The Erwin Schrodinger International Boltzmanngasse

ESI Institute for Mathematical Physics A Wien Austria

Motivations and Physical Aims

of Algebraic QFT

Bert Schro er

Vienna Preprint ESI August

Supp orted by Federal Ministry of Science and Research Austria

Available via httpwwwesiacat

Motivations and Physical Aims of Algebraic QFT

Bert Schro er

Freie Universitat Berlin

Institut fur Theoretische Physik

Arnimallee Berlin

email schro erphysikfub erlinde

July

Abstract

We present illustrations which show the usefulness of algebraic QFT

In particular in lowdimensional QFT when Lagrangian quantization is

not available or is useless eg in chiral conformal theories the algebraic

metho d is b eginning to reveal its strength

This work is dedicated to the memory of Claude Itzykson

History and Motivation

One characteristic feature which distinguishes algebraic QFT from the various

quantization approaches to relativistic quantum physics based on classical La

grangians canonical pathintegral is its emphasis on lo cality and its insistence

in separating lo cal from global prop erties The former reside in the algebraic

structure of lo cal observables whereas the latter usually enter through states

and the suitably constructed representation spaces of lo cal observables

The idea that the Global constitutes itself from the Lo cal is of course

the heartpiece of classical electromagnetisms as formulated by Faraday Maxwell

and Einstein

Algebraic QFT is more faithful to physical principles than to particular for

malisms As such it has more in common with the KramersKronig disp ersion

relations of the s as a test of Einstein causality than with the p ost Feynman

developments of functional formalisms Its conceptual and mathematical arsenal

is however signicantly richer than the QFT underlying the derivation of the

Kramers Kronig disp ersion relations

When physicists rst analyzed the new concepts of standard quantum theory

they payed little attention to lo cal structure since their main aim was to under

stand the new mechanics For example in von Neumanns early general account

of the mathematical framework of quantum physics in terms of observables and

their measurement lo cality and causality did not enter at all Even in the subse

quent renements of Wick Wightman and Wigner concerning limitations on

the sup erp osition principle lo cality was not used

The idea that by combining the sup erselection principle with relativistic lo

cality one may obtain a p owerful framework in which global sup erselected charge

sectors including their asymptotic particle structure together with the statis

tics could emerge from the more fundamental nets of lo cal observables can b e

traced back to the work of Rudolf Haag in the late s

More than a decade b efore EP Wigner had successfully demonstrated

that it is p ossible to investigate fundamental issues of relativistic quantum physics

without any recourse to quantization Here we are of course referring to his

famous representation theory for the Poincare group which ended the longlasting

but somehow academic discussion ab out relativistic linear eld equations

By the end of the s there also existed an axiomatic framework the

Wightman framework for p ointlike covariant elds which if enriched by the

LSZ asymptotics gave a unied viewp oint linking the Heisenb ergPauli

1

canonical commutation approach with the Wigner representation theory

2

under one ro of

1

We will whenever p ossible refer to textb o oks or reviews where the content of historical

pap ers is presented from a viewp oint similar to ours

2

Behind these elegant schemes which were based on a few principles was of course the sweat

and tears of the pioneers of renormalized p erturbation theory

The b eginning of what is nowadays called algebraic QFT is usually identied

with the pap er of Haag and Kastler although there were prior imp ortant

developments notably by Borchers and Araki Borchers showed among other

things that the scattering matrix is an invariant shared by a whole lo cal class of

elds the Borchers equivalenc class and Araki demonstrated that the lo cal

observable algebras of algebraic QFT have no minimal pro jectors ie they allow

no maximal measurements even for the case of free elds In contemp orary

mathematical terms they are typ e III hyp ernite von Neumann factors a fact

1

which as will b e commented on later has deep physical relations to such diverse

lo oking issues as the existence of antiparticles and the HawkingUnruh eect

The derivation from the physical principles of lo cal observables of the prop erties

of chargecarrying elds including their statistics and internal symmetries was

an imp ortant result of the early s and is nowadays referred to as the DHR

theory Its sp ecic adaption to lowdimensional QFT with d which

leads to braid group statistics is of a more recent vintage temp eratur physics

For reasons of completeness we should add the remark that nontrivial su

p erselection rules generalized angles can also b e obtained without lo cal

observables in global quantum mechanics However in that case strictly sp eak

ing one has to go b eyond the Heisenb ergWeylSchrodinger theory and consider

C algebras with nite degrees of freedom which like the particle on a circle

the rotational C algebra or the rigid top C SO group algebra are not

simple and hence admit inequivalent irreducible representations In all those

global cases the sup erselection rules are not fundamental but rather result from

a physical overidealization Take the example of the stringlike AharonovBohm

solenoid This is of course the idealization of a nitely extended physical solenoid

in Schrodingertheory It so happ ens that in the singular limit when the math

ematics simplies the C algebra b ecomes the rotational algebra which is con

ceptually dierent from the Heisenb ergWeyl algebra The fact that there is a

single C algebra with inequivalent irreducible representations is often overlo oked

in the dierential geometric bre bundle framework which tends to interpret

dierent representations as dierent theories and not as sup erselected sectors of

just one C algebra

Sup erselection rules in quantum mechanics have more similarity to the seized

vacuum structure generalized angles than to the charge sup erselections struc

ture of QFT Only in lo cal theories with innite degrees of freedom ie quantum

eld theories also in the thermo dynamic limit of lattice theories one may en

counter fundamental sup erselection rules as a natural consequences of the lo cal

observable structure and the various globalizations it naturally leads to In the

3

present context it is more imp ortant to stress the dierences of lo cal quantum

3

To view lo cal quantum physics as just a particular extension of quantum mechanics or a

quantized extension of classical eld theory is in my view somewhat misleading It is rather a

new realm of physics with novel concepts

physics with entirely new concepts with resp ect to quantum mechanics than

some similarities b etween them

The assumptions on which the algebraic metho d is founded can b e separated

into two groups

The lo cal algebraic net prop erties for the observables

They are formalized in terms of a coherent map A of nite spacetime regions

O into von Neumannalgebras

A O AO

It is sucient to dene this net ie family of von Neumann algebras indexed

by spacetime regions for the smallest family of regions stable under Poincare

transformations which is the family K of diamonds or double cones It is

generated from double cones centered symmetrically around the origin by apply

ing Poincare transformations The lo cal algebras AO are op erator algebras in

an auxiliary Hilb ertspace Among the coherence prop erties they satisfy is the

4

spacelike commutativity Einstein causality Often the auxiliary Hilb ertspace

is chosen to b e physical the representation space of the vacuum representation

The net of von Neumann algebras lends itself to two globalizations the C

algebra of quasilo cal observables the inductive limit for families K directed to

wards innity A or the larger universal observable algebra A which

q uasi uni

corresp onds to an algebraic analogue of geometric spacetime compactication

spacetime

The global quantum asp ects of states

Since the globalized net allows for many inequivalent representations one

is forced to work with a more general concept of states than that of Schrodinger

quantum mechanics In QFT states are linear p ositive functionals on the

observable net which satisfy certain continuity prop erties with resp ect to this

net Among the physical state there is the distinguished vacuum state Behind

the vacuum requirement or the use of the closely related KMS temp erature

states is the stability idea

At this p oint it is helpful to alert the reader against p otential misunderstand

ing of the algebraic approach as some form of axiomatics or do ctrine on lo cal

quantum physics As already in case of von Neumanns concepts of observables

the idea of a net of lo cal observables is nothing more than a useful ordering

principle of the mind in order to separate a part of reality ab out which we can

safely apply our intuitive lo cality and causality principles from useful additional

constructions like eg chargecarrying elds with weaker and less immediate lo

calization prop erties For the latter our intuition is less reliable since it is based

4

This locality principle was and continues to b e see also section on chiral conformal eld

theories the heartpiece of the algebraic approach and lo cal quantum physics

more on exp erience with formalisms and less on physical principles Whether

one extends observable nets or makes them smaller do es not signicantly in

uence the prop erties of the complete theory in as much as in the analogous

problem of Heisenb ergs cut b etween the system and the observer in the pro cess

of measurement the exact p osition of the cut is exible as long as there is a cut

somewhere

Lo oking at the dichotomy b etween observables and states the question where

are the covariant p ointlike elds will invariably arise This question is b est

answered through an analogy with dierential geometry The elds serve as a

co ordinatization of the lo cal algebras They play the role of a kind of p ointlike

generators and very dierent lo oking systems of elds may generate the same

lo cal algebras The intrinsic physics resides in the lo cal net and its physical states

but for explicit mo del calculations it is convenient to use elds On the other

hand the structural prop erties of the theory are more appropriately analyzed

within the net formalism Certain co ordinates are however distinguished eg

lo cal No ether currents and charge carrying elds with the smallest dimension

the relevant variables in the language of critical phenomena in their resp ective

sup erselection class

The deemphasis on elds in favour of lo cal nets is typical for the algebraic

metho d as compared to quantization approaches Once a prop erty has b een

understo o d within the algebraic setting it is clear that one is dealing with an

intrinsic asp ect of the physical system ie an asp ect which do es not dep end on the

formalism by which the theory has b een constructed Not all prop erties obtained

by applying the various quantization approaches are intrinsic in this sense as

examples will show later on The notion intrinsic which I use throughout

this article has an analogous signicance as op erational or observable in

the early days of quantum mechanics when Gedankenexp eriments were used

for clarication Here its meaning is a bit more technical Prop erties which

can b e reconstructed by just using the physical correlation functions are called

intrinsic and structures which dep end on the path of construction and cannot b e

unambiguously deduced from the observables example lo cal gauge invariance

axial anomalies are extrinsic

Among the imp ortant discoveries which originated from a quasiclassical or

p erturbative view ab out QFT that were successfully incorp orated and extended

by the algebraic metho d are the GoldstoneNambu mechanism of sp ontaneous

symmetry breaking and SchwingerHiggs screening mechanism in ab elian gauge

theories In algebraic QFT these ideas are elevated to the level of structural

theorem on equal fo oting with the TCP and Spin and Statistics Theorem

Unlike the Wightman framework the algebraic metho d as previously men

tioned do es not make any a priori assumptions on the form of commutation re

lations b etween charge carrying elds The representation theory sup erselection

theory of observable nets is p owerful enough to determine those commutation re

lations mo dulo Klein transformations which are then xed by invoking lo cality

Even p erturbation theory if lo oked up on in the spirit of algebraic QFT

presents new and interesting problems Here the recent discovery of the microlo

cal sp ectrum condition following an idea of Radzikowski has b een instru

mental in the resurging interest in p erturbation theory with algebraic metho ds

It is interesting to note that this lo cal form of sp ectrum condition was discovered

