# sQFT – an autonomous explanation of the interactions of quantum particles

K.-H. Rehren1, L.T. Cardoso2, C. Gass3,
J.M. Gracia-Bondía4,5, B. Schroer6, J.C. Várilly7<EMAIL_ADDRESS>(corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
1 Institut für Theoretische Physik, Georg-August-Universität Göttingen, 37077
Göttingen, Germany.
2 Coordenadoria Acadêmica, Universidade Federal de Santa Maria, Cachoeira do
Sul, RS 96503-205, Brazil.
3 Department of Mathematical Methods in Physics, University of Warsaw, 02-093
Warszawa, Poland.
4 Centro de Astropartículas y Física de Altas Energías, Universidad de
Zaragoza, 50009 Zaragoza, Spain.
5 Laboratorio de Física Teórica y Computacional, Universidad de Costa Rica,
San Pedro 11501, Costa Rica.
6 Institut für Theoretische Physik, Freie Universität Berlin, 14195 Berlin,
Germany.
7 Escuela de Matemática, Universidad de Costa Rica, San Pedro 11501, Costa
Rica

###### Abstract

Successful applications of a conceptually novel setup of Quantum Field Theory,
that accounts for all subtheories of the Standard Model (QED, Electroweak
Interaction and Higgs, Yang-Mills and QCD) and beyond (Helicity 2), call for a
perspective view in a broader conceptual context. The setting is “autonomous”
in the sense of being intrinsically quantum. Its principles are: Hilbert
space, Poincaré symmetry and causality. Its free quantum fields are obtained
from Wigner’s unitary representations of the Poincaré group, with only
physical and observable degrees of freedom. A “quantization” of an
“underlying” classical theory is not needed. It allows renormalizable
perturbation theory with interactions whose detailed structure, and in some
cases even the particle content, is predicted by internal consistency. The
results confirm and extend observable predictions for the interactions of the
SM without assuming a “principle” of gauge invariance.

##  1 The autonomous quantum approach to QFT

We review “string-localized quantum field theory (sQFT)” by contrasting its
conceptual ambitions with standard approaches to perturbative QFT. In later
sections, we shall explain in a largely untechnical way how sQFT proceeds, and
report the results it has produced by now in the context of the Standard Model
of particles (SM).

#### Conceptual issues of QFT.

Standard textbook approaches to QFT start from classical field theory plus
“quantization”. At the level of free fields, the strategy is to use a
classical free Lagrangian and “derive” canonical commutation relations that
have to be fulfilled in a covariant way. This initial step is beset with
problems: In the case of the Dirac field, the canonically conjugate momenta
are functions of the field itself; and one has to replace equal-time CCR by
CAR ad hoc. In the case of the Maxwell field, one has to take the unobservable
gauge potential as “canonical” variables, one has to change the dynamics by a
“gauge fixing term” in order to get a canonically conjugate momentum for all
components of the potential, and one gets CCR that can only be represented on
a Fock space with indefinite inner product (a Krein space), which contains
unphysical “photon states”. We have learned how to circumvent such trouble by
ingenious tricks (Dirac’s theory of constraints, the Gupta-Bleuler method to
eliminate unphysical states with negative or zero norm square, and the normal-
ordering prescription to make the vacuum energy finite). But we would rather
prefer to avoid this first step altogether.

When it comes to adding an interaction Lagrangian, one usually invokes gauge
invariance, a postulate on unobservable gauge-dependent quantities.111 In a
famous panel dicussion among P.A.M. Dirac, S. Ferrara, H. Kleinert, A. Martin,
E.P. Wigner, C.N. Yang, and A. Zichichi, recorded in [38], it was consented
that gauge theory does not refer to a “symmetry” of Nature. Its main role is
the selection of renormalizable interactions that lie at the basis of the
Standard Model of particles, without constituting a symmetry of the final
quantum theory. In order to make the previous quantization methods survive the
interaction, in many cases of interest one has to introduce ghost fields and
promote the Gupta-Bleuler condition to BRST invariance.

Apart from its need of unphysical degrees of freedom in gauge theories, one
may object against “quantization” at a more fundamental level. Classical
physics is a limiting regime in a quantum world, applicable in situations
where the uncertainty bound is largely exceeded. A prescription to “read back”
the true theory from its limit precisely in the regime where the latter is not
applicable, seems a hazardous endeavour. It goes a long way towards explaining
the “need” for artful but artificial devices like the ones just exposed.

The autonomous quantum approach, advocated in this paper, instead starts from
quantum principles: Hilbert space, unitary Poincaré symmetry and Causality. In
the words of P. Jordan in 1929 [18]: it shows how QFT can be formulated
“without the crutches of classical correspondences”. Along the way, it will
show that the Wightman axioms (postulating causality or anti-causality for all
Wightman fields, as an extrapolation from free fields) fall too short for
interacting charged fields. sQFT tells us how interacting charged quantum
fields behave instead.

The autonomous approach proceeds as follows. One or several of Wigner’s
unitary representations of the Poincaré group [35] for one’s favorite physical
particles give rise to a Fock space, which is a Hilbert space. The generators
of the Poincaré symmetry of the one-particle space lift to the Fock space, and
their lifts can be expressed as integrals over densities that are already
normal-ordered from the outset. Covariant free fields may be defined with
Weinberg’s method [34], intertwining momentum space Fock space operators with
position space local fields, and their linear equations of motion follow from
the construction. Their two-point functions and commutators can be computed
directly, and causality is automatic for fields associated with particles of
integer spin. (In the massless case, one has to combine helicities $+h$ and
$-h$.)

The spin-statistics theorem is the observation that in the case of half-
integer spin, causality would conflict with the positivity of two-point
functions viewed as inner products of one-particle state vectors, and the best
one can achieve in this case is anti-locality – and this forces one to use a
fermionic Fock space. Equal time (anti-)commutation relations are
consequences, and they may not be “canonical” at all. Because anti-locality
means maximal violation of causality, fermionic free fields are not observable
fields. Their role is to create fermionic particle states, and to allow the
construction of observable conserved currents. Their fate when an interaction
is turned on has to be analyzed case by case.

For interactions, one adds an interaction Hamiltonian density, and appeals to
time-dependent perturbation theory with a spacetime cutoff function $\chi(x)$
to begin with. In the end the adiabatic limit $\chi(x)\to 1$ has to be taken.
This limit may be a tough problem, corresponding to IR problems popping up
when one works without a cutoff function from the outset. The mathematically
rigorous setup for perturbation theory with a cutoff, in which causality is
under control, is known as “causal perturbation theory”, or as the “Epstein-
Glaser renormalization scheme (EG)” [11].

Common to both approaches is the resulting formula for the S-matrix in the
vacuum sector

$\displaystyle S=Te^{i\int d^{4}x\,L_{\rm int}(x)},$ (1.1)

where $L_{\rm int}=-H_{\rm int}$ is a Wick polynomial in free fields. The
S-matrix has to be computed perturbatively, by expansion of the exponential.
In the EG approach, one replaces the coupling constant $g$ in $L_{\rm int}(x)$
by the test function $g(x)=g\chi(x)$, and regards the resulting “local
S-matrix” $S[g]$ as an operator-valued distribution with properties that can
be axiomatized (and that may be established by a perturbative construction).

Worth mentioning is the absence of a “free Lagrangian $L_{0}$” in this
formula222Path integral approaches use the “free Lagrangian” only for the
formal definition of a (Euclidean) “Gaussian measure”, the counterpart of the
free Fock space whose inner product is the two-point function in real time.,
which signals that its troublesome use for the mere purpose of “quantization”
can (and should) be avoided altogether if possible. Indeed, the autonomous
approach starts from the free fields, intrinsically defined in terms of the
Wigner representations of the Poincaré group.

In the standard approach, one has to postulate BRST invariance of the
S-matrix, which otherwise cannot be unitary. This is a nontrivial issue,
because $\delta_{\rm BRST}L_{\rm int}$ is (at lowest order) a total derivative
which in general cannot be ignored because time-ordering does not commute with
derivatives, see Sect. 2. This failure causes “obstructions” against BRST-
invariance of the S-matrix, that have to be taken care of. In the form of a
“principle of Perturbative Gauge Invariance (PGI)” [2, 31], the BRST method
was developed to a mature tool to predict the structure of interactions,
compatible with all sectors of the Standard Model. It is expanded in G.
Scharf’s book [31]. The technical (but not the conceptual!) aspects of this
method are in fact a precursor of the autonomous approach, which ultimately
makes the BRST method obsolete altogether.

#### String-localized quantum fields.

On the Wigner Fock space for helicity 1 (photons), there exist fields that one
may oxymoronically call “gauge-invariant gauge potentials”. The latter share
formal properties with gauge potentials, but they do not contain unphysical
modes, see Sect. 2.

