Localization and the interface between quantum mechanics, quantum field theory and quantum gravity I (The two antagonistic localizations and their asymptotic compatibility)

Lo calization and the in terface b et w een quan tum mec hanics, quan tum ﬁeld theory and quan tum gra vit y I (The t w o an tagonistic lo calizati o ns and their asymptot i c compatibilit y) dedicated to the memory o f Rob Clifton Studies in History and Philosophy of Mo dern Physics 41 (2010) 104–127 Bert Sc hro er CBPF, Rua Dr. Xa vier Siga ud 150 22290-1 80 Rio de Janeiro, Brazil and Institut f uer Theoretisc he Ph ysik der FU Berlin, German y Octob er 2 9, 2018 Con ten ts 1 In tro ductory remarks 2 2 Historical remarks on the in terface b et w een Q M and QFT 9 3 Direct particle in teractions, rel ativistic QM 11 4 First brush with the in tricacies of the particles-ﬁeld probl ems in QFT 16 5 More on Born v ersus cov arian t lo calization 19 6 Mo dular lo cali zation 29 7 Algebraic asp ects of mo du l ar theory 35 8 String-lo calization and gauge theory 45 9 Building LQP via p ositioning of monads in a H i lb ert space 52 1 10 63 Abstract It is show n th at there are signiﬁcan t conceptu al diﬀerences betw een QM and QFT whic h make i t diﬃcult to view the latter as j ust a relativistic extension of the principles of Q M. At the root of this is a fundamental dis- tiction b etw een Born-lo calization in QM (which in the relativistic context changes its name to Newton-Wigner lo calization) and mo dular lo c alization whic h is the localization underlying QFT, after one separates it from its standard presentation in terms of ﬁeld co ordinates. The ﬁrst comes with a probabilit y notion and pro jection operators, whereas the latter describes causal propagation in Q FT and leads to thermal asp ects of lo cally re- duced ﬁnite en ergy states. The Born-Newton- Wigner localization in QFT is only app licable asymptotically and the cov arian t correlation b etw een asymptotic in and out lo calization pro jectors is th e b asis of the existence of an inv arian t scattering matrix. In this ﬁrst part of a tw o p art essay the modular localization (the intrinsi c content of ﬁeld lo calization) and its philosophical consequences take th e center stage. Imp ortant physical consequences of vacuum polar- ization will b e the main topic of p art I I. Both parts t ogether form a rather comprehensive p resenta tion of k now n consequ ences of th e tw o antagonis- tic localization concepts, including the those of its misunderstandings in string theory . When the ﬁrst version of this pap er wa s submitted to hep- th it w as immediately remov ed and against its QFT conten t placed on phys. gen without any p ossibility to cross list. 1 In tro duc tory remarks Ever since the discov ery o f quan tum mechanics (QM), the conceptual diﬀer- ences b etw een classica l theory and Q M hav e b een the sub ject o f fundamental inv estigations with pro fo und physical and philosophica l consequences. But the conceptual relation betw een quantum ﬁeld theory (QFT) a nd QM, whic h is at least as challenging and r ich of surprises, has not received the same amount of attention and scrutiny , and often the s ubsuming o f Q FT under ”r elativistic QM” nour ished prejudices and preven ted a c r itical foundationa l debate. Apart from some admirable work on the signiﬁcant changes which the theory of mea- surements must undergo in order to be consis tent with the structure of QFT, which emanated fro m p eople who are or hav e b een aﬃliated with the Philo sophy of Science Departmen t of the Univ ers ity o f Pittsbur g h [1][2][3], as well as s ome related deep mathematical a nd conceptual work from q uantum ﬁeld theo rists [4][97][98], this sub ject has remained in the mind of a few individua ls working on the fo unda tions of QFT and is still far from b eing pa rt of the collective knowledge of the foundation of QT co mmu nity . Often results of this kind which inv olve adv a nced knowledge of Q FT do no t attract muc h attention ev en when they hav e b earing s on the foundations of QT 2 as e.g. the issue of Bel l states in lo c al quantu m physics (LQP 1 ) [6] or the imp or- tant relations b etw een caus al dis joint edness with the existence o f unco rrelated states as well as the issue to what e xtent ca usal indep e ndence is a consequence of statistical independence [7]. The reason is no t so muc h a lack of in tere s t but rather that QFT is often thought to b e just a kind of relativistic q uantum mechanics and that po ssible diﬀerences are of a tec hnical nature. This may explain why there has b een an ama zing lack of balance b etw een the very de- tailed and sophisticated liter ature ab out interpretational a sp e cts of Q M and its relation w ith qua n tum information theory (aiming s ometimes at some very ﬁne, if not to say a c ademic/metaphoric p oints e.g. the multiw or ld int erpr etation), and the almost complete lack of profound interpretive a ctivities ab out our most fundamen tal q ua nt um ﬁeld theor y of matter. A lthoug h the name QT us ually app ears in the title of foundationa l pa p er s, this mostly hides the fact that they deal exclusively with concepts from Q M leaving out QFT. If on the other ha nd some foundational motiv ated quantum theorist be c ome aw are of the deep conce ptua l diﬀerences between particles a nd ﬁelds, they tend to lo ok at them as antagonistic and create a battleground; the fact that they are fully compatible where for physical reaso ns they must a gree, namely in the asymptotic r e g ion of s cattering theory , remains often uncommen ted. The aim of this ess ay is to show that at the r o ot of these diﬀer e nces there are tw o lo caliza tion concepts: the quantum mechanical Bor n-Newton-Wigner lo calization and the mo dular lo calization of LQP . The BNW lo ca lization is not Poincar´ e cov ariant but attains this prop erty in a ce r tain a symptotic limit namely the o ne o n which scattering theory is founded. Modula r lo calization on the o ther hand is caus al and cov ariant at all dis tances but pr ovides no pr o jectors on subspaces as they arise from s p ectr al decomp ositions of selfadjoin t o r unitary op erator s , instea d the linear spaces of lo c alized states are usually dense in the Hilber t spa ce of all states . One of the a ims o f this article is to collect some facts which, somewhat ov ers impliﬁed, show that b esides sharing the notio n of Hilber t space, op er a tors, sta tes and Planck’s constant ℏ , Q M and QFT are conceptually worlds apar t 2 and yet they har monize p erfectly in the asymptotic region of scattering theory . F or a long time the subtle distinction b etw een the non- cov aria n t BNW lo- calization (based on the exis tence of a p osition o p er ator) and the autonomous cov aria n t lo ca lization concept of QFT was insuﬃciently understo o d. It has b een claimed (pr iv ate communication by Rudolf Haag) that the reaso n for Wigner , who together with Jo rdan s igniﬁcantly enriched QFT, to b ecome later disen- chan ted with this theo ry was that he failed to obtain a cov ariant lo calization 1 W e use this terminology instead of QFT if we wan t to dir ect the reader’s atten tion aw ay from the textbo ok Lagrangian quan tization to wards the underlying pri nciples [5]. QFT (the con ten t of QFT textbo oks) and LQP deal wi th the same physical pri nciples but LQP is less comitted to a particular f ormalism (Lagrangian quan tization, functional int egrals) and rather procures alwa ys the m ost adaequate mathematical concepts for their implement ation. It includes of course all the results of the standard p erturbative Lagrangian quantizat ion but presen ts them in a concept ually and m athematically more satisfactory