in the quantum eld theory in curved spacetime where a go o d substitute for

the in that case meaningless global vacuum reference state in form of a new

principle b ehind the recip e of the Hadamard condition or the adiabatic

vacuum prescription had b een particularly pressing More comments on these

very recent results can b e found in section

As far as lo cal gauge theory is concerned the diculties which algebraic QFT

had in attributing an intrinsic meaning to an elusive lo cal gauge principle have

deep ened Comments on this problem can b e found in many pap ers on algebraic

QFT and there are examples of low dimensional gauge theories which allow for

an alternative formulation without lo cal gauge elds and anomalies thus showing

that the lo cal gauge formulation is at b est an option but not a principle

A recent criticism of the gauge principle app ears in the work of N Seib erg

in sup ersymmetric gauge theories However there remains a dierence b e

tween the Seib ergWitten interpretation of the asymptotic freedom in terms the

infrared limit regime as phases of the theory versus that of algebraic QFT where

eg the short distance theory reveals itself as another theory which is related

to the original one in a very interesting way and generically has more degrees

of freedom than the original This enhancement of global symmetry

enhancement of degrees of freedom is unfortunately not typical for lo cal gauge

theories It happ ens abundantly in two dimensional QFT example Ising eld the

ory where the chiral limit has signicantly more sectors than its massive parent

theories The phenomenon is related to the emergence of nontrivial dual order

disorder charges in the scaling limit whereas in gauge theory it corresp onds to

shortdistance deconnement

With all these dierences in interpretation and formalism it should come as

no surprise that algebraic QFT also leads to a dierent view on the Schwinger

Higgs screening mechanism First some historical remarks are in order In an

interesting early note Schwinger p ointed out the p ossibility that charged

matter interacting through electromagnetism may suer a charge screening ac

companied by the app earance of massive photons He did not think in terms of

condensates formed by the matter elds Since he had no rigorous argument in

d he invented the d Schwinger mo del in order to illustrate

the p ossibility of a screened phase which he imagined to o ccur in d

theories for strong Maxwellian couplings

Later Higgs apparently b eing unaware of Schwingers work studied

scalar electro dynamics and found a p erturbative regime with a condensate of

the matter eld and a massive photon He phrased his ndings within the cur

rent framework of lo cal gauge theories and thought ab out the massive photon as

originating from a twostep pro cess rst a sp ontaneous symmetry breaking with

a Goldstoneb oson and then the photon swallowing this b oson and in the pro

cess b ecoming massive Lowenstein and Swieca made Schwingers arguments

in the d mo del more transparent as well as rigorous and showed that there

is a chiral symmetry breaking which indeed leads chiral to condensates Since

there is no GoldstoneNambu mechanism in d one notes that this mo del

do es not allow the aforementioned twostep interpretation

As already mentioned b efore Schwingers ideas were vindicated in a b eautiful

structural theorem proved by Swieca The theorem assumes that the charge

currents are the sources of a Maxwell eld but uses no lo cal gauge principle

or condensates For this reason it is very much in the spirit of algebraic QFT

since the latter are less intrinsic concepts But it seems that the more p ertur

bativly manipulative but physically less intrinsic condensate viewp oint b ecame

more p opular than Schwingers screening picture

Algebraic QFT views the would b e charge carriers not as p ointlike covariant

and hence gauge dep endent elds but rather as semiinnitly extended lo cally

gauge invariant ob jects whose lo calization regions are spacelike cones in the

singular limit spacelike semiinnite Mandelstam strings Although they are

lo cally gauge invariant they carry a global gauge charge

In the SchwingerHiggs phase of scalar electro dynamics these lo cally gauge in

variant extended elds condense into the vacuum and in this way they improve

their lo calization prop erties by lo osing their infrared photon clouds but only at

the exp ense of birth of dual magnetic ob jects with even more severe infrared

structures At this p oint it would have b een nice to illustrate this within the

pathintegral p erturbative setting and the Faddeev Popp ov program including

the gauge xing But I was unable to nd an appropriate reference Apparently

the problem of conversion of the pathintegral manipulations into the physically

clearer language of op erator algebras did not enjoy much p opularity it cannot b e

done by dierentialgeometric techniques However the Schwinger mo del pro

vides at least a nice rigorous op erator illustration of these nonlo cal ob jects in

d

In sections and I will discuss the physical aims of the algebraic metho d

in the context of low dimensional QFT where they app ear presently in their

clearest form through the braidgroup statistics problem It has b een known

for some time that chiral conformal QFT furnishing the simplest analytic illus

tration of braid group statistics The exchange algebras carrying this statistics

with the Rmatrix structure constants follow from the aforementioned princi

ples in a natural way They represent a sp ecialization of a more general

and abstract concept of algebraic QFT In recognition of the fact that the

richness of chiral sup erselection rules and braidgroup statistics results from the

aforementioned lo cality and stability principles and not just from sp ecial ad ho c

algebras as KacMo o dy algebras current algebras etc we called one of our

contributions Einstein causality and Artin Braids However I also should

not hide the fact that my attempts to understand certain global prop erties as eg

the mo dular structure in the sense of Kac and Wakimoto in the framework

of algebraic QFT were less than successful Since an understanding of these

mo dularity prop erties as a consequence of the principles p erhaps with the help

5

of the p owerful more general TomitaTakesaki mo dular theory is a physically

fascinating challenge and p erhaps b esides the braid structure the only physical

reason for b eing interested in chiral theories I will certainly try a second time

It is encouraging that at least a partial result in this direction has b een obtained

very recently

The reason b ehind this analytic simplicity of chiral conformal QFT is that on

the right or left light cone a genuine interaction is not p ossible a kind of algebraic

Huygens principles The charged chiral conformal elds should b e viewed as the

freest elds which fulll physically admissible chargefusion rules together with

the uniquely aliated and measurable plektonic statistics or more precisely R

matrix commutation relations since chiral conformal QFT has no Wignerparticle

structure These new fusion laws unlike those of compact symmetry group

representations are not compatible with the Lagrangian formalism In fact these

elds strictly sp eaking are not even Wightman elds since they come equipp ed

with source and range pro jectors The problem of whether one can rescue the

Lagrangian formalism by combining right and left chiral elds to lo cal elds is

in my view of an academic nature since all the computational p ower resides

in the nonLagrangian chiral elds The Lagrangian quantization approach is a

useful to ol whenever it applies but certainly not a concept required by physical

principles

In d plektonic theories the charges and their fusion laws b ecome also

attributes of particles Like in d a given set of sup erselection rules admit

many solutions which are distinguished by coupling constants which may b e

thought of as dynamical deformation parameters There the idea of freeness may

b e used to select a distinguished eld The b est way to implement this idea seems

to pro ceed by an extension of the Wigner representation theory as in section

Again it seems that Lagrangians play no useful computational role as so on as

plektonic statistics app ears The situation for d is presently less clear

An rened intrinsic notion of interaction would lead to two options mentioned

later One uses the relation b etween the scattering op erator S and the mo dular

asp ects of the TCP op erator whereas the other is based on the microlo cal

sp ectrum condition Both attempts are severely limited by the present lack of

understanding lo calization infrared prop erties of dual electric and magnetic

5

This structure app ears precisely at the place of the NelsonSymanzik symmetry for the

Euclidean interchange of space and time ie at one of the most profound discoveries of the

ies

charged ob jects On the other hand there have b een many new insights into the

relation of temp erature with groundstate QFT relativistic KMS condition fate

of sup erselection rules fate of sp ontaneously broken symmetries In order not

to overburden this article with to o many issues I will not present these latter

developments but rather limit myself to what I think could b e attractive to a

pragmatically minded reader with some amount of curiousity ab out structural

prop erties of QFT

We nally would like to attempt to incorp orate algebraic QFT into the history

of ideas on relativistic quantum physics The most inuential metho d of thinking

which has dominated QFT for almost three decades is that of Dirac which is based

on geometric intuition formal mathematical elegance and the p ower of analogies

The square ro ot of the KleinGordon op erator the lling of the Dirac sea

in order to implement the idea of stability in relativistic quantum physics and

the geometric construction of magnetic monop oles may serve as illustrations of

the Diracian mo de of thinking

Even if we nowadays see things in a slightly dierent way nob o dy can deny

that this approach to fundamental problems has b een extremely successful It

app ears to me that the discovery of the electroweak phenomenological theory

has b een the last real big physics conquest made in this geometric spirit All at

tempts to go b eyond this discovery GUT strings sup ersymmetry and probably

also most of the more recent inventions were not met with success This failure

which by the end of this century has develop ed into a profound crisis of particle

physics QFT and even of adjacent areas may b e the result of an exaggerated

unbalanced use of the Diracian approach rather than an inherent weakness of

this metho d

On the other hand there has also b een a tradition which emphasizes the im

p ortance of physical principles as the condensated form of all our past exp erience

This mo de of thinking always confronts actual ndings with these principles of

ten by the intervention of Gedankenexp eriments It develops its full strength if

this confrontation leads to an antinomy or paradoxon One nds this mo de of

thinking particularly in the works of Bohr and Heisenb erg in fact it was essential

for the discovery of quantum mechanics by Heisenb erg Schrodinger was p erhaps

a bit more on the Diracian side Also the later contributions of von Neumann

Jordan Pauli and Wigner serve as go o d illustrations For the purp ose of our

presentation let us call this mo de of thinking BohrHeisenb ergian A more

recent example for this approach is algebraic QFT

I do not know a more appropriate illustration for the dierence b etween the

Diracian and BohrHeisenb ergian way of doing lo cal quantum physics than

the two dierent ways in which extended ob jects in particular with stringlike

ob jects enter lo cal quantum physics

The Diracian physicist go es b eyond p ointlike or compactly lo calized ob jects

for reasons of esthetical pleasure related to mathematical consistency and b eauty

