Foundations of Physics

, Volume 45, Issue 3, pp 219–252 | Cite as

A Hilbert Space Setting for Interacting Higher Spin Fields and the Higgs Issue



Wigner’s famous 1939 classification of positive energy representations, combined with the more recent modular localization principle, has led to a significant conceptual and computational extension of renormalized perturbation theory to interactions involving fields of higher spin. Traditionally the clash between pointlike localization and the the Hilbert space was resolved by passing to a Krein space setting which resulted in the well-known BRST gauge formulation. Recently it turned out that maintaining a Hilbert space formulation for interacting higher spin fields requires a weakening of localization from point- to string-like fields for which the d = s + 1 short distance scaling dimension for integer spins is reduced to d = 1 and and renormalizable couplings in the sense of power-counting exist for any spin. This new setting leads to a significant conceptual change of the relation of massless couplings with their massless counterpart. Whereas e.g. the renormalizable interactions of s = 1 massive vectormesons with s \(<\) 1 matter falls within the standard field-particle setting, their zero mass limits lead to much less understood phenomena as “infraparticles” and gluon/quark confinement. It is not surprising that such drastic conceptual changes in the area of gauge theories also lead to a radical change concerning the Higgs issue.


String-localized massive vectormesons and Hilbert space positivity Coupling massive vectormesons to Hermitian scalar fields Induced H-selfinteraction versus symmetry-breaking Higgs mechanism. 

1 Improved Short Distance Scaling from String-Localization and Massive Vectormesons

It is well-known that in d = 1 + 3 spacetime dimensions only a finite number of renormalizable couplings between pointlike fields exist. In fact only interactions among spin s = 0 and s = 1/2 fields permit a pointlike covariant renormalization theory in a Hilbert space setting. Another characteristic property of such low spin couplings is that they maintain their particle interpretation in the massless limit, which manifests itself in the absence of zero mass infrared divergencies.

This situation changes for \(s\ge 1\) (referred to as “higher spin”); a renormalizable pointlike formulation in Hilbert space is not possible, and infrared divergencies in the massless limit indicate a breakdown of the standard field-particle relation. The traditional Lagrangian quantization parallelism to classical field theories leads inevitably to quantum fields in an indefinite metric space (more precisely a Krein space), as it is well-known from the Gupta–Bleuler formalism of quantum electrodynamics (QED) and its more sophisticated successor, the BRST setting [1].

The advantage of this gauge formalism is that, in spite of its later refinements, it uses basically a renormalization formalism which has been known since the discovery of covariant perturbative renormalization theory in the context of QED by Tomonaga, Feynman, Schwinger and Dyson. It is the best compromise between two opposing properties, the singular nature of quantum fields on the one hand, and on the other hand the canonical aspects of the classical Lagrangian field formalism. The Hilbert space structure of quantum field theory (QFT) played no role in the discovery of perturbative renormalization; but with the contribution of Gupta and Bleuler, the awareness about the importance of Hilbert space positivity and its relation to quantum gauge invariance gradually grew.

The path from the formulation of QED to the modern BRST setting of nonabelian gauge theory passed through several stages among which the adjustment of renormalization to Yang–Mills couplings by ’t Hooft–Veltman, the functional Faddeev–Popov reformulation and the Slavnov identities are important landmarks.

Whereas in the classical Maxwell theory the introduction of vectorpotentials associated with field strengths was achieved without problems and their use in quantum mechanical problems in external electromagnetic fields became indispensable (the quantum mechanical Aharonov–Bohm effect), their quantization raised serious conceptual questions, well-known from the Lagrangian quantization setting of QED. The origin of these problems is not the particular method of quantization but rather the clash between pointlike localization of \(s\ge 1~\)massless “potentials” with the Hilbert space positivity. This is well-known from the \(s=1\) vectorpotential \(A_{\mu }(x)~\)associated with the \(F_{\mu \nu }~\)field strength; its s = 2 counterpart is the second degree symmetric tensor potential \(g_{\mu \nu }(x)\) associated the the field strength \(R_{\mu \nu \kappa \lambda }\) (with the symmetry properties of a linearized Riemann tensor) and for \(s=n\) there are symmetric tensorpotentials of degree \(n~\)and associated multi-indexed field strengths with mixed symmetry properties.1

There is another more systematic way of looking at this problem. In the setting of Wigner’s positive energy representation theory of the Poincaré group the Hilbert space positivity is taken care of by the unitarity of the group representations. The Wigner representation space for the massless finite helicity representations \((m=0,h)\) contains covariant wave functions of the field strength type; the representation space contains however no covariant pointlike vectorpotentials \(A_{\mu }(x).\) In the \(s=1\) massive case both, pointlike covariant field strengths as well as vectorpotentials \(A_{\mu } ^{P}(x)\) (Proca potential) exist. Using the functorial relation between covariant wave functions and free quantum fields these findings have corresponding free field counterparts. In the next section we will return to this issue.

The lack of a Hilbert space description in a quantum theory is a almost a “contradictio in adjecto” since the Hilbert space is the foundational property of any quantum theory be it quantum mechanics (QM) or QFT. Without the Hilbert space positivity there will be no quantum probability and none of the structural properties of QFT (LSZ scattering theory, the connection between spin and statistics, TCP, ...) can be derived, nor would it be possible to formulate a physically meaningful causality and localization property. This is the only property which has no classical counterpart and hence is not part of the (Lagrangian) quantization parallelism between classical and quantum field theory. As already mentioned, the problem starts with zero mass quantum fields of spin s = 1 and becomes worse with higher spin. As the Wigner representation theory shows, it cannot be blamed on a particular quantization procedure but rather results from a quantum incompatibility between pointlike localization of (\(m=0,s\ge 1\)) potentials and the Hilbert space positivity. For massive pointlike \(s\ge 1~\)fields this incompatibility is hidden behind the more subtle relation between pointlike nonrenormalizability and breakdown of pointlike localizability (see later).

The simplest way to see that this problem disappears if instead of massless pointlike potentials one introduces their covariant stringlike counterpart is to start from a massless field strength and define a massless covariant vectorpotential which, instead of being localized on a point, “lives” on a semi-infinite straight spacelike string
$$\begin{aligned} A_{\mu }(x,e)&=\int F_{\mu \nu }(x+\lambda e),~~e\cdot e=(e,e)=-1\\&localized~on~x+\mathbb {R}_{+}e\nonumber \end{aligned}$$
By definition \(A_{\mu }(x,e)\) it is covariant (\(e\) transforms as a vector) and two stringlocal potentials commute if they are mutually spacelike separated. This vectorpotential lives in the same Hilbert space as the field strength and both fields are in the same localization class. The choice of straight strings guaranties the covariance. Such potentials are not Euler–Lagrange fields and hence cannot be obtained by Lagrangian quantization.

The Hilbert space positivity which requires their introduction is not an issue in classical physics; apart from the gauge freedom in the \(A_{\mu }-F_{\mu \nu }\) relation, the potential is a classical fields as any others. Quantizing classical potentials requires the use of the more subtle quantum gauge theory whose primary purpose is to recover the Hilbert space positivity. The disadvantage of such a description is that in the presence of interaction with matter the gauge-invariant local observables do not contain the physical matter fields which one needs to generate physical states from the vacuum.

In the massive case the pointlike (Proca) potential is in an interesting way related to its stringlocal counterpart. Starting from a Proca potential \(A_{\mu }^{P}(x)\) we define
$$\begin{aligned} F_{\mu \nu }(x)&\equiv \partial _{\mu }A_{\nu }^{P}(x)-\partial _{\nu }A_{\mu } ^{P}(x),~~A_{\mu }(x,e)\equiv \int _{0}^{\infty }d\lambda F_{\mu \nu }(x+\lambda e)e^{\nu },~e\cdot A=0~\\ \phi (x,e)&\equiv \int _{0}^{\infty }e^{\mu }A_{\mu }^{P}(x+\lambda e)d\lambda ,~\curvearrowright A_{\mu }(x,e)=A_{\mu }^{P}(x)+\partial _{\mu } \phi (x,e)\nonumber \\ \partial ^{\nu }F_{\mu \nu }&=m^{2}A_{\mu }=:j_{\mu }^{Max},~~Q^{Max}:=\int j_{o}^{Max}d^{3}x=0\nonumber \end{aligned}$$
where the linear relation between the stringlocal potential, the Proca potential and the scalar stringlocal \(\phi \) is an algebraic consequence. This relation between three fields in the same localization (Borchers) class2 will be important in later sections, but for the present purpose the relevant property is the observation that the Proca field has the short distance scaling dimension \(d_{sd}=2\) whereas the two stringlocal fields \(A_{\mu },\phi \) have \(d_{sd}=1.\) The stringlocal scalar field will be referred to as the “intrinsic escort” of \(A_{\mu }(x,e);\) the reason for this terminology will become clear in the next section.

The third line defines the identically conserved free Maxwell current whose associated charge vanishes. This property continues to hold for interacting massive vectormesons coupled to matter, independent of whether the matter is charged (massive spinor or scalar QED) or neutral (a Hermitian field \(H\)). It is known as the Schwinger–Swieca screening of the Maxwell charge. Here the \(H\) does not only stand for Hermitian but also for Higgs. This screening property and not the metaphor of a spontaneous mass-creating symmetry breaking referred to as the “Higgs mechanism” is the true intrinsic property of all massive vectormesons. Complex (charged) matter has also a conserved charge-anticharge counting current (which is absent for \(H\)-matter). The two currents coalesce in the massless limit (in which \(H\)-interactions vanishes), More in Sect. 5.

As the spacetime point \(x,\) the string direction \(e~\)is a spacetime parameter in which the field fluctuates, in fact it may be viewed as a spacetime position in the unit \(d=1+2~\)de Sitter space of spacelike directions. These fluctuations relieve the pointlike fluctuations of the Proca field; formally the \(d_{sd}=2~\)derivative of the intrinsic escort field compensates the leading short distance singularity of the Proca field at the price of a mild weakening of localization from point- to stringlike. The stringlocal potential is the only covariant localized vectorpotential which is consistent with the Hilbert space positivity and localizability in the presence of interactions (renormalizability\(\simeq \)pointlike localizability), in fact it “lives” together with \(F_{\mu \nu }~\)and its mutually local Proca sibling in the same Hilbert space. It is the only covariant potential which survives in the massless limit; the zero mass radiation gauge potential which lives in the same Hilbert space is neither covariant nor localized. All these \(m>0~\)potentials, including the intrinsic escort \(\phi \) are linear combinations of the three \(-1\le s_{3}\le 1\) Wigner creation/annihilation operators \(a^{\#}(p,s_{3})~\)with different \(u\)-\(v~\)intertwiners.

It is rather straightforward to generalize the idea of stringlocal \(d_{sd} =1~\)(independent of \(s\)3); in this case the spin \(s~\) tensorpotentials (conveniently described in terms of a symmetric tensor of degree \(s\)) which generalizes the Proca potential has \(d_{sd}=s+1~\)is accompanied by \(s\) intrinsic escort fields with spins between zero and \(s-1\) which iteratively peel off the leading short distance behavior of the pointlike \(d_{sd}=s+1~\)tensor potential. In this paper we will present the dynamical use of this idea for \(m>0,~s=1\) which results in a renormalizable interacting stringlocal formulation in Hilbert space which, different from the BRST gauge theory, contains in addition to the local observables the physical matter fields and permits to pass to \(m\rightarrow 0\) where only correlation functions of stringlocal physical matter fields survive.

Although stringlocal fields have the \(s\)-independent short distance \(d_{sd}=1~\)and hence permit the construction of interactions within the power-counting restriction for any spin, only those interactions for which there exists a pointlike generated vacuum sector of local observables are of physical interest. In the new setting stringlocal fields are the basic fields in terms of which the perturbative interactions are formulated; the pointlike local observables are pointlocal composites of these fields. The main physical role these stringlocal physical (= acting in Hilbert space) matter fields play in models of massive vectormesons is that their application to the vacuum generates states whose large time behavior in the setting of the LSZ scattering theory approach multi-particle scattering states. On the level of particles the difference between point- and string-like disappears; The \(e\)-dependence of the fields simplifies in the large time limit where it only leaves its trace in the particles states from where it can be removed by a change of normalization.4 Note that all this changes in the limit \(m\rightarrow 0~\)of massless vectormesons where scattering theory and together with the relation between fields and Wigner particles breaks down.

One of the strongest nonperturbative results which relates particles to stringlocal fields is a theorem by Buchholz and Fredenhagen which states that in theories with a mass-gap and a nontrivial vacuum sector (generated by the local observables) all particles (including those which are outside the vacuum sector) and their scattering states can be generated by operators which are localized in arbitrary narrow spacelike cones [4].

The theorem does not In fact our results show that massive QED are the simplest models Since LSZ scattering theory the proof of this theorem uses the Hilbert space positivity in an essential way, none of the theorems of QFT can be expected to hold in a Krein space. In the local quantum physics (LQP) setting in which this theorem was proven the arbitrary narrow spacelike cones correspond to semi-infinite spacelike stringlocal fields (they live on the core of such cones); pointlocal fields correspond in LQP to the core of arbitrary small double cones and are a special case of (e-independent) stringlocal fields.