This miracle is possible because one admits these potentials to have a weaker
localization than usual: they may be “localized along a string”. A “string” is
a line or a narrow cone extending from $x$ to spacelike infinity, and
“localization” is understood in the sense of commutation relations. The string
can be labelled by some auxiliary parameter $c$ (a function), as will be
explained in Sect. 2. This feature is the origin of the name “string-localized
quantum field theory (sQFT)”. String-localized potentials also exist in the
massive case: they have a better UV behaviour than local massive vector
fields, because the weaker localization tames their vacuum fluctuations. This
allows power-counting renormalizable interactions333We use the term
“renormalizable” in the sense of power-counting throughout. The actual
renormalization of sQFT has not been achieved sofar, but the main result of
[13] indicates that power-counting still is a valid criterion. that are
otherwise forbidden.

The string-localized fields are free fields, used to build interactions

$L_{\rm int}(c)=gL_{1}(c)+\hbox{\large{$\frac{g^{2}}{2}$}}L_{2}(c)+\dots$

as a Wick polynomial (or possibly a power series) on the physical Hilbert
space.

#### How sQFT determines interactions.

(i) Through the string-localized potentials, $L_{\rm int}(c)$ depends on an
auxiliary function $c(e)$, characterizing the “string” which has no direct
physical meaning, see Sect. 2. The point is that the S-matrix must not depend
on this function $c$ (“string-independence”). At lowest order (in $g$), this
implies that when $c$ is varied by a function $\delta c(e)$, then
$\delta_{c}L_{1}(c)$ must be a total derivative:

$\displaystyle\delta_{c}L_{1}(x,c)=\partial_{\mu}Q_{1}^{\mu}(x,c,\delta c)$
(1.2)

by virtue of the equations of motion of the free fields. We call (1.2) an
“$L$-$Q$ pair”. The $L$-$Q$ pair condition selects admissible first-order
interactions, to begin with. They are typically cubic in the free fields. The
first noticeable fact is that all known cubic interactions of the SM are among
them.

Even more, in all cases of sQFT pertaining to SM interactions, there is also a
point-localized interaction density $K_{\rm
int}=gK_{1}+\frac{g^{2}}{2}K_{2}+\dots$ to compare with. When the Hilbert
space where $L_{\rm int}(c)$ is defined is suitable embedded into the possibly
indefinite state space where $K_{\rm int}$ is defined, it holds

$\displaystyle L_{1}(x,c)=K_{1}(x)+\partial_{\mu}V^{\mu}_{1}(x,c).$ (1.3)

We call (1.3) an “$L$-$V$ pair”. It is the point of departure for an analysis
[26, 29] to establish that $L_{\rm int}$ and $K_{\rm int}$ produce the same
S-matrix, see below.

(ii) Once an initial $L$-$Q$ pair (or $L$-$V$ pair) is selected, it turns out
that the higher orders of the S-matrix built with $L_{1}(c)$ will in general
depend on the auxiliary function $c$: there arise “obstructions against
string-independence” at higher order of the perturbative expansion. Their
origin, as well as the strategy to cancel them is quite analogous to the
failure and recuperation of BRST invariance in the PGI approach: even if at
first order (where there is no time-ordering)

$\delta_{c}\int dx\,L_{1}(x,c)=\int dx\,\partial_{\mu}Q_{1}^{\mu}(x,c,\delta
c)=0$

by virtue of (1.2), the second-order contribution

$\displaystyle\delta_{c}\int
dx\,dx^{\prime}\,T[L_{1}(x,c)L_{1}(x^{\prime},c)]=\int
dx\,dx^{\prime}\,T[\partial_{\mu}Q_{1}^{\mu}(x,c,\delta
c)L_{1}(x^{\prime},c)+(x\leftrightarrow x^{\prime})]$ (1.4)

will in general not vanish because the derivatives cannot be taken out of the
time-ordered product. It must be cancelled by a second-order interaction
$L_{2}$, which is only possible if the obstruction has a suitable form. In
that case, it is said to be “resolved” by $L_{2}$, which is called the
“induced” interaction. Restricting the analysis of the obstruction to tree
level, it follows from Wick’s theorem that if $L_{1}$ is cubic in the fields,
then $L_{2}$ will be quartic. The second noticeable fact is that the induced
interactions thus determined coincide with the known quartic interactions of
the SM.

(iii) It may happen that obstructions are non-resolvable. A model with non-
resolvable obstructions would have an S-matrix depending on the auxiliary
string variable $c$ without physical meaning, and must be abandoned as
inconsistent with fundamental principles. The third noticeable fact is that
the non-resolvable obstructions of several first-order interactions can cancel
each other if all cross-channels are considered. We call this a “lock-key
scenario”: Pieces of the SM will be inconsistent, while the whole becomes
consistent. Examples are given in Sect. 3.

(iv) The method to compute induced interactions follows a recursive scheme,
whose systematics has been elucidated in [29]. In principle, it may continue
to arbitrary order. The fourth remarkable fact is that in all cases
investigated so far, the recursion terminates at the power-counting bound for
renormalizability – without renormalizability having been imposed.444The case
of perturbative interactions involving massless particles of helicity 2
(“gravitons”) is an exception, see Sect. 3.5: the lowest orders of induced
interactions, that have been computed so far, reproduce the Einstein-Hilbert
action which is known to be power-counting non-renormalizable already at first
order, and there is no hint for a termination.

#### Interacting fields and observables.

It is worth remembering that interaction densities $L_{\rm int}$ are just the
field-theoretical tool to produce the corresponding S-matrix of interactions
among (asymptotic) particles, by formula (1.1). As long as the S-matrix
computed with string-localized $L_{\rm int}(c)$ is independent of the string,
it appears that one is done.

One should, however, object that a QFT is more than an S-matrix: it is defined
by its interacting fields whose LSZ limit at $t\to\pm\infty$ is described by
that S-matrix. Interacting fields are perturbatively defined by Bogoliubov’s
formula

$\displaystyle\Phi|_{L_{\rm int}}(x)=(Te^{i\int L_{\rm
int}})^{*}(T[\Phi(x)e^{i\int L_{\rm int}}]),$ (1.5)

to be expanded as a series of retarded integrals over free fields. In gauge
theory, interacting charged fields are not BRST invariant and are therefore
not defined on the physical Hilbert space which is defined as the BRST
quotient space (2.2). In the BRST approach, one must be content with an
interacting quantum field theory lacking fields of major interest, that would
generate charged states.

In sQFT, interacting fields can be constructed by the same formula (1.5).
Along with $L_{\rm int}(c)$ being defined on the physical Hilbert space, so
are the interacting fields. The point is, however, that a non-local (string-
localized) interaction density will jeopardize locality of the perturbative
interacting fields. Fortunately, there is a powerful tool (the “magic formula”
(1.6) below). It proceeds by relating the interacting fields with interaction
$L_{\rm int}(c)$ to interacting “dressed” fields with a local interaction
$K_{\rm int}$ as in the standard frameworks. The free “dressed” fields are
much easier to compute than interacting fields, and their localization can be
directly read off. The virtue of the magic formula is that (because $K_{\rm
int}$ is local) the interacting dressed fields have the same localization as
the free dressed fields, and this determines the localization of the
interacting Hilbert space fields $\Phi|_{L_{\rm int}(c)}$ even if $L_{\rm
int}(c)$ is string-localized.

Before we shall explain the working of the “magic formula”, let us pause for a
reflection about the meaning of “localization” in quantum field theory; and
explain how sQFT allows to assess the localization of interacting fields.

Localization in quantum field theory is an algebraic feature – not a geometric
one as the term might suggest (and of course not a matter of just labelling
either). An operator is localized “somewhere” if it commutes with all
observables at spacelike distance from “somewhere”. Thus, localization refers
to causality of the fields, rather than locality of the interaction.

A more precise formulation would be that localization is a relative algebraic
property because it refers to the localization of observables. E.g., in
perturbation theory, interacting fields are represented as power series of
retarded integrals over free fields. One may therefore not even expect them to
be localized relative to the free fields, and Borchers [4] has shown that they
cannot be so if there is nontrivial interaction.

But they can be localized relative to each other. This property selects
interacting observables. Perturbed fields which fail to be localized relative
to the interacting observables may still be present in the theory as non-
observable fields, and they are welcome because they create states carrying
some charge with non-local features [8]. One may call them “charged fields”.
Notice that anti-local fields (even free ones) violate causality in the
strongest possible way. This is why only quadratic expressions like the Dirac
current $j^{\mu}=\,\colon\\!\\!\overline{\psi}\gamma^{\mu}\psi\\!\colon$ have
a chance to qualify as observables, while the Dirac field $\psi$ creates
charged states.

In sQFT, string-localized free fields (notably the “gauge-invariant vector
potentials” mentioned before) arise by integrating a free observable along a
string emanating from $x$. These appear in $L_{\rm int}(c)$, and define the
perturbation theory. They are of little interest for the resulting interacting
observables, but they leave their traces on the interacting charged fields.

