# The role of positivity and causality in interactions involving higher spin
Nuclear Physics B 941 (2019) 91-144

Bert Schroer
CBPF, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil
permanent address: Institut für Theoretische Physik, FU-Berlin,
Arnimallee 14, 14195 Berlin, Germany
email<EMAIL_ADDRESS>
(June 15th 2018)

###### Abstract

It is shown that the recently introduced positivity and causality preserving
string-local quantum field theory (SLFT) resolves most No-Go situations in
higher spin problems. This includes in particular the Velo-Zwanziger causality
problem which turns out to be related in an interesting way to the solution of
zero mass Weinberg-Witten issue. In contrast to the indefinite metric and
ghosts of gauge theory, SLFT uses only positivity-respecting physical degrees
of freedom. The result is a fully Lorentz-covariant and causal string field
theory in which light- or space-like linear strings transform covariant under
Lorentz transformation.

The cooperation of causality and quantum positivity in the presence of
interacting $s\geq 1$ particles leads to remarkable conceptual changes. It
turns out that the presence of $H$-selfinteractions in the Higgs model is not
the result of SSB on a postulated Mexican hat potential, but solely the
consequence of the implementation of positivity and causality. These
principles (and not the imposed gauge symmetry) account also for the Lie-
algebra structure of the leading contributions of selfinteracting vector
mesons.

Second order consistency of selfinteracting vector mesons in SLFT requires the
presence of $H$-particles; this, and not SSB, is the raison d’être for $H.$

The basic conceptual and calculational tool of SLFT is the S-matrix. Its
string-independence is a powerful restriction which determines the form of
interaction densities in terms of the model-defining particle content and
plays a fundamental role in the construction of pl observables and sl
interpolating fields.

###### Contents

  1. 1 Introduction and history of the problem
  2. 2 String-local tensor potentials and conservation laws
    1. 2.1 Massless string-local potentials
    2. 2.2 A brief interlude, relation with concepts of algebraic QFT
    3. 2.3 Massive string-local potentials
    4. 2.4 Infinite spin revisited
  3. 3 Causality and the Velo-Zwanziger conundrum
    1. 3.1 Recalling the solution of the Weinberg-Witten problem and the associated local charges
    2. 3.2 The V-Z conundrum arises from an incorrect implementation of causality
  4. 4 Particle wave functions and causal localization
    1. 4.1 Newton-Wigner localization and its causality-providing modular counterpart
    2. 4.2 Mathematical properties of modular localization
    3. 4.3 A critical perspective based on modular localization
  5. 5 Renormalization in the presence of massive sl vector mesons
    1. 5.1 Remarks on scalar QED; induced interactions and counterterms
    2. 5.2 The perturbative S-matrix in the SLFT setting
    3. 5.3 External source models
    4. 5.4 Hermitian $H~{}$coupled to a massive vector potential
    5. 5.5 Selfinteracting vector mesons
    6. 5.6 The pair condition for higher spins
  6. 6 Dynamical string-localization of interacting fields
  7. 7 Resumé, loose ends and an outlook

Dedicated to Klaus Fredenhagen on the occasion of his 70th birthday

##  1 Introduction and history of the problem

The positivity property of quantum states guaranties the probabilistic
interpretation of quantum theory. It enters the mathematical formalism through
the identification of states with unit rays in a Hilbert space on which the
quantum observables act as operators. In quantum field theory (QFT), or more
generally for models with infinitely many degrees of freedom, it is often more
appropriate to identify states with positive linear functionals on operator
algebras. Thanks to the existence of a canonical construction111The
”reconstruction theorem” in [1] is a special case of the more general Gelfand-
Naimark-Segal (”GNS”) reconstruction theorem [2]. this formulation in terms of
expectation values permits a return to the more common Hilbert space setting.

Its validity in quantum mechanics is guarantied by Heisenberg’s canonical
quantization of positions and momenta in conjunction with the von Neumann
uniqueness theorem which insures that irreducible representations of the
Heisenberg commutation relations are unitarily equivalent to the Schrödinger
representation. Born’s identification of the absolute square of the
Schrödinger wave function with the probability density for finding a particle
at a particular position connects positivity with spatial localization.

This situation undergoes significant changes in relativistic QFT where the
positivity of field-quantization looses its ”von Neumann protection” in the
presence of higher spin $s\geq 1$. The first such clash with positivity was
noticed by Gupta and Bleuler who observed that quantized massless vector
potentials are incompatible with Hilbert space positivity. In the absence of
interactions it is straightforward to restore positivity by passing from
potentials to field strengths, but the use of local gauge invariance to
preserve at least part of positivity in the presence of interactions leads to
a loss of important physical operators and states.

This includes in particular all interacting fields which interpolate charge-
carrying particles in the sense of large time scattering theory. Such
interpolating fields play an indispensable role in connecting the causal
localization- and quantum positivity- principles of QFT with observed
scattering properties of particles. Their absence in quantum gauge theory (GT)
is accompanied by a loss of mathematical tools of functional analysis. The
proofs of structural properties as TCP and Spin&Statistics theorems use
Hilbert space positivity in an essential way and have no substitute in
indefinite metric Krein spaces. This reduces the use of GT to perturbative
rules for dealing with indefinite metric- and ghost- degrees of freedom (the
BRST formalism)

Positivity-obeying massive tensor potentials and their spinorial counterpart
are provided by Wigner’s unitary representation theory of positive energy
particle representations of the (covering of the) Poincaré group, but they
come with an increase of their short distance scale dimension222It is most
conveniently obtained from property of the field’s 2-ptfct
$x\rightarrow\lambda x$ for $\lambda\rightarrow 0.$ with spin
$d_{sd}=s+1~{}$which prevents their use in renormalized perturbation theory
involving fields with higher spin $s\geq 1.$ It turns out that this worsening
of short distance behavior with increasing spin is accompanied by a $m^{-s}$
divergence for small masses. Hence a formulation of QED in terms of
positivity-maintaining point-local potentials is not possible.

In his well-known monograph Weinberg presents a systematic construction of the
intertwiner functions which relate Wigner’s spin $s$ momentum space particle
creation and annihilation operators $a^{\\#}(p,s)$ associated with the unitary
($m,s$) representations with covariant pl free fields which act in the Wigner-
Fock Hilbert space of the Wigner operators [3]. This interesting section in
his book remained a torso since the (with increasing $s$) worsening short
distance scale dimension of point-local fields prevents their use in
renormalized perturbation theory as soon as $s\geq 1$.

In the main part of his book Weinberg uses the positivity-violating (but
renormalizability-improving) gauge theoretic setting as obtained by Lagrangian
quantization in which a perturbative inductive argument secures the positivity
of gauge invariant operators. For this reason one does not find GT in
presentations of nonperturbative QFT.

The independence of short distance dimensions of quantized gauge fields from
spin/helicity is a consequence of the spin independence of the classical
dimension $d_{cl}=4$ of Lagrangians. For $s=0,1/2$ these fields agree with
those obtained from the Wigner-Weinberg construction, but for $s\geq 1$ the
equality of the short distance dimension with the classical dimension in terms
of mass units (”engineering” dimension), namely $d_{sd}=d_{cl}=1$ for integer
and $3/2$ in case of half-integer spin, comes with an improvement of
renormalizability at the price of the presence of unphysical degrees of
freedom.

Whereas in older work [4] positivity problems for propagators of higher spin
fields in GT have been at least partially addressed, more recent publications
([5] [6] and papers cited therein) are mainly concerned with classical
geometric aspects of the Lagrangian gauge formalism for which these problems
can be ignored.

The setting of string-local quantum field theory (SLFT) in the present article
overcomes this conceptual gap between GT and costructions of fields based on
Wigner’s representation theory by providing a positivity maintaining causal
perturbative QFT formalism which includes the important physical interpolating
fields of particles whose large-time properties account for a unitary S-matrix
and which are missing in GT. After almost 70 years of GT this amounts to a
paradigmatic shift which does not only affect renormalized perturbation theory
but also requires to extend the nonperturbative setting of ”axiomatic QFT” as
presented in [1].

A convenient starting point is to recall the construction of positivity
obeying quantum fields $\Psi_{a}~{}$in Weinberg’s intertwiner formulation (for
simplicity for massive tensor potentials):

$\displaystyle\Psi_{\alpha}(x)$
$\displaystyle=\int(\sum_{s_{3}=-s}^{s}e^{ipx}v_{\alpha,s_{3}}(p)a^{\ast}(p,s_{3})+\hbox{h.c.})d\mu_{m}(p)+h.c.,~{}~{}$
(1) $\displaystyle with~{}\
d\mu_{m}(p)=\theta(p_{0})\delta(p^{2}-m^{2})d^{4}p$

The intertwiner functions $v(p)$ convert the Wigner creation/annihilation
operators $a^{\\#}(p,s_{3})$ into covariant fields $\Psi_{\alpha}$; their
calculation uses only group theory [3]. They come with two indices, the
$s_{3}$ which runs over the $2s+1$ values of the third component of the
physical spin, and a tensor index $\alpha=(\mu_{1},..\mu_{s})$ which refers to
the $4s$ dimensional tensor representation of the Lorentz group of tensor
degree $s$. The extension to fermions is straightforward but not needed for
the problems addressed in the present work.

The momentum space Wigner creation/annihilation operators $a^{\\#}(p,s_{3}),$
and hence also the covariant fields $\Psi_{\alpha}.$ act in the Wigner-Fock
Hilbert space obtained from the 1-particle Wigner representation by ”second
quantization”333Note the difference to the standard use of ”quantization” (in
the words of Ed Nelson: ”second quantization is a functor, wheres quantization
is an art”). . Looking at the explicit form of the intertwiners and
calculating the two-point function (2-ptfct) of $\Psi$ one finds that the
latter scales as $\lambda^{-2(s+1)}$ for $x\rightarrow\lambda x$ in the limit
of small distances $\lambda\rightarrow 0$ which leads to assigning the short
distance dimension $d_{sd}=s+1$ to the point-local (pl) $\Psi.$

This construction via intertwiners permits much more flexibility than
Lagrangian or functional integral quantization in converting the unique
$(m,s)$ Wigner operators into fields with different prescribed covariance- and
causal localization- properties than quantization; includes in particular
string-local (sl) covariant quantum fields with improved ($0<d_{sd}<s+1$)
short distance properties. such fields are localized on causally separable
(i.e. allowing relative causal positioning) semiinfinite space- or light-like
strings (rays) $\mathcal{S}=x+\mathbb{R}_{+}e,$ $e^{2}=-1$ or $0.$ Whereas in
Weinberg’s construction the covariance under Lorentz transformations is
sufficient since causality , the construction of sl fields requires the direct
use of causal localization. The resulting covariant fields extend the linear
part of the pl relative causality class to sl (with pl considered as a special
case of sl) and Wick-ordered products thereof constitute the nonlinear
members.

This huge set of sl free quantum fields associated to an irreducible Wigner
representation contains in particular a sl tensor fields which is linearly
associated with its pl counterpart. This sl tensor field appears together with
$s$ escort fields with lower tensor degrees [7][45]. Escorts are reminiscent
of negative metric Stückelberg fields in gauge theory, except that they do not
add unphysical degrees of freedom to the physical $a^{\\#}(p,s_{3})$ Wigner
operators but only differ in their intertwiner functions.

Positivity and hence the unitarity of the S-matrix in the resulting string-
local QFT (SLFT) is automatic444The causal separation properties of sl fields
are more than enough for deriving linked cluster fall-off properties and
insure the $e$-independence of the (on-shell) S-matrix. (no Nobel-prize worthy
hard work as in gauge theory) and the chances to solve age-old infrared
problems (large time scattering theory in QED, QCD confinement,..) are
significantly enhanced 555Recently Rehren showed that infinite spin fields can
be obtained in terms of appropriatly defined Pauli-Lubanski limits of finite
spin escort fields [8].. One prerequisite is the substitution of nonexistent
positivity-maintaining pl potentials by sl counterparts and the according to
the Weinberg-Witten No-Go theorem [9] missing $h\geq 1~{}$sl current and
stress-energy tensors in [11] by suitably defined conserved sl substitutes.
The smooth passing from massive sl two-point functions with $2s+1$ degrees of
freedom to their massless two-component helicity counterpart leads to a
profound (indefinite metric- and ghost-free) understanding of the D-V-Z
discontinuity problem [12, 13].

An important step in the development of (SLFT) was the construction of fields
for the class of massless infinite spin Wigner representations for which
Yngvason’s 1970 No-Go theorem excluded pl fields [14]. For this class
Weinberg’s group theoretic method is without avail; one rather had to resort
to ideas from modular localization [34, 35, 15]. This paved the way for the
construction of the simpler finite spin sl free fields, including the use of
their short distance lowering and hence renormalization improving properties
in interactions.

In the same work it was also realized that finite spin/helicity sl fields can
also be obtained in a more direct way by integrating pl fields along semi-
infinite lines. This direct construction is particularly useful for those sl
potentials and their escorts which are linearly related to the pl spin $s\geq
1$ potentials. Different from the pl potentials which diverge in the massless
limit, the corresponding sl potentials pass to corresponding finite helicity
potentials which have no pl counterpart.

An important support for a string-extended QFT comes from work by Buchholz and
Fredenhagen who used the setting of algebraic QFT [16] to show that models
with particle states separated by spectral gaps which fulfill certain
consistency properties with respect to the local observables always contain
interpolating operators localized in arbitrarily narrow spacelike cones (whose
cores are strings). Perturbative SLFT is more specific by showing that in the
presence of $s\geq 1$ particles positivity together with causal localizability
leads to noncompact causal localization whose tightest localized covariant
generating fields are string-local.

The combinatorial nature of perturbation theory per se does not require
positivity and works also for gauge theory, but without positivity provided by
a Wigner-Fock Hilbert space the quantum theory remains incomplete. SLFT
reveals among other things that several limitations of gauge theory which are
the cause of certain No-Go theorems of which the best known is the
aforementioned Weinberg-Witten No-Go theorem (for a recent survey see [17])
are converted into Yes-Go statements in SLFT [11].

SLFT is the only formulation in which state-creating interpolating fields are
separated from observables by spacetime localization properties. Whereas in
the absence of interactions the localization of free fields associated to a
Wigner representation (”kinematic localization”) may be chosen at will, that
of interacting fields in SLFT is determined by the particle content of the
interacting theory: observables are pl and interpolating fields are sl.

The space- or light-like interpolating fields can be placed in spacelike
separated positions which is a prerequisite for the application of the LSZ
scattering theory. The particle states in which the expectation values of
observables are measured are constructed in terms of suitably defined large
time asymptotic limits to the vacuum. A theorem of large-time scattering
theory insures that the dependence on the interpolating operator in the large
time limit is contained in its vacuum-to-one particle matrixelement which is
then removed by passing to the correctly normalized particle states. This
implies in particular that the $e$-dependence of sl fields does not affect
particles and their scattering matrix.

The so-called cluster decomposition property of correlation functions of
fields plays an important role in the derivation of scattering properties. It
is a consequence of a mass gap and the existence of an arbitrary large number
of sl fields in relative spacelike position. This applies to spacelike strings
and (with some stretch of geometric imagination) also holds for lightlike
strings; it is however violated for timelike strings. In the present work the
terminology ”nonlocal” is avoided since its historical connotation may hinder
to see that these fields can be brought into causally separated positions.

The reader is also reminded that the terminology ”interaction density” instead
of ”interacting part of a Lagrangian” is not nitpicking; apart from
interactions between $s<1$ particles, interaction densities constructed from
sl Wigner fields are never interacting parts of Lagrangian.