wa y . Most of the sub jects in this article are outside of textbook QFT. 2 F or more illustrations of this p oint see the concluding remaks in part II. 3 concept whic h w as able to directly connect his repr e sentation theory of the Poincar´ e group with QFT 3 . The application of the no n-cov ar ia nt and hence frame-dep endent BNW lo ca lization to ﬁnite distances leads to incorrect results in particular to supe rluminal phenomena. But only after the publication of an article [8], in which it was claimed tha t F ermi’s re s ult a b out his tw o-a tom Gedankenexperiment contradicts understanding s ab out spacetime lo ca liz a tion and signa l propaga tion, ca me the iss ue of BNW- and mo dular - lo calization to a climax. The editor of Natur e at that time wro te an article, in w hich among other asp ects of superluminal propaga tion, the issue of time machines was discussed. After this there was no holding back; an a v alanche of ar ticles ab out sup er- luminal journeys and time ma chines entered the worldpress; the fact that the original article a pp ea red in a highly reputable jo urnal, and its conten t was r e - inforced a nd repr o cessed for a scientiﬁcally interested public by a well-known scientiﬁc journalist, made the topic irr esistible to the general public, esp ecially since it had the right mixture o f trustw or th y origin with sensationa l co nt ent. F ortunately this was not the end of this episo de. The sa me journal which published the ar ticle based on the use o f BNW lo calizatio n accepted a second article [9] in which F ermi’s conclus ions a b o ut ﬁnite propaga tion sp eed were reinforced on the basis of mo dular lo c alization 4 . This episode underlines the subtlet y of lo calization in QFT and most of the conten t of b oth parts of this essay will consis t in explaining why this is such a delicate problem which led to many misunders ta ndings. It is not our inten tion to present a new axiomatic setting (for an older pre- sentation of the e x isting one s ee [5]). Such a go al would b e to o ambitious in view of the fact that w e are confr o ntin g a theor y where, in c o ntradistinction to QM, no conceptual clo sure is yet in sig ht. Although there ha s b een some r e- mark able nonp erturbative pro gress concerning c o nstructive control (i.e. so lving the existence problem) of mo dels based on mo dular theory , the main knowledge ab out mo dels of Q FT is still limited to numerically succe s sful, but nevertheless diverging p ertur ba tive ser ie s . Here the mo re mo dest aim is to collect some either unknown or little k nown facts whic h co uld present so me foo d fo r thoughts a bo ut a more inclusive mea- surement theory , including all of quantum theory (QT) end not just QM. On the other hand one would like to improv e the under standing ab out the in terface betw een QFT in CST c urved spacetime (CST) and the still e lusive quantum gravit y (QG). Since both expres s ions QFT and LQP are us ed do denote the same theory , let me emphasize aga in that ther e is no diﬀerence in the physical aims since LQP originated from QFT a nd incorp orated all concepts and c o mputational results of QFT including renorma lized p er turbation theory; LQP is used in- stead of Q FT whenever the conce ptual level o f the pr esentations g e ts be yond what the reader is able to ﬁnd in standa rd textb o oks o f QFT, mor e sp eciﬁcally 3 This is precisely what mo dular l ocalization achiev es (section 6). 4 In the publications during the 9 0s, this terminology wa s not yet a v ailable. It w as suﬃcient to si mply think in terms of the kind of lo calization which i s intrinsic to pointlik e co v ariant ﬁelds. 4 whenever one is interested in no nper turbative mathematica lly controlled co n- structions of mo dels in terms of intrinsic (”ﬁeld-co or dinatization indep endent”) structures. There is one recommendable exception, namely Rudolf Haa g’s bo o k ”Lo cal Quantum P h ys ic s” [5]; but in a fast developing area of particle physics t wo decades (referr ing to the time it w as written) ar e a lo ng time. The pa pe r consis ts of tw o parts, the ﬁr s t is entirely dedicated to the ex po si- tion of the diﬀerences be tw een (rela tivistic 5 ) Q M and LQP and their co existence at large time separa tions within the setting o f s cattering theory . The second part, which will app ear as a se parate contribution, deals with thermal and en- tropic consequences of v a cuum p olariza tio n ca used b y causa l loc a lization, as well as some conse quences fo r Q FT in CST. A Q G theory do es not yet ex ist, but a pro found understanding of those foundational asp ects is exp ected to b e impo rtant to arrive at one. The sections of the paper a t hand are as follows. The ﬁr st s e c tions presents the little known theor y of dir e ct p article inter actions (DPI), a framework which incorp ora tes all those prop erties of a rela tiv is tic theo ry which one is a ble to formulate solely in terms of r elativistic particles; some of them already app eared in the pre F eynman S-matr ix work of E.C.G. St ¨ uck elbe rg. In cont r a distinction to nonrelativistic QM where the cluster factoriza tion follows from the additiv- it y of t wo-particle interactions, its enforce men t in DPI r equires more r eﬁned arguments. As a clo s ely rela ted result, DP I do es not a llow a second qua n ti- zation presentation, ev en though it is a p erfect legitimate multiparticle theory in which n-particles are linked to n+1 particles by cluster factoriza tion. Most particle physicists tend to belie ve that a r elativistic par ticle theory , consistent with macro-causa lit y and a Poincar´ e-inv ar iant S-matrix, m ust be equiv alent to QFT 6 , therefor e it may b e helpful to show that this is not co rrect. DPI the- ories fulﬁll all the physical r equirements which o ne is able to formulate so lely in ter ms r elativistic pa r ticles without r ecourse to ﬁelds, as Poincar´ e cov ariance, unitary and macr o -causality of the r esulting S-matrix (which includes clus ter factorization). In this w ay one learns to appre ciate the fundamental diﬀerence b etw een quantum theories which hav e no a lgebraica lly built-in ma ximal velocity and those which ha ve. As a qua nt um mec hanica l theory DPI only leads to s ta- tistical ” eﬀective” ﬁnite velocity propag ation for asymptotically large time-like separatio ns b etw een lo calized even ts a s they o ccur in scattering theory . With other words the c a usal propaga tion b etw een Bo rn-lo calized even ts is a macro - scopic phenomenon for which, in ana logy to the ac o ustic velocity in QM, the large time b ehavior of dissipating wa ve pac kets is imp ortant, whereas in Q FT the ma ximal velocity is imprinted into the alg ebraic str uc tur e. DPI do es no t po ssess cov ar iant lo cal op era tors, the only cov a riant o b ject is the Poincar´ e in- 5 In order to show, that b y making QM r elativistic, one do es not remov e the fundamental diﬀerences with QFT , the next section wi ll b e on the relativistic setting of ”direct particle int eractions”. 