of $A_{\mu}^{P}$ in
the calculation maintains the predictive power of the model. If needed one can
convert the sl setting with the help of $\phi(x,e)$ to a $A_{\mu}^{P}$. In
contrast to the directly calculated $A_{\mu}^{P}$ this via sl determined pl
potential is ”better”.

Passing from external source to external potential problems the differences
between the direct pl results and those obtained via the sl detour are much
stronger (section 3).

###  5.4 Hermitian $H~{}$coupled to a massive vector potential

The coupling of a vector potential to a Hermitian scalar matter field $H$
comes with a new phenomenon. In addition to a change of the time-ordered
product of the $H$-field there is now a genuine induction of
$H$-selfinteractions.

The ”germ” of an interaction density (the ”ignition”) for an $A_{\mu},H$ field
content is the $mA\cdot AH$ coupling, where the vector meson mass factor $m$
accounts for the classical dimension $d_{eng}=4$ and also indicates that the
model has no nontrivial Maxwell limit (the reason why it was discovered a long
time after QED). Its sl operator dimension is $d_{sd}=3,$ hence the germ is a
superrenormalizable sl interaction density. The first order $L,Q_{\mu}$ pair
property ($Q_{\mu}=d_{e}V_{\mu}$) requires the presence of the escort $\phi$
also in $L~{}$and leads to ($L,Q_{\mu}$ relation easy to check)

$\displaystyle
L=m\left\\{A\cdot(AH+\phi\overleftrightarrow{\partial}H)-\frac{m_{H}^{2}}{2}\phi^{2}H\right\\}+U(H),~{}U(H)=mc_{1}H^{3}+c_{2}H^{4}$
$\displaystyle V_{\mu}=A_{\mu}\phi
H+\frac{1}{2}\phi^{2}\overleftrightarrow{\partial_{\mu}}H,~{}~{}\
d_{e}(L-\partial^{\mu}V_{\mu})=0,\text{ }L^{P}=L-\partial^{\mu}V_{\mu}.$ (79)

A systematic determination of this first order pair $L,V_{\mu}$ pair starting
from the simplest coupling (the ”germ”) $gmA\cdot AH~{}$of the $A$-$H$
particle content and a general ansatz for $L$ and $V_{\mu}$ containing all
kinematically possible $d_{sd}\leq 4~{}$terms (19 terms in $L$) which can be
formed from $H,A_{\mu}$ and its escort $\phi$ shows that (79) is (up to
$\partial_{\mu}~{}$divergence terms and exact $d_{e}$ differentials) is the
unique solution [89]. However a verification that the $L,V_{\mu}$ pair
satisfies the pair condition requires only the use of free field equations and
the relations between $A_{\mu}$ and its escort and will be left to the reader.

The first order pair condition does not determine the strength of the
$H$-selfinteractions since $e$-independent contributions to $L$ simply pass
through the pair condition. The necessity of their presence which includes the
determination of the $c_{i}~{}$in (79) is seen in second and third order. This
”induction” of additional contributions with well-defined numerical
coefficients is a new phenomenon of SLFT; there is a formal similarity with
the imposition of the second order BRST gauge invariance on the $S$-matrix
[64] but the essential difference is that the $e$-independence of $S$ is a
consequence of the positivity and causal localization principle of QFT.

For the S-matrix one only needs the second order tree component to the
obstruction $O^{(2)}$ in (75) In addition to a second order change of the time
ordering of the propagator involving derivatives of $H~{}$which parallels that
in (70) one now encounters a genuine second order induction (76)

$L_{2}=g[(m_{H}^{2}+3c_{1}m^{2})H^{2}\phi^{2}-\frac{m_{H}^{2}}{4}\phi^{4}+c_{2}H^{4}]$
(80)

Finally the vanishing of the third order tree contribution fixes the values of
$c_{1},c_{2}$ in terms of the three physical parameters of its field content
which were already present in the germ namely $g,m,m_{H}.$ To allow for a
comparison with the Higgs mechanism we write the result in the form

$L_{tot}^{(2)}=mA\cdot(AH+\phi\overleftrightarrow{\partial}H)-V(H,\phi),~{}V=g\frac{m_{H}^{2}}{8m^{2}}(H^{2}+m^{2}\phi^{2}+\frac{2m}{g}H)^{2}-\frac{m_{H}^{2}}{2g}H^{2}\quad$
(81)

where $L_{tot}^{(2)}=L+\frac{g}{2}L_{2}.$ The appearance of a quadratic mass
term is the result of writing the interaction density as if it would be part
of a classical Lagrangian of gauge potentials. The reader may fill in the
details of the straightforward calculations by himself or look up the more
detailed presentation in [89].

Apart from a mass contribution the $V~{}$looks like a field-shifted Mexican
hat potential. But different from the Higgs mechanism it has not been obtained
by postulating a Mexican hat potential and subjecting it to a shift in field
space. It is rather induced by a renormalizable $A,H$ field content and it is
the unique renormalizable QFT with this field content. There is simply no room
for imposing a Mexican hat potentials since the induction of the $H$ and
$\phi$ selfinteractions is a consequence of $e$-independence of the S-matrix
which in turn is a consequence of scattering theory involving $d_{sd}=1$
causally separable space- or light-like strings.