According to this theorem models of QFT with a mass gap are generated by stringlocal fields \(\Psi (x,e);\) pointlocal fields correspond to \(e\)-independent \(\Psi .\) Fields which are nonrenormalizable in the pointlike sense may be renormalizable after converting their first order pointlike interaction into a suitably defined stringlike form (Sect. 3). This conversion will be explicitly presented for in the \(s=1\) massive QED; it shows that the pointlike nonrenormalizabilty is caused by the singular nature of the fields which cannot be localized by smearing with finite supported testfunctions in the sense of Wightman (they are not operator-valued Schwartz distributions). A formal pointlike localization can only be attained in Krein space gauge setting caused at the price the physical meaning of causal localization (which is only recovered in the vacuum sector of gauge invariant observables). This shows the powerful role of Hilbert space positivity in relating the breakdown of pointlike renormalizability with a weakening of localization. It limits the use of pointlike localization to renormalizable interactions between \(s<1\) fields.

The use of Krein spaces as in the BRST gauge formulation is limited to formal manipulations in perturbation theory, without Hilbert space positivity locality is physically void and the known properties of the connection between fields and particles (e.g. spin&statistics, LSZ scattering theory,..); The properties of QFT are limited to the gauge invariant vacuum sector for which the Hilbert space setting is recovered. Quantum gauge symmetry is not a physical symmetry but rather a mathematical trick to rescue the Hilbert space of the vacuum sector which is generated by gauge invariant local observables from an unphysical Krein space setting. In classical field theory the Hilbert space positivity is not an issue; vectorpotentials are classical fields and the gauge transformations define a symmetry which transforms between vectorpotentials associated with a fixed field strength.

In the massless QED limit the the singular pointlike fields disappear and the strings become rigid; the appearance of infinitely extended photon clouds along the e-direction convert particles into infraparticles for which the \(e\)-direction is “frozen”, namely there are no unitaries which change the direction \(e\) (as it was still possible in the massive case) which causes the spontaneous breakdown of Lorentz invariance [3]. These facts can be derived from the appropriately formulated quantum Gauss law [6].

Interactions involving stringlocal higher spin (\(s\ge 1\)) fields with first order interaction densities of short distance dimensions \(d_{sd}^{int}\le 4\) are renormalizable in the sense of the power counting criterion, whereas their pointlike analogs are nonrenormalizable since \(d_{sd}^{int}(s)>4~\)invreases with \(s\). Here “nonrenormalizability” refers to subsumes two properties, on the one hand the large momentum behavior is not polynomially bounded but rather increases with the perturbative order; this means that the corresponding fields cannot be Wightman fields (operator-valued Schwartz distributions); if they exist at all they are extremely singular with unbounded short distance dimension \(d_{sd}=\infty .\) What makes nonrenormalizable theories rather useless (apart from phenomenological applications) is that the growth of the polynomial degree in momentum space is accompanied by an increase of free counter-term coupling parameters; a theory which depends on infinitely many parameters looses its predicitive power and raises doubts about its mathematical consistency.

A renormalizable stringlocal situation of a theory which is pointlike nonrenormalizable cannot improve the singular pointlike behavior, but at least it presents the unbounded increase of parameters; the pointlike field is a singular coordinatization of the same theory with the same finite set of coupling parameters as those which parametrize the stringlocal renormalizable description. It will be shown that the higher order stringlocal interaction densities allow to construct pointlike counterparts for which a direct construction fails; the so constructed pointlike densities have the expected bad high energy behavior but contain no new parameters. Since the pointlike S-matrix can be shown to be identical to its stringlike definitions, this has the interesting consequence that an S-matrix may have a better high energy behavior than one would expect on the basis of the LSZ reduction formula which relates it to the mass shell restriction of singular pointlike fields.

Although in the present work we will limit our constructions to the S-matrix, the construction of stringlocal fields suggests that the matter counterpart of the additive relation between massive point- and stringlocal fields (2) is a multiplicative formula of the exponential form5
$$\begin{aligned} \psi (x)=e^{-ig\phi (x,e)}\psi (x,e) \end{aligned}$$
Here the singular (nonrenormalizable) pointlike (spinor, scalar) matter field is defined in terms of its stringlike sibling; \(\phi \) is the stringlocal scalar escort field and \(g~\)is the\(.\)coupling of the massive vectormeson to the matter field of (scalar, spinor) massive QED. Such exponential relations are known to convert Wightman fields into singular fields, in fact the standard illustration is the exponential of a scalar free field [7]. Whereas the fields \(\psi ,\phi \) on the right hand side are Wightman fields (i.e. can be smeared with Schwartz testfunctions \(f(x)h(e)\)), this property is lost in the normal product; such a nonlinear relations do not allow to convert testfunction spaces.

The relations (2) together with (3) looks like a gauge transformation. But the conceptual role is different; whereas gauge transformations are a tool which one needs in order to extract physical properties (local observables) from an unphysical Krein space description, their present role is to relate the renormalizable stringlocal description with its singular pointlike counterpart. Although the calculations are done in the renormalizable stringloca field (SLF) setting, the relation to the pointlike objects is important for the construction of the first order stringlocal interactions and to secure the \(e\)-independence of the scattering amplitudes in higher perturbative orders.

Proposals to define gauge invariant matter fields begun to appear shortly after the dicovery of field quantization in the conrext of QED. Contrary to gauge invariant local observables (field strength, conserved currents) the requirement of gauge invariance imposed on matter fields led inevitably to stringlocal expressions un terms of gauge-variant fields
$$\begin{aligned} \psi (x,e)=\psi ^{K}(x)e^{ig\int _{0}^{\infty }A_{\mu }^{K}(x+\lambda e)d\lambda },~~(e,e)=-1, \end{aligned}$$
Here the superscript \(K\) refers to the fact that quantum gauge theory is formulated in an indefinite metric Krein space and only gauge invariant fields can be accommodated in a Hilbert space. There is a formal similarity between (3) and (4) but for our purpose it is more instructive to emphasize the conceptual differences. In the gauge theoretic representation one defines a stringlocal physical (gauge invariant) object in terms of pointlike gauge-variant fields. Our relation (4) on the other hand defines a very singular field (well-defined in every order with increasing \(d_{sd}\)) in terms of a renormalizable stringlocal field. The singular nature is the price for staying in a Hilbert space. The reason for stressing this difference is that the similarity of the two formula hides an enormous conceptual difference which one is not aware of if one remains in the perturbative realm. Perturbation theory is mainly combinatorics + Feynman loop integration, but for functional analytic calculations proofs in QFT one needs the Hilbert space setting, not to mention the probability interpretation of QT. Formulas as (4) which involve nonlocal composite are outside computational control whereas the stringlike fields in (4) are the basic objects of a new Hilbert space based renormalized perturbation theory; this new setting also permits to calculate pointlike fields which fulfill (3) in every order perturbation theory.

The new setting leads to a radical change in the way the relation between massless and massive gauge theories is viewed. In the past, and to a certain extend even at present, the massless models are viewed as being simpler than their massive counterpart. This has historical roots, since QED was the first QFT and models involving massive vectormesons (viz the Higgs model) appeared much later. But this alleged simplicity of massless models only refers to formal perturbative properties. The problem starts when one tries to extract physical properties (infraparticles, confinement). In that case one runs into nearly intractable “infrared problems” under the rug of perturbation theory which are related to fundamental changes in the field-particle relation. There is no such problem in the presence of mass gaps; in this case all the textbook knowledge about the field-particle relations applies; the LSZ scattering theory leads to the standard relations between fields and Wigner particles provided the operators act in a Hilbert space. Only in a Hilbert space setting the (necessarily stringlocal) electron field is physical and can explain the infrared properties in terms of long distance behavior of stringlocal matter fields. The conceptual simplicity of theories with a mass gap suggests to turn the historical massless-massive relation from its head to its feet by studying massless interacting vectormesons in terms of massless \(s=1~\)of their simpler massive counterparts.

A convincing illustration of this new point of view is provided by the Higgs model6 which in the new setting is simply the renormalizable coupling of a massive vectormeson to a scalar Hermitian matter field. There are three types of couplings of massive vectormesons: couplings to charge (complex) matter (massive QED), to neutral (Hermitian) matter and self-couplings (unconfined massive gluons and quarks). Even if this is not evident from the way in which the Higgs model is constructed in the literature, it must be one of these couplings because the list is exhaustive. Its standard construction starts from the two-parametric massless scalar QED (besides the vectorpotential coupling \(g\) there is a renormalization-induced \(\left| \varphi \right| ^{4}\) self-coupling with an independnet coupling strength \(c\)). The symmetry (the gauge symmetry!) is spontaneously broken by a numerical shift \(\varphi \rightarrow \varphi +d\) in field space of the gauge-variant scalar \(\varphi \)-field; the result is the famous Mexican hat potential. The parameters \(c\) and \(d\) are then converted into more physical mass parameters of the vector meson and \(H\)-mass \(m,m_{H}~\)and the result is presented as a a result of the Higgs-mechanism namely a “mass creation by spontaneous symmetry-breaking”.

The correct presentation, consistent with the conceptual properties of QFT,7 is in terms of a first order trilinear \(gA\cdot AH\) coupling. Usually one does not count the masses of the interaction-defining fields as parameters; this unusual bookkeeping is only necessary if one wants to compare the resulting theory with the afore mentioned symmetry breaking description based on the Mexican hat potential which arises from two-parametric scalar QED through a field shift. Implementing however the gauge invariance of the S-matrix up to second order \(g^{2}\) in the BRST nilpotentent \(s\)-formalism (\(sS=0\)) for the \(A\)-\(H\) model, one finds that the this implementation induces quadrilinear self-couplings with induced coupling strengths which are mass ratios of the two involved masses \(m_{H}^{2}/m^{2}~\)which can be written in terms of a Mexican hat potential. This shows that behind the alleged mass generation in terms of a metaphoric (gauge !) symmetry breaking through a shift in field space of (two-parametric) scalar QED there is something quite different. Instead of gauge symmetry breaking by an imposed field shift in scalar QED (the Mexican hat potential) the implementation of BRST gauge symmetry on a \(A\)\(H\) coupling of a massive vectorpotential to a Hermitian field induces8 a potential of that form.

There are several episodes in theoretical physics in which (the conceptual/mathematical) nature asserted itself against human misconceptions, but the Higgs mechanism of mass creation is one of the strongest illustration because it endured more than 4 decades. The second order calculation presented in Sect. 5 confirms this observations: the Mexican hat potential is not the result of a gauge symmetry breaking mechanism in scalar QED but is induced from the implementation of BRST gauge invariance of the S-matrix in a renormalizable \(A\)\(H\) coupling of a massive vectormson to a Hermitian field.

It is an interesting question to ask why this much simpler description has been overlooked for such a long time. The answer may be that this coupling has no massless long range classical limit; the \(m\rightarrow 0~\)limit decouples the two fields, so the standard quantization construction starting from a Maxwell setting was not available. The unfortunate idea that the simpler massive \(s=1~\)interactions have to be understood in terms of the allegedly simple massless models accounts for the rest.

Whereas a Goldstone’s spontaneous symmetry breaking is characterized by a conserved current whose associated charge diverges at long distances as the result of a coupled zero mass Goldstone boson9 [8], massive vectormesons interacting with matter are characterized by the Schwinger–Swieca screening of the massive Maxwell current. Both cases represent very different manifestations of conserved currents, the normal case of a finite nontrivial charge is in the middle between these opposite extremes. The claimed special mass-giving role (including its own mass) of a distinguished Higgs particle (the “God particle”) is simply the result of a misunderstanding.

From a philosophical viewpoint the removal of a distinguished particle reestablishes the “nuclear democracy” between particles which carry the same superselection charge; in the case at hand particle democracy means that there is no hierarchy between a Higgs particle, a scalar bound state of the escort field \(\phi \) or a scalar massive “gluonium” bound state. The SLF Hilbert space setting which confirms these results extends this democracy to quantum fields; instead of having gauge theories in addition to “standard” QFTs, the Hilbert space setting places all models under the conceptual roof of the causal localization principle and decides in what cases pointlike field interactions have to be replaced by their stringlike counterparts (namely for all ms \(\ge 1~\) interactions).

The formulation of the new Hilbert space setting requires the standard field-particle relation which is guarantied in the presence of a mass gap (absence of zero mass fields). Problems of zero mass vectorpotentials must be approached by taking massless limits of correlation functions involving interacting stringlocal massive vectormesons which induce stringlocal matter fields. The singular pointlike matter fields and self-interating massive gluons disappear in the massless limit and only their stringlocal counterparts remain. In this SLF setting confinement correspond to the vanishing of massless limits of all correlation functions which besides pointlike observables contain also stringlocal gluon and quark fields.10 A zero mass situation with peculiar properties is the noncompact “stuff” associated to Wigner’s infinite spin representation class. Its properties are very different from matter as we know it and only fit the properties which are ascribed to dark matter. More on this in the next section.

The paper is organized as follows. The next section presents the construction of stringlocal fields from Wigner’s representation theory of the Poincaré group with special emphasis of the noncompact “stuff” associated to the third Wigner class. Section 3 explains the concept of the intrinsic escort field \(\phi ~\)in more detail, including its role in the model-defining first order interaction density, as well as its use in the construction of a string-independent S-matrix. The fourth section contains the simplest nontrivial second order illustration of the Hilbert space setting of stringlocal fields (SLF) which is provided by the model of massive scalar QED. Section 5 presents the analogous second order calculation of coupling of a massive vectormeson to Hermitian instead of charged matter and its relation to the Higgs model. The end of Sect. 5 contains additional remarks on confinement as well as a contrasting juxtaposition of confinement with the properties of the inert noncompact stuff (dark matter?) associated to the third Wigner class.

