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Positivity and causal localization in higher spin

quantum field theories

Bert Schroer

CBPF, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil

permanent address: Institut fur Theoretische Physik, FU-Berlin,

Arnimallee 14, 14195 Berlin, Germany

email: schroer@zedat.fu-berlin.de

March 9, 2018

Contents

1 Introduction 2

2 SL massless and massive vector potentials and their use in con-

servation laws 11

3 Relation between the Weinberg-Witten problem and the Velo-

Zwanziger conundrum 26

4 Particle wave functions and causal localization 33

5 New aspects of QFT from string-local renormalization 44

5.1 Scalar QED, induced interactions and counterterms . . . . . . . . 44

5.2 The perturbative S-matrix in the SLFT setting . . . . . . . . . . 47

5.3 External source models . . . . . . . . . . . . . . . . . . . . . . . . 50

5.4 Hermitian H coupled to a massive vector potential . . . . . . . . 51

5.5 Selfinteracting vector mesons . . . . . . . . . . . . . . . . . . . . 53

6 The role of the pair condition in the construction of interacting

fields 54

7 Resume, loose ends and an outlook 59

Abstract

1

It is shown that the recently introduced positivity and causality pre-serving string-local quantum field theory (SLFT) resolves most No-Gosituations in higher spin problems. This includes in particular the Velo-Zwanziger causality problem which gets in an interesting way related tothe solution of zero mass Weinberg-Witten issue. In contrast to GaugeTheory SLFT uses only physical degrees of freedom and as a result ob-tains a simpler and clearer view of many relations. This includes theimportant non-covariant lightcone gauge which becomes replaced by a co-variant string field with a lightlike string directions which, different fromgauge fixing parameters, participate in the in the Lorentz transformation.It reveals that the Lie algebra structure of s=1 selfcouplings is not im-posed by quantization but rather follow from the causality principles inconjunction with positivity and in many provides more profound expla-nations than those of Gauge Theory, in particular for the raison d’etre ofthe Higgs particle.

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

1 Introduction

The positivity property of quantum states is an indispensable requirement whichguaranties the probabilistic interpretation of quantum theory. It enters themathematical formalism through the identification of quantum states with unitrays in a Hilbert space on which the quantum observables act as operators. Inquantum field theory (QFT), or more generally for models with infinitely manydegrees of freedom, it is often more appropriate to identify states with positivelinear functionals on operator algebras. Thanks to the existence of a canonicalconstruction1 this formulation in terms of expectation values permits a returnto the more common Hilbert space setting.

Its validity in quantum mechanics is guarantied by Heisenberg’s canonicalquantization of positions and momenta in conjunction with the von Neumannuniqueness theorem which insures that irreducible representations of the Heisen-berg commutation relations are unitarily equivalent to the Schrodinger repre-sentation. Born’s identification of the absolute square of the Schrodinger wavefunction with the probability density for finding a particle at a particular posi-tion connects positivity with spatial localization.

This situation undergoes significant changes in relativistic QFT where thepositivity of field-quantization looses its ”von Neumann protection” in the pres-ence of higher spins s ≥ 1. The first such clash with positivity was noticedby Gupta and Bleuler who observed that quantized massless vector potentialsare incompatible with Hilbert space positivity. In the absence of interactionsit is straightforward to restore positivity by passing from potentials to fieldstrengths, but the use of local gauge invariance to regain positivity in the pres-

1The ”reconstruction theorem” in [1] is a special case of the more general Gelfand-Naimark-Segal (”GNS”) reconstruction theorem [2].

2

ence of interactions leads to a loss of important physical operators and putativephysical states.

This includes in particular all fields2 which interpolate charged particles inthe sense of large time scattering theory. These interpolating fields play an in-dispensable role in connecting the causal localization- and quantum positivity-principles of QFT with observed properties of particles in form of analytic struc-ture of their scattering matrix. The absence of interpolating physical fields andcharge-carrying states which they create from the vacuum is accompanied by aloss of some mathematical tools of functional analysis. The proofs of importantstructural properties, which in particular includes the TCP and Spin&Statisticstheorems, use Hilbert space positivity in an essential way and have no substitutein indefinite metric Krein spaces.

Positivity obeying massive tensor potentials and their spinorial counterpartare provided by Wigner’s unitary representation theory, but it comes with anincrease of their short distance scale dimension with spin that prevents their usein renormalized perturbation theory for s ≥ 1. Even worse, the massless limit ofthese Wigner fields diverges; in fact there exist no covariant positivity-obeyingpoint-local (pl) fields which are the potentials of massless field-strengths.

This paper presents a new idea which overcomes this apparent dilemma andin this way leads to a wealth of new results, which in particular in connec-tion with higher spin particles go beyond what one has achieved by using thepositivity violating gauge theoretic setting.

In his well-known monography Weinberg presents the systematic construc-tion of the intertwiner functions which relate Wigner’s spin s creation and an-nihilation operators associated with the unitary (m, s) representations with co-variant pl free fields acting in a Wigner-Fock Hilbert space [3]. However thisinteresting section remained a torso since the worsening short distance scaledimension dsd = s+ 1 with increasing spin s prevents their use in renormalizedperturbation theory so that in the main part of his book Weinberg uses thepositivity-violating gauge theoretic setting.

On the other hand the gauge theoretic formulation of higher spin equationsfits naturally into the Lagrangian formalism since the integer spin short distancedsd = 1 gauge potentials match the classical dimension dcl = 1 of fields in termsof mass units which complies with the dcl = 4 dimension of classical actiondensities. Higher spin QFT, apart from Weinberg’s constructions of free fieldsin terms of intertwiners, has been traditionally formulated in the gauge theoreticLagrangian setting. But whereas in the older work [4] the positivity problemhas been at least addressed by showing that it can be partially satisfied byintegrating propagators in both ends with suitable classical source functions,one looks in vain for arguments which connect classical Lagrange gauge theorywith positivity-obeying quantum fields in more recent work as [5] [6] and paperscited therein.

The gauge theoretic localization of matter fields is not the physical localiza-

2Except the photon whose interpolating field is the observable field strength.

3

tion in their role as interpolating fields of particles; in gauge theory only thegauge invariant local observables are positivity obeying and causally localizable.There are different stampings of gauge theory and usually the terminology refersto the higher spin Fronsdal-Vasiliev (F-V) setting.

The representation of positivity obeying quantum fields Ψa in Weinberg’sintertwiner formulation for integer spin reads 3

Ψα(x) =

∫

(s

∑

s3=−s

eipxvα,s3(p)a∗(p, s3)+h.c.)dµm(p), dµm(p) = θ(p0)δ(p

2−m2)d4p

(1)The intertwiner functions v(p) convert the Wigner creation/annihilation opera-tors a#(p, s3) into covariant fields Ψα. They have two indices, the s3 which runsover the 2s + 1 values of the third component of the physical spin and tensorindex (tensor-spinor index in case of half-integer spin) which refers to the 4sdimensional tensor representation of the Lorentz group of tensor degree s. Hereand in the remainder of the paper all constructions refer to integer spin unlessstated otherwise. The extension to fermions is generally straightforward but theequations become a bit longer.

The momentum space Wigner creation/annihilation operators a#(p, s3) andhence also the covariant fields Ψα act in the Wigner-Fock Hilbert space obtainedby ”second quantization” from the 1-particle Wigner representation. The quo-tation mark refers to Ed Nelsons: ”second quantization is a functor, whereasquantization is an art”. Looking at the explicit form of the intertwiners or cal-culating the two-point function (2-pfct) of Ψ one finds that the latter scales asλ−(s+1) for x→ λx in the limit of small distances λ→ 0 which is the meaningbehind assigning dsd = s+ 1 to Ψ.

This construction permits a generalization to string-local (sl) covariant quan-tum fields with better short distance properties. In this construction one hasto make explicit use of causality and admit a more general form of Lorentzcovariance. The momentum space intertwiners vα,s3(p, e) depend on an addi-tional spacelike unit vector e, e2 = −1 and the resulting fields Ψα(x, e) arecausally localized on covariant semiinfinite spacelike or lightlike strings (rays)S = x+R+e. Among the resulting string-like (sl) fields of higher spin s ≥ 1 thereare also scalar (α = 0, s ≥ 1) sl fields. More generally the range of Lorentztensor indices α becomes decoupled from s which is impossible with pl fieldsfor which α = (µ1..µs). Each of these different sl spin s ”escort” fields with alesser than s number of tensor indices accounts for the full content of the (m, s)Wigner representation.

Historically it was in fact the prior construction of scalar sl fields for infinitespin Wigner representations which overcame Yngvason’s 1970 No-Go theorem[21] for pl fields which paved the way for the construction and use of finite spinsl fields (see below). For this Weinberg’s method was without avail; one rather

3Unless otherwise stated s ≥ 1 is chosen integer since; this covers the physically importantapplications.

4

had to use ideas from modular localization [7, 8, 41]. Fortunately it turned outthat one does not need such ”heavy guns” for finite spin; in that case all sl freefields can be obtained by simply integrating pl fields along semiinfinite lines.

Besides preserving positivity the construction of fields through intertwinershas unlike the Lagrangian F-V quantization gauge formalism the advantage thatthere are only physical degrees of freedom right from the start; hence there isno need to control them through differential conditions which in the presence ofinteractions causes severe problems and in many cases leads to No-Go situations. The main aim of the present work is to present new results which go beyondthose obtained in the gauge theoretical setting [9].

The mathematics used in perturbation theory is basically limited to com-binatorics; it works for s ≥ 1 gauge theory in a similar way as for positivity-preserving interactions involving only s < 1 interacting fields. This changes ifone use gauge theory beyond perturbation theory and attempts to derive non-perturbative theorems as TCP, Spin&Statistics and others or tries to establishthe existence of QFTs beyond the perturbative combinatorics. This would re-quire functional-analytic and operator-algebraic methods for which positivity isan indispensable prerequisite.

All presently existing books on QFT are based on the assumption that thestandard setting of point-local quantum fields remains valid in the presence ofinteractions independent of the value of spin of the involved particles. The No-Go situations which positivity leads to in the presence of higher spin interactingsl fields raises the question whether such books on a nonperturbative formulationof QFT in the presence of higher spin s ≥ 1 particles may not be addressing anempty set.

Buchholz and Fredenhagen have rigorously (outside perturbation theory)established4 the string-local nature of interpolating fields in models with localobservables and a mass gap [10]; attempts to extend this result to pl failed. Be-fore these nonperturbative insights it had been already known that the tightestlocalization of interacting electric charge-carrying fields in QED which is con-sistent with the quantum formulation of the Gauß law is causal localization onsemi-infinite spacelike strings (the cores of spacelike cones which one obtains bydirectional smearing from strings) [11].

These results represent the strongest nonperturbative support for the string-local nature of interpolating fields. Recent perturbative insights obtained withinthe past 3 years revealed the more specific picture that, apart from interactionsbetween exclusively s < 1 fields, all positivity-preserving renormalizable inter-action densities which involve s ≥ 1 fields are only consistent if these free fieldsenter the model-defining interaction density as sl fields; in this case the higherorder perturbative contribution spread the string-localization to the s < 1 fieldswhich entered the interaction density as pl fields. This leads to an intrinsic

4Those authors work in the setting of algebraic QFT and consequently they show theexistence of operators localized in arbitrary narrow spacelike cones whose core is a spacelikestring.

5

quantum division into particle-interpolating sl fields and pl local observableswhich has no counterpart in classical field theory where particles and their in-terpolating sl fields have no place5 and all fields are observable.

In such a setting the local observables measured in counters localized infinite spacetime regions are mathematically described by field strengths, con-served currents or other pl composites. But the particle states in which theseobservables are measured correspond to the large time limit of vectors obtainedby applying interpolating fields localized on spacelike semi-lines S = x + R+e,e2 = −1 (or their lightlike e2 = 0 counterpart) to the vacuum state. Thee-dependence in the large time limit is contained in a vacuum-to-one-particlematrixelement and can (and should) be removed by a trivial re-normalization.

It is important to realize that there still exists a continuous infinity of string-localized fields placed in relative spacelike position so that all the consequencesof causality, including the cluster factorization property which insures the largetime asymptotic convergence in the sense of scattering theory remain valid.Hence space- or light-like string localization remains causal and one should becareful with the use of the terminology ”non-local”.

This works for space- and light-like e , but obviously not for timelike stringswhich permit no causal separation. These sl interpolating quantum fields andthe states they create from the vacuum are the carriers of the causal localizationprinciples of QFT and hence indispensable for establishing analytic properties ofscattering amplitudes (e.g. dispersion relations and crossing symmetry). UnlikeString Theory sl quantum field theory (SLFT) is not a playful invention oftheorists but rather the tightest covariant localization which is consistent withcausality and the preservation of positivity in the presence of interacting higherspins s ≥ 1. Particles in SLFT remain what they always were since Wignerclassified them in his famous 1939 paper; there is no sl mark on them. Morecomments on this point will be deferred to section 4.

The string-local QFT (SLFT) is implemented within a new perturbativesetting which reveals that in the presence of s ≥ 1 sl fields all fields in the in-teraction density, even those which enter the first order density as s < 1 pl freefields, are converted into higher order sl fields. Only observable field strengthsand certain pl localized composites of sl fields represent local observables. Themain role of string-localization is the improvement of the short distance prop-erties of the interpolating fields with the aim to preserve the power-countingbound dsd(L) ≤ 4 of renormalizability. Compact spacetime localizability is onlymandatory for local observables.

To show how causality and positivity cooperate in a harmonious way it isnot enough to merely modify some technical details of the existing formalism;one rather needs to avoid the use of the quantization parallelism to classicalLagrangian fields altogether. The terminology ”interaction density” instead of”interacting part of a Lagrangian” serves to highlight the independence of SLFT

5By classical we mean classical in Nature (Maxwell, Einstein) and not classical in thesense of a Lagrangian which has been red back from QFT into the classical realm.

6

from any quantization parallelism to classical field theory. All the physical pa-rameters of the interaction-defining field content (including the physical masses)are contained in the interaction, a conversion of the content of a Wigner repre-sentation into ”free Lagrangians” is not needed. This relieves the conversion ofthe particle content into an interaction density from the nontrivial (and unnec-essary) conversion of Wigner particles into Lagrangians.

The new setting confirms the on-shell perturbative results of gauge theo-ries which cover a substantial part of the Standard Model, but it also exposesits limitations concerning off-shell properties; above all it leads to surprisingnew physical insights. The abandonment of paraphernalia of gauge theoriesas indefinite metric- and ghost- degrees of freedom in favor of strict inherenceto only physical degrees of freedom has been metaphorically referred to as theapplication of ”Occam’s razor” to gauge theory [12, 13].

The longing for an intrinsic formulation of QT in which the umbilical chordto classical field theory provided by quantization has been cut is almost as oldas QFT itself. In the first (still pre-renormalization) presentation of quantumelectrodynamics at an international conference in 1929 [14] Pascual Jordan ex-pressed this in the form of a plea for an intrinsic understanding of QFT whichavoids the use of ”(quasi)classical crutches” and a decade later his former collab-orator Eugen Wigner took the first step in his famous classification of relativisticparticles [15].

One generally expects that a theory with a more restricted range of observa-tional validity re-appears as a limiting case of a more foundational new theory.But it seems unnatural to hope for a general magical ”quantization key” whichconverts classical systems into their more fundamental quantum counterpart,although for quantum mechanics such a key nearly exists.

With the arrival of the covariant formulation of quantized electrodynamicsin the early 50s, Jordan’s dictum and its partial realization in Wigner’s classifi-cation of noninteracting particles faded into the background; the new covariantcomputational rules of quantized electrodynamics took a firm hold, so the firstcovariant QFT was a positivity-violating gauge theory.

The surprising aspect of these first covariant renormalized calculations wasthe contrast between the precision of the experimentally verified perturbativeresults and the robustness of the calculational rules against the use of quitedifferent cutoff- and regularization- prescriptions or even different ways of im-plementing Lagrangian quantization (Gell-Mann–Low, Feynman path integrals,Bogoliubov’s generating S-functional).

Lagrangian quantization causes a partial loss of quantum positivity, butalso this did not create serious concern as long as the prescriptions for gaugeinvariant scattering amplitudes led to accurate experimentally verifiable results.On the conceptual/philosophical level the situation became however somewhatschizophrenic in view of the fact that gauge theory can not provide those in-terpolating physical fields in terms of which the at that time newly discoveredLSZ scattering theory connects the causal localization principles of QFT with

7

the analytic properties (dispersion relations) of scattering amplitudes. Subse-quent attempts to extend these properties with the help of crossing symmetryand to extract a pure on-shell S-matrix theory without referring to fields werenot successful. Such ideas entered the somewhat free-lance Regge-trajectoryformalism and were reprocessed in the form of the dual model and finally led tothe superstring formalism and all its derivatives.

With QCD and the Standard Model the interest in gauge theory returnedand the main attention was focussed on how to extract a unitary S-matrix fromperturbative gauge theory without using gauge-dependent interpolating fields.Occasionally there were also attempts to construct gauge invariant interpolatingfields by modifying gauge fields. Actually the oldest proposal was made byJordan a long time before renormalized QED [16] and consisted in replacingthe gauge dependent matter field by the formally gauge invariant string-localcomposite field

Ψ(x) = ψK(x) exp ig

∫ ∞

0

AKµ (x0, . . . , x3 + λ)dλ (2)

where the K refers to the gauge dependent operators in the indefinite metricKrein space. He successfully used this for an algebraic derivation of Dirac’sgeometric magnetic monopole quantization [17].

For Mandelstam this formula served as an inspiration in his attempts toconstruct a perturbation theory which avoids the use of potentials in favor ofthe gauge invariant field strength [18]. Steinmann [19] studied the problem ofdefining this composite field in higher order renormalization theory. Such con-structions of composite fields which do not occur in the standard perturbationtheory in addition to those operators which appear in the gauge theoretic La-grangian perturbation theory requires a lot of additional work and is of littleinterest unless it leads to new insights. There were also attempts to recoverparts of positivity by imposing other topologies on Krein spaces [20].

The later development of the electro-weak Standard Model led to the com-putational efficient well-known BRST prescription for perturbatively unitaryscattering amplitudes which moved the positivity problem into the background.It is the aim of this paper to show how the SLFT formulation of QFT, whichplaces causality and positivity under a common conceptual roof, is presentlyrevolutionizing the area of QFT hitherto covered by gauge theory.

The basic idea goes beyond Weinberg’s construction of pl fields from covari-ant intertwiners functions [3] for pl fields and admit instead covariant sl fields.This cannot be done within Weinberg’s solely group theoretical setting but onealso has to use causality. An important new conceptual impulse came fromthe solution of the almost seven decades lasting problem of finding a causallocalization respecting field-theoretic description of Wigner’s infinite spin rep-resentation class. That this was is not possible with pl fields is the result of analready mentioned important but largely overlooked publication in the 70s [21].When Weinberg wrote his book [3] he did not know about this but it would besurprising if he did not try to construct an intertwiner with his group-theory

8

based method and realized that this is problematic; in his book one merely findsthe remark that ”Nature does not use” this representation.

What finally led to the solution was a new concept referred to as ”modularlocalization”. It permits to control causal localization properties already withinthe Wigner positive energy representation space before passing via intertwinersto quantum fields [22]. In view of the fact that causal localization of fieldsneeds both sign of energies this is at first sight somewhat surprising. In thepresent context it turns out to be particularly useful for the more general classof covariant intertwiners which one needs in the new renormalization theory.