The SSB picture of the Higgs model also reveals another common
misunderstanding, this time about SSB. The Mexican hat potential together with
the shift in field space is not the definition of SSB but rather a way to
implement such a situation when it is possible. The definition of SSB is
rather the existence of a locally conserved current whose global charge
diverges. This is only possible in the presence of massless Goldstone bosons
and all verbal attempts to make SSB consistent with a mass gap (a photon
becoming fattened to a vector meson by eating a Goldstone) only obscure the
interesting correct understanding.

QFT is not a theory which can create the masses of its model-defining field
content. In particular SSB is not about creating finite masses from an
initially massless situation; to the contrary it is about how to place a
massless particle (the Goldstone boson) into an interaction density so that
the current conservation remains that of a symmetric theory but some local
charges are prevented to converge in the infinite volume limit to a finite
global charge (the definition of SSB). The only known The prescription of a
field shift on a Mexican hat potential as the ”Higgs mechanism” has to be seen
in a historical context; it helped to overcome the formal problems which one
faces when one tries to extend Lagrangian quantization from Maxwell’s theory
of charge-carrying fields to a situation in which a vector potential couples
to a Hermitian matter fields. There are numerous historical illustrations of
situations for which important discoveries were made through formal
manipulations which were later replaced by a derivation which is consistent
with the principles of QFT. Incorrect placeholder are useful but only up to
the discovery of the real reasons.

A model of QFT is uniquely fixed in terms of its field content. The SLFT
setting (which seems to be the only one consistent with all principles of QFT)
for a $A_{\mu},H$ field content starts with a $A_{\mu}A^{\mu}H$ as the
simplest coupling and the rest is done by induction using the $L.Q_{\mu}$ pair
property which converts the heuristic physical content of the ill-defined pl
interaction density into the physically superior SLFT setting where the
”induction” resulting from the implementation of the pair property to all
orders unfolds the full content of SLFT.

###  5.5 Selfinteracting vector mesons

It is straightforward to check that there is no renormalizable $L,Q_{\mu}$
pair for a self-coupled singlet $(A\cdot A)^{2}.$ The principles of QFT as
embodied into the pair condition admit however selfinteractions between
multiplets (”colored”) of vector potentials while imposing strong restrictions
on the ”multi-colored” coupling parameters. In this case the germ is a $FAA$
selfinteraction and the general ansatz for the construction of a
$L,V_{\mu}~{}$pair which includes the ”colored” escorts is of the form

$L=\sum(f_{abc}F_{c}^{\mu\nu}A_{a,\mu}A_{b,\nu}+h_{abcd}A_{a,\mu}A_{b,\nu}A_{c}^{\mu}A_{d}^{\nu})+\,\hbox{terms
in}\,A_{a,\mu}\,\hbox{and}\,\phi_{a}^{\prime}s$ (82)

where the couplings and the masses of the vector mesons are initially freely
variable parameters but, as expected, the first and second order pair
condition places strong restrictions on them [89], among other things the $f$
and $h$ are interrelated in the same way (Jacobi identities of reductive Lie-
algebras) as in gauge theory [64]; in particular the $A$-$\phi$ and
$\phi$-$\phi$ couplings depend also on the masses of the vector mesons. The
main distinction to gauge theory is that these properties are direct
consequences of the principles of QFT and do not arise in the course of the
gauge theoretic extraction of physics from a unphysical (positivity-violating)
description through the imposition of gauge invariance.

The most interesting aspect of the SLFT formulation is that there remains a
renormalizability destroying second order induced selfinteraction which, if
left uncompensated, destroys renormalizability even though the interaction
density fulfills the power counting restriction of renormalizability. The way
to overcome this is to compensate this $d_{sd}=5~{}$term with a second order
contribution from a $A$-$H$ interaction with a scalar Higgs field313131A
$s\geq 1$ field would worsen the second order short distance behavior.. This
is a totally different situation from the abelian $A$-$H$ interaction for
which such all second order terms stay within the power counting bound.
Neither case bares any physical resemblance to spontaneous symmetry breaking
since in both cases the field shifted Mexican hat potential is second order
induced.

The idea of short distance compensations between contributions from different
spins arose in connection with supersymmetry. Although not invented for this
purpose, SUSY does improve the short distance behavior somewhat but not enough
to guaranty the renormalizability and preservation of supersymmetry in higher
orders. The situation of selfinteracting vector mesons is different, in that
the preservation of renormalizability is the raison d’être for the $H.$ Nature
does not have to decide between a symmetry and its SSB, rather the existence
of the $H$ is directly connected to the preservation of its positivity and
causality principles or in other words a massive $A_{\alpha}^{\mu}$ field
content by itself is not consistent.

Gauge symmetry is not a physical symmetry so there is nothing to break; all
these physically incorrect pictures evaporate if one maintains causality and
positivity which is perturbation theory is only possible by starting with an
sl $L,V_{\mu}$ or $Q_{\mu}~{}$pair property. The fibre bundle like Lie
structure of the $f_{abc}$ couplings is not the result of an imposed symmetry
it rather arises from the string-independence of the S-matrix which in turn is
a result of LSZ scattering theory of interacting causally separable positivity
obeying quantum fields; hence the situation is very different from the
superselection structure of unitary representation classes of observable
algebras which leads to the notion of inner symmetries. This shows that
quantum causality is much more fundamental than its classical Faraday-Maxwell-
Einstein counterpart.