2 Wigner Representations and Stringlocal Fields

In the development of the ideas which led up to the present SLF setting, the Wigner representation theory of the Poincaré group played a prominent role [9]. There are three classes of positive energy representation,11 the mass \(m>0~\)class and two zero mass classes: the finite helicity class and Wigner’s “infinite spin” class. With the exception of the third class, the connection of irreducible positive energy Wigner representations with interaction-free quantum fields is well-known and has been explicitly presented in many articles; the most detailed exposition can be found in the first volume of Weinberg’s book [10]. Written in terms of \(u,v~\)intertwiners the result isMerkel hat
$$\begin{aligned} \psi ^{A,\dot{B}}(x)=\frac{1}{\left( 2\pi \right) ^{3/2}}\int (e^{ipx} u^{A,\dot{B}}(p)\cdot a^{*}(p)+e^{-ipx}v^{A,\dot{B}}(p)\cdot b(p))\frac{d^{3}p}{2p_{0}} \end{aligned}$$
The intertwiners for \(m>0\) are rectangular \((2A+1)(2B+1)\cdot (2s+1)\) matrices which intertwine between the unitary \((2s+1)\)-component unitary Wigner representation and the covariant spinorial representation, and the \(a,b\) refer to particle and antiparticle creation/annihilation operators. For the m = 0 representations the formula is the same except that dot stands for the inner product in a two-dimensional space (the space of the two helicities \(\pm \left| h\right| \)). Another difference is the range of possible spinorial indices for a given physical spin \(s~\)the range of spinorial (half)integer spinorial representation indices of the homogeneous Lorentz group indices is restricted by
$$\begin{aligned} \left| A-\dot{B}\right|&\le s\le A+\dot{B},~~m>0\end{aligned}$$
$$\begin{aligned} \left| A-\dot{B}\right|&=\left| h\right| , \quad m = 0 \end{aligned}$$
the second formula shows that the the vector representation \(A=1/2=B\) does not occur for m = 0 i.e. pointlike covariant vectorpotential are not consistent with the Hilbert space positivity of quantum theory.

The recollection of these facts is helpful for the presentation of the stringlocal infinite spin field which can be written in the same way (5) except that the intertwiner \(u\) is not point-but rather stringlike and that the Hilbert space to which the inner product refers is an infinite dimensional representation space of the so-called little group (the full two-dimensional noncompact Euclidean group \(E(2)\) which leaves a lightlike momentum \(p\) invariant. In this case the group theoretical method for the calculation of intertwiners (as used by Weinberg) fails. The solution came from the calculation of a string-dependent intertwiner using the concept of “modular localization”12 [14]. This intrinsic (independent of field-coordinatization) way of formulating quantum localization (which had been used before in the construction of integrable models of QFT [15, 16] and led to existence proofs [17]), adds a new aspect to Wigner’s representation theory. The positive energy property implies the existence of a nontrivial dense subspaces of the Wigner space (a kind of ome-particle Reeh-Schlieder property) which describe spacelike cone-localized wave functions. Whereas for the \(m>0\) and the \(m=0\) finite helicity representations the localization can be tightened to (dense localization spaces for) arbitrary small double cones (which can be shown to be generated by testfunction smearing of pointlike covariant wavefunctions), all compact localized subspaces of the third class are trivial. As a consequence this representation class corresponds to noncompact “stuff” (in order to distinguish from matter as we know it) which is generated by stringlocal covariant fields whose field class does not contain pointlocal composites i.e. the third class Wigner stuff has no local observables).

Since this noncompact localized stuff cannot be approximated by normal (compact localizable) matter, there is no way to produce it from collisions of compact matter, nor can it be detected (localized) in (necessarily compact ) earthly counters.13 Apart from its stability and its coupling to gravitation (both properties are shared by all positive energy representations) it is completely inert. Its natural arena would be galaxies and its inertness makes it a candidate for dark matter [18]. Attempts to define interactions by coupling these fields lead to intractable infrared problems; unlike interacting zero mass couplings of normal matter there is no way of resolving infrared problems as limiting cases of models with a mass gap (as in the case of interacting zero mass vectorpotentials).

The main role of the construction of stringlocal massive fields for \(s\ge 1\) is to avoid the use of singular pointlike fields which are the cause of nonrenormalizability in the pointlike Hilbert space setting. Massive stringlocal fields have some unusual properties. For example stringlocal scalar fields can linearly (no composites necessary) interpolate particles of any integer spin, whereas for pointlike massive fields the spin is related to the spinorial indices (6). Their unavoidable appearance as intrinsic escorts \(\phi \) of massive vectormesons in stringlike interactions makes them competitors of extrinsic Higgs fields \(H\).14 As in the quantum mechanical condensed matter description of superconductivity where the change of long ranged vectorpotentials to their corresponding short ranged counterparts is achieved without the additional degrees of freedom, the intrinsic escorts \(\phi \) permit the existence of massive vectormesons without adding additional (Higgs) degrees of freedoms.

There is a radical change for zero mass potentials which only exist in the form of stringlocal fields with appropriate spinorial indices [19]. Whereas their associated pointlike field strengths obey (7), the stringlike localization of zero mass potentials permits the wider relation (6) between \(\left| h\right| \) and the spinorial indices. This shows that the algebraic changes in the zero mass limit are very drastic indeed; the objects on the right hand side of (2) disappear15 and only the stringlocal potential on the left hand side survives. The representation of the fields changes and the limit can only be taken for correlations functions from where a Wightman reconstruction [2] permits the construction of a new operator formulation. Another noteworthy property of stringlocal zero mass vectorpotentials which shows their superiority over the standard indefinite metric pointlike potentials even in the absence of interactions is the fact that the use of Stokes theorem in the derivation of the QFT Aharonov–Bohm effect16 in the gauge theoretic theoretic setting leads to a zero result, whereas the stringlocal potential in Hilbert space gives to the correct effect [20]. In terms of localization properties the A–B effect is the special helicity \(h=1\) case of a general violation of Haag duality for multiply connected spacetime regions.

Introducing a semiinfinite line integral \(\phi (x,s)\) over the Proca field along the same spacelike line, one obtains the important linear relation between the three free fields in the same localization class which share the same Wigner creation/annihilation operators \(a^{\#}(p,s)\) but have different intertwining functions.

Their two-point functions (including the mixed ones) are consequences of the properties of the massive Proca field and the above definitions. They can be computed via the intertwiners, or directly in terms of the above definitions and the well known two-point function of the Proca field
$$\begin{aligned} \left\langle A_{\mu }^{P}(x)A_{\mu ^{\prime }}^{P}(x^{\prime })\right\rangle&=\frac{1}{(2\pi )^{3/2}}\int e^{-ipx}M_{\mu \mu ^{\prime }}^{A^{P}}(p)\frac{d^{3}p}{2p_{0}} \nonumber \\ M_{,\mu \mu ^{\prime }}^{A^{P}}(p)&=-g_{\mu \mu ^{\prime }}+\frac{p_{\mu } p_{\mu ^{\prime }}}{m^{2}} \end{aligned}$$
Whereas the short distance scale dimension of the Proca field is \(d_{sd}^{P}=\)\(2\) (too big for obtaining interactions within the power-counting bounds of renormalizability), that of the stringlocal potential \(A\) is \(d_{sd}^{S}=1\)
$$\begin{aligned} M_{\mu \mu ^{\prime }}^{A}(p;e,e^{\prime })&=-g_{\mu \mu ^{\prime }}-\frac{p_{\mu }p_{\mu ^{\prime }}(e\cdot e^{\prime })}{(p\cdot e-i\varepsilon )(p\cdot e^{\prime }+i\varepsilon )}+\frac{p_{\mu }e_{\mu ^{\prime }}}{(p\cdot e-i\varepsilon )}+\frac{p_{\mu }e_{\mu ^{\prime }}^{\prime }}{(p\cdot e^{\prime }+i\varepsilon )}\\ M^{\phi }(p;e,e^{\prime })&=\frac{1}{m^{2}}-\frac{e\cdot e^{\prime }}{(p\cdot e-i\varepsilon )(p\cdot e^{\prime }+i\varepsilon )}\nonumber \end{aligned}$$
where in the second line is the 2-pointfunction of the Stückelberg field and the \(\varepsilon ~\)notation refers to the definition of distributions with positive energy spectrum in terms of boundary values of analytic functions.17
The mixed 2-pointfunctions \(M^{A,A^{P}},M^{,A,\phi },M^{A^{P},\phi }\) also follow directly from the definition (2)
$$\begin{aligned} M_{\mu \mu ^{\prime }}^{A,A^{P}}(p;e)&=-g_{\mu \mu ^{\prime }}+\frac{p_{\mu }e_{\mu ^{\prime }}}{(p\cdot e-i\varepsilon )}\nonumber \\ M_{\mu }^{,A,\phi }(p;e,e^{\prime })&=\frac{1}{i}\left( \frac{e_{\mu }^{\prime } }{(p\cdot e^{\prime }+i\varepsilon )}-\frac{p_{\mu }e\cdot e^{\prime }}{(p\cdot e-i\varepsilon )(p\cdot e^{\prime }+i\varepsilon )}\right) \\ M_{\mu }^{A^{P},\phi }&=i\left( \frac{p_{\mu }}{m^{2}}-\frac{e_{\mu }^{\prime } }{(p\cdot e^{\prime }+i\varepsilon )}\right) \nonumber \end{aligned}$$
The most convenient and systematic way is to use the representation of the three fields in terms of the \(u\)-intertwiners which was introduced in [9] and used for the derivation of (2) in [22]. From the Wightman two-point functions one obtains the stringlocal propagators which will be used in the construction of the \(e\)-independent S-matrix.
The definition of \(d_{sd}=1\) stringlocal tensor fields \(A_{\mu _{1}..\mu _{n} }\) in terms of their pointlike \(d_{sd}=n+1,\)\(s=n\) siblings leads to \(s\) intrinsic \(\phi \)-escorts
$$\begin{aligned} A_{\mu _{1}..\mu _{n}}(x,e)=A_{\mu _{1}..\mu _{n}}^{P}(x)+\partial _{\mu _{1}} \phi _{\mu _{2}..\mu _{n}}+\partial _{\mu _{1}}\partial _{\mu _{2}}\phi _{\mu _{3} ..\mu _{n}}+\cdots +\partial _{\mu _{1}}\ldots \partial _{\mu _{n_{n}}}\phi \end{aligned}$$
The left hand side represents a stringlocal spin \(s=n\) tensor potential associated to a pointlike tensor potential with the same spin. The \(\phi ^{\prime }s\)\(s=n-i,~i=1,..,n\) tensorial stringlocal fields of dimension \(d=n-i+1\); all the fields belong to the same localization (Borchers) class, in fact they are linear combinations of the same Wigner creation/annihilation operators with different intertwiners i.e. the linear relation can be written as one between intertwiners. Each \(\phi \) “peels off” a unit of dimension so that at the end one is left with the desired spin \(s\) stringlocal \(d_{sd}=1\) counterpart of the tensor analog of the Proca field. All intrinsic escorts appear in the first order stringlocal interaction density and play an important role in the \(e\)-independence of the S-matrix.

Such relations may be important in attempts to generalize the idea of gauge theories in terms of SLF couplings involving massive \(s>1~\)fields. In this paper we will stay with \(s=1.\)

3 Interactions Involving Stringlocal Field

In this section the idea of stringlike massive vectormeson fields in the same localization class as their pointlike Proca counterpart is used in order to convert nonrenormalizable pointlike interactions into stringlike renormalizable ones. For concreteness take the model of massive (spinor or scalar) QED. The pointlike interaction density is (all operator products are Wick-ordered)
$$\begin{aligned} L^{P}=gj^{\mu }A_{\mu }^{P},~~j^{\mu }=\bar{\psi }\gamma ^{\mu }\psi ~~or~~j^{\mu }=\varphi ^{*}\overleftrightarrow {\partial ^{\mu }}\varphi \end{aligned}$$
Since the short distance scaling dimension of the massive (Proca) vectorpotential is \(d_{sd}(A^{P})=2,\) the interaction is above the power-counting limit 4 since \(d_{sd}(L^{P})=5.\) Now we use (2) to rewrite the pointlike interaction in terms of its stringlike counterpart \(L\)
$$\begin{aligned} L^{P}=L\mathcal {-\partial }^{\mu }V_{\mu },~~V_{\mu }\equiv j_{\mu }\phi \end{aligned}$$
The stringlike interaction density \(L\) involves the \(d_{sd}(A)=1~\)stringlocal potential \(A_{\mu }(x,e)\) and is therefore renormalizable in the sense of power-counting. It results from the nonrenormalizable \(L^{P}\) by “peeling off” one unit of scaling dimension (for this reading one should bring the derivative term to the other side) so that \(L\) has instead of \(5\) only \(d_{sd}=4\). The rewriting of \(d_{sc}=5\) interaction densities into stringlike renormalizable densities with \(d_{sc}=4\) is our construction principle; it secures in addition to the validity of the power-counting restriction also the preservation of the physical content which one intuitively associates with pointlike interaction. In the BRST gauge setting the pointlike renormalizability is preserved at the price of a loss of Hilbert space. The Hilbert space positivity which is behind the physical short distance properties is recovered for gauge invariant operators whereas physical matter fields of charged particles are outside the range of gauge theory.

Integrating (13) with a test function \(g(x)~\)and taking the adiabatic limit \(g(x)\rightarrow g,\) the divergence term becomes a surface term at infinity which vanishes in massive models (in the sense of bilinear form between localized states). Formally the resulting integral represents the first order S-matrix; since the pointlike and the stringlike expressions coalesce in the adiabatic limit, this first order S is \(e\)-independent.