#### The “magic formula” settles trouble with non-locality.

A first reaction on sQFT is often the (in general justified) alarm that a non-
local interaction will spoil locality of the interacting fields. We sketch
here how locality is taken care of when there is an $L$-$V$ pair available.

There exists an all-orders analysis [29] starting from an initial $L$-$V$ pair
(1.3) which provides the prerequisites to investigate whether $K_{\rm int}$
and $L_{\rm int}(c)$ produce the same S-matrix (at tree level at all orders).
This indeed turns out to be the case for all interactions of the SM. Then,
even if $K_{\rm int}$ may be non-renormalizable, e.g., because it contains
couplings of massive vector bosons, the S-matrix $Te^{i\int dx\,K_{\rm
int}(x)}$ will be unitary and renormalizable – because $Te^{i\int dx\,L_{\rm
int}(x,c)}$ is. Conversely, the latter will be string-independent because the
former is.

An offspring of the $L$-$V$ analysis is the “magic formula”

$\displaystyle\Phi\big{|}_{L_{\rm int}(c)}(x)=\Phi_{[g]}\big{|}_{K_{\rm
int}}(x),$ (1.6)

where the formula to compute the “dressed fields” $\Phi_{[g]}$ from the
$L$-$V$ pair is given in [29]. This formula may transfer some of the degrees
of freedom of the string-localized potentials onto the unperturbed field
$\Phi$. The dressed fields may therefore depend on $c$ and are manifestly
relatively local or string-localized relative to the observable free fields
(the latter can be characterized by the property that $\Phi^{\rm obs}_{[g]}$
is local – e.g., if $\Phi^{\rm obs}_{[g]}=\Phi^{\rm obs}$). By Borchers’
result [4], this means that $\Phi_{[g]}$ are still not interacting – but their
localization can be just read off. Then, because $K_{\rm int}$ is a local
interaction density, standard results of causal perturbation theory assure
that $\Phi_{[g]}\big{|}_{K_{\rm int}}$ have the same relative localizations
among each other as $\Phi_{[g]}$. Therefore, $\Phi|_{L_{\rm int}}$ is
relatively local resp. string-localized to the interacting observables. This
is all one needs to know for the physical discrimination between “observable”
or “charged” fields.

Some dressed fields, and consequently also the corresponding interacting
fields are indeed local (and string-independent), and qualify as the local
observables of the perturbative QFT. Some others (like the Dirac field of QED)
become dynamically string-localized.

This bug is a feature: First, it is unproblematic because charged fields are
not observables. Second, it is even to be expected in theories like QED,
because charges can be measured and have effects at infinite spacelike
distance by the Gauss Law. Notice that the Gauss Law does not violate
causality, as the “creation of a charge” is not a physical process. The
effects at spacelike infinity record the behaviour of the charges in the past.

In massive theories, the need to admit string-localized interacting fields was
anticipated by an analysis (based on the axioms of algebraic quantum field
theory referring exclusively to local observables) [8] of the causal support
where charged superselection sectors are distinguishable from the vacuum
sector. The result can be interpreted as a statement on putative fields that
would create such sectors from the vacuum, and conforms exactly with the
weaker localization of the interacting fields in sQFT.

##  2 String-localized quantum field theory

We give here several complementary motivations for string-localized QFT. The
first one starts from gauge theory and BRST and arises as an attempt to
eliminate all obstructions against BRST invariance, as would appear in the PGI
approach, from the start. Only the next two are truly in the autonomous
spirit. They are enhancements towards applications in the SM of the original
motivation [27] which was rooted in Algebraic QFT and “modular localization”.
This background will also be explained briefly below.

For simplicity, and in order to best exhibit the complementarity, we shall
concentrate on the prototypical case of QED, with comments on other cases
where appropriate.

#### String-localized quantum fields as the “reverse side of BRST”.

If one comes from gauge theory, one would start with a classical interaction
density like

$\displaystyle L_{1}(x)=A_{\mu}(x)j^{\mu}(x)$ (2.1)

for QED. After covariant quantization, the field $A_{\mu}(x)$ is defined on a
Fock space that contains states of negative norm square (“timelike photons”)
and states of zero norm (“longitudinal photons”). One eliminates these
unphysical states by the Gupta-Bleuler method, or equivalently by the more
general BRST method, which is the method of choice in non-abelian gauge
theories. The latter extends the indefinite Fock space by another indefinite
Fock space of “ghost” particles. On the extended Fock space $\mathcal{F}$
there is defined a nilpotent fermionic BRST operator $Q$ with positive-
semidefinite kernel. Its graded commutator $[Q,\cdot]_{\mp}$ defines the BRST
variation $\delta_{\rm BRST}=:s$. Physical states (in QED: photons with only
the two physical helicities, and no ghosts) are those annihilated by $Q$
modulo states in the range of $Q$, which are of norm zero. In mathematical
language, the physical Hilbert space is the quotient space (“cohomology”)

$\displaystyle\mathcal{H}=\mathrm{Ker}(Q)/\mathrm{Ran}(Q).$ (2.2)

In order to be defined on this Hilbert space, observables must be BRST-
invariant ($s(X)=0$), including the S-matrix.

One finds that

$\displaystyle s(A_{\mu})=\partial_{\mu}u,$ (2.3)

where $u$ is a ghost field. Consequently,

$\displaystyle
s(L_{1})=\partial_{\mu}uj^{\mu}=\partial_{\mu}(uj^{\mu})=:\partial_{\mu}P^{\mu}$
(2.4)

is a total derivative. In particular, $L_{1}$ itself is not BRST invariant –
only its classical action integral $\int d^{4}x\,L_{1}(x)$ is BRST invariant.
But then the same is not automatically true for the perturbative S-matrix
(1.1) because time-ordering does not commute with derivatives. E.g.,

$\displaystyle
s(T[L_{1}(x)L_{1}(x^{\prime})])=T[\partial_{\mu}P_{1}^{\mu}(x)L_{1}(x^{\prime})]+(x\leftrightarrow
x^{\prime})$ (2.5)

is in general not a derivative, in which case the integral will not vanish.
This failure is called an “obstruction” against BRST invariance.

Fortunately, QED is particularly good-natured in that this obstruction
actually does not occur. (It is particularly bad-natured by its IR problems,
though. The IR aspects of QED will be addressed separately in Sect. 3.1.) But
the obstruction does occur in many other cases of interest, including the non-
abelian case if $L_{1}$ is only the minimal interaction and the cubic part of
the (self-)interaction. The quartic part of the self-interaction precisely
cancels the obstruction. In [2, 31], the attitude is taken (and proven) that
this cancellation mechanism is a justification for gauge invariant Lagrangians
even if one does not assume gauge invariance from the start.

How would string-localized quantum fields enter the scene?

Because of (2.3) and (2.4), one seeks a field $\phi$ whose BRST variation
would be

$\displaystyle s(\phi)=-u.$ (2.6)

Then, one could replace $A_{\mu}$ by the gauge-equivalent but BRST-invariant
field $A_{\mu}+\partial_{\mu}\phi$, and $L_{1}$ by the BRST-invariant
interaction density $q(A_{\mu}+\partial_{\mu}\phi)j^{\mu}$.

Such a field $\phi$ is usually not “in the list” of BRST variations. But one
can easily produce it: the string-localized integral

$\displaystyle\phi(x,e):=\int_{x}^{\infty}dx^{\prime\mu}\,A_{\mu}(x^{\prime})=\int_{0}^{\infty}ds\,e^{\mu}A_{\mu}(x+se)$
(2.7)

does the job, with any string direction $e$.

One notices that (by virtue of the homogeneous Maxwell equations)

$\displaystyle
A_{\mu}(x,e):=A_{\mu}(x)+\partial_{\mu}\phi(x,e)=\int_{0}^{\infty}ds\,F_{\mu\nu}(x+se)e^{\nu},$
(2.8)

which makes the BRST-invariance of $A_{\mu}(x,e)$ more transparent: it is just
a string integral over the BRST-invariant field strength. (2.8) shows that
$A_{\mu}(x,e)$ creates and annihilates only physical photon states with
helicity $\pm 1$, both manifestly (because $F$ does so) and by counting
degrees of freedem (namely, (2.8) implies
$\partial^{\mu}A_{\mu}(x,e)=e^{\mu}A_{\mu}(x,e)=0$).