Having thus strengthened the conceptual understanding of interactions between
vector mesons in the Standard Model one may ask whether SLFT contains also
messages about their coupling to matter. In recent work [90] it was shown that
SLFT does not only restrict the selfcouplings between vector mesons and
requires the presence of a Higgs particle in the presence of selfinteracting
massive mesons but it also restricts their coupling to the Fermion currents
and their chirality properties. This is of particular interests for massive
$W_{\pm}.Z$ vector mesons and the photon, a case for which the authors explain
the restrictions from SLFT in detail.

###  5.6 The pair condition for higher spins

The extension of SLFT S-matrix construction to that of interacting higher
spins $s\geq 2~{}$is an important issue about which one presently knows
little. There have been quite extensive investigations in a gauge theoretic
equivalent of the pair condition by Scharf [64]. In view of formal
similarities with SLFT it is interesting to take a closer look at some of his
results.

Scharf looked at the simplest $s=2~{}$selfinteraction which is of a cubic form
$trh^{3}$ where $h_{\mu\nu}$ is the $s=2$ massless tensor field. The physical
interest in this model is connected with the use of $h_{\mu\nu}~{}$as a linear
approximation of the gravitational $g_{\mu\nu}~{}$field. As in SLFT, the short
distance dimension of integer spin gauge fields is equal to their classical
dimension in terms of mass units namely $d_{sd}=1.$In [64] it was shown that
there exists no gauge theoretic trilinear selfinteraction $L^{K}$ with
$d_{sd}(L^{K})=3$ without involving derivatives of $h_{\mu\nu},$ its trace
$h_{\mu}^{\mu}~{}$as well as ghost fields and their anti-ghost. He found a
cubic interaction density of $d_{sd}(L^{K})=5$ which is above the power-
counting bound of renormalization, but still presents a huge reduction from
the $d_{sd}(L^{P})=11$.

Taking into account that gravitational coupling carries a dimension and
expanding the Einstein-Hilbert Lagrangian in a suitable way using
$\kappa=\sqrt{32\pi G}$ as an expansion parameter, he arrived at a formal
connection of the classical expansion with the quantum-induced correction up
to second order; this was later extended to all tree orders [64, 91]. The
agreement of tree approximations with classical perturbation theory is not
unexpected in itself, but in the present context it relates two competing
ideas, one being of classical geometric origin (the Einstein-Hilbert action)
and the other the gauge theory of selfinteracting $h=2$ particles .

Christian Gaß showed recently (private communication) that SLFT provides a
simpler version of such a cubic selfinteractions in the form

$\displaystyle L$
$\displaystyle=\kappa(2\partial_{\rho}h^{\kappa\lambda}\partial_{\sigma}h_{\kappa\lambda}+4\partial_{\beta}h_{\rho}^{\alpha}\partial_{\alpha}h_{\sigma}^{\beta})h^{\rho\sigma},~{}~{}h_{\mu\nu}:=A_{\mu\nu}^{(2)}$
(83) $\displaystyle
d_{e}A_{\mu\nu}^{(2)}=\partial_{\mu}a_{\nu}+\partial_{\nu}a_{\mu},~{}$ (84)

where $h_{\mu\nu}=A_{\mu\nu}^{(2)}$ is the sl helicity 2 potential from (25)
section 2 (which already played a role in solving the D-V-Z discontinuity
problem [11]). Using the relation between $d_{e}$ and $\partial_{\mu}$ of the
second line one easily verifies that $d_{e}L$ is of the form
$\partial^{\mu}Q_{\mu}$ i.e. the above $L$ belongs to a $L,Q_{\mu}~{}$pair.
Since massless $h\geq 1$ fields are intrinsically sl, the corresponding
minimal models are expected to be ”ultra-distributions” which are localizable
in spacelike cones.

For $h=1$ there exists no colorless selfinteraction whereas for $h=2$ the
situation seems to be reverse since the existence of colored cubic
selfinteractions can be excluded [92]. A proof based on Scharf’s S-matrix
gauge formalism can be found in [93].

The fact that there are no renormalizable $s=2$ selfcouplings does not exclude
the possibility to find sl $L,Q_{\mu}$ pairs of interactions between sl
$h_{\mu\nu}$ with lower spin fields as $H$ or/and $A_{\mu}$. An ansatz for $L$
which generalizes the $A_{\mu},H$ particle content of the abelian Higgs model
would be of the form ($h_{\mu\nu}$ massive)

$L=mgh_{\mu\nu}h^{\mu\nu}H+\,U(H,h,\phi)$ (85)

where the first term represents the ”ignition” i.e. the simplest
renormalizable ($d_{sd}=3$) interaction associated with a $h,H$ particle
content and $U$ contains all the remaining possible at most quadrilinear
couplings between $h_{\mu\nu},$ its 5 escorts $\phi_{\mu},\phi$ and $H$. Their
coupling strengths are determined from the first or second order (”induction”)
pair condition.

The $L,Q_{\mu}$ pair may be uniquely determined, but it is rather improbable
that $d_{sd}(L)\leq 4.$ It would be premature to dismiss $L,V_{\mu}$ pairs
with $d_{sd}(L)>4.$ The example of pl models with $d_{sd}(L^{P})=5,~{}$which
in the standard pl renormalization theory leads to a with perturbative order
increasing number of renormalization parameters but under the guidance of an
S-matrix-equivalent sl pair turns into an improved formalism. This upgraded
$L^{P}$ description contains $d_{sd}\rightarrow\infty$ pl fields but shares
its parameters, the S-matrix and its pl local observables with the SLFT
renormalization theory.