The idea is now to generalize this peeling process (13) and the subsequent adiabatic disposal of high \(d_{sd}\) derivative terms in the adiabatic limit. Since \(e\) is a fluctuating variable as \(x,\) the number of \(e\)-variables increases together with the number of \(x.~\)The main difference is that there is no integration over \(e^{\prime }s,~\)instead the construction is done in such a way that the \(e\)-dependence drops out in the adiabatic limit.

For pointlike fields the connection between fields and particles is given in terms of LSZ scattering theory which leads to the LSZ reduction formula in which the scattering amplitudes are expressed in terms of mass-shell restrictions of time-ordered functions of fields. The derivation uses only the mass-gap property and the Hilbert space positivity i.e. the LSZ theory cannot be derived in a Krein space BRST gauge setting. It would be extremely cumbersome to use this formula for perturbative calculations; for that purpose one uses the Stückelberg–Bogoliubov formula which expresses the n\(^{th}\) order S-matrix in terms of the time-ordered n-fold product of the first order interaction density
$$\begin{aligned} S(gL)\equiv \sum _{n}\frac{i^{n}}{n!}T_{n}(L,\ldots ,L)(g,\ldots ,g)=:Te^{i\int L(x)g(x)},~S_{scat}=\lim _{g(x)\rightarrow g}S(gL) \nonumber \\ \end{aligned}$$
Here \(g(x)\rightarrow g\) is the adiabatic limit in which the spacetime dependent coupling in the Bogoliubov operator functional \(S(g)\) approaches the physical coupling constant and the S-matrix become Poincaré invariant. There is no direct relation between the LSZ reduction formula and the perturbative Bogoliubov formula. The derivation of the latter uses the time-development \(U(t,s)\) in the interactions picture. Whereas the propagation operator at fixed times is formal (ill-defined for interacting fields), the Bogoliubov S-matrix formula is free of these drawbacks.
In order to formalize the idea of \(e\) independence of certain objects we rewrite the relation (13) in terms of a differential form calculus in the unit \(d=1+2~\)de Sitter space of spacelike directions where \(L\) and \(V_{\mu }\) are zero forms and \(Q_{\mu }=d_{e}V_{\mu }\) and \(u=d_{e}\phi \) are exact one-forms
$$\begin{aligned}&d_{e}(L\mathcal {-\partial }^{\mu }V_{\mu })=0~~or~L\mathcal {\ }d_{e} L=\partial ^{\mu }Q_{\mu },~~Q_{\mu }=d_{e}V_{\mu }\\&S^{(1)}=\int L^{P}d^{4}x=\int Ld^{4}x~~or~L^{P}\overset{AE}{\simeq }L\nonumber \end{aligned}$$
i.e. two operators are \(d_{e}\)-equivalent of their difference is a zero one-form, and two interactions are adiabatic equivalent (AE) if their adiabatic limits of their first order S-matrices agree. the two interactions are adiabatically equivalent (AE).
The multidimensional aspect of this differential calculus appears in higher orders of the S-matrix. The differential form of the multidimensional higher order \(e\)-independence is:
$$\begin{aligned}&d_{e}(TLL^{\prime }-\partial ^{\mu }TV_{\mu }L^{\prime })=0,~~d_{e^{\prime } }(TLL^{\prime }-\partial ^{\mu }TV_{\mu }^{\prime })=0\end{aligned}$$
$$\begin{aligned}&dTLL^{\prime }-d_{e}\partial ^{\mu }TV_{\mu }L^{\prime }-d_{e^{\prime }} \partial ^{\prime \mu }TLV_{\mu }^{\prime }=0,~d:=d_{e}+d_{e^{\prime }} \end{aligned}$$
where for simplicity of notation, we illustrate the basic idea for n = 2 and use the notation \(L^{\prime }\) for \(L(x^{\prime },e^{\prime }) \). The second line is the manifest symmetric form of (16). These relations are extensions of (15) which account for the noncommutance of derivatives with the time-ordering at point- or string- crossings.
This suggests to go one step further and write (with the same reasoning)
$$\begin{aligned}&d_{e^{\prime }}(\partial _{\mu }TV^{\mu }L^{\prime }-\partial _{\mu }\partial _{\nu }^{\prime }TV^{\mu }V^{\nu \prime })=0=d_{e}(\partial _{\nu }TLV^{\prime \nu }-\partial _{\mu }\partial _{\nu }^{\prime }TV^{\mu }V^{\nu \prime })\end{aligned}$$
$$\begin{aligned}&TL^{P}L^{\prime P}:=(TLL^{\prime }-\partial _{\mu }TV^{\mu }L^{\prime } -\partial _{\nu }TLV^{\prime \nu }+\partial _{\mu }\partial _{\nu }^{\prime }TV^{\mu }V^{\nu \prime }),\text { }\curvearrowright dTL^{P}L^{\prime P}=0 \end{aligned}$$
where the last line is obtained from adding the first line in (18) to the first line in (16) and writing the result as \(d(..)=0\) where \(d=d_{e}+d_{e^{\prime }}~\)is applied to the content of the bracket in the second line.

For the proof of string-independence of the S-matrix \(dS=0~\)in second order\(,~\)the validity of (17) is sufficient. But the relation (19) does more; it defines a second order pointlike interaction, whose direct definition would run into the standard problems of pointlike fields with nonrenormalizable interactions. The important point here is that, different from the direct pointlike renormalization theory, the conversion of the well-behaved stringlike to the more singular pointlike interaction density does not add new undetermined parameters. Another important message is that the high energy behavior of scattering amplitudes of \(s\ge 1\) interactions is better than what one expects from the mass shell restriction of nonrenormalizable time-ordered correlation functions. The on-shell improvement by “peeling off” high derivative terms which the adiabatic limit removes remains hidden in momentum space.

It is an interesting question whether this idea to start with a pointlike nonrenormalizable interaction and rewrite it the in terms of stringlike fields and the divergence of a \(V_{\mu }\) term also works for \(s>1,\) The precondition, namely the existence of a linear relation between a pointlike potential with \(d_{sd}=s+1\) and its stringlike \(d_{ds}=1~\)counterpart together with \(s~\)intrinsic escort \(\phi ^{\prime }s\) of spin zero up to \(s-1~\)(11) is certainly fulfilled.

Returning to the \(s=1\) case, the computation of the S-matrix starts by expanding the \(T_{0}\) product of fields in the bracket (16) into Wick-products. Our main interest is the 1-contraction component (the tree approximation). It consists of sums over terms where each term is a time-ordered propagator two fields multiplied with the Wick-ordered product of the remaining uncontracted fields. Although the tree contribution in terms of the \(T_{0}\) ordering is a well-defined expression, the above \(e\)-independence relations (16) are not fulfilled. The violation is called an “anomaly”
$$\begin{aligned} A_{e}&=\!d_{e}(T_{0}LL^{\prime }\!-\!\partial ^{\mu }T_{0}V_{\mu }L^{\prime } )_{1},\quad A_{e^{\prime }}\!=\!d_{e^{\prime }}(\ldots ),\quad A_{e^{\prime }}(x,e;x^{\prime },e^{\prime })\!=\!A_{e}(x^{\prime },e^{\prime };e,x)\\ \!-\!A_{e}&\!=\!d_{e}(N_{e}\!+\!R_{e}\!+\!\partial ^{\mu }N_{\mu ,e}),\quad \!-\!A_{e^{\prime } }\!=\!d_{e^{\prime }}(\ldots ),\quad N,R,N_{\mu }\text { }are~local~ T_{0}LL^{\prime }|_{1}\rightarrow \nonumber \\&TLL^{\prime }|_{1}\!=\!T_{0}LL^{\prime } |_{1}\!+\!N_{e}\!\!+\!\!R_{e},\quad T_{0}V_{\mu }L^{\prime }|_{1}\rightarrow TV_{\mu }L^{\prime }|_{1}\!=\!T_{0}LV_{\mu }^{\prime }|_{1}\!+\!\!N_{\mu }\nonumber \end{aligned}$$
Here local means that they are products of \(\delta (x-x^{\prime })~\) functions multiplied with Wick-products of four (point-or string-local) free fields; the subscript \(1~\)referes to the 1-contraction component. The notation \(N,N_{\mu }\) indicates that they are normalization terms which can be encoded into a change of time-ordering. All regular (non delta) terms cancel since for those one can take the derivative inside the time-ordered product and use the relation in the second line of (15)

The remaining \(R\) is quadrilinear in terms of scalar fields, including the scalar stringlocal intrinsic escort \(\phi \) of the vectormeson. It is not present in massive QED but it appears in models in which massive vectormesons are coupled to Hermitian scalar fields where it leads to a delta function modification of the second order \(TLL^{\prime }\rightarrow TLL^{\prime } +\delta (x-x^{\prime })L_{2}\). In fact the \(L_{2}\) turns out to have the form of the Mexican hat potential except that it is not part of the defining interaction but induced from the requirement of string-independence of the S-matrix. The main point in Sect. 5 is to show that behind the incorrect symmetry-breaking mechanism of the Higgs model there is the coupling of a massive vectormeson \(A_{\mu }\) to a Hermitian scalar field\(~H\). The implementation of the BRST gauge invariance for the S-matrix (or the \(e\)-independence in the Hilbert space formulation) induces a \(L_{2} ~\)quadrininear normalization term in \(H\) which has the form of a Mexican hat potential.

From a conceptual viewpoint the difference could not be bigger: instead of an imposed gauge-symmetry violating Mexican hat potental the implementation of BRST gauge invariance (or better the \(e\)-independence) on the second order S-matrix for a \(A\)\(H\) interaction induces quadrilinear Mexican hat like \(H\)-selfinteraction. There are only 3 types of renormalizable massive vectormeson couplings: couplings to complex (charged) matter (massive QED), their neutral counterpart (the \(H\)-coupling) and self-couplings (massive Y–M) which leaves no place for a mysterious mass-creating Higgs-mechanism. This confirms an important point made in the operator setting of the BRST gauge formulation more than 20 years ago by a group at the university of Zürich [23] [24]; for a more recent account see [25].

Perturbative calculations of the (on-shell) S-matrix in massive \(s\ge 1\) stringlike QFT are simpler than those of fields and their (off-shell) correlation functions; for calculation of interacting fields one has to extend the Stückelberg, Bogoliubov, Epstein-Glaser (SBEG) formalism for the S-matrix (14) to fields [26]. The matter fields enter the interaction density of massive QED only in the form of pointlike free currents. The stringlocal interaction transfers the string-locality of the vectorpotentials to that of the higher order matter fields \(\varphi (x,e).\)

The perturbative calculation of the matter field confirms the exponential relation (3) between the renormalizable stringlocal matter field and its singular pointlike partner. The analogy with exponential composites of a free scalar field suggests the exponentials of Wightman fields belong to a singular class of fields which do not permit a localization by smearing with arbitrary compact supported Schwartz testfunctions. Such singular pointlike fields are expected to belong to a field class studied by Jaffe [7] who succeed to formulate previous ideas [27] about a connection between non-renormalizability and breakdown of the standard localization property in a mathematical precise way. Their connection with renormalizable stringlocal fields in the present work assigns to them an interesting role in perturbation theory. A more detailed perturbative discussion of these singular pointlike objects (2) in terms of stringlocal Wightman fields will be given in separate work.

The conceptual content of (2) and (3) is, despite the resemblance with gauge transformations, very different from the role of the latter. These equations formalize the adherence of pointlike fields and their stringlike siblings to the same localization class; they have nothing to do with a gauge symmetry in Krein space whose only purpose is to rescue physical local quantum observables from an unphysical indefinite metric setting. Whereas the BRST gauge setting is consistent with Lagrangian quantization, the SLF Hilbert space setting does not support a quantization parallelism to classical fields theory; stringlike fields are not solutions of Euler-Lagrange equations and cannot be used in functional integral representations. Fortunately perturbation theory in the form of “causal perturbation” [26] does not depend on a quantization parallelism to classical field theory; an interaction density can be defined in terms of any form of free fields. The umbilical quantization cord of \(s\ge 1~\)QFT with classical fields is cut because the clash between the Hilbert space positivity with pointlike localization which leads to SLF has no counterpart in classical field theory. The present work shows that this has its strongest impact in gauge theory.

SLF is what the foundational causal localization principle leads to if one does not force it to pass through the quantization parallelism to the less fundamental Lagrangian quantization. Classical field theory shares many analogies with QFT; after all this is the reason why Lagrangian quantization, patched up by renormalization theory and some additional hindsight, turned out to be useful. But it looses its guiding power when it comes to structures which are in contradiction with positivity requirements of the Hilbert space setting of QT and this includes all \(s\ge 1\) interactions.

The new SLF Hilbert space setting is particularly simple for models with a mass gap since in such cases the standard scattering relation between fields and particles holds and the connection with (singular) pointlike fields is not completely lost. The S-matrix turns out to be a global \(e\)-independent invariant of the local equivalence class of fields which includes renormalizable stringlocal fields and singular pointlike fields. As mentioned before the new view inverts the conceptual relation between massive and zero mass interactions of vectormesons with matter fields and puts it from its head to its feet. Theories with mass gaps are by far the simpler models since they follow the standard spacetime LSZ asymptotic relation between fields and particles, whereas most of the not understood properties, as gluon/quark confinement and a spacetime description of the momentum space recipe of photon-inclusive cross sections for collisions between charged particles in QED, refer to massless vectormesons. Physical matter fields coupled to zero mass vectorpotentials are inherently stringlocal. The only known way to overcome the problem of the radical change of the Wigner–Fock space in the massless limit is to study the massless limit of correlation functions und use the Wightman reconstruction to return to an operator description in an appropriate Hilbert space. This leads to a much clearer idea of what confinement is about and suggests perturbative resummation techniques to prove it (end of Sect. 5).