6 The related folklore one ﬁnds in the literature amoun ts to the dictum: r elativistic quan tum theory of particles + cluster f actorization prop erty = QFT. Apparen tly this conjecture go es bac k to S. W ein b erg. 5 v ariant S-matrix ; from this viewp o int DPI is an S-matrix theor y . If one uses it outside o f asy mptotic propa gation, o ne of course ﬁnds the vio la tion of causal- it y which led to misunder standing of F er mi’s cla im [8] which w as subsequently corrected with the (implicit) use of modular theory [9]. A t the ro o t of the QM-Q FT (particle-ﬁeld) antagonism is the existence of t wo very diﬀeren t concepts of lo calizatio n namely the Born lo c aliza tion 7 (whic h is the only lo caliza tion for QM), and the mo dular lo c alization which underlies the causal lo cality in QFT. The Bo r n lo calization and the rela ted p osition o p e rator has b een a dapted to the cov ariant nor malization of relativ is tic wa ve functions in a pap e r b y Newto n and Wigner [11] (whereup on it bec o mes frame-dep endent) and will hencefo rth b e refer red to as the BNW lo ca lization. Whereas r elativistic QM p er mits o nly the BNW lo caliz ation, Q FT needs b oth, the mo dular lo cal- ization 8 in order to implement causal pro pagation a nd the BNW lo calization to g et to the indisp ensable asymptotic scattering pr obabilities (cross sectio ns). Without the BNW localiza tion QFT w ould remain a beautiful mathematical construct with no a c cessible physical conten t; on the other ha nd without mod- ular lo ca lization QFT w ould no t have in tera ction-induced v acuum polar ization and its description of reality at ﬁnite dista nc e s would contain acausa l po ltergeist- daemons of the kind ment ioned a b ove. Note that we av oid the phrase ”particle- wa ve dualism” b e cause in our understanding this issue has be e n s olved in the transformatio n theor y showing that Schroedinge r ’s w av e function for malism is equiv alen t to the alg ebraic formulation in terms of p.q operator s or in the rela- tivistic context that Wigner’s particle p ositive energy repr esentation approa ch to particles is uniquely functorially related to fre e ﬁelds or the r elated spa c etime lo calized sytem of a lgebras (sections 6,7). The particle- ﬁeld pr o blem starts o nly in the presence of in teracting QFTs. Particles are ob jects with a well-deﬁned o nt olo gical status, whe r eas (basic and comp os ite) ﬁelds form an inﬁnite set of c o ordinatizations w hich gener ate the lo cal algebra s . By this we mea n that particles ar e the truly real and unique ob jects which a re sub ject to direct observ ations a nd indep e nden t of any ”ﬁeld- co ordinatiza tio n”, a prop erty which is not deroga ted b y the fact that their existence is only an a symptotic one. What is referr ed to as a n (as y mptotic) n- particle state is a state in which n well separated coincidence counters (in a world cobbled with counters) click simultaneously and a part from the lo caliza tion of the co unters, the num b er n do es not change at later times and the par ticle cr oss section is frame indep endent. 7 It is interesting to note that Born introduced the probability conce pt in QM in the con- text of the Born approximation of what we call now ada ys the cross section and not of the Sc hro edinger wa ve function [10]. With other w ords he introduced it in the asymptotic re- gion where it is indidp ensible and where the BNW lo calization b ecomes indep enden t of the reference frame. 8 Mo dular lo calization is the same as the causal locali zation inheren t in QFT after one lib erates the letter f rom the cont ingencies of particular selected ﬁelds. It is a pr operty of the lo cal equiv alence class of relative ly lo ca; ﬁelds (the Borchers class) or of the associated sysrem of lo cal algebras. If one considers, as it is done i n algebraic QFT, the lo cal ﬁelds as coordinatizations of the lo cal algebras, mo dular lo calization is independent of the ”ﬁeld coordinatization”. 6 Quantum ﬁelds on the other hand have a more ﬂeeting and les s individual existence and ther e a re alw ays inﬁnitely many ﬁelds which are asso ciated with one particle. This ﬁnds its express ion in the terminolog y ”int er p o lating ﬁelds” used in the L SZ scattering theor y of the 50 s . B ut a s epistemic entities ﬁelds or lo cal a lgebras are indisp ensa ble since a ll our intuit ion about loca l interac- tions and their causal lo ca lization prop erties is injected on the level of ﬁelds or directly int o the lo cal observ able algebras whic h they g e nerate. Without Born lo calization and the ass o ciated pro jecto r s, there would b e no scattering theo ry leading to cross sections and hence QFT would be reduce d to just a physically anemic mathematical pla yg round. In contradistinction to DPI, in int er acting QFT there is no wa y in which in the presence of interactions the notion of p articles at ﬁnite times can be sav ed. T he statement that an isola ted relativistic particle cannot b e lo calized below its Compton wa ve length refers to the (Newton-Wigner adaptation of the) Born lo calization and, as all s tatement s inv olving Bor n lo caliza tion, it is meant in an eﬀe ctive pro ba bilistic sense. Only in the timelike a symptotic limit betw een t wo even ts the BNW lo calization b ecomes a sharp geometric rela tions in terms o f momenta with c b eing the maximal velocity which is indep endent of the r eference frame; fortunately this is precisely what one nee ds to obtain a Poincar´ e in v ariant macro causa l S-matrix . The maximal v elo city in the sense of asymptotic exp ectations in suitable states of rela tivistic particle theor ies pla ys a similar r ole a s acoustic velo city in nonrela tivistic QM which leads to (materia l-dep endent) acoustic velo cities. Placing our interpretation in the co n text of prior work on this sub ject [2][1 2], our conclusion is that neither ”Reeh-Sc hlieder defea ts Newton-Wigner ”, nor do e s Newton-Wigner ” mee t” Reeh-Schlieder in the nonaymptotic spacetime re gion, rather b oth indisp ensable lo c alization schemes approximate each other asymp- totically at t → ±∞ wher e the Newton-Wigner lo calization b ecomes co v ariant, the results of macro-ca usality coa le sce with those of micr o-causa lity and the mo dular lo calization share s the asymptotic pro bability notion with BNW, i.e. no defeat o f either one, but harmony at the only place where they can meet a nd remain faithful to their principles. The next section co nt ains some r emarks a bo ut the history of the growing aw areness abo ut prop er ties whic h sepa rate QM from Q FT. This is follow ed in section 3 with the pr esentation of a little known consistent setting of interacting relativistic particles without ﬁelds: the direct particle interaction theor y (DPI) by Co ester and Polyzou. Sectio ns 4 and 5 foc us on the radica l diﬀerence b etw een the Newton-Wigner (NW) lo ca lization (the name for the Born loca lization after the ada ptation to the r elativistic pa rticle setting) and the lo caliza tio n which is inherent in QFT, which in its intrinsic form, i.e. libera ted from singular p oint- like ”ﬁeld co ordinatizatio ns ”, is referr ed to as mo dular lo c alization [13][14][15] . The terminology has its origin in the fact that it is back ed up by a ma the- matical theor y within the setting of op era tor algebr as which bea rs the name T omita-T akesaki 9 mo dular the ory. Within the setting of thermal QFT, physi- 9 T omita was a Japanese mathematician who discov ered the main prop erties of the theory 7 cists indep endently discov ere d v arious asp ects o f this theory [5]. Its r elev ance for causal lo calizatio n w as only sp otted a decade later [1 7] and the apprec iation of its role in problems of therma l b ehavior at causal- and e vent- horizons and black hole ph ysics had to wait another deca de [1 8]. Sections 6-1 0 are all centered around an in-depth exp os ition of v ario us as- pec ts of mo dular lo ca lization, starting fro m the mo dular lo ca liz ation of states and pa ssing to its mor e restrictive a lgebraic count er part. Among its very recent application is the notion of semiinﬁnite spa celike string lo calization which on the one hand settled the age old problem of the a ppropriate loca lization for behind the Wigner inﬁnite spin represe n tation but also shows that the ob ject of string theory is