Presently our understanding of the consequences of the higher order SLFT pair
requirements is too scarce to say anything credible about $~{}L,V_{\mu}$ pairs
with $d_{sd}(L)>4$. A clarification of this important issue will be left to
future research.

##  6 Dynamical string-localization of interacting fields

Free massive pl fields can not only be converted into their sl counterparts by
integration along strings but the directional $e^{\mu}\partial_{\mu}$
differentiation on sl free fields permits also the return to its pl form.
Together with their Wick-ordered composites they form the local equivalence
(sl extended Borchers-) class $\mathcal{B}$ of free fields (pl fields are
viewed as special cases of sl).

Recall that for the construction of the S-matrix corresponding to a prescribed
particle content one uses pl fields for $s<1$ and those special $s\geq 1$
massive sl potentials which were constructed in section 2.3 by ”fattening”
their uniquely defined sl massless counterpart. Together with the uniquely
defined pl Proca potential and a scalar sl field referred to as the escort
they constitute a triple of relatively causally localized fields which act in
the massive $s=1$ Wigner-Fock space. They fulfill a linear relation which is
the basis for the construction of renormalizable sl interaction densities with
string-independent S-matrices.

This ”kinematic” sl localization of $s\geq 1~{}$free fields is important for
the construction of the S-matrix ala Bogoliubov. But it does not account for
the physical localization of the interacting fields which is not in the hands
of the calculating physicists but is determined by the particle content of the
model. To distinguish between the two the localization of the interacting
fields will be referred to as ”dynamic localization”.

To understand this important point it is helpful to recall the form of the
Bogoliubov map which relates the pl or sl Wick-ordered free fields from the
local equivalence class of free fields $\mathcal{B}$ to that of normal ordered
interacting fields $\mathcal{B}|_{L}$ (46). For pl gauge theoretic
interactions densities $L^{K}$ this problem has been studied in [61].

One important result is that this perturbatively defined linear Bogoliubov map
preserves the relative causality of fields but not the algebraic structure.
This is in agreement with algebraic QFT which is based on the idea that the
full physical content of QFT in the presence of interactions is contained in
the net of spacetime localized algebras [2]. What is shared between
$\mathcal{B}$ and $\mathcal{B}|_{L}~{}$in case of massive vector potentials is
the Wigner-Fock Hilbert space in which these fields act.

This transfer of pl causality undergoes significant changes in the presence of
sl fields. As in the calculations in the previous section one uses a lightlike
$e$, in this case no directional testfunction smearing is necessary.

For the understanding of changes in localization caused by the Bogoliubov map
it is not necessary to enter the details of perturbative renormalization. It
suffices to understand the relations between free fields in $\mathcal{B}$
which result from the assumption that their interacting images of the
Bogoliubov map into the target spaces $\mathcal{B}|_{L_{tot}\text{ }}$and
$\mathcal{B}|_{L_{tot}^{P}\text{ }}$coalesce. Hence one may omit the
prefactors $S^{-1}$ in the Bogoliubov maps and write

$\displaystyle S(g(x)L_{tot}^{P}+\lambda
f\varphi_{g})|_{\lambda=0}\overset{a.l.}{\simeq}S(g(x)L_{tot}+\lambda
f\varphi)|_{\lambda=0}$ (86)
$\displaystyle\varphi_{g}|_{L_{tot}^{P}}=\varphi|_{L_{tot}},~{}~{}\varphi_{g}(x,e)=\varphi(x,e)+\sum_{k=1}^{N}\varphi_{k}(x,e)$
(87)

where the $\varphi_{g}|_{L_{tot}^{P}}$ refers to the interacting image of
$\varphi_{g}$ under the $L_{tot}^{P}$ Bogoliubov map.

The information about the localization of an interacting field
$\varphi|_{L_{tot}}$ is contained in the left hand side323232The important
point is that the $L^{P}$ Bogoliubov map preserves localization; hence one can
use it to find out whether the image of $\varphi$ under $L~{}$is sl or pl.
whereas its renormalizability status (finite or infinite $d_{sd}$) can be red
off on the the right hand side. Fields which are renormalizable and at the
same time pl in the $L^{P}$ setting represent observables whereas
renormalizable fields which are sl on the $L_{tot}^{P}$ side are sl
interpolating fields.

The formal combined map of $\mathcal{B}$ into itself is highly non-linear and
generally changes localization properties; this is the price for the
preservation of renormalizability. The $\varphi_{k}(x,e)$ in (87) are
determined by the induction

$\varphi_{k+1}(x,e)=ig\int
T(L_{tot}^{P}(x^{\prime})-L_{tot}(x^{\prime},e))\varphi(x,e)=ig\int[\partial^{\prime\mu}T]V_{tot,\mu}(x^{\prime},e)\varphi_{k}(x,g)d^{4}x^{\prime}$
(88)

where $[\partial^{\prime\mu},T]$ denotes the difference between the
$\partial~{}$acting outside and inside the time-ordering which either vanishes
or contributes a $\delta$-term.