The new formulation for interacting vectormesons leads to many conceptual and computational changes which cannot be accomodated in Feynman rules; this includes the process of induction of interactions from string-independence of scattering amplitudes. The biggest surprise arises from the application of the new ideas to the interaction of a massive vectormeson with a Hermitian scalar matter field i.e. the neutral counterpart of massive scalar QED. All interactions of massive vectormesons in a Hilbert space setting involve intrinsic escort fields \(\phi \) which explicitly participate in the interaction\(.~\)It looks like a curious accident that these fields have many properties which are incorrectly ascribed to \(H\)-fields. They are inexorably connected with massive vectormesons and disappear in the massless limit. But they do not add new degrees of freedom to massive vectormesons nor do they break any symmetry and create masses. Their appearance is a new phenomenon of massive \(s\ge 1\) interactions in a Hilbert space formulation i.e. their existence is a result of the powerful Hilbert space positivity which has only partially been taken care of in the Krein space gauge setting.

The SLF Hilbert space formulation does not only reinstate the particle democracy of the old S-matrix setting (no distinguished mass-giving “God particle”), but it also removes the apartheid between gauge theories and non gauge theories by collecting all QFT, independent of their spin, under the conceptual roof of the foundational causal locality principle of QFT.

A gauge theoretic formulation for \(s>1\) interactions does not seem to be known. On the other hand the higher spin analogs of the linear relation between stringlocal free fields and their pointlike partners in the same localization class are rather straightforward; instead of a single scalar stringlike Stückelberg field one obtains a linear relation involving a family of intrinsic tensor Stückelberg fields for all spins up to \(s-1\) [12]. These are the prerequisites for an extension to higher spin interactions. The difficult problem is to find interactions with lead to local observables. Since in the present paper our main interest are interactions involving massive vectormesons, we will not comment on \(s>1.\)

Since the gauge theoretic setting is an established part of particle theory, it may be instructive to compare the stringlocal Hilbert space setting with the gauge theoretic BRST formulation in Krein space in somewhat more detail. It can be presented in a formally similar manner as (2), namely as18
$$\begin{aligned} A_{\mu }^{K}(x)\simeq A_{\mu }^{P}(x)+\frac{1}{2m}\partial _{\mu }\phi ^{K}(x),\quad \curvearrowright \partial ^{\mu }A_{\mu }^{K}(x)-m\phi ^{K}(x)\simeq 0 \end{aligned}$$
where the superscript \(K\) refers again to Krein space i.e. in an indefinite metric space which is obtained from a Hilbert spaces by changing the metric in terms of Hermitian operator \(\eta \) [23, 25]. Here the reduction of the short distance scale dimension \(d=2\) of the Proca potential is achieved not by changing the physical localization but rather by “brute force” namely by compensating the renormalizability-preventing scale dimension by a free scalar field with the two-point function of the opposite sign so that the resulting \(d=1\)\(A_{\mu }^{K}~\)potential acts also in Krein space. As a result of the interaction of this pointlike potential with a \(s<1\) matter field, the indefinite metric creeps into all fields and renders them unphysical. The difficult part of this formalism is to find an “operator gauge requirement” which filters out a subalgebra generated by local observable fields which applied to the vacuum create a Hilbert space.
As well known to the many practitioners of the BRST formalism, this cannot be done directly since the above relations (21) cannot be formulated as operator equations; as they stand they only express equivalences. In order to obtain a manageable operator formalism one must extend the above set of fields in terms of anti-commuting ghost operators \(u^{K},\hat{u}^{K}.\) The result is the famous BRST setting in which one can formulate a certain (unphysical) gauge symmetry in terms of a nilpotent \(s\)-operation whose only purpose is to describe the content in terms of local observables as gauge-invariants
$$\begin{aligned} sA_{\mu }^{K}=\partial _{\mu }u^{K},\quad s\phi ^{K}=u^{K}, \quad su^{K}=0, \quad s\hat{u} ^{K}=-(\partial A^{K}+m^{2}\phi ^{K}) \end{aligned}$$
The bracket in the last relations vanishes on physical states. Unlike physical symmetries gauge symmetries by their very nature cannot be broken (neither explicitly nor spontaneous) since this would wreck their only purpose, namely filtering out physics from an unphysical description.

In the above presentation the analogy to a Hilbert space formulation has been highlighted by choosing the same notation for those operators which permit a formal correspondence. But the analogy falls apart on the conceptual physical level since the purpose of abstract cohomological BRST formalism based on a nilpotent \(s\)-operation in Krein space is totally different from that of the differential form calculus \(d=d_{e_{1}}+..d_{e_{n}}~\)in the space of string directions (d=1+2 de Sitter spacetime) whose purpose is to relate the stringlike description to its pointlike counterpart in the same localization class. The SLF setting retains the pointlike fields \(A_{\mu }^{P}\) and the definition (3) of the singular \(\varphi ^{P}(x)\) in terms of its stringlike sibling, whereas the pointlike physical objects, apart from local observables, are lost in the BRST setting.

A curious side result of this formal analogy is the fact that the SLF differential form calculus in terms of the \(Q_{\mu }\) and \(u,\) which strengthens the formal similarity of the \(d\) differential calculus with the more abstract nilpotent \(s\) operation, also allows a better formal \(m\rightarrow 0\) limit behavior. Such formal considerations are however no replacement for the explicit construction of the expectation values of stringlocal massless theory as massless limits of their stringlocal massive counterparts. Such constructions, which remain outside the gauge formalism, provide the safest way of constructing stringlocal physical matter fields in QED.

Another point which remains insufficiently understood is to what extend one needs gauge symmetry in order to obtain e.g. the relation between the quadratic second order \(A_{\mu }\) coupling and the first order coupling. The answer is; one does not need it at all; the Hilbert space SLF setting determines the relation between these two couplings (next section). This is particularly important in Y–M QFT; in that case the typical form of the nonabelian gauge interaction between stringlocal massive gluons of equal mass results from locality in conjunction with the Hilbert space positivity. The relation between the first order defining interaction and the induced local part of the second order interaction density, which is the epitome of classical gauge theory, is the conceptual implication of quantum locality and Hilbert space positivity. Local quantum physics can stand on its own feet; it does not need the quantization crutches of classical fibre bundle mathematics. The Lie-algebraic form of the selfinteraction between gluons needs no classical imposition; it is the result of the foundational causal localization principle [36].

The main purpose of the following two sections is the explicit illustration of these new concepts of SLF in Hilbert space and their computational results in second order perturbation theory. For a more detailed and mathematically rigorous presentation of the modifications of renormalized perturbation theory in the presence of stringlike fields; see the forthcoming work [22, 34].

4 S-matrix of Massive Scalar QED

According to the traditional view massless scalar QED is a pointlike model with two coupling parameter19; it is known to be renormalizable in the pointlike BRST Krein space setting. Unlike its classical counterpart, this quantum gauge description is severely restricted; the positivity requirements of the Hilbert space clash with the pointlike localization and quantum gauge theory is the result of a compromise. the description is limited to local observables which constitute the gauge invariant part, whereas the formally gauge-variant charge-carrying operators and physical charge-carrying states remain outside the pointlike BRST formalism.

Massiven vectormesons are described in terms of Proca potentials \(A_{\mu }^{P}\) whose short distance dimension \(d_{sd}=2~\)is too high for the construction of interactions below the power-counting limit. The BRST gauge setting attains renormalizability by replacing the Proca potentials by \(d_{sd}=1\) vectorpotentials \(A_{\mu }^{K}\) in Krein space; again the price to be paid is the rather small physical range of such a description which includes the gauge invariant local observables but contains no computational accessible construction for physical matter fields and the states which they create from the vacuum. In a Hilbert space description the existence of a mass gap would insure the validity of scattering theory, but without the powerful Hilbert space positivity the physical compass for navigating through the network of field-particle relations is lost.

It is the main purpose of the present paper to change this situation by proposing a Hilbert space setting which maintains the \(d_{sd}=1~\)short distance property for any integer spin20 and as a result permits to extends renormalization theory in Hilbert space. Here we are primarily interested in a Hilbert space setting which replaces the BRST gauge formalism. Since the mass gap property in Hilbert space setting guaranties the validity of scattering theory and the standard field-particle relations it seems to be reasonable to construct massless models as massless limits of their massive counterpart. The important difference to the BRST gauge setting is that all fields are physical; there is no point is studying such limits for pointlike gauge-variant fields. This way of studing massless models as limits of their massive counterparts is contrary to the standard approach which in its most extreme form claims that masses of vectormesons should be viewed as arising from a spontaneous symmetry-breaking from massless vectorpotentials.

The alleged simplicity of massless models refers to formal aspects of renormalized perturbation theory and ignore the unsolved problems of “infraparticles” in QED [28], not to mention the unsolved millennium problem of confinement in QCD.

In this section the model of massive scalar QED will be used as a nontrivial testing ground for the new SLF Hilbert space formalism. The idea is to use the short-distance improvement of stringlike potentials in Hilbert space, while mainting the pointlike nature of observables which in the massive case includes the string independence of the S-matrix. It will be shown how the quadratic term in the vectorponetial arises from the locality of the second order S-matrix.

The defining first order stringlocal interaction density of massive scalar QED
$$\begin{aligned} L(x,e)&=gA_{\mu }(x,e)j^{\mu }(x)=L^{P}+\partial ^{\mu }V_{\mu }\\ j^{\mu }&=\varphi ^{*}\overleftrightarrow {\partial ^{\mu }}\varphi ,~V_{\mu }=\phi j_{\mu }\nonumber \end{aligned}$$
is according to (15) \(d_{e}\)-equivalent to its pointlocal counterpart \(L^{P}\). This secures the \(e\)-independence of the first order S-matrix in the AE limit. In these equivalences the stringlocal Stückelberg field \(\phi ,\) which appears explicitly in \(V_{\mu },\) play an essential role. Whereas the first order relation is a result of the definition of a “stringlocal” interaction, the second order relation (16) is a nontrivial restriction on the renormalization.
One defines a reference time-ordering \(T_{0}\) of two-pointfunctions of derivatives of the complex scalar field \(\varphi \) by taking the derivatives outside the two-point function e.g.
$$\begin{aligned} \left\langle T_{0}\partial _{\mu }\varphi ^{*}(x)\partial _{\nu }^{\prime }\varphi (x^{\prime })\right\rangle =i\frac{\partial _{\mu }\partial _{\nu } ^{\prime }}{\left( 2\pi \right) ^{4}}\int d^{4}pe^{-ipx}\frac{1}{p^{2} -m^{2}+i\varepsilon } \end{aligned}$$
On the other hand the renormalized time ordering in Epstein and Glaser’s renormalization uses normalization terms whose delta function degree is determined by the short distance scaling degree of the object to be renormalized. For the degree 4 propagator of the derivative of a free scalar field the E–G renormalization leads to the modified \(T\)-product
$$\begin{aligned} \left\langle T\partial _{\mu }\varphi ^{*}(x)\partial _{\nu }^{\prime } \varphi (x^{\prime })\right\rangle =\left\langle T_{0}\partial _{\mu } \varphi ^{*}(x)\partial _{\nu }^{\prime }\varphi (x^{\prime })\right\rangle -aig_{\mu \nu }\delta (x-x^{\prime }) \end{aligned}$$
where \(a\) is a free parameter.
If we were to treat the defining first order interaction \(A_{\mu }j^{\mu }\) in terms of pointlike \(A_{\mu }\) field in a Krein space the interaction is renormalizable in the perturbative inductive Epstein–Glaser setting where it leads to two counterterms. The first counterterm (24) appears in the second order tree approximation and amounts to a modification of the interaction through a second order contact term (all operator products are meant to be Wick-ordered)
$$\begin{aligned} aA_{\mu }(x)A^{\mu }(x)\varphi ^{*}(x)\varphi (x) \end{aligned}$$
with an independent coupling parameter \(a.\) There is an additional quadrilinear counterterm with a coupling parameter of the form
$$\begin{aligned} b\left( \varphi ^{*}(x)\varphi (x)\right) ^{2} \end{aligned}$$
which appears for the first time in 4th order (the box graph); these two counterterm exhaust the possibilities of E–G counterterm structures (primitively divergent contributions in the Feynman graph setting), which means that the renormalized theory is 3-parametric, the first order coupling and \(a,b.\)

To recuperate local observables acting in a Hilbert space (at the expense of charge-carrying matter fields which remain unphysical fields in Krein space) one has to extend the Krein space formulation by the ghost operators; in this way one arrives at the BRST gauge formulation which fixes the parameter \(a\) in (25) to a numerical value \(a=2\) by the requirement that the second order S-matrix fulfills \(sS=0,~\)where \(s~\)is the nilpotent BRST s-operation (the BRST implementation of quantum gauge invariance) according to the rules of a formal “gauge symmetry”. By itself this term (25) has no direct physical interpretation apart from its role in the extraction of local observables from an unphysical description. An elegant way to deal with this second order \(A\cdot A~\)dependent term is to encode it into the change \(T_{0}\rightarrow T\) (25) of the time-ordered product; in that case the second order tree-contribution rewritten in terms of \(T_{0}~\)reproduces this term; in fact the encoding into \(T~\)covers the tree component of all orders. The standard gauge formalism leaves the quadratic contribution and combines it (the gauge-invariant extension) with the first order term but this does not change the fact that only the sum of \(T_{0}LL\) and the pointlike contribution to the second order \(S\) is \(s\)-invariant.