Because of the distributional nature of quantum fields, one must smear the
string direction $e$ in (2.7), (2.8) with a suitable smooth function $c(e)$
and a measure $d\sigma(e)$, thus defining $\phi(x,c)$ and $A_{\mu}(x,c)$.555In
contrast to the smearing in $x$, we consider the smearing of the string
variable as a label of the field $A(\cdot,c)$. If $\int d\sigma(e)\,c(e)=1$
(“unit weight”), then

$\displaystyle A_{\mu}(x,c)=A_{\mu}(x)+\partial_{\mu}\phi(x,c)$ (2.9)

is still a potential for $F_{\mu\nu}$:

$\displaystyle\partial_{\mu}A_{\nu}(x,c)-\partial_{\nu}A_{\mu}(x,c)=F_{\mu\nu}(x).$
(2.10)

One may then work with the BRST-invariant interaction density

$\displaystyle
L_{1}(x,c)=A_{\mu}(x,c)j^{\mu}(x)=A_{\mu}(x)j^{\mu}(x)+\partial_{\mu}\big{(}\phi(x,c)j^{\mu}(x)\big{)}.$
(2.11)

(2.11) is an $L$-$V$ pair for QED. The S-matrix computed with $L_{1}(c)$ will
be manifestly BRST-invariant, In the case of QED, one can establish that the
derivative term does not alter the S-matrix. In more general cases, one may
have to add higher-order interactions, as indicated after (1.4).

So, it seems that nothing is gained: instead of obstructions against BRST-
invariance, there arise obstructions against string-independence, that must be
resolvable. But on the contrary:

Because $L_{\rm int}(c)=gL_{1}(c)+\frac{g^{2}}{2}L_{2}(c)+\dots$ is BRST-
invariant, it is defined on the physical Hilbert space (2.2), and so will be
the S-matrix and the interacting fields (1.5). One has a (perturbative) QFT on
a Hilbert space from the outset, including charged fields that can generate
charged states from the vacuum, whereas interacting charged fields in the BRST
approach are not defined on the BRST Hilbert space. See the next paragraph.

#### The autonomous motivation for massless string-localized quantum fields.

One can spare the detour through BRST by starting directly from the integral
over $F$ in (2.8)

$\displaystyle A_{\mu}(x,c):=\int
d\sigma(e)\,c(e)\int_{0}^{\infty}ds\,F_{\mu\nu}(x+se)e^{\nu}$ (2.12)

as an autonomous definition of the string-localized potential satisfying
(2.10), where $F$ is defined on the Wigner Fock space. As in (2.11), the
interaction is

$\displaystyle L_{1}(x,c)=A_{\mu}(x,c)j^{\mu}(x),$ (2.13)

with the distinction that there is no split (2.9) making it an $L$-$V$ pair,
because neither $A$ nor $\phi$ are defined on the Wigner Fock space. Instead,
one has the $L$-$Q$ pair of QED

$\displaystyle\delta_{c}(L_{1}(x,c))=\partial_{\mu}w(x,\delta
c)j^{\mu}(x)=\partial_{\mu}\big{(}w(x,\delta
c)j^{\mu}(x)\big{)}=:\partial_{\mu}Q_{1}^{\mu}(x,\delta c).$ (2.14)

Namely, one can show that $\delta_{c}(A_{\mu}(x,c))$ is the derivative of a
quantity defined on the Wigner Fock space, that we call $w(x,\delta
c)$.666Because $c$ must have unit weight, a variation must be of the form
$\delta c(e)=\partial^{e}_{\kappa}b^{\kappa}(e)$. The field $w(\delta c)$ is
then computed by integrating $\partial^{e}_{\kappa}$ by parts onto the
$s$-integral in (2.8) where it produces a derivative $\partial^{x}_{\kappa}$,
and using the homogeneous Maxwell equations. The resulting precise form of $w$
is not of interest here. The field $Q_{1}$ contains a factor $w$, and may in
more general models than QED also involve other string-localized fields.

The next steps are analogous to the PGI approach [31], with the string
variation $\delta_{c}$ instead of the BRST variation: Perturbation theory with
an interaction $L_{1}(c)$ will in general exhibit an obstruction against
string-independence. In order to resolve the obstruction, one seeks a second-
order interaction $L_{2}$ that makes

$\displaystyle\delta_{c}\big{(}Te^{i\int
dx\,(gL_{1}+\frac{g^{2}}{2}L_{2}+\dots)}\big{)}$ $\displaystyle\\!=\\!$
$\displaystyle ig\int dx\,\delta_{c}(L_{1}(x,c))$
$\displaystyle+\hbox{\large{$\frac{(ig)^{2}}{2}$}}\Big{[}\iint
dx\,dx^{\prime}\,\delta_{c}\big{(}T[L_{1}(x,c)L_{1}(x^{\prime},c)]\big{)}-i\int
dx\,\delta_{c}(L_{2}(x,c))\Big{]}+\dots$

vanish. The first-order term vanishes thanks to (1.2). The condition that the
second-order term vanishes at tree level is conveniently formulated as

$\displaystyle O^{(2)}(x,x^{\prime})\big{|}_{\rm
tree}\stackrel{{\scriptstyle!}}{{=}}\delta_{c}(L_{2}(x,c))\cdot
i\delta(x-x^{\prime})+i\mathfrak{S}_{2}\partial^{x}_{\mu}Q_{2}^{\mu}(x;x^{\prime},c,\delta
c)$ (2.16)

where $\mathfrak{S}_{2}$ is the symmetrization in two arguments, and

$\displaystyle
O^{(2)}(x,x^{\prime}):=2\mathfrak{S}_{2}\big{(}T[\partial_{\mu}Q_{1}^{\mu}(x,c,\delta
c)L_{1}(x^{\prime},c)]-\partial^{x}_{\mu}T[Q_{1}^{\mu}(x,c,\delta
c)L_{1}(x^{\prime},c)]\big{)}.$ (2.17)

(2.17) is analogous (with $Q_{1}$ replacing a conserved current) to the
violation of a Ward identity. By Wick’s theorem, its tree-level part can be
computed in terms of differences of time-ordered two-point functions
(propagators)

$\displaystyle O_{\mu}(\varphi(x),\chi(x^{\prime})):=\langle
T[\partial_{\mu}\varphi(x)\chi(x^{\prime})]\rangle-\partial_{\mu}\langle
T[\varphi(x)\chi(x^{\prime})]\rangle$ (2.18)

of the linear free fields of the model. The subtractions of explicit
derivative terms bring the advantage that (2.18) and the numerical
distributions in (2.17) automatically exhibit delta functions and string-
integrated delta functions.

The existence of $L_{2}$ and $Q_{2}$ resolving the obstruction (2.16) is a
nontrivial feature of a model. In particular, all string-localized delta
functions must be part of the derivative term $\partial Q_{2}$. If non-
resolvable obstructions arise, the model must be discarded, unless they can be
cancelled (in the “lock-key scenario”) by the inclusion of further first-order
interactions. In general, one may have to proceed to higher orders.

The interacting fields $\Phi|_{L_{\rm int}(c)}$ are computed with Bogoliubov’s
formula (1.5). In order to assess their localization with the help of the
“magic formula” (1.6), one embeds the Wigner Fock space into a suitable
indefinite state space777 In QED, the embedding can be made in such a way that
the split (2.9) holds on a positive-definite subspace with a third degree of
freedom besides the two polarizations of the photon. Upon interaction, this
extra degree of freedom is transferred onto the interacting Dirac field – a
feature whose importance has been stressed in [7]. (The photon continues to
have two physical states.), where there exists an $L$-$V$ pair with a local
interaction $K_{\rm int}$. Then, $\Phi|_{L_{\rm int}(c)}$ are defined on the
Hilbert space, but in general localized in the cone spanned by the strings $e$
in the support of $c$.

In QED, the interacting Maxwell field turns out to be local, while the Dirac
field is localized along the string, see Sect. 3.1. This feature resolves the
conflict between the Gauss Law (the total electric charge equals the total
electric flux at spacelike infinity), the Noether property (the total electric
charge operator generates the $U(1)$ symmetry of the charged field), and
locality (the electric flux at infinity should commute with the Dirac field at
$x=0$). There is no such conflict when the last property is dynamically
invalidated. sQFT produces this effect intrinsically.

#### The autonomous motivation for massive string-localized quantum fields.

A second beneficial feature of the formula (2.8) arises with massive vector
fields. The Proca field in the Wigner representation has scaling dimension 2:
in its two-point function and propagator, $-\eta_{\mu\nu}$ is replaced by
$-(\eta_{\mu\nu}+m^{-2}p_{\mu}p_{\nu})$. The momentum-dependent term secures
positivity but increases the UV scaling dimension, which is a measure for the
size of vacuum fluctuations. This jeopardizes the use of the Proca field in
perturbation theory: the minimal coupling to a current has dimension 5 beyond
the renormalizability bound.

One way to deal with this is to just drop the term $m^{-2}p_{\mu}p_{\nu}$
which brings down the dimension to $1$. The resulting field is a “massive
gauge field”, whose two-point function violates positivity. One then needs
BRST to return to a Hilbert space. BRST requires gauge invariance, but the
mass violates gauge invariance. So the usual method of choice is to replace
the massive gauge field by a massless one and invoke the Higgs mechanism to
make it “behave as if it were massive”.