Another important point which requires attention is the fact that the
kinematical sl localizations of free fields in terms of line integrals over
$s\geq 1$ pl fields only serves to construct interaction densities whose
S-matrix is independent on string directions. As mentioned the physical
localization of the corresponding interacting fields is dynamical and
generally different from that of their free counterpart used in the
construction of the interaction density and the S-matrix.

A surprising property of SLFT is that in the presence of $s\geq 1$ particles
its positivity and localization properties determine a unique model in terms
of its particle content whenever such a theory exists. This is a result of the
strong restriction which the string-independence of the S-matrix exerts on
interaction densities.

The quest for an intrinsic formulation in which the umbilical chord to
classical field theory provided by quantization has been cut is almost as old
as QFT. In the first (still pre-renormalization) presentation of quantum
electrodynamics at an international conference in 1929 [18] Pascual Jordan
expressed this in the form of a plea for an intrinsic understanding of QFT
which avoids the use of ”(quasi)classical crutches”; a decade later his former
collaborator Eugen Wigner took the first step in his famous classification of
relativistic particles [19].

The second step was taken two decades later by Rudolf Haag [2] when he
proposed an intrinsic formulation of QFT in terms of ”causal nets of algebras”
in which Wightman fields at best play the role of ”coordinatizations” (in
analogy to the use of coordinates in geometry).

With the arrival of the covariant formulation of quantized electrodynamics in
the 50s, Jordan’s dictum and its partial realization in Wigner’s
classification of noninteracting particles faded into the background; the new
covariant computational rules of quantized electrodynamics took a firm hold
and as a result the first covariant QFT was a positivity-violating gauge
theory.

A somewhat unexpected aspect of these first successful calculations was the
contrast between the precision of the experimentally verified perturbative
results and the robustness of the calculated results against the use of quite
different cutoff- and regularization- prescriptions, or even against different
ways of implementing Lagrangian quantization (Gell-Mann–Low, Feynman path
integrals, Bogoliubov’s generating S-functional).

The presence of gauge theoretic indefinite metric degrees of freedom in
interaction densities involving $s\geq 1$ particles led to conceptual
problems. A formal proposal to overcome these shortcomings was made by Jordan
[20]. It consisted in replacing the gauge dependent matter field666Jordan used
these fields for a pure algebraic derivation of Dirac’s geometric magnetic
monopole quantization [21]. by the formally gauge invariant string-local
composite field

$\Psi(x)=\psi^{K}(x)\exp
ig\int_{0}^{\infty}A_{\mu}^{K}(x_{0},\dots,x_{3}+\lambda)d\lambda$ (2)

where the $K~{}$refers to the gauge dependent Lagrangian field which acts in
an indefinite metric Krein space which in addition to physical degrees of
freedom contains also indefinite metric quanta (scalar and longitudinal
photons).

After the discovery of renormalized perturbation theory Mandelstam used such
representation as a starting point in his attempt to construct a perturbation
theory which avoids the use of potentials in favor of working directly with
gauge invariant fields [22]. Subsequently Steinmann [23] studied the problem
of recovering positivity by constructing such fields $\Psi~{}$in higher order
perturbation theory. Different proposals to recover positivity can be found in
[24]. The constructions of such formally gauge invariant composite fields and
their renormalization requires a lot of additional work and is of little
interest unless it leads to new physical insights.

The SLFT perturbation theory in the present paper uses sl potentials with the
$s$-independent short distance dimension $d_{sd}=1$ which ”live” in a physical
Wigner-Fock particle space$.~{}$The starting point is the observation that
there exist sl vector potentials $A_{\mu}(x,e)$ localized on causally
separable spacelike strings $\mathcal{S}=x+\mathbb{R}_{+}e$ which together
with their scalar sl ”escorts” $\phi(x,e)$ are linearly related to their pl
counterpart simple illustration is provided by the interaction density
$L^{P}=A_{\mu}^{P}j^{\mu}$ of massive QED which is related to its pl
counterpart as $A_{\mu}(x,e)=A_{\mu}^{P}(x)+\partial_{\mu}\phi(x,e)$.

Its use in an interaction density of e.g. massive QED
$L^{P}=A_{\mu}^{P}j^{\mu}$ results in a relation $L^{P}=L-\partial^{\mu}\phi
j_{\mu}$ in which the sl density $L(x,e)~{}$has an improved short distance
dimension $d_{sd}(L)=4$ (instead of $d_{sd}(L^{P})=5$) and accounts for the
first order contribution to the (on-shell) $S$-matrix in the adiabatic limit
$S=\int L$ to which the boundary term from $V_{\mu}=\phi j_{\mu}$ does not
contribute.

This is in a nut-shell a perturbative implementation of the aforementioned
abstract Buchholz-Fredenhagen theorem; it secures the existence of
interpolating sl fields whose directional smearing provides the B-F operators
localized in arbitrary narrow spacelike causally separable cones and insures
that their large-time scattering limits results in $e$-independent Wigner
particles and their S-matrix.

The extension of this first order $L$ to higher orders involves time-ordered
products in the interaction densities $L~{}$respective $L^{P}$ and leads to
new powerful normalization conditions which insure that the two different
interaction densities lead to the same S-matrix. As a result of the with
perturbative order growing number of counterterms, the $L^{P}$ theory by
itself is physically useless; but being ”guided” by the sl $L,V_{\mu}$ pair it
becomes a well-defined physically useful companion which shares not only its
parameters but also its S-matrix and local observables with the $d_{sd}(L)=4$
SLFT. Its only memory about its ”unguided past” is the with perturbative order
increasingly singular $d_{sd}\rightarrow\infty$ short distance dimension of
its interpolating fields.

SLFT is an S-matrix theory in the sense that the particle content together
with the string-independence of $S$ determines (in all cases studied up to now
uniquely) the form of the interaction density. In a second step the
construction of the S-matrix is extended to that of pl and sl interacting
fields.

Different from Lagrangian quantization the SLFT formalism does not prefer
certain fields. All interacting fields which act in the same Wigner-Fock space
and are members of the same causality class are on equal footing; which
particle they interpolate depends only on the nontriviality on their vacuum-
to-one-particle matrix elements.

Often new theoretical insights are the result of accidental observations. SLFT
is not of this kind; what led to it is the rather deep connection of sl
localization with modular localization theory. The terminology ”string” used
in quantizations of classical actions (Nambu-Goto actions, world-sheets,..)
bears no relation to the causal localization of string-local quantum fields in
the present work.

A definition of causal localization which avoids such misunderstandings is
that in terms of modular localization. In fact modular localization permits to
identify a pre-form of causal localization already within the Wigner positive
energy representation space [25] before ”second quantization” converts it into
the algebraic form of Einstein causality in QFT. This idea paved the way for
the construction of the QFT behind Wigner’s infinite spin representation.

Modular localization theory can be traced back to the Tomita-Takesaki modular
theory of operator algebras of the 60’s. It is one of a few mathematical
theories to which physicists working on problems of statistical mechanics of
open systems [26] made important contributions. It made its first appearance
in the context with causal localization in the Bisognano-Wichmann theorem [27]
which deals with modular properties of wedge-localized algebras. Modular
operator theory and modular localization requires positivity and hence cannot
be applied to GT and Lagrangian quantization.

As the result of accommodating thermal aspects and causal localization under
one conceptual roof, it led to profound insights (thermal properties of ”event
horizons”) into Hawking’s black hole radiation [28]. A first survey about its
history enriched by new results was presented by Borchers [29].

Modular localization also played an important role in the construction of QFTs
from S-matrices of integrable models in $d=1+1$ dimensions [30] [31].
Presently ideas from modular operator theory are being successfully applied to
obtain a foundational understanding of entanglement entropy caused by causal
localization (for a survey see [32] [33] and references therein).

Modular localization theory permits to extend Weinberg’s intertwiner
construction to Wigner’s infinite spin representations and to obtain explicit
expressions for the associated sl fields [34, 35, 36][37]. More recently these
fields reappeared as ”Pauli-Lubanski limits” of finite spin sl fields [8].
This made it possible to investigate physical properties of quantum matter
through the study of its positivity-obeying causal localization structure and
look for theoretical reasons why certain types of matter can not be seen in
counters [38].

One should also mention a series of more recent publications [39] [40] in
which covariant wave functions were constructed, but the much stronger result
in the aforementioned work was overlooked. Relativistic wave equations for
infinite spin appeared already in Wigner’s 1948 paper ([41] 12.1-12.4). These
different wave functions describe different covariant bases in Wigner’s
irreducible representation space. But for studying physical manifestations of
matter one needs to know its causal localizability which in case of infinite
spin does not follow from covariance and needs the use of modular localization
theory as used in the cited 2006 papers. In this way the 1970 No-Go theorem
[14] which excluded point-like localization was replaced by a sl Yes-Go
theorem.

In fact the exclusion of linear pl fields is part of a more general No-Go
theorem which rules out the possibility of constructing pl composites from the
linear sl infinite spin fields. In its most general form the theorem excludes
the existence of operator algebras localized in finite spacetime regions [42].

These theorems against pl localization of infinite spin matter may be seen as
an extreme counterpart of the Weinberg-Witten No-Go theorem against the
existence of higher helicity conserved currents and energy-momentum tensors.
The difference is that there still exist W-W local charges, whereas in case of
infinite spin there are no nontrivial operators localized in finite regions.

Recall that the raison d’être of a relativistic quantum field theory (for the
difference between QFT and relativistic QM, see section 3 in [43]) is the
realization of the ”Nahewirkungsprinzip” (action in the neighborhood
principle) of Faraday and Maxwell which culminated in Einstein’s concept of
relativistic covariance and causal localization. The positivity requirement of
quantum probability turns the construction of models of QFT into a challenging
problem which gauge theory did not solve.

It is the aim of this work to show how the recent SLFT formulation solves
problems which have remained outside the range of GT (for a review of such
problems see [17]). In [10] [11] this was already achieved for the problem
behind the Weinberg-Witten (W-W) No-Go theorem [9] and the $s=2$ van Dam-
Veltman-Zakharov (D-V-Z) discontinuity [12, 13]. Here we add the causality
problems raised by Velo and Zwanziger (V-Z) [44].

SLFT’s central point is however the presentation of a sl-based perturbation
theory which in contrast to gauge theory preserves the Hilbert space
positivity (no indefinite metric- and ghost-degrees of freedom) without
destroying the causal separability of fields. It leads in particular to
interesting different physical interpretations of interactions between vector
mesons and Hermitian fields (Higgs models, but no SSB Higgs mechanism) [45].

The content of this paper is organized as follows.

The next section recalls and extends recent (partially already published)
results concerning the construction of causally separable string-local free
fields. It consists of 4 subsections which includes the construction of sl
massless vector potentials and their canonically related massive counterpart.

The third section addresses the problem of interactions with external
potentials. It is shown that the origin of the Velo-Zwanziger causality
problem is the incorrect expectation that by modifying free field equations by
adding linear couplings to external potential one preserves causality in the
sense of causal propagation of Cauchy data. The solution of the V-Z problem
has a close formal proximity to the solution of the Weinberg-Witten problem in
[11].

Section 4 provides some background about modular localization. Its aim is to
show that causal localization is incompatible with any form of quantization
but important for understanding properties of causally localized quantum
matter.

In section 5 the SLFT renormalization theory is applied to calculation of the
S-matrix in various models involving vector mesons. including some speculative
remarks on $s\geq 2$ interactions.

Section 6 addresses problems of interacting sl fields in particular the model-
dependent distinction between pl observables and sl interpolating fields.

The concluding remarks in section 7 summarize the new insights and present an
outlook.

##  2 String-local tensor potentials and conservation laws

This section provides the kinematical prerequisites of SLFT i.e. the
construction of those sl free fields which are used in later sections for the
calculation of the S-matrix and interacting fields. The kinematic localization
of free fields is not the same as the dynamic localization of their
interacting counterparts (section 6).

###  2.1 Massless string-local potentials

The fact that even in the absence of interactions massless gauge potentials
have no positivity-maintaining pl counterpart led to a more foundational re-
thinking regarding the relation between positivity and causal localizability
for which the solution of the massless infinite spin problem in terms of sl
fields served as a role-model [34, 35]. The finite helicity problem is simpler
since in this case their exists only one covariant family of sl potentials
$\hat{A}_{\mu}$ in the Wigner-Fock helicity Hilbert space whose field strength
is the pl field strengths

$\partial_{\mu}\hat{A}_{\nu}(x)-\partial_{\nu}\hat{A}_{\mu}(x)=F_{\mu\nu}(x)~{}~{}$
(3)

They have the form of a semi-infinite line integrals (strings, rays)

$\displaystyle\hat{A}_{\mu}(x)=A_{\mu}(x,e):=\int F_{\mu\nu}(x+\lambda
e)e^{\nu}=:(I_{e}F_{\mu\nu})(x)e^{\nu}$ (4) $\displaystyle
U(a,\Lambda)A_{\mu}(x,e)U(a,\Lambda)^{\ast}=(\Lambda^{-1})_{\mu}^{\nu}A_{\nu}(\Lambda
x,\Lambda e)~{}$

with $e$ representing a space- time- or lightlike vector which participate in
the transformation under the homogenous Lorentz group. Causal localizability
requires the possibility of placing an arbitrary large number of such sl
fields in relative spacelike separated positions (denoted as $\times$ ). This
excludes the timelike case but permits space- and light-like strings777This is
less obvious in the lightlike case.

$\left[A_{\mu}(x,e),A_{\mu}(x^{\prime},e^{\prime})\right]=0,~{}x+\mathbb{R}_{0}^{+}e\,{\lower
1.0pt\hbox{\LARGE$\times$}}\,x^{\prime}+\mathbb{R}_{0}^{+}e^{\prime}$ (5)

Spacelike unit vectors $e$ with $e^{2}=-1~{}$are points on the $d=1+2$ unit de
Sitter space, whereas lightlike vectors $e$ with $e^{2}=0~{}$may be identified
with points on the two-dimensional celestial sphere. A closer examination
shows that line integrals of massless field strengths along lightlike lines
are ill-defined (see below) but well-defined (as distributions in $e$) for
spacelike $e$; time-like lines would violate causal separability.

The derivation of nonperturbative theorems (PCT, Spin&Statistics, cluster
properties, LSZ scattering theory,..) does not need pl fields; what is
important is the preservation of causal separability i.e. the fact that one
can place an arbitrary number of sl fields into relative spacelike position.