really an inﬁnite comp onent ﬁeld (section 7). The p enultimate section pr esents LQ P as the result of r elative p ositioning of a ﬁnite (and r a ther sma ll) num b er o f monads within a Hilb ert spa c e ; her e we a re using a terminolog y which Leibniz in tro duce d in a philoso phical context but which makes a p erfect match w ith the co nce ptual structure of LQP once one go es b eyond the class ic a l quantization para lle lis m. On the other ha nd this underlines the enormous conceptual distance betw een to QM for which such concepts a re no t av aila ble. Wher eas a single monad als o app ea rs in diﬀere nt contexts, e.g. KMS sta tes on o pen quantum systems (even in Q M) and the infor- mation theoretical in terpr e tation o f bipartite spin algebras in suitable singular states [4][19 ], the mo dular positio ning of several copies is totally c har acteris- tic for LQP . Although its physical and mathematical co nt ent is quite diﬀerent from Mermin’s [20] new lo ok (the ”Ithaca -interpretation” of QM) a t quantum mechanical reality exclusively in terms of corr elations b etw een subsystems, the t wo concepts sha re the asp ect o f viewing reality in relatio nal terms 10 . Ma the- matically a mo nad in the sense o f this article is the unique hyperﬁnite type I I I 1 factor algebr a to which all lo cal alge br as in LQP ar e isomorphic, so a ll concr ete monads are copies of the abstr act monad. Naturally a mo nade in iso lation is an abstract entit y without structure, the r eality emer ges fro m r elations b etw een monads within the same Hilbert space. Whereas for Newton physical reality consisted of matter moving in a ﬁxed space according to a universal time, rea lit y for Leibniz emerg es from inter- relations b etw een monads with spacetime serving as an ordering device. The mo dular p os itioning of monads go es one step further in that even the Minko wski spacetime together with its inv ar iance gr oup (the Poincar´ e group) app ear s as a consequence of p ositioning in a more abstra ct sense namely of a ﬁnite n umber of monads in a join t Hilb ert space (subsection 7 ) [93 ]. F or actual constructions of interacting LQP mo dels it is ho wever adv antageous to star t with o ne monad and the action of the Poincar´ e group o n it [3 0][74]. The a lgebraic str ucture of QM on the o ther hand, relativis tic or not, has no such monad structure; the glo bal a lgebra as w ell as all Born-lo calized subalge- bras in gr ound states ar e a lwa ys o f type I i.e. either the algebra of a ll bounded in the ﬁrst half of the 60s, but it needed a lot of p olishing by T akesaki in or der to b e accept ed. 10 Mermin’s relational idea remains how ever somewhat v ague compared to the mathemati- cally very precise and ph ysically complete c haracterozation of QFT mo dels via mo dular posi- tioning. 8 op erator s B ( H ) in an appropriate Hilb e rt space or multiples thereo f. Corr e- lations are ch ar acteristic features of quan tum mechanical s tates, wher eas for the characterization of a Q M system global o p e rators as the Hamilto nian ar e indispe nsable. Part I of this essay closes w ith a s e ction o n the split inc lus ion which shows how in the ubiquitous presence of v acuum p ola rization so me of the notion known from QM (tenso r factorization of disjoint subsystem, entanglemen t) can b e r e- cupe r ated. The second part will present many mor e applications of mo dular lo calization and the split pro p e rty notably those rela ted to ther mal and entropic prop erties which are of po ten tial astrophysical and cosmological relev ance. 2 Historical remarks on the in terface b et w een QM and QF T Shortly after the disco very of ﬁeld quantization in 19 25 [21][2 2], there w ere tw o opp osed viewp oints ab out its co nten t and purp os e of r elativistic QT repr esented by Dirac and Jo rdan [2 3]. Dirac maintained that quantum theor y should stand for quantizing a classic al r e ality which meant ﬁeld q ua nt iza tio n for e lectromag- netism and quantization of cla ssical mechanics for particles. Jorda n, on the other hand, pro p o sed an uncompromising ﬁeld qua nt izatio n p oint of view; his guiding theme was that a ll what can b e quantized should b e qua ntized, inde- pendent of whether there is a classica l reality or not 11 . The more ra dical ﬁeld quantization including particles ﬁnally won the argument, but ironically it w as Dirac’s par ticle setting (the hole theo ry) and not Jor dan’s version of ”Mur- ph y’s law” (”everything which can be quantized must b e quantized”) to a ll ﬁeld ob jects which co n tributed the richest structura l pr op erty to QFT, namely charge-antic harge symmetry le a ding to the necessa ry pr esence of antiparticles. It was als o Dirac’s hole theory setting in whic h the ﬁrst p ertur bative QE D computations (which entered the textb o oks o f Heitler and W enzel) were done, befo re it w as recognized that this setting w as no t r eally consistent. This in- consistency show ed up in problems in volving renor malization in which vacuum p olari zation plays the essential r ole. The succ e ssful p e r turbative re normaliza- tion of QED in the charge symmetric descr iption was a lso the end of hole theory as well as the start of Dirac’s la te acceptance of QFT as the genera l s etting for relativistic pa r ticle physics (at the be ginning o f the 50s). This shows that b e- fore ser ious erro r s misled pa rticle physics into the pr esent crisis [31] there were fascinating error s whose profound understanding led to o a deep enrichmen t and belo ng to our precious heritage. V acuum p olarizatio n is a very p eculiar phenomenon whic h in the sp ecia l context of currents a nd the as s o ciated lo cal charges of a complex free Bo se ﬁeld was noticed alrea dy in the 30s by Heisenberg [24]. But only when F urr y and 11 His radicalit y about sub jecting ev erything which mathematically can be quan tized (in- cluding De Brogli e matter wa ve s) to the new quant um recip e was also met with in certain cases justiﬁed critici s m.esp ecially when he together with Klein second quant ized the Schroedinger equation (wh y quan tize s omething again whic h was alr eady ”quantum”) . 9 Opp enheimer [2 5] studied perturba tive in tera ctions of Lagra ngian ﬁelds and bec ame aw ar e to their amaze men t that the Lag rangian ﬁeld applied to the v ac- uum created inevita bly par ticle- antiparticle pairs in addition to the expected one-particle s tate, the subtlet y o f the particle-ﬁeld relatio n within interacting QFT b egun to b e noticed. The num ber o f thes e pairs increas e with the p er - turbative order, p o inting tow ar ds the fact that in case of sharp lo calization (”banging” with sha r ply lo calized op era tors ont o the v acuum) one has to deal with inﬁnite p olar ization clouds containing arbitrary high energ y comp onents. Since there is no po sition op er ator in QFT, there is a fortiori no Heisen b erg uncertaint y re lation. As a QFT subs titute one ma y consider the un b ounded increase of lo ca lization entrop y E nt ( O , ε ) where O is the spacetime lo ca lization region and ε the split distance which creates a ”fuzzy” sur fa ce. When ε → 0 the en tropy diverges as A ( H ( O )) ε 2 l n q A ( H ( O )) ε 2 where A ( H ( O )) is the are a of the causal horizon of O , i.e . the sharp er the loca lization the bigger the lo ca lization ent ro py [26][27]. Whenever one tries in an in tera cting theor y to cre a te particles v ia lo cal disturbances of the v ac uum, the v acuum po la rization clouds corrupt the ob- serv ation o f thos e particles which one intends to crea te, but a fter a suﬃcient amount of time the particle conten t separates from the po larization cloud. In the pr esence o f interactions