In massive QED this conversion (87) has no effect on pl observables; fields as
$A_{\mu}^{P}$ and$\ F=\operatorname{curl}A^{P}$ simply pass through since with
$V_{\mu}=\phi j_{\mu}$ and $\varphi_{0}=A_{\mu}^{P}~{}$the right hand side
(88) vanishes and hence

$A_{\mu}^{P}(x)|_{L^{P}}=A_{\mu}^{P}(x)|_{L},~{}~{}F_{\mu\nu}|_{L^{P}}=F_{\mu\nu}|_{L}\quad$
(89)

The idea underlying such conversions was first used by Mund [94] in the
context of massive spinor QED. He calculated higher orders for the charge-
carrying $\psi$ (spinor or complex scalar) and found consistency with

$\displaystyle\psi(x)|_{L^{P}}=e^{ig\phi(x,e)}\psi(x)|_{L}$ (90)
$\displaystyle\psi(x)|_{L}=e^{-ig\phi(x,e)}\psi(x)|_{L^{P}}$ (91)

The formula is reminiscent of gauge transformation, however its physical
content is quite different.

A particularly interesting application of the conversion formalism arises in
the Higgs model. Different from massive QED, neither the $s=1$ field$\
A_{\mu}^{P}|_{L^{P}}$, nor $H_{L^{P}}$ are local observables. Using the form
of $V_{\mu}~{}$in (79) one finds that $H$ is transformed into $H_{1}$

$H_{1}(x,e)=-\int[\partial^{\prime\mu},T]V_{\mu}(x^{\prime})H(x)d^{4}x^{\prime}=\frac{1}{2}:\phi^{2}(x,e):$
(92)

i.e. $H$ is against naive expectations not an observable but rather represents
a sl interpolating field. The same holds for $A^{P}$ or its
$F=\operatorname{curl}A^{P}.$

Allowing additive composite modifications $H\rightarrow H+polyn(H,A^{P})$
which preserve the asymptotic scattering state of the $H$-particle does not
change the situation. The same holds for the $A_{\mu}^{P}$ or
$F=\operatorname{curl}A^{P}$ Hence both fields which are linearly related to
the particle content of the model are interpolating fields and do not
represent observables. The fact that $A_{\mu}^{P}|_{L^{P}}$ and $F$ are
observables333333The line integral over an observable commutes with ”switching
on” the interaction and does not represent an interpolating fields. in massive
QED but not in the Higgs model shows that the observable-interpolating field
dichotomy is not a kinematic property.

Hence fields representing local observables in the Higgs model are necessarily
composite. A composite field which exists in every model is the interaction
density $L_{tot}^{P}|_{L_{tot}^{P}}=L_{tot}|_{L_{tot}}.$ The right hand side
was calculated in (81) and the computation of $L^{P}~{}$will be contained in a
forthcoming publication [89].

A better understanding about the singular structure of pl fields may shed new
light on the localizability of $d_{sd}=\infty$ fields which arose in
connection with summing up graphical structures in certain nonrenormalizable
models [95]. This problem was taken up by Arthur Jaffe [66][67] who discovered
a new class of singular distributions which still permit smearing with a dense
set of compact supported Schwartz testfunctions. These Jaffe distributions had
no impact on QFT because the unguided pl nonrenormalizability with its
infinite number of renormalization counterterm parameters does not present a
well-defined arena for physical applications. Such $d_{sd}=\infty~{}$pl fields
are probably too singular to generate operator algebras, but they may still
create physical states in the SLFT-guided $L^{P}$ formalism.

In section 3.2 we sketched the application of this formalism to interactions
with external potentials. Such interactions do not lead to loop contributions.
This simplicity of only induced contribution promises an interesting
mathematically controllable playground for the study of the pl localization
properties of observables and that of sl interpolating fields.

SLFT is presently the last step in a process of dissociating QFT from its
historic ties with Lagrangian quantization. When shortly after the discovery
of renormalized QED Arthur Wightman presented his ”axiomatic” formulation of
QFT in terms of pl fields, it appeared to be the most appropriate intrinsic
formulation which can be extracted from Lagrangian quantization and Wigner’s
representation theory [1]. In his algebraic formulation of Local Quantum
Physics (LQP) Rudolf Haag proposed a setting of QFT based on a net of
localized operator algebras representing observables which removed the last
vestiges of quantization [2].

The next step was taken in the 80s by Buchholz and Fredenhagen who showed that
the existence of observable algebras and suitably defined particle states
guaranties the presence of operators localized in arbitrary narrow spacelike
wedges (whose cores are strings) which create these particle states from the
vacuum [16]. These constructions were too far removed from the exigencies of
renormalized perturbation theory in order to have a direct impact on
calculations.

As a result it tooks more than 30 years to incorporate these observations into
a new sl perturbation theory in whose discovery the understanding of the
noncompact localization of Wigner’s infinite spin matter was an important
catalyzer [35]. Fortunately one does not have to go through the details of
this history in order to do perturbative calculations. But what may be
interesting to note is that, different from Wightman’s extraction of his
axiomatic setting from what one learned from the mathematically rather ill-
defined rules of Lagrangian quantization, the construction of SLFT took the
opposite path by converting ideas from LQP into perturbatively accessible
computations.

Its most remarkable physical property is that observables are distinguished
from interpolating fields in terms of localization, which is of course to be
expected in a theory based on causal localization, but which GT could not
accomplish.