The mathematical status of sl fields requires a more careful look at their
singularity structure. For this purpose it is convenient to compute their
2-point function (2-pfct). Starting from that of the field strengths

$\displaystyle\left\langle
F_{\mu\nu}(x)F_{\kappa\lambda}(x^{\prime})\right\rangle$ $\displaystyle=\int
e^{-ip(x-x^{\prime})}M^{F_{\mu\nu},F_{\kappa\lambda}}(p)d\mu_{0}(p),\text{ \
}d\mu_{m}(p)=\frac{d^{3}p}{2\sqrt{\vec{p}^{2}+m^{2}}}$ (6) $\displaystyle
M^{F_{\mu\nu},F_{\kappa\lambda}}(p)$
$\displaystyle=-p_{\mu}p_{\kappa}g_{\nu\lambda}+p_{\mu}p_{\lambda}g_{\nu\kappa}-p_{\nu}p_{\kappa}g_{\mu\lambda}+p_{\nu}p_{\lambda}g_{\mu\kappa}$

and, using the fact that the $\lambda$-integration amounts to the Fourier
transform of the Heavyside function and hence leads to distribution
$(pe)_{i\varepsilon}^{-1}=\lim_{\varepsilon\rightarrow
0}(pe+i\varepsilon)^{-1}$ as boundary values of analytic functions, one
obtains [11]

$\displaystyle\left\langle
A_{\mu}(x,-e)A_{\nu}(x^{\prime},e^{\prime})\right\rangle=\int
e^{-ip(x-x^{\prime})}M^{A_{\mu},A_{\nu}}(p,e,e^{\prime})d\mu_{0}(p)$ (7)
$\displaystyle
M^{A_{\mu},A_{\nu}}(p,e,e^{\prime})=E_{\mu\nu}(-e,e^{\prime})=-\eta_{\mu\nu}+\frac{p_{\mu}e_{\nu}}{(pe)_{i\varepsilon}}+\frac{e_{\mu}^{\prime}p_{\nu}}{(pe^{\prime})_{i\varepsilon}}-\frac{(ee^{\prime})p_{\mu}p_{\nu}}{(pe)_{i\varepsilon}(pe^{\prime})_{i\varepsilon}}$

where the tensor $E_{\mu\nu}$ turns out to be an important building block of
higher helicity 2-pfcts. The scaling degree $d_{sd~{}}$ is defined as the
leading short distance contribution $\lambda^{-2d_{sd}\text{ }}$of the 2-ptfct
under the scaling $\xi\rightarrow\lambda\xi,$ $\xi=x-x^{\prime}$ for
$\lambda\rightarrow 0~{}$and can be directly read off from the large momentum
behavior. Whereas $d_{sd}(F)=2,$ the line integration lowers the degree to
$d_{sd}(A)=1$.

A more detailed study shows that sl potentials and their 2-pfcts are well-
defined as distributions in $e,~{}e^{2}=-1$ (the unit de Sitter space) and
$x.$ All operators and correlation function are of homogeneous degree zero and
hence the de Sitter differential can be written in the covariant form
$d_{e}=de_{\mu}\frac{\partial}{\partial e_{\mu}}.$ For lightlike $e^{\prime}s$
and $m>0$ the last term in (7) vanishes for $e^{\prime}=$ $-e$ and the
distributional dependence of $A_{\mu}(x,e)~{}$on $e$ changes to that of a
function so that a directional testfunction smearing in $e\ $is not necessary.
The identification of $e^{\prime}s$ in products of fields leads to a
significant notational simplifications in perturbative calculations. The
existence of momenta for which $p$ is parallel to $e$ excludes however
massless limits lightlike strings.

For the timelike directions the denominators never vanish and no smearing is
needed, but the causality requirement, namely the existence of an arbitrary
number of causally separated sl fields, cannot be satisfied. Hence the choice
$e=e_{0}=(1,0,0,0)$ leads to the nonlocal Coulomb- (or radiation-) potential
with $A_{0}^{C}=0$ and spatial components

$M^{A_{i}^{C}A_{j}^{C}}=\delta_{i,j}-\frac{p_{i}p_{j}}{\mathbf{p}^{2}}$ (8)

”Freezing” this timelike string direction destroys the covariant
transformation and one obtains a noncovariant inhomogeneous transformation law
in which only the rotations and translations maintain their covariant
appearance (see 12 below). Full covariance can be restored by letting the
timelike direction participate in the Lorentz transformation, but the loss of
causal localization remains. The Coulomb potential is used in quantum
mechanics where relativistic covariance and causality play no role.

It is interesting to note that the Coulomb potential results also from
averaging a spacelike string over spatial directions in the $t=0$ plane
orthogonal to the timelike $e_{0}~{}$vector. There is no direct way to undo
this directional averaging; one rather has to return from $A^{C}$ to its
covariant field strength $F$ and obtain the associated sl potential as in (4).
This directional averaging reveals a close formal connection between the
axial- and Coulomb- ”gauge”. Both potentials exist in the same Wigner-Fock
helicity space, but only the covariant sl potential (4) is manifestly causal.

The use of sl potentials turns the so-called noncovariant axial- and
lightcone-gauges into better manageable covariant Einstein-causal fields which
act in a positivity maintaining Hilbert space.

It should be mentioned that in the literature the terminology ”gauge” is used
with two different meanings. In the covariant setting of QED perturbation
theory it refers to a formal symmetry whose generator is a ”gauge charge”
which depends unphysical indefinite metric degrees of freedom. On the other
hand the Coulomb- or axial- gauge contains only the two helicity $h=\pm 1$
degrees of freedom and there is no symmetry-implementing gauge charge,
although the additive contribution to the Lorentz transformation looks like a
non-covariant gauge transformation (12) re-expressing the Lorentz-transformed
$e=\Lambda e_{0}$ in terms of original $e_{0}$.

It is not the aim of this work to change historically grown terminology. Here
the terminology ”gauge” is exclusively used the situation in which unphysical
degrees of freedom provide a covariant ”gauge symmetry”. Quantum gauge
symmetry is not a physical symmetry (and consequently there is no physical
sense in which it can be broken) but rather a formal tool to extract a
physical theory as a subtheory from an unphysical formalism.

The large momentum behavior of the 2-pfct determines the short distance
behavior of the field whereas the distributional behavior in $e$ depends on
the dimensionality of spacelike $e$-directions on which $pe$ vanishes. The
case of lightlike $e^{\prime}s$ is a bit more tricky. For massive $p$ the $pe$
denominator does not vanish since $p$ and $e$ only touch at lightlike infinity
and as a result the sl fields are functions in $e$.

This changes in a radical way for massless $p;$ in that case for each $e$
there are lightlike $p^{\prime}s$ on which $pe$ vanishes and as a result
massless fields localized on lightlike strings do not even exist in the sense
of distributions 888I am indebted to Henning Rehren for drawing attention to
the nonexistence of massless lightlike string localized fields.. Lightlike sl
fields have an interesting connection with light-cone quantization. In the
massless case they reveal in a much clearer way the problematic nature of
”lightcone quantization” [46].

The main purpose of this work is to offer a positivity- and causality-
preserving alternative to gauge theory which avoids the use of the
quantization parallelism to classical field theories by starting from Wigner’s
manifestly positivity-preserving particle representation theory. The important
point is that spacetime localization properties already exist in the pre-form
of modular localization within Wigner’s particle theory. They can be used to
construct pl or sl intertwiner functions which convert Wigner’s creation and
annihilation operators into covariant pl or sl free fields.

The perturbative construction of the S-matrix and of interacting sl fields
does not need modular localization theory. For problems as localization
entropy [33] [32] and nonperturbative constructions [31] its use is however
indispensable. In the context of the present paper its importance is based on
pinning causal localization to quantum positivity; whereas Lagrangian
quantization of fields allows the presence of unphysical degrees of freedom,
modular localization excludes them. More remarks on modular localization will
be deferred to section 4.

The construction of sl potentials in terms of pl field strengths (4) permits
an iteration to a scalar potential $\Phi$

$A_{\mu}(x,e)-A_{\mu}(x,e^{\prime})=\partial_{\mu}\Phi(x,e,e^{\prime}),~{}\Phi=(I_{e^{\prime}}I_{e}F_{\mu\nu})(x)e^{\mu}e^{\prime\nu}~{}$
(9)

The $\Phi$ represents a field which is localized on the 2-dimensional conic
region $\lambda e+\lambda^{\prime}e^{\prime},~{}\lambda,\lambda^{\prime}\geq
0.$ In the massless limit this flux $\Phi$ is logarithmically divergent. The
logarithmic divergence is expected to lead to an $e,e^{\prime}$ dependent
continuous set of superselection rules which extend the Wigner-Fock helicity
space.

This is reminiscent of the behavior of the exponential of a massive scalar
free field in $d=1+1$ in the massless limit999This infrared behavior was first
observed in the coupling of a $d=1+1$ current to the derivative of a massless
scalar field (”infraparticle” [47] ). which played an important role in the
work on ”bosonization” of massless fermions and anyons [48]. In that case the
massless limit of the properly mass-normalized exponentials leads to the
superselection property

$\left\langle e^{ia_{1}\varphi(x_{1})}\dots
e^{ia_{n}\varphi(x_{n})}\right\rangle=0\quad\hbox{if}~{}~{}\sum_{1}^{n}a_{i}\neq
0.$ (10)

corresponding to $a_{i}$-”charge” conservation.

The ”photon cloud” in the $e$-direction associated with $\exp ig\varphi$ is
expected to cause a directional superselection rule which appears in the form
of $e^{ig\varphi}\psi$ in the large time behavior of electric charge carrying
fields and causes the modification of LSZ scattering theory. In this way one
may hope to obtain a genuine spacetime understanding of the infrared momentum
space recipes in [49].

The interest in this problem is also motivated by the existence of rigorous
results derived from an appropriate formulation of the quantum Gauss law [50].
This theorem states that interacting electric charge-carrying operators $\psi$
are accompanied by spacelike extended ”photon clouds” whose different
asymptotic conic directions correspond to a continuum of superselection
sectors within the same charge-carrying sector. This is the cause a
spontaneous breaking of Lorentz symmetry [51].

The existence of a continuum of superselection sectors for free photons would
suggest the existence of large time asymptotic charge-carrying matter fields
of the form $\psi_{0}e^{ig\Phi}$ with $\psi_{0}$ a free matter field. Their
large time asymptotic behavior is expected to play an important role in a
future spacetime understanding of infrared properties which is outside the
physical range of gauge theories.

For many applications it is useful to encode change in $e$ (9) into changes of
the Lorentz transformation law. A differential relation which is the basis for
such conversion has the form ([11] Corollary 3.3)101010As mentioned therein
this remains well defined since sl fields and their correlation functions are
homogeneous functions of degree zero in $e$ and $p.$

$d_{e}A_{\mu}(x,e)=\partial_{\mu}u(x,e),\
~{}d_{e}=\sum_{i}d_{e_{i}}\partial^{e_{i}}$ (11)

where $u$ is an exact de Sitter one-form $u=d_{e}\phi$. This conversion of
directional de Sitter differentials into $x$-derivatives plays an important
role in passing from interactions in the presence of a mass gap to their
massless limit.

In the present context the formula for the change of $e^{\prime}s$ can be used
to compute the additive change which is necessary in order to maintain the
timelike $e_{0}$ direction of the Coulomb potential
$A_{i}^{C},~{}A_{0}^{C}=0$. The resulting affine transformation formula

$U(a,\Lambda)A_{i}^{C}(x)U(a,\Lambda)^{\ast}=(\Lambda^{-1})_{i}^{~{}l}A_{l}^{C}(\Lambda
x+a)+(\Lambda^{-1})_{i}^{~{}\mu}\partial_{\mu}\chi(x)$ (12)

is equivalent to that obtained by starting from the Wigner helicity
representation and using transverse polarization vectors [3].

A similar situation arises if one fixes an ”axial” direction as e.g.
$e=(0,1,0,0).$ In this case the causal localizability is preserved in both
descriptions. Ignoring the spacetime localization aspect and treating the
axial direction as a noncovariant gauge misses the necessity of directional
smearing (smearing around a point in de Sitter space) and probably contributed
to the abandonment of the ”axial gauge fixing”. But what became a curse in the
axial gauge fixing turns out to be a blessing in the covariant SLFT setting.

Covariant gauges as used in covariant perturbation theory always require the
presence of ghost-extended indefinite metric BRST degrees of freedom setting
which reduces the physical range. SLFT cuts the umbilical cord between
perturbative Lagrangian quantization and classical gauge theory and restores
positivity.

###  2.2 A brief interlude, relation with concepts of algebraic QFT

The simplest illustration of the interplay between positivity and causality is
provided by the Aharonov-Bohm effect. To see this recall that Einstein
causality is the statement that the algebra of operators localized in the
causal complement $\mathcal{O}^{\prime}$ of a spacetime region $\mathcal{O}$
belong to the commutant $\mathcal{A}(\mathcal{O})^{\prime}$ algebra (the von
Neumann algebra which consists of all operators which commute with
$\mathcal{A}(\mathcal{O})$)

$\displaystyle\mathcal{A}(\mathcal{O}^{\prime})$
$\displaystyle\subseteq\mathcal{A}(\mathcal{O})^{\prime}\quad\hbox{or}\quad\mathcal{A}(\mathcal{O})\subseteq\mathcal{A}(\mathcal{O}^{\prime})^{\prime},\quad\hbox{Einstein
causality}$ (13) $\displaystyle\mathcal{A}(\mathcal{O}^{\prime})$
$\displaystyle=\mathcal{A}(\mathcal{O})^{\prime},\quad\hbox{Haag duality}$

The second line defines the somewhat stronger Haag duality which states that
an operator which commutes with all operators localized in the causal
complement of $\mathcal{O}~{}$must belongs to $\mathcal{A}(\mathcal{O})$.

Einstein causality is a defining property of relativistic QFT, but Haag
duality may be violated. In the absence of interactions such a violation can
be excluded for massive QFT’s but it does occur in the massless case when the
$2s+1$ spin degrees of freedom are converted into the $\pm h$ helicities. As
observed in [52] (unpublished) Haag duality, which holds for simply connected
spacetime regions, is violated for multiply connected regions as (genus one)
tori.

In their proof the authors carefully avoid the use of gauge potentials.
associated to $m>0$ massive is a property ($g\geq 1~{}$tori). This violation
is an intrinsic property of the operator algebra generated by the field
strength $F_{\mu\nu}~{}$of the $h=1~{}$Wigner representation. But if one wants
to understand this in terms of vector potentials one must use the positivity-
maintaining sl potentials which preserve a somewhat hidden topological
properties of Wilson loops which cause the breakdown of Haag duality while it
upholds Einstein causality [45]. The indefinite metric potentials cannot
distinguish between the two; only the localization in the presence of
positivity is physical.

The cause of ”eeriness” about the Aharonov-Bohm effect [53] (but also of its
popularity) is that we erroneously interpret the intuitively accessible
geometric Haag duality with the more abstract Einstein causality, thus
forgetting that the latter also admits operators which have no unambiguous
causal localization region (e.g. the magnetic flux through a surface with a
fixed boundary). The ideal solenoid in the A-B setup closes at spacelike
infinity, which in the conformal Wigner-Fock helicity world is a circle. In
case of a finite tube one must place the electric circuit into a region of
little magnetic backflow from north- to south-pole.

This is a strong reminder that it is not possible to separate causality from
positivity and a warning not to confuse the ”fake localization” of gauge
dependent objects with genuine causal localization of quantum matter. It
points to a potential source of misunderstanding involved in transferring the
perfectly reasonable classical notion of local gauge symmetries to QFT by
attributing a physical meaning to the formal observation that quantum gauge
charges are ”more local” than those corresponding to internal symmetries. It
is also a reminder to rethink the physical meaning behind the terminology
”gauging a model”.

From (11) it follows that a Wilson loop111111By convoluting with a test
function one can convert the Wilson loop integral into into an operator
localized on a solid torus. formed with $A_{\mu}(x,e)$ is independent of the
choice of the direction $e~{}$[45]$.~{}$However it retains a topological
memory of the string directions of the integrand which prevents a naive
materialistic identification with a localization in a torus. One can choose
$e^{\prime}s$ in such a way that this extension is spacelike with respect to
any simply connected convex compact region121212In case the solenoid has open
ends the Wilson loop should avoid the region of the north-south magnetic
backflow.. Yet it is not possible to completely forget that the vector
potential has a directional $e$-dependence. An elegant formulation of this
$h\geq 1~{}$topological phenomenon directly based on field strengths and their
duals in terms of ”linking numbers” can be found in [54].