On the other hand he is able to b e somewhat lighthearted ab out the underlying

physics viz the sudden transition of the string formulated dual mo del of strong

interactions to alleged semiclassical approximation of quantum gravity of mo d

ern string theory The development of string theory from the dual mo del was

certainly a Diracian contribution

A Diracian physicist likes inventions sup ersymmetry sup erstrings quan

tum groups and sometimes even takes them for discoveries

Quite dierently the BohrHeisenb ergian physicist generally shuns away

from invention and tends to b e less impressed by mathematical b eauty unless

it arises as a result of comp elling conceptual insight If stringlike ob jects are

necessary in order to fully explore a physical principle he is able to fullheartedly

embrace such ob jects

This was indeed precisely the way how stringlike ob jects entered algebraic

QFT At the b eginning of the s Buchholz and Fredenhagen proved

a structural theorem following up some problems raised in Swiecas work

which established the arbitrary thin spacelike cone semiinnite stringlike

lo calization of chargecarrying elds as the a priori b est p ossible lo calization

resulting from the physical principles of algebraic QFT together with the mass

gap hyp othesis Later it b ecame clear that in lowdimensional theories this

more general lo calization really o ccurs in connection with braid group statistic

theories eg in d braid group statistics exhausts this semiinnite

stringlike lo calization in the sense that any b etter lo calization ie compact or

p ointlike necessarily leads back to p ermutation group statistics In other words

only chargecarrying op erators with this semiinnite extension are able to carry

plektonic charges fullling the new fusion laws which are not covered by the

representation theory of compact internal symmetry groups Such elds do

not app ear naturally in Lagrangian quantization although attempts have b een

made to describ e such situations via a ChernSimons extension of the standard

quantization formalism

In in the BohrHeisenb ergian mo de of thinking the discovery of the con

nection b etween lo calization and statistics is interpreted as an extension of the

Einsteincausality principle and the stability principle b ehind the p ositive energy

condition b eyond Lagrangian quantization

In the construction of these new charge carriers which result from old prin

ciples one has to leave the Lagrangian quantization framework and use the

metho ds oered by algebraic QFT One needs many new mathematical concepts

and it testies to the conceptual strength of the algebraic metho d that in de

veloping the algebraic framework of sup erselection rules Doplicher Haag and

Rob erts whose pap ers were written a decade b efore the impressive mathemat

ical work relating subfactors and symmetries by Vaughan Jones anticipated

in a more limited physical context some concepts of the latter This profound

connection was deep ened in the context of lowdimensional QFT and braid group

statistics

Lo oking at the rather long time of physical stagnation in the p ost electro

weak era one gets the impression that the Diracian metho d has b een much more

fruitful for mathematics than for physics

Most of the more recent scientic contributions of Claude Itzykson to whose

memory I dedicate this article are Diracian with geometrical ideas in the fore

front

I rememb er very spirited conversations on the present state and the p ossible

future role of QFT the last such conversation over dinner in Berlin in late autumn

With sympathy and resp ect it is easy to enjoy also those conversations

which do not lead to an identical viewp oint Claude listened to my account of

the deep crisis in physics notably in the area of QFT Finally he suggested to me

not to worry ab out these inevitable developments to o much and instead enjoy

the b eautiful mathematical structure evolving from physical Diracian ideas

Although Claude always kept a critical distance to the physics of everything

he enjoyed certain fashionable ideas b ecause they gave him a chance to work on

those mathematical structures which he found so app ealing without having to

ap ologize p ermanently for doing mathematics instead of physics In that resp ect

times were more lenient than two decades b efore when eg one had to present

an ap ologetic b ehaviour for b eing interested in d pathologies

Standard d = 3 + 1 versus LowDimensional

QFT

One of the most impressive structural achievements in d algebraic QFT

is the elab oration of the DHR sup erselection theory of lo cal nets which culmi

nates in the DRtheorem on the b osonfermion statistics alternative and its

inexorably linked internal compact symmetry group as computable attributes of

the observable algebra One very physical way of understanding its content is in

terms of a twostep pro cess

i The derivation of parab oson or fermion statistics mixed Young tableaux

in the sense of Green from the algebraic net prop erties and appropriate state

lo calization prop erties This rst step yields chargecarrying elds ele

ments of the exchange algebra or the reduced eld bundle with Rmatrix

structure constants The latter dene a representation of the innite p ermu

2

tation group S and therefore fulll the relation R in addition to the Artin

B braid group relations The assumed state lo calization prop erties are those

suggested by Wigners analysis of particles as irreducible p ositive energy repre

sentations of the Poincare group Not all sectors of p otential interest charged

infraparticles ie physical electrons innite energy quark states are however

covered by Wigners theory

ii The mixed Youngtableaux lead to statistical weights which may b e en

co ded into a multiplicity enlargement of the Hilb ertspace and the construction

of multicomp onent elds numb er of comp onents statistical weight These

tensor elds are b osonic or fermionic and transform under a compact internal

symmetry group which is computable from the structure of the observable net

Clearly this kind of insight cannot b e obtained from a Lagrangian approach

rather it constitutes the prerequisite for the applicability of such an approach

Indeed the paraon elds although having the same lo cal observables and the

same Smatrix as the tensorial elds are more noncommutative Rmatrix in

stead of lo cal b osonic or fermionic commutation relations and less lo cal than the

tensorial elds They also do not have a quasiclassical limit and hence unlike

their tensorial counterparts do not t into a Lagrangian framework There

fore the formal requirement to work with elds which can b e thought of as b eing

obtained through a quantization gives preference to the tensorial description

In d QFT one still can carry out the rst step i by using the string

lo calization resulting via the BFtheory from the mass gap hyp othesis

2

But whereas in d this would still lead to exchange elds with R this

additional relation which forces the braidgroup to coalesce to the p ermutation

group is absent in this case The statistical dimensions apart from the ab elian

braid group situations are now noninteger numb ers and the DR construction

of enco ding these statistical weights into multiplicities of tensorial elds ie the

step ii do es not work

It turns out that chiral conformal eld theories which have much simpler

analytic prop erties than d massive theories lead to the same family of

admissible sup erselection rules and braid group representations In b oth cases

one observes as a substitute for ii some very fundamental lo oking dierences

from the standard d theories The observable nets now allow for a

natural compactication in terms of a global universal algebra This rather

big observable algebra contains charge measuring op erators which turn out to

b e dual to the chargecreating endomorphisms of the DHR theory In addition

the new situation is selfdual in the sense that the Verlinde matrix S which in

algebraic QFT just represents the value of the various charges in the various

sup erselection channels turns out to b e symmetric with resp ect to its charge

measuring chargecreating entries In the case of nite group symmetry this

only happ ens for ab elian groups but for the case at hand one do es not know

an internal symmetry concept which explains the symmetry of S and the non

integer statistical dimensions in terms of a multiplicity concepts

One observes that the universal observable algebra A contains also a kind

uni

of chargeinduction op erators which change charges but only in the presence

of other charges These op erators if inserted b etween basis elements of the

orthogonal intertwiner basis break the orthogonality and lead to generaliza

tion of the matrix S which are asso ciated to the representation theory of socalled

mapping class groups So it seems that the latter form an imp ortant asp ect of

the illundersto o d new symmetry concept a universal mapping class group al

gebra The DHR C observable algebras do not contain such interesting central

charges

Presently there is no go o d comprehension of why in low dimensional QFT the

internal symmetry gets so inexorably linked with the spacetime symmetry

whereas in higher dimensions one was unable to marry them in a nontrivial

way apart from that physically rather uninspiring marriage via extended su

p ersymmetry Wittens top ological QFT in terms of functional integrals as well

as the algebraic approach through lo calized endomorphism and their intertwiners

app endix of ref describ e eg manifolds invariants in a eld theoretic setting

but do not really explain the aforementioned phenomenon The q deformation

theory of quantum groups do es not cast any light on this problem either even

worse such attempts are inconsistent with the p ositivity of quantum theory and

their only useful role seems to b e to furnish admissible Rmatrices at the ro ots

of unity The latter also app ear in a much more natural and physical way com

patible with quantum theory via the DHR or the Jones subfactor theory

Although algebraic QFT denies the existence of a lo cal gauge principle

with a direct physical meaning some of its attributed asso ciated structures as

the description of charge carriers of top ological charges in the sense of in

terms of semiinnite Mandelstamlike spacelike strings or as the app earance

of additional degrees of freedom at short distances the asymptotic freedom

prop erty of certain gauge theories are naturally accounted for as an extended

realization of known physical principles Since algebraic QFT was intro duced

with predominant emphasis on structural problems of QFT it is particularly

gratifying to note that its metho ds have proved to b e increasingly useful for the

actual construction of ob jects as anyons and plektons which hardly t into a

Lagrangian quantization framework

Before we present some partial results in the next sections it is worthwhile to

take notice of the fact that the issue of braid group statistics is essentially a

quantum eld theoretical problem for which the recourse to quantum mechanics

despite particle numb er conservation do es not yield a signicant simplication

Consider for example Wilczeks idea of using the AharanovBohm eect

in order to obtain an interaction b etween pairs of dyons Since this interac

tion is extremely long range one do es not apriori know the kind of b oundary

conditions to b e used in the stationary scattering formalism there is a corre

sp onding problem in the timedep endent formulation For physical reasons the

multiparticle Smatrix must however full l the standard cluster property If for

n particles one particle is converted into a sp ectator ie shifted to spatial

innity then the n particle S matrix has to coalesce into the n particle

S matrix This consistency condition although trivially satised for shortrange

interactions gives rise to a b o otstrap problem in the present longrange case

The transcription of this problem in terms of quantum mechanical pathintegrals

do es not seem to solve this consistency problem with the cluster decomp osition

The nparticle b oundary condition is only implicitly dened by this cluster

requirement which couples all channels Therefore the main advantage of the

nonrelativistic limit namely the decoupling of these channels through particle

conservation is lost and one may as well face the QFT problem of plektons right

away and then construct the nonrelativistic limit afterwards since in QFT the

cluster prop erties are safely built in

The braid group statistics problem has a vague resemblance with relativistic

integrable mo dels of QFT eg the SineGordon theory in the sense that al

though there is no creation of real particles the S matrix is purely elastic

the virtual particle structure is quite complex

More on Physical Aims

After we to ok notice of the fact that the problem of plektonic statistics cannot b e

simply solved by referring rst to a plektonic quantum mechanics but requires

a more careful eld theoretic treatment it is worthwhile to ask ab out the physical