Readers interested in mathematical and conceptual aspects may be curiousto know that modular localization theory can be traced back to the modularTomita-Takesaki theory of operator algebras. It is one of a few mathematicaltheories of which important properties were simultaneously discovered by math-ematicians and physicists working in the algebraic setting of QFT and statisticalmechanics [23, 24]. It permits to formulate equilibrium statistical mechanics di-rectly in the thermodynamic infinite volume limit, and its (later noticed) relationto causally localized subalgebras in QFT [25] led to a profound insight aboutthe origin of thermal-like KMS properties of causal localization-horizons whichincludes in particular thermal manifestations of event horizons related to Hawk-ing’s thermal radiation [26]. In the present context it owes its importance tothe fact that it can be used to convert in the absence of interactions ”modu-lar localized subspaces” of Wigner’s one particle spaces into localized operatorsubalgebras.

In this way it became possible to extend Weinberg’s intertwiner constructionto Wigner’s infinite spin representations and to obtain explicit expressions forthe associated sl fields [8, 28] and to construct sl tensor potentials associatedto pl field strengths [29]. The possibility that these sl field algebras contain plsubalgebras was excluded by arguments which revealed the impressive powerof modular localization [27]. More recently these fields reappeared as ”Pauli-Lubanski limits” of finite spin fields [30].

This made it possible to investigate physical properties of matter through thestudy of its positivity-obeying causal localization structure and find out whetherthere are theoretical reasons which justify to exclude infinite spin fields as objectsof particle physics [31]. Of particular interest with relation to dark matter istheir apparent lack of reactivity with respect to normal matter in combinationwith their possible contribution to gravitational backreaction (section 6).

At this point one should also mention a series of publications in which a(necessarily pl) gauge theoretical F-Z gauge kind of construction was attempted[33] [32]. Initially the authors were not aware about the existence of a No-Gotheorem. In later papers the old work excluding pl was cited and its solutionin [8] was referred to, but the authors did not explain how their gauge theorywhose gauge invariants are physical pl fields can be consustent with a theorywhich has no pl fields at all (not even gauge invariant ones).

Here one should recall that the main purpose of a field theory is the re-

9

alization of the ”Nahewirkungsprinzip” (action in the neighborhood principle)of Faraday and Maxwell which culminated in Einstein’s concept of relativisticcovariance and causal localization. Any physical theory which realizes theseprinciples in a (positivity maintaining) quantum theory is by definition a QFT.The cited gauge theoretic attempts to construct a QFT for infinite spin doesnot address this QFT defining causality property of QFT; constructing covari-ant wave functions and associated fields is not enough and the approach of theauthors does not face the issues which they raise in their paper.

It is the aim of this work to show the power of the SLFT formulationin solving those problems which in recent reviews of higher spin QFT as [9]have remained outside the range of Fronsdal-Vasiliev gauge theory either in theform of No-Go theorems (as the Weinberg-Witten (W-W) theorem [34], or thes = 2 van Dam-Veltman-Zakharov (D-V-Z) discontinuity [35, 36] and the Velo-Zwanziger (V-Z) causality conundrum [37]. The present paper addresses higherspin problems for which gauge theory offered no convincing solution. After hav-ing contributed to the solution of the first two problem [38] [39] the presentpaper presents the author’s understanding of the V-Z causality problem and itssolution.

To achieve this in a way which does not loose the reader, the next sec-tion recalls and comments on very recent but largely already published resultsconcerning string-local free fields and their important properties. This sectionincludes properties of degrees of freedom counting in passing from the spin ofs ≥ 1 massive fields to their massless helicity limit (solution of the D-V-Z prob-lem) and the construction of higher spin sl energy-momentum tensors and theirsmooth massless helicity limit (conversion of the pl No-Go W-W problem intoa Yes-Go situation) and a similar phenomenon for higher spin electric currents.The second section also explains how the in higher spin gauge theory importantnoncovariant lightcone gauge passes to a much simpler covariant sl field withlightlike string direction.

The third section addresses the problem of interactions of currents withexternal potentials and shows that the origin of the Velo-Zwanziger causalityproblem is the naive expectation that by modifying free field equations by addinglocal couplings to an external potential one preserves causality in the sense ofpropagation of Cauchy data. The simplest way to solve this problem is toconvert it into a problem of an interaction of a quantum field with an externalpotential and pass to the V-Z classical field by taking expectations values of thequantum field in coherent Wigner-Fock states.

Section 4 provides some background about modular localization which is theintrinsic property of causal localization of quantum matter. This is importantfor understanding that String Theory bears apparently no relation to string-localized quantum fields in spacetime. Modular localization theory also helps tounderstand the passing from pl to sl in the absence of interactions as a changeof ”field-coordinatization”.

Section 5, which consist of 5 subsections, introduces the concept of induced

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interactions which is then applied to different models involving vector mesons.

Section 6 presents the problems one faces and their partial solution whenone extends the pair condition of SLFT to interacting models involving s ≥ 2interactions.

The concluding remarks in section 7 summarize the new concepts and resultsand also present additional ideas concerning the solution of old problems andexpectations about the future of SLFT.

2 SL massless and massive vector potentials and

their use in conservation laws

The traditional formulation for models of QFT is that of Lagrangian quantiza-tion, either in its simple form of canonical quantization or in the form of moresophisticated covariant form of (euclidean) path integrals based on a euclideanform of the Lagrangian. These quantization methods work for QM and leadto renormalizable Feynman rules for QFT for models for spin s < 1 fields, butpositivity and renormalizability start to clash for spin s = 1 and the situationworsens with increasing spin.

The positivity problem of massless helicity h ≥ 1 free fields is well-known.Intertwiners which convert Wigner’s operators a#(p,±h) and into massless pltensor potentials or their spinorial counterparts do not exist. The only pl mass-less intertwiners are those associated to field strengths, they can neither be usedfor the construction of composites as conserved currents nor for constructingrenormalizable interaction densities.

SLFT solves the positivity problem in a way which is consistent with thecausality principles of QFT. For the construction of a massless sl vector potentialone starts from the pl field strength. The problem is to find a covariant solutionAν of

∂µAν(x)− ∂νAµ(x) = Fµν(x) (3)

There are precisely three localization classes of covariant solutions which havethe form of a semi-infinite line integral

Aµ(x) = Aµ(x, e) :=

∫

Fµν(x+ λe)eν =: (IeFµν)(x)eν (4)

U(a,Λ)Aµ(x, e)U(a,Λ)∗ = (Λ−1)νµAν(Λx,Λe)

with e representing a spacelike Lorentz covariant unit vector which participatesin the transformation under the homogenous Lorentz group. These vectors ewith e2 = −1 are points on the d = 1+2 unit de Sitter space. In case one woulduse timelike unit vectors e2 = 1 on the d = 3 unit mass hyperboloid one stillmaintains covariance but looses causality. A third possibility is to work withlightlike e i.e. e2 = 0.

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In the timelike case it is not possible to causally separate two strings whereasfor space- or light-like e′s there exists a continuum of possibilities for spacelikeseparated (denoted as× ) strings (half-lines)6

[Aµ(x, e), Aµ(x′, e′)] = 0, x+ R+

0 e×x′ + R+0 e

′

The richer mathematical structure of sl fields requires a more careful look attheir singularity structure. For this purpose it is convenient to compute their2-point function (2-pfct). Starting from that of the field strengths

〈Fµν(x)Fκλ(x′)〉 =

∫

e−ip(x−x′)MFµν ,Fκλ(p)dµ0(p), (5)

MFµν ,Fκλ(p) = −pµpκgνλ + pµpλgνκ − pνpκgµλ + pνpλgµκ

and using the fact the λ-integration can be carried out in momentum space interms of the Fourier transform of the Heavyside function one obtains

〈Aµ(x, e)Aν (x′, e′)〉 =

∫

e−ip(x−x′)MAµ,Aν (p, e, e′)dµ0(p) (6)

MAµ,Aν (p, e, e′) = Eµν(−e, e′) = ηµν +pµeν(pe)iε

+e′µpν

(pe′)iε− (ee′)pµpν

(pe)iε(pe′)iε

where the tensor Eµν will be an important building block of higher helicity2-pfcts M.

The formula simplifies significantly for lightlike e′ = −e since in that case thelast contribution vanishes and the singularities at pe = 0 for lightlike p requiresp ∼ e where the two middle terms cancel. This means that for spacelike e theassociated fields need directional smearing (pl smearing on d = 1 + 2 de Sitterspace) in addition to the standard endpoint smearing (a stump of a cone),whereas in the lighlike case a directional smearing is unnecessary (a lightliketube). The physically important property of quantum fields is that they can beplaced into relative spacelike positions; this is easy to vizualize for spacelike e,but it also holds in the lightlike case.

For the timelike case the denominators never vanish and such fields areintrinsically nonlocal (no relative causal positioning possible). For the noncausalchoice e = e0 = (1, 0, 0, 0) the 2-pfct (6) is easily seen to be that of the Coulomb(or radiation) potential with AC

0 = 0 and the spatial components

MACi AC

j = δi,j −pipjp2

(7)

”Freezing” the timelike string direction destroys the covariant transformationand converts it into a noncovariant inhomogenous transformation law (12) inwhich only the rotations and translations maintain their covariant appearance.Letting the timelike direction participate in the Lorentz transformation recov-ers covariance but nonlocality remains. Such fields may be used in quantum

6This is less obvious in the lightlike case.

12

mechanics which is intrinsically nonlocal7. The same potential results from av-eraging a spacelike string over all directions in the t = 0 plane orthogonal to thetimelike e0 vector. There is no direct way to undo this directional averaging; theonly way is to return to the pl field strength and pass to the covariant potential(4)

The directional averaging reveals a close formal connection between the axial-and the Coulomb- ”gauge”. Both potentials exist in a covariant and a non-covariant form in the same Wigner-Fock helicity space, but only the covariantspace- or light-like sl potential (4) is manifestly causal.

It should be mentioned that in the literature the terminology ”gauge” is usedin two different meanings. In the covariant setting of QED perturbation theoryit refers to a formal symmetry whose generators (”gauge charges”) depend onthe unphysical indefinite metric degrees of freedom. On the other hand theCoulomb- or axial- gauge contains only the two helicity h = ±1 degrees offreedom; they live in the same Wigner-Fock helicity space and are distinguishedby the choice of e from different directional classes and there is also no unphysicalgauge symmetry implementing degrees of freedom.

The noncovariant form of their transformation law arises if one preventsthe homogeneous form of the Lorentz transformation by keeping the e fixed(12). The present work tries to avoid conceptual confusions by using the word”gauge” exclusively for the situation in which unphysical degrees of freedomprovide a covariant ”gauge symmetry”. Quantum gauge symmetry is not aphysical symmetry but rather a formal tool to extract a physical subtheoryfrom an unphysical description. One cannot change a traditional terminology,but one should at least be aware of these physical fundamentally different waysof using it.

Pl quantum fields are operator-valued distributions in x and one expectsthat the appearance of pe factors in the denominators of sl fields (6) also requiresmearing with test functions on the directional d = 2 + 1 unit de Sitter spaceor on the 2 dim. celestial sphere. As mentioned the singular behavior in e isdetermined by the dimensionality of the tangential subspace of p′s orthogonalto e in the sense of the Minkowski metric. This does not only depend on thedirectional spaces but there is also a difference between massive and masslesspotentials. In the massive case this dimension of is 3 for spacelike e′s and 0in the lightlike case. This accounts for the fact that a directional smearing isrequired in the spacelike case whereas it is not needed on the celestial sphere.

The lightlike sl formulation has an interesting connection with the light-conegauge field theory. The latter treats e = (1.0, 0, 1) as a fixed gauge parameterand sacrifices Lorentz covariance which becomes replaces by a complicated in-homogeneous transformation law of the form of (12). The relation between thetwo is obtained by turning back Λe in the covariant law to e using a differentialrelation between de and ∂µ (11). The simplification coming from the use of thecovariant sl setting in constructing cubic self-interacting interaction densities

7The ”Born-localization” of QM is not intrinsic (see section 4).

13

for higher helicity particles are substancial (section 6).

The main purpose of this work is to present a positivity- and causality-maintaing alternative to gauge theory which avoids the use of the quantizationparallelism of classical field theories altogether by starting from Wigner’s man-ifestly positivity-preserving representation theory. The important point here isthat the spacetime localization properties do not have to be added via quantiza-tion; they exist already in the form of modular localization in Wigner’s positiveenergy representations theory and are waiting to be converted into the net ofnoninteracting causally localized operator algebras and their covariant causalquantum field coordinatizations needed to define interaction densities (section4) for the purpose of constructing the nets of interacting fields and their operatoralgebras which share the same S-matrix.

Modular localization theory played an important role in the recent concep-tual development of QFT. In particular it revealed the subtle manner in whichcausal localization is already contained in Wigner’s representation theory (sec-tion 4). It is this aspect of representation theory which plays an analogouspositivity-conserving role as Heisenberg’s quantization in terms of canonicalcommutation relations whose positivity maintaining nature is secured by thevon Neumann unicity theorem.

It should cause no suprise that such a starting point deviates significantlyfrom the way in which higher spin gauge theories are treated in Lagrangianquantization. Even the implementation of gauge theory theory itself looks dif-ferent if formulated outside the quantization parallelism to classical field theory.Some of the problems on runs into if one first sets up free higher spin Lagrangianand afterwards tries to modify them by inserting interactions are homemade anddisappear upon leaving the confines of quantization (section 3).

The remainder of this section presents two additional interesting conceptualrefinements resulting from the use of positivity maintaining sl potentials. Thedefinition in terms of the field strengths (4) permits to calculate properties forwhich the use of sl vector potentials leads to additional conceptual insights.

For the difference between two sl potentials with different string directionsone obtains

Aµ(x, e)−Aµ(x, e′) = ∂µΦ(x, e, e

′), Φ = (Ie′IeFµν)(x)eµe′ν (8)

The Φ represents a field which is localized on the 2-dimensional conic partof a planar surface λe + λ′e′, with λ, λ′ ≥ 0. These Φ(x, e, e′) fluxes Φ arelogarithmically divergent but their exponential exp igΦ exists and represent anew field whose Hilbert space extends the helicity Wigner Fock space and whichdescribes superselected photon clouds localized on this conic part with differentclouds belonging to different superselection sectors.

The situation is reminiscent to that of the infrared divergent massless fieldϕ in d = 1 + 1 which is known to play an important role in ”bosonization” offermions and the creation of ”anyonic” charge sectors. In that case they are the

14

logarithmic divergent potential of a conserved current:

ϕ(x, e) =

∫ ∞

0

jµ(x+ λe)eµdλ, jµ = εµν∂νϕ (9)

⟨

eia1ϕ(x1,e) . . . eianϕ(xn,e)⟩

= 0 if

n∑

1

ai 6= 0. (10)

and the superselection sectors are labeled by the ”charges” ai (without lossof generality one may take e = (0.1)). The logarithmic divergence of ϕ is theprerequisite for the interpretation of ai as charges (([42]) section 3). This super-selection rule owes its existence to coherence properties of ”infrared ϕ -clouds”.This was was first noticed in the d = 1+1 model of a derivative coupling jDµ ∂

µϕ

of a massless scalar ϕ to the current jDµ of a massive Dirac particle [43]. The

solution = eigϕψ0 with ψ0 a free Dirac field of mass m has a 2-pfct in which theexponential infrared cloud of ϕ-quanta weakens the mass-shell singularity of the

ψ0. This is most conveniently seen in the ψ propagator in which the(

p2 −m2)−1

mass-shell pole of the ψ0 propagator is changed into a milder(

p2 −m2)−1+f(g)

coupling strength dependent branch-cut singularity whose expansion in the cou-pling g leads the well-known logarithmic infrared divergencies: in this toy modelthe Dirac particle turns into an ”infraparticle”.

In the present context of sl potentials and their composites these exponentialsexp igΦ remain causally separable but they are expected to act in an infraredextended Wigner-Fock Hilbert space which transcends the range of Wigner’srepresentation theory8. Infrared photon clouds for different directions are uni-tarily inequivalent (spontaneous breaking of Lorentz invariance [44]). For morecomments see the concluding remarks.

For many applications it is useful to cast the change in e (8) into a differentialform setting on the 2 + 1 spacelike directional de Sitter space ([39] Corollary3.3)9

deAµ(x, e) = ∂µu(x, e), de =∑

i

dei∂ei (11)

where u is an exact de Sitter one-form. This conversion of directional differen-tials into x-derivatives plays an important role in passing from interactions inthe presence of a mass gap to their massless limit.

In the present context the formula for the change of e′s can be used tocompute the additive change which is necessary in order tomaintain the timelikee0 direction of the Coulomb potential

U(a,Λ)ACi (x)U(a,Λ)∗ = (Λ−1) l

i ACl (Λx+ a,Λe0) + (Λ−1) µ

i ∂µΦ(x,Λe0, e0)(12)

8I acknowledge many discussions over an extended period of time on this matter with JensMund.

9As mentioned therein this remains well defined since sl fields and their correlation functionsare homogeneous functions of degree zero in e and p.

15

The resulting affine transformation formula is equivalent to that obtained bystarting from the Wigner helicity representation and using transverse polariza-tion vectors [3]. This construction reveals how the covariant transformationof the timelike string is related to the inhomogeneous transformation of theCoulomb potential; its nonlocality remains as an unchangeable intrinsic prop-erty.

A similar situation arises if one fixes an ”axial” direction as e.g. e1 =(0, 1, 0, 0). In this case the causal localizability is preserved in both descrip-tions. Ignoring the spacetime localization aspect and treating it as if it wouldbe obtained by noncovariant gauge fixing is the cause of those intractable renor-malization aspects which led to the abandonment of the ”axial gauge fixing”.What became a curse in the axial gauge fixing turns out to be a blessing in thecovariant SLFT setting.

This limits the terminology quantum gauge theory as used in covariant per-turbation theory to the covariant ghost-extended indefinite metric pl BRSTsetting in which the existence of formal symmetry creating gauge charges is di-rectly related to the presence of unphysical indefinite metric and ghost degreeof freedoms. SLFT cuts the umbilical cord by which quantization maintainsa kind of parallelism between classical Lagrangian field theory and the morefoundational local quantum physics (LQP).

Even results which have been with us for decades, as the Aharonov-Bohmeffect, reveal an interesting refinement if presented in terms of a positivity pre-serving sl potentials for which the h = ±1 helicity degrees of freedom are notcontaminated by s = 0 negative metric residues of gauge theoretical potentials.

To see this recall that Einstein causality is the statement that the algebra ofoperators localized in the causal complement O′ of a spacetime region O belongto the commutant A(O)′ algebra (the von Neumann algebra which consists ofall operators which commute with A(O))

A(O′) ⊆ A(O)′ or A(O) ⊆ A(O′)′, Einstein causality (13)

A(O) = A(O′)′, Haag duality

The second line defines the somewhat stronger Haag duality which states that anoperator which commutes with all operator localized in the causal complementof O also belongs to A(O).

Einstein causality is a defining property of relativistic QFT but Haag du-ality may be violated. The cause of our ”eeriness” about the Aharonov-Bohmeffect (see remarks below) is that without looking at details we erroneously in-terpret the violation of the more specific Haag duality intuitively as a violationof Einstein causality. Fact is that in QFTs associated with massless h ≥ 1 fieldsHaag duality breaks down for operators localized in multiply connected regions(g ≥ 1 tori) [45].