###  2.3 Massive string-local potentials

Before passing to the construction of massive sl fields it is helpful to
recall the construction of their pl intertwiner functions $v(p)$ which convert
the$~{}m>0\ $Wigner creation and annihilation operators $a^{\\#}(p,s)~{}$into
covariant [3]. For the $s=1$ Proca field they are the three polarization
vectors $v_{\mu}(p,s_{3})$ obtained by applying a rotation-free Lorentz boost
to the spatial coordinate unit vectors. By definition they are Minkowski-
orthogonal to $p_{\mu}$ and hence correspond to the 3 polarization vectors $v$
obeying the completeness relation

$\displaystyle\,\sum_{s_{3}=--1}^{1}v_{\mu}(p,s_{3})v_{\nu}(p,s_{3})=-\eta_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{m^{2}}$
$\displaystyle
M^{A_{\mu}^{P},A_{\nu}^{P}}(p)=-\pi_{\mu\nu}(p),\quad\pi_{\mu\nu}(p)=\eta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{m^{2}}$

where the $\pi_{\mu\nu}$ of the momentum space 2-pfct which also turns out to
be the basic building block of all higher spin massive tensor potentials.

With a pl Proca potential $A_{\mu}^{p}~{}$one may associate two sl fields, the
scalar sl field $\phi~{}$defined in terms of a line integral $I_{e}$ of the
$e$-projected Proca field $A_{\mu}^{P}e^{\mu}$ along $e$ starting from the
point $x$

$\phi(x,e)=(I_{e}A_{\mu}^{P})(x)e^{\mu},~{}~{}a(x,e)=-m\phi(x,e)$ (14)

and the sl vector potential $A_{\mu}$ in terms of the field strength of the
Proca potential

$A_{\mu}(x,e)=(I_{e}F_{\mu\nu})(x)e^{v},\text{
}F_{\mu\nu}(x)=\partial_{\mu}A_{\nu}^{P}-\partial_{\nu}A_{\mu}^{P}$ (15)

whose massless limit coincides with (4).

The multiplication with $m$ in (14) restores $d_{cl}=1$ and removes the mass
singularity, so that the $m=0$ limit is an $e$-independent massless scalar
$d_{sd}(a)=1$ free field. In fact for all massive or massless sl tensor
potentials $d_{sd}=1$ and $3/2$ for halfinteger $s$ whereas their pl
counterparts increase linearly as $d_{sd}=s+1~{}$or $d_{sd}=s$ and diverge
like $m^{-s}$ for $m\rightarrow 0$

In particular the momentum space 2-pfct of the massive field strength and its
massless associated sl vector potential are identical to their massless
counterparts. This permits to lower the number of degrees of freedom by
passing from $p\in H_{m}^{\uparrow}$ to$~{}p\in V^{\uparrow}$ (and its
”fattening” inversion, see below). In the pl setting this is not possible or
can only be achieved in the presence of indefinite metric degree of freedom
(the DVZ discontinuity, the WW problem).

It is instructive to look at this degrees of freedom balance in more detail
[10, 11]. With the help of a $p$-dependent 4-matrix $J$ (complex conjugation
changes the sign of $e$, $tr~{}$= transposed)

$\displaystyle J_{\mu}^{~{}~{}\nu}(p,e)=\eta_{\mu}^{\ \
\nu}-\frac{p_{\mu}e^{\nu}}{(pe)_{i\varepsilon}},~{}\overline{J(p,-e)}=J(p,e)$
(16) $\displaystyle M^{A_{\mu}(-e),A_{\nu}(e)}=:E_{\mu\nu}(e,e)=(J\pi
J^{tr})_{\mu\nu}$

the in $e$ diagonal momentum space 2-pfct takes the form of the second
line131313Taking the same $e$ would lead to the distributionally ill-defined
denominator $pe_{-i\varepsilon}pe_{i\varepsilon}.$. It shows that the
positivity of the sl 2-pfct is inherited from the pl positivity. The rank of
the $E$-matrix accounts for the degrees of freedom is 3 as a result of
$J^{tr}e=0$ and the additional relation $E_{\mu\nu}p^{\nu}=0$ for $p\in
V^{+}~{}$leads to a reduction from the three spin component to the two
helicities $h=\pm 1$. This degrees of freedom counting breaks down in the
presence of indefinite metric.

This descend from $p\in H_{m}^{\uparrow}$ to $V^{+}$ permits an inversion,
namely by continuous passing from momenta $p\in V^{+}$ to the mass shell
$H_{m}^{+}$ (”fattening”) one creates a new physical degree of freedom which
together with former $\pm 1$ accounts for the $3$ degrees of freedom of spin
$s=1$. Such ”magical” conversion of the particle content of two inequivalent
Wigner representations can neither be achieved in terms of pl fields (no
massless limit) or become contaminated by the presence of indefinite metric
causing and ghost degrees of freedom of gauge theory. This is of particular
interest in case of $s=2$ [10, 11] (see below). The use of sl fields is even
more important for passing to the massless limit in the presence of
interactions involving higher spins.

In the literature the terminology ”fattening” had been used in connection with
the Higgs model which describes the interaction between a massive vector meson
with a massive real scalar field $H$ as the result of spontaneous breaking of
gauge symmetry (the Mexican hat potential). This idea contains two conceptual
misunderstandings (which will be commented on in section 5 and the concluding
remarks).

The real power of SLFT emerges in models of selfinteracting massive vector
mesons where the preservation of 2nd order renormalizability requires the
compensatory presence of a coupling to a Hermitian scalar $H$ (Higgs)
field141414This (and not SSB) is the raison d’être for the $H$ (section 5).
and imposes a Lie-algebra structure on the leading terms in the
$A_{\mu}~{}$self-interactions. In section 5 we will provide the arguments.

An important property of the previously introduced pl and sl vector potential
and its scalar escort $\phi(x,e)$ is their linear relation

$A_{\mu}(x,e)=A_{\mu}^{P}(x)+\partial_{\mu}\phi,\quad\phi=-\frac{1}{m}a$ (17)

This property justifies to call the $\phi^{\prime}s$ ”escorts” of the sl
potential, they share the same degrees of freedom. The appearance of the
escort in form of a derivative is a consequence of Poincaré’s lemma. The
linear relation between fields corresponds to that between intertwiners ($J$
as before):

$J_{\mu}^{\ \ \nu}v_{\nu}(p)=v_{\nu}-p_{\mu}\frac{(ve)}{(pe)_{i\varepsilon}}$
(18)

which follows directly from the definition (15). Each field contains the full
information of the ($m,s=1$) Wigner representation; the encoding of $s=1$ into
a scalar is only possible within sl.

It is not accidental that the massive vector potentials which result from
”fattening” their unique massless counterpart play a distinguished role in the
new SLFT renormalization theory. Their smooth connection represents the higher
spin analog of the smooth relation between $s<1$ massless fields and their
massive counterpart. The weakening of localization is necessary to preserve
this smoothness in the presence of change of the number of degrees of freedom.

In the massless limit the $A_{\mu}^{P}(x)$ and $\phi(x,e)$ diverge as
$m^{-1}~{}$whereas the $A_{\mu}(x,e)$ and $a(x,e)$ stay infrared finite. The
relation ($u~{}$was introduced in (11))

$\displaystyle\partial^{\mu}A_{\mu}$
$\displaystyle=-ma,~{}~{}d_{e}A_{\mu}=\partial_{\mu}u$ (19) $\displaystyle u$
$\displaystyle=-m^{-1}d_{e}a$ (20)

leads to a divergence-free massless vector potential (Lorentz
condition151515Note that this is an operator identity and not an imposed gauge
condition.) and a relation between two massless 1-forms in the de Sitter space
of spacelike directions (that which remains of (17)). The purpose of the mass
factors is to preserve the relation $d_{sd}=d_{cl}~{}$for all sl fields. The
massless limit of $u$ is logarithmically divergent.

Escorts (whose number increase with $s$) do not contain new degrees of freedom
since, as the pl $A^{P},~{}$they are linear in the Wigner $s=1$
creation/annihilation operators $a^{\\#}(p,s_{3})$ and only differ in their
intertwiners. Rearrangements of degrees of freedom are quite common in quantum
mechanical many-body problems161616A well-known case is the appearance of
Cooper pairs encounters in passing to the low temperature superconducting
phase. Without this rearrangement classical vector potentials would not become
short range inside a superconductor (the London effect).. Escorts are
rearranged $s=1$ degrees of freedom which carry the full content of ($m>0,s$)
Wigner representations.

For $s\geq 2$ the sl fields lead to new properties. As a result of a possible
relation with gravitation the case $s=2~{}$is of special interest. The
intertwiner of spin $s$ Proca potentials (the $P$ in $A_{...}^{P}$ refers to
Proca or alternatively to pointlike) must be a divergence- and trace-free
symmetric tensor; this is a consequence of the way the $2s+1$ component
subspace of spin is embedded in the $3s$-fold tensor product. Hence the
intertwiners $v_{\mu_{1}\dots\mu_{s}}(p,s_{3})\ $convert the symmetric trace-
free $s$-fold tensor product of three-component spin $1~{}$polarization
vectors into covariant tensors of tensor-degree $s.$

For the momentum space $s=2~{}~{}$2-pfct one obtains

$M^{A_{\mu\nu}^{P},A_{\kappa\lambda}^{P}~{}}(p)=\frac{1}{2}\left[\pi_{\mu\kappa}\pi_{\nu\lambda}+\pi_{\mu\lambda}\pi_{\nu\kappa}\right]-\frac{1}{3}\pi_{\mu\nu}\pi_{\kappa\lambda}$
(21)

where the numerical factors have their combinatorial origin in the symmetry
and tracelessness and hence depend on the degrees of freedom. The sl 2-pfcts
are of the same algebraic form and result by substituting
$\pi_{\mu\nu}\rightarrow E_{\mu\nu}(-e,e)$ [10, 11]. As for $s=1~{}$this can
be seen by passing from the Proca potential to the field strength ($as$ stands
for antisymmetrisation)

$F_{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}=\underset{\mu\leftrightarrow\nu}{as}~{}\partial_{\mu_{1}}\partial_{\mu_{2}}A_{\nu_{1}\nu_{2}}^{P}$
(22)

and using the two-fold momentum space $I$ operation to pass from the field
strength to the potentials. Note that the symmetry of the Proca potential
reduces the anti-symmetrization to a pairwise operation
$\mu_{i}\leftrightarrow\nu_{i}.$ The resulting permutation properties of the
resulting $F$ are those of the linearized Riemann tensor.

The new phenomenon for $s>1$ is that the massless limit of this field strength
is not the same as that obtained directly from the massless $h=\pm 2$ Wigner
representation. Correspondingly the sl potential associated with the massless
limit of $F^{s=2}$ is different from that of$~{}F^{\left|h\right|=2}$

$\displaystyle A_{\nu_{1}\nu_{2}}(x,e)$
$\displaystyle=(I_{e}^{2}F_{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}^{s=2})(x)e^{\mu_{1}}e^{\mu_{2}}$
(23) $\displaystyle A_{\nu_{1}\nu_{2}}^{(2)}(x,e)$
$\displaystyle=(I_{e}^{2}F_{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}^{\left|h\right|=2})(x)e^{\mu_{1}}e^{\mu_{2}}$
(24)

This means in particular that the massive $s=2$ sl potential obtained by
fattening the $A^{(2)}~{}$is not the same as $A$ although both account
correctly for the $2s+1$ spin degrees of freedom and share their Wigner-Fock
Hilbert space. The massless limit of $A$ splits into the direct sum of the two
$\left|h\right|=2$ degrees of freedom and the $h=0$ contribution which is the
remnant of the $s_{3}=0$ component. Conserved currents and stress-energy
tensors preserve the number of degrees of freedom by converting the $\pm
s_{3}$ components into $\left|h\right|=s_{3}$ helicities of a Wigner-Fock
tensor product space.

In order to show how these results are related to the van Dam-Veltman-Zakharov
discontinuity problem one must look at some details. Whereas fattening and
taking the massless limit connect the 2-pfct of the 2-component massless
helicity $\left|h\right|=2$ potential$~{}A^{(2)}$ with that of its 5-component
$s=2$ by deforming the momenta of the 2-pfct between $H_{m}^{\uparrow}$ and
$V^{\uparrow},$ the massless limit of $A$ is a cul de sac from which a return
to the original massive pure $s=2$ tensor potential is not possible.

The relation between the massless limit of $A$ with that of $A^{(2)}$ are
easily seen to have the following form

$\displaystyle A_{\mu\nu}^{(2)}(x,e)$
$\displaystyle=A_{\mu\nu}(x,e)+\frac{1}{2}E_{\mu\nu}(e,e)A^{(0)}(x,e)$ (25)
$\displaystyle E_{\mu\nu}(e,e)$
$\displaystyle=\eta_{\mu\nu}+(e_{\mu}\partial_{\nu}+e_{\nu}\partial_{\mu})I_{e}+e^{2}\partial_{\mu}\partial_{\nu}I_{e}^{2}$

where the momentum space $E_{\mu\nu}~{}$has been rewritten as an integro-
differential operator acting on a scalar sl field and the massless limit of
$A^{(0)}~{}$is a (properly normalized) scalar escort. Combining this relation
with that between the $s=2~{}$pl field $A^{P}$, its sl counterpart $A$ and the
derivatives of escorts (the $s=2$ analog of (17)) one obtains

$A_{\mu\nu}^{P}=A_{\mu\nu}+\hbox{derivatives of escorts}$

one concludes that in the adiabatic limit the interaction between ”massive
gravitons” and a trace-free energy-momentum tensor source $T_{\mu\nu}$ is [35]

$\displaystyle\lim_{m\rightarrow 0}\int A_{\mu\nu}^{P}T^{\mu\nu}$
$\displaystyle=\lim_{m\rightarrow 0}\int
A_{\mu\nu}T^{\mu\nu}=\int(A_{\mu\nu}^{(2)}-\frac{\eta_{\mu\nu}}{\sqrt{6}}\varphi)T^{\mu\nu}$
(26) $\displaystyle where\text{ ~{}}\varphi(x)$
$\displaystyle=\sqrt{\frac{3}{2}}\lim_{m\rightarrow 0}a^{(0)}(x,e)$ (27)

The independence of the integrated massless $A^{(2)}$ contribution from the
string direction follows from

$\displaystyle\partial_{e_{\kappa}}A_{\mu\nu}^{(2)}$
$\displaystyle=m^{-1}(\partial_{\mu}A_{\kappa\nu}^{(2)}+\partial_{\nu}A_{\mu\kappa}^{(2)})$
(28) $\displaystyle\partial_{e_{\kappa}}J_{\mu}^{~{}~{}\nu}$
$\displaystyle=-\frac{p_{\mu}}{(pe)_{i\varepsilon}}J_{\kappa}^{~{}~{}\nu}$

which in turn follows from the identity in the second line (for more details
see [11]) and represents the $s=2$ counterpart of the relation between de
Sitter space 1-forms in (19).

The result confirms the van Dam-Veltman-Zakharov discontinuity: the massless
limit of massive gravity differs from the result obtained directly with
massless gravitons. But different from Zakharov’s calculation which identifies
this contribution as being the relic of a unphysical gauge theoretical degrees
of freedom, the present calculation shows that it is really the massless
footprint of the physical $s_{3}=0$ spin component. For the traceless stress-
energy tensor of photons the last contribution vanishes whereas for couplings
to matter (mercury perihelion) it remains.