aims one hop es to achieve with a theory of plektonic elds

Most mathematicallyminded eldtheorist are aware of the rich applications

of standard d QFT to elementary particle physics and therefore consider

lowdimensional QFT as a kind of theoretical lab oratory As a result of ab

sence of genuine interactions on half the lightcone Huygens principle in d

leads to chiral factorization and rules out any genuine interaction chiral con

formal QFT is a kinematical family of QFTs which realizes the complete set of

admissible plektonic statistics analogous to dimensional free elds

As a generalization of analyticstructural arguments ab out p oint functions

one nds for each admissible family of plektonic statistics Hecke algebra

plektons and BirmanWenzl algebra plektons two families for their eld

theoretic realizations In terms of standard terminology in chiral conformal eld

theory they corresp ond to Wmo dels observable algebra without continuous

group symmetry and currentalgebra mo dels There are also go o d arguments

that the former can b e obtained from the latter by the DHR invariance

principle ie the application of a conditional exp ectation through averaging over

the group The arguments that these families apart from isolated exceptional

cases exhaust all plektonic p ossibilities are somewhat weaker

It is physically very protable to take notice of the fact that these families

b eyond their role as theoretical lab oratories explain and classify the critical

indices of a large class of d classical statistical mechanic mo dels in terms of the

statistical phases related to braids and knots of the asso ciated noncommutative

chiral conformal QFT Even though some of the Euclidean eld theories which

describ e the critical limit of statistical mechanics may mo dulo renormalization

problem p ermit an interpretation in terms of Lagrangian and pathintegrals

the noncommutative chiral theory which carries all the computational p ower

is not describable in such terms the analytically continued chiral theory is as

noncommutative as the original real time theory The aforementioned analytic

determination of p oint functions based on their plektonic content do es not

use quantization ideas There is the realistic hop e that the critical b ehaviour

of a large class of classical statistical mechanics mo dels p erhaps even all may

b e understo o d in terms of the asso ciated noncommutative nets and their braid

group statistics which is derived and classied solely on the basis of the lo cality

principle and the stability requirement mentioned in the intro duction

Although this deep relation of the systematics of d critical indices with

the sup erselection charges of chiral conformal theories has all the attributes of

a miracle this mystery can b e traced back to the still somewhat mysterious

discovery of the relation b etween commutative ab elian von Neumann algebras

euclidean and noncommutative causal nets real time eld theory The concep

tual frameworks including the interpretation of lo calization in b oth theories is

totally dierent a fact often not appreciated by physicist who use the Euclidean

formalism for other purp oses eg numerical calculations than structural inves

tigations of QFT Whereas in d massive theories the analytically contin

ued correlation functions have maximally two physical b oundaries namely the

Minkowski b oundary exp ectation values of noncommutative op erators and

the Euclidean b oundary sto chastic exp ectation values of commuting variables

in the sense of classical statistical mechanics the restriction of the continued

chiral correlation functions to any real one dimensional boundary dene a posi

tive denite Wightman theory This curious prop erty makes chiral theories more

geometric than others and p ermits the asso ciation to Riemann surfaces It is in

my view the origin of the high spacetime symmetry Mobiusinvariance dif

feomorphism covariance and the inexorable and mysterious link of spacetime

symmetry with internal symmetry in chiral theories also exp ected for d

The Riemann surfaces app ear where in standard Wightman theories one had the

BWH analyticity domain of vacuum correlation function For eg genus one the

original real time QFT lives on one cycle and the Euclidean on the orthogonal

cycle The p osition of these cycles is not distinguished and what was called Eu

clidean is now as noncommutative as the original theory In passing I mention

that the algebraic approach also suggest an interesting intrinsic meaning for two

expressions whose historical origin dates back from a time when chiral conformal

QFT was thought of as separated from the rest of QFT One is holomorphic

eld whose meaning the literature is very unprecise on such concepts seems

to b e lo cal chiral It would b e nicer to use it for the ab ove very curious and

distinguished prop erty of admitting a continuous family of physical b oundaries

in the sense of p ositivity In this meaning the prop erty of b eing holomorphic

explains the high dieomorphism convenience of the chiral theories The second

expression is vertex op erators which at rst sight just seems to b e another

name for the go o d old lo calizable eld Linking this expression however with

the previously mentioned app earance of nonWightman eld which carry cen

tral source and range pro jectors it b ecomes a very useful terminology which

6

transcends chiral conformal QFT

Plektonic theories in d are physically more imp ortant physics of

quantum layers but analytically more dicult than chiral conformal theories

The main reason is that their chargecarriers have a semiinnite spacelike string

lo calization and their sup erselection structure do es not uniquely determine them

rather they have physical deformation parameters coupling constants These

interaction sup erselected charges may b elong to d Wigner particles whose

nonrelativistic limits are quasiparticles of condensed matter physics which retain

their plektonic statistics in the nonrelativistic limit

Our present understanding of plektonic prop erties already supplies us with

two families of numb ers of great p otential physical signicance

Algebraic QFT condensed matter physics

statistical dimensions amplication factors rel size of degree of freedom

statistical phases em prop erties of plektons fractional Hall eect

A particle physicist would think of amplication factors as the analogs of

Casimirs multiplicity factors I in cross sections But such weights for

degrees of freedom also enter and mo dify the BCS relation b etween critical

temp erature T and the gap at zero temp erature

c

2

const T d

c

Here d is the statistical weight of the plekton carrying the sup erselection

charge For d the relation is the standard BCS relation The quadratic

dep endence on d can b e made plausible for a hyp othetical SU fermionic BCS

mo del linear dep endence on the Casisimir eigenvalue Anyons ab elian

plektons have d and change only the statistical phases but do not amplify

the weights

Note that nonab elian symmetry groups in condensed matter physics can

only app ear in an articial way ie by making ad ho c approximations on the

U invariant manyparticle problem On the other hand since plektons are as

selfconjugate as the sup erselection structure of ab elian group symmetry a transi

tion from a U liquid fermi phase to a plektonic phase is p erfectly conceivable

It cannot b e stressed enough that the issue of relativistic lo calization is only

imp ortant in the classication of physically admissible plektonic statistics As

for Fermions and Bosons the results remain valid in the nonrelativistic domain

Since as already mentioned the relation b etween the sup erselection data and the

elds in d is not unique we need a principle which selects one realization

among all dynamically p ossible ones ie a substitute for a Lagrangian Such a

principle based on the idea of freeness will b e abstracted from the longrange

cluster prop erties of the Smatrix in what follows

6

I am indebted to DI Olive for this suggestion

In d the S matrix fullls the wellknown cluster prop erties which in

the extreme cluster limit all pairs b ecome innitely separated in the sense of

their wavepacket lo calization takes the form

cluster

S

limit

It is b elieved that the Borchers classes of the various free eld exhaust the p os

sibilities of lo cal interp olating elds with trivial scattering however a solid pro of

only exists for zero mass eld If true this would select the various stan

dard SU N or SO N selfconjugate free elds of d QFT as the freest

Bosons or Fermions

For d this limit generically lead to S In particular a more

lim

detailed analysis shows that for d one obtains as the leading long

distance contribution an elastic energy rapidity dep endent S matrix which

7

fullls the YangBaxter equations as a consistency condition The only surprise

is that such a limiting S matrix is again the S matrix of a lo calizable QFT the

starting p oint of the socalled b o otstrap program for the the construction of

relativistic integrable d massive mo dels

In d S seems to b e piecewise constant in momentum space with

lim

matrices which jump from one Rmatrix representation of a word in the braid

group to another as so on as the particle velo cities go through a coalescent degen

erate conguration The cross sections of such piecewise constant S matrices are

exp ected to vanish and there is some ambiguity in what part of the asymptotics

should b e accounted for as a change of inner pro ducts for in and outstates

and what part should b elong to the prop er S matrix In any case the lo calized

elds which may corresp ond to such a situation have not yet b een computed In

section we use an extension of the representation metho d of Wigner which is

however limited to anyons in order to shed some light on the lo calization and

statisticprop erties of free plektonic particles and elds

Remarks on Perturbation Theory and Inter

actions

Standard p erturbation theory aims at the p erturbation expansion for the vacuum

exp ectation values of time ordered pro ducts of p ointlike quantum elds The

most convenient presentation is via Feynman rules which can b e formally derived

either by op erator or by functional metho ds The rules for renormalizing those

formal expressions have remained complicated compared to the simplicity of the

Feynman rules themselves and an elegant incorp oration of renormalization into

7

It is a pleasure to acknowlede that A Zamolo dchikov had similar ideas on an intrinsic

notion of integrability private communication It testies to the naturalness of this idea

the op erator or functionalmetho d do es not exist for go o d reasons see b elow

The separation of the lo cal algebraic asp ects from the global state prop erties can

in principle b e done at the end assuming that the p erturbation series converges

in some sense with the help of the suitably adapted GNS reconstruction But

in practice this remains a very dicult task and this last reconstructive step is

usually left out sometimes in the erroneous b elief that functional integrals based

on classical actions will at least on a formal level always dene quantum eld

theories

The algebraic approach to p erturbative interactions even on a formal level

organizes things in a slightly dierent way The rst step is the construction

of a net in terms of free elds in Fo ckspace The choice of the Fo ckspace is

determined by what kind of free elds one wants to couple For implementing

interaction one uses Bogoliub ovs op erator S g h in Fo ckspace which has

formal unitary representation terms of time ordered pro ducts

R

4

(x))g (x)+ (x)h i (W ( (x))d x

I

0 0

S g h T e

Here x stands for the collection of free elds and W is a Wickordered L

I

o

invariant p olynomial coupling S was used by Bogoliub ov to intro duce causal

elds in Fo ckspace

1

S g h x S g h

g

hx

h=0

Here g is a Schwartz test function which is chosen constant on a large double

cone C

It is then a simple consequence of Bogoliub ovs causality prop erty that

x is indep endent of the values of g outside the past light cone V C subtended

g

by C According to a recent observation by Fredenhagen private communication

to b e published a change of g inside V C which leaves the constant value in C

unchanged can b e implemented by a unitary transformation U in Fo ckspace

1

g x x C g xU U

1 g

1

g

C

Since the lo cal net within C do es not change under a common unitary transfor

mation of all of its memb ers the elds x are formal candidates for eld

g

co ordinates of a net in C C can b e made arbitrarily large the p ossible diver

gence of the U s in p ose no problem for the net and hence one obtains a net

of algebraic QFT this time not in the vacuum representation but rather in an

auxiliary Hilb ert space States on this net in particular the vacuum state lead

to physical representations of the global C algebra which are inequivalent to the

auxiliary Fo ckspace representation

Several comments on this pro cedure are helpful In order to get away from

the formal p erturbative time ordered expressions Bogoliub ov axiomatized the

prop erties of the testfunction dep endent unitary op erators S g h in Fo ckspace