This is an intrinsic property of the operator algebra associated with the plfield strength Fµν of the h = 1 Wigner representation which does not requirethe use of potentials [45]. But if one wants to understand this in terms of vector

16

potentials one must use the positivity-maintaining sl potentials which preservethe somewhat hidden topological properties of Wilson loops [13] whereas thepl Gupta-Bleuler potentials misses them. This is a strong reminder of the in-exorable connection between positivity and causal localization and a warningnot to confuse the ”fake localization” of gauge dependent objects with genuinecausal localization of quantum matter. It points to a potential source of seri-ous misunderstanding involved in transferring the perfectly reasonable classicalnotion of local gauge symmetries to QFT by attributing a physical meaningto the formal observation that quantum gauge charges are ”more local” thanthose corresponding to internal symmetries in s < 1 QFTs. It is also a strongreminder to rethink the physical meaning behind the saying ”gauging a model”.

Some additional comments may be helpful. From (11) it follows that a Wil-son loop10 formed with Aµ(x, e) is independent of the choice of the directione [13]. However the toroidal causal localization of this object retains a topologicalmemory of a former presence of a spacelike cylindrical e-extension which pre-vents a naive materialistic identification with a localization in a torus. One canchoose e in such a way that this extension is spacelike with respect to any simplyconnected convex compact region which remains spacelike separated from thespacetime torus. Yet it is not possible to completely forget that the integrandof the loop contained a direction e. This plays an important role for a properunderstanding of the Aharonov-Bohm effect [13].

In order to avoid misunderstandings, it should be emphasized that the break-down of Haag duality is not a result of the use of a particular field coordina-tization but rather an intrinsic property of the generated net of local algebrasand can be traced back with the help of modular localization to a correspond-ing property in Wigner’s helicity representations. The use of sl potentials andWilson loops in a positivity maintaining formulation instead of only workingwith field strengths is not necessary, but it facilitates the verification of thissomewhat hidden causality property.

The indefinite metric gauge potential AKµ (x) complies with Einstein causal-

ity but cannot account for the breakdown of Haag duality. Since our heuris-tic perception tends to identify causality with the stronger Haag duality, theAharovov-Bohm effect often causes an eerie feeling of a possible causality vio-lation (which is also the reason of its popular attraction).

When working with gauge theory one tends to overlook the strong connectionbetween positivity and causal localization. From a physical point of view gaugedependent quantities are unphysical because they do not describe the physicalspacetime localization. This is the foundational reason why the interactingLagrangian matter fields of gauge theory cannot play the role of interpolatingfields of particles.

The intertwiner functions v(p) which convert the m > 0 Wigner creationand annihilation operators a#(p, s) into covariant fields are well known. For the

10By convoluting with a test function one can convert the Wilson loop integral into into anoperator localized on a solid torus.

17

s = 1 Proca field they are of the form of polarization vectors eµ(p, i) obtainedby applying a rotation-free Lorentz boost to the coordinate unit vectors11 eµ(i)(i = 2− s3, s3 = −1, 0, 1)

vµ(p, s3) = eµ(p, i),∑

i

eµ(p, i)eν(p, i) = −ηµν +pµpνm2

MAPµ ,AP

ν (p) = −πµν(p), πµν(p) = ηµν −pµpνm2

where the πµν of the momentum space 2-pfct also turns out to be the basicbuilding block of all higher spin massive tensor potentials.

With a massive pl Proca potential Apµ one may associate two sl fields. One

is the scalar sl field φ defined terms of a line integral along e projected into thedirection e

φ(x, e) = (IeAPµ )(x)e

µ, a(x, e) = −mφ(x, e) (14)

The line integration has changed its classical dimension deng in terms of massunits and in order to maintain deng = 1 for all scalar and tensor fields it isconvenient to use the escort a instead of φ.

The classical dimension has to be distinguished from the quantum field theo-retic short distance dimension which is the number dsd(A) which appears in theleading short distance scaling behavior of the 2-pfct of a field A (or equivalentlyin the large momentum behavior of its Fourier transform). For pl fields

〈A(λx)A(λx′)〉 λ→0∼ λ−2dsd(A), MA,A(λp)λ→∞∼ λ2dsd(A)−2 (15)

〈A(x)A(x′)〉 =∫

eip(x−x′)MA,A(p)dµ(p)

The engineering dimension connects dsd with the infrared mass dimension dinfr =dsd − deng which measures the strength of the m→ 0 singularity m−infr.

Whereas deng = 1 for tensor potentials and 3/2 for their half integer spincounterparts, dsd and dinfr of pl fields increase linearly with the spin s as dsd =s + 1, the classical dimension remains at the value deng = 1 (3/2) for integer(half-integer) spin fields. Except for deng these relations change in passing frompl to sl fields. The line integral operation Ie lowers dsd and the zero massdivergence degree dinfr by one unit. Hence the sl scalar field a(x, e) (14) as wellas the sl vector potential

Aµ(x, e) = (IeFµν)(x)ev , Fµν(x) = ∂µA

Pν − ∂νAP

µ (16)

have dsd = 1 and dinfr = 0 whereas the deng remains unchanged.

In particular the momentum space 2-pfct of the massive field strength andits associated sl vector potential are identical to their massless counterparts,apart from the fact that in the massive case p ∈ H↑

m and for m = 0, p ∈ V ↑

which permits a continuous deformation from the forward mass-hyperboloid to

11Not to be confused with the components of string directions.

18

the mantle of the forward light cone (and its ”fattening” inversion, see below).In a pl description such a smooth passing between the three s = 1 spin degreesof freedom and the two helicities h = ±1 is impossible.

It is instructive to look at this degrees of freedom balance in more detail[38, 39]. With the help of the momentum space 4-matrix J (complex conjugationchanges sign of e, tr = transposed)

J νµ (p, e) = η ν

µ − pµeν

(pe)iε, J(p,−e) = J(p, e) (17)

MAµ(−e),Aν(e) =: Eµν(e, e) = (JπJ tr)µν

the in e diagonal momentum space 2-pfct takes the form of the second line. Itshows that the positivity of the sl 2-pfct follows from that of its pl localization.The rank of the E-matrix accounts for the degrees of freedom; J tre = 0 and theadditional relation Eµνp

ν = 0 for p ∈ V ↑ leads to a reduction from the 3 spincomponent to the two helicities h = ±1. This picture becomes blurred in thepresence of indefinite metric and ghost degrees of freedom of the gauge theoreticdescription.

A particularly interesting phenomenon is the continuous passing from themassless helicity situation to that of spin s = |h| by sliding from V ↑ to H↑

m.Although the massive and massless higher spin/helicity Wigner representationsand their associated fields correspond to inequivalent representations, the sl mas-sive 2-pfct is obtained by ”fattening” its massless counterparts. The fatteningproperty extends the smooth passing between massless and massive correlationfunctions known from s < 1 fields to higher spins/helicities [38, 39].

In the literature the terminology ”fattening” was used in order to interpretthe massive vector meson in the abelian Higgs model as the result of ”masscreation” by spontaneous symmetry breaking (SSB) of the gauge symmetry intwo-parametric scalar QED with a tachyonic mass term (the ”Mexican hat”potential). But this contains a misunderstanding. The correct understandingof the raison d’etre for the H in the presence of selfinteracting vector mesonsin which positivity and causal localization plays an important role turns out tobe much deeper. Its revelation is an important result which shows the power ofthe implementation of causal localization in the context of quantum theoreticalpositivity; we will return to this problem in section 7.

The use of correlation functions of sl fields is important in the study of theinfrared problems which one encounters in passing to massless limits (the inverseof fattening). In the absence of interactions the use of sl localization permits tocontinuously relate quantum field theories which belong to unitary inequivalentrepresentations (helicity versus spin). More important is its expected futurerole in unravelling unsolved infrared problems in the presence of interactions.Whereas in the presence of a mass gap the Wigner-Fock space retains its physicalrelevance in the presence of interactions, it does not pass to its helicity counter-part in the limit of vanishing Aµ mass. In that case the physical structure ofthe Hilbert space is not known (”infraparticles”, confinement). Although there

19

exist observational successful on-shell momentum space prescriptions, there isas yet no field theoretic (off-shell) spacetime understanding of infrared phenom-ena. An infraparticle Hilbert space is expected to reveal why the large-timelimits of LSZ scattering of charged particles with a finite number of outgoingphotons vanish, and how a spacetime scattering theory which leads directly tosoft photon-inclusive cross sections should be formulated.

The most important property of the previously introduced sl scalar escortφ(x, e) or a(x, e) (14) is their appearance in the linear relation

APµ (x) = Aµ(x, e) + ∂µφ, φ = − 1

ma (18)

This property justifies to call them ”escorts” of the sl potential. The appearanceof the derivative of a scalar field is a consequence of Poincare’s lemma andthe concrete form of the intertwiner of a can be directly obtained from thecorresponding relation between intertwiners (J as above)

J νµ vν(p) = vν − pµ

(ve)

(pe)iε(19)

which follows directly from the definition (16). They contain the full informationof the (m, s = 1) Wigner representation.

In the massless limit the dinfr = 1 of APµ (x) and φ(x, e) diverge whereas the

Aµ(x, e) and a(x, e) stay infrared finite. The relation

∂µAµ = −ma, deAµ = ∂µu (20)

where u = −m−1dea was introduced in (11), leads to a divergence-free (Lorentzcondition12) massless vector potential and a relation between two massless 1-forms in the de Sitter space of spacelike directions (that which remains of (18)).The purpose of the mass factors is to obtain the s-independent relation dsd =deng = dinfr + 1 for all sl fields.

Escorts (s for spin s. see next section) do not contain new degrees of freedomsince as the pl AP they are linear in the Wigner s = 1 creation/annihilationoperators a#(p, s3) and only differ in their intertwiners. Rearrangements ofexisting degrees of freedom are quite common in quantum mechanical many-body problems. A well-known case is the appearance of Cooper pairs which oneencounters in passing to the low temperature superconducting phase. Withouttheir presence as a result of rearrangement of existing degree of freedom theirpresence vector potentials would not become short range inside a superconductor(the London effect). In the field theoretic context escorts which are rearrangeds = 1 degrees of freedom are necessary in order to rewrite the ”fattening” ofmassless 2-pfcts in terms of pl fields. Each of these fields carries the full contentof (m > 0, s) Wigner representations.

For s ≥ 2 the sl fields lead to new properties. As a result of a possiblerelation with gravitation the case s = 2 is of special interest. The intertwiners

12But note that this is an operator relation and not a gauge condition within gauge theory.

20

of spin s Proca potentials13 must be a divergence- and trace-free symmetrictensor; this is a consequence of the way the 2s + 1 component subspace ofspin is embedded in the degree 3s-fold tensor product. Hence the intertwinersvµ1...µs

(p, s3) convert the symmetric trace-free s-fold tensor product of three-component spin 1 polarization vectors into covariant tensors of tensor-degrees.

For the momentum space s = 2 2-pfct one obtains

MAPµν ,A

Pκλ (p) =

1

2[πµκπνλ + πµλπνκ]−

1

3πµνπκλ (21)

where the numerical factors have their combinatorial origin in the symmetry andtracelessness and hence depend on the degrees of freedom. The sl 2-pfcts are ofthe same algebraic form and result by substituting πµν → Eµν(−e, e) [38, 39].As for s = 1 this can be seen by passing from the Proca potential to the fieldstrength (as means antisymmetrisation)

Fµ1ν1µ2ν2 = asµ↔ν

∂µ1∂µ2

APν1ν2 (22)

and using the two-fold momentum space I operation to pass from the fieldstrength to the potentials. Note that the symmetry of the Proca potential re-duces the antisymmetrization to a pairwise operation µi ↔ νi. The permutationproperties of the resulting F are those of the linearized Riemann tensor.

The new phenomenon is that (unlike for |h| = 1), the massless limit of thisfield strength is not the same as that obtained directly from the massless h = ±2Wigner representation. Correspondingly the sl potential associated with F s=2

is different from that of F |h|=2

Aν1ν2(x, e) = (I2eFs=2µ1µ2ν1ν2)(x)e

µ1eµ2 (23)

A(2)ν1ν2(x, e) = (I2eF

|h|=2µ1µ2ν1ν2)(x)e

µ1eµ2 (24)

This means in particular that the massive s = 2 sl potential obtained by fat-tening the A(2) is not the same as A although both account correctly for the2s+ 1 spin degrees of freedom and share their Wigner-Fock Hilbert space. Themassless limit of A splits into the direct sum of the two |h| = 2 degrees of free-dom and the h = 0 contribution which is the remnant of the s3 = 0 component.Conserved composite objects as currents and stress-energy tensors preserve thenumber of degrees of freedom by converting the ±s3 components into |h| = s3helicities.

In order to show how these results are related to the van Dam-Veltman-Zakharov discontinuity problem one must look at some details. Whereas fat-tening and taking the massless limit connect the 2-pfct of the 2-componentmassless helicity |h| = 2 potential A(2) with that of its 5-component s = 2 bydeforming the momenta of the 2-pfct between H↑

m and V ↑, the massless limit

13The P of the notation AP... refers to Proca or alternatively to pointlike.

21

of A is a cul de sac from which a return to the original massive s = 2 tensorpotential is not possible.

The relation between the massless limit of A with that of A(2) are easily seento have the following form

A(2)µν (x, e) = Aµν(x, e) +

1

2Eµν(e, e)A

(0)(x, e) (25)

Eµν(e, e) = ηµν + (eµ∂ν + eν∂µ)Ie + e2∂µ∂νI2e

where the momentum space Eµν has been rewritten as an integro-differentialoperator acting on a scalar sl field and the massless limit of A(0) is a (properlynormalized) scalar escort. Combining this relation with that between the s =2 pl field AP , its sl counterpart A and the derivatives of escorts (the s = 2analog of (18)) one obtains

APµν = Aµν + derivatives of escorts

one concludes that in the adiabatic limit the interaction between ”massive gravi-tons” and a trace-free energy-momentum tensor source Tµν is

limm→0

∫

APµνT

µν = limm→0

∫

AµνTµν =

∫

(A(2)µν −

ηµν√6ϕ)T µν (26)

where ϕ(x) =

√

3

2limm→0

a(0)(x, e) (27)

The independence of the integrated massless A(2) contribution from the stringdirection follows from

∂eκA(2)µν = m−1(∂µA

(2)κν + ∂νA

(2)µκ ) (28)

∂eκJν

µ = − pµ(pe)iε

J νκ

which in turn follows from the identity in the second line (for more details see[39]) and represents the s = 2 counterpart of the relation between de Sitterspace 1-forms in (20).

The result confirms the van Dam-Veltman-Zakharov discontinuity: the mass-less limit of massive gravity differs from the result obtained directly with mass-less gravitons. Different from Zakharov’s calculation which identifies this con-tribution as being the relic of a unphysical gauge theoretical degrees of freedom,the present calculation shows that it is really the massless footprint of the physi-cal s3 = 0 spin component. For the traceless stress-energy tensor of photons thelast contribution vanishes whereas for couplings to matter (mercury perihelion)it remains. Thus we obtain the discontinuity in the massless limit.

This calculation permits a straightforward extension to any spin. The rela-tion between the Proca potential, its sl counterpart and the associated sl escortsreads

APµ1...µs

= Aµ1...µs+ sym.(∂µ1

φµ2...µs+ ∂µ1

∂µ2φµ3... + · · ·+ ∂µ1

. . . ∂µsφ) (29)

22

where the φµ1...µiis an s − i fold iterated line integral along e of the spin

s Proca potential and the symmetrization is over all indices and the φ arealready symmetric by construction. For our purposes it is more convenient touse other escorts which are obtained in terms of descending from the sl Aµ1...µs

in terms of forming divergencies

APµ1...µs

= Aµ1...µs− sym.(∂µ1

ma(s−1)µ2...µs

+∂µ1

∂µ2

m2a(s−2)µ3... + · · ·+ ∂µ1

. . . ∂µs

msa(0))

(30)

ma(s−r)µr...µs

= −∂µa(s−r+1)µµr ...µs

, a(s)µ1...µs:= Aµ1...µs

The second line shows that the a escorts start from the sl potential and descendby differentiation instead of descending from AP by line-integration. The a havethe same dimension dsd = 1 = deng , dinfr = 0, and are linear combinations ofthe φ escorts. For m > 0 each escort carries the full content of the Wigner spins representation.

Although the a′s have a massless limit they do not de-couple. The vanDam-Veltman-Zakharov discontinuity shows that for s = 2 the |h| = 2 andh = 0 contributions stay together and have to be separated with the help ofan integro-differential operation (25). The analogous situation in the generalcase is that the even and odd s3 remain coupled and can only be split in termsof their helicity content by the use of such integro-differential operations [39].Naturally one can obtain a spin s vector potential from fattening a masslesshelicity h potential if h = s.

The tensor vµ1..µ|h|(q, e) which appears in the relation of the helicity h tensor

field A(|h|)µ1..µ|h|

and the Wigner operator a#(q, h) (which extends the construction

of A(2)µν in (25) to arbitrary helicity h). This e-dependent polarisation tensor

v..(q, e) replace the only up to re-gauging defined polarization tensor. If used inWeinberg’s scattering argument for the soft scattering contribution of a masslessparticle with momentum q to a scattering process among n massive particleswith momenta pi, i = 1, ..n ([9, 4.1]), one obtains the same conclusions exceptthat the gauge theoretic argument is replaced by the e-independence of themodified S-matrix which follows in first order from the fact that the directionalderivative with respect to e on these polarisation tensors can be written as aspacetime derivative ∂µ acting on such a tensor. The use of sl polarizationtensors is demanded by positivity, and shows that the argument has nothing todo with any imposed gauge symmetry.

The weakness of Lagrangian constructions of conserved currents and stress-energy tensors is that with the exception of low spins there is no guarantythat the so obtained classical expressions have the correct commutation rela-tions with the quantum fields. It is much safer and easier to start from thecommutation relations between Wigner’s generators of the Poincare group withthe Wigner operators particle operators a#(p, s3) and rewrite this noncovariantparticle-form of the commutation relations with the help of the intertwiners intocovariant commutation relations of the stress-energy tensor [30]. In this way the

23

correct covariant commutation relations are satisfied by construction.

This construction provides a simple illustration of the new spirit of ”intrinsicquantum” and it is worthwhile to take a quick look. One starts from the expres-sions of the infinitesimal generators of translation Pµ and Lorentz generatorsMµν in terms of the Wigner operators a#(p, s3)

Pµ =

∫

∑

s3

a∗(p, s3)pµa(p, s3)dµ(p) (31)

Mµν = −i∫

(δs3s′3p ∧ ∂p + d(ω)ts3s′3)µνa∗(p.s3)a(p, s

′3)dµ(p) (32)

The first step is two rewrite the contribution of the spin component ss to Pµ as

Pµ =

∫ ∫

dµ(p)dµ(p′)∑

s3,s′3

(pµa∗(p, s3)δs3s′3(2π)

3δ(p− p′)(p10 + p20)a(p′, s′3)

(33)

(2π)3δ(p− p′) =

∫

e−i(p−p′)xd3x =

∫

e−i(p−p′)xd3x (34)

where in the second line used the cancellation of the p0 components.