This calculation permits a straightforward extension to any spin. The relation
between the Proca potential, its sl counterpart and the associated sl escorts
reads

$A_{\mu_{1}\dots\mu_{s}}^{P}=A_{\mu_{1}\dots\mu_{s}}+sym.(\partial_{\mu_{1}}\phi_{\mu_{2}\dots\mu_{s}}+\partial_{\mu_{1}}\partial_{\mu_{2}}\phi_{\mu_{3}\dots}+\dots+\partial_{\mu_{1}}\dots\partial_{\mu_{s}}\phi)$
(29)

where the $\phi_{\mu_{1}\dots\mu_{i}}~{}$is an $s-i$ fold iterated line
integral along $e$ of the spin $s~{}$Proca potential and the symmetrization is
over all indices and the $\phi$ are already symmetric by construction. For our
purposes it is more convenient to use a different basis of escorts which are
obtained by descending from the sl $A_{\mu_{1}\dots\mu_{s}}$ in terms of
divergencies

$\displaystyle
A_{\mu_{1}\dots\mu_{s}}^{P}=A_{\mu_{1}\dots\mu_{s}}-sym.(\frac{\partial_{\mu_{1}}}{m}a_{\mu_{2}\dots\mu_{s}}^{(s-1)}+\frac{\partial_{\mu_{1}}\partial_{\mu_{2}}}{m^{2}}a_{\mu_{3}\dots}^{(s-2)}+\dots+\frac{\partial_{\mu_{1}}\dots\partial_{\mu_{s}}}{m^{s}}a^{(0)})$
(30) $\displaystyle
ma_{\mu_{r}\dots\mu_{s}}^{(s-r)}=-\partial^{\mu}a_{\mu\mu_{r}\dots\mu_{s}}^{(s-r+1)},~{}\
~{}~{}a_{\mu_{1}\dots\mu_{s}}^{(s)}:=A_{\mu_{1}\dots\mu_{s}}$

The second line shows that the $a$ escorts start from the sl potential and
descend by differentiation instead of descending from $A^{P}$ by line-
integration. The $a$ have the same dimension $d_{sd}=1=d_{eng},~{}d_{infr}=0$,
and are linear combinations of the $\phi$ escorts. As long as $m>0~{}$each
escort carries the full content of the Wigner spin $s$ representation.

Although the $a^{\prime}s~{}$have a massless limit they still do not decouple.
The van Dam-Veltman-Zakharov discontinuity shows that for $s=2$ the
$\left|h\right|=2$ and $h=0$ contributions stay together and have to be
separated with the help of an integro-differential operation (25). The
analogous situation in the general case is that the even and odd
$s_{3}~{}$contributions remain coupled among themselves and can only be split
in terms of their helicity content by the use of such integro-differential
operations [11]. Naturally one can obtain a spin $s$ vector potential from
fattening a massless helicity $h$ potential if $h=s.$

The tensor $v_{\mu_{1}..\mu_{\left|h\right|}}(q,e)$ which appears in the
relation of the helicity $h$ tensor field
$A_{\mu_{1}..\mu_{\left|h\right|}}^{(\left|h\right|)}$ and the Wigner operator
$a^{\\#}(q,h)$ (which extends the construction of $A_{\mu\nu}^{(2)}~{}$in (25)
to arbitrary helicity $h).$ This $e$-dependent polarization tensor
$v_{..}(q,e)~{}$replace the only up to re-gauging defined polarization tensor.
If used in Weinberg’s soft scattering limit of a massless particle with
momentum $q$ scattering on $n$ massive particles with momenta $p_{i},i=1,..n$
[17, 4.1], one obtains the same conclusions except that the gauge theoretic
argument is replaced by the $e$-independence which follows in first order from
the fact that the directional derivative with respect to $e$ on these
polarization tensors can be written as a spacetime derivative $\partial_{\mu}$
acting on such a tensor in analogy to (11). The use of sl polarization tensors
instead of gauge symmetry is required by using positivity which guaranties the
exclusive appearance of physical degrees of freedom.

The weakness of Lagrangian constructions of conserved currents and stress-
energy tensors is that with the exception of low spins there is no guaranty
that the so obtained classical expressions have the correct commutation
relations with the quantum fields. It is much safer and easier to start from
the commutation relations between Wigner’s generators of the Poincaré group
and the Wigner particle operators $a^{\\#}(p,s_{3})$ and to rewrite them with
the help of the intertwiners into covariant commutation relations.

###  2.4 Infinite spin revisited

A simple illustration of such an ”intrinsic quantum” construction of the
stress-energy tensor has been recently presented in [8]. One starts from the
expressions of the infinitesimal generators of translation
$\mathbf{P}_{\mu}~{}$and Lorentz generators $\mathbf{M}_{\mu\nu}~{}$in terms
of the Wigner operators $a^{\\#}(p,s_{3})$

$\displaystyle\mathbf{P}_{\mu}$
$\displaystyle=\int\sum_{s_{3}}a^{\ast}(p,s_{3})p_{\mu}a(p,s_{3})d\mu(p)~{}$
(31) $\displaystyle\mathbf{M}_{\mu\nu}$
$\displaystyle=-i\int(\delta_{s_{3}s_{3}^{\prime}}p\wedge\partial_{p}+d(\omega)_{s_{3}s_{3}^{\prime}}^{t})_{\mu\nu}a^{\ast}(p.s_{3})a(p,s_{3}^{\prime})d\mu(p)$
(32)

The first step is two rewrite the contribution of the spin component $s_{s}$
to$~{}\mathbf{P}_{\mu}\,$as

$\displaystyle\mathbf{P}_{\mu}=\int\int
d\mu(p)d\mu(p^{\prime})\sum_{s_{3},s_{3}^{\prime}}(p_{\mu}a^{\ast}(p,s_{3})\delta_{s_{3}s_{3}^{\prime}}(2\pi)^{3}\delta(\mathbf{p}-\mathbf{p}^{\prime})(p_{10}+p_{20})a(p^{\prime},s_{3}^{\prime})$
(33) $\displaystyle(2\pi)^{3}\delta(\mathbf{p}-\mathbf{p}^{\prime})=\int
e^{-i(\mathbf{p}-\mathbf{p}^{\prime})x}d^{3}x=\int
e^{-i(p-p^{\prime})x}d^{3}x$ (34)

where in the second line used the cancellation of the $p_{0}~{}$components.

What remains to do is to convert the Wigner operators via intertwiners into
the covariant fields. For this one uses their completeness relation in order
to write the unit operator in spin space as

$g^{MN}\nu_{Ms_{3}}\overline{\nu_{Ns_{s}^{\prime}}}=\delta_{s_{3}s_{3}^{\prime}}$

where $M$ and $N~{}$represent the multi-tensor indices of the intertwiner.
What remains is to use the Fourier transform (34) and pass from the Wigner
operators to the fields. Using the fact that the $a^{\ast}a^{\ast}\ $and $aa$
contributions vanish as a result of the presence
of$~{}\overleftrightarrow{\partial}_{0}$ and that $aa^{\ast}$ terms are absent
in Wick-ordered products one verifies that

$\mathbf{P}_{\mu}=\int\tilde{T}_{\mu
0}(x)d^{3}x,~{}~{}~{}\tilde{T}_{\mu\nu}(x)=-\frac{1}{4}\int:A_{\mu_{1}..\mu_{s}}^{P}(x)\overleftrightarrow{\partial}_{\mu}\overleftrightarrow{\partial}_{0}A^{P,\mu_{1}..\mu_{s}}(x):$
(35)

where $\tilde{T}_{\mu\nu}$ is a contribution to the stress-energy tensor.

The full tensor density which generates all Poincaré transformations is of the
form

$T_{\mu\nu}=\tilde{T}_{\mu\nu}+\partial^{\rho}\Delta_{\mu\nu,\rho}$ (36)

To compute the second contribution, which is also a bilinear expression in the
$A^{P}~{}$tensor fields, one starts from the bilinear expression for
$M_{\mu\nu}$ in terms of the $a^{\\#}~{}$Wigner operators which also contains
a contribution the infinitesimal part of Wigner’s little group. The
representation of the Poincaré group generators in terms of pl stress-energy
tensors may be rewritten in terms of their sl counterparts [11]. For recent
results about constructing infinite spin fields and their E-M tensors as
Pauli-Lubanski limits we refer to [8].

Rehren’s construction of infinite spin quantum fields in terms of the Pauli-
Lubanski limit is the most natural one; it corresponds to the use of the
distinguished tensor potentials obtained by fattening its unique massless
counterpart at fixed spin, except that it goes into the opposite direction at
fixed P-L parameter171717In this way it selects a unique countable family of
fields within the equivalence class of all relatively causal fields
constructed in [35].. The tensor field disappears in this limit and what
remains (after appropriate adjustments) is the infinite family of escorts with
arbitrary high tensor degree.

The nonexistence of the infinite spin tensor potential
$A_{\mu_{1}\mu_{2}...\infty}$ accounts for the absence of a relation which
converts the differential $d_{e}$ into a spacetime divergence as well as the
absence of a gauge theoretic formulation. This is the reason why infinite spin
matter cannot interact with ordinary quantum matter [38].

Of physical relevance is the existence of conserved currents and energy-
momentum tensor in the sense of bilinear forms [8]. Hence expectation values
of E-M tensors and possible gravitational backreaction remain physically
meaningful.

##  3 Causality and the Velo-Zwanziger conundrum

The Velo-Zwanziger conundrum is an alleged causality paradox which arose from
the naive expectation that $s\geq 1$ quantum fields, whose free field
equations are modified by linear pl couplings to external potentials, maintain
their causal propagation. Formally it is closely related with the Weinberg-
Witten No-Go theorem which excludes the existence of higher helicity conserved
pl currents. This connection turns out to be useful for the solution of the
V-Z conundrum.

###  3.1 Recalling the solution of the Weinberg-Witten problem and the
associated local charges

In [11] it was shown that for massive $s\geq$ $1$ free field one can construct
sl tensor potentials whose associated conserved sl currents have finite
massless limits even when according to the Weinberg-Witten (W-W) theorem
physical (gauge-invariant) pl currents do not exist.

In the massive case both the pl and sl currents are members of the same local
equivalence class which consist of all Wick-ordered composites of pl fields
and their related sl counterparts. Their relative causality reads

$\left[j_{\mu}^{P}(x),j_{\nu}(x^{\prime},e)\right]=0\text{~{}}for\text{
}x\,{\lower
1.0pt\hbox{\LARGE$\times$}}\,\mathcal{S}(x^{\prime},e),~{}\mathcal{S}(x^{\prime},e)=x^{\prime}+\mathbb{R}_{+}e,~{}e^{2}=-1\text{
}or~{}0$ (37)

Their charge-densities differ by spatial divergencies and hence they share the
global $U(1)~{}$generators. In the massless limit the sl spin potentials pass
continuously to their massless counterpart (not possible with pl potentials)
which act in the conformally covariant helicity Wigner-Fock space. The sl
currents are bilinear in the charge carrying sl potentials[11].

The two currents (37) share the same ”engineering” dimension (classical
dimension in terms of mass units) $d_{cl}=3,$ but possess different short
distance scaling dimensions $d_{sd}(j_{\mu}^{P})=2(s+1)+1$ and
$d_{sd}(j_{\mu})=d_{cl}=3;~{}$this accounts for the fact that the sl $j_{\mu}$
allows a massless limit whereas $j_{\mu}^{P}$ diverges as
$j_{\mu}^{P}\overset{m\rightarrow 0}{\sim}m^{-2s}\ $(the W-W obstruction). As
expected, the sl $j_{\mu}(x,e)$ admits a massless limit in which the $2s+1$
spin degrees of freedom decompose into a direct sum of $s$ helicity and one
scalar contribution so that the Wigner-Fock space turns into a tensor product
of helicity spaces.

The presentation concerning the relation between pl and sl conservation laws
in [11] was mainly focussed on the stress-energy tensors (SET); in the
following we present the corresponding problem for conserved currents. A
convenient illustration is provided by the sl current with the lowest W-W
helicity $h=1$ as follows.

Using the linear relation
$A_{\mu}^{P}(x)=A_{\mu}(x,e)-\partial_{\mu}\phi(x,e)~{}$between pl and its
canonically associated sl field and the gradient of its escort derived in the
previous section (17) one finds that the pl and sl currents are related as
(omitting Wick-ordering)

$\displaystyle
j_{\mu}^{P}=iA^{P\nu}(x)^{\ast}\overleftrightarrow{\partial_{\mu}}A_{\nu}^{P}(x)=j_{\mu}(A(x,e))+j_{\mu}(a(x,e))+\partial^{\kappa}C_{\kappa\mu}$
(38) $\displaystyle
a(x,e)=m\phi(x,e),~{}C_{\kappa\mu}=iA_{\kappa}^{\ast}\overleftrightarrow{\partial}_{\mu}\phi~{}+i\phi^{\ast}\overleftrightarrow{\partial}_{\mu}\partial_{\kappa}\phi+h.c.$

The first two contributions are conserved sl currents whose massless limit
correspond to the current of the complex $s=1~{}$sl field $A_{\nu}(x,e)$
(which replaces the nonexistent pl W-W current), and that of a complex scalar
field $a(x)=\lim_{m\rightarrow 0}a(x,e).$ The $m^{-2}$ W-W obstruction $C$
does not contribute to the global charge.

The ”obstructing” contribution $\partial^{\kappa}C_{\kappa\mu}$ carries both
the leading short distance dimension $d_{sd}=5~{}$and and the $m^{-2}$
divergence which is the culprit for the W-W problem. This kind of
decomposition into $s~{}$conserved$~{}d_{sd}(j)=3~{}$sl currents, which pass
for $m\rightarrow 0$ to $s$ sl helicity currents and a pl current of a scalar
particle, exists for every spin $s\geq 1.$

Using the free field equation for $A_{\mu}$ and $\phi$ one verifies that
$C$-contribution is of the form of a spatial divergence and hence does not
contribute to the infinite volume limit of the charges [11]

$Q(A^{P})=Q(A)+Q(a)$ (39)

i.e. the massless limit decomposes the three spin degrees into the $\pm 1$
helicities of $A_{\mu}$ and $h=0$ carried by $a.~{}$Before this limit both sl
fields $A_{\mu}$ and $a$ account for the three $s=1$ degrees freedm.

For pl currents there exists extensive literature on the problem of relation
between conserved currents, local charges, and their global limits
[55][56][57][58]. The basic idea is to start from a conserved current and
define

$\displaystyle Q$
$\displaystyle=\lim_{R\rightarrow\infty}Q(f_{R},f_{d}),~{}~{}Q(f_{R},f_{d}):=j_{0}(f_{R},f_{d})$
(40) $\displaystyle f_{R}(\mathbf{x})$
$\displaystyle=\left\\{\begin{array}[c]{c}1\ \ \ \ \ \ \
~{}\left|\mathbf{x}\right|<R\\\ \ \ 0~{}~{}\ \
\left|\mathbf{x}\right|>R+r\end{array}\right.~{}$ (43) $\displaystyle
f_{d}(x_{0})$ $\displaystyle\geq
0.~{}~{}suppf_{d}\subset\left|t\right|<d,~{}\int f_{d}(x_{0})dx_{0}=1$ (44)

One then uses the conservation law of the current to show that the commutator
$\left[Q(f_{R},f_{d}),A\right]$ for $A$ $\mathcal{\in A(O})~{}$is independent
of the choice of the smearing function $g(x)=f_{R}(\mathbf{x})f_{d}(x_{0})$ as
long as $\mathcal{O}$ remains inside their timelike extended shell structure
(43).

The local charge $Q(g)$ which measures the charge of an operator $A$ localized
in $\left|\mathbf{x}\right|<R$ converges towards the generator $Q~{}$of the
global $U(1)$ symmetry. The concept of a local charge content becomes
problematic in case of sl currents since the use of a rigid spacelike
direction $e$ does not allow the causal separation of $j_{0}(f_{R},f_{d})$
from the localization region of $A.$ The heuristic idea for achieving such a
separation would be to ”comb” the strings emanating from the shell between $R$
and $R+r$ into different directions so that they remain causally separated
from $A$ $\mathcal{\in A(O}).$ But then the strings emanating from poins
inside the shell would have to move to spacelike infinity outside the larger
sphere and violate the localization inside the larger sphere

In view of a recent proof of the so-called split property [59], which is known
to secure the local implementation of global symmetries in massless $h\geq 1$
[60], there is no problem with the existence of local charges for QFT’s with
global symmetries; what is not clear is whether such charges can be described
in terms of conserved currents.