the no tion of pa rticles in lo ca l regions a t a ﬁx e d time is, s trictly sp eak ing, meaningless b ecause even the ﬁeld with the ”mildest” v ac- uum po larization tak en from the class of all po s sible relative lo cal ﬁelds which all in terp ola te the s ame pa rticle still genera tes an inﬁnite v acuum p o larization cloud which sticks ins eparably to the pa rticle of int er e st 12 . It is somewhat ironic that pa rticles, which are the ma in bridge be t ween QFT and its lab or atory real- it y (and which are the basic ob jects o f QM), hav e o nly an asymptotic ex is tence as incoming and outgoing asymptotic particle conﬁgurations. In the next s ubsection it will b e shown that relativistic QM in the for m of DPI, in c o ntradistinction o f wha t most pa r ticle theorists b elieve, can b e consistently for m ulated [3 3] and this setting can even b e extended to incorp or a te creation and annihilation channels [34]. T his g o es along w ay to vindicate Dir ac’s relativistic particle viewpoint. But it does not vindicate it completely , since theories which sta rt a s particle theories but then lea d to v acuum po larization as Dirac’s hole theor y ar e a t the end inconsis ten t unless one co nv erts their conten t int o a charge symmetric ﬁeld theor etic s etting (in which cas e the connection with Dirac ’s whole theory is lost). By contrasting QFT with DPI, one obtains a b etter a ppreciation o f the con- ceptual depth of QFT, in particular one beco mes aware of its still unexplored regions. DPI is basically a r elativistic p article setting i.e. it deals only with prop erties which c a n b e formulated in terms o f particles; this limits caus ality prop erties to macr o-c ausality i.e. s pacelike cluster factorization a nd timelike 12 Only if one allows noncompact lo calization r egions one is able to ﬁnd ”PFGs” i.e. opera- tors whic h applied to the v acuum generate one-particle states wi thout p olari zation admixture. W edge r egions in Mi nk owski space l ead to the b est compromise b et ween particles and ﬁelds and play a fundamental role i n recen t mo del constructions [ 28 ][29][30] and i s at the r oot of the crossi ng property [31]. F or a philosophical viewpoint s ee [32] . 10 causal resca ttering. Apar t from the fact that the multi-particle representation theory of the Poincar´ e gro up is incompatible with the additivity of interaction terms which complicates the implementation of the clus ter factoriza tion prop- erty and preven ts an eleg ant seco nd q uantization des cription in Wigner-F o ck space, the DPI setting is a s well understo o d as nonr e lativistic Q M. In con- trast nob o dy who has studied QFT b eyond a textb o ok level w ould cla im that QFT is an ywher e nea r its clo sure. The last section illustrates this p oint by an unexp e cted new abstract characterization o f QFT which is diﬀerent from any previous axio matic attempt. 3 Direct particle in teractions, relativistic QM The Co ester -Polyzou theory o f dir e ct p article inter actions (DPI) (where ”di- rect” mea ns ”not ﬁeld- mediated”) is a relativistic setting in the sense of repre- sentation theory of the Poincar´ e gr oup which, among other things, leads to a Poincar´ e in v ariant S-matr ix . E very pro p er t y which can b e form ulated in ter ms of particle s , a s the cluster factoriza tio n into systems w ith a lesser num b er of particles and o ther timelike asp ects of macr o causality , can b e implemented in this setting. The S-matrix do es how ever not fulﬁll such ana lyticity prop erties as the cr ossing [31] pro per ty whose deriv ation relie s on the existence of lo cal int erp o lating ﬁelds. In contradistinction to the more fundamental lo cally c ov ariant Q FT, DPI is primarily a phenomenolo gical setting, but o ne which is consistent with ev - ery pro per ty which can b e e x pressed in terms of relativistic particles o nly . So instead of approximating nonp erturbative Q FT in a metapho ric w ay outside conceptional-ma thema tical co nt ro l, the idea of DP I is to a rrange phenomeno- logical calculatio ns in such a way that at lea st the principles of relativistic mechanics a nd macro-causa lity a re maintained [33]. F or the interaction of tw o rela tivistic particles the intro duction of r elativistic int era ctions amounted to add to the free mass op erato r (the Hamiltonia n in the c.m. sy stem) an interact which dep ends o n the relative p osition and momen- tum. The e x igencies of repr esentation theory of the P oincar´ e gr o up ar e then fulﬁlled and the cluster pr op erty stating that S → 1 fo r large spatial sepa ration is a consequence o f the short rang ed interaction. Ass uming for simplicit y iden- tical sc a lar Bosons , the c.m. inv aria nt energy op era to r is 2 p p 2 + m 2 and the int era ction is intro duced by adding an in tera ction term v M = 2 p ~ p 2 + m 2 + v , H = q ~ P 2 + M 2 (1) where the inv ariant po tent ial v dep ends on the relative c .m. v ariables p, q in a n inv ariant manner i.e. such that M commutes with the Poincar´ e generators of the 2- particle system which is a tensor pr o duct of t wo o ne-particle systems. One ma y follow Bak amjian a nd Thomas (BT) [35] and choose the Poincar ´ e generator s in their wa y so that the interaction only app ea rs in the Hamiltonian. Denoting the interaction-free generators by a subscript 0 , one arrives at the 11 following system of tw o-particle ge ne r ators ~ K = 1 2 ( ~ X 0 H + H ~ X 0 ) − ~ J × ~ P 0 ( M + H ) − 1 (2) ~ J = ~ J 0 − ~ X 0 × ~ P 0 The interaction v ma y b e taken a s a lo c al function in the re lative co ordinate which is c o njugate to the rela tive momentum p in the c.m. system; but since the scheme anyhow do es not lea d to local diﬀeren tial equations, there is not m uch to be ga ined fro m such a choice. The Wigner canonical spin ~ J 0 commutes with ~ P = ~ P 0 and ~ X = ~ X . 0 and is related to the Pauli-Lubanski vector W µ = ε µν κλ P ν M κλ . As in the nonrela tivistic s e tting, shor t ra nged int era ctions v lead to Møller op erator s and S-matrices via a con verging sequence of unitaries formed from the free and interacting Hamiltonian Ω ± ( H, H 0 ) = lim t →±∞ e iH t e − H 0 t (3) Ω ± ( M , M 0 ) = Ω ± ( H, H 0 ) (4) S = Ω ∗ + Ω − The identit y in the second line is the conseq uence of a theor em which say that the limit is not aﬀected if instead of M one ta kes ta ke a positive function of M (4) as H ( M ) , a s long as H 0 is the same function of M 0 . This insures the the asymptotic fr ame-indep endenc e of obje cts as the Mø l ler op er ators and the S-matrix but not necessar ily that of semi asy mptotic op erator s a s formfactor s of lo ca l op erator s betw een ket in and bra out particle sta tes . Apart from this identity for op er ators and t heir p ositive functions (4) which is no t needed in the nonrelativistic scattering , the rest be haves just as in nonrelativistic scatter ing theory . As in standard QM, the 2-particle cluster prop e rty is the s tatement that Ω (2) ± → 1 , S (2) → 1 , i.e. the scattering formalism is iden tical. I n particular the t wo particle cluster prop erty , which says that for short r ange interactions the S-matrix approaches the identit y if one separa tes the ce nter o f the wa ve pa ck ets of the t wo inco ming particles , holds also for the r elativistic ca se. The implemen tation of cluster ing is more delicate for three particles as can be seen fro m the fact that the ﬁrst attempts were sta rted in 196 5 by C o ester [36] a nd consider a bly later generalized (in collab ora tion with Polyzou [3 3]) to an arbitrar y high particle num b e r. T o anticipate the result b elow, DPI leads to a consistent sch eme which fulﬁlls cluster factoriz a tion but