Hence the \(s\)-invariance reduces the original 3-parametric pointlike model to two a 2-parameter model, the quadratic second order counterterm is “induced” by operator gauge invariance. In contrast to classical gauge theory, quantum gauge symmetry is a technical trick which permits to extract the physics from a Krein space description; in particular it cannot be spontaneously broken.

In the SLF Hilbert space setting the \(e\)-independence of the S-matrix induces the correct value of \(a\) from the model-defining first order \(A\cdot j~\)interaction; it is simply the result of the implementation of locality in Hilbert space setting. No additional principle as gauge symmetry has to be invoked in order to fix \(a\) to its correct numerical value. The induction mechanism exists only for higher spins \(s\ge 1;\) for lower spins the renormalization theory is the well-known counterterm formalism with freely varying coupling strengths.

For the case at hand \(a\) is calculated as follows. From the results in the previous section we know that the second order locality requirement for the S-matrix in the presence of stringlike fields amounts to the vanishing of the \(d_{e}~\)operation on the renormalized tree (\(1\)-contraction) component
$$\begin{aligned}&d_{e}(TA\cdot jA^{\prime }\cdot j^{\prime }-\partial ^{\mu }T\phi j_{\mu }A^{\prime }\cdot j^{\prime })_{1-con}=0\\ -A_{e}&:=d_{e}(T_{0}A\cdot jA^{\prime }\cdot j^{\prime }-\partial ^{\mu } T_{0}\phi j_{\mu }A^{\prime }\cdot j^{\prime })_{1-con}=N_{e}+\partial ^{\mu }N_{_{e},\mu }\nonumber \end{aligned}$$
and a similar expression in which the unprimed and primed \(x,e\) are interchanged. Both \(N\) are products of a delta function \(\delta (x-x^{\prime })~\)with a Wick polynomial of degree 4. The simplicity of the model allows us to take a short cut which bypasses the calculation of the \(N^{\prime }s\) in the anomaly. By inspection on sees that the choice \(a=1~\)in the definition of the “renormalized” \(T\) (24) solves the problem of the anomalies from \(\varphi \)-contractions; as a consequence of the identity \(d_{e}\partial ^{\mu }\phi =d_{e}A^{\mu }\) there are no contributions from \(\phi \)-\(A_{\nu }\) contractions. This renormalized \(T~\)product is characterized by the absence of the propagator anomaly for the derivative of the \(\varphi \)-field.
$$\begin{aligned} \partial ^{\mu }\left\langle T\partial _{\mu }\varphi ^{*}(x)\partial _{\nu }^{\prime }\varphi (x^{\prime })\right\rangle =-i\partial _{\nu }^{\prime } \delta (x-x^{\prime })-ia\partial _{v}\delta (x-x^{\prime })+reg=reg~~if~a=1 \end{aligned}$$
The \(N_{e}\) and \(N_{e,\mu }\) can be red off from the difference between the \(T\) and \(T_{0}\) in (27)\(.\)
$$\begin{aligned} N_{e}=\delta \varphi ^{*}\varphi A\cdot A^{\prime },\quad N_{e,\mu }=\delta \varphi ^{*}\varphi \phi A_{\mu }^{\prime } \end{aligned}$$
The \(N_{e^{\prime }},~N_{e^{\prime },\mu }~\)result from \(e\leftrightarrow e^{\prime }~\)and the \(N\) and \(N_{\mu }~\)are the sums of the primed and unprimed \(Ns.\)
As expected from gauge theory, \(a=1~\)leads to a \(N~\)which is quadratic in the vectorpotential of the form21
$$\begin{aligned} 2\varphi ^{*}A_{\mu }\delta (x-x^{\prime })\varphi ^{\prime }A^{\mu \prime } \end{aligned}$$
There is a small but nevertheless important difference to the corresponding gauge theoretic result; the two \(A~\)have independently fluctuating string directions. This is quite different from the use of the axial gauge in gauge theory where the coalescing gauge parameter causes intractable renormalization problems which led to the abandonment of this gauge. Together with the contribution from \(N_{e^{\prime }}\) with \(x,e\longleftrightarrow x^{\prime },e^{\prime }\) one finds the \(e\)-\(e^{\prime }~\)symmetric form
$$\begin{aligned} TLL^{\prime }&=T_{0}LL^{\prime }+2i\delta (x-x^{\prime })L_{2},~L_{2} =2\varphi ^{*}(x)\varphi (x)A\cdot A^{\prime }\\ S&=ig\int (L+\frac{-i}{2}gL_{2})-g^{2}\frac{1}{2}\int \int T_{0}LL^{\prime }+higher~orders \end{aligned}$$
The last line is the gauge theoretic way of writing the result up to second order. But the preferable notation in the SLF setting is to encode the \(L_{2}\) term into a modified \(T\)-product. The reason is that only the sum in the second formula leads to a \(e\)-independent second order S-matrix. The \(T\)-encoding instead of the \(T_{0}\) has the additional advantage that it takes care of all the higher order tree contributions. In a formal pointlike setting all counterterm couplings are independent; the induction of \(a\) which reduces the number of freely varying counterterm parameters is the result of \(e\)-independence of the second order S-matrix.

Another important difference to the pointlike setting is the possibility to use the relation (19) of the previous section to define a pointlike second order interaction density. Such a calculation is more involved than that for the S-matrix since one also has to calculate the “renormalized” derivative terms (the \(N_{\mu }\) terms). As mentioned before the SLF Hilbert space setting induces pointlike densities; the standard problem of nonrenormalizability of having an ever increasing number of counterterms with new couplings is evaded although the presence of derivative terms increases the high energy behavior. Since the derivative terms drop out in the adiabatic S-matrix limit the scattering amplitudes inherent the improved high energy behavior of the stringlocal interaction density. This may serve as a warning against inferring the presence of additional particles on the basis of Feynman graphs for pointlike interactions.

5 Maxwell-Currents, Charge-Screening and the Higgs Issue

Additional information about the stringlocal setting for massive QED can be obtained from the extension of the perturbative formalism to the construction of stringlocal fields and their possibly pointlike composites. They reveal aspects which in the global S-matrix remain hidden. One such observable is the identically conserved Maxwell-current \(j\) which is defined as the divergence of the field strength
$$\begin{aligned}&\partial ^{\nu }F_{\mu \nu }=gj_{\mu },~~F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu },~~j_{\mu }=\partial ^{\nu }F_{\mu \nu }\\&J_{\mu }=\frac{1}{2}i\varphi ^{*}D_{\mu }\varphi +h.c.,~Q=\int J_{0} (x)d^{3}x,Q^{Max}=\int j_{0}(x)d^{3}x\nonumber \end{aligned}$$
where the integral over the Maxwell-current \(j_{\mu }\) defines the Maxwell charge \(Q_{M}.\) In the massless case this charge coalesces with the particle-antiparticle “counting charge”, but it deviates in a physically significant way in the presence of massive vectormesons. This difference finds its physical expression in the charge-screening theorem which confirms a conjecture by Schwinger [29]; already the free Proca Potential leads as a result of \(j_{\mu }=m^{2}A_{\mu }^{P}\) to a screened charge \(Q_{M}=0\).


(Swieca 1976 [30, 32]) In the presence of a mass gap, the identically conserved current associated with an antisymmetric tensor \(F_{\mu \nu }~\) leads to a screened charge \(Q_{M}=\int j_{0}d^{3}x=0.\)

In order to avoid any confusion, QFT in this paper always refers to a theory of quantum fields (or localized nets of algebras [5]) in Hilbert space; the proof of this and the following structural theorems depend on the Hilbert space positivity in an essential way; no structural theorem in QFT can be derived without the Hilbert space positivity. There exists another structural theorem which has a close historical connection to the screening theorem and which is equally interesting in the context of massive gauge theories and the \(H\)-coupling.


(Buchholz–Fredenhagen 1982 [4]) In a QFT with local observables and a mass gap, charge-carrying matter can be localized in arbitrary narrow spacelike cones (whose cores are semi-infinite spacelike strings).

This theorem states that, under the mentioned restriction, a QFT can be generated from operators which are members of operator algebras localized in arbitrary narrow spacelike cones i.e. objects which are localized on spacelike surfaces are not needed as building blocks of such a QFT. The possibility of generating a particular model in terms of objects localized in arbitrary small double cone22 regions (whose core is a point) is a special case covered by the theorem.

It is believed that this localization property continues to be valid in massless models for \(s\ge 1\). In the case of QED there exists a structural proof based on the appropriately formulated quantum Gauss law [6]. But as a result of infinitely extended photon-clouds in the \(e\)-direction, the charged stringlocal matter fields in QED are more rigid than their massive counterparts. Whereas the string directions of massive strings can be changed by Lorentz transformations, the Lorentz invariance in QED is spontaneously broken [3] and the string directions define a continuous set of superselection rules within which the countable set of charge superselection rules can be unhinged [28].

These zero mass localization aspects are inexorably linked with the occurrence of perturbative logarithmic divergencies in scattering amplitudes whose summation of the leading logarithmic is an important tool in the study of the physics behind perturbative infrared divergencies. Their summation before taking the massless limit is known to lead to the well-known vanishing of scattering amplitudes for scattering of charged particles with a finite number of photons in the in/out states [33]. The physical aspects of collision theory in such cases are described in terms of photon-inclusive cross sections which involves a summation over infinitely many infrared photons.

Another spacetime interpretation of this result is based on the idea that charged “infraparticles” have time-ordered correlation functions which, instead of the usual mass-shell poles, have milder coupling-dependent cut singularities. The latter are too mild for being able to compensate the large time spreading of wave packets in the LSZ scattering limits and consequently lead to a vanishing \(t\rightarrow \infty \) limit. On the other hand zero mass interaction of pointlike fields with \(s<1\) maintain the particle structure and are consistent with the standard time-dependent scattering theory.

It is useful for later purpose to complete this list of structural theorems by adding a third theorem about spontaneous symmetry breaking which is a precise version of Goldstone’s idea which he exemplified on Lagrangian models.


(Ezawa–Swieca 1967 [8, 31]) The large-distance divergence of the charge associated to a conserved current (i.e. the intrinsic definition of a spontaneous symmetry breaking) \(Q_{G}=\infty \) is the result of the presence of a massless Goldstone boson which couples to the current.

Emboldened by the successful test for the stringlocal renormalization setting of massive scalar QED in the previous section, we now consider the coupling of a massive vectormeson to a neutral field\(H,\) where as before \(H\) stands for Hermitian or (as it will become clear later on) Higgs. We could call such a coupling massive “chargeless QED” if it would not be for the fact that its massless limit is trivial since the interaction disappears and hence it is not part of classic Maxwell theory.

As already indicated before, the renormalization theory for a chargeless (Hermitian) coupling will lead to more induced (even and odd) terms since the evenness from (particle-antiparticle) charge conservation is now absent. We start the induction from the most general pointlike interaction of engineering dimension \(d_{en}=3\) between a Proca potential and a scalar Hermitian field (omitting the coupling strength \(g\))
$$\begin{aligned} L^{P}=m(gA_{\mu }^{P}A^{P\mu }H+bH^{3}) \end{aligned}$$
where the presence of the factor \(m~\)(the vectormeson mass) maintains the engineering dimension to be that of an interaction density, namely \(d_{eng}=4.\) Since the short distance scale dimension of the Proca field is \(d=2,\) the operator dimension of the interaction density is \(d=5;\) hence the pointlike model is nonrenormalizable, as expected. A third possible \(d=5~\)trilinear term \(A_{\mu }^{P}\partial ^{\mu }HH\) does not contribute since as a result of \(\partial ^{\mu }A_{\mu }^{P}=0\) it turns out to be a total derivative. We will not add a quadrilinear term \(cH^{4}\) but we will later see that the string-independence of the S-matrix induces such a term (the induced Mexican hat potential).
The “peeling formula” for \(L^{P}\) i.e. its decomposition into a stringlocal \(L\) and an on-shell disposable “surface term” is straightforward and leads to
$$\begin{aligned}&L^{P}\!=\!L\mathcal {-\partial }^{\mu }V_{\mu },~with~L\!=\!m\left( A_{\mu }A^{\mu }H\!+\!A^{\mu }\phi \overleftrightarrow {\partial }_{\mu }H-\frac{m_{H}^{2}}{2}\phi ^{2} H\!+\!bH^{3}\right) \\&and~V_{\mu }=m\left( A_{\mu }\phi H+\frac{1}{2}\phi ^{2}\overleftrightarrow {\partial }_{\mu }H\right) ,~Q_{\mu }=d_{e}V_{\mu }=m\left( A_{\mu }uH+u\phi \overleftrightarrow {\partial }_{\mu }H\right) \nonumber \end{aligned}$$
where in returning from \(L\) to \(L^{P}\) the \(m_{H}^{2}\) in \(L\) is compensated by the mass term in the Klein-Gordon equation which results from the divergence of \(V_{\mu }.\) Let us now look at the second order relation (16) which expresses the independence from \(e~\)(with a corresponding relation for \(d_{e^{\prime }}\))
$$\begin{aligned} d_{e}(T_{0}LL^{\prime }-\partial ^{\mu }T_{0}V_{\mu }L^{\prime })_{1-con} =(d_{e}T_{0}LL^{\prime }-\partial ^{\mu }T_{0}Q_{\mu }L^{\prime })_{1-con} \ne 0 \end{aligned}$$
where \(T_{0}\) is defined in the same way as previous (take all derivatives in front of the time-ordering). The violation of the \(e~\)and \(e^{\prime } ~\)independence terms in the bracket are well-defined, and as in (20) of the third section the anomalies lead us to the necessary \(N,R\) and \(N_{\mu }\) modifications where again \(N\) contains the quadratic in \(A_{\mu }\) contributions. The \(L~\)of the neutral model has more terms than massive scalar QED which consists of only one term. In fact it will turn out that the requirement of second order string-independence of the S-matrix induces self-interacting \(H\)-contributions; in fact \(R\) stand for such induced \(A_{\mu }\)-independent contributions which did not occur in the massive QCD model. Despite the presence of more terms, the \(H\) model depend, just as its charged counterpart in the previous section, only on the massive vectormeson-\(H\) coupling \(g.~\)In short: all terms beyond the basic \(A\cdot AH\) interactions are “induced” by the locality and positivity (Hilbert space) requirements of QFT and hence depend only on the masses of the \(A_{\mu }\) and \(H\) fields.