Meanwhile K.-H. Rehren informed me that his student M. Heep constructed local
charges from sl currents by appropriate use of conformal transformations. The
idea is to construct a local charge operator localized in a half-space, that
is then mapped to a sphere by a conformal transformation. In this way the
strings become ”curled” and end in the north pole.

Hence the W-W No-Go theorem excludes pl currents, but does not affect the
causal localizability of charges in arbitrary small spacetime regions.

###  3.2 The V-Z conundrum arises from an incorrect implementation of
causality

A simple class of models for a critical examination of the V-Z conundrum is
provided by linear couplings of conserved currents to external vector
potentials the relevant property of the sl current is its lowered short
distance dimension. A suitable setting for such problems is obtaind in terms
of Bogoliubov’s definition of the S-matrix and interacting local fields in
terms of adiabatic limits of the Bogoliubov $S(g)$-functional181818Our use of
the Bogoliubov’s formalism is close to that in [61][62][63]

$\displaystyle S:=\lim_{g(x)\rightarrow g}S(g),~{}~{}~{}S(g)=T\exp
i{\displaystyle\int}g(x)L_{int}(x)d^{4}x$ (45) $\displaystyle
A(x)|_{L_{int}}=\lim_{g(x)\rightarrow g}\frac{\delta}{i\delta
f(x)}S^{-1}(gL)S(g(x)L_{int}(x)+fA)|_{f=0}$ (46)

Here the interaction density $L_{int}$ is a Wick-ordered product of not more
than 4 free fields from the class of Wick-ordered composites of free fields
and $A|_{L_{int}}$ the interacting counterpart of $A(x)~{}$which is either a
free field or a Wick-ordered product of free fields (the terminology ”free” is
used for linear fields and Wick polynomials). The interacting field has the
form of a power series in $g$ with retarded products of $n$ $L^{\prime}s$
which is retarded in $A(x).$ The linear Bogoliubov map $A\rightarrow
A|_{L_{int}}$ does not preserve the algebraic structure but it maintains the
property of causal separability. Hence fields constructed in this way maintain
causality, and the solution of the V-Z problem consists in the proper
computation of the interacting fields via (44).

The class of interactions with external potentials to be studied is of the
form $L_{int}=L^{P}=U^{\mu}j_{\mu}^{P}$ with $U^{\mu}$ a external (classical)
vector potential and $j_{\mu}^{P}$ a conserved current as before. For the
current of a scalar complex free field $\varphi$ there is no problem; its
conserved current has $d_{sd}(j_{\mu}^{P})=3$ and hence (with
$d_{sd}(L_{int})=3$) renormalizable. This is the model on which Velo and
Zwanziger base their propagation picture: namely the scalar field obeys a
linear field equation which is linear191919Here and in the sequel linear
stands for linear in $U_{\mu}$ and its derivtives. in $U^{\mu}$ and they
expect (erroneously, as will be seem) that this holds independent of spin.

For $s=1$ the $d_{sd}(L^{P})=5$ and hence the pl model is nonrenormalizable.
To reduce the $d_{sd}$ from 5 to 4 one uses the relation (38) which rewritten
in terms of the interaction density reads

$\displaystyle
L^{P}=j_{\mu}^{P}U^{\mu}=L-\partial^{\kappa}V_{\kappa},~{}C_{\kappa\mu}=iA_{\kappa}^{\ast}\overleftrightarrow{\partial}_{\mu}\phi~{}+i\phi^{\ast}\overleftrightarrow{\partial}_{\mu}\partial_{\kappa}\phi+h.c$
(47) $\displaystyle
L:=j_{\mu}^{s}U^{\mu}-C_{\kappa\mu}\partial^{\kappa}U^{\mu},~{}~{}V_{\kappa}=-C_{\kappa\mu}U^{\mu},\text{
}S^{(1)}\overset{a.l.}{=}\int L^{P}d^{4}x=\int Ld^{4}x$

where the decomposition (38) of $j_{\mu}^{P}$ was used. Since
$d_{sd}(C_{\kappa\mu})=4$ the power counting bound $d_{d}(L)=4$ holds, the
model is renormalizable and its first order S-matrix $S^{(1)}$ (the adiabatic
limit of the interaction density) is the same for the two densities and hence
string-independent (the suitably defined adiabatic limit of the $\partial V$
vanishes).

The decomposition of $j_{\mu}^{P}$ (38), which previously served to solve the
W-W problem (by converting the pl current into its for $m\rightarrow 0$
regular sl counterparts and a $C$-term, which carries the $m^{-2}$ mass
divergence but does not contribute to the global charge202020Using conformal
invariance of massless helicity representations one can also show the
existence of local charges (see remarks in previous subsection).), is now used
to solve the V-Z causality problem. To achieve this one uses the fact that the
$C$-term is a 4-divergence and disappears in the adiabatic limit which
represents the S-matrix.

The two interaction densities $L^{P}$ and $L$ share the same S-matrix; whereas
the pl $L^{P}(x)$ side insures that the S-matrix is that of a causal
interaction, the sl $L(x,e)$ guaranties the renormalizability of $S^{(1)}.$
The $L(x,e)~{}$together with $V_{\mu}(x,e)$ forms what will be referred to as
a $L,V_{\mu}$ pair. The first order S-matrix (47 second line), which is the
adiabatic limit of the interaction density, is the same for $L^{P}$ and $L~{}$
referred to as the linear relation (47). The $L^{P}$ represents the heuristic
content of the interaction, but as a result of its bad short distance behavior
it is not suitable for perturbative calculations. The short distance improved
$L$ weakens the localization but retains enough of it to keep fields causally
separated and to maintain scattering theory.

The remaining problem is the extension of this idea to higher order. For
convenience of notation one uses a differential formulation of pl localization
in the form of $e$-independence in the form $d_{e}(L-\partial V)=0.$ It is
convenient to use lightlike $e^{\prime}s$ since in this case no smearing is
needed. The problem in higher order is the time-ordering. For the
$e$-independence of the S matrix one needs the $\partial$ to act outside the
time-ordering e.g.

$\displaystyle
d_{e}(TL(1)L(2)-\partial_{1}^{\mu}TV_{\mu}(1)L(2)-\partial_{2}^{\nu}TL(1)V_{\nu}(2)+\partial_{1}^{\mu}\partial_{2}^{\nu}TV_{\mu}(1)V_{\nu}(2)\overset{?}{=}0$
(48) $\displaystyle~{}d_{e}T(L(1)-\partial V(1))(L(2)-\partial V(2))=0$

and higher order extensions involving one $V_{\mu}$ and $n-1$ $L^{\prime}s.$

This is generally not possible without creating ”obstructions” of the form of
delta contributions of the form
$\delta(x_{1}-x_{2})d_{e}L_{2}(x_{1},e)~{}$which are quadratic in $U_{\mu}$
and its derivatives. Higher order violations may lead to contributions of
higher polynomial degree in $U_{\mu}$ and derivatives; it is a characteristic
property of obstructions in models of external potential interactions that all
obstructions remain bilinear in the quantum fields.

These obstructions are absorbed in the form of induced contributions into a
modified Bogoliubov formalism by defining

$L_{tot}=L+\frac{1}{2}L_{2}+..\frac{1}{n!}L_{n}+..$ (49)

where $L_{n}$ is of polynomial degree $n$ in $U_{\mu}$ and its derivatives and
remains quadratic in free fields. Note that induced contributions do not
increase the number of parameters and hence must be distinguished from
counterterms of pl renormalization theory.

In the $s=1$ model (47) the $L_{2}$ contribution can alternatively be encoded
into a redefinition of time ordering

$\displaystyle
T_{0}\partial_{\mu}\phi(x_{1})\partial_{\nu}\phi(x_{2})\equiv\partial_{\mu}\partial_{\nu}T_{0}\phi(x_{1})\phi(x_{2})$
(50) $\displaystyle
T\partial_{\mu}\phi(x_{1})\partial_{\nu}\phi(x_{2})=T_{0}\partial_{\mu}\phi(x_{1})\partial_{\nu}\phi(x_{2})+icg_{\mu\nu}\delta(x_{1}-x_{2})$
$\displaystyle\partial^{\mu}T\partial_{\mu}\phi(x_{1})\partial_{\nu}\phi(x_{2})-T\partial^{\mu}\partial_{\mu}\phi(x_{1})\partial_{\nu}\phi(x_{2})=(1+c)\partial_{\nu}\delta(x_{1}-x_{2})$

and the validity of (48) for the $T\,\ $time-ordering requires to set $c=-1$.
For $s>1$ The kinematic $T_{0}$ time-ordering contains more derivatives and
one has accordingly more $c^{\prime}s$ which must be numerically adjusted in
such a way that the $T$ time-ordering satisfies the higher order pair
requirements beyond (48).

The following side-remark maybe helpful for the later extension of SLFT to a
full QFT. The second order $A_{\mu}A^{\mu}\varphi^{\ast}\varphi$ term of
scalar QED within the new SLFT can either be viewed as induced or encoded into
a change of time-ordering for the derivatives of the complex scalar field. But
not all obstructions can be absorbed in this way. The $H$-selfinteractions of
the Higgs model is a genuine second order induced term which results
exclusively from the implementation of the positivity and causality principle
of QFT (rather than from an imposed Mexican hat interaction).

The verification of the higher order pair relations will be deferred to a more
complete treatment of external potential problems. The expected result is:

Conjecture Couplings of conserved currents to external potentials fulfill the
higher order time-ordered $L,V_{\mu}~{}$relation. For $s=1$ the resulting
field equations are quadratic in $U_{\mu}$ whereas for $s>1$ they are of
infinite order (expected since $d_{sd}(L)\geq 5$ ).

The form of the linear causal field equations (in particular the higher order
$U$ contributions) is determined by the form of the induced contributions.

The external potential formalism and its formal connection with the solution
of the W-W problem works in an analogous way for $s=2.$ The sl potential
$A_{\mu\nu}(x,e)$ has two escorts, a vector $a_{\mu}$ and a scalar escort $a$
which can be chosen in such a way that the operator dimension for all fields
is identical to their classical dimension $d_{cl}=1$ (or $3/2$ for half-
integer spin)

$\displaystyle
A_{\mu\nu}^{P}=A_{\mu\nu}+m^{-1}(\partial_{\mu}a_{\nu}+\partial_{\nu}a_{\mu})+m^{-2}\partial_{\mu}\partial_{\nu}a$
(51) $\displaystyle
j_{\mu}^{P}(x)=iA_{\kappa\lambda}^{P\ast}\overleftrightarrow{\partial_{\mu}}A^{P\kappa\lambda}=j_{\mu}^{s}(x,e)+\partial^{\kappa}C_{\kappa\mu}$
$\displaystyle
j_{\mu}^{s}(x,e)=iA_{\kappa\lambda}^{\ast}\overleftrightarrow{\partial_{\mu}}A^{\kappa\lambda}-2ia_{\kappa}^{\ast}\overleftrightarrow{\partial_{\mu}}a^{\kappa}+ia^{\ast}\overleftrightarrow{\partial_{\mu}}a$

In this case $d_{sd}(L^{P})=7$ and $d_{sd}(j^{\mu})=3~{}$and hence $L^{P}$ is
by $3$ units beyond the power-counting bound $d_{sd}=4$. For $s=1$ the
$\partial^{\kappa}C_{\kappa\mu}$ carries the highest $d_{sd}=7$ contribution.
After an additional linear disentanglement between $A_{\mu\nu}$ and $a$ one
arrives at a decomposition of $j_{\mu}^{s}$ which in the massless limit
represents the $h=2,1,0$ helicity contributions [11].

The use of this decomposition in the rewriting of
$L^{P}=j_{\mu}^{P}(x)U^{\mu}$ for $s=2~{}$as a sl pair with $L^{P}=L-\partial
V$ leads to a $d_{sd}(C_{\kappa\mu})=6~{}$ contribution which contains
bilinear in $\phi=m^{-2}a$ terms with more than 2 derivatives. In analogy to
counterterms in every order in a nonrenormalizable full pl QFTs one expects to
find induced terms in arbitrary high orders.

It is worthwile to mention that there is also a gauge theoretic formulation in
which the linear operator relation between the sl potential and its pl
counterpart is replaced by the relation
$A_{\mu}^{K}(x)=A_{\mu}^{P}(x)+\partial_{\mu}\phi^{K}$ where the $K$ refers to
the Krein space and the esort $\phi^{K}$ is the Stückelberg negative metric pl
scalar which adds additional unphysical degrees of freedom to the indefinite
Krein space. This is the formulation of the Uni Zürich group [64][65] adapted
to the presentation used in the present paper.

The gauge theoretic analog to the pair relation is
$L^{K}=L^{P}+\partial^{\mu}V_{\mu}^{K}.$ The model has a formal similarity
with SLFT, but its pl interpolating fields are unphysical; positivity obeying
interpolating fields are simply inconsistent with pl localization. The pair
property works the same way, one only has to replace the $d_{e}$ in (48) by
the BRST $\mathfrak{s}$.

However negative metric degrees of freedom lead to an unphysical realization
of appropriately nonlinear modified causal pl V-Z equation and should be
rejected inasmuch as gauge dependent pl currents have been discarted by W-W in
their No-Go theorem. Classical field theory is free of positivity requirements
and gauge theoretic causal propagation is perfectly compatible with its
principles. But the nonlinear dependence on external potential which was
overlooked by V-Z is also needed for the classical propagation of Cauchy date.

Recently there have been attempts to solve the V-Z conundrum in terms of
String Theory [68] [17]. These authors extract a system of pl equation in
$d=3+1~{}$via dimensional reduction from the Virasoro algebra in 10 dimensions
and found nonlinear modifications in case of constant external fields.

But the fact that there is nothing stringy about their pl equations raises the
old question: what do string-theorist really mean when they claim that their
objects are stringy in spacetime. Does their use of the terminology ”string”
perhaps refer to a circular structur in a 10-dimensional target space whose
Fourier components correpond to the irreducible Wigner components of the
highly reducible superstring representation?

Their strings bear no relation with causal localization in spaceteime but
rather seem to refer to Born’s quantum mechanical localization related to the
spectral decomposition of the x operator arises. Their use of world-sheet and
Nambu-Goto actions point into this direction and the way in which they think
of their localized objects as vibrating in space strengthens this presumption.
Causal localization in spacetime is very different (for more see next
section).

The next section explores important aspects of causal localization which,
although known to some experts, remained outside the conceptual radar screen
of most particle physicists.

##  4 Particle wave functions and causal localization

There is no concept in particle physics which led to more misunderstandings
than that of causal localization in spacetime. The strings of String Theory
obtained e.g. from quantized world-sheet or Nambu-Goto actions bear no
relation causal localization. A concept which reveals such misunderstandings
and corrects them in the clearest possible way is ”modular localization”.