it ha s no useful s e c ond quantized fo rmulation so it ma y stand accused of lack of elega nce; one is inclined to view less elegant theor ies also a s less fundamental. It is also mor e nonlo cal and nonlinear than Ga lilei-incariant QM, This had to b e exp ected since adding int era cting pa rticles do es no t mean adding up interactions as in Schroedinger QM. The BT form for the gene r ators can be a chiev ed inductively for an arbitra ry nu mber of pa r ticles. As will be seen, the adv antage of this form is that in passing from n- 1 to n-particles the interactions add after appropria te Poincar´ e 12 transformatio ns to the jo in t c.m. system and in this way one ends up with Poincar´ e group generator s for an interacting n-particle sys tem. But for n > 2 the aforementioned subtle pr oblem with the cluster pro pe r ty arises; whereas this iter ative co nstruction in the nonrelativis tic setting complies with cluster separability , this is not the case in the relativ is tic context. This problem s hows up for the ﬁr st time in the presence of 3 par ticles [3 6]. The BT iteration from 2 to 3 particles gives the 3 - particle mass opera tor M = M 0 + V 12 + V 13 + V 23 + V 123 (5) V 12 = M (1 2 , 3) − M 0 (12; 3) , M (12 , 3) = q ~ p 2 12 , 3 + M 2 12 + q ~ p 2 12 , 3 + m 2 and the M ( ij, k ) result from cyc lic permutations. Here M (12 , 3) denotes the 3-particle inv a riant mass in case the third particle is a “sp ectator ”, which by deﬁnition do es not in teract with 1 and 2 . The momentum in the last line is the relative moment um b etw een the (12)-cluster and particle 3 in the joint c.m. system and M 12 is the asso ciated tw o-particle mass i.e. the inv a riant ener gy in the (12) c.m. system. W ritten in terms of the origina l tw o-pa rticle in teractio n v , the 3-par ticle mas s term a ppea rs nonlinea r. As in the no nr elativistic case, o ne can always add a totally c o nnected con- tribution. Setting this contribution to ze ro, the 3 -particle mass op erato r only depe nds on the tw o- particle interaction v . But contrary to the nonrelativistic case, the BT genera tors cons tr ucted with M as it stands do es not fulﬁll the cluster sepa rability r equirement. The latter demands that if the interaction b e- t ween t wo clusters is remov ed, the unitary r epresentation factor izes into that of the pro duct of the t wo clusters. One expects that shifting the third particle to inﬁnit y will render it a spec- tator and r e sult in a factoriza tion U 12 , 3 → U 12 ⊗ U 3 . Unfortuna tely what re- ally happ ens is that the (12) in terac tio n also gets switched o ﬀ i.e. U 123 → U 1 ⊗ U 2 ⊗ U 3 . The r eason for this violation of the cluster sepa rability prop erty , as a simple ca lculation (using the trans fo rmation formula fro m c.m. v ar ia bles to the or iginal p i , i = 1, 2 , 3) shows [3 3]), is that although the spatial tra ns lation in the orig inal system (instead of the 12 , 3 c.m. sys tem) do es r emov e the third particle to inﬁnity as it s hould, unfortunately it also drives the tw o- particle mass op erator (with whic h it does not commute) tow ar ds its free v alue which vio lates clustering. In other w or ds the B T pro duces a Poincar´ e cov ar iant 3- pa rticle in teraction which is a dditiv e in the resp ective c.m. in tera ction terms (5), but the Poincar´ e representation U of the r e sulting system will not b e cluster-separ able. How ever this is the time fo r in tervention of a saving g race: sc att ering e quivalenc e . As shown ﬁr s t in [36], even thoug h the 3- particle representation of the Poincar´ e group arrived at by the above ar guments violates clustering, the 3 - particle S-matrix computed in the additive BT scheme turns out to hav e the cluster factorization prop erty . But without implementing the correct cluster factorization also for the 3-particle Poincar´ e gene r ators there is no chance to pro ceed to a clustering 4 -particle S-matr ix . 13 F ortunately there always exist unitar ies which tr a nsform BT systems int o cluster-separ able systems without aﬀe cting the S-matrix . Such transfor mations are called sc attering e quivalenc es. They were ﬁrst intro duced into QM by Sokolo v [3 7] and their in tuitiv e con tent is related to a cer tain insens itivit y o f the sca ttering op er ator under quasilo cal changes of the quantum mechanical de- scription a t ﬁnite times. This is reminisce nt of the insens itivit y of the S-ma tr ix against lo ca l changes in the interpo lating ﬁeld-co or dinatizations 13 in QFT by e.g. using comp osites instea d o f the La grangia n ﬁeld. The notion of sca ttering e q uiv alences is conv eniently describ ed in ter ms of a subalg ebra of asymptotic al ly c onstant op er ators C deﬁned by lim t →±∞ C # e iH 0 t ψ = 0 (6) lim t →±∞  V # − 1  e iH 0 t ψ = 0 where C # stands for b oth C and C ∗ . These oper ators, which v anish on diss ipat- ing free wa ve pack ets in co nﬁguration space, form a *-suba lgebra whic h extends naturally to a C ∗ -algebra C . A s cattering equiv a lence is a unitary mem b er V ∈ C which is asymptotically equal to the identit y (the conten t of the second line). Applying this asymptotic equiv alence rela tion to the Møller op era tor one obtains Ω ± ( V H V ∗ , V H 0 V ∗ ) = V Ω ± ( H, H 0 ) (7) so that the V cancels out in the S-matrix . Scattering equiv alences do how- ever change the interacting representations of the Poincar ´ e group accor ding to U (Λ , a ) → V U (Λ , a ) V ∗ . The upshot is that there exists a clustering Hamiltonian H clu which is uni- tarily related to the BT Hamiltonian H B T i.e. H clu = B H B T B ∗ such that B ∈ C is uniquely determined in terms o f the s cattering data computed from H B T . It is precisely this clustering of H clu which is needed for obtaining a c lus- tering 4-pa rticle S- ma trix which is cluster-as so ciated with the S (3) . With the help of M clu one deﬁnes a 4 -particle interaction following the additive B T pre - scription; the subsequent scattering for ma lism leads to a clustering 4-par ticle S-matrix and again one would no t b e able to go to n=5 without passing from the BT to the cluster-fa ctorizing 4-pa rticle Poincar´ e gro up r epresentation. Co es ter and Polyzou show ed [33] that this pro cedure can b e iterated and doing this o ne arrives at the following s tatement Statemen t : The fr e e dom of cho osing sc attering e quivalenc es c an b e use d to c onvert the Bakamija n-Thomas pr esentation of mu lti-p article Poinc ar´ e gener a- tors into a cluster-factorizing r epr esentation. In this way a cluster- factorizing S-matrix S ( n ) asso ciate d to a BT r epr esentation H B T (in which clustering mass op er ator M ( n − 1) clu was use d) le ads via the c onstruction of M ( n ) clu to a S-matrix 13 In ﬁeld theoretic terminology this means chan ging the poi n tlike ﬁeld by passing to another (composite) ﬁeld in the same equiv alence class (Borc hers class) or in the setting of AQFT by pic king another op erator from a lo cal operator algebra. 