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

gMNνMs3νNs′s = δs3s′3

where M and N represent the multi-tensor indices of the intertwiner. Whatremains is to use the Fourier transform (34) and pass from the Wigner operatorsto the fields. Using the fact that the a∗a∗ and aa contributions vanish as a result

of the presence of←→∂ 0 and that aa∗ terms are absent in Wick-ordered products

one verifies that

Pµ =

∫

Tµ0(x)d3x, Tµν(x) = −

1

4

∫

: APµ1..µs

(x)←→∂ µ←→∂ 0A

P,µ1..µs(x) : (35)

where Tµν is a contribution to the stress-energy tensor. The full tensor densitywhich generates all Poincare transformations is of the form

Tµν = Tµν + ∂ρ∆µν,ρ (36)

To compute the second contribution, which is also a bilinear expression in theAP tensor fields, one starts from the bilinear expression for Mµν in terms of thea# Wigner operators which also contains a contribution the infinitesimal partof Wigner’s little group.

The representation of the Poincare group generators in terms of pl stress-energy tensors may be rewritten in terms of their sl counterparts. For this we

24

refer to [30, 39]. It is not the purpose of the present paper to re-hash resultswhich need no improvement but rather to explain how they fit into a projectwhose aim is to move away from quantization and place QFT on its own feet.

The last topic in this section is the construction of the infinite spin fieldsas Pauli-Lubanski limits. A direct construction of a scalar infinite spin fieldwas presented in [8]. As the result of the intrinsic noncompact causal localiza-tion structure of infinite spin matter [27] the tightest field localization whichpreserves causality is sl. For the explicit construction of these fields it was nec-essary to use methods of modular localization. As recently shown by Rehren[30] it is also possible to obtain such fields as P-L limits of higher spin escorts.Since this construction extends the old result and broadens the understandingof this matter it may be helpful to the reader to present some of these ideas.

The starting point are the a-intertwiners and their recursive interconnec-tions (30). Since the P-L limit requires a variable spin, one needs to label thea′s by two indices as a(s,r)(x.e), r ≤ s. It turns out that the properly renormal-ized a(s,0) converges towards the scalar infinite spin fields φκ(0) at value κ of theP-L invariant which were previously computed with modular methods [8, 28].Even more: the properly renormalized a(s,r) converge in the P-L limit at fixedinvariant κ against infinite spin tensor fields φκ(r) of tensor degree r [30]

P -L limN (s)(1 −m√

−e2Ie)sa(s,r)(x, e) = φκ(r)(x, e) (37)

where N (s) is an r-independent combinatorial normalization factor [30].

This limit has very different properties from the m→ 0 limit. Whereas thea(s,r) de-couple in the massless limit [39] this does not happen in the P-L limit;the φκ(r) for different tensor degree r remain coupled in terms of differentialoperators. In fact, although they are massless, they behave in a certain senseas if they were massive. Hence it is not surprising that, as already mentionedbefore, they are also not conformal covariant.

Infinite spin matter shares its string-like nature with sl interacting finitespin/helicity fields but the cause of their stringiness is very different. Whereasthe sl nature of infinite spin free fields and their Wick composites originatesfrom the presence of infinite degrees of freedom (it is the only elementary matterwith infinite degrees of freedom), interpolating interacting fields are genuine14 slbecause this is the only way in which positivity and causality can be preservedin the presence of s ≥ 1 interactions. This difference also accounts for theimpossibility of infinite spin matter to reach equilibrium, a fact already observedby Wigner and extended in [30].

The absence of a highest tensor degree from which all escorts result by de-scending as in (30) has the consequence that infinite spin matter cannot interactwith normal matter (section 6).

14Unlike sl finite spin free fields they are not line integrals over interacting pl fields.

25

3 Relation between the Weinberg-Witten prob-

lem and the Velo-Zwanziger conundrum

In previous work [39] it was shown that for massive s ≥ 2 free field one canconstruct sl tensor potentials whose associated sl E-M tensors Tµν(x, e) differfrom their pl counterpart TP

µν by divergence terms which do not contribute, sothat their associated separately conserved momentum- and Lorentz- densitieslead to the same global Poincare ”charges”. The pl and sl fields in terms ofwhich the pl and sl energy-momentum tensors are written lie in the same localequivalence class as the pl and sl fields from which they are constructed i.e.the tensor fields commute if x is spacelike with respect to the (space- or light-like) string S(x′, e)[

TPµν(x), Tκλ(x

′, e)]

= 0 for x×S(x′, e), S(x′, e) = x′ + R+e, e2 = −1 or 0

(38)

The two tensors naturally share the same ”engineering” dimension (the di-mension in terms of mass units) deng = 4 but posses very different short dis-tance dimensions dsd(T

P ) = 2(s + 1) + 2 and dsd(T ) = deng = 4 which ac-counts for the fact that only the sl T allows a massless limit whereas TP di-

verges as TP m→0∼ m2−2s (the W-W obstruction). As expected, the masslesslimit of sl T turned out to be a direct sum of the 2s+ 1 helicity contributions|h| = 0, 1, ..s acting on the tensor product of Wigner-Fock helicity spaces [39].

A similar but somewhat simpler situation arises with conserved pl electriccurrents of massive complex s ≥ 1 free fields. In this case dsd(j

Pµ ) = 2(s+1)+1

and jPµm→0∼ m1−2s, so that the degree of the W-W ”zero mass badness” of

jP at the same value of the spin is by one unit worse than that for TP . Inorder to see the connection with the V-Z causality conundrum it is interestingto to first consider a conserved s = 1 current coupled to an external potentialLP = jPµ U

µ which with dsd(AP ) = 2 and hence dsd(j

P ) = 5 violates the power-counting bound of renormalizability but is in the indefinite metric pl gaugesetting as well as in the positivity-maintaining sl formulation renormalizabledsd(j

g,s) = 4.

Whereas for s = 0 (s = 1/2) there is neither a gauge nor a sl descriptionand the pl coupling coalesces with what V-Z (and everybody) would write forthe Klein-Gordon (Dirac) equation in an external potential Uµ, there is a milddiscrepancy for s = 1 between what they naively would write down and theclassical counterpart of the causality-preserving quantum situation.

Using the relation between pl and its canonically associated sl field and the

26

gradient of its escort APµ (x) = Aµ(x, e)− ∂µφ(x, e) one finds15

jPµ = iAPν(x)∗←→∂µA

Pν (x) = jµ(A) + jµ(a) + ∂κCκµ (39)

Cκµ = iA∗κ

←→∂ µφ+ h.c.+ iφ∗

←→∂ µ∂κφ

Here the first two contributions are the conserved currents of the complex s =1 sl field Aν(x, e) and its scalar sl escort a(x, e) = m−1φ(x, e)16, whereas theobstructing contribution ∂κCκµ, which carries the m → 0 and short distancedivergencies, appears in the form of a 4-divergence. Both sl fields Aµ and a havenontrivial zero mass limits; the limit of Aµ(x, e) is a helicity h = ±1 potentialwhich is the positivity obeying sl counterpart of the indefinite metric masslessgauge potential, whereas that of a is the e-independent massless scalar fieldwhich carries the remains of the s3 = 0 spin component.

The first two contributions in (39) account fully for the charge. To seethis one must show that the spacetime divergence of the time component ∂κC0µ

can be transformed into a spatial divergence; the argument uses the free fieldequation of the contributing fields and is the same as that used to connect theconserved E-M tensor with the global Poincare ”charges” in [39]. The obstruct-ing ∂κC0µ is disposed of in the infinite volume limit. The conserved currentsof both sl free fields have dsd = 3 instead the dsd = 5 for the original current,hence the short distance and m → 0 limiting behavior of the remaining chargedensity is not worse than that of a charge-carrying s < 1 and hence we obtainfor the charge

Q(AP ) = Q(A) +Q(a)

jPµ (x) ≃ jsµ(x, e) = jµ(A) + jµ(a)

The equivalence≃ is meant to express the fact that the currents on both sides aremembers of the same sl local equivalence class and, since their zero componentsdiffer only by total divergence, their charges are equal.

This covers the use of the sl currents in ordrt to overcome the zero massWeinberg-Witten problem. Their use in solving the V-Z causality conundrumstarts from the pl interaction density LP = jPµ U

µ

It is important to understand that as long as m > 0 both fields Aµ(x, e) andits scalar escort a(x, e) carry the full content of the massive s = 1 QFT, it isonly in the m = 0 limit that the Hilbert space splits (tensor-factorizes) into thedifferent helicity components. Hence the situation of the currents is analogousto that of the E-M tensor with the charges corresponding to the generators ofthe Poincare group [39]. In both cases the contributions in terms of sl fields canbe written in such a way that the terms on the right hand side corresponds tothe helicity split in the massless limit.

15In the gauge setting the degrees of freedom preserving escort field is replaced by the d.o.f.increasing indefinite metric Stuckelberg field.

16The change of normalization of the escort maintains its classical dimension and nontrivialmassless limit.

27

The existence of a massless limit as a result of the weakening of the causallocalization from pl to sl is connected with the improvement of short distanceproperties. Hence the sl cure of the W-W obstruction [34] in the massless limitof pl currents is expected to contain informations about the V-Z obstruction [37].

The simplest family of V-Z models correspond to interaction densities of theform LP = gjPµ U

µ in which conserved spin s currents jPµ are coupled to anexternal potential Uµ. To solve the classical V-Z problem one uses its tightconnection to its quantum counterpart. This is motivated by the experiencethat quantum causality problems (Einstein causality) are somewhat simpler tohandle than their classical counterpart (causal propagation of Cauchy data). Inparticular a quantum field obeying a linear field equations is directly related tothe space of classical solutions of the same field equation by forming expectationvalues of the quantum field in coherent Wigner-Fock states so the relation isrigorous and explicit.

Recall that for s = 0 with dsd(jP ) = 3 the V -Z classical field theory associ-

ated with this superrenormalizable external potential LPmodel is not covariantbut remains causal in the sense of Einstein causality and classical hyperbolicpropagation. The field equation is precisely of the kind considered by V-Znamely a local modification of the free equation by a term linear in coupling ofthe field to Uµ. Our main message will be that for s ≥ 1 causal couplings willbe more involved than such linear in Uµ additions.

The first clouds on the causality horizon appears at s = 1. In this casedsd(j

P ) = 2+2+1 = 5 and a linear in Uµ dependence on the external potentialsuffers from the V-Z causality syndrome. The message from the solution of theW -W problem is to pass to the sl dsd(j) = 3 coupling. But whereas passingfrom the pl current to its sl counterpart is a change within the relative causalitymaintaining equivalence class (a change of field coordinatization in the senseof section 4) which preserves the global charge, this change does not maintainthe quantum action

∫

LPd4x.At this point it it is useful to recall the relationbetween the quantum action and the S-matrix.

The perturbative definition of S in terms of the interaction density is in termsof the adiabatic limit of Bogoliubov’s operator-valued generating S-functionalS(g(x))

S = limg(x)→g

S(g(x)), S(g(x)) := T exp i

∫

g(x)LP (x)d4x (40)

S(g(x)) = 1 +

∞∑

n=1

in

n!

∫

..

∫

g(x1)..g(xn)TLP (x1)..L

P (xn)d4x1..d

4xn (41)

We will bypass well-known technical issues as the definition of time-orderedproducts in terms of a list of properties and get immediately to the point inwhich this formulation differes from standard ones.

This consists in realizing that there is a certain amount pf freedom for chang-

28

ing interaction densities LP → L without changing S. As long as

LP − L = ∂µVµ,

∫

∂µVµg(x)d4x

g(x)→g→ 0 (42)

and L is an operator in the same Wigner-Fock space the contributions fromthe divergence produces a boundary term which vanishes at spacetime infinity(verified by sandwiching ∂V between Wigner-Fock particle states). We refer tothis relation as the L, Vµ pair relation.

For the case at hand this flexibility can be used to convert the pl interactionsdensity into its better behaved sl counterpart L and a dsd = 5 and m−1 singulardivergence part

LP = jPµ Uµ = L− ∂κVκ, Vκ = −CκµU

µ (43)

L := jsµUµ + Cκµ∂

κUµ

where C has been previously defined in the relation between jPµ and its sl coun-terpart (39). Since dsd(C) = 4 the sl interaction density stays within the power-

counting bound of renormalizability, hence (analogy to the second orderA·A |ϕ|2in scalar QED) one expects an additional second order contribution.

Before returning to this point it is helpful to introduce a useful notation.There is an elegant way of formulating this physical equivalence in terms of thedifferential calculus on the d = 1+2 de Sitter space of spacelike string directionse, e2 = −1 as (using the previously introduced differential notation (11))

deLP = de(L − ∂µVµ) = 0, V µ = −Cµ

κUµ, de = deµ∂

∂eµ(44)

This L, Vµ pair requirement guaranties that the first order S-matrix which ap-pears in the adiabatic limit

limg(x)→1

∫

g(x)LP (x) = S(1) = limg(x)→1

∫

g(x)L(x) (45)

remains the same for both interactions.

The use of this differential calculus on de Sitter space is particularly conve-nient for time-ordered products of interaction densities as they appear in higherorder perturbations of S. This places a condition on time-ordered productswhich enter the higher order contributions to S and amounts for n = 2 to therequirement17

TLP (1)LP (2) = TL(1)L(2)− ∂(1)µ TV µ(1)L(2)− ∂(2)µ TL(1)deL(2)+ (46)

+ ∂(1)µ ∂(2)ν TV µ(1)V ν(2)

or de(TL(1)L(2)− ∂(1)µ TV µ(1)L(2)− ∂(2)µ TL(1)V µ(2) + ...) = 0

17In order to use the same e for all fields one must use a lightlike e (see remarks on relationbetween lightcone quantization and lightlike strings in previous section).

29

and its obvious extension to arbitrary n. The derivatives outside the time-ordering are necessary in order to be allowed to omit these terms in the adiabaticlimit of the S-matrix and in this way preserve the higher order invariance of theS-matrix under the LP → L change of the interaction density.

The validity of the higher order pair condition may be disturbed by localδ-contributions (”obstructions”) which result from taking the ∂µ inside wherethe pair condition leads to a vanishing, typically

∂µ 〈T∂µϕ∗(x)∂νϕ(x′)〉 − 〈T∂µ∂µϕ∗(x)∂νϕ(x

′)〉 ≃ ∂νδ(x − x′) (47)

as higher order changes of the the interaction density L → Ltot = L + gL(2) +..and refers to them as induced terms where Ltot is the new interaction densityto be used in the g expansion of the Bogoliubov formula. The induction phe-nomenon is very different from the counterterms of the pl setting which comewith new coupling parameters.

In the present s = 1 case the induction can be shown to stop at secondorder with dsd(L

(2)) = 3. This leads to quadratic Uµ contribution in the fieldequation which is not what Y-Z would write down. This contains a warningagainst confusing the naive locality of addition of linear couplings in the fieldequations with the preservation of physical causality of quantum fields or theirclassical counterparts obtained by taking expectation values of the quantumfield in coherent Wigner-Fock particle states.

There exists a general definition of a quantum field and a derivation of itsfield equation which does not require the interaction density to be quadratic inthe quantum fields or the use of the Lagrangian quantization formalism but canbe directly formulated in terms of the interaction density [69]. It describes aquantum field as the adiabatic limit of the following expression

A(x)|L = limg(x)→g

δ

iδf(x)S−1(g(x)L)S(g(x)L+ f(x)A)|f=0 (48)

where again S(g(x)L) and L is either LP = gjPµ Uµ or the sl L with and the field

on the left hand side is the interacting field in the L setting which for L = L isstring-local. Different from the on-shell S-matrix the off-shell fields depend onthe choice of the pair-related interaction densities. But the fields correspondingto different Ls live in the same Hilbert space and remain within the same localequivalence class i.e. they commute if points and strings are spacelike separated.

In fact there are explicit formulas which connect the interacting field in onedescription to a power series in g of composites in the pair related setting (seesection 6). Hence in contrast to the S-matrix the quantum fields differ but re-main mutually related in the sense that one field is a power-series in the couplingparameter with coefficients which are composites (normal products) in the re-spectively other description. In the massless limit the L, Vµ pair setting breaksdown and the S-matrix is lost, but the definition of the fields still survives ifL is the sl L and Vµ is replaced by Qµ = deVµ with the pair condition being

30

deL = ∂µQµ. The reason is that the appearnce of S−1 in (48) leads to com-pensations of divergent terms with the numerator in the adiabatic limit. Thisopens the possibility to rigorously study infrared phenomena caused by exter-nal potentials. In the massless limit the only surviving positivity-maintainingformulation is the string-local setting.

Returning to the V-Z issue, the so obtained sl quantum fields and their rig-orously associated classical sl field are string-local so that causality is preservedbut in a weaker sense than that which one needs in order to formulate causalpropagation of Cauchy data. But since as a result of the new normalizationcondition (46) which relates the pl fields in a unique way to their sl counterpart,the former have a much better status than without this relation there is the pos-sibility to pass from sl to pl fields. The resulting pl fields cannot be Wightmanfields (operator-valued Schwartz distributions). They are ultra-distributions inthe sense Jaffe [46]. An illustrative example associated with the local equiva-lence class of free fields is the Wick-ordered exponential of a free scalar field:exp gA(x) :. Jaffe showed that their infinite short distance dimension dsd =∞does not necessarily destroy their causal localizability; their testfunction spacesmay still contain a dense set of O localized smearing functions. But at thattime an idea of how to use this extended notion of causal localization in actualcomputations in nonrenormalizable theories was missing, the only indication inthis direction were some ”rainbow approximations” in certain nonrenormaliz-able models [81]. The new concepts nourish the hope that the new sl settingmay extend the range of perturbative QFT and turn many of the ”effective”models into respectable QFTs. A particular simple class for the mathematicalstudy of such questions are external potential models.

Formally one may try to save the V-Z situation by giving up positivityand work with gauge theory. In this case the Wigner-Fock space is replacedby a Gupta-Bleuler Krein space and instead of a LP , L pair one would nowhave a LP , LK pair and the s = 1 model will the pointlike nature and thecausal propagation of the field equations will be preserved but the nonlineardependence on the external potential remains.

We remind the reader that classical counterparts of quantum fields corre-sponding to quadratic interaction densities are obtained in the same way asfor free fields: the linear space of classical solutions is spanned by expectationvalues of the basic quantum field in coherent Wigner-Fock states.

As expected the formal pl perturbation theory is afflicted with the (withperturbative order) increasing number of counterterms and associated unde-termined coupling parameters. The reason is that the powerful normalizationcondition (46) is not available in a pure pl setting, it only enters on the sl sideand the pl formulation inherits its physical consequences through the pair rela-tion. For the s = 1 external potential coupling this pair relation cannot preventthe with perturbative order worsening momentum space increase of pl fields butat least it can prevent the increase of the number of free parameters.

Formally this idea works also in gauge theory; in this case the scalar negative

31

metric Stuckelberg field φK(x) plays an analogous role as the escort field apartfrom the fact that it adds an unphysical degree of freedom which enlarges theWigner-Fock space to the indefinite metric Gupta-Bleuler Krein space

AKµ (x) = AP (x) + ∂µφ

K(x) (49)

LP = LK − ∂µV Kµ

Hence the form of the first order LK , V Kµ pair condition and the second order

induction is completely analogous; the main difference is conceptual in that thepl localization of gauge dependent fields is not the physical localization. But alsoin this setting the (formally) causal field equations would have a more involvedUµ dependence than that the linear Uµ coupling in the field equation proposedby V-Z. The missing positivity is a quantum phenomenon which has no classicalcounterpart. This shows that causality is not simply a property of adding linearlocal couplings in field equations.