###  4.1 Newton-Wigner localization and its causality-providing modular
counterpart

Wigner’s theory of positive energy representations presents an interesting
meeting ground of two very different localization concepts. On the one hand
there is the quantum mechanical localization of dissipating wave packets whose
center moves on relativistic particle trajectories. Its formulation in terms
of quantum mechanical Born probabilities leads to the so-called Newton-Wigner
localization [70]. For a scalar $m>0$ particle

$\displaystyle(\psi,\psi^{\prime})=\int\bar{\psi}\overleftrightarrow{\partial_{0}}\psi^{\prime}d^{3}x=\int\bar{\psi}_{NW}\psi_{NW}^{\prime}d^{3}x$
(52)
$\displaystyle\hbox{hence}\quad\tilde{\psi}_{NW}(\mathbf{p})=(2p_{0})^{-1/2}\tilde{\psi}(\mathbf{p})$

Hence an improper N-W eigenstate of the position operator $\mathbf{x}_{NW}$
has a mass-dependent extension of the order of a Compton wave length. In
scattering theory, where only the large-time asymptotic behavior matters, such
ambiguities in assigning relativistic quantum mechanical positions at finite
times are irrelevant; the centers of wave packets of particles move on
relativistic velocity lines and the probability to find a particle dissipates
as $t^{-3}$ along these lines for all inertial observers. In fact Wigner never
thought of his Poincaré group representation theory as an entrance into causal
QFT; for him it remained part of relativistic QM212121This perhaps explains
why Wigner, inspite of his overpowering role in the development of $20^{th}$
century quantum theory, never participated in the QED revolution and its QFT
aftermath. For him his representation theory always remained part of
relativistic quantum mechanics (the quantum mechanical Newton-Wigner
localization in section 3). An interesting discussion can be found in Haag’s
memoirs [71] page 276..

The more recent discovery of modular localization shows that causality
properties are dormant within Wigner’s positive energy representation theory;
they are reflected in properties of dense subspaces obtained by applying
algebras of local observables $\mathcal{A(O}$ to the vacuum state
$\mathcal{H(O})=\mathcal{A(O)}\Omega~{}$and projecting the so obtained dense
set of states of a QFT to the one-particle subspace
$\mathcal{H}_{Wig}(\mathcal{O})=E_{1}\mathcal{H(O}).$ That such spaces are
dense in the Hilbert space (and consequently their projection in the one-
particle subspace) is a special case of a surprising discovery made in the
early in the early 60s (the Reeh-Schlieder theorem [1] [2]) which showed that
the omnipresence of vacuum polarization confers to QFT a very different notion
of localization from that of Born’s quantum mechanical setting based on
position operators.

The projection $\mathcal{H}_{Wig}\mathcal{(O)}~{}$has the remarkable property
that it can be constructed without the assistance of QFT solely in terms of
data from Wigner’s representation theory and that in the absence of
interaction one can even revert the direction and obtain the net of causally
localized subalgebras directly from that of modular localized Wigner subspaces
[25].

In this way one does not only gain a more profound understanding of QFT but
one also learns that Weinberg’s pure group theoretic construction of
intertwiners starting from Wigner’s representation theory is part of a much
more general setting which, if properly used, leads to an extension of
perturbative renormalizability. This important concept of modular localization
was not available during Wigner’s lifetime (see remarks in the introduction).

The simplest way to see that the quantization of a relativistic classical
particle associated with the action $\sqrt{-ds^{2}}$ does not lead to a
covariant quantum theory is to remind oneself that there exists no operator
$\mathbf{x}$ which is the spatial component of a covariant 4-vector [35]. The
conceptual problem one is facing is better understood by first showing that
causal localization bears no relation to Born’s probabilistic quantum
mechanical definition.

Starting from the quantum mechanical projectors $E(R)$ for
$R\subset\mathbb{R}^{3}$ which appear in the spectral decomposition of
$\mathbf{x}_{op}$

$\mathbf{x}_{op}=\int\mathbf{x}dE(\mathbf{x})$ (53)

one has

$E(R)E(R+\mathbf{a})=0\quad\hbox{for}\quad R\cap(R+\mathbf{a)}=0$

Define $E(R+a)=U(a)E(R)U(a)^{\ast}$ for $a\in\mathbb{R}^{4}.$ Assuming that
this orthogonality relation has a causal extension in the sense that
$E(R)E(R+a)=0$ for spacelike separated$~{}R{\lower
1.0pt\hbox{\LARGE$\times$}}R+a~{}$leads immediately to clash with positivity
of the energy. This follows from the fact that the positivity of the energy
leads to the analyticity of expectation values $(\psi,E(R)E(R+a)\psi)$ for
$\operatorname{Im}a_{0}>0$ which in turn implies their identical vanishing
(the Schwarz reflection principle) and
with$~{}\left|\left|E(R)\psi\right|\right|^{2}=0$ the triviality of such
projectors $E=0.$

A slight extension of the argument reveals that it can be dissociated from the
position operator of quantum mechanics. It then states that in models with
energy positivity it is not possible to describe causal localization (”micro-
causality”) in terms of projectors and orthogonality of subspaces [72]. A
profound intrinsic understanding of causal localization in QFT points into a
very different direction from that obtained from the quantization of actions
describing classical world lines or world sheets and Nambu-Goto action. The
problems become insurmountable of one tries to construct actions in the
presence of several of such objects and their distance.

Before presenting the relation to QFT it is worthwhile to mention a little
known fact: it is perfectly possible to construct a relativistic description
of interacting particles in relativistic QM build on macro-causality. For two
particles this amounts to Poincaré group preserving modification in the centre
of mass system, but for more particles it is more complicated ([43] section
3). Apart from the fact that it leads to a Poincaré-invariant S-matrix, it
does (unlike Schrödinger theory) not permit a description in terms of second
quantization.

###  4.2 Mathematical properties of modular localization

To prepare the ground for causal localization it is helpful to start with some
mathematical concepts concerning relations between real subspaces $H$ (linear
combination with reals) of a complex Hilbert space
$H\subset\mathcal{H}\mathbf{.}$ The symplectic complement $H^{\prime}~{}$of a
real space is defined as the closed real subspace
($\overline{H^{\prime}}=H^{\prime}$) defined in terms of the imaginary part of
the scalar product in$~{}\mathcal{H}$

$\displaystyle H^{\prime}$ $\displaystyle=\left\\{\xi\in\mathcal{H};\
\operatorname{Im}(\eta,\xi)=0~{}\forall\eta\in H\right\\}$ (54) $\displaystyle
H_{1}$ $\displaystyle\subset H_{2}\text{ }\Rightarrow H_{1}^{\prime}\supset
H_{2}^{\prime}$ (55)

which turns out to be the real orthogonal space on the real $iH$ (only real
linear combinations)

A closed real subspace $H$ is called ”standard” if it is both cyclic and
separating

$\displaystyle H\quad\hbox{cyclic:}\quad$
$\displaystyle\overline{H+iH}=\mathcal{H}$ (56) $\displaystyle
H\quad\hbox{separating:}\quad$ $\displaystyle H^{\prime}\cap
H=\left\\{0\right\\}$ $\displaystyle\left(H+iH\right)^{\prime}$
$\displaystyle=H^{\prime}\cap iH^{\prime}$

Cyclicity and the separation property have a dual relation in terms of
symplectic complements as written in the third line.

It is quite easy to obtain such standard spaces from covariant free fields. In
the simplest case of a scalar field the Hilbert space $\mathcal{H}$ is the
closure of the 1-particle Wigner space defined by the two-point function of
the smeared fields

$\displaystyle\left(f,g\right)$ $\displaystyle=\left\langle
A(f)^{\ast}A(g)\right\rangle=\int\tilde{f}^{\ast}(p)\tilde{g}(p)d\mu(p)$ (57)
$\displaystyle\left[A(f)^{\ast},A(g)\right]=-i\operatorname{Im}\left(f,g\right)$

where $d\mu\ $is the invariant measure on the positive mass hyperboloid.
According to the Reeh-Schlieder theorem [2] the one-particle projection of the
dense subspace of causally $\mathcal{O}$-localized states222222States
localized in the spacetime region $\mathcal{O}$ are defined as the dense Reeh-
Schlieder subspace obtained as $\mathcal{A(O)}\left|0\right\rangle$ where
$\mathcal{A(O)}$ is the $\mathcal{O}$ localized subalgebra of $\mathcal{A}$.
is dense in the one-particle Wigner space.$\mathcal{\ O}$-localized real
testfunctions define a dense real subspace $H(\mathcal{O)}$ and causal
disjointness corresponds to ”symplectic orthogonality” and produces a closed
real subspace.

$H(\mathcal{O}^{\prime})=H(\mathcal{O)}^{\prime}$ (58)

As a side remark we mention that the same construction applied to a higher
halfinteger spin field leads to a corresponding situation

$ZH(\mathcal{O}^{\prime})=H(\mathcal{O})^{\prime},\ \
Z=\frac{1+iU(2\pi)}{1+i}$ (59)

where the unitary ”twist” operator $Z$ which is related to the factor $-1$ of
the $2\pi$ rotation. The use of the twist operator allows to treat bosons and
fermions under one common roof.

The important step for an intrinsic understanding of QFT (i.e. without the use
of P. Jordan’s ”(quasi-)classical crutches” of quantization) is to invert the
previous construction: find a relation within Wigner’s representation theory
which permits to define $\mathcal{O}$-localized real subspaces which have the
correct covariant transformation properties under Poincaré transformations
[25]. For this purpose it is helpful to reformulate the above properties so
that they take the form known from the mathematical Tomita-Takesaki theory of
operator algebras which permits a direct connection of positive energy Wigner
representations with a ”local net of operator algebras”. It provides a unified
view in which Weinberg’s pl intertwiner formalism and its sl extension are
seen as two ways of generating the same free field theory. As shown in
previous sections the improvement of short distance properties is the basis
for a new perturbative renormalization setting.

To achieve this one needs an additional mathematical tool. The first step
consists in a suitable extension of the modular concepts. A standard subspace
$H$ comes with a distinguished operator. With $D(Op)$ denoting the domain of
definition of an operator one defines

###### Definition 1

A Tomita operator $\mathcal{S}$ is a closed antilinear densely defined
involutive operator $D(\mathcal{S})\subset\mathcal{H}$

In physics one encounters such ”transparent” operators only in QFT. It is easy
to see that there exists a 1-1 correspondence between Tomita operators and
standard subspaces $H;$ $H\leftrightarrow\mathcal{S}$. This follows from the
definition $\mathcal{S}(\xi+i\eta)=\xi-i\eta,~{}\xi,\eta\in H,$ whereas the
opposite direction is a consequence of the definition
$H=\ker\left\\{\mathcal{S}-1\right\\}.$

As a result of involutiveness, the full content of Tomita operators is
contained in their dense domains which coincides with their range
(”transparency”). Hence modular theory may be alternatively formulated in
terms of subspaces (56) $Dom\mathcal{S}$ of Tomita operators.

The polar decomposition of $\mathcal{S}=J\Delta^{1/2}$ of $\mathcal{S}$ into
an anti-unitary $J$ and a positive operator $\Delta\ $with
$D(S)=D(\Delta^{1/2})$ leads to the unitary modular group $\Delta^{it}$ acting
in $\mathcal{H}$ and preserving the standard subspace $\Delta^{it}H=H,$
whereas the modular conjugation maps into the symplectic complement $J$
$H=H^{\prime}$

A Tomita operator appears in a natural way in Wigner’s representation theory
of positive energy representations of the Poincaré group $\mathcal{P}$. It is
obtained by defining $\Delta_{W_{0}}^{it}$ in terms of the Lorentz boost
operator which leaves the wedge
$W_{0}=\left\\{x;~{}z>0,\left|t\right|<z\right\\}$ invariant

$\displaystyle\Delta_{W_{0}}^{it}=U(\Lambda_{W_{0}}(-2\pi t))$
$\displaystyle\mathcal{S}_{W_{0}}=J_{W_{0}}\Delta_{W_{0}}^{1/2},\ \
J_{W_{0}}=TCP\cdot R_{\pi}$

together with an anti-unitary $J$ obtained by multiplying the TCP reflection
$TCP$ with a $\pi$-rotation $R$ in the $x$-$y$ plane as written in the second
line, taking $W_{0}~{}$into its causal complement.

The charge conjugation $C~{}$maps an irreducible Wigner particle space into
its charge conjugate and may need a doubling of the Wigner space whereas $TP$
corresponds to the spacetime inversion $x\rightarrow-x$. The preservation of
energy requires the time-reversal $T$ to be anti-unitary. Massless
representations need a helicity doubling $\pm h.$

Unbounded operators $\mathcal{S}$ whose dense domain is stable (”transparent”
in the sense domain=range) are somewhat unusual in quantum physics; they
appeared first in quantum statistical mechanics [2] and later in searches for
an intrinsic understanding of causal localization of QFT (without referring to
Lagrangian quantization) [27]. In the present context the dense subspace of
the Wigner space (possibly doubled by charge conjugation) corresponds to wave
functions which are ”modular localized” in the wedge $W_{0}$. Modular
localization of Wigner wave functions is closely related to causal
localization of fields and provides an extension of the Weinberg intertwiner
formalism which includes Wigner’s infinite spin class [35].

The construction proceeds as follows: start from Wigner’s positive energy
representation theory, define the Tomita operator $S_{W_{0}}~{}$in the way
described before, use the Poincaré transformations to construct a net of
modular localized real subspaces $H(W)$ and use the second quantization
functor (the Weyl or CAR functor) to pass to an interaction-free net of
standard wave functions spaces $H(W)$ to causally localized operator algebras
$\mathcal{A(}W)$ acting in a Wigner-Fock Hilbert space.

One also may directly construct real dense subspaces $H_{\mathcal{O}}$ and
their complexified counterparts
$D(\mathcal{S}_{\mathcal{O}})=H_{\mathcal{O}}+iH_{\mathcal{O}}\subset\mathcal{H}$
corresponding to more general causally complete convex spacetime regions
$\mathcal{O}_{c}$ as intersections:

$\displaystyle H_{\mathcal{O}_{c}}$
$\displaystyle=\bigcap_{W\supset\mathcal{O}_{c}}H_{W}$ (60) $\displaystyle
H_{\mathcal{O}}$
$\displaystyle=\bigcup_{\mathcal{O}_{c}\subset\mathcal{O}}H_{\mathcal{O}_{c}}$

whereas for more general regions the standard space is defined in terms of
exhaustion from the inside (second line). For details we refer to [25].

The energy-positivity of the massive and the $\pm\left|h\right|$ massless
Wigner representation classes plays an important role in establishing the
isotony and causal localization of the ”net of modular localized standard
spaces”

isotony: $\displaystyle H_{\mathcal{O}_{1}}\subset
H_{\mathcal{O}_{2}}\quad\hbox{if}\quad\mathcal{O}_{1}\subset\mathcal{O}_{2}$
(61) causality: $\displaystyle H_{\mathcal{O}_{1}}\subset
H_{\mathcal{O}_{2}}^{\prime}\quad\hbox{if}\quad\mathcal{O}_{1}{\lower
1.0pt\hbox{\LARGE$\times$}}\mathcal{O}_{2}$

where $\times$ denotes spacelike separation.

In the absence of interactions the passage from the spatial modular theory to
its algebraic counterpart is almost trivial. One passes to the Wigner Fock
space created by symmetrized tensor products and defines the
$\mathcal{O}$-localized operator algebra as the von Neumann algebra generated
by the Weyl operators

$\displaystyle\mathcal{A(O})=\left\\{e^{iA(h)};h\in
H_{\mathcal{O}}\right\\}^{\prime\prime},~{}\left[A(h_{1}),A(h_{2})\right]=i\operatorname{Im}(h_{1},h_{2})$
(62) $\displaystyle\quad\hbox{with}\quad
A(h)=\int\sum_{s_{3}=-s}^{s}(h(p,s_{3})a^{\ast}(p,s_{3})+h.c.)d\mu(p),~{}h\in
H_{\mathcal{O}}$ (63)

in terms of the Wigner creation/annihilation operators232323The ”second
quantization” counterparts of the Wigner wave functions. $a^{\\#}(p,s_{3})$ or
their helicity counterparts. For half-integer spin or helicity the presence of
the twist operator (59) leads to fermionic commutation relation. In both cases
the second quantization functor maps the spatial Tomita operator into its
algebraic counterpart which acts in the (”second quantized”) Wigner Fock space
$\mathcal{H}$ associated with the Wigner one-particle space $H$

$\mathcal{S}_{\mathcal{O}}^{alg}A\Omega=A^{\ast}\Omega,~{}A\in\mathcal{A(O})$

In the sequel only the algebraic Tomita-Takesaki theory will be used and the
superscript $alg~{}$will be omitted for convenience.