Alternatively (see [31]), one may as well introduce a (positive) Stückelberg
field which allows to adapt the BRST method without gauge invariance and
ghosts. Also in this case, the BRST variation of $L_{1}$ is a derivative,
causing obstructions of the S-matrix, which can be resolved only if the vector
potential is coupled to a scalar field with the quartic interaction of the
Higgs field.

The autonomous approach, in contrast, secures power-counting renormalizability
by string-localization. The field strength of the Proca field of dimension 2
also has dimension 2 because the exterior derivative kills the “dangerous”
momentum-dependent term in the two-point function. Then, because the
subsequent integration decreases the dimension, the string-integral (2.12)
with the massive Proca field strength (instead of the Maxwell field strength
$F$) is a massive vector potential of dimension 1 on the Wigner Fock space of
the Proca field. When this field is coupled to a current, the
renormalizability bound is respected.

For more on this treatment of the massive vector bosons of the electroweak
interaction, including its ensueing prediction of chirality and the shape of
the Higgs self-coupling, see Sect. 3.4.

#### Modular localization.

Localization in QFT is an algebraic feature: observables at spacelike
separation must commute with each other. Usually, one constructs free fields
$\phi(x)$ whose commutator function is manifestly local. Upon perturbation
with local scalar interactions, the relative localization is preserved.

In these approaches, the localization of an operator is encoded in the test
function with which a field $\Phi(x)$ is smeared. In contrast, “modular
localization” is a construction of algebras of local observables that are
distinguished by the relevant causal commutativity, without specific
commutation relations when the separation is not spacelike. It proceeds with
Modular Theory.

Modular Theory is an abstract theory about von Neumann algebras $M$ on a
Hilbert space with cyclic and separating vectors $\Omega\in\mathcal{H}$. From
such a pair $(M,\Omega)$ it allows to extract its “modular data”: a one-
parameter group of automorphisms of $M$ and an anti-linear conjugation $j$
taking $M$ to its commutant $M^{\prime}$. These data have functorial
properties which can be exploited for manifold applications of Modular Theory
in QFT (by encoding locality in terms of commuting algebras), see [5].

“Modular localization” also starts from the Fock space over unitary Wigner
representations. On this Hilbert space, one can construct von Neumann algebras
$M$ with the vacuum as a common cyclic and separating vector, in such a way
that their modular data coincide with boost subgroups of the Poincaré group
and PCT transformations [3]. (For the latter to be possible, one must combine
the massless representations of helicities $+h$ and $-h$.)

Exploiting the correspondence between wedge regions of Minkowski spacetime and
boost subgroups that preserve the regions, together with the functorial
properties of modular data allow to consistently define algebras of local
observables in wedge regions. Namely, this definition complies with the
inclusion and commutation properties that are required for such an
interpretation. In other words, the “localization” arises from Poincaré
symmetry and modular data. Algebras of local observables in smaller regions
can be defined as intersections of algebras for all wedge regions that contain
the smaller regions.

Modular localization has not least opened the way to novel non-Lagrangian
constructions of QFT models with interactions (in two dimensions sofar), by
deformations of von Neumann algebras [20].

The method applies as well to the “infinite spin representations” of the
Wigner classification, for which local free fields do not exist [37]. Only in
the last step, the intersections of wedge algebras defining putative algebras
of local observables in bounded regions turn out to be trivial. The smallest
intersections of wedge regions that admit nontrivial local observables are
spacelike cones extending to infinity [27, 21].

This construction suggests that free fields for the infinite spin
representations exist despite [37], provided they are allowed to be localized
in cones. Indeed, the authors give concrete examples for such fields in [27].
They then notice that constructions like (2.12) are possible for every finite
spin and helicity, and already observe their possible usefulness for
perturbation theory, as outlined in the previous motivations.

The present paper intends to show that sQFT lives up to these expectations.

#### String-localized quantum field theory in action.

In Sect. 3 we shall look at concrete realizations of sQFT in the Standard
Model of elementary particles.

A main emphasis will be on the fact that interactions are predicted by the
need to resolve obstructions, i.e., they are ultimately consequences of the
underlying fundamental principles of quantum field theory: Positivity,
covariance and locality, as explained in Sect. 1. The strategy is always to
find first-order (cubic) interactions complying with the condition of string-
independence (1.2), and study their obstructions to determine the higher
interactions that make the S-matrix string-independent by “resolving the
obstructions” as in (2.16).

Since string-independence is a necessary condition, it is legitimate to study
the obstructions and their resolution only at tree level. Higher-order
obstructions at tree level can always be reduced to expressions generalizing
the form (2.17), which in turn can be computed, with the help of Wick’s
theorem, in terms of propagators of the linear free fields of the model. The
systematics at higher orders is fairly simple when one starts from $L$-$Q$
pairs, and more involved with $L$-$V$ pairs, see [29] for details.

The interactions determined by tree-level string-independence are then the
starting point for the UV renormalization of the loop contributions, as in all
other approaches. The power-counting bound being satisfied for $L_{\rm
int}(c)$ is of course instrumental for this step.

The recurrent “lock-key scenario” has already been mentioned: The obstructions
of certain first-order interactions cannot be resolved separately. One has to
extend the model by other first-order interactions whose obstructions cancel
each other. E.g., the minimal interactions in non-abelian models require the
cubic self-interactions of the vector bosons; and the non-abelian self-
interactions of massive vector bosons require interactions with a Higgs field.
Weak interactions of massive leptons require also Yukawa couplings to the
Higgs field. Notice that these interactions are not added “in order to make
massless particles massive”, but they are required by consistency of
interactions involving massive particles from the outset.

In this way, all the interactions otherwise known from gauge theory are
accounted for. The resulting orthodox point of view of sQFT is that one should
abandon “gauge invariance” altogether as a physical principle (not least
because it exclusively refers to unobservable entities). However, we thank D.
Buchholz for the comment that also imaginary numbers are “not real”, yet they
simplify life enormously. This is true beyond any doubt, and the same seems to
be true for gauge theory methods as compared to sQFT (even if ghost fields may
also be technically awkward). But the usefulness of gauge theory should not be
mistaken as a “principle of Nature”.

##  3 Examples and achievements so far

We give a brief overview of the manifold achievements of sQFT (some of them
work in progress). Their relevance for the Standard Model of elementary
particles will become apparent.

We shall present the relevant $L$-$Q$ pairs and some of the resulting
obstructions. The treatment of $L$-$V$ pairs is technically more intricate,
cf. [30], and will not be presented here (except for some results concerning
dressed fields).

For detailed accounts, we refer to the cited literature.

###  3.1 QED

The initial $L$-$Q$ and $L$-$V$ pairs of QED were given in (2.14), (2.11)
above. It turns out that obstructions of the S-matrix, as illustrated at
second order in (2.17), vanish at all orders, and no induced higher-order
interaction is needed.

Among the interacting fields, the Maxwell field tensor and the Dirac current
remain local at all orders.

On the other hand, the dressed Dirac field is easily computed and turns out to
be string-localized, so that also the interacting Dirac field is dynamically
string-localized. The dressed Dirac field has the (superficially) simple form

$\displaystyle\psi_{[g]}(x,c)=\,\colon\\!\\!e^{ig\phi(x,c)}\\!\colon\psi(x),$
(3.1)

where $\phi(x,c)$ was defined in (2.9). The function $c(e)$ must be supported
within the unit sphere of a spacelike plane.888The smearing with $c(e)$ is
needed because of the distributional nature of quantum fields. It must be
restricted to a spacelike plane in order not to spoil positivity. Quantities
of the form (3.1) were previously considered by pioneers of QED [19, 9, 22] as
classical expressions with the motive to quantize only gauge invariant
quantities.

However, what looks like an innocent classical gauge transformation, has
drastic consequences in quantum theory: the exponential of the infrared
divergent quantum field $\phi$ turns the free Wigner Dirac field with a sharp
mass into an infrafield which admits no mass eigenstates: its spectrum is only
bounded from below. This spectral manifestation of the long-distance behaviour
of QED was anticipated in a two-dimensional model [32], and later recognized
as a consequence of the Gauss Law of QED in four dimensions [6].

Steinmann [33] also used quantum expressions of the form (3.1) as a starting
point to formulate perturbation theory. His idea is very close to ours, with
the distinction that he chose (3.1) (not least because it is gauge-invariant),
while sQFT derives it.

The quantum field $\phi(c)$ in the exponent (3.1) is IR singular because of
the integration over the massless vector potential involved in (2.7). Its
exponential can be defined non-perturbatively with the help of an IR
regularization [25]; and one finds that the string-smearing function $c(e)$
aquires a physical meaning: it describes the profile of the asymptotic
electric flux density at infinity in the direction $e$, measured in states
created from the vacuum by the field (3.1). Loosely speaking, it is the “shape
of the photon cloud” of an electron.