It seems that for the distinction b etween dierent interaction p olynomials

one needs some smo othness of these S in terms of their dep endence of f and

h Even presently it is not known if there exist such op erators in Fo ckspace

which corresp ond to a given W ie whether the Bogoliub ov axiomatics has

I

such solutions

The second step namely the construction of the vacuum and other repre

sentation is related to the socalled adiabatic limit This twostep separation

avoids the pitfall related to Haags theorem which can also b e circumvented

by the conceptually less attractive intro duction of an infrared cuto and the

construction of the thermo dynamic limit the adiabatic limit The chances for

existence of the net together with a vacuum state on it may even improve if one

restricts the Bogoliub ov metho d to a smaller algebra eg the algebra generated by

neutral comp osite elds An example is the coupling of a massive vector meson to

a conserved spinor current for which the neutral subalgebra stays renormalizable

It is interesting to note that even on a formal level it is not clear that

gauge theories t into this two step p erturbative construction The dicult

p oint here is that the gauge condition which are necessary for the denition of

op erator algebras eg the GuptaBleuler condition are mostly global even in

ab elian gauge theories The net formulation however is only consistent with lo cal

conditions

One may hop e that one should b e able to bypass the existence problem of

the Bogoliub ov axiomatics by constructing interacting wedge algebras This

hop e is based on the recent observation that the free elds algebras b elonging to

the Rindler or BisognanoWichmann according to upbringing and taste

wedge region can b e directly constructed without reference to lo cal elds

This raises the exp ectation that there should also b e a truly algebraic way to

intro duce an interaction for this preferred lo calization region Supp ort for this

hop e comes from the distinguished representation of the S matrix in terms of

mo dular conjugations

S J J

W 0W

Here J is the mo dular conjugation of the interacting wedge algebra whereas

W

J has the same relation to the incoming interactionfree wedge algebra

0W

This formula is only a transcription of the representation of S in terms of the

corresp onding TCP op erators

S

0

Here we used the BisognanoWichmann relation b etween and J which are

W

only dierent by a rotation around the xaxis

U Re J W txwedge

x W

as well as the action of on and

in

x x

1

hence x x

out in

x x

0 0

in out

and hence up to a phase factor which may b e absorb ed into the s Therefore

a natural distinguished equivalence transformation b etween the interacting and

the free incoming wedge algebra would b e given by a square ro ot of S which is

exp ected to b e a kind of eld theoretic Mller op erator Unfortunately we do not

have the vacuum representation but rather a representation of the wedge algebra

in some auxiliary Fo ckspaces which is related to the vacuum representation by

another unknown unitary equivalence transformation A p ossible construction of

interacting nets in the Hilb ertspace of incoming states based on this distinguished

role of the Smatrix as a mo dular invariant has b een sketched in

It is very gratifying to notice that the von Neumann algebras of this p ertur

bative net ie induced by p erturbative couplings b etween free elds b elongs to

a folium of states the reference state and all the mixed states obtained by den

sity matrices constitute a folium on the lo cal algebras double cones and wedges

which allows an elegant characterization in terms of a microlo cal sp ectrum con

dition Whereas the usual sp ectrum condition vacuum and p ositive energy

condition is a global requirement the microlo cal sp ectrum condition is a con

dition which can b e formulated directly on the lo cal algebras It is on the one

hand weaker it characterizes a whole folium of states instead of a unique vacuum

ground state and it do es not distinguish sup erselection sectors but on the other

hand it promises to lead to an intrinsic lo cal characterization of interactions

As mentioned in the intro duction this microlo cal sp ectrum condition was re

cently discovered in the quantum eld theory in curved spacetime for which the

ground state or vacuum requirement b ecomes meaningless apart from some

sp ecial cases which admit a timelike Killing vector As far as the curved space

time theories are concerned the microlo cal prop erty not only led to an under

standing of the Hadamard recip e and its equivalence to the socalled adiabatic

reference state construction but it also p ermits an incorp oration of inter

acting matter nonlinear equation of motions and a natural extension of the

Wickpro ducts and the denition of an energy momentum tensor op erator The

latter is needed in a quasiclassical interpretation of the classical Einstein equa

tions

Hence the algebraic metho d suggests new ways of lo oking at old problems

In this way one also hop es to get a b etter understanding of which diculties in

p erturbation theory already o ccur on a lo cal level ie in the construction of

nets and which of those are more related to global obstructions ie related to

states The free elds used for the construction of the interaction and the lo cal

net b ecome more disso ciated from the physical particle content of the mo del so

that incorrect pictures ab out particles b eing closely related to elds are avoided

The quantization approach via functional integrals is entirely global since it is

physically meaningless to functionally integrate over the eld space b elonging to a

lo cal piece of Minkowskispace In addition such a formulation is only meaningful

in the framework of a sub class of Euclidean theories and one has to keep in mind

that the Euclidean lo calization in the sense of statistical mechanics is extremely

nonlo cal with resp ect to the real time lo calization

The end of this section is also the natural place to make some more detailed

comments ab out the most serious problem in contemp orary QFT and elementary

particle physics the profound crisis of the last twenty ve years Even for some

b o dy not familiar with the content of the various fashions the crisis is already

evident from the observation that nothing of any physical relevance happ ened af

ter the discovery except of the electroweak interactions its detailed exp erimental

verication This stagnation is very remarkable in view of the fact that the rst

three quarter centuries were so rich in discoveries A glance at the selected pa

p ers edited by J Schwinger Selected Pap ers on Quantum Electro dynamics

edited by J Schwinger Dover Publications or even the later pap ers edited

by Imai and Takahasi QFT I and II in Series of Selected Pap ers in Physics

removes any doubt Even pap ers on the most central issues of contemp orary

physics as the recently rep orted even in the international press breakthrough

in QCD connement based on the fascinating ideas concerning electricmagnetic

duality seems to b e without much computational consequences and creates the

feeling of lo oking at a faint shadow of a theory Since eective Lagrangians are

just names for certain momentum space Taylor co ecients of correlation func

tions the relation to lo cal quantum physics of these global observations remains

8

unclear More sp ecically eective Lagrangian seem to b e ineective in exp os

ing imp ortant infrared and lo calization structures of the charged dual electric

or magnetic ob jects This state of aairs which is so starkly dierent from the

aforementioned situation cries out for an explanation

It would b e to easy to blame this crisis on the marketing of string theory

9

although statements of some of the prop onents at conferences may tempt one

to do so But things are not that easy String theory is not the cause for the

crisis but rather one of its most remarkable manifestations In my view the ro ots

of the present crisis can b e treated back to the aftermath of the early renormal

ization theory ie to one of the most imp ortant cross roads of mo dern physics

The innities which app eared in QED are not intrinsic they are rather the

result of the repair job one has to p erform on a slightly incorrect quantization

8

In one of those pap ers the author seems to suggest that in order to secure these

duality observations one may need a dierent conceptual framework from that in which these

observations were made

9

The string theory allows to understand everything but compute nothing David Grosss

answer to the question of an exp erimentalist ab out the computational asp ect of TOE

Lagrangian starting p oint The idea of Kramers concerning the distinction of

physical from naked parameters the selfenergy problem which came from

the classical Poincare Lorentz electron theory was extremely imp ortant in or

der to disentangle the physical renormalized answers However these ideas

b ecomes somewhat of a burden like many go o d ideas if misread as a kind of par

allelism b etween classical and quantum eld theoretical ideas ab out the electron

In QFT particles cannot b e intro duced in addition to elds but they are conse

quences of lo cal relativistic elds Wigners characterization of particles as irre

ducible representations of the Poincare group is already an inexorable part of the

unitary representations coming with quantum elds or algebraic nets Whereas

in the classical case one had to add the particle structure via those partial mo d

els of Poincare and Lorentz in lo cal quantum theory there is no such p ossibility

b ecause it is already there Therefore there should b e an alternative formulation

which allows to avoid those innities Indeed the split p oint metho d rst used

by Schwinger and Johnson can b e used to accomplish this As one can dene

free eld Wick pro ducts instead of using global frequency ordering also by lo cal

limits through formulas involving p oint split pro ducts one may similarly dene

comp osite normal pro duct elds which enter the eld equations as the interacting

terms An illegitimate interchange of the limit with the additive and multiplica

tive comp onents of such a normal pro duct expression will bring back the innities

Since the interaction graphical rules are not as elegant there are several typ es

of twop oint functions and not just one typ e as the time ordered propagators

in Feynmans scheme such renormalization schemes never app eared on a su

cient general level in the literature only in simple twodimensional mo dels as the

Thirring or Schwinger mo del they have b een used In general it is much faster to

compute with Feynman rules and repair the illegitimate start by regularization

and renormalization

The wrong turn came ab out by interpreting the subtle and deep relation

b etween classical statistical mechanics and noncommutative QFT to o naively

Whereas statistical mechanics has natural physical cutos this is not the case

in QFT In order to appreciate this remark without prejudices it is helpful to

take a lo ok at the fascinating history of attempts at relativistic nonlocal theo

s

ries and at the notion of elementary length In the it was thought that by

putting form factors into the interaction Lagrangian one could avoid the ab ove

innities and encounter a less singular b ehaviour in correlation function But

it was so on realized Kristensen that through their iteration such form factors

have a disastrous eect on macrocausality their presence wrecks the physical

interpretability of the theory Later Lee and Wick made the prop osal to change

Feynman rules by pairs of complex conjugates p oles which would also soften

the light cone singularities again with the same unacceptable result of acausal

precursors as was shown later The subtle relation b etween euclidean theory for

which certain cutos have physical meaning and QFT holds only after removal

of the cuto This is also in agreement with the previous remark converning

the complete relative nonlo cality b etween lo calization concepts in statistical me

chanics and QFT If a cuto in relativistic QFT would b e more than a formal

intermediate device without any intrinsic physical signicance this would have

b een a truly sensational event The increasing numb er of nontrivial massive lo cal