Modular localized spaces $H(\mathcal{O})$ and their associated noninteracting
field algebras $\mathcal{A(O})$ are by construction ”causally complete”
$\mathcal{A(O})=\mathcal{A(O}^{\prime\prime})$ and hence a fortiori Einstein
causal (section 2 13). The causal completion $\mathcal{O}^{\prime\prime}$ is
obtained by taking twice the causal complement $\mathcal{O\rightarrow
O}^{\prime}.$

For wedge regions the modular group coincides with the unitary Wigner
representation of the wedge-preserving Lorentz group; for all other regions
the modular groups in $m>0$ Wigner representations acts in a non-geometric
(”fuzzy”) way. It is believed that this nongeometric action becomes
asymptotically (near the boundary of the causal completion) geometric.

Massless finite helicity theories have a larger set of regions related to
Huygen’s principle in which modular groups act in a geometric way; this
includes all regions which are images of wedges under the action of conformal
transformation as e.g. finite double cones. Sl potentials whose semiinfinite
spacelike lines remain inside wedges may under suitable conformal
transformation pass to potentials localized on finite elliptic curves which
connect two points on different edges of double cones; they can be viewed as
substitutes for the nonexistent pl coordinatization of double cones.

It may also happen that the standard spaces for compact spacetime regions
$\mathcal{O}$ are trivial $H_{\mathcal{O}}=\left\\{0\right\\}.$This occurs
precisely for the Wigner’s zero mass infinite spin representation for which
the tightest localized nontrivial spaces correspond to modular localization in
arbitrary narrow spacelike cones. On the other hand zero mass finite helicity
spaces are the most geometric representation since their modular groups
continue to act geometrically even for double cones; in fact they correspond
to conformal transformations which preserve double cones. Unlike finite
helicity fields infinite spin representations are massless but not conformal.

Modular localization plays also an important role in the understanding of
topological peculiarities of massless $h\geq 1~{}$free QFT in connection with
toroidal regions (thickened Wilson loops). Last not least without their use it
would not have been possible to discover the intrinsic noncompact localization
of Wigner’s infinite spin matter and the string-like nature of its generating
causally localized fields.

This raises the question if modular theory preserves its constructive power in
the presence of interactions. It turns out that in that case one is required
to use the stronger operator algebraic modular theory. Interacting theories
share the same modular groups $\Delta_{\mathcal{W}}^{it}$ with their
noninteracting counterpart which is solely determined by the particle content.

The interaction enters through the dependence of the $J_{\mathcal{W}}$ on the
interaction which, using the fact that the incoming TCP is related with its
outgoing counterpart through the S-matrix $S$ [69] [30], amounts to

$J_{\mathcal{W}}=J_{\mathcal{W}}^{(in)}S$ (64)

Again the starting point is a von Neumann operator algebra $\mathcal{A}$
acting in a Hilbert space $\mathcal{H}~{}$which contains a vector $\Omega$
which is cyclic and separating under the action of $\mathcal{A}$ (in QFT the
Reeh-Schlieder property for $\mathcal{A(O)}$)

cyclic: $\displaystyle\mathcal{A(O)}\Omega\quad\hbox{is dense
in}\quad\mathcal{H}$ (65) separating: $\displaystyle
A\Omega=0\quad\Rightarrow\quad A=0,~{}~{}A\in\mathcal{A(O)}~{}$

The definition$~{}H$ $=\overline{\mathcal{A}_{sa}(\mathcal{O})\Omega}$ ($sa=$
selfadjoint part) or $\mathcal{H=}\overline{\mathcal{A(O)}\Omega}$ connects
the algebraic modular theory with its previously presented spatial
counterpart. But there remains an important difference: the map between
standard subspaces and algebras is generally not injective whereas the
algebraically generated subspaces (65) always are .

The interaction-free situation remains exceptional in that there exists a
functorial relation between modular localized Wigner subspaces and
interaction-free causally localized subalgebras defined in terms of the Weyl
map (62). This functorial map is lost in the presence of interactions in which
case the relation between modular theory and particles becomes more involved.

###  4.3 A critical perspective based on modular localization

Most of what is presently known about modular theory comes from the Bisognano-
Wichmann theorem [27] which clarifies the modular properties of wedge-
localized subalgebras $\mathcal{A}(W)$. In models with a complete particle
interpretation one can use the modular theory of free fields to derive the B-W
theorem in the presence of interactions [81]. Of special interest is the
relation of the modular conjugation $J$ with the TCP operator which is known
to be connected to the S-Matrix [69]. This is particularly useful in $d=1+1$
integrable models whose S-matrix is known (64).

For integrable models without bound states this led to an interpretation of
the Zamolodchikov-Faddeev algebra in terms of modular localization in which
the concept of ”vacuum-polarization-free generators” (PFG)242424The weakest
assumptions under which PFG’s exist were determined in [74]. plays an
important role [30][73]. This in turn led to existence proofs for certain
$d=1+1$ integrable models in terms of operator algebraic constructions based
on modular theory [76]. For a recent account with many references to previous
publications see [31]. These results complement those obtained in terms of the
”bootstrap-formfactor” program in [75].

The important role of the S-matrix for modular localization in wedges has
triggered attempts to reconstruct a full causal QFT from its ”on-shell
footprint” in form of its S-matrix [77, 78]. These ideas are presently too
weak for constructions in higher dimensions. Among the unexpected results of
modular theory is the proof that models in $d=1+2$ with anyonic statistics (or
its nonabelian ”plektonic” counterpart) have always nontrivial S-matrices
[80].

Whereas the idea that a nonperturbative QFT is uniquely determined by its
S-matrix remains still part of folklore, the S-matrix based perturbative SLFT
construction in the previous section is the basis of the new perturbation
theory. Different from the standard approach based on Lagrangian quantizations
in which the S-matrix is obtained from the mass-shell restriction (the LSZ
reduction formula) of time-ordered correlation functions of fields, SLFT
inverts this situation by encoding the model-defining particle content into an
interaction density of a perturbative defined S-matrix. This part of the
construction uses exclusively pl or sl free fields which are directly related
to the particles. Interacting (off-shell) quantum fields are constructed in a
second step in which the higher order contributions to the interaction density
(which were induced in the construction of $S$) serve as an input (46).

Each pl or sl field from the equivalence of free fields and their Wick-ordered
composites has an interacting counterpart. Interacting fields which have
nonvanishing matrixelements between the vacuum and a one-particle state have
after appropriate normalization the same large-time in- and out- limits.

Finally we come to an important point whose clarification was promised at the
beginning of this section namely the possible connection of sl quantum fields
with String Theory. String theorists attribute a string-like localization to
their objects without providing arguments in favor of causal localizability.
Ideas based on worldlines, worldsheets, Nambu-Goto actions or strings start
from classical relativistic mechanics and hope that quantization preserve
these properties. This is evident from the way in which the covariant world-
line action $\sqrt{-ds^{2}}$ in [83] is used to prepare the ground for the
subsequent presentation of world-sheets and Nambu-Goto actions. The
impossibility to place two of such vibrating quantum mechanical strings into a
relative spacelike position reveals the problems which ST has with causal
localization in spacetime.

Fact is that, apart from Lagrangian quantization of $s<1$ fields, only the
Wigner representation theory (which cannot be accessed by quantization)
contains the seed for causal localization. Modular localization as a pre-form
of causal localization needs positivity and is inconsistent with gauge theory.
One cannot declare an object as ”stringy” because its classical action
suggests this. This type of misunderstanding is clearly visible in
Polchinski’s use of the action of a relativistic particle as a preparatory
step for relativistic worldsheet and Nambu-Goto actions.

This does not exclude the use of the quantum mechanical Newton-Wigner
localization to describe the dissipation of wave-packets. One may even
construct a macro-causal Poincaré-invariant multi-particle theory which
satisfies cluster decomposition properties and leads to a Lorentz-invariant
scattering matrix ([43] section3). Such a construction disproves the
conjecture that relativistic particles and cluster properties alone will lead
to QFT.

Proposals which avoid quantization of classical actions and try to find causal
quantum matter in the target space of certain models of $d=1+1~{}$conformal
QFTs are less easy to dismiss. One such proposal is based on the Virasoro-
algebra of a supersymmetric 10-component chiral conformal current model (the
”superstring”). Its target space contains an algebraic structure which leads
to a highly reducible unitary Wigner representation of the Poincaré group
[84].

This is primarily a group theoretical observation which (apart from the ten
spacetime dimensions) fits well into Majorana’s 1932 project of finding
algebraic structure which, in analogy to the $O(4,2)$ hydrogen spectrum,
describes wave functions of families of higher spin/helicity particles. But
why should one believe that the corresponding fields are sl in the absence of
any supporting argument? Is the terminology perhaps related to picturing the
tower of representations containing different masses and spins as Fourier
components of a kind of internal circle? In that case the use of ”string” for
something bears no relation to spacetime and would be misleading.

Looking at the ST literature one gets frustrated about the disproportionate
relation between its conceptual poverty as compared to its mathematical
richness which its vague physical pictures lead to in the hands of
mathematicians. The word ”string” should be more than an ”epitheton ornans”
for a physically insufficiently understood mathematical formalism.

With additional conceptual care one can also avoid a widespread
misunderstanding in the physical interpretation of the $AdS_{n+1}$-$CFT_{n}$
isomorphism. It was certainly consequential to complement the observation of
equality of the symmetry groups of the two spacetimes by the verification of a
stronger Einstein causality-preserving isomorphism between the two QFTs. But
unlike classical field theory the timelike completion property252525The causal
completion $\mathcal{O}^{\prime\prime}$ is the result of taking twice the
causal complement. in QFT $\mathcal{A(O}^{\prime\prime})=\mathcal{A(O}),$
(which roughly speaking describes causal propagation) is not a consequence of
the spacelike Einstein causality. Formally it is equivalent to Haag duality
$\mathcal{A(O}^{\prime})=\mathcal{A(O})^{\prime}$ from which it results by
taking the commutant on both sides and rewriting
$\mathcal{A(O}^{\prime})^{\prime}$ by applying Haag duality to the (generally
noncompact) region $\mathcal{O}^{\prime}.$

In the old days [87][88] it was shown that Einstein causality and the causal
dependency property (formally equivalent to Haag duality) are independent
requirements. Causality without Haag duality occurs if there are ”too many”
degrees of freedom as in the case of the generalized free field; a phenomenon
which has no classical analog (since the notion of quantum degrees of freedom
has no classical counterpart).

This manifests itself in a sort of ”poltergeist effect” in that there may be
more quantum degrees of freedom streaming into the dependency region as time
moves on than there were in the original appropriately defined initial
(”Cauchy”) data; in operator-algebraic notation
$\mathcal{A(O})\varsubsetneq\mathcal{A(O}^{\prime\prime})$ while Einstein
causality$\mathcal{A(O})\subset\mathcal{A(O}^{\prime})^{\prime}$ is preserved.

In fact it is quite easy to construct Einstein-causal models for which this
completion property is violated. Generalized free fields with a suitably large
$\kappa$ behavior of their Kallen-Lehmann spectral functions (containing a
much larger cardinality of degrees of freedom than free fields) were used at
the beginning of the 60s262626I am sometimes asked about the origin of this
terminology. The answer is simple, it accounts for the fact that Haag had this
idea already before I entered the collaboration; my contributions consisted in
providing calculations involving generalized free fields. to show that the
causal shadow property represents a separate requirement (initially called
”the time-slice property”).

The heuristic picture is that ”squeezing” a QFT with the natural cardinality
of degrees of freedom corresponding to a n+1-dimensional QFT into an n
dimensional spacetime (”holographic image”) causes an ”overpopulation”; this
is precisely what happens in the $AdS_{n+1}$-$CFT_{n}$ case [86]. The simplest
illustration is obtained by projecting a free AdS field and noting that the
resulting conformal field is a generalized free field of the kind used in
[87][88]. In the opposite direction i.e. starting from a ”normal” CFT one
expects an ”anemic” degree of freedom situation on the AdS side. As shown in
[85] this is precisely what happens; in fact there are no degrees of freedom
at all in compact AdS regions (double cones); to find any one has to pass to
infinitely extended wedge-like regions in AdS.

The overpopulation of degrees of freedom distinguishes ”holographic
projection” from ”normal” lower dimensional QFT and this explains the
quotation marks in ”pathology”; a holographic projection maintains the degrees
of freedom of the original QFT and in this way prevents the conformal side to
be a normal QFT. This is a point which had been ignored in most post Maldacena
work.

A helpful viewpoint concerning such ”overpopulated” models which one obtains
by dimension reducing projections is to not consider them as autonomous QFTs
but to view them rather as stereographic projections of the original QFT.

The degree of freedom issue is not limited to the AdS-CFT isomorphism but
affects all attempts to extend (quasi)classical Kaluza-Klein dimensional
reductions of lowering extra dimensions to full-fledged causal QFT. To the
extend that such pictures are compatible with the principles of
QFT272727Massaging Lagrangians is not the same as passing from one QFT to
another. they correspond to perform a stereographic projection on the original
theory and not to pass to a lower dimensional QFT. This poses the question
whether the fashion around extra dimensions would have occurred with a higher
conceptual awareness about the subtle nature of positivity and causal
localization of relativistic quantum matter which forbid the use of naive
quasiclassical arguments.

A breakdown of Haag duality for entirely different reasons occurs in local
nets generated by massless helicity $h\geq 1~{}$free field strengths. This
”topological duality violation” was mentioned in section 2 where its relation
to the Aharonov-Bohm effect and linking numbers was explained. It is lost in
the positivity-violating gauge theoretical setting which cannot distinguish
between the Haag duality and the somewhat coarser Einstein causality.

In his book on Local Quantum Physics Haag proposes an interesting extension
from the established one-fold duality for observable algebras to causally
separated two double cone localized algebras. In fact he explores the
possibility of the existence of a homomorphism from the orthocomplemented
lattice of causally complete regions in Minkowski space into that of von
Neumann algebras of observables ([2], Tentative Postulate 4.2.1). For a region
which consists of two spacelike separated double cones $K_{1},K_{2}$ this
requires $\mathcal{A}(K_{1}\vee
K_{2})=\mathcal{A}(K_{1})\vee\mathcal{A}(K_{2}).$

Haag notices with a certain amount of disappointment that duality is violated
for doubly localized observable algebras associated to the conserved currents
of free fields. This follows from the existence of a pair
$\psi(f)\psi(g)^{\ast}~{}$with $supp~{}f\subset K_{1},~{}supp~{}g\subset
K_{2}~{}$which commutes with $\mathcal{A}(K_{1}\vee K_{2})~{}$but is not in
$\mathcal{A}(K_{1})\vee\mathcal{A}(K_{2}).~{}$He viewed this as a shortcoming
of free fields which he expected to disappear in the presence of interactions.
He uses the idea of a ”gauge bridge” between $\psi(f)$ and$~{}\psi(g)^{\ast}$
as a hint that a future positivity-maintaining replacement of gauge theory may
satisfy this strengthened form of causality (”Haag duality”) may.

The string-bridges of SLFT do precisely this i.e. they prevent that Haag