This profile is not necessarily uniform in all directions (because physically
it depends on the trajectory of the electron in the past), and it can be
“engineered” by choosing the function $c(e)$ of unit weight.

States generated by the dressed Dirac field are orthogonal to the Fock space
of the free fields because the coherent state created by the exponential
factor has infinite particle number. In fact, the cloud function $c$ defines a
discrete superselection sector structure on an uncountably extended Hilbert
space such that states with different $c$ belong to different sectors. More
precisely, for states with several electrons and positrons created by fields
with different $c$, the sum (resp. difference) of their cloud functions is
superselected.

While this is manifestly true for the dressed fields, the uncountable
superselection structure survives for the interacting Dirac fields, but seems
to be “dynamically deformed” by the interaction $K_{\rm int}$ in (1.6) [25].

It was stressed in [7] that it is impossible to satisfy the Gauss Law with
only two photon degrees of freedom, see Footnote 7. Indeed, unlike the
potential $A_{\mu}(c)$, the field $\phi(c)$ is not defined on the photon Fock
space. By restricting the support of $c(e)$ to a spacelike plane, it remains
positive but contributes one additional physical degree of freedom which is
then transferred as a dressing to the Dirac field (3.1). It is the exponential
dressing factor that turns the free Dirac field into an infrafield. Notice
that the field $\phi(c)$ itself is not part of the interacting theory because
of its IR singularity.

###  3.2 Yang-Mills

A cubic $L$-$Q$ pair involving several massless vector potentials
$A^{a}_{\mu}(c)$ is necessarily of the form

$\displaystyle
L_{1}(c)=-\hbox{\large{$\frac{1}{2}$}}\sum_{abc}f_{abc}F^{a\mu\nu}A^{b}_{\mu}(c)A^{c}_{\nu}(c),\qquad
Q_{1}^{\mu}(c)=-\sum_{abc}f_{abc}F^{a\mu\nu}w^{b}(c)A^{c}_{\nu}(c).$ (3.2)

A most general ansatz shows that the numerical coefficients $f_{abc}$ must be
totally antisymmetric (string-independence at first order) and must satisfy
the Jacobi identity (second order) [14]. Thus, they are necessarily the
structure constants of an (unspecified) Lie algebra $\mathfrak{g}$. (The same
result was also found within the PGI approach, see [2].)

If $\mathfrak{g}$ is non-abelian, then there is a second-order obstruction
that can be resolved by the induced quartic interaction

$L_{2}(c)=-\hbox{\large{$\frac{1}{2}$}}\sum_{abcde}f_{abe}f_{cde}A^{a}_{\mu}(c)A^{b}_{\nu}(c)A^{c\mu}(c)A^{d\nu}(c),$

and higher-order interactions do not appear.
$gL_{1}(c)+\frac{g^{2}}{2}L_{2}(c)$ is precisely the string-localized version
of the Yang-Mills interaction.

###  3.3 QCD

One may add to (3.2) an $L$-$Q$ pair of minimal quark-gluon interactions
(copies of (2.14) with colored currents
$j_{a}^{\mu}=\overline{\psi}\gamma^{\mu}\tau_{a}\psi$). The minimal
interaction, if taken separately, has a non-resolvable obstruction at second
order:

$\displaystyle-\sum_{abc}f_{abc}w^{a}(x,c)A^{b}_{\mu}(x,c)j_{c}^{\mu}(x)\cdot\delta(x-x^{\prime}),$
(3.3)

which arises from the violation of the Ward identity for the non-abelian
currents. It is resolved by the same obstruction with the opposite sign
appearing in the cross contributions when the self-interaction (3.2) is
included. In particular, there is no induced interaction in the quark sector.
This is the first appearance of the “lock-key scenario” mentioned in Sect. 2.

The computation of the dressed quark field is much more difficult than in QED,
because of the non-abelianness. We do not have a closed expression like (3.1),
but the first three orders [30] indicate the onset of a path-ordered
exponential

$\displaystyle\psi_{[g]}(x,c)=\,\colon\\!\\!Pe^{ig\phi(x,c)}\\!\colon\cdot\psi(x).$
(3.4)

Recall that $\phi(e)$ is a Lie-algebra-valued line integral over $A$. The
path-ordering orders the Lie-algebra generators, while the field operators are
Wick-ordered. Path-ordering is well defined for a sharp string ($c(e)$
supported in a single direction) and the resulting Wilson operator makes (3.4)
invariant under classical gauge transformations that are trivial at infinity.
For smooth functions $c(e)$ (which are needed because of the distributional
nature of quantum fields), “path-ordering” is not defined in an obvious way,
but (3.4) still enjoys gauge-invariance (to third order). To give a flavour of
the result [30]:

$\displaystyle\psi_{[g]}(x,c)=\,\colon\\!\\!\exp
i\big{[}g\phi(c)+\hbox{\large{$\frac{g^{2}}{2}$}}I_{c}(i[\phi(c),2A+\partial\phi(c)])+\hbox{\large{$\frac{g^{3}}{6}$}}\phi^{(3)}(c)+\dots\big{]}(x)\\!\colon\cdot\psi(x),\quad$
(3.5)

where $A=A(c)-\partial\phi(c)$ is the Lie-algebra-valued local potential in
Feynman gauge999The expression (3.5) is not manifestly positive. But it is
positive, because it was computed with the help of an $L$-$V$ pair on a
positive-definite subspace of the Feynman gauge Fock space, see Footnote 7.,
$i[\cdot,\cdot]$ is the Lie algebra commutator, and $I_{c}(Y)$ is short-hand
for the string integral operation on a vector field

$(I_{c}^{\mu}Y_{\mu})(x):=\int
d\sigma(e)\,c(e)\int_{0}^{\infty}ds\,e_{\mu}Y^{\mu}(x+se),$

appearing already in (2.12), so that $\phi(x,c)=(I_{c}A)(x)$. The third-order
term in (3.5) is

$\phi^{(3)}=3I_{c}(i[I_{c}(i[\phi,2A+\partial\phi]),A+\partial\phi])+I_{c}(i[\phi,i[\phi,3A+\partial\phi]])-\hbox{\large{$\frac{3}{2}$}}i[\phi,I_{c}(i[\phi,2A+\partial\phi])].$

For sharp strings, the nested string integrations of commutators in the
exponent take care of the path-ordering along the string.

As in QED, the string-integrated field $\phi=I_{c}A$ is infrared divergent,
and its appearance in the exponent requires a non-perturbative definition. We
speculate that analytic confinement criteria such as [36] or [12] could then
be established already at the level of the dressed quark field (3.4); but for
this one would need a closed formula at all orders which is presently not
available.

###  3.4 Electroweak interactions

Turning to weak interactions, one may as well generalize the $L$-$Q$ pair
(3.2) to massive vector bosons, including self-coupling terms proportional to
the masses [17]. One may again add $L$-$Q$ pairs for minimal couplings to
vector or axial currents $J^{\mu}=c_{V}j_{a}^{\mu}+c_{A}j_{a}^{5\mu}$. For
massive fermions (because the axial current is not conserved), one also needs
couplings to scalar and pseudoscalar currents with coefficients determined at
second order.

The fermionic sector with the exact field content and masses of the SM
(including the Higgs field) has been analyzed, and its details are reported in
[17]. The vector and axial minimal couplings exhibit an obstruction of the
same form as (3.3) with $j^{\mu}$ replaced by
$(c_{V}^{2}+c_{A}^{2})j_{a}^{\mu}+2c_{V}c_{A}j_{a}^{5\mu}$, which need the
self-interaction to be resolved. The main emphasis is that the maximal
chirality ($c_{A}=\pm c_{V}$) of the weak coupling of the $W$-bosons is a
necessary consequence of this resolution of obstructions. Chirality is of
course well-known as an empirical fact and used as a constitutive feature of
the GSW model, but has no apriori explanation from gauge theory. Also the
double-role of the Weinberg angle specifying both the ratio of weak and
electromagnetic coupling constants and the mass ratio $M_{W}/M_{Z}$, the
electric charges of the particles, and the mass-dependence of the Yukawa
couplings are all consequences of string-independence at first and second
order.

The bosonic sector is presently under investigation [16]. The most important
feature is that the bosonic Higgs couplings are necessary in order to resolve
a non-resolvable obstruction of the self-interaction.