It is helpful for the reader (and also matter of historical correction) to give credit to previous important work which pointed to the misunderstandings of the Higgs symmetry-breaking mechanism [23, 24]. Their work, which was based on the operator formulation of the BRST gauge setting, unfortunately remained unnoticed.

In the BRST setting one starts from the counterpart of (34) in terms of the abstract nilpotent \(s\)-operation replacing the concrete differential calculus on the directional de Sitter space. The basic first order relation which corresponds to (33) is
$$\begin{aligned} sL^{K}=\partial ^{\mu }Q_{\mu }^{K} \end{aligned}$$
where the \(s\)-operation on the field (including the ghost fields) was mentioned at the end of the third section. In terms of this \(s\) operation the BRST anomaly is defined23
$$\begin{aligned} A^{K}:=(sT_{0}L^{K}L^{\prime K}-\partial ^{\mu }T_{0}Q_{\mu }^{K}L^{\prime K}-\partial ^{\prime \mu }T_{0}L^{K}Q_{\mu }^{\prime K})_{1-con} \end{aligned}$$
where \(K\) refers to the Krein space and the violation of the \(s\)-invariance comes again from delta function contributions which arise from the application of the wave operator to time-ordered propagators. The \(L^{K}\) and \(Q_{\mu } ^{K}~\)in [23] are (for simplicity of notation we omit the superscript \(K\) for Krein space on the individual fields \(A_{\mu },\phi ,u\)) of the BRST gauge setting
$$\begin{aligned} L^{K}&=m\left( A\cdot AH-H\overleftrightarrow {\partial }\phi \cdot A-\frac{m_{H}^{2}}{2m^{2}}H\phi ^{2}+bH^{3}+u\tilde{u}H\right) \\ Q_{\mu }^{K}&=m\left( uA_{\mu }H-\frac{1}{m}u\phi \overleftrightarrow {\partial }_{\mu }H\right) \nonumber \end{aligned}$$
Apart from the appearance of the \(u\tilde{u}~\)ghost contribution and the different engineering dimension of the negative metric Stückelberg field \(\phi ^{K}\) as compared to the stringlocal escort \(\phi ~\)(\(m\phi \sim \phi ^{K}\)) the formulas are identical, despite the big difference between the conceptual aspects of their derivation.
There are now two propagators involving derivatives of fields whose divergence leads to delta functions.
$$\begin{aligned} \partial ^{\mu }\left\langle T_{0}\partial _{\mu }H\partial _{\nu }^{\prime }H^{\prime }\right\rangle&=-i\partial _{\nu }^{\prime }\delta (x-x^{\prime })+reg,~~\partial ^{\mu }\left\langle T_{0}\partial _{\mu }HH^{\prime }\right\rangle =-i\delta (x-x^{\prime })+reg~\end{aligned}$$
$$\begin{aligned} \partial ^{\mu }\left\langle T_{0}\partial _{\mu }\phi \partial _{\nu }^{\prime } \phi ^{\prime }\right\rangle&=i\partial _{\nu }^{\prime }\delta (x-x^{\prime })+reg,~~\partial ^{\mu }\left\langle T_{0}\partial _{\mu }\phi \phi ^{\prime }\right\rangle =i\delta (x-x)+reg \end{aligned}$$
The negative sign in the second line comes from the Krein field \(\phi \) whose two-point function has the opposite sign (the negative metric Stückelberg field). Again we use the freedom of normalization.
$$\begin{aligned} \left\langle T\partial _{\mu }H\partial _{\nu }^{\prime }H^{\prime }\right\rangle&=\left\langle T_{0}\partial _{\mu }H\partial _{\nu }^{\prime }H^{\prime }\right\rangle -a_{H}g_{\mu \nu }\delta (x-x^{\prime })\\ \left\langle T\partial _{\mu }\phi ^{K}\partial _{\nu }^{\prime }\phi ^{\prime K}\right\rangle&=\left\langle T_{0}\partial _{\mu }\phi ^{K}\partial _{\nu }^{\prime }\phi ^{\prime K}\right\rangle +a_{\phi }g_{\mu \nu }\delta (x-x^{\prime })\nonumber \end{aligned}$$
As in the previous section, the contribution to the \(N\) of the anomaly coming from contractions of \(\partial _{\mu }H~~\)and \(\partial _{\mu }\phi ~\)with the first two terms in \(L\) can be absorbed in a redefinition \(T_{0}\rightarrow T~\)made to vanish by choosing \(a_{H}=1=a_{\phi }.~\)These terms lead to a nontrivial \(N\)-contribution to the anomaly (20).
But in contrast to the massive QED case, the story does not end here. There are two \(A_{\mu }\)-independent quadrilinear remaining delta anomaly terms which result from contractions of the second term in \(Q_{\mu }^{K}\) with the third and fourth term in \(L^{K}.\) They lead to a “potential” in the two scalar fields
$$\begin{aligned} R=-i\delta (x-x^{\prime })\left\{ -\left( \frac{m_{H}^{2}}{m^{2}}+3b\right) \phi ^{2} H^{2}+\frac{m_{H}^{2}}{4m^{2}}\phi ^{4}\right\} \end{aligned}$$
As in the case of massive QED we omit the calculation of the \(N_{\mu }\) which renormalize the time order products \(Q_{\mu }L^{\prime }\) and \(LQ_{\mu }^{\prime }.\) At this point the \(b\) is still a free coupling parameter.
In order to get from the \(R\)-potential to the Mexican hat form, we follow Scharf [23] and observe that the tree approximation of the third order has a nontrivial anomaly which comes from the time ordered product the first order \(Q_{\mu }^{k}~\)with the second order potential \(T_{0}Q_{\mu } R.~\)Without adding the before mentioned \(cH^{4}\) to the induced potential \(R~\)it is not possible to get rid of this anomaly. However with this term one finds compensation for the following values of \(b,c\).
$$\begin{aligned} b&=-\frac{m_{H}^{2}}{2m^{2}},~c=-\frac{m_{H}^{2}}{4m^{2}}\end{aligned}$$
$$\begin{aligned} R_{ind}^{K}&=-i\delta (x-x^{\prime })\frac{m_{H}^{2}}{4m^{2}}(H^{2}+\phi ^{2})^{2} \end{aligned}$$
As in the case of massive QED one may combine the induced second order \(R_{ind}~\)potential with the \(\phi ,H~\)dependent part of the first order and write the result in the form
$$\begin{aligned} V_{1}^{K}&=g\frac{m_{H}^{2}}{2m}(H\phi ^{2}+H^{3}),~V_{2}=g^{2}\frac{m_{H}^{2}}{4m^{2}}(H^{2}+\phi ^{2})^{2}\\ V^{K}&=V_{1}^{K}+\frac{1}{2}V_{2}^{K}=\frac{m_{H}^{2}}{8m^{2}}\left( H^{2} +\phi ^{2}+\frac{2m}{g}H\right) ^{2}-\frac{m_{H}^{2}}{2}H^{2}\\ S^{K}&=1+i\int gA\cdot AH-\int \int \left( \delta (x-x^{\prime })V+\frac{g^{2}}{2}TLL^{\prime }\right) +higher~order. \end{aligned}$$
An obvious c-number field shift in the \(H\)-field leads to the symmetric form of the Mexican hat potential (the \(V\) without the mass term). But there is no physical reason for writing the induced potential in this form; in fact this way of writing feigns a \(1/g\) dependence which has been artificially created by this way of writing. The rewriting of this gauge-induced quartic potential into the form of a symmetry-breaking Mexican hat potential can only lead to confusions.

The important message here is that the requirement of second order gauge independence of \(S\) in the form of \(sS^{(2)}=0~\)uniquely determines the changes of \(T_{0}LL^{\prime }~\)which must be done in order to achieve this task. There is no symmetry-breaking or mass generation, rather the model is defined in terms of the original trilinear interaction between the massive vectorpotential and a Hermitian field. The Mexican hat potential is induced by this elementary interaction; it is not part of the definition of the model as in the case of the unphysical Higgs mechanism.

Second order calculations contain no information about quadrilinear self-couplings which are not induced (counterterms whose couplings define new parameters). One would expect that the 4th order box-contributions lead to additional self-couplings \((\varphi ^{*}\varphi )^{2}\) in in pointlike QED and \(H^{4}~\)in the H-model. Unlike induced potentials, such contributions are genuine undetermined renormalization parameters. This problem will be taken up in a separate publication [35].

The correct formulation of the Higgs model in terms of a \(A\)\(H~\)coupling shows that there is a irreconcilable difference of a massive vectormeson leading to Schwinger–Swieca–Higgs screening with a Goldstone spontaneous symmetry breaking. Whereas the conserved current in the Goldstone situation leads to a diverging charge \(Q_{G}=\infty \) (this is the intrinsic definition of spontaneous symmetry breaking), the only conserved charge in the Higgs model is screened \(Q_{H}=0\). In fact its screening is a special case of a property shared by all theories which couple massive vectormesons to any matter (the Schwinger–Swieca screening) including \(H\) matter. Between these two extremes there is the normal symmetry situation in which the \(Q\) is finite and nontrivial.

As mentioned before the critique of the Higgs mechanism points to a radical change in the conceptual relation between massive and massless interacting fields. A much stronger indication that massless \(s\ge 1\) interactions should be constructed as limits of massive models with their clear field-particle relation comes from the still unsolved problems of QCD confinement and the only partially understood problem of QED infraparticles. Here the stringlocal Hilbert space setting is essential since the unphysical pointlike objects of the BRST gauge setting are too removed from the physical matter fields which are necessarily stringlocal.

In the following we will assume that we are in the zero mass situation where singular pointlike descriptions of matter do not exist. In this case it is reasonable to distinguish between reducible and irreducible stringlocal fields. The former are stringlocal fields which can be described as long distance limits of finite localized objects, whereas for irreducible stringlocal fields this is not possible. Among free fields only the noncompact third Wigner positive energy class is irreducible: no stringlocal field associated to such a representation (not even composites [9]) can be represented as integrals over pointlike fields. Interacting abelian potentials are integrals over observable field strengths, but it is not possible to represent interacting Y–M gluon fields in this way. For such inherently noncompact objects their creation from collisions of compact matter leads to problems with the principle of causal localization. Whereas causality forces third class Wigner matter, in case it occurs in our universe, to be “inert” cosmological staff without any possibility to interact apart from gravity (dark matter?, see Sect. 2), it prevents interacting gluons from emerging from a collision of ordinary matter in a compact region.

Formulated in terms of massless limits of correlations of massive stringlike gluons this strongly suggests to define gluon confinement as the vanishing of the limiting correlations functions which contain in addition to pointlike composites also gluon or quark operators; in this way the mentioned causality problems disappear. For quarks in QCD which carry a charge, one expects the only exception for such configurations in which the string direction \(e~\)of \(q\)\(\bar{q}\) are parallel to the spacelike distance of the endpoints so that the string is really a finite bridge. The benefit of this definition of confinement is that it suggests a way to prove it by summation of the leading logs in the limiting correlation functions in which the \(m\) is used as a natural infrared regularization parameter.

The historically learned reader will recall that a similar summation idea was used by Yenni–Frautschi–Suura in order to show that the scattering amplitude for charged particles with only a finite number of photon vanish (the only nontrivial scattering data a photon-inclusive cross sections). These calculations were done in terms of summing leading logarithmic infrared contributions of infrared regularized scattering amplitudes; the present setting suggests to re-do these calculations in the stringlocal Hilbert space setting by using the vectormeson mass as natural covariant infrared regulator. The before defined gluon/quark confinement should result from similar summation of leading infrared terms in off-shell massive gluon/quark correlations for \(m\rightarrow 0.\) which plays a similar role as the ad hoc regularization parameter in the YFS argument. The difference is that the QED infrared problem is on-shell (the electron string is reducible), whereas confinement is an off-shell phenomenon (the gluon string is irreducible).

The cited work in [23, 24] which the Higgs mechanism was replaced by a \(A\)-\(H~\)coupling, as well as the present Hilbert space formulation of higher spin \(s\ge 1~\)field interactions are rather late attempts to point at problems which, despite their more than 40 years of existence, need more conceptual attention. Recalling that Swieca’s effort to direct the focus of attention away from symmetry-breaking by using the terminology “Schwinger–Higgs” in most of his publications [13] (“Schwinger” for the screening idea and “Higgs” for the neutral \(H\)-coupling) got lost in the maelstrom of time, one cannot be optimistic about the success of the present attempts to shed additional critical light on these old problems. This is particularly difficult if incomplete results, which are still not anywhere near to their closure, have been sanctioned by Nobel prizes.