In fact this implementation of gauge theory had been successfully applied tovarious s = 1 couplings involving vector potentials by Scharf and co-worker ([40]and references therein). It was only formulated for the S-matrix but permits anextension to fields which is similar to the above pair construction. This way offormulating gauge theory is generally different from the Fronsdal-Vasiliev wayof introducing interactions in the setting of gauge invariant Lagrangian quan-tization and avoids certain problems of the latter. In this context it is worthmentioning that Mund recently discovered what he calls a hybrid implementa-tion of the pair property (to appear). In a Gupta-Bleuler setting without ghostshe succeeds to unite the different S-matrix preserving conditions under one roof.

A new situation arises from s = 2; the rewriting of the pl current into its slcontributions and a more singular remainder reads

APµν = Aµν +m−1(∂µaν + ∂νaµ) +m−2∂µ∂νa (50)

jPµ (x) = iAP∗κλ

←→∂µA

Pκλ = jµ(x, e) + ∂κCκµ

jµ(x, e) = iA∗κλ

←→∂µA

κλ − 2ia∗κ←→∂µa

κ + ia∗←→∂µa

In this case dsd(LP ) = 7 and dsd(j

µ) = 3 with LP being by 2 units beyond thepcb dsd = 4; the ∂κCκµ carries the dsd > 3 contributions. After an additionallinear disentanglement the three terms in the last line correspond in the masslesslimit of the h = 2, 1, 0 helicity contributions. This has the same origin as thevan Dam-Veltman-Zakharov helicity mismatch whose helicity disentanglementhas been discussed in [39]. In contrast to a full QFT the perturbation theory ofan external potential coupling has only induced contributions from tree-terms;there are no contributions from loops and hence there are no new countertermparameters.

The obstructions against the implementation of (46) are again convertedinto higher order induced contributions to the interaction density L → Ltot =

32

L + L(2) + L(3) + ...so that the interaction density to be used in the S-matrixarising in the adiabatic limit of the Bogoliubov S-functional is Ltot and not L;this leads to a regrouping of nth order terms (see section 5.2).

A direct s = 2 pl perturbation calculation would miss the use of the powerfulnormalization condition (46) and the resulting induced terms and instead leadback to the number of parameter-increasing counterterms of nonrenormalizabletheories. Hence it is mandatory to pass to the pl setting via the sl formalismusing the conversion between sl and pl mentioned after (48) and explained inmore detail in section 6.

For s = 1 the short distance dimensions of fields in the sl description remainfinite whereas the aforementioned S-matrix-preserving conversion into pl leadsto dsd = ∞ fields in agreement with the pl nonrenormalizability. In s ≥ 2interactions even the sl fields are dsd = ∞ ultra-distributions. Causal localiz-ability in such a situation requires the existence of a dense set of testfunctionssupported in spacelike cones or in lightlike tubes18; the affiliated pl fields areeven more singular but may still be localizable in finite spacetime regions Osince there is a hierarchy of causally localizable but short distance worseningultra-distributions. External potential models provide a simple playground forstudying localization properties of ultra-distributions.

In the present work the external potential problems and the V-Z causalityconundrum plays the role testing ground for new concepts. This topic deservesa more detailed and profound presentation. The purpose of the present sectionwas the more modest aim to convince the reader that V-Z arrived at theircausality violation by their incorrect belief that local modification of free fieldequations by adding linear coupling to external potentials Uµ preserve preservecausal localization.

There has been an attempt to solve the V-Z conundrum in terms of StringTheory [48] [9]. The problem with this is that the issue of causal localizationhas no natural place in string theory which are solely based on properties ofthe Virasoro algebra in a special 10-component conformal current model. Thephysical meaning of ”string” in ST remains a mystery (see next section).

4 Particle wave functions and causal localization

Most of the content of this section is known to those who are familiar with theconcepts of local quantum physics [2]. But since without knowledge of some ofthese concepts it is difficult to understand the aim and scope of the new set-ting of perturbation theory, its brief presentation may provide additional help-ful background. In the last part of this section it will be used to explain whyString Theory despite its name bears bears no resemblance to string-localizationof quantum fields. Modular localization is also helpful for obtaining a better

18These are the regions obtaine from finite smearing in x and e. Ultra-distributions whichremain localizable in cones have been investigated in [47].

33

understanding about in what sense quantum fields, unlike their classical coun-terparts which can be directly measured, play the role of ”coordinatization” ofthe operator content of QFT, similar to the use of coordinates in the descriptionof geometry.

Wigner’s theory of positive energy representations presents an interestingmeeting ground of two very different localization concepts. On the one handthere is the quantum mechanical localization of dissipating wave packets whosecenter moves on relativistic particle trajectories. Its formulation in terms ofquantum mechanical Born probabilities leads to the so-called Newton-Wignerlocalization [49]. For a scalar m > 0 particle

(ψ, ψ′) =

∫

ψ←→∂0ψ

′d3x =

∫

ψNWψ′NW d3x (51)

hence ψNW (p) = (2p0)−1/2ψ(p)

Hence an improper N-W eigenstate of the position operator xNW has an meanextension of the order of a Compton wave length in the relativistic description.In scattering theory, where only the large-time asymptotic behavior matters,such ambiguities in assigning relativistic quantum mechanical positions at finitetimes are irrelevant; the centers of wave packets of particles move on relativisticvelocity lines and the probability to find a particle dissipates as t−3 along theselines for all inertial observers.

On the other hand the idea (known by a lesser number of particle physicists)behind modular localization is related to properties of dense subspaces obtainedby applying algebras of local observables A(O) ⊂ A of a QFT to the vacuumstate: H(O) = A(O)Ω where O is some spacetime region. That such subspacesare dense in the Hilbert space was a surprising discovery in the early 60s (theReeh-Schlieder theorem [1] [2]) which showed that the omnipresence of vacuumpolarization confers to QFT a very different notion of localization from thatrelated to Born’s quantum mechanical localization based on position operators.

The projection of H(O) onto the one-particle Wigner space HWig(O) =E1H(O) has the surprising property that it can be constructed solely in terms ofdata from Wigner’s representation theory and that in the absence of interactionone can even recover the net of causally localized subalgebras directly from thatof modular localized Wigner subspaces [22].

In this way one does not only gain a more profound understanding of QFTbut one also learns that Weinberg’s pure group theoretic construction of in-tertwiners starting from Wigner’s representation theory is part of a more gen-eral setting which leads to an extension of perturbative renormalizability. Thisimportant concept of modular localization was not available during Wigner’slifetime 19.

The simplest way to see that the quantization of a relativistic classical par-

19From later conversations between Wigner and Haag [50] it is clear that Wigner looked for away to connect his particle representation theory directly with QFT and became disappointedwhen he realized that N-W localization is not what he was looking for.

34

ticle associated with the action√−ds2 does not lead to a covariant quantum

theory is to remind oneself that there exists no operator x which is the spatialcomponent of a covariant 4-vector. The conceptual problem one is facing is bet-ter understood by showing that causal localization bears no relation to Born’sprobabilistic quantum mechanical definition.

Starting from the quantum mechanical projectors E(R) for R ⊂ R3 whichappear in the spectral decomposition of xop

xop =

∫

xdE(x) (52)

one hasE(R)E(R+ a) = 0 for R ∩ (R+ a) = 0

Define E(R+a) = U(a)E(R)U(a)∗ for a ∈ R4. Assuming that this orthogonalityrelation has a causal extension in the sense that E(R)E(R+a) = 0 for spacelikeseparated R×R + a leads immediately to clash with positivity of the energy.This follows from the fact that the expectation value (ψ,E(R)E(R + a)ψ) isanalytic for Im a0 > 0 and hence its vanishing on an open set on the real a0axis implies its identical vanishing (the Schwarz reflection principle) leads andusing the analyticity of in the time component Im a0 > 0, which follows from thepositivity of the energy spectrum, one finds that this expectation value cannotvanish for nonoverlapping regions at a fixed time without vanishing identically.Hence ||E(R)ψ||2 = 0 which implies the triviality of such projectors E = 0.

A slight extension of the argument reveals that it can be dissociated fromthe position operator of quantum mechanics. It then states that in modelswith energy positivity it is not possible to describe causal localization (”micro-causality”) in terms of projectors and orthogonality of subspaces [51]. A pro-found intrinsic understanding of causal localization in QFT points into a verydifferent direction from that of quantization of actions describing classical worldline, world sheets or the Nambu-Goto action to name just a few.

To prepare the ground it is helpful to start with some mathematical conceptsconcerning relations between real subspaces H (linear combination with reals)of a complex Hilbert space H ⊂ H. The symplectic complement H ′ of a realspace is defined as the closed real subspace (H ′ = H ′) in terms of the imaginarypart of the scalar product in H

H ′ = ξ ∈ H; Im(η, ξ) = 0 ∀η ∈ H (53)

H1 ⊂ H2 ⇒ H ′1 ⊃ H ′

2 (54)

which turns out to be the real orthogonal space on the real iH (real means onlyreal linear combinations are allowed)

A closed real subspace H is called ”standard” if it is both cyclic and sepa-

35

rating

H cyclic: H + iH = H (55)

H separating: H ′ ∩H = 0(H + iH)′ = H ′ ∩ iH ′

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

It is quite easy to obtain such standard spaces from covariant free fields.In the simplest case of a scalar field the Hilbert space H is the closure of the1-particle Wigner space defined by the two-point function of the smeared fields

(f, g) = 〈A(f)∗A(g)〉 =∫

f∗(p)g(p)dµ(p) (56)

[A(f)∗, A(g)] = −i Im (f, g)

where dµ is the invariant measure on the positive mass hyperboloid. Accord-ing to the Reeh-Schlieder theorem [2] the one-particle projection of the densesubspace of causally O-localized states20 is dense in the one-particle Wignerspace. O-localized real testfunctions define a dense real subspace H(O) andcausal disjointness corresponds to ”symplectic orthogonality” and produces aclosed real subspace.

H(O′) = H(O)′ (57)

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

ZH(O′) = H(O)′, Z =1 + iU(2π)

1 + i(58)

where the unitary ”twist” operator Z which is related to the factor −1 of the2π rotation. The use of the twist operator allows to treat bosons and fermionsunder one common roof.

The important step for an intrinsic understanding of QFT without the useof ”(quasi)classical crutches” of quantization is to invert the previous construc-tion: find a relation within Wigner’s representation theory which permits to de-fine O-localized real subspaces which have the correct covariant transformationproperties under Poincare transformations [22]. For this purpose it is helpfulto reformulate the above properties so that they take the form known from themathematical Tomita-Takesaki theory of operator algebras. This theory per-mits a direct connection of positive energy Wigner representations with a ”localnet of operator algebras”.

It has the advantage of providing a unified view in which Weinberg’s plintertwiner formalism and its sl extension are seen as two ways of generating

20States localized in the spacetime region O are defined as the dense Reeh-Schliedersubspace obtained as A(O) |0〉 where A(O) is the O localized subalgebra of A.

36

the same free field theory. The better short distance properties of higher spins ≥ 1 sl fields is the starting point of a new renormalization theory in which,unlike for free fields, pl and sl is not a matter of free choice but rather leads to theintrinsic physical distinction between pl local observables and sl interpolatingfields of particles. But before we get there we need to close the gap betweenmodular localization of wave functions and the causal localization of operatorsin QFT.

For this task one needs additional mathematical tools. The first step consistsin a suitable extension of the modular concepts. A standard subspace H comeswith distinguished operators. With D(Op) denoting the domain of definition ofan operator one defines

Definition 1 A Tomita operator S is by definition a closed antilinear denselydefined involutive operator with dense D(S) ⊂ H

It is easy to see that there exists a 1-1 correspondence between Tomitaoperators and standard subspaces H ; H ↔ S. This follows from the definitionS(ξ + iη) = ξ − iη, ξ, η ∈ H, whereas the opposite direction is a consequence ofthe definition H = ker S − 1 .

As a result of their involutiveness, the full content of Tomita operators iscontained in their dense domains. Hence modular theory may be alternativelyformulated in terms of standard subspaces (55) or dense complex subspacesDomS in terms of Tomita operators.

The polar decomposition of S = J∆1/2 of S into an anti-unitary J and apositive operator ∆ with D(S) = D(∆1/2) leads to the unitary modular group∆it acting in H and preserving the standard subspace ∆itH = H, whereas themodular conjugation maps into the symplectic complement J H = H ′

A Tomita operator appears in a natural way in Wigner’s representationtheory of positive energy representations of the Poincare group P . It is obtainedby defining ∆it

W0in terms of the Lorentz boost operator which leaves the wedge

W0 = x; z > 0, |t| < z invariant

∆itW0

= U(ΛW0(−2πt))

SW0= JW0

∆1/2W0, JW0

= TCP ·Rπ

together with an anti-unitary J obtained by multiplying the TCP reflectionTCP with a π-rotation R in the x-y plane as written in the second line, takingW0 into its causal complement.

The charge conjugation C maps an irreducible Wigner particle space intoits charge conjugate and may need a doubling of the Wigner space. The anti-unitary TP corresponds to the total spacetime inversion x → −x. The preser-vation of energy requires T to be anti-unitary. Massless representations alwaysneed a helicity doubling ±h and hence an antiunitary J whose geometric actioncorresponds to a map of W0 →W ′

0.

37

Unbounded operators S whose dense domain is stable (”transparent” inthe sense domain=range) are very unusual. In physics they appeared first inquantum statistical mechanics and more recent attempts to obtain an intrin-sic understanding of causal localization of QFT (one which does not refer toLagrangian quantization). In the present context the dense subspace of theWigner space (possibly doubled by charge conjugation) corresponds to wavefunctions which are ”modular localized” in the wedgeW0. As already suggestedby the previous observation (56), modular localization of Wigner wave functionsis closely related to causal localization of fields and turns out to provide a deeperinsight into ways to extend the Weinberg intertwiner formalism for the purposeof including Wigner’s infinite spin class [8].

The construction proceeds as follows: start from Wigner’s positive energyrepresentation theory, define the Tomita operator SW0

in the way described be-fore and use the Poincare transformations to construct a net of modular localizedsubspaces H(W ) and finally utilize the second quantization functor (the Weyl orCAR functor) to pass to an interaction-free net of of standard wave functionsspaces H(W ) to causally localized operator algebras A(W ) acting in a Wigner-Fock Hilbert space.

Standard subspaces HO and their complexified dense subspaces D(SO) =HO + iHO ⊂ H corresponding to more general causally complete convex space-time regions Oc can be obtained by intersecting modular localization spaces ofwedges

HOc=

⋂

W⊃Oc

HW (59)

HO =⋃

Oc⊂O

HW

whereas for more general regions O the standard space is defined in terms ofexhaustion from the inside (second line). For details we refer to [22].

The energy-positivity of the massive and the two massless Wigner repre-sentation classes plays an important role in establishing the isotony and causallocalization of the ”net of modular localized standard spaces”

isotony: HO1⊂ HO2

if O1 ⊂ O2 (60)

causality: HO1⊂ H ′

O2if O1×O2

where× denotes spacelike separation.

In the absence of interactions the passage from the spatial modular theory toits algebraic counterpart is almost trivial. One passes to the Wigner Fock space(the bosonic Fock space built from the 1-particle Wigner space) and definesthe O-localized operator algebra as the von Neumann algebra generated by the

38

Weyl operators

A(O) =

eiA(h);h ∈ HO

′′

, [A(h1), A(h2)] = i Im(h1, h2) (61)

with A(h) =

∫

∑

s3

(h(p, s3)a∗(p, s3) + h.c.)dµ(p), h ∈ HO

The second quantization functor maps the spatial Tomita operator into its al-gebraic counterpart which acts in the Wigner Fock space associated with theWigner space H

SalgO AΩ = A∗Ω, A ∈ A(O)In the sequel only the algebraic part referring to operator algebras of the Tomita-Takesaki theory will be used and the superscript alg will be omitted for conve-nience.

Modular localized spaces H(O) and their associated noninteracting field al-gebras A(O) are by construction Einstein causal and ”causally complete” i.e.A(O) = A(O′′

) where the causal completion O′′ is obtained by taking twice thecausal complement O → O′.

For wedge regions the modular group coincides with the unitary Wignerrepresentation of the wedge-preserving Lorentz group, but for all other regionsthe modular groups in massive Wigner representations acts in a non-geometric(”fuzzy”) way. It is believed that the nongeometric action becomes asymptoti-cally geometric near the boundary of the causal completion.

Massless finite helicity theories have a larger set of regions in which modulargroups act in a geometric way; this includes all regions which are images ofwedges under the action of conformal transformation as e.g. double cones. Thisis interesting because it means that the sl potentials, whose semiinfinite spacelikelines remain inside wedges, pass to potentials localized on finite elliptic curveswhich connect two points on different edges; they can be viewed as substitutesfor the nonexistent pl coordinatization of double cones.

It may also happen that the standard spaces for compact spacetime regionsO are trivial HO = 0 .This occurs precisely for the Wigner’s zero mass infinitespin representation for which the tightest localized nontrivial spaces correspondto modular localization in arbitrary narrow spacelike cones. On the other handzero mass finite helicity spaces are the most geometric representation since theirmodular groups continue to act geometrically even for double cones ; in fact theycorrespond to double cone preserving conformal transformations.

The massless infinite spin representations do not share these modular prop-erties of finite helicity fields. In view of the somewhat unexpected properties ofPauli-Lubanski limits in the previous section this is not surprising.

Modular localization plays also an important role in the understanding oftopological peculiarities of helicity h ≥ 1 free massless QFT as mentioned in con-nection with toroidal Wilson loops. Last not least without its use it would nothave been possible to discover the intrinsic noncompact localization of Wigner’s

39

infinite spin matter and the string-like nature of its generating causally localizedfields.

This raises the question if modular theory preserves its constructive powerin the presence of interactions. It turns out that in that case one is required touse the stronger operator algebraic modular theory. Interacting theories sharethe same modular groups ∆it

W with their noninteracting counterpart which issolely determined by the particle content of incoming or outgoing particles. Theinteraction enters through the dependence of the JW on the interaction which,using the fact that the incoming TCP is related with its outgoing counterpartthrough the S-matrix S [70] [54], amounts to

JW = J(in)W S (62)

Again the starting point is a von Neumann operator algebra A acting in aHilbert space H which contains a vector Ω which is cyclic and separating underthe action of A (in QFT the Reeh-Schlieder property for A(O))

cyclic: A(O)Ω is dense in H (63)

separating: AΩ = 0 ⇒ A = 0, A ∈ A(O)

The definition H = Asa(O)Ω (sa = selfadjoint) or H =A(O)Ω connects thealgebraic modular theory with its previously presented spatial counterpart. Butthere remains an important difference: the map between standard subspaces andalgebras is not injective.

The interaction-free situation remains exceptional in that there exists a func-torial relation between modular localized Wigner subspaces and interaction-freecausally localized subalgebras defined in terms of the Weyl map (61). Thisfunctorial map is lost in the presence of interactions.

In interacting models with a complete particle interpretation (for which theHilbert space can be identified with an asymptotic Wigner-Fock particle space)modular localization theory attributes a special dynamical role to the modularreflection J of the wedge-localized algebra A(W ) by relating it to the S-matrixin the form (62). Hence modular theory attributes to the S-matrix a dominantrole in the construction of a QFT. But on the other hand S is the physicallymost important object (Heisenberg: ”the crown of the theory”) and the causalquantum fields are indispensable for its construction, an at first sight hopelesslooking ”catch 22 situation”21.