There remains the question how GT with its lack of quantum positivity for
interpolating fields achieves to be such an amazingly successful description.
This will be commented on in the concluding remarks.

    Erratum: the claim that the interacting Proca field $A_{\mu}^{P}$ of the abelian Higgs model is an sl interpolating
    field is incorrect. The correct propagator is transverse which implies that $(A_{\mu}^{P})_{1}=0$ i.e. $A_{\mu}^{P}$ remains a pl
    observable.

##  7 Resumé, loose ends and an outlook

SLFT is a formulation of QFT in which renormalizable interacting fields
maintain the tightest possible localization which is compatible with quantum
positivity and causality. In contrast to gauge theory its physical range is
not limited to local observables and the S-matrix but also includes string-
local interpolating fields which mediate between the causality principles of
QFT and the string-independent scattering properties of particles. All degrees
of freedom are provided by Wigner’s particle representation theory.

As described in the introduction the discovery of SLFT was triggered by the
construction of sl free fields associated to Wigner’s positive energy infinite
spin representation [35]. Yngvason’s 1970 No-Go theorem [14] precluded the
existence of pl fields. It turned out that Wigner’s massless infinite spin
representation presents a much stronger barrier against pl localization than
that observed by Weinberg and Witten in massless finite helicity
representations. The Weinberg-Witten No-Go theorem excludes the existence of
conserved higher helicity pl currents and energy-momentum tensors; in view of
the absence of massless pl vector potentials and the fact that the existence
of pl massless limits depends on the short distance dimension $d_{sd}$ this is
hardly surprising .

The infinite spin case excludes the existence of pl composites; more general:
the causal localization of infinite spin matter is necessarily noncompact [42]
in concordance with smearing sl fields with directionally compact localized
test functions $f(x,e),~{}e^{2}=-1.$ Closely related is that infinite spin
matter cannot interact with ordinary (finite spin) quantum matter , but
through its energy-momentum tensor its backreaction on classical gravity may
lead to a noncompact form of gravity. Quantum inertness combined with
gravitational reactivity are properties attributed to dark matter [38]. Since
the sl infinite spin energy-momentum tensor is known as a bilinear form [8]
such a calculation appears feasible.

The existence of sl infinite spin field with finite $d_{sd}$ suggested that
the renormalizability destroying $d_{sd}=s+1$ increase of short distance
dimension can be avoided by using sl fields. This was the start for the
construction of sl potentials for finite $s,h$ which provided the positivity
preserving (the Gupta-Bleuler degrees of freedom avoiding) $d_{sd}=1$
potentials. As mentioned in section 3 the absence of pl currents does not
exclude the existence of local charges which are localized in arbitrary small
spacetime regions.

The weakening of causal localization in SLFT should not be misunderstood as
(what is commonly referred to as) ”nonlocal”343434The authors of [10] had
problems with referees who rejected the work with the argument that SLFT is
nonlocal.. The use of covariant semi-infinite space- or light-like half-lines
does not get into conflict with the causality prerequisites of scattering
theory (namely the possibility of placing an arbitrary number of fields in
relative spacelike positions), nor is the derivation of important structural
theorems (TCP, Spin&Statistics) impeded.

Among the continuously many sl potentials only one for each $s$ plays a role
in SLFT perturbation theory. The key observation for its construction is that
the equation $\operatorname{curl}A=F$ for a sl massless field $A_{\mu}(x,e)$
acting in the Wigner-Fock helicity space associated to the ($m=0,h=\pm 1$)
Wigner representation has a unique solution which replaces the positivity
violating pl potential of GT.

By a process referred to as ”fattening” (section 2) this solution selects
among the many possible massive sl potentials (which act in the $s=1$ Wigner-
Fock Hilbert space of the unique pl Proca potential) a distinguished sl vector
potential. Together with a canonically constructed scalar sl potential
$\phi(x,e)$ (the escort) one obtains a triple of linear related fields
$A_{\mu}-\partial_{\mu}\phi=A_{\mu}^{P}$ which act in the $s=1$ Wigner Fock
space and belong to the linear part of the causal equivalence class of (Wick-
ordered) free fields associated to the Wigner representation $(m>0,s).$

The string independence expressed as the pair relation
$d_{e}(A-\partial\phi)=0$ is the basis for constructing a renormalizable
interaction density $L(x,e)$ which couples the $s=1~{}$sl $A$ and $\phi$
fields to lower spin free fields which remain pl. Together with a suitably
defined vector density $V_{\mu}~{}$ one arrives at the pair relation
$d_{e}(L-\partial V)=0$ which insures the string-independence of the S-matrix
which is obtained by taking the adiabatic limit of time-ordered product of the
interaction density. The lowest order pair relation may need an extension by
induced terms which result from the implementation of higher order pair
conditions. This is a new phenomenon which has no counterpart in the old pl
perturbation theory.

The interacting quantum fields associated to this S-matrix are constructed in
terms of the Bogoliubov map which converts pl or sl fields from the causal
equivalence class of Wick-ordered free fields into their normal ordered
interacting counterpart. The restriction to pl $s<1$ and sl $s=1$ free fields
is only necessary in the construction of the S-matrix; the Bogoliubov map can
be applied to any (pl or sl, elementary or composite) field in the free field
class.