6 Resumé and Outlook

New concepts, which shed light on insufficiently understood old problems, usually lead to new questions, and the extension of QFT to string-localized fields is no exception. The clarification of the old controversies about spontaneous symmetry breaking and mass generation in this paper was obtained with a rather modest computational effort within a new setting. The new concepts used to achieve this show that Hilbert space positivity requires a quite different formalism from that of pointlike fields. Whereas in the latter case the perturbative systematics can be encoded into Feynman rules for which the different type of vertices represent independent couplings, such graphical presentations loose their utility in the presence of induced normalization contributions with computable coupling strengths.

Our new Hilbert space setting sheds very different light on open problems related to the Higgs model. Its systematic mathematical presentation requires a nontrivial extension of the locality-based inductive Epstein–Glaser renormalization formalism [26] from point to string crossings [21].

The fact that there exist stringlocal fields with the minimal short distance dimension for all spins24 (d = 1 for integer spin and d = 3/2 for fermionic strings) permits to define interactions which remain within the power-counting criterion of renormalization theory for all spins. The appearance of a \(s=0\) Higgs-like intrinsic escort field \(\phi \) for \(s=1~\)is a special case of a new phenomenon, namely the presence of \(s\) stringlocal intrinsic escort fields of lower spin which are inexorably linked to the massive stringlocal spin \(s\) field and appear explicitly in its interaction. The presence of pointlocal observables is an additional physical restriction on interactions. It is truly surprising that for the rather small prize of weakening locality from point- to stringlike one is able to open a whole new world of \(s\ge 1\) renormalizable models whose interactions stay below the power counting limit.

There are important problems for \(s=1\) which cannot be properly addressed in the BRST gauge setting. The matter fields of the gauge setting are unphysical, the only renormalizable physical matter fields25 are stringlocal and hence outside the range of gauge theory (this also includes physical self iinteracting Y–M fields). The new Hilbert space setting as presented in this paper addresses problems of massive \(s\ge 1~\)fields; massless situations have to be approached by taking massless limits of massive correlation functions; the latter can then be used to reconstruct a zero mass operator QFT [2]. There are a good physical reasons for approaching massless situations from the massive side; the physics behind massless interaction remains largely unknown since the standard field-particle relation breaks down and the use of a multi-particle Fock space is lost. It hides phenomena as gluon/quark confinement as well as incompletely understood infrared aspects of charged infraparticles. The new setting creates favorable conditions for their solution in that the stringlike physical fields incorporate the very restrictive Hilbert space positivity.

Looking back at history and recalling that the idea of the Higgs mechanism originated in a time in which massless models, as QED, were considered to be simpler than their massive counterpart, the new message supports the opposite view. Whereas renormalizable couplings of massive vectormesons to charged or neutral matter (as well as Y–M self-couplings) lead to standard field-particle picture backed up by scattering theory, all this breaks down for \(s\ge 1\) in the massless limit. Massless models present the real challenge, and there is the good chance to understand them in the new SLF Hilbert space setting in terms of massless limits of physical objects.

Taking a more philosophical stance, one may say the new setting de-mystifies the gauge principle in favor of substituting it by the foundational causal localization principle in Hilbert space; in this way all models of QFT, independent of the spin of their fields, are unified under the shared conceptual roof of the causal localization principle.


  1. 1.

    Starting at s = 3/\(\;\not 2 \;\) there are corresponding zero mass “spinor-potentials” and their associated “spinor field strengths”. In the present paper only integer spins will be considered.

  2. 2.

    Fields in the same Borchers class are known to describe the same physics; they represent different “field-coordinatizations” of the same QFT [2, 5].

  3. 3.

    In the case of halfinteger \(s\) the s-independent dimension of the stringlocal field is \(d_{sd}=3/2.~\)In the present work \(s\) is always integer.

  4. 4.

    This is similar to the change of one-particle normalizations in the Haag–Ruelle scattering which is necessary to extract \(\left| p,\cdot \right\rangle \) particle components from states generated by local observables.

  5. 5.

    All operator products are formal; their precise meaning in terms of normal products is part of the problem of their perturbative construction.

  6. 6.

    For our critical view it is enough to consider the simplest (abelian) Higgs model.

  7. 7.

    The principles of QFT are expected to determine the masses of bound states which are generated by composite fields, but the masses of the interaction-defining “fundamental” fields are part of the definition of a model.

  8. 8.

    Induced couplings are counterterms with numerical coefficients which are uniquely determined in terms of the first order data.

  9. 9.

    A shift in field space on a symmetric \(SO(n)\) invariant quadrilinear selfinteractig n-component scalar field is a mnemonic device to obtain an illustrative example. but the intrinsic definition is the diverging charge of a conserved current and not the shift in field space.

  10. 10.

    The only expected surviving \(q\)\(\bar{q}\) configuration is that in which the string direction is parallel to the spacelike distance between the endpoints of the \(q\) and \(\bar{q}~\)strings.

  11. 11.

    The positivity of energy is the stability requirement on relativistic quantum matter.

  12. 12.

    This new concept is not only important for the ongoing research in QFT [11], but it also permits to understand old problems in a new light [12].

  13. 13.

    Based on the standard idea that a click in a counter localizes a particle in the counter (with rapidly decreasing vacuum polarization caused tails).

  14. 14.

    Extrinsic fields add degrees of freedom whereas intrinsic escorts don’t.

  15. 15.

    Interestingly not only the \(\phi ~\)but also the interacting \(H\) disappears; the zero mass limit of the abelian Higgs model are \(A_{\mu },H\) free fields.

  16. 16.

    Usually this terminology is used for situations of QM in external electromagnetic fields.

  17. 17.

    This and the following e-dependent two-pointfunctions have been computed by use of the intertwiner functions of the corresponding fields [9] by Mund [21].

  18. 18.

    Such formulas do not appear in the work of the University of Zürich group [23]; they appear for the first time in [21].

  19. 19.

    The electromagnetic coupling and a parameter related to a counterterm-induced quadrilinear scalar field self-coupling.

  20. 20.

    A similar construction with \(d_{sd}=3/2~\)is possible for fermions.

  21. 21.

    We remind the reader that all operator products are Wick-products.

  22. 22.

    Narrow spacelike cones are the smallest noncompact causally closed regions whereas the smallest compact such regions.

  23. 23.

    Although we use the same notation \(V_{\mu },Q_{\mu }=sV_{\mu },\) they are very different operators in Krein space. Their role with respect to the cohomological \(s\) is analogies to the differential cohomology in de Sitter space.

  24. 24.

    d = 1 for integer spin and d = 3/2 for fermionic strings.

  25. 25.

    As explained in this paper, one can define “singular pointlike” fields in terms of the renormalizable stringlike fields which are well-defined in every order, but they are not renormalizable in the standard sense.



I am indebted to Jens Mund for numerous discussions and for making his notes available prior to publication and last not least for reading my manuscript and suggesting improvements. Although for the time being we have split the problems of the \(s\ge 1\) interactions in Hilbert space into mathematical/conceptual and historical/conceptual part, it is our intention to unite them in a future project and in this way to lessen the present deep schism between LQP and theoretical problems of Standard Model particle theory. I also thank Raymond Stora for having accompanied this new development with great interest and challenging questions which led to better formulations of some important points.


  1. 1.
    Becchi, C., Rouet, A., Stora, R.: Renormalization of the abelian Higgs–Kibble model. Commun. Math. Phys. 42, 127 (1975)CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics, and All That. W. A. Benjamin, Inc., New York (1964)MATHGoogle Scholar
  3. 3.
    Fröhlich, J., Morchio, G., Strocchi, F.: Infrared problem and spontaneous breaking of the Lorentz group in QED. Phys. Lett. 89B, 61 (1979)CrossRefADSGoogle Scholar
  4. 4.
    Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982)CrossRefADSMATHMathSciNetGoogle Scholar
  5. 5.
    Haag, R.: Local Quantum Physics. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  6. 6.
    Buchholz, D.: Collision theory for massless. Phys. Lett. B174, 331 (1986)CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Jaffe, A.: High energy behavior in quantum field theory, strictly localizable fields. Phys. Rev. 158, 1454–1461 (1967)CrossRefADSGoogle Scholar
  8. 8.
    Ezawa, H., Swieca, J.A.: Spontaneous breakdown of symmetries and zeromass states. Commun. Math. Phys. 5, 330–336 (1967)CrossRefADSMATHMathSciNetGoogle Scholar
  9. 9.
    Mund, J., Schroer, B., Yngvason, J.: String-localized quantum fields and modular localization, CMP 268 (2006) 621, math-ph/0511042Google Scholar
  10. 10.
    Weinberg, S.: The Quantum Theory of Fields I. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  11. 11.
    Schroer, B.: The ongoing impact of modular localization on particle theory. SIGMA 10, 085 (2014). arXiv:1407.2124 MathSciNetGoogle Scholar
  12. 12.
    Schroer, B.: Modular localization and the holistic structure of causal quantum theory, a historical perspective, to be published in SHPMPGoogle Scholar
  13. 13.
    Schroer, B., Jorge A.: Swieca’s contributions to quantum field theory in the 60s and 70s and their relevance in present research. Eur. Phys. J. H. 35, 53 (2010), arXiv:0712.0371
  14. 14.
    Brunetti, R., Guido, D., Longo, R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759 (2002)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Schroer, B.: Modular wedge localization and the d = 1 + 1 Formfactor program. Ann. Phys. 295, 190 (1999). and references thereinCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Schroer, B.: The foundational origin of integrability in quantum field theory. Found. Phys. 43, 329 (2013). arXiv:1109.1212
  17. 17.
    Lechner, G.: Towards the construction of quantum field theories from a factorizing S-matrix. Prog. Math. Phys. 251, 175 (2007). arXiv:hep-th/0502184 MathSciNetGoogle Scholar
  18. 18.
    Schroer, B: Dark matter and Wigner’s third positive-energy representation class, arXiv:1306.3876
  19. 19.
    Plaschke, M., Yngvason, J.: Massless, string localized quantum fields for any helicity. Journal of Math. Phys. 53, 042301 (2012)CrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Schroer, B.: An alternative to the gauge theoretic setting. Found. Phys. 41, 1543 (2011). arXiv:1012.0013 CrossRefADSMATHMathSciNetGoogle Scholar
  21. 21.
    Mund, J.: The Epstein–Glaser approach for string-localized field, in preparationGoogle Scholar
  22. 22.
    Mund, J.: String-localized massive vector Bosons without ghosts and indefinite metric: the example of massive QED, to appearGoogle Scholar
  23. 23.
    Scharf, G.: Quantum Gauge Theory. A True Ghost Story. Wiley, New York (2001)Google Scholar
  24. 24.
    Aste, A., Scharf, G., Duetsch, M.: On gauge invariance and spontaneous symmetry breaking. J. Phys. A30, 5785 (1997)ADSGoogle Scholar
  25. 25.
    Duetsch, M., Gracia-Bondia, J.M., Scheck, F., Varilly, J.C.: Quantum gauge models without classical Higgs mechanism. Eur. Phys. J. C. 69, 599–621, arXiv:1001.0932
  26. 26.
    Epstein, H., Glaser, V.: The role of locality in perturbation theory. Ann. Inst. Henri Poincaré A XIX, 211 (1973)MathSciNetGoogle Scholar
  27. 27.
    Bardackci, K., Schroer, B.: Local approximations in renormalizable and nonrenormalizable theories II. J. Math. Phys. 7, 16 (1966)CrossRefADSGoogle Scholar
  28. 28.
    Buchholz, D., Roberts, J.: New light on infrared problems: sectors, statistics, symmetries and spectrum. Commun. Math. Phys. 330, 935–972. arXiv:1304.2794
  29. 29.
    Schwinger, J.: Trieste Lectures, 1962, p. 89. I.A.E.A, Vienna (1963)Google Scholar
  30. 30.
    Swieca, J.A.: Charge screening and mass spectrum. Phys. Rev. D 13, 312 (1976)CrossRefADSGoogle Scholar
  31. 31.
    Swieca, J.: Goldstone’s theorem and related topics. Cargese Lectures in Physics 4, 315 (1970)Google Scholar
  32. 32.
    Buchholz, D., Fredenhagen, K.: Nucl. Phys. B 154, 226 (1979)CrossRefADSGoogle Scholar
  33. 33.
    Yenni, D., Frautschi, S., Suura, H.: Ann. of Phys. 13, 370 (1961)ADSGoogle Scholar
  34. 34.
    J. Mund, String-localized quantum fields, modular localization, and gauge theories, New Trends in Mathematical Physics (V. Sidoravicius, ed.), Selected contributions of the XVth Int. Congress on Math. Physics, Springer, Dordrecht, 2009, pp. 495Google Scholar
  35. 35.
    J. Mund and B. Schroer, Massive vectormesons coupled to Hermitian scalars and the Higgs mechanism, in preparationGoogle Scholar
  36. 36.
    J. Mund and B. Schroer, Renormalization theory of string-localized self-coupled massive vectormesons in Hilbert space, in preparationGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikFU-BerlinBerlinGermany
  2. 2.CBPFRio de JaneiroBrazil

Personalised recommendations