QFTs Ising QFT SineGordon QFT which have b een constructed via the

nonp erturbative formfactor program starting from factorizing elastic SMatrices

have signicantly reduced the space for folkloric ideas on renormalization More

10

sp ecically the following ideas turned out to b e prejudices

i quantum gravity has to b e invoked in order to make QFT welldened

ii Light cone singularities which go signicantly b eyond the canonical b e

haviour threatens the existence of quantum elds

Anyb o dy who b elieves that a cuto has a physical signicance in a real time

QFT is kindly invited to mo dify one of those lowdimensional mo dels by the im

plantation of a relativistic cuto without wrecking the physical interpretability of

the mo del He will so on get an impression of the scop e of such a problem All

structural investigations of QFT tell us that without Einstein causality the basis

for the physical interpretation of the theory is lost eg there would b e no basis for

the validity of cluster prop erties scattering theory etc Time and again we had

to learn that notions like a little bit nonlo cal as well as a little bit nonuni

tary are as nonsensical as a little bit pregnant at least within the accepted

framework of QFT Only if one go es completely outside standard QFT into the

direction of noncommutative spacetime one nds a chance for replacing causality

by some new structure whose consistency is presently under investigation

It is interesting to note that for certain families of lattice theories cluster

prop erties and hence the existence of multiparticle multimagnon scatter

ing states can b e shown but those are of course not relativistic and moreover

the pro ofs are much more involved than the corresp onding derivations in lo cal

QFT Needless to say that my critical remarks naturally also include one of my

own pap ers I nowadays consider my contribution on the expansion to the

enhancement of physical knowledge as b eing smaller than any preassigned

I will not follow here the path which leads from the original sin of neglect

of the intrinsicness of concepts in QFT to the unscrupulous treatment of es

tablished physical principles in string theory which in my view the path into the

present crisis

Rather I would like to mention some mo dest successes outside this way of

thinking Closely related to the ambitious program of nding nonGaussian xed

p oints of the renormalization group is the more mo dest program of discovering

the physical concepts which determine the sp ectrum of admissible critical indices

in d ie the generalization and extension of Onsagers work on the Ising

mo del Starting from Kadano s Coulomb Gas Representations of critical

10

The idea that theories which are nite in the sense of Feynman p erturbation theory may

also b e physically preferred sup ersymmetry string theory results from this prejudice

indices for sp ecic family of mo dels and going through the discoveries of Belavin

Polyakov and Zamolo dchikov of the minimal mo dels asso ciated to the Virasoro

algebra this program has now reached a certain conceptional depth through

the recognition of critical indices as b eing the numerical values of sup erselection

charges related to the statistical phases of algebraic QFT The latter relates the

sp ectrum to braids and knots It seems that the issue of braid group statistics

is susceptible to a complete classication at least in rational chiral theories

the analogy with the classication of nite depth Jones subfactors is more than

sup ercial

The mo dern p oint of view which analyses chiral conformal QFT directly from

the lo cal eld theoretical principles without the intervention of global Fourier

algebras as the Virasoro or KacMo o dyalgebra remained remarkably close to

the QFT framework in which the present author and collab orators came across

conformal QFT in an observation which I already elab orated in ref

My construction in of the chiral energy momentum tensor as a Lie

eld was inspired by older work on Lie elds starting with Greenb erg and

culminating in the work of Lowenstein These Lie elds in present ter

minology are not necessarily conformal Welds ie conformal causal elds for

which mo dulo a cnumb er Schwinger term the commutators close The case of

a onecomp onent scalar eld was most extensively studied but the exam

ples although structurally close to current algebras bilinear in free elds were

physically not as interesting as the chiral energy momentum tensor not to

mention the more recent development on Walgebra With the additional knowl

edge of the dierential identities b etween elds and current corresp onding

to the mo dern KnisknikZamolo dchikov identities and the old conformal decom

11

p osition theory the question arises why was the richness of chiral conformal

eld theories minimal mo dels etc not seen already at the b eginning of the s

Partly the reason was in my opinion that the condence with which physicist

investigate low dimensional QFT nowadays was lacking at that time Contribu

tions in lowdimensional eld theory were often buried in conference rep orts

since they did not enjoy much appreciation Another p erhaps even more imp or

tant reason is that a systematic mathematical study of representation of innite

dimensional algebras was not yet available Although mathematicians use more

global metho ds Fourier co ecients of Ts and js the global Verma mo d

ule approach to unitary representations was imp ortant also for lo cal quantum

physicist since it generated the missing mathematical condence The rupture

b etween the old physically oriented and the new geometrically inspired QFT

often more the result of neglect of the past rather than deep dierences in phi

losophy o ccurred during the second half of the s This in my view unique

th

phenomenon in the history of century physics is one of contributing factors

to the present crisis

11

For a comparitative study of the old versus the new approach see ref

Recently a clever strategy of p erturbing chiral conformal QFT has b een

found by Zamolo dchikov Here p erturbation is not taken literally otherwise

one would have to handle terrible infrared divergencies but rather serves as

the starting p oint for consistency arguments which select the integrable mas

sive representatives These representatives are probably identical to the theories

which corresp ond to the large distance limit of the Smatrix in the sense of section

There remains the interesting question of the numb er of massive sup erselec

tion classes according to section equal to the numb er of factorizable mo dels

which corresp ond to one conformal limit For example it is known that the con

formal Ising mo del allows for two dierent integrable p erturbations and there are

indications that these two p ossibilities are already preempted through two dier

12

ent interpretations of the Ising partition functions These are op en structural

problems where one exp ects algebraic eld theory to contribute

It is interesting to note that Zamolo dchikovs construction picture runs

opp osite to the renormalization group extrinsic approach Our starts in a com

pletely intrinsic matter from a classication of rather simple theories as sketched

ab ove and moves away towards more complicated theories by applying structural

selfconsisting arguments on relevant p erturbing elds which are chosen from

the list of comp osite elds of the chiral theory It seems that for practical pur

p oses the way from simple mo dels to more complicated ones is more constructive

than the renormalization group approach The latter is to o structureless for I

have not seen any construction of a nonGaussian xed p oint except the illfated

expansion But on the other hand the algebraic QFT p oint of view and

Zamolo dchikovs approach which is very close is presently limited to d

even though it uses concepts which are quite general There is no reason to b elieve

that the sup erselection structure will determine critical indices in d

Some of the remarks made in this section should b e viewed as a qualication of

my critical comments made at a round table discussion Physics and Mathemat

th

ics XI International Congress of Mathematical Physics International Press

Daniel Iagolnitzer Editor

Duality and Mo dular Prop erties of Chiral

Field Algebras

It has b een known for some time in algebraic QFT that certain nontrivial in

clusions of observable algebras in the vacuum representation contain valuable

informations ab out sup erselected charges In the following we will describ e three

dierent typ e of inclusions

i The coronainclusion III of ref in d massive QFT with

12

I am indebted to AA Belavin and A Fring for discussions on this p oint

a mass gap

AK AK AC

0 1 0

Here K is a centrally lo cateddouble cone which is surrounded by a double

1

conering C and K is the ringlike wedge region consisting of all p oints which

are spacelike relative to C and K With other words K K is causally

1 1

disjoint from the ring C The op erator notation is standard is the vacuum

0

representation and the upp er dash denotes the commutant of an op erator algebra

Note that the inclusion is prop er ie there is an obstruction against equality

Haag duality can only b e recovered by extending the algebras in the case of the

algebra lo calized on the corona this presumably can b e achieved by dening a

suitable halfspace radial disorder op erator In case of the existence of only a

nite numb er of DHR sup erselection sectors the DR theorem tells us that this

inclusion is equivalent to that dened by the xed p oint algebra under a double

of a nite group G F is the DR eld algebra

GG

AK AK F K F K j H

0 1 0 1 0

D iag (GG)

AC F K F K j H a

0 1 0

GG D iag (GG)

F K F K j H F K F K j H b

1 0 1 0

In other words the DR group symmetry of d chargecarrying elds

is already visible on the level of neutral observables through The equality

holds if instead of observables we use eld algebras in particular for free elds

ii The quarter circle inclusion of d chiral conformal QFT

AI AI AI AI

0 1 0 3 0 2 4

Here I i are four intervals obtained by equipartitioning the circle ie the

i

13

lo calization space of chiral conformal theories This inclusion is selfdual in the

sense that it is equivalent to the inclusion obtained by exchanging I and I with

1 3

I and I Therefore it is not surprising that apart from nite ab elian groups

2 4

we again restrict our interest to the case of a nite numb er of sup erselection

rules the quantum symmetry of sup erselection sectors of chiral theories is not

covered by groups This is the case which we will treat in this pap er

The theories with a mass gap d fulll a similar formula with the

intervals b eing replaced by spacelike cones

13

It is the chiral analogue of the corona inclusion

iii The emcorona inclusion for d Maxwel lianl ike free elds

The formula is the same as ie

AC AC

0 0

but now AC is the algebra generated by zero mass neutral spin elds electro

magnetic elds ie the obstruction is characteristic for m and s In

particular the chargeless free electromagnetic elds leads to a prop er inclusion

without nontrivial sup erselection sectors which one could blame for this duality

obstruction The equality sign can b e formally recovered by using indenite met

ric gauge p otentials but the use of unphysical gauge elds in indenite metric

spaces with BRST conditions is against the spirit of algebraic QFT The latter

prefers to deal directly with the cohomological problem hidden b ehind the prop er

inclusion instead of articially removing it at the exp ense of intro ducing vector