For two-dimensional integrable models without bound states with an explic-itly known S-matrix this has led to a modular localization interpretation of theZamolodchikov-Faddeev algebra [54] and its ”vacuum-polarization-free gener-ators” (PFG) [55]. The problem to what extend interacting fields applied tothe vacuum can be prevented to create in a addition to a one-particle state a

21The existence of d = 1 + 1 models with canonical (free field like) short distance behaviorhad decades before been established in a measure theoretical path integral setting in [52]without referring to the S-matrix. For a recent more operator-algebraic construction see [53].

40

vacuum-polarization admixture was also studied in [56]. This in turn paved theway to the first existence proofs for certain d = 1 + 1 models with nontrivialshort distance behavior in terms of modular localization-motivated operator al-gebraic methods [58]. . For a recent account with many references to previouspublications see [57] and citations therein.

The important role of the S-matrix for modular localization in wedges hastriggered attempts to reconstruct a full causal QFT from its ”on-shell footprint”in form of its S-matrix [59, 60]. These ideas, based on the concept of emulationof states generated by application of interacting operators to the vacuum interms of those generated in terms of incoming free fields, are presently tooweak for constructions in higher than two dimensions. Among the establishedunexpected results is the nontriviality of the S-matrix in d = 1 + 2 for anyons(or their nonabelian ”plektonic” counterpart) which excludes the existence offree fields [61].

Modular localization suggests to view quantum fields as objects which ”coor-dinatize” nets of causally localized algebras. In the absence of interactions fieldsform local equivalence classes and with certain restrictions it does not matterwhich coordinatization one uses for generating the net of local algebras in termsof testfunction-smearing as long as these fields act cyclic on the vacuum state.But when one uses such fields in interaction densities in order to construct theirinteracting counterparts one faces the problem of preventing the change of freefield coordinatization from changing the physical content (the S-matrix). Thisis particularly important when trying to improve short distance properties byusing sl fields instead of their pl counterparts. This is the origin of the paircondition which plays a crucial role in the new perturbative setting and leadsto the important concept of higher order induced contributions of the previoussection.

Finally we come to an important point whose clarification was promisedat the beginning of this section: how sl quantum fields are connected withString Theory. String theorists attribute sl spacetime localization properties totheir favorite objects which are claimed to lead up to a theory of everything(TOE). The short answer is that ST has nothing to do with quantum causallocalization in spacetime; its claimed covariant string localization based on thequantization of classical Lagrangian as world sheets, Nambu-Goto surfaces etc.,bears no relation to the kind of causal localization which Einstein identified asthe spacetime structure which underlies the ”action at a neighborhood principle”of the Faraday-Maxwell theory and which in the form of spacelike commutationrelations became a defining property of QFT.

The misconceptions about causal localization which led string theorists toclaim that their objects are covariant stringy entities become clearer by lookingat their constructions which are based on the quantization of actions describingworld sheets, Nambu-Goto surfaces or Polyakov strings. Apparently there existsa believe that the quantization of classically covariant actions lead to covariantcausal quantum objects. At least this is what one must conclude if one looksat their use of the classical relativistic action

√−ds2of a classical particle which

41

serves as a trailer to the mentioned surface actions [63] and motivates theirFeynman graph inspired reading which replaces propagator lines by splittingand recombining surfaces as graphical visualizations of quantizations. In thisway it becomes clear that the terminology ”string” is motivated by classicalpictures and does not refer to intrinsic properties of relativistic QFTs.

On the other hand structure which contain such intrinsic properties asWigner’s representation theory cannot be attained by quantization of classi-cal actions; hence the start from classical actions is somewhat treacherous andself-deceiving.

In later publications string theorists avoided quantization by implement-ing their idea about strings directly in terms of 2-dimensional 26- or the 10-component conformal QFTs in d = 1 + 1. thanks to a theorem [64] whichestablished the existence of (highly reducible) Wigner particle representationsin 26 respectively 10 spacetime dimensions. This at least made contact with rela-tivistic quantum particles which via modular localization is related to spacetimelocalization. These numbers came from the underlying Virasoro algebra struc-ture of conformal QFT; in particular from a model of a free supersymmetric10-component conformal current whose algebra contains a the highly reducibleso-called superstring representation of the Poincare group in the form of aninfinite tower of Wigner particles.

This is primarily a group theoretical observation which probably would havepleased Majorana (apart from the 10 dimensions) who, in analogy to the O(4, 2)hydrogen spectrum was looking for ”natural” algebraic structures which con-tains infinite component relativistic fields as far back as 1932. The 10 dimen-sional representation is such a (supersymmetric) positivity-obeying Wigner par-ticle representation referred to as the ”superstring” representation. Exceptingits validity the next question is what led string theorist to believe that the fieldsassociated with these particles are stringy? Is it their reading as a mapping ofthe conformal light ray as a source space into a 10 component target space?

The rather subtle solution of the Velo-Zwanziger causality conundrum shouldserve as a warning against too naive geometrical pictures about quantum causallocalization.

The simplest way to answer this question would consist in showing that sucha source-target interpretation can be reconciled with causal space-, light-like orcurled strings. The unitary superstring Wigner representation can certainly bepictured as an infinite tower of irreducible representation. So does the wordstring perhaps refer to a picture in which these irreducible representations com-ponents are Fourier components of some loop in inner space over each spacetimepoint? The attribution of vibration could also be taken as a hint that the local-ization refers to the nonintrinsic quantum mechanical Born localization.

The word ”string” in ST remained an ”epitheton ornans”. In case the termi-nology would have a material quantum content, the fields of a free superstringcannot be anything else than string-local fields in the sense of the present paper;this is the only possible non-metaphoric meaning of the word ”string”.

42

Without the conceptional dulling by ST one probably would have been morecareful with respect to the physical interpretation of the AdSn+1-CFTn isomor-phism. It was certainly important to complement the old observation of equalityof the symmetry groups of the two spacetimes by the much stronger verificationof an Einstein causality preserving isomorphism. But in QFT Einstein causal-ity is only the spacelike manifestation of local causality; there is also a timelikecausality property which corresponds to the classical picture of propagation ofsay t = 0 Cauchy data. In QFT this is Haag duality A(O′′) = A(O) (section2 (13)) where O is a thin time slice subtended from a t = 0 spatial sphere andO its causal completion i. e. the double cone in spacetime subtended by thesphere.

In [68] it was shown that, unlike classical field theory, Einstein causality andHaag duality in case of such s spacetime region are independent assumptions.In other words it is possible to construct models of QFTs which are Einsteincausal but violate Haag duality. This happens in particular with Einstein causalfields which have ”too many” degrees of freedom. This manifests itself in a sortof ”poltergeist effect” in that there may be more quantum degrees of freedomstreaming into a causal shadow region (the spacetime region within which clas-sical radiation is completely determined in terms of Cauchy data associatedwith a compact region at a fixed time) than there were in the original appro-priately defined initial value (”Cauchy”) data; in operator-algebraic notationAcon(O) Acon(O′′

) where the causal completion O′′ is the causal complementtaken twice.

In fact it is quite easy to construct Einstein-causal models for which thiscausal completion property is violated. Generalized free fields with a suitablylarge κ behavior of their Kallen-Lehmann spectral functions (which contain amuch larger cardinality of degrees of freedom than free fields) were used at thebeginning of the 60s [68] to show that the causal shadow property does not followfrom Einstein causality, it rather represents a separate requirement (althoughin many later publications of the ”axiomx” this was not mentioned).

”Squeezing” the natural cardinality of degrees of freedom of an n+1-dimensionalQFT into one of lower say n dimensions one heuristically expects an ”overpopu-lation” of degrees of freedom and this is precisely what happens in the AdSn+1-CFTn case [67]: a free AdS field is mapped into a generalized free conformalfield which has that causality pathology of a too high cardinality of degrees offreedom. In the opposite direction i.e. starting from a CFT which satisfies bothrequirements one expects an ”anemic” degrees of freedom behavior on the AdSside. This is indeed the case; such an AdS theory has no degrees of freedom atall in compact regions (double cones) and one has to pass to infinitely extendedwedge-like regions to find any [66].

A situation in which the breaddown of Haag duality leads to a refined under-standing of the Aharonov-Bohm effect had been mentioned in section 2. Thisrefinement is lost in the the positivity-violating gauge theoretical setting.

The degree of freedom issue is not limited to the AdS-CFT isomorphism

43

but affects all attempts to extend the (quasi)classical Kaluza-Klein constructionof reducing extra dimensions to causal QFT. It poses the question whether thefashion of extra dimensions would continue with the same intensity if peoplewould have been more aware about the subtle nature of causal localization andpositivity problems of relativistic quantum matter (and have taken a more crit-ical view of results based on ”massaging” Lagrangians). This does of course notaffect the use of such arguments as formal tricks to construct a lower dimen-sional Lagrangian from a known higher dimensional one by use of Kaluza-Kleinmanipulation (disposing of one dimension by ”rolling it up” and passing to thelimit of zero radius (as in [9] section 2.3.3) .

Part of the problem is that QFT itself even after more than 80 years remainsan unfinished project. The fault line of the present schism separates the STcommunity and their belief in extra dimensions from those who are working onthe incredibly successful but still largely unfinished project of QFT in terms ofits intrinsic causality and positivity principles is precisely the issue of causalityand its refinements.

5 New aspects of QFT from string-local renor-

malization

5.1 Scalar QED, induced interactions and counterterms

An important new concept from sl renormalization theory is that of inducedinteractions. In section 3 it was shown that an interaction density which de-scribes a coupling of currents of higher spin sl fields to an external potentialleads generally to a higher order modification of the sl interaction density Lby induced contributions. In contrast to the counterterms in pl renormalizationtheory these terms do not come with new coupling parameters, hence the modelis uniquely fixed in terms of the original first order L, Vµ pair.

It is precisely the increasing number of undetermined counterterm in pairswith dsd(L) > 4 which reduce nonrenormalizable theories to the lower status of”effective” QFT, an euphemistic expression for the idea that they may revealat best low energy features of an unknown nonperturbative QFT.

In section 3 it was shown that the sl perturbation of external potentialmodels comes with only induced terms; in the present section it will be shownthat interactions between sl quantum fields lead to a perturbation theory inwhich the number of parameter enhancing counterterms occur together withinduced terms. This suggests a new conceptual view about nonrenormalizableperturbative QFT which will be illustrated in the model of scalar massive QED.

The point-local coupling LP = jµ(x)APµ (x) has a short distance dimension

dsd(LP ) = 3 + 2 = 5 which is one unit above the power counting bound and

hence the pl model treated as a conventiona pl model is nonrenormalizable withinfinitely many new counterterm parameters. The remedy is to convert this

44

pl coupling LP into a better behaved dsd(L) = 4 sl interaction density L =jµ(x)Aµ(x, e) and a divergence ∂µVµ as LP = L − ∂µVµ with Vµ = jµφ. Thisproves the e-independence deS = 0 of the first order S-matrix in the adiabaticlimit (see (45) in section 3).

Higher order calculations as well as the notation become somewhat simplerif one uses Qµ = deVµ = jµdeφ instead of Vµ; then the second order L,Qµ paircondition reads

deTLL′ − ∂µTQµL

′ − ∂′µTLQ′µ = 0 (64)

where the time-ordering T is not necessarily identical to the ”kinematic” timeordering T0 which is obtained by attaching a i(p2 − m2) denominator to themomentum space 2-ptfct. In x-space this amounts to addin||g a δ-functioncontribution, e.g.

〈T∂µϕ∗∂′νϕ′〉 = 〈T0∂µϕ∗∂′νϕ

′〉+ icgµνδ(x− x′) (65)

The initially free c is fixed by the second order requirement in (64). Thischange is necessary since as a result of the singular (distributional) nature ofquantum fields, the differentiation ∂ does generally not commute with the time-ordering. This provides the chance to compensate the delta contributions whichresult from the differentiation

∂µ 〈T∂µϕ∗∂′νϕ′〉 − 〈T∂µ∂µϕ∗∂′νϕ

′〉 = i(−1 + c)∂′νδ(x− x′) (66)

∂µ 〈Tϕ∗∂′νϕ′〉 − 〈T∂µϕ∗∂′νϕ

′〉 = −icgµν∂′νδ(x − x′)

against that which results from the change of T0 with T in the TLL′ terms in(64) where it amounts to an addition of a cδ(x− x′) |ϕ|2 AµA

µ term. For c = 1and the use of deAµ = ∂µdeφ the second order pair relation is fulfilled. In thiscase the obstruction does not induce higher order changes of L but remainshidden in the T0 → T change which expressed in terms of the old kinematicalT0 amounts to

TLL′ = T0LL′ +AµA

µ |ϕ|2 (67)

Hence the quantum origin of the second order quadratic in Aµ contributionfollows without referring to classical fibre of Lagrangian quantization directlyfrom the e-independence of the S-matrix

S(2) ≃∫ ∫

TLL′d4xd4x′, deS(2) = 0

The pair requirement in the L,Qµformulation remains valid in the zero masslimit since the divergence of the de Sitter one-form ∂µQµ remains finite whereasthe Qµ itself is logarithmically infrared divergent (and Vµ suffers from a m−1

mass divergence).

The 4th order sl setting shares with gauge theory the appearance of thec(ϕ∗ϕ)2 counterterm from the 4th order box diagram with which it is expected

45

to share the short distance behavior. The advantage of the sl setting is thatthe implementation of the higher order pair property permits a clear distinctionbetween sl induced contributions and new parameter generating countertermswhich is solely based on quantum positivity and causality.

The separation of higher order contributions to L into uniquely determinedinduced terms and new coupling parameter generating counterterms is a conse-quence of the new normalization property from the higher order implementationof the pair condition; it has no counterpart in a exclusive pl setting. A pl settingwhich via pair condition remains connected with sl has however markedly im-proved properties. In case of scalar QED the with perturbative order worseninghigh energy behavior of the charge-carrying field cannot be prevented but atleast the appearance of an ever increaing number of new coupling parametersis avoided and the aforemented c remains the only additional renormalizationparameter.

Another difference between induced terms and counterterms is that the off-shell composite fields associated with on-shell counterterms are e-independentlocal observables as the above (ϕ∗ϕ)2 in scalar QED. Varying c one thereforeobtains a continuum of models which share the same first order L,Qµ pair butlead to different scattering matrices S. Among this continuum of models sharingthe first order pair there is a ”minimal” one for which all counterterm couplingsvanish so that its is uniquely fixed in terms of this pair and its parameters. Thefact that it fulfills the pair requirement in every order provides an amount ofconsistency which a pure LP theory does not have.

This ”minimal” scalar QED model is the one which physicists implicitly usewhen they study electromagnetic properties of e.g. charged π mesons; c plays norole since experiments cannot clearly separate electromagnetic from hadronic π-π scattering contributions. However the question whether the distinction of sucha minimal model has a physical basis remains open. In the external potentialproblems for charged s ≥ 1 particles in section 3 this problem did not arise sincethanks to the sl pair setting all higher order changes of L are induced and hencethe problem of minimal versus non-minimal models did not arise. A direct plsetting without the sl detour would bring back the counterterm problem.

Those who use the Fronsdal-Vasiliev gauge setting for higher spin interac-tions apparently believe in the existence of a distinguished model [9]. In particu-lar the study of cubic dsd = 5 sl selfcouplings (which for h = 2 are candidates forquantum gravity, see section 6) would be purely academic without a selectionprinciple. Encouraged by the absence of counterterm parameters in externalpotential problems we take the minimal model with only induced contributionsas a reference for studying Jaffe [46] or Soloviev [47] type causal localizationproblems of nonrenormalizable models.

This opens the possibility to attribute a physical role to nonrenormalizablecouplings. The ”minimal solution” with all counterterm parameters set to zero isa perturbativly consistent SLFT for the prescribed particle content; its S-matrixis uniquely fixed by the first order L,Qµ pair condition. The explicit construction

46

of its higher order pl off-shell ”Jaffe fields” as highly singular interpolating fieldsof particles is an interesting topic for future work.

Massive spinor QED is a model which in the sl setting has neither inducedcontributions nor on-shell counterterms. as ”massive spinor QED” the differ-ence between the nonrenormalizable direct pl coupling (i.e. without converting itvia pair condition into sl ) and its renormalizable sl counterpart is considerable.In the standard perturbative setting (based on the off-shell Gell-Mann–Low for-mulation or path integrals), where one first calculates the off-shell correlationsof fields and then uses the LSZ scattering formulas to obtain the on-shell S-matrix, one has no access to the L, Vµ pair condition as a constructive tool ofperturbation theory.

The gauge theoretical pair construction is formally similar but pl quantumpositivity is only recoverd after passing to the positivity obeying pl setting(see remarks in section 6). For dsd(L

P ) = 5 this provides concrete models forJaffe’s pl scheme whereas for dsd(L

P ) > 5 the ultra-distribution aspect appearsalready in SLFT. In this case the off-shell fields are expected to belong to themore general wedge- or lightlike-tube localized dense testfunction spaces studiedby Soloviev [47]. Mathematically similar situations are expected in the gaugetheoretical pair setting.

After having exemplified the main difference between gauge theory andSLFT in a particular model of scalar QED, the focus of the following subsectionswill be directed to all dsl(L) ≤ 4 models which involve vector mesons. The firstsubsection presents the SLFT S-matix formulation in more detail whereas thecontent of the second one are the conceptual changes of the abelian Higgs modelin the SLFT setting. The third subsection focusses on the significant conceptualchanges of the role of the H-particle in the presence of Aµ-self-interactions.

5.2 The perturbative S-matrix in the SLFT setting

The appropriate formalism for the direct perturbative calculation of the on-shellS-matrix is based on the adiabatic limit of Bogoliubov’s operator-valued time-ordered S(g) functional. The relevant definitions had already been mentionedin (40) section 3.

The time-ordering is characterized in terms of properties among which thecausal factorization is the physically most important one. These are strongenough to inductively determine the nth order in terms of the model-defininginteraction density within the Epstein-Glaser setting [62] which does not referto Lagrangian quantization. The properties which characterize time orderingdefine the time ordering of the nth contribution inductively T up to countertermson the x1 = · · · = xn diagonal in terms of a minimality restriction imposed onits short distance scaling degree and renormalizable models are those which onlypossess a finite set of counterterm structures.

The E-G perturbation theory can be extended to include the constructionof fields. The defining formula had also been given in section 3 (48). The

47

conversion of pl interaction densities into better behaved sl ones based on thepair conditions works in the same way as in the case of external potentials insections 3. But as already mentioned in the previous section there is a fly in theointment: different from the appearance of only induced contributions in theexternal potential couplings in section 3 there are now in addition to inducedterms also counterterms. The simple case of scalar QED where both kindsoccur was presented in the previous section. This is typical for dsd(L) = 4.From dsd(L) ≥ 5 onward the situation worsens and there are infinitely manydifferent counterterms and hence an infinite parametric S-matrix which renderssuch a situation useless.

The pragmatic proposal in such a case was to keep only the uniquely fixedinduced contributions. This minimal theory is certainly a consistent solutionand even if one does not find physical arguments it would be useful to takeone solution and explore its properties in order to obtain some insight intononrenormalizable theories. Note that this cannot be made in a pure pl setting;one needs the higher order pair properties ( section 3) since without them thepure pl setting would lack an important consistency condition which renders ituseless already for dsl(L) = 4.