Its interacting target fields have in general a different localization from
their source fields. The target localization has to be determined in the
$L^{P}$ setting (see previous section). Renormalizable ($d_{sd}<\infty$)
fields in the $L$ target space (independent of their pl or sl localization)
represent observables if their $L^{P}$ source fields are pl; otherwise they
represent interpolating sl fields.

SLFT has been applied up to second order to all models in which vector mesons
interact with themselves or with $s<1$ particles. The by far conceptually most
demanding and interesting QFT is the Higgs model which in its most simple
(abelian) form is the QFT in which a vector meson interacts with a $s=0$
Hermitian field. The first order pair turns out to be uniquely fixed and its
second order implementation induces a $H$ selfinteraction which looks as if it
would come from spontaneous symmetry breaking on a postulated Mexican hat
selfinteraction. The conceptual difference to SLFT is enormous.

A similar but somewhat more elaborate second order calculation for
selfinteracting massive vector mesons reveals that the coupling structure of
the leading $d_{sd}=4$ contributions up to second order is that of a reductive
Lie-algebra. The surprise is that, different from gauge theory, this apparent
symmetry in the $d_{sd}$ leading contribution has not been imposed. In fact it
is not even a symmetry in the sense in which this terminology is used to
describe unitary implemented inner symmetries.

Whereas symmetries and their spontaneous or complete breaking of
selfinteracting scalar particles can be freely imposed, there are strong
restrictions from first principles on the form of $s\geq 1~{}$SLFT
selfinteractions which leave no such freedom; the form of selfinteractions in
the presence of $s\geq 1~{}$is fully determined by the particle content of the
model and not at the disposition of the calculating theorist. The use of the
positivity violating gauge symmetry obscures this important insight. The
chirality theorem [90] shows that these principles also affects the coupling
of selfinteracting vector potentials to Dirac fermions.

Another somewhat unexpected property is that renormalizable interaction sl
densities $L$ may produce second order sl $d_{sd}=5$ contributions which, if
left uncompensated, destroy the $e$-independence of $S$ as well as
renormalizabilty. The only way to save such a model is to enlarge its particle
content by a $A$-$H~{}$interaction which induces a compensating second order
$A$ selfinteraction. This, and not SSB, is the raison d’être for the presence
of an $H$-particle in models of selfinteracting massive vector mesons.

The application of the SLFT perturbation theory to the Higgs model leads to
other foundational questions whose answer may be trendsetting for the
development of QFT. The basic interaction density $A\cdot AH$ for a
$A_{\mu},H$ particle content (the ”ignition” from which the $L,V_{\mu}$ pair
requirement uniquely induces all other contributions) is superrenormalizable
since $d_{sd}(AAH)=d_{cl}=3$. One does not expect that interactions induced by
superrenormalizable couplings lead to higher order counterterms with new
coupling parameters. A $4^{th}$ order confirmation of this expectation does
presently not exist (neither in SLFT nor in the gauge theoretic SSB setting).

Even more important is to find out if $SLFT$ permits an extension to $s\geq
2.$ The remarks in section $5.6~{}$on $s=2$ selfinteractions show that
$d_{sd}(L)=5.$ To conclude that the theory is useless because it violates the
power-counting bound is premature since (previous section) the main reason for
dismissing interaction densities is that they lead to a with perturbative
order increasing number of coupling parameters and not the fact that there are
fields with an increasing short distance scaling degree. For the acceptance of
a model it suffices that its S-matrix is well-defined and that its observables
remain pl with bounded $d_{sd},$ independent of whether the $d_{sd}$ of its sl
interpolating fields increase with perturbative order.

Candidates with $s=2$ potentials $A_{\mu\nu}~{}$and superrenormalizable
”ignition” of the form $A_{\mu\nu}A^{\mu\nu}H~{}$or
$A^{\mu\nu}A_{\mu}A_{\nu}~{}$and induced $L,V_{\mu}$ pairs couplings with
$d_{sd}(L)=5~{}$are expected to exist. As long as the number of counterterm
coupling parameters does not increase with perturbative order and the physical
predictability is maintained there is no obvious reason for their exclusion of
such $L,V_{\mu}$ pairs. Only further research can resolve these challenging
problems.

Can SLFT shed some light on the perplexing question why GT inspite of its
obvious conceptual shortcomings353535Positivity is an indispensable property
which secures the probability interpretation of quantum theory.  remained such
an amazingly successful theory? This paradigmatic question may have a positive
answer. The $L^{K},V_{\mu}^{K}~{}$pair property is a consequence of the
relation $A_{\mu}^{P}=A_{\mu}^{K}+\partial_{\mu}\phi^{P,K}$ where $K$ refers
to the Krein space of GT and $\phi^{P,K}$ is a scalar pl ”hybrid” escort which
mediates between the $P$ and $K$ formalism [97].

In the BRST formulation used in [64] the fields act in a Stückelberg- and
ghost- extended BRST space. The physical space, to which the action of
$A_{\mu}^{P}$ can be restricted, is defined in terms of BRST cohomology and
observables are defined as objects invariant under the BRST operation
$\mathfrak{s}$ ($\mathfrak{s}O=0$ for observables and $\mathfrak{s}S=0$ for
the S-matrix).

The advantage of the hybrid formulation proposed by Mund [97] is that,
different from the formalism used in [64], Stückelberg- and ghost- degrees of
freedom are not needed. Instead of spaces which are embedded in the sense of
BRST cohomology one deals with factorization through Gupta-Bleuler subspaces.