14

p otentials For more details on this inclusion see ref It is b elieved

that the existence of nontrivial electric and magnetic charges ie Maxwelllike

interactions will remove this obstruction against Haag duality It is interest

ing to note that the orderdisorder duality structure in low dimensional QFT

do es not p ermit the simultaneous absence of order and disorder charges Due to

the unsolved infrared problems of algebraic QFT this sub ject has not yet b een

thoroughly investigated in d theories

I b elieve that a lo cal algebraic approach to electro magnetic duality ie

b etween F and F as in the Seib ergWitten mo dels and in particular the

construction of the e and m charge carrying op erators which are resp onsible

for this duality should start from these observations Sup erselection rules arising

from F and F are exp ected to b e very p eculiar and related to the problematization

of the notion of magnetic elds and quark connement The charge carriers

of such dual charges must b e essentially dierent from those in the DHR theory

The lo calization prop erties are exp ected to b e so weak that standard covariant

eld description are presumably not applicable We will not pursue this matter

here

We now return to the quarter circle situation ii In that case one proves the

following theorem

Theorem The vacuum representation of the abelian current algebra mul

ticomponent Weyl algebra violates Haag duality for the quarter circle inclusion

and full ls instead the fol lowing lattice duality



I I A I I A

2 4 L 1 3 L

The von Neumann algebras on b oth sides are generated from Weyl algebras over

1

real test functions of subspaces of C S V Let S I I denote the subspace

L 1 3

14

Although vector p otentials can b e avoided in all structural arguments of algebraic QFT

they do enter in the Lagrangian quantization approach and in the intermediate step of the

ensuing standard p erturbation theory of physical observables

consisting of real test functions which are constant in I and I and fulll the

2 4

condition f z f z L z I where L even lattice in V Then

2 4 24 24

A I I Al g fW f f S I I g

L 1 3 L 1 3



I I with L b eing the dual lattice to L and similarly for A

2 4 L

Note that only in the case of a selfdual lattice one obtains the standard form

of Haag duality

The content of the theorem and its pro of b ecomes clearer if one maximally

extends the multicomp onent Weyl algebra Here maximally means that no

further b osonic lo cal extension is p ossible It is wellknown that such maximal

extensions corresp ond to even lattices in the sense that the functions f in the

1

Weyl generators W f fulll quasip erio dic lattice b oundary conditions on S

instead of b eing p erio dic This more general ane set of functions still maintains

ext

the causal net structure but the extended net A has a nite numb er namely

L

Lj sup erselection sectors instead of continuously innitely many Collect jL

ing the results in terms of theorems we have the following

ext

full ls the fol lowing Theorem The maximal ly extended observable net A

L

extended duality relation

(0)

ext

I I A I I F



2 4 1 3 L

L

L

Here the left hand side is the commutant of the Lextended observable algebra

in its natural vacuum representation The right hand side is the eld algebra

L

restricted to the chargeanticharge split total charge neutral indicated by the

sup erscript subalgebra and pro jected onto the vacuum sector

The pro of of the second theorem is based on the rst one Both pro ofs are

easy and natural if one adapts the framework of Buchholz Mack and To dorov

and will not b e given here This extended duality is the prerequisite for a

geometric form of the TomitaTakesaki mo dular theory for the quarter circle sit

uation Here we will only present an application of that theory to the case of no

nontrivial sup erselection rules ie for selfdual lattices or for the CAR fermion

algebra Let us take the sp ecial case of chiral CAR algebrasThen we have

Theorem based on Arakis analysis of CAR algebra

iht

Let U t e be a one parametric group of dieomorphism acting as Bogoliubov

1

automorphisms on the wave function space underlying the CAR S C algebra

Then there exists an associated KMS state If the dieomorphism has a nite

number of xed points then this KMS state has a natural factorial III restriction

1

to a subalgebra

In this way one exp ects to obtain a geometric mo dular theory for this sub

algebra and a suitable state ie one hop es there exists a state which together

with the subalgebra has the given dieomorphism group as the TomitaTakesaki

automorphism group

Instead of showing how a pro of can b e abstracted from Arakis work we

mention two physically interesting sp ecial cases of dieomorphisms

z D il tz as the dieomorphism

In this case the situation of the theorem coalesces for with the mo dular

theory of the semicircle CAR algebra and the vacuum state vector For

the restricted KMS state b elongs to another folium of the simple C

algebra CAR dierent from the vacuum folium

1

2 1

z T D il tT with T z z covering of S

2 2

2

This dieomorphism has four xed p oints and b elongs to the quartercircle situa

tion Instead of the semicircle CAR algebra one now considers the one b elonging

to I and the state vector is not the vacuum but it is rather dened implicitly by

1

the GNS construction within Arakis CAR theory If the vacuum folium contains

a KMS state for this dieomorphism group acting on CAR I then the

1

state must b e the so constructed one So in order to maintain a geometric mo d

ular theory for general dieomorphisms one should allow reference states which

are dierent from the vacuum and outside the vacuum sector

To recover Haag duality in its original form we must construct the eld

1



S for the extended current algebra mo del This algebra contains algebra F

L

all lo calizable charge carries and it fullls the following twisted Haag duality

tw

 

I I F Theorem F

L L

Here the sup erscript on the right hand side denotes the twisted commutant In



our case of Z ab elian mo dels the twist is a generalization of the fermionic

jL Lj

2 iL

0

twist namely a suitable square ro ot of the rotation e

Up to now we have b een dealing with ab elian anyonic representations

of braid group statistics level multicomp onent current algebras How do es

one construct plektons Since a quantum symmetry generalizing the compact

group symmetry do es not yet exist a conservative substitute for the previously

discussed ab elianization of paraones which leads to b osons and fermions ie

the DR theorem would b e to try to anyonize plektons In the following we

will sketch such an attempt

Let us start again from the multicomp onent current algebra It is wellknown

that the current algebra in the Weyl form denes an orthomo dular functor

from real Hilb ert subspaces H of the multicomp onent complex wave function

R

2 1

space H L S V into von Neumann subalgebras N of B H

F ock

H N H a

R R

N H b N H H

R

R R

denotes the symplectic complement of H with the symplectic form Here H

R

R

f g b eing given by the imaginary part of the scalar pro duct

Z

2

f g I mf g g f d

0

H is called standard if H iH and H iH is dense in H In that

R R R R R

case N H turns out to b e a factor in standard position with the vacuum state

R

vector in Fo ckspace b eing a cyclic and separating reference vector

If the real subspace consists of real multicomp onent functions lo calized in an

1

interval I S the TT mo dular theory is geometric and the mo dular auto

morphism is the dilation which leaves the endp oints xed whereas the mo dular

conjugation is an antilinear reection of the algebra N H I onto N H I

R R

Both mo dular op erations are the functorial image of op erators in H Note that

localized real subspaces are automatically standard This is the wave function

preversion of the ReehSchlieder theorem for the lo calized Weylsubalgebras act

ing on the vacuum It also may b e viewed as a prop erty of coherent states in

Fo ck space aliated with lo calized real wave functions

The previous maximal lo cal extension by an even lattice amounts to an ex

tension of this functor to real ane spaces

af f

H H L

R

Here L is an even lattice which app ears in the quasip erio dic b oundary conditions

of the real multicomp onent wave function The extended symplectic bilinear form

is

R

2

1 1

f hf g i hf g i hf g g i hf g id

0

4 4

f f g g

f g

where L is a lattice vector which is determined by the quasip erio dic b oundary

conditions

So the original Hilb ertspace H b ecomes augmented by a lattice and the ex

tended functor denes a map from real ane subspaces to von Neumann algebras

which are subalgebras of the charged eld algebra The latter is a kind of crossed

pro duct b etween the previous neutral observable algebra and an ab elian group

algebra representing L The lo calized subalgebras A I corresp ond to rather

L

complicated real ane wave function subspaces which are not linear sub

spaces

In this sense the aforementioned maximally extended multicomp onent current

algebras are still interpretable as extended Weyl algebras

af f

We again emphasize that H is not simply the tensor pro duct of the CCR

with the rotational lattice C algebra but rather the lattice acts on the lo cal

ization function via b oundary condition

Within this functorial calculus one can form diagonal generators in the k fold

tensor pro duct

(k ) ext ext

W W f W f

ext(k )

They generate a subalgebra A of the k fold tensor pro duct

L

It is wellknown that the level one SU N lo op group representation is gener

ated by the extended Weyl algebra for a particular lattice L the ro ot lattice of

SU N Let us therefore reinterpret the ab ove tensorpro duct formula for those

sp ecial group lattices as

(k ) (1) (1)

W W g W g with g lo opgroup

(1) (k ) (k )

with W g lo op group representation of level Then W W

By reduction we now could construct higher level lo op vacuum representation

which are known to lead to plektonic nonab elian nets But such an ad ho c pro

cedure invoking the lo op group and its asso ciated net is not quite in the spirit of

algebraic QFT I b elieve that at least for nets with a nite numb er of sup erselec

tion sectors rational chiral theories there exists a natural plektonic extension

(k )

of the tensor pro duct algebra generated by the W s in formula My con

jecture is based on the observation that the lo op group nets and the W nets are

the only irreducible plektonic families which implement the classied statistics

families a remark which generalizes the structural discussion of plektonic p oint

functions in

We ap ologize to the reader for these complicated technicalities They were

only meant to supp ort the hop e that chiral conformal QFT can b e classied by

purely intrinsic QFT principles valid in any dimension without reference to

particular algebras as KacMo o dy or W algebras Such messages are imp ortant

if one b elieves that the main physical role of chiral theories is that of a theoret

ical lab oratory for general QFT Algebraic QFT is designed to cop e with nets

which are far away from the standard CCR and CAR situations and their p er

turbative interactions The latter can b e considered as function spaces with a

nonab elian C structure ie as lo calizable functors of Hilb ertspaces into von

Neumann algebras Algebraic nets are supp osed to cover the area beyond such

lo calizable function algebras ie situations in which the lo cality principle is not

implemented through functions but in a more noncommutative fashion But in

order to explicitly construct examples b eyond CCR and CAR one has to start

from the latter b ecause these are the only nets which we thoroughly know

Constructive Attempts for d = 2 + 1 Anyons

The most natural idea for the construction of free elds with braid group

statistics is the adaption of Wigners representation theory of the Poincare group

to this problem Since the little group is the ab elian U the Wigner wave

function are onecomp onent functions on the mass hyp erb oloid which transform

under Lorentztransformations as

is (p) 1

W

U p e p

where the Wigner phase p is a computable real function which as a result

W

g

of R SO represents an element of the covering of the Lorentzgroup In

case of charged particles one has to double the wavefunction space in order to

obtain a representation of TCP

In the next step the Wigner theory has to b e extended by a lo calization

concept which was not known to Wigner ie dierent from the NewtonWigner

lo calization One considers real subspaces H of H similar to those of chiral

R

current algebra of the previous section The relevant concept of causally disjoint

lo calization is then dened in terms of the symplectic complement using the

symplectic form

anti

f g I mf g H H H

H W

W

Without lo cal elds it is generally very dicult to explicitly nd these lo cal

subspaces H O which are asso ciated to regions example double cones O in

R

d Minkowski space

However there is one exception the Rindler or BisognanoWichmann wedge

regions For wedges the mo dular TomitaTakesaki theory b ecomes geometric On

the Wigner space this theory admits a kind of pre or rstquantizedversion

it

Instead of the mo dular conjugation J and the mo dular group op erators

it

in Fo ckspace we have op erators j and acting geometrically on the Wigner

wave function space The preTomita op erator

12

s j

2

is an unb ounded closable involutive s op erator on H which serves to

dene a real subspace H wedge as the eigenspace of s

R

H w edg e closure of real linear span f H s sg

R

The s is explicitly known since can b e computed from the Lorentzb o ost which

conserves the wedge and j is obtained from the TCP op erator on H by splitting

o a rotation around the xaxis in case of the standard t x wedge

In the standard d Wigner theory this idea was recently used to

dene lo cal nets in a direct fashion ie without lo cal elds as intermediaries

Since double cones can b e obtained as intersection of wedges the double cone

algebras can b e dened in terms of intersections of wedge algebras and the isotonic

and causal structure of this so dened net follows without reference to lo cal elds