As the (ϕ∗ϕ)2 in the case of scalar QED in the previous subsection coun-terterms are expected to correspond to composite scalar local observables. Themain advantage of such a ”minimalistic” situation is that, as in the case ofthe simpler external potential problem, one can study problems concerning thecausality of such nonrenormalizable sl models. Their associated pl fields cannotbe Wightman fields, rather they are ultra-distribution fields which hopely admita dense set of test functions supported in lightlike (taking e2 = 0) supportedtubes. Their formally corresponding pl fields are even more singular but theymay still be local in the sense of Jaffe since there is no worst space of ultra-fieldwhich contains all Jaffe fields. This would be a project in its own right.

For the rest of the paper we will stay with models which are sl renormalizabledsd(L) ≤ 4 which includes all couplings whose particle content includes s = 1coupled among themselves or to s < 1. The reader is again reminded that forzero mass h = 1 the S-matrix either looses its meaning as the result of infrareddivergencies (QED, Yang-Mills) or the interaction becomes trivial (the abelianHiggs model). In the infrared case the L, Vµ must be replace by the weakerL,Qµ conditions deL = ∂µQµ which in many cases turns out to be sufficient.For the rest of this subsection we will explain the conversion of second orderobstructions into induced higher order changes of L so that in the subsequentsubsections one only has to adapt the expressions to the different models.

Let us recall the form of the L,Qµ pair condition

deLP = 0 = deL− ∂µQµ, Qµ := deV

µ (68)

and remember that this form of pair conditions holds also in gauge theory.But as shown in [40] in that case it is not sufficient to work in the Gupta-Bleuler Krein space but one also has to extend this space by the BRST ghostoperators and the spacetime differential form de operation is replaced by the

48

BRST s. SLFT uses only physical degrees of freedom. The BRST formalismdoes not lead to interpolating fields of particles whose large-time scattering limitconnects causal fields with particles.

This form of gauge theory as compared with that of Lagrangian quantizationthe advantage of avoiding certain problems, but it does not provide interpolatingfields for particles so that the perturbatively unitary S-matrix remains discon-nected from the scattering theory of interpolating fields of particles. The fullgauge formalism needs an additional extension by unphysical degrees of freedom:the BRST ghosts and anti-ghosts22.

The sl setting on the other hand is free of these problems and its physicalinterpolating sl fields fulfill all the causality requirements which a necessary forestablishing the cluster properties which in turn lead to the e-independent pl inand out fields of scattering theory; at this point the theory contains all physicalfields and provides in particular the physically correct off-shell long distancebehavior.

Note that the Epstein-Glaser setting in whose spirit the SLFT perturba-tion theory is formulated SLFT does (unlike the Gell-Mann–Low perturbationtheory or euclidean path integrals) not refer to Lagrangian quantization. Butthe development of the full higher order E-G formalism for sl fields is still aconstruction site [84]. One expects that a complete sl time-order formalism forthe S-matrix and its retarded counterpart for fields [69] will be available sometime in the future.

Already in second order the implementation of the pair condition may leadto δ obstructions of the kind already encountered in external potential problems(section 3) which in the simpler L,Qµ pair setting (68) reads (in case of lightlikee)

O(2) := deTLL′ = T∂µQµL

′ − ∂µTQµL′ + TL∂′µQ′

µ − ∂′µTLQ′µ (69)

O(2) = δ(x− x′)deL2(x, e)

Encoding them into interaction density one obtains

Ltot := L+ gL2, S(g) = T exp

∫

ig(x)Ltot(x, e)d4x (70)

This change of bookkeeping which converts higher order obstruction intoinduced contributions Ln amounts a change of L → Ltot in the BogoliubovS(g) is important. It affects the higher orders; the third order obstruction isnow

O(3)(g, g, g) = de

[

TL(g)L2(g2) +

i

3TL(g)L(g)L(g)

]

(71)

In models of interacting s = 1 vector mesons as the Higgs model or scalarmassive QED the third order obstruction vanishes in the adiabatic limit and

22A complete description of gauge theory in this setting can be found in [40].

49

the induced contributions account for the Mexican hat potential. As a conse-quence the terms in this potential are induced and not postulated . This willbe explicitely verified in the following subsections.

The L,Qµ pair condition and its higher order extension within the sl Bogoliubov-Epstein-Glaser setting is also meaningful for dsd(L) > 4. The before mentioned”minimal” models contain only induced contributions but their number in-creases with the perturbative order. By definition of minimal there are no higherorder counterterm parameters so the model depends only on those parameterswhich are already present in the interaction density L.

The conceptual and mathematical superior aspects of SLFT poses the ques-tion whether it is possible to pass directly from SLFT to pl ultra fields, thusavoiding the pl counterterm formalism. This problem will come up again inconnection with with cubic h = 2 selfinteractions in the next section (80).

5.3 External source models

To get a first glimpse of how this works we consider a vector potential coupledto an conserved classical current jµ. The S-matrix and the interacting vectorpotential are

LP = APµ j

µ = Aµjµ − ∂µ(φjµ), L = Aµj

µ, Vµ = φjµ (72)

Se0(g) = T exp i

∫

g(x)L(x, e0)g(x)→g→ S = exp ig

∫ ∫

jµi∆F j′µ : exp ig

∫

APµ j

µ :

Aretµ (x, e, e0) = Se0

−1(g)−iδ

δfµ(x, e)Se0(g, j → j + f)f=0 = Aµ(x, e) +

∫

Gretµµ′jµ

′

Gretµµ′ (x, e;x′, e0) = (−ηµµ′ − ∂µeµ′Ie + e0µ∂µ′I−e0 + (ee0)∂µ∂µ′IeI−e0)G

ret(x− x′)

A direct use of the LP according to the rules of perturbative renormalizationwould have led to a delta function counterterm contribution in the time-orderedProca propagator i∆F → i∆F + c

m2 δ. The current conservation leads to ae0-independent retarded potential Aret

µ (x, e) with a well-defined massless limitand a string-independent observable retarded field strength F ret

µν (x). The plfield coordinatization on the other hand would add a renormalization parameterwhich worsens the short distance behavior and diminishes the predictive power.A s = 2 potential hµν coupled to a conserved Tµν source leads to a similarresult. In both cases the pl setting is different from its sl counterpart by theappearance of an undetermined counterterm parameter.

Passing from external source to external potential problems the differencebetween pl and sl is much stronger. This problem has already been addressedin section 3 in connection with the Velo-Zwanziger causality problem.

50

5.4 Hermitian H coupled to a massive vector potential

The coupling of a vector potential to a hermitian scalar matter field H comeswith a new phenomenon. In addition to a change of the time-ordered productof the H-field there is now a genuine induction of terms

The ”germ” of an interaction density for an Aµ, H field content is the mA ·AH coupling, where the vector meson mass factor m accounts for the classicaldimension deng = 4 and also indicates that the model has no nontrivial QEDlimit (the reason why it was discovered a long time after QED). Its sl operatordimension is dsd = 3, hence the germ is a superrenormalizable sl interactiondensity. The first order L,Qµ pair property (Qµ = deVµ) requires the presenceof the escort φ also in L and leads to

L = m

A · (AH + φ←→∂ H)− m2

H

2φ2H

+ U(H), U(H) = mc1H3 + c2H

4

Vµ = AµφH +1

2φ2←→∂µH, de(L− ∂µVµ) = 0, LP = AP ·APH = L− ∂µVµ.

(73)

A systematic determination of the first order pair (73) starting from the simplestcoupling (the ”germ”) gmA·AH of the A-H particle content and a general ansatzfor L and Vµ containing all dsd ≤ 4 terms which can be formed from H,Aµ andits escort φ shows that the numerical coefficients in (73) are determined in termsof coupling g and the two massesm andmH [73] whose couplings are determinedin second order. A verification that the L, Vµ pair satisfies the condition requiresonly the use of the Klein-Gordon equations for H and is left to the reader.

The first order pair condition does not determine the strength of the H-selfinteractions since e-independent contributions to L simply pass through thepair condition. The necessity of their presence which includes the determinationof the ci is seen in second and third order. This ”induction” of additionalcontributions with well-defined numerical coefficients is a new phenomenon ofSLFT; there is a formal similarity with the imposition of the second order BRSTgauge invariance on the S-matrix [40] but the essential difference is that thee-independenc of S is a consequence of the positivity and causal localizationprinciple of QFT.

For the S-matrix one only needs the second order tree component to theobstruction O(2). In addition to a second order change of the time ordering ofthe propagator involving H derivatives which parallels that in (66) one nowencounters a genuine second order induction (70)

L2 = g[(m2H + 3c1m

2)H2φ2 − m2H

4φ4 + c2H

4] (74)

Finally the vanishing of the third order tree contribution fixes the values of c1, c2in terms of the three physical parameters of its field content which were alreadypresent in the germ namely g,m,mH . To allow for a comparison with the Higgs

51

mechanism we write the result in the form

L(2)tot = mA·(AH+φ

←→∂ H)−V (H,φ), V = g

m2H

8m2(H2+m2φ2+

2m

gH)2−m

2H

2gH2

(75)

where L(2)tot = L + g

2L2. The appearance of a quadratic mass term is the resultof writing the interaction density as if it would be part of a classical Lagrangianof gauge potentials. The reader may fill in the details of the straightforwardcalculations by himself or look at [73].

Apart from a mass contribution the V looks like a field-shifted Mexican hatpotential. But different from the Higgs mechanism it has not been obtained bypostulating a Mexican hat potential and subjecting it to a shift in field space. Itis rather induced by the germ of a renormalizable A,H field content and it is theunique renormalizable QFT with this field content. There is simply no room forimposing a Mexican hat potentials; the induction of the H and φ selfinteractionsis a consequence of e-independence of the S-matrix which in turn is guarantiedby fields localized in causally separable space- or light-like (but not timelike)strings.

The SSB picture of the Higgs model also reveals another common misun-derstanding, this time about SSB. The Mexican hat potential together with theshift in field space is not the definition of SSB but rather a way to implementsuch a situation whenever it is possible. The definition of SSB is rather theexistence of a locally conserved current whose global charge diverges. This isonly possible in the presence of massless Goldstone bosons and all verbal at-tempts to make SSB consistent with a mass gap (a photon becoming fattenedto a vector meson by eating a Goldstone) only obscure the interesting correctunderstanding.

QFT is not a theory which can create the masses of its model-defining fieldcontent. The prescription of a field shift on a Mexican hat potential as the”Higgs mechanism” has to be seen in a historical context; it helped to overcomethe formal problems which one faces when one tries to extend Lagrangian quan-tization from Maxwell’s theory of charge-carrying fields to a situation in whicha vector potential couples to a hermitian matter fields. There are numeroushistorical illustrations of situations for which important discoveries were madethrough formal manipulations which were later replaced by a derivation whichis consistent with the principles of QFT. Incorrect placeholder are useful butonly up to the discovery of the real reasons.

A model of QFT is uniquely fixed in terms of its field content. The SLFTsetting (which seems to be the only one consistent with all principles of QFT) fora Aµ, H field content starts with a AµA

µH as the simplest coupling and the restis done by induction using the L.Qµ pair property which converts the heuristicphysical content of the ill-defined pl interaction density into the physically su-perior SLFT setting where the ”induction” resulting from the implementationof the pair property to all orders unfolds the full content of SLFT.

52

5.5 Selfinteracting vector mesons

It is straightforward to check that there is no renormalizable L,Qµ pair fora self-coupled singlet (A · A)2. The principles of QFT as embodied into thepair condition admit however selfinteractions between multiplets (”colored”)of vector potentials while imposing strong restrictions on the ”multi-colored”coupling parameters. In this case the germ is a FAA selfinteraction and thegeneral ansatz for the construction of a L, Vµ pair which includes the ”colored”escorts is of the form

L =∑

(fabcFµνc Aa,µAb,ν + habcdAa,µAb,νA

µcA

νd) + terms inAa,µ andφ

′as (76)

where the couplings and the masses of the vector mesons are initially freelyvariable parameters but, as expected, the first and second order pair conditionplaces strong restrictions on them [73], among other things the f and h areinterrelated in the same way (Jacobi identities of reductive Lie-algebras) as ingauge theory [40]; in particular the A-φ and φ-φ couplings depend also on themasses of the vector mesons. The main distinction to gauge theory is that theseproperties are direct consequences of the principles of QFT and do not arisein the course of the gauge theoretic extraction of physics from a unphysical(positivity-violating) description through the imposition of gauge invariance.

The most interesting aspect of the SLFT formulation is that there remainsa renormalizability destroying second order induced selfinteraction which, if leftuncompensated, destroys renormalizability even though the interaction densityfulfills the power counting restriction of renormalizability. The way to overcomethis is to compensate this dsd = 5 term with a second order contribution from aA-H interaction with a scalar Higgs field23. This is a totally different situationfrom the abelian A-H interaction for which such all second order terms staywithin the power counting bound. Neither case bares any physical resemblenceto spontaneous symmetry breaking since in both cases the field shifted Mexicanhat potential is second order induced.

The idea of short distance compensations between contributions from differ-ent spins arose in connection with supersymmetry. Although not invented forthis purpose, SUSY does improve the short distance behavior somewhat but notenough to guaranty the renormalizability and preservation of supersymmetry inhigher orders. The situation of selfinteracting vector mesons is different, in thatthe preservation of renormalizability is the raison d’etre for the H. Nature doesnot have to decide between a symmetry and its SSB, rather the existence ofthe H is directly connected to the preservation of its positivity and causalityprinciples or in other words a massive Aµ

α field content by itself is not consistent.

Gauge symmetry is not a physical symmetry so there is nothing to break;all these physically incorrect pictures evaporate if one maintains causality andpositivity which is perturbation theory is only possible by starting with ansl L, Vµ or Qµ pair property. The fibre bundle like Lie structure of the fabc

23A s ≥ 1 field would worsen the second order short distance behavior.

53

couplings is not the result of an imposed symmetry it rather arises from thestring-independence of the S-matrix which in turn is a result of LSZ scatter-ing theory of interacting causally separable positivity obeying quantum fields;hence the situation is very different from the superselection structure of unitaryrepresentation classes of observable algebras which leads to the notion of innersymmeries. This shows that quantum causality is much more fundamental thanits classical Faraday-Maxwell-Einstein counterpart.

Having thus strengthened the conceptual understanding of interactions be-tween vector mesons in the Standard Model one may ask whether SLFT containsalso messages about their coupling to matter. In recent work [74] it was shownthat SLFT does not only restrict the selfcouplings between vector mesons andrequires the presence of a Higgs particle in the presence of selfinteracting mas-sive mesons but it also restricts their coupling to the Fermion currents and theirchirality properties. This is of particular interests for massive W±.Z vectormesons and the photon, a case for which the authors explain the restrictionsfrom SLFT in detail.

6 The role of the pair condition in the construc-

tion of interacting fields

The extension of SLFT to interactions involving higher spins s ≥ 2 is an im-portant issue about which one presently knows little. There have been quiteextensive investigations in the gauge theoretic equivalent of the pair conditionby Scharf [40]. In view of formal similarities with SLFT it is interesting to takea closer look at some of his results.

Scharf looked at the simplest s = 2 selfinteraction which is of a cubic formtrh3 where hµν is the s = 2 massless tensor field. The physical interest inthis model is connected with the use of hµν as a linear approximation of thegravitational gµν field. As in SLFT, the short distance dimension of integerspin gauge fields is equal to their classical dimension in terms of mass unitsnamely dsd = 1.In [40] it was shown that there exists no gauge theoretic trilinearselfinteraction LK with dsd = 3 without involving derivatives of hµν , its tracehµµ as well as ghost fields and their anti-ghost. He found a cubic interactiondensity of dsd = 5 which is above the powercounting bound of renormalizationbut still amounts to a huge reduction from the dsd(L

P ) = 11 to that of its gaugetheoretic counterpart in Krein space.

Taking into account that gravitational coupling carries a dimension and ex-panding the Einstein-Hilbert Lagrangian in a suitable way using κ =

√32πG

as an expansion parameter, he arrived at a formal connection of the classicalexpansion with the quantum-induced correction up to second order which waslater extended by Dutsch to all tree orders [40, 76]. The agreement of treeapproximations with classical perturbation theory is not unexpected, but thereare two competing ideas one being of classical geometric origin (the Einstein-

54

Hilbert action) and the other is related to the gauge theory of selfinteractingh = 2 particles .

Rehren showed recently (private communication) that SLFT provides a sim-pler version of such a cubic selfinteractions which

L = κ(2∂ρhκλ∂σhκλ + 4∂βh

αρ∂αh

βσ)h

ρσ, hµν = A(2)µν (77)

deA(2)µν = ∂µaν + ∂νaµ, (78)

where A(2)µν is the sl h = 2 potential from (25) section 2 which already played a

role in solving the D-V-Z discontinuity problem. Using the relation between deand ∂µ in the second line one easily verifies that deL is of the form ∂µQµ i.e. theabove L belongs to a L,Qµ pair. Since massless h ≥ 1 fields are intrinsically slthe corresponding minimal models are expected to be ultra-distributions whichare localizable in lightlike tubes. One expects that similar cubic selfinteractionexists for all h > 2.

For h = 1 there exists of course the sl dsd = 4 color-carrying Yang-Millsinteraction and a colorless dsd = 5 selfinteraction is not expected it describes acolorless (abelian) dsd = 5 selfinteraction. In the literature one finds a calcu-lation which excludes the existence of selfinteracting ”colored” gravitons” [77].A gauge theoretic proof in favor of such a No-Go result which has a close for-mal connection to the above construction has also been given within Scharf’sformulation of h = 2 gauge theory [78].

The fact that there are no renormalizable s = 2 selfcouplings does not ex-clude the possibility to find sl dsd(L) ≤ 4 interactions of sl hµν fields with lowerspin fields. An ansatz for L which generalizes the Aµ, H particle content of theabelian Higgs model would be of the form

L = m(c1hµνAµAν + c2h

2H + c4hµµH

2) + contr. from escorts +U(H,A) (79)

where U contains hµν-independent interactions and m is the mass of the hµνparticle. There are altogether three fields and three escorts, two for hµν andone for Aµ. With a similar ansatz for the vector Qµ the fulfillment of thedeL − ∂µQµ = 0 pair relation and the higher order induction is much moreelaborate than that for the s = 1 Higgs model and necessary calculations havenot been done. One expects that L,Qµ pairs only exist if one allows dsd(L) ≥ 5.

The problems of fulfilling the pair condition grow with increasing spin. Par-ticles for which there exists no L,Qµ pair which couples their spin s fields amongthemselves or with lower spin fields are referred to as inert [31]. Up to knowthe only known inert particles are those belonging to the Wigner infinite spinrepresentation; their string-localization is intrinsic and since the de acting onsuch string fields cannot be converted into a ∂µ. These fields have also inertkinematical properties in that infinite spin matter is cannot reach thermal equi-librium [30]. The only possible interactions are those with external fields whichpermits gravitational backreactions.

An important issue is the construction of massless fields. Whereas the S-matrix for massless h ≥ 1 particles is infrared divergent, the off-shell massless

55

fields exist but have no pl counterpart (not even in the sense of Jaffe). Thedifference between on and off shell is that in the definition of fields (??) theS−1 cancels the infrared divergencies of the numerator in the adiabatic limit.The off-shell sl fields become important for the understanding of long distancebehavior which includes a spacetime description of QED scattering theory andthe QCD quark-gluon confinement. In case of massless vector potentials theseinclude problems of infrared divergencies and the quark-gluon confinement.