14 S ( n +1) which clusters in t erms of al l the pr eviously determine d S ( k ) , k < n. The use of sc attering e quivalenc es pr events t he existenc e of a 2 nd quantize d formal- ism. F or a pro of we refer to the o riginal pap ers [33][34]. In passing we mention that the minimal extension, (i.e. the one determined uniquely in terms of the t wo-particle interaction v ) from n to n+1 for n > 3 , c ontains c onne cte d 3-and higher p article int er actions which are nonlinear ex pressions (inv olv ing nested ro ots) in ter ms of the o riginal t wo-particle v . This is ano ther unexpec ted phe- nomenon as compared to the nonrelativis tic ca se. This theorem s hows that it is p ossible to construct a r elativistic theory which only uses par ticle concepts, thus cor recting an old folklor e which says relativity + clustering = QFT. Whether one s ho uld c all this DPI theor y ”relativis tic QM” or just a r elativistic S-matrix theor y in a Q M setting is a matter of taste; it depe nds on what signiﬁcanc e one attributes to those un usual sca ttering equiv- alences. In a ny case it deﬁnes a r elativistic S - matrix sett ing with the corr e ct particle b ehavior i . e . all prop erties which one is able to for m ulate in terms of particles (without the use of ﬁelds) as unitarity , Poincar´ e inv ariance and macro- causality ar e fulﬁlled. In this context o ne sho uld a lso mention that the S-matr ix bo otstrap approa ch never a ddr essed these macro -causality problems of the DPI approach; it was a grand self-deluding des ig n for a unique theory of all no n- gravitational interactions in which imp ortant physical details were arr ogantly ignored. As mentioned a b ov e Co ester and Polyzou also show ed that this rela tivistic setting can b e ex tended to pro cesses which ma int ain cluster factorization in the pr esenc e of a ﬁnite numb er of cre ation/annihilatio n channels , thus demon- strating, as mentioned b e fore, that the mer e pr esenc e of p article cr e ation is not char acteristic for Q FT but rather the pr esence of inﬁnite v a cuum pola rization clouds fro m ”banging” with lo calized o pe r ators onto the v acuum (see section 7 ). Diﬀerent from the nonr elativistic Sc hro eding er QM, the supers election rule for masses of particles which results fro m Galilei inv ar iance for no nr elativistic QM do es not car ry ov er to the relativis tic setting; in this resp ect DPI is les s res tr ic- tive than its Galilei-inv a riant QM counterpart where such creatio n pr o cesses ar e forbidden. One may co nsider the DPI setting o f Co ester and Polyzou as tha t scheme which r esults from implementing the mentioned particle prop erties within a n- particle Wigner r epresentation setting in the presence o f interaction [33], it is the only re la tivistic QM which is consistent with macr o causality 14 . Appar ently the work of these mathematical nuclear physicists has not bee n noticed by particle ph ysic ists probably since the authors ha ve published most of their res ults in nu clea r physics journa ls. What mak es it worth while to mention this w ork is that even physicists of g reat renown as Steven W einberg did not b elieve that such a theor y exists b ecause o therwise they would not have conjectured that the implemen tation of cluster factorization pro pe r ties in a relativistic setting lea ds 14 Macrocausality consists of s pacelik e clustering and tim el i ke causal rescattering (St¨ uck elberg); it is the only causalit y whi ch one is able to fulﬁll in terms of particles only (without ﬁelds). 15 to QFT [38]. Certain pr op erties which are conse q uences of lo ca lity in QFT and can b e formulated but not derived in a pa rticle s e tting as the TCP symmetry , the spin-statistics connection and the existence of ant i-pa rticles, ca n b e added ”b y hand” to the DPI setting. Other proper ties which are on-shell relics of lo cality which QFT imprints on the S-matrix and which r equire the notion of analytic contin uation in particle momenta (as e.g. the cr o ssing pr op erty fo r formfactors) cannot b e implemen ted in the Q M setting of DPI. 4 First bru sh with the in tricacies of the particles- ﬁeld prob lems in QFT In co nt ra st to QM (Schr¨ odinger- QM or relativis tic DPI), interacting QFT do es not admit a par ticle interpretation at ﬁnite times 15 . If it would not b e for the asymptotic scattering interpretation in ter ms of incoming /outgoing par ti- cles asso ciated with the free in/out ﬁelds, ther e w ould be hardly anything of a non-ﬂeeting mea s urable nature. In QFT in CST and thermal Q FT where even this asymptotically v alid par ticle c oncept is mis sing, the set of conceiv able mea - surements is essentially r educed to energy- a nd entropy- densities in thermal states and in black hole states with ev ent horiz ons as well a s of cosmological states descr ibing the microw av e background radiation. Since the notion of particle is often used in a more genera l sense than in this pap er, it may b e helpful to have a brief interlude on this issue. By particle I mean an asymptotica lly stable ob ject which lea ds through its n-particle tensor pro d- uct structure to a n asymptotically complete description of the Hilber t space of a Minko wski spac etime QFT. It is precisely this concept which furnishes Q FT with a (LSZ, Ha ag-Ruelle) co mplete asymptotic particle in terpre ta tion 16 , so that the Hilber t space of such an interacting theor y has a F o ck spa ce tensor structure . The ph ysics b e hind is the ide a [39] that if w e were to ”cobble” the asymptotic spacetime reg ion with counters which monitor co incidences/anticoincidences of lo calization events (lo cal deviations from the v acuum) a fter a collisio n of t wo incoming particles has taken place, then the deﬁning pro per ty of an outgoing n- particle state is the stable n-fold co incidence/anticoincidence (the latter in or der to ins ur e tha t we reg istered a ll par ticles) betw een n counters. The intuit ive idea is that after some time the n would not c hange and the n-fold loca l excitations from the v acuum w ould mov e along tra jectories of free rela tivis tic particles. would even tually r emain sta ble b ecause the far re moved loca liz ation c e n ters would hav e ceased to int er a ct and fr o m there on move fr e e ly . The o ccurrence rate of these coincidences as well as their correlation with that of the inco ming 15 Although the one-particle states and their m ultiparticle count erparts ar e global states in the Hi lb ert space, they are not accessible by acting lo cally on the v acuum. Scattering theory is the only known nonlo cal inte rven tion. 16 The asymptotic completeness property wa s for the ﬁrst time established (together with a recen t existence pro of ) in a family of f actorizing tw o-dimensional mo dels (see the section on modular lo calization) with non trivial scattering. 16 coincidences is indep endent of the frame of refer e nc e even though BNW lo cal- ization at ﬁnite spacetime reg ions is frame dep endent. In p opular textb o oks this is expr essed as. the BNW lo caliza tion b ecomes ”eﬀectively” cov aria nt for dis- tances b eyond a Compton w av elength (exactly cov ariant only in the large time limit.) The Newton-Wig ner adaptation of the Born p osition oper ator w ould lead to g enuinely Poincar´ e inv aria nt frame-indep endent tr ansition pro babilities betw een incoming and outgoing Newton-Wigner localiz ation even ts. The par ticle concept in Q FT is ther efore precisely a pplicable where it is needed, namely for asymptotica lly sepa rated B NW-lo calized even ts fo r which the proba bilit y interpretation and cov ar iance b eco me compatible. In fac t the use of the B NW lo ca lization for ﬁnite distances is known to lead to tro uble in form of unph ysica l sup erluminal eﬀects; in that cas e one s ho uld formulate the problem in the setting of the mo dula r lo ca lization whic h ha s instead o f probabilities a nd pro jectors dense subsets of states (the Reeh- Schlieder pro pe rty [5]). Tying the particle c oncept in QFT to asymptotically stable coincidences of counters can b e traced back to a seminal pap er by Haag and Swieca [39]. These authors noticed for the ﬁrst time that the phase s pace degr ee of freedom density in QFT, unlike that in QM, is not ﬁnite, rather its cardinality is mildly inﬁnite (the phase space is nucle ar ). The lar ger num b er of degrees of fr eedom in form of an enhanced phase s pace density