An explicit expression for the mixed hybrid escort $\phi^{P,K}$ was recently
calculated by Mund (private communication). The Proca potential lives in the
transverse subspace to the mass shell $p^{\mu}\psi_{\mu}(p)=0$ which is
embedded in the Krein space whereas the living space of fields is the full
4-component Krein space. The triple relation can the be used to define a
$L^{K},V_{\mu}^{K}$ pair which is S-matrix-related to the physical $L^{P}$
formulation.

A favorable situation for studying infrared phenomena arises from the hybrid
triple
$A_{\mu}^{sl}(x,e)=A_{\mu}^{K}(x)+\partial_{\mu}\phi^{sl,K}(x,e).~{}$The
$A_{\mu}^{S}~{}$(without) lives on the physical subspace of the Gupta-Bleuler
Krein space. All three contributions have a massless limit, but $\phi^{s,K}$
without the derivative has the typical logarithmic infrared divergence known
from scattering theory of charge-carrying particles. Its exponential $\exp
ig\phi^{S,K}(x,e)$ seems to provide the kind of directional superselection
rule of photon ”clouds” whose presence is required by a theorem [50]. This
picture is a closer analog of (10) than the $\exp ig\Phi(x,e,e^{\prime})$
constructed in (9).

This hybrid pair description does not only explain the close relation of a
(from ghosts and negative metric Stückelberg fields liberated) Gupta-Bleuler
GT with the positivity preserving SLFT, but it also shows that GT plays a
useful constructive role for a better understanding of SLFT in QED. The hybrid
relation reveals that the physical origin of quantum gauge theory is that one
cannot squeeze causally spacetime localized pl vector potentials into the
Wigner momentum space; this is only possible by permitting a noncompact but
still causally separating localization.

In particular it contains informations about the change of the Wigner particle
space for the $\mathcal{B}|_{L}$ operators (previous section) in the massless
limit. Whereas the fields in $B$ live in a Wigner-Fock helicity space, their
interacting images in $\mathcal{B}|_{L}$ act on a larger space for whose
construction one needs to form line integrals on indefinite Gupta-Bleuler
potentials $A_{\mu}^{K}(x)$ (still indefinite) and convert them into complex
exponential fields (the photon clouds) whose associated Hilbert space is
expected to show a similar infrared structure as the exponentials $\exp
ig\varphi(x)$ of the indefinite logarithic divergent massless $d=1+1$ scalar
fields (the $\varphi$-clouds).

The hope is to obtain a spacetime understanding of infrared phenomena
including the large-time behavior which replaces that of the LSZ scattering
theory. This includes the vanishing of scattering between charge-carrying
particles with only a finite number of outgoing photons.

This cannot be described solely in terms of free matter fields, rather the
exponential sl dependent photon cloud fields must play an important role.
Similar to the $\varphi$-clouds in a two dimensional model (10) they are
expected to ”soften” the mass-shell singularity and account for the zero
probability for the emission of a finite number of photons in collisions of
charged particles whereas a perturbative expansion which ignores this change
of the mass-shell leads to the logarithmic infrared singularities. As often,
the devil is in the details.

SLFT also suggests that behind confinement there could be a more radical off-
shell perturbative logarithmic infrared divergence of massless selfinteracting
gluons. Such off-shell divergences are absent in covariant gauges of
nonabelian GT, but off-shell long distance singular behavior of self-
interacting gluons in SLFT may be stronger than in GT [45][38].

Most problems of SLFT remain unsolved; on particular the present state of
knowledge about higher order perturbative renormalization is insuffient. Its
strengths is that the new ideas passed many tests and that the promise to
transcend the conceptual limitations of GT is too tempting to resist.

This may be attributed in part to the fact that its underlying ideas are in
embryo and the number of researchers who know about their existence and
decided to study them is still very small. There is no lack of researchers
working on foundational problems of QFT extending the pioneering work of
Wightman, Haag and others. Most theoreticians use the existing gauge theoretic
formalism to solve problems of high energy particle physics or cosmology.
During the last 5 decades a lot of time has been invested in reseach on
speculative ideas as String Theory, Multiverses, Supersymmetry and alike; the
incentive was obviously to continue the success of the first three decades of
QFT in which such speculative way of proceeding was very successful and which
led to most of our by now household goods.

The lack of any tangible results of these attempts led meawhile to feelings of
somberness. The SLFT raises the question why loose time with speculative ideas
if we still know so little about our most successful theory?

Acknowledgements. This work is part of a joint project whose aim is to develop
a positivity-preserving formulation of causal perturbation theory QFT which,
instead of using indefinite metric and ”ghosts”, is based on only physical
degrees of freedom. The people presently involved in it are José Gracia-
Bondía, Jens Mund, Karl-Henning Rehren and Joseph Várilly and their students.
For intense and profitable contacts I owe thanks to all of them. I am also
indebted to Michael Dütsch for helpful comments.

I acknowledge a stimulating correspondence with Edward Witten which started
with his curiosity about the history of Haag duality in connection with a
recent upsurge of interest in the role of causality in problems of
entanglement. SLFT has similar conceptual roots and particle physics would
profit if it could divert some of this new interest in fundamental properties
of QFT. Last not least I thank the referee for his constructive proposals.

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