Free pl fields can not only be converted into their sl counterparts but theeµ∂µ differentiation on sl free fields permits also the return to their pl form.This is generally not possible in the presence of interactions. This is wherethe distinction between observables and interpolating fields becomes important.Local observables remain pl, whereas interpolating fields of particles (except theFµν for photons and the Lorentzian Riemann tensor for gravitons) are sl. Butwhat is an observable and when does an operator represent an interpolatingfield depends on the model.

In order to understand this point in more detail it is helpful to start fromthe defintion of fields (??). The equality of the S-matrix on for the LP and slL interaction density suggests to omit the identical S−1 factors and write

S(g(x)LP + λfϕ)a.l.≃ S(g(x)L + λfϕg)|λ=0,a.l. (80)

in words: calculating an interacting field ϕ|L associated to a free field ϕ inthe LP setting is in the adiabatic limit the same as starting with a coupling-dependent composite fields in the local equivalence class of the free field ϕ inthe L-setting. In order to extract a concrete formula in terms of an ansatz

ϕg(x, e) = ϕ0(x) +∑

gkϕk(x, e) (81)

It is necessary that LP and L are pair related so that the iterative applicationof the pair relation leads to

ϕ1(x, e) = i

∫

T (LP − L)ϕ(x) = i

∫

∂′µTVµ(x′)ϕ(x)d4x′ (82)

ϕ2(x, e) = −i∫

TLϕ1(x, e) +1

2

∫ ∫

T (LL− LPLP )ϕ(x), etc. . .

with the stipulation that all derivatives are outside the time-ordering as writtenin the first line.

The result is a series of Wick-ordered composite sl free fields (for brevitythe Wick-ordering sign is omitted). This field ϕg(x, e), which is a member ofthe sl causal equivalence class of the free field ϕ(x), determines the associatedinteracting fields in the LP (or inversely in the L setting)

ϕ(x)|LP = ϕg(x, e)|Le(83)

ϕ(x)|Le= ϕ−g(x, e)|LP (84)

56

where the subscript refers to the interacting counterpart of the free field ac-cording to the Bogoliubov formalism (??). Hence the interacting fields in twosettings which share the same S-matrix are mutually related in terms of a seriesin g whose accompanying operators are interacting composites. Such a relationhad been already mentioned in section 3.

It is important to emphasize that the LP calculations are not done in theold pl counterterm formalism which would lead to a worthless infinite parametertheory and destroy the idea of a uniquely defined S-matrix; rather they are basedon the connection pair connection between LP and Le as

LP = Le − ∂µVµ (85)

TLPL′P = TLeL′e − ∂µTVµL′

e − ∂′νTLeV′ν + ∂µ∂′νTVµV

′ν , etc.

which in case of dsd(L) ≤ 4 leads automatically to the ”minimal” pl QFT inwhich number of parameter-increasing counterterm contributions do not occur.Whereas the sl theory remains a Wightman field theory, its pl description re-quires the ultra-distribution setting of Jaffe or Soloviev.

This idea had goes back to Mund who recently applied it to massive spinorQED [79] for the verification of the expected relation

ψg(x.e)|LP = eigφ(x,e)ψ(x)|L (86)

ψ(x, e)|L = (e−igφ(x,e)ψ(x))LP (87)

in low orders. An identical relation is expected in massive scalar QED.

For observables and their line integrals as Fµν , Aµ(x, e), APµ (x), φ(x, e) in

scalar or spinor QED one expects that they and their line integrals simply passthrough

Fµν |LP = Fµν |Le, Aµ(x, e)|LP = Aµ(x, e)|Le′

etc. (88)

In other words the sl formalism preserves the line integral connection of sl poten-tials with their observable field strengths, and in this way maintains the linearconnection between the pl Proca potential and its sl counterpart including theescort. This is easy to see in lowest order that the divergence terms does notcontribute so that ϕ1 = 0; the vanishing of the higher terms is less obvious.

The interacting complex or hermitian matter fields are however irreduciblystring-dependent (and not just integrals of pl fields along e). This true string-dependence results from δ terms from the action of ∂µ on propagators. Theindependence of observables from the choice of pl or sl coordinatization extendsthat of the S-matrix. In fact one may turn this around: those interacting fieldswhich remain invariant under LP → L are observables and those which changeare interpolating fields. genuine in the sl setting fields are interpolating fieldsof particles. In particular those operators which, in contast to induced terms,appear as higher order counterterms are observables in the sense of the previousdefinition.

This idea opens the possibility to define a pl spinor field within the SLFTsetting as long as the vector meson remains massive. Such a construction can-not undo the (with perturbative order) unbounded increase of the short distance

57

dsd, but it prevents at least the unlimited increase of the number of couplingparameters from counterterms which would leave a direct pl perturbative ex-pansion without predictive power. Similar observations were made in the olddays when conserved spinor currents were coupled to massive vector mesons ina gauge theoretic inspired setting (still without BRST ghosts) and the spinorfield in the ”unitary gauge” turned out to have dsd = ∞ [80]. As mentionedin section 3 the mathematical aspect of this question received a partly positiveanswer in the work on ultra-distributions by Jaffe [46].

A more mundane application of the sl-pl connecting relation is to find outwhether the Higgs field H is a local observable or an sl interpolating field. Toidentify it as an interpolating matter field it would suffice to show that H1 in(82) for ϕ0 = H is nontrivial. The second line in (82) specialized to the abelianHiggs model for which Vµ has the form (73) results in the first order contribution

H1(x, e) =1

2: φ2(x, e) : (89)

Different from charge-carryingmatter fields the HermitianH(x, e) does not seemto have pl composites since in contrast to massive scalar QED where ϕ∗ϕ is anobservable there seems to be no Wick polynomial of H for which the first orderin (82) vanishes.

This suggests that in some not completely understood sense the sl nature ofthe interacting Higgs field is very different (much stronger) than that of charge-carrying interpolating fields (86). Whereas the latter have quadratic observablesin the form of conserved pl currents it may very well be that the algebra ofrenormalizable Higgs fields have contains no pl composites. These properties(and not the metaphorically ascribed ability to create masses of vector mesonsby spontaneous symmetry breaking) distinguish H from other scalar fields asthose of scalar QED..

Interestingly the same fate affects the Proca field; whereas in massive (scalaror spinor) QED interacting Proca fields as well as their field strength and slcompanions are local observables24, their counterparts in in the Higgs model aregenuinly e-dependent (i.e. they are not just line integrals over pl observables).This stands in an interesting contrast to charge-carrying matter coupled tovector potentials.

Gauge invariant currents in gauge theories are formally constructed from(positivity-violating) products of gauge-bridged unphysical matter fields withthe help of the point split method25. For example the observable spinor currentjµ = ψγµψ, which is perturbatively defined as a composite field with ϕ =ψγµψ being inserted into Bogoliubov’s functional machine (77), can also beobtained from the interacting spinor field in terms of a point split limit: onestarts from ψ(x)γµψ(x

′) with a gauge bridge between the for space- or light-like

24Integrating local observables along lines commutes with turning on interactions so thate · ∂ inverts the line-integration in the presence of interactions.

25I am indebted to Detlev Buchholz for a disscussion on this point.

58

separated points and takes the x′ → x limit after having subtracted certain inthe x = x′ limit diverging bilocal expressions.

The sl formalism suggest to proceed in a similar way, but the point splitprocedure is now more subtle. If one takes the same e-directions in both fieldsand chooses e to be parallel to the spacelike separation between x and x′ oneexpects that the infinite part of the ψ string compensates that of the conjugateψ so that only a compact localized ”string-bridged” pair remains

jµ(x) = p.s. lim ψ(x, e)γµ..ψ(x′, e), e =

x′ − x√

−(x′ − x)2(90)

where ψ(x, e)γµ..ψ(x′, e) := ψ(x)γµ..ψ(x

′)|Le

With light-like e′s behaving in the most regular way one should perhaps for-mulate the point split limit in lightlike direction. Only additional work on thisproblem can resolve the appropriate formulation of the point split method inSLFT.

In contrast to previous sections, in which most of the results have been(or can be) backed up by methods of mathematical physics or are supportedby explicit model calculations, the content of the present section is of a morespeculative nature; many of the necessary calculations still need to be done. Thisis not surprising in an area of particle physics which is very recent and whichrequires more familiarization with a new physical philosophy and computationalmethods.

7 Resume, loose ends and an outlook

SLFT is characterized as that formulation of QFT in which fields maintain thetightest possible localization compatible with quantum positivity. Unlike freefields for which point- or semiinfinite string-like localization remains (except forhelicity tensor/spinor potentials and their currents and stress energy tensorswhich only exist as sl potentials) a matter of choice, their localizability in thepresence of interactions depends on the kind of interaction; e.g. the field strengthin jµA

µ interactions is a pl observable but looses this attribute in models ofselfcoupled A′

µs.

The relations between classical and quantum field theory as embodied in(canonical or functional) Lagrangian quantization which are still maintained inGauge Theory are lost in SLFT; both theories still share the notion of observ-ables but GT contains no physical objects which play the role of interpolatingfields which connect the causal localization principles with scattering amplitudesof Wigner particles; as a consequence its gauge dependent charge-carrying mat-ter fields do not only violate quantum positivity but, as a result of the tightconnection between positivity and causal localization, they also fail on the issueof localization apart from that of local observables.

The construction kit of SLFT contains no Lagrangians and hence no La-

59

grangian quantization. The start is the specification of a model-defining Wignerparticle content which with the help of intertwiner functions is first convertedinto covariant pl fields are used to form a pl interaction density. This LP (x) car-ries the heuristic content of the interaction but in case it contains a s ≥ 1 fieldsits short distance dimension is beyond the power-counting bound dsd(L

P ) >4 and hence not suited to be used in a renormalized perturbation theory. Byconverting it into a better behaved sl density L(x, e) with dsd(L) ≤ 4 in such away that the difference between LP and L has the form of a divergence ∂µVµ(the ”L, Vµ pair condition”), one insures that this does not affect the adiabaticS-matrix limit. The extension of this condition to higher orders comes witha new powerful structure referred to as induction; different from countertermsinduced contributions only depend on the parameters in L (coupling strengths,masses of fields).

This new setting has been worked in low perturbative orders for all interac-tions involving s = 1 fields coupled to s < 1 matter fields. The main difference ascompared to gauge theory is an improved understanding of the relation betweenfields and particles.

The resulting renormalization theory comes with a new formalism whichis more physical but also more elaborate than its perturbative indefinite met-ric gauge theoretic counterpart; many properties which in the gauge settingwere the result of the unphysical ”gauge symmetry” as the Lie-structure ofAµ selfcouplings or the second order A · A |ϕ|2 contribution from the classical∂ → D = ∂ − igA fibre bundle structure are now directly connected with thephysical positivity and causal localization principles of QFT. The SLFT settingalso shows that the raison d’etre for the Higgs particles in massive selfinter-actions bears no relation to the alleged mass creation by symmetry breakingbut is rather related to a second order renormalizability saving short distancecompensation.

Little is known about interactions involving s = 2 fields. As the result ofpossible connections with quantum gravity massless selfinteractions are of par-ticular interest. Here SLFT confirms the existence of cubic selfinteractions withdsd = 5. Different from gauge theoretic results, the SLFT formalism containsno unphysical degrees (indefinite metric, ghosts). There is however the problemof renormalizability. In the gauge theoretical setting particle physicists referto ”effective field theory” which is a speculative idea claiming that under cer-tain condition one can extract physically relevant results if the energy range ofinterest is small compared to a suitably defined high energy cutoff.

The causality and positivity principles do not support such a concepts in-volving cutoffs. Instead they suggests that among infinitely many higher orderS-matrices associated to the first order L,Qµ pair the minimal one, which isuniquely determined by this pair and contains only higher order induced termsbut no counterterms plays a special role. From a mathematical point of viewthe minimal model is useful as a reference model for the study of localizationproblems of ultra-distributions.

60

An important question is whether in analogy to the appearance of the classi-cal Lie algebra structure from the sl quantum principles in s = 1 selfinteractionsand their coupling to matter one can also expect a geometric connection forh = 2. More concretely: can the classical geometric Einstein-Hilbert structureof general relativity be reconciled with the properties of sl cubic selfinteractingh = 2 quantum fields and their coupling to matter s ≤ 1 matter ? This is ofcourse the famous question of whether the geometric Einstein-Hilbert theorycan be reconciled with the principles of QFT. Some researchers present argu-ments that this may not be possible in a rather forceful way, in particular theyclaim that the problem of dark matter is one of Einstein’s theory not shared byh = 2 QFT [75], but this could still turn out to be premature.

Besides low order perturbative results SLFT has shown to be capable to solveimportant problems which are already present in the absence of interactionsas the W-W No-Go theorem, the closely linked D-V-Z problem of degrees offreedom counting of s ≥ 2 fields in them→ 0 helicity limit and last not least theV-Z causality conundrum. Surprisingly the use of sl instead of gauge potentialsleads also to a refinement of the causality aspects of the age old Aharonov-Bohmeffect and several other problem which uphold Einstein causality but violateHaag duality in a very interesting way. It also requires to do more homeworkon String Theory before attributing a physical meaning to the terminology.

Interestingly the applications of SLFT also covers problems of chirality ofspinor currents coupled to selfinteracting nonabelian massive and massless vec-tormesons which show that the Standard Model is closer related to the founda-tional principles than hitherto expected [74]. The authors in [84] have initiatedwork which aims at a systematic extension of the intrinsic Epstein-Glaser for-mulation of perturbation theory to sl fields. This is quite challenging from theviewpoint of causal quantum physics since certain concepts as ”chronologicalordering” require refinements in the form of ”string-chopping”.

One expects that even though gauge fields are quite different from their slcounterparts, their asymptotic short distance properties coalesce so that theasymptotic freedom property also holds in SLFT. In particular the Callan-Symanzik equations with its beta function and the short distance describinggamma functions should be e-independent and agree with their gauge theoreticcounterparts. But this should be verified (”control is better than trust”).

One of the oldest problems of QFT is Bloch and Nordsieck’s 1937 observa-tion that the scattering of charge-carrying particles in QED does not fit into thestandard collision formalism. Decades later Yennie, Frautschi and Suura [86]incorporated these ideas into renormalized QFT and showed that there exists aprescription which absorbes the logarithmic divergences in a (unfortunetely non-covariant) momentum space prescription for a infrared-photon-inclusive scatter-ing probability (cross sections) instead of a scattering amplitude.

There is presently no spacetime understanding in terms of a t →∞ behav-ior in the presence of massless h = 1 fields which matches that which LSZ andHaag-Ruelle established as the basis of the connection between fields and the

61

S-matrix in the presence of a mass gap. But there exists a formal observationwhich promises to play an important role in a foundational spacetime under-standing of infrared problems. This is the appearance of infrared superselectionsectors obtained from converting Wigner’s helicity representations into covariantfields depending on space- or light-like string directions namely the exponentialspacetime fields exp igΦ(x; e, e′) with a g-dependent short distance dimension(section 2).

In massive models these fields would be conventional Wick-ordered compos-ites associated to the local equivalence class of the s = 1 representation actingin the corresponding Wigner-Fock space of massive vector mesons, but in themassless limit they create new superselection sectors ”on top” of the WignerFock representation; in fact the related photon clouds they spontaneously breakLorentz invariance (section 2). They accompany in some asymptotic sense thet→∞ electrically charged matter fields and dissolve the mass shell of the latterby sucking it slightly (depending on g) into the soft photon continuum. In as = 1+1 toy model this amounts to a weakening of the mass-shell δ singularity ofthe charged particle to (p2−m2)−g branch-cut singularity so that the weakened

large time behavior of the interacting fields looses against the t−3

2 dissipation ofthe wave packets in the LSZ formula and a perturbative expansion in g accountsfor the logarithmic dibergencies. As often the devil is in the details, but it isclear that the natural Hilbert space of such a situation does not contain a h = 1Wigner-Fock space.

Whereas gauge theory and SLFT are believed to share the same short dis-tance behavior, this is certainly not expected on the other end of the scale;here one expects significant off-shell differences to gauge theory. This affectsin particular the second oldest unsolved problem namely confinement in QCD.In contrast to infrared problem of QED which is basically an on-shell prob-lem (the scattering amplitudes diverge but there is no off-shell divergence).theconfinement problem is directly related on the properties fields of massless self-interacting multiplet of vector potentials (gluons) and their interacting withs = 1/2 quark matter.

A perturbative signal of confinement should manifest itself as a logarithmicinfrared long distance divergence in the massless limit of the vector meson massof those correlation functions which contain at least one gluon or quark and an-tiquark fields which are not connected by a string-bridge 26. Since the presenceof massless multiplet of self-interacting gluons is the expected cause of such adivergence it seems reasonable to first look for such divergencies in gluon cor-relation functions without quarks. The perturbative construction of nonabeliangauge theory in covariant gauges leads to infrared finite correlations [87], whichin view of the fact that within gauge theory causal localization becomes maxi-mally incorrect for long distances is of no physical relevance.

As a result of the local directional fluctuations of the Aµ(x, e) gluon fields

26The strings must be semiinfinite since for a string-bridged q-q pair one expects no diver-gence.

62

in SLFT one expects a strong manifestation of long distance behavior for self-interacting massless vector potentials so the problem of finding a signal of con-finement boils down to the study of higher order 2-pfct of the gluon field; theappearance of logarithmic off-shell infrared divergences would be the first signof confinement. One would start from string-local massive selfinteracting vectormesons (naturally together with the renormalizability preserving Higgs particle)and look for logarithmic off-shell divergences in the massless limit which wouldbe the first signal that the fundamental fields disappear and their only tracesare in e-independent gluonium composites.

To nail down this speculative idea one must show that although the shortdistance behavior of the gauge theoretic and the sl short distance behavior ofthe propagators of vector potentials agree, the long distance behavior of the slpropagator is sufficiently different to cause a second order off-shell logarithmicinfrared divergence in the adiabatic limit of correlation functions.

There are still many open problems in the SLFT setting of QFT; some ofthose addressed in this article remain unsolved. This may partly be attributedto the fact that the SLFT idea is very recent and the number of researchers whonoticed its existence is still quite small.

Acknowledgements. This work is part of a joint project whose aim isto develop a positivity-preserving formulation of QFT which instead of usingindefinite metric and ”ghosts” is based on only physical degrees of freedom.The people presently involved in it are Jose Gracia-Bondıa, Jens Mund, K.-H.Rehren and Joseph Varilly. For intense and profitable contacts I owe thanks toall of them. I am also indebted to Michael Dutsch for helpful comments.

I also acknowledge an interesting correspondence with Edward Witten whichstarted with his interest in the history of Haag duality whose motivation Iunderstood after I received his preliminary notes on his new project: ”Noteson some Entanglement Properties of Quantum Field Theories”. It is certainlyencouraging to notice the increasing interest in the powerful methods of operatoralgebras. Together with the attempt by a (still) small group of researchers withinLocal Quantum Physics (which presently consist of the names mentioned beforetogether with the author) who are taking the opposite direction, namely to usethe causality and positivity principles without to the (quasi)classical crutches ofLagrangian quantization, this raises hope that the somewhat fractured situationof the last 4 decades may come to an end.

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