is yet a nother line o f unexp ected diﬀerent consequences [4 0][41] r esulting fro m the diﬀerent lo calization concepts in QM and QFT, but this in teresting topic will no t be pursued her e. Not all particles comply with this deﬁnition; in fact a ll electrically charged particles ar e infr ap articles i.e. ob jects which a re asymptotica lly stable but in contrast to Wigner par ticles they ar e inexo rably attached to an unobserved cloud of inﬁnitely many infrared pho tons leave a mark of their pre s ence even in the low energy part of the inclusive QE D cross s ection for charged infraparti- cles. The existence of an electron a s a Wigner pa rticle ass o ciated with a sharp mass hyperb olo id on top of a photon background is a ﬁctio n which is incom- patible with QED. Rather electrically c harg ed particles hav e instead of a mas s shell delta function in their Kallen-Lehmann t wo´point function a cut whic h starts at p 2 = m 2 which makes a precise description of such infr ap articles [42] and their sca ttering theory mo re inv olved. There is a diﬀere nce b etw een the direct application to a theory for whic h the pole structur e has b een replaced by the infrapar ticle cuts and the p erturbative calculation which pro ceeds a s if the mass shell r e striction makes sense (using ad ho c infra red cutoﬀs to co m bat divergencies). Since the p ositivity o f the K ¨ allen-L ehmann measur e for c es the cut sing ularity to be milder than the mass shell delta function, the LSZ limits of the charged ﬁelds v anish. O n the other hand the result of the ca lculation of the inclusive infrapar ticle cro ss sec tio ns in the Bo ch-Nordsiek mo del a nd the p ertur- bative summation of leading infrared singular ities in [7 8] lead to nonv anishing results only if the photon r esolution is kept ﬁnite; in pa r ticular one o btains a v anishing cross section for a ﬁnite num ber of pho tons which is c o nsistent with the trivial LSZ limits. Such successful recip es hide the fact that the ro ot of the problem is a radica l change of the pa rticle conc e pt which entails a fundamental adjustment [43]. These asymptotic attempt in momentum spac e hide the fact 17 that g auge theo r ies are only useful fo r po intlik e g enerated ﬁelds (ﬁeld streng th, current) but not for the even more imp or tant charge ﬁelds which turn out to b e semiinﬁnite str ing -like genera ted. In co ntrast to Wigner particles whic h are repre s entation theoretical ob jects of the P oinca r´ e g roup, infra particles exist only in QED-like interacting theorie s in whic h the charge ob eys a quantum Gauss law ho lds. The most dramatic diﬀerences be t ween infraparticles and Wigner particles show up in loca lization asp ects. Whereas Wigner par ticles ”are p ointlik e” i.e. hav e p ointlik e generating wa ve functions, the sharp est lo ca liz ed gener ators for infr a particles a re semiin- ﬁnite stringlike. O n a forma l level this has been known for a long time as expressed in the Dira c-Jor dan-Mandelstam formulas in which a Dir ac spinor is m ultiplied by a n exp one ntial semiinﬁnite line integral ov er the vectorp otential (33). Their mo dern exp osition would b e an impo rtant part o f an es say abo ut v arious lo calization concepts. Ho wev er the description of string-lo ca lized infr a - particles is to o subtle and would req uire a presentation which go es much beyond the conten t o f this es say . W e hope to return to issue in a separ ate pap er. It is the asymptotic p article st ructur e which lea ds to the obser v ational rich- ness of QFT. Once we leave this setting by g oing to cur ved spacetime o r to QFT in KMS thermal representations, or if we restrict a Minko wski spacetime theory to a Rindler w edge with the Hamiltonia n b eing now the b o ost op erato r with its tw o-s ided sp ectrum, in a ll these cases we a re lo os ing not only the setting o f scattering theor y but a lso the v ery no tion of particles a s elementary systems with resp ect to the Poincar´ e group. With it also most of the observ ational wealth related to scattering theo ry is lost. Any devia tion fr o m Poincar´ e cov ar i- ance also endang ers the exis tence of a v acuum. The restrictio n to the Rindler world preserves the F o ck space particle structure of the free ﬁeld Minkowski QFT, but it lo oses its intrinsic physical signiﬁcance with resp ect to the Rindler situation 17 . Since the Minkowski v acuum restricted to the Rindler world is now a ther mal KMS state, there is no pa rticle scattering theo r y in the ”b o ost time” in such a ther mal situation. The remaining observ able phenomena ar e Hawking-lik e [4 7] ra diation densities a nd their ﬂuctuatio ns i.e . observ ables such as they a re pres e n tly studied in the co smic ba ckground r a diation. Some of the conceptual problems related to the Unruh eﬀect [4 6] hav e been addressed in the philoso phically or iented liter ature [4 4][45 ]. Q ua nt um ﬁelds ar e not directly accessible to mea surements 18 and therefore the pr o blem what happ ens to the wealth of pa rticle ph ysics in suc h QFT requires mo re res earch. F ormally the loca l cov ar iance principle for c es the cons truction of a QFT on all causally complete manifolds and their submanifolds at once. So the QFT in Mink owski spacetime with its particle interpretation is a lwa ys part of the solution. What o ne would like to have is a mor e dir ect physical co nnection e.g. 17 There is of course the mathematical p ossibili t y of choosing a groundstate representat ion for a Rindler world instead of restri cting the Minko wski v acuum and to hav e a ﬁnite num b er of ”quan ta” (excitations). But there is no reason for b elieving that these ob jects fall in to the range of v alidity of the Haag-Ruelle scattering theory which is the hallmark of particle physics as we know it. 18 An opp osing opinion to this ”in terp olating ﬁeld point of view” can be found in [48]. 18 a particle concept in the tangent spa ce or something in this direction. The conceptual diﬀerences betw een a DPI relativistic QM and QFT are enormous, but in or der to p e rceive this, one ha s to get aw ay from the shar ed prop erties of the qua ntization forma lism, a step which with the exc e ption o f [5] is usua lly not undertaken in textbo oks whose prime o b jective is to ge t to cal- culational r ecip es with the lea s t conc e ptual inv estment. It is the main purpos e of the following s e ctions to highlight thes e contrasts by g oing more de eply into QFT. There are c e rtain folkloric statements ab out the relatio n QM–Q FT whose refutation does not require m uch conceptual sophistication. F or exa mple in trying to make QFT more sus ceptive to new comer s it is s ometimes said that a free ﬁeld is no thing more than a collection of inﬁnitely ma ny coupled oscillators . Although not outrig ht wrong, this characteriza tion miss es the most imp ortant prop erty of how spacetime e nters as an ordering pr inciple into QFT. It would not help any new comer who knows the quantum oscillator , but has not met a free ﬁeld b efore, to construct a free ﬁeld fro m such a verbal descr iption. Even if he manages to write down the fo r mula of the free ﬁeld he w ould still hav e to appreciate that the most imp ortant asp ect is the c a usal lo caliza tion and no t that what o scillates. This is s omewhat reminiscent of the alle g ed vir tue fro m equating QM via Schr¨ odinger ’s formulation with classic a l wa ve theo ry . What may b e gained for a newcomer by app ealing to his computatio na l abilities acquired in classical electro dynamics , is mor e than lost in the conceptual pr oblems which he confronts later when facing the subtleties of entanglemen t in q uantum physics. 5 More on Born v ersus co