As a “proof of concept”, we have completed the analysis of the simpler Higgs-
Kibble model with massless fermions and vector bosons of equal mass and no
photon [30]. The $L$-$Q$ pair for the self-coupling is (with summation over
$abc$ understood)

$\displaystyle L_{1,{\rm self}}(c)$ $\displaystyle=$
$\displaystyle\hbox{\large{$\frac{1}{2}$}}f_{abc}\big{(}F^{a\mu\nu}A^{b}_{\nu}(c)-m^{2}\phi^{a}(c)B^{b\mu}\big{)}A^{c}_{\mu}(c),$
$\displaystyle Q_{1,\rm self}^{\mu}(c)$ $\displaystyle=$ $\displaystyle
f_{abc}\big{(}F^{a\mu\nu}A^{b}_{\nu}(c)-\hbox{\large{$\frac{1}{2}$}}m^{2}\phi^{a}(c)B^{b\mu}\big{)}w^{c}(c)$

where $B_{\mu}=A_{\mu}(c)-\partial\phi(c)$ is the local Proca field. With
massless fermions there is, besides $C_{A}=\pm c_{V}$, also a solution of the
fermionic obstruction with purely vector currents ($c_{A}=0$).

The self-coupling has a non-resolvable obstruction at second order of the form

$\displaystyle m^{2}\cdot f_{abe}f_{cde}\cdot
w^{a}(x,c)A^{b}_{\mu}(x,c)\phi^{c}(x,c)B^{d\mu}(x).$ (3.6)

This obstruction can be cancelled by another non-resolvable obstruction
arising from the coupling to a scalar field (the Higgs field of mass $m_{H}$)
with $L$-$Q$ pair (summation over $a$ understood)

$\displaystyle L_{1,{\rm Higgs}}(c)$ $\displaystyle=$
$\displaystyle\mu\cdot\big{(}A^{a}_{\mu}(c)B^{a\mu}H+A^{a}_{\mu}(c)\phi^{a}(c)\partial^{\mu}H-\hbox{\large{$\frac{1}{2}$}}m_{H}^{2}\phi^{a}(c)\phi^{a}(c)H\big{)},$
$\displaystyle Q_{1,{\rm Higgs}}(c)$ $\displaystyle=$ $\displaystyle\mu\cdot
w^{a}(c)\big{(}B^{a\mu}H+\phi^{a}(c)\partial^{\mu}H\big{)}.$

Its non-resolvable obstruction has the form

$\displaystyle-4\mu^{2}\cdot(\delta_{ab}\delta_{cd}-\delta_{ad}\delta_{cb})\cdot
w^{a}(x,c)A^{b}_{\mu}(x,c)\phi^{c}(x,c)B^{d,\mu}(x).$ (3.7)

The cancellation of (3.6) against (3.7) requires the structure constants
$f_{abc}=\varepsilon_{abc}$ of $\mathfrak{su}(2)$ and $\mu=\frac{1}{2}m$. (The
overall relative factor $\frac{1}{2}$ as compared to the abelian model [25]
corresponds to the normalization convention of the generators
$\frac{1}{2}\sigma_{a}$ of $\mathfrak{su}(2)$.) Other Lie algebras are
presumably possible when several Higgs fields are admitted.

The resulting induced second-order interaction in the Higgs sector is

$L_{2,{\rm
Higgs}}(c)=\mu^{2}\big{(}-\hbox{\large{$\frac{m_{H}^{2}}{m^{2}}$}}\phi^{a}(c)\phi^{a}(c)H^{2}-\hbox{\large{$\frac{1}{4}$}}(\phi^{a}(c)\phi^{a}(c))^{2}\big{)}.$

Now, there will be two non-resolvable obstructions at third order (multiples
of $\phi^{a}\phi^{a}H^{3}$ and $(\phi^{a}\phi^{a})^{2}H$). They can be
cancelled if one adds $-\frac{1}{4}\frac{m_{H}^{2}}{m}H^{3}$ to $L_{1}$, and
$-\frac{1}{16}\frac{m_{H}^{2}}{m^{2}}H^{4}$ to $L_{2}$. These are precisely
the interaction terms that one also would get from the shifted-symmetric
double-well potential

$V(H)=\hbox{\large{$\frac{m_{H}^{2}}{2}$}}H^{2}\big{(}1+\hbox{\large{$\frac{g}{4m}$}}H\big{)}^{2}.$

(Details of this analysis [30] are almost identical with the abelian model as
reported in [25].)

Notice that sQFT predicts a Higgs particle along with its self-interaction,
but without a “Higgs mechanism” because the vector bosons are massive from the
start.

###  3.5 Helicity 2

In [15] we have pursued an analysis of interactions of massless particles of
helicity $\pm 2$ (“gravitons”) with massive or massless matter of spin $0$,
$\frac{1}{2}$ or $1$. The latter do not need sQFT, but the former do, because
there is no local covariant Hilbert space field to serve as a “metric
deviation” $h_{\mu\nu}$. There is, however, a string-localized field
$h_{\mu\nu}(c)$, built from the “linearized Riemann tensor field” on the Fock
space over the unitary Wigner representations of helicity $\pm 2$, by a
formula analogous to (2.12) (with two string integrations).

The analysis of possible cubic $L$-$Q$ pairs results in unique couplings
$\frac{1}{2}h_{\mu\nu}(c)\Theta^{\mu\nu}_{\rm matter}$ to the conserved matter
stress-energy tensors, and in a unique (up to total derivatives) cubic self-
coupling. The matter couplings separately have non-resolvable obstructions,
which are all cancelled in the cross-channels with the self-coupling. This is
another appearance of the lock-key scenario. More remarkably, the induced
interactions (computed only at second order) coincide with the classical
expansions of the Einstein-Hilbert action and of the standard generally
covariant matter couplings, upon substitution of the classical metric
deviation field $h_{\mu\nu}$ by the quantum field $h_{\mu\nu}(c)$.

General covariance was not assumed. It arises instead as a consequence of the
quantum principles underlying sQFT – a feature that we call “quantum general
covariance”.

Clearly, the expansion is not power-counting renormalizable. Yet, the strong
constraints underlying the resolution of obstructions may even raise the hope
that the renormalized sQFT of gravity has no free parameters besides the
coupling constant.

A similar result was obtained in the PGI setting [10]: Consistency with BRST,
which in turn is needed to secure positivity, is secured by the expansion of
the Einstein-Hilbert self-interaction in all orders.

###  3.6 Beyond the Standard Model?

There are many proposals that “dark matter” might consist of particles of spin
or helicity beyond 2. We want to discuss what could be possible with sQFT in
this respect.

Recall that by the Weinberg-Witten theorem, particles of helicity $2$ or
higher do not admit local and covariant stress-energy tensor fields on a
Hilbert space. This is often taken as an argument that massless matter of
higher helicity cannot couple gravitationally, see a discussion in [28].
Stress-energy tensors of massive matter of higher spin exist but have
increasing UV scaling dimension, leading to “more and more non-renormalizable”
interactions.

On the contrary, there exist string-localized stress-energy tensors
$\Theta^{\mu\nu}_{\rm matter}(c)$ of UV dimension 4 for every integer spin and
helicity [23]. The self-coupling of the helicity-2 field mentioned in Sect.
3.5 is a small modification of
$\frac{1}{2}h_{\mu\nu}(c)\Theta^{\mu\nu}_{h=2}(c)$ (which would be part of an
$L$-$Q$ pair only if $\Theta_{h=2}$ were string-independent). This observation
raises the hope that sQFT might also admit $L$-$Q$ pairs for graviton
couplings to massless matter of helicity beyond 2. Unfortunately, the
systematic search turned out no candidates beyond the known ones with
$h=0,1,2$. The analysis of $L$-$Q$ pairs coupling gravitons to particles with
half-integer helicity and to massive higher-spin particles remains to be done.

One may also wonder whether sQFT allows new interactions among known
particles. One such proposal is [1]. The authors observed that with an
interaction $L^{\prime}$ involving the $Z_{\mu}$ field multiplying two
dressing factors $\,\colon\\!\\!\exp ig\phi(e)\\!\colon$ and
$\,\colon\\!\\!\exp-ig\phi(e^{\prime})\\!\colon$ with $e\neq e^{\prime}$, one
can produce the decay of a $Z$-boson into two photons, otherwise forbidden by
the Landau-Yang theorem. (The latter uses Lorentz invariance which is
manifestly broken by the choice of the strings.)

The proposed interaction term $L^{\prime}$ is not part of an $L$-$Q$ or
$L$-$V$ pair, so the resulting scattering matrix has no reason to be string-
independent. This may not even be intended by the authors; but what weighs
heavier is that as a consequence, there is no “magic formula” which would
ensure any kind of localization for quantum fields subjected to an interaction
$L^{\prime}$, along the lines discussed in Sect. 1.101010The authors use an
adhoc IR and UV regularization which in fact violates positivity. Smearing the
strings with $c(e)$ would solve the UV issue. The positivity-preserving IR
regularization used in [24] produces the superselection structure mentioned in
Sect. 3.1, and would actually make the amplitude in [1] vanish whenever $c\neq
c^{\prime}$.

We do not know what is possibly “allowed” as string-localized interactions
beyond $L$-$Q$ or $L$-$V$ pairs and higher-order interactions induced from
them, and how much flexibility is gained for interactions beyond the SM. E.g.,
$L$-$Q$ pairs that are quartic in the free fields may have slipped our
attention. But we strongly feel that one cannot choose any interaction merely
because it seems convenient. Locality and positivity are precious goods that
must be taken care of.