It is very interesting to extend these ideas to d anyonic spin

representations In this case the statistical phase b elonging to the Wigner

spin s integer shows up as a mismatch b etween the j transformed real subspace

the symplectic complement and the geometric opp osite wedge obtained by a

spatial rotation

j H wedge H oppwedge

R R

The op erator which removes this mismatch is related to a Kleintransformation

In case of eg Z anyons this Kleintransformation T is again a suitable square

N

ro ot of the rotational in Theorem

A signicant dierence b etween the semiinteger spin case and genuine frac

tional spin app ears if one investigates the p ossibility of having sharp er lo cal

ization than just wedge lo calization It turns out that for the fractional case no

compact lo calization is p ossible ie certain intersection of H wedgespaces are

R

empty In those cases one exp ects the semiinnite spacelike string lo calization

to b e the b est p ossibility Therefore one lo oks for stringlike lo calized elds

The standard way to obtain elds from the Wigner wave functions is to fac

torize Wigners wave functions into covariant wave functions times a generaliza

tion of the u and v spinors Candidates for covariant wave functions in d

were rst prop osed by Mund and Schrader

1

pg F L pg p

cov w

g

with F a yet unsp ecied kinematic function on the covering group SO

which has the following equivariance law with resp ect to the little group

isw

g

F w g e F g w R S

Indeed this equivariance leads to the covariant transformation

1

U g p g g p g g

cov cov

Intro ducing momentum space creation and annihilation op erator and covariant

op erators in xspace according to

Z

2

d p

ipx ipx

x g e ap g e a p g

w

we have

+

U g x g U g g x g g

and

Z

2

d p

1 1 ip(x x )

1 2

x f x g F L pg F L pg e

1 1 2 2 1 2

w

Cho osing x and g opp osite to x g ie

2 2 1 1

x R x R rotation by

2 1

g R g

2 1

one obtains the statistics factor in agreement with our previous wedge lo caliza

tion

2 is

x g x g e x g x g

1 1 2 2 2 2 1 1

In order to obtain elds which have a stringlike lo calization one has to sp ec

ify appropriate functions F As Fredenhagen and Gab erdiel showed the

following sp ecication leads to lightlike strings

ig (0)s

F g e

g

1

with g u R for u R S SO and g acting in the natural way as a

fractional SO transformation

g w

iu

e

w u w ar g

iu

e

These lightlike strings should b e the singular limit of more natural spacelike

strings but a detailed investigation of the letter is still missing Presently it

is not clear if the previous twop oint function which was obtained from the

Wigner theory b elongs to a spacelike cone lo calized QFT But even if the two

p oint function of free anyons has a supp ort b eyond the mass shell the mo dular

lo calization metho d should reveal why this is so

A precise comparison with the results obtained on the basis of a ChernSimons

picture ie a ChernSimons vector p otential interacting with spinor free matter

is dicult b ecause the Chern Simons approach has not b een p erused far enough

in order to obtain op erators fullling braid group statistics However on a very

formal level it app ears that the mo died spinor matter elds are of the following

typ e

+

bilin( )

0

0

e

0

This is similar to the formula of the order parameter in the d massive

Ising eld theory Such a structure would have a more complicated combinatorics

than the previous Wigneranyons Whereas the latter are elementary ob jects

the ChernSimons anyons are composite represented in the Fermion Fo ckspace

I presently have no comment on this discrepancy except the obvious remark

that ChernSimons anyons are p erhaps not the simplest p ossible realization of

d braid group statics

The problem of constructing genuine d free plektons by algebraic

metho ds is at least as complicated as that anyonization of conformal plektons

of the previous section and there are presently no constructive ideas This area

of research promises to b e interesting lively and controversial for some time to

come

Concluding Remarks

In this work we argued that algebraic QFT presently plays its most constructive

role in d QFTs with plektonic statistics were Lagrangian metho d fail

For d QFTs the algebraic metho d suggests two areas of p otential

progress a reinvestigation of p erturbative problems in the sense of section

including an intrinsic understanding of interaction indep endent of the use of

sp ecial eld co ordinates and a b etter understanding of the physical principles

b ehind gauge theories

Algebraic QFT always maintained a critical distance to an alleged lo cal gauge

principle Therefore there is no particular reason for an algebraic eld theorist

to react towards some new paradoxical lo oking situations eg the alleged clash

b etween the Seib ergWitten duality and the lo cal gauge principle with

surprise To the contrary a critical evaluation from an algebraic rather than

a dierential geometric standp oint would lead one to b elieve that among all

attempts to press a new physical reality into the quasiclassical straightjacket

of Lagrangian quantization lo cal gauge theories is the most useful one The

formal gauge idea suggests a new p otentially fruitful concepts of algebraic QFT

the enhancement of degrees of freedom in asymptotic limit theories asymptotic

freedom

Earlier attempts to problematize the notion of magnetic eld in terms of

op erator cohomology on the same level of depth as the problematization of the

notion of charge in the theory of sup erselection sectors which can b e reinter

preted as lo cal op erator cohomology have not met with much success

In the light of recent observations on the em duality in sup ersymmetric QFTs

the main problem from the algebraic p oint of view is the understanding of a lo cal

physical prop erty of the net generated by the eld strength F which excludes

the free phase ie excludes the simultaneous absence of electric as well as

magnetic charges The free electromagnetic eld has as we have seen a p eculiar

obstruction not shared by other massive free elds It violates Haag duality for

top ologically nontrivial regions It is b elieved that F s with nontrivial charges

v

may not show this obstruction For dual orderdisorder situations in d

theories such a criterion would not b e necessary since the case that b oth dual

variables leave the vacuum sector invariant ie do not create charges is au

tomatically excluded But a classication of sup erselection rules coming from

Maxwellianlike e and m charges remains dicult since the bad infrared b e

haviour prevents a lo calization of charges in arbitrarily thin semiinnite spacelike

cones

The lesser p opularity of algebraic QFT as compared to the geometric quan

tization approaches seems to b e a result of its conservative attitude with resp ect

to inventions

However underneath the conservative surface of the algebraic metho d hides a

revolutionary new way of lo oking at QFT a change of paradigm with resp ect to

Lagrangian quantization In more philosophical terms the latter complies with

the Newtonian way of understanding physical reality a spacetime manifold with

a material content The net p oint of view of algebraic QFT with its relations

and inclusions b etween isomorphic lo cal algebras hyp ernite III von Neumann

1

factors on the other hand harmonizes much more with Leibniz view of reality

as coming ab out through the relations of monades the hyp ernite III von

1

Neumann factors representing the lo cal algebras This is of course also the

underlying philosophy ab out mathematics in the Jones subfactor theory

Lo oking at present QFTrelated theoretical issues it is not dicult to nd a

rough division into three topics Renormalization group QFT String theory and

Algebraic QFT

The rst one has led to a philosophy that one should b e happy with cuto

Lagrangians The new asp ect of the philosophy underlying the renormalization

group in QFT is not that it states that reality is like an innite onion each layer

has its principles only to b e sup erseded by the next one Physics was always like

this and old principles were always eventually rendered limiting cases of new ones

Rather it resides in the message that to strife for conceptual and mathematical

consistency in each layer is not required any more since our lack of knowledge

ab out the next layer can b e summarily taken care of by cutos even though

this violates our present principles With other words the principles with which

this century had such a glorious start are an illusion eective theories with

cutos are the new goal This ideology lo oks mo dest apart from its tendency to

lab el reductionist viewp oints as the one in this article as scientic arrogance

page To me it app ears as the physicists analogue of Fujikawas so cio

economical ideology

The second topic namely string theory will have to make a big eort in order

not to end as the mathematically most useful physical phlogiston theory of the

th 15

century Rampant mathematics unbridled by physical principles

and concepts do es not seem to b e the way out of the most profound crisis

Areas close to ones own research one usually criticizes more mildly However

15

With the following qualications added I partially share this p essimistic outlo ok Although

the emduality concepts are touching up on very deep physical issues I think that the global

quasiclassical metho ds as well as the more recent dierential and algebrogeometric consistency

approach are limited to some very sp ecial global features of sup ersymmetric duality without

any presently visible relation to the problem of emduality in d lo cal quantum physics

In this article I alluded to the latter problem in terms of the more noncommittal terminology

as the problematization of the magnetic eld on the same level of conceptual profoundness

as the successfull problematization of charge in the theory of sup erselection sectors

In a very optimistic p oint of view the aforementioned global attempts parallel the global

Kramers Wannier lowdimensional orderdisorder duality and the lo cal understanding can b e

hop ed for as arising eg from progress in zero mass lo calization concepts in QFT telling us

why we should go b eyond the semiinnite space like string lo calization of the rst section

But lo oking at the present so ciology in physics it is very dicult to entertain such optimism

The fact that string theory has created a new theoretical realm which can b e understo o d

without profound knowledge of the development of quantum eld theory may b e the reason

why it attracts so many young physicists and has led to a globalization of the market with its

cleansing eects on long term intellectual investments One may b e afraid that there is little

ro om left for thinkers in the Bohr Heisenb ergian spirit who are prepared to make that big

investment which could reconquer the lost conceptual terrain

it is not a secret that algebraic eld theorist are a bit like religious preachers

Their everyday life teaching courses on QFT by using a year old formalism

is often quite apart from those high conceptual values they preach My hop e

is that this will change in the near future Algebraic QFT is attractive to me

b ecause it remained a family activity and one do es not feel comp elled to make

unfulllable claims

There is an interesting historical precedent that a conservative approach ie

one which is faithful to physical principles but revolutionary in their implemen

tation may at the end b e the more successful one I am again referring to the

discovery of renormalization theory in which did not use any of the

many inventions of those days but just consisted in an elab oration of a conserva

tive ideas and the pursuit of its intrinsic logic For a very informative account I

refer to Weinb ergs new textb o ok

Acknowledgements

Most of this work was carried out at the Universidade Federal do Espirito

Santo Brazil I am indebted to Prof JC Fabris for the invitation and to my

colleagues at the UFES for their helpful attitude I also feel obliged to thank for

the hospitality at the ESI in Vienna where I enjoyed many stimulating discussions

ab out the the content of this work in particular with Prof R Stora

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