##  4 Conclusion and outlook

String-localized quantum field theory (sQFT) is a new framework to address the
relation between particles and quantum fields. For free fields, it is just
“more flexible” than local QFT. This flexibility can be used to overcome
problems with Hilbert space positivity due to canonical quantization of
massless vector fields, and problems with renormalizability due to the
increased scaling dimension of vacuum fluctuations of massive vector fields.

sQFT deploys its power when applied to perturbation theory: it strongly
constrains admissible interactions by fundamental quantum principles (notably
Hilbert space). In fact, sQFT can be understood as a scheme to predict the
interactions of the SM in an “autonomous” (purely quantum) way.

A prominent lecture that sQFT teaches us is that perturbed quantum fields,
apart from the observables of the model, must not be expected to be local (or
anti-local). Their dynamical string-localization is a physical feature (e.g.,
related to the Gauss Law of QED), and it still allows “sufficient
commutativity” to conceive a scattering theory in terms of correlations at
asymptotic times. However, the prevalent methods of scattering theory have to
be properly adjusted.

The dressed Dirac and quark fields (3.1) and (3.4) seem to be instances of an
issue that calls out for a deeper understanding: given that sQFT a priori
“knows nothing” about gauge theory, why do (classically) gauge-invariant
quantities play a distinguished role in sQFT, and why seem the observables of
sQFT to correspond to (classically) gauge-invariant local fields? Is perhaps
“gauge-invariant and local” just another characterization of “string-
independent”?

The focus in this paper was on the relevant analysis to determine
interactions, which proceeds at tree-level. The resulting interactions are
power-counting renormalizable, and the main result of [13] indicates that the
power-counting bound is sufficient for UV renormalization of all loop
diagrams, also in sQFT.

The development of an efficient UV renormalization program is the main open
agenda of sQFT. An adaptation of the EG program [11] to the string-localized
context might be most promising. However, the equivalence results at tree-
level, presented in this paper and [29], let us expect that sQFT will not
produce other results than local QFT with its “negative probabilities plus
BRST”. One may therefore do the actual loop calculations in the usual
framework, while the message of sQFT is that the interactions of the SM can be
understood without gauge fields and ghosts, and are rather direct consequences
of fundamental principles of quantum field theory: Hilbert space positivity,
covariance and locality.

#### Acknowledgments:

We thank J. Mund and J. Yngvason who contributed many substantial ideas (some
of them unpublished) in earlier stages of the line of research reported here.
We thank D. Buchholz for interesting comments. KHR thanks his former students,
M. Chantreau (MSc), I. Hemprich (MSc), T. Joswig (BSc), L. Kersten (BSc), N.
Münch (BSc), M. Pabst (BSc) and F. Tippner (MSc), who with their thesis works
have been involved in various explicit computations in some of the models of
Sect. 3 and [30]. CG was funded by the National Science Center of Poland under
the grant UMO-2019/35/B/ST1/01651.

## References

  * [1] M. Asorey, A.P. Balachandran, M. Arshad Momen, B. Qureshi: Violation of the Landau-Yang theorem from infrared Lorentz symmetry breaking, JHEP 10 (2023) 028.
  * [2] A. Aste, G. Scharf: Non-abelian gauge theories as a consequence of perturbative quantum gauge invariance. Int. J. Mod. Phys. A14 (1999) 3421–3432.
  * [3] R. Brunetti, D. Guido, R. Longo: Modular localization and Wigner particles. Rev. Math. Phys. 14 (2002) 759–785.
  * [4] H.-J. Borchers: Über die Mannigfaltigkeit der interpolierenden Felder zu einer kausalen S-Matrix (in German). Nuovo Cim. 15 (1960) 784–794.
  * [5] H.-J. Borchers: On revolutionizing quantum field theory with Tomita’s modular theory. J. Math. Phys. 41 (2000) 3604–3673.
  * [6] D. Buchholz: Gauss’ Law and the infraparticle problem. Phys. Lett. B 174 (1986) 331–334.
  * [7] D. Buchholz, F. Ciolli, G. Ruzzi, E. Vasselli: On string-localized potentials and gauge fields. Lett. Math. Phys. 109 (2019) 2601–2610.
  * [8] D. Buchholz, K. Fredenhagen: Locality and the structure of particle states. Commun. Math. Phys. 84 (1982) 1–54.
  * [9] P.A.M. Dirac: Gauge-invariant formulation of Quantum Electrodynamics, Canadian J. Phys. 33 (1955) 650–660.
  * [10] M. Dütsch: Proof of perturbative gauge invariance for tree diagrams to all orders. Ann. d. Phys. (Leipzig) 14 (2005) 438–461.
  * [11] H. Epstein, V. Glaser: The role of locality in perturbation theory. Ann. Inst. H. Poincaré A19 (1973) 211–295.
  * [12] K. Fredenhagen, M. Marcu: Confinement criterion for QCD with dynamical quarks, Phys. Rev. Lett. 56 (1986) 223–224.
  * [13] C. Gass: Renormalization in string-localized field theories: a microlocal analysis. Ann. Henri Poincaré 23 (2022) 3493–3523.
  * [14] C. Gass, J.M. Gracia-Bondía, J. Mund: Revisiting the Okubo–Marshak Argument, Symmetry 13 (2021) 1645\.
  * [15] C. Gass, J.M. Gracia-Bondía, K.-H. Rehren: Quantum general covariance. Class. Qu. Grav. 40 (2023) 195016.
  * [16] J.M. Gracia-Bondía, K.-H. Rehren, J.C. Várilly: An autonomous account of the full electroweak interaction, in preparation
  * [17] J.M. Gracia-Bondía, J. Mund, J.C. Várilly: The chirality theorem. Ann. Henri Poincaré 19 (2018) 843–874.
  * [18] P. Jordan: Der gegenwärtige Stand der Quantenelektrodynamik (in German). In: Talks and discussions of the theoretical-physical conference in Kharkov (May 19-25, 1929), Physik. Zeitschrift XXX (1929) 700–712.
  * [19] P. Jordan: Zur Quantenelektrodynamik (in German). Zeitschr. Phys. 95 (1935) 202–209.
  * [20] G. Lechner: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277 (2008) 821–860.
  * [21] R. Longo, V. Morinelli, K.-H. Rehren: Where infinite spin particles are localizable. Commun. Math. Phys. 345 (2016) 587–614.
  * [22] S. Mandelstam: Quantum electrodynamics without potentials. Ann. Phys. 19 (1962) 1–24.
  * [23] J. Mund, K.-H. Rehren, B. Schroer: Helicity decoupling in the massless limit of massive tensor fields. Nucl. Phys. B924 (2017) 699–727.
  * [24] J. Mund, K.-H. Rehren, B. Schroer: Gauss’ Law and string-localized quantum field theory. JHEP 01 (2020) 001.
  * [25] J. Mund, K.-H. Rehren, B. Schroer: Infraparticle quantum fields and the formation of photon clouds. JHEP 04 (2022) 083.
  * [26] J. Mund, K.-H. Rehren, B. Schroer: How the Higgs potential got its shape. Nucl. Phys. B987 (2023) 116109.
  * [27] J. Mund, B. Schroer, J. Yngvason: String-localized quantum fields and modular localization. Commun. Math. Phys. 268 (2006) 621–627.
  * [28] M. Porrati: Old and new No Go theorems on interacting massless particles in flat space. In: 17th International Seminar on High Energy Physics (Quarks 2012), arXiv:1209.4876.
  * [29] K.-H. Rehren: On the effect of derivative interactions in quantum field theory. arXiv:2405.nnnnn.
  * [30] K.-H. Rehren: Nonabelian models in sQFT. arXiv:2405.nnnnn.
  * [31] G. Scharf: Gauge Field Theories: Spin One and Spin Two (Dover) 2016.
  * [32] B. Schroer: Infraparticles in quantum field theory. Fortsch. Phys. 11 (1963) 1–32.
  * [33] O. Steinmann: Perturbative QED in terms of gauge invariant fields. Ann. Phys. 157 (1984) 232–254.
  * [34] S. Weinberg: Feynman rules for any spin. Phys. Rev. 138 (1964) 1318–1332.
  * [35] E.P. Wigner: On unitary representations of the inhomogeneous Lorentz group. Annals Math. 40 (1939) 149–204.
  * [36] K. Wilson: Confinement of quarks. Phys. Rev. D10 (1974) 2445–2459.
  * [37] J. Yngvason: Zero-mass infinite spin representations of the Poincaré group and quantum field theory, Commun. Math. Phys. 18 (1970) 195–203.
  * [38] Panel discussion, recorded in: “Gauge Interactions: Theory and Experiment”, A. Zichichi (ed.), pp. 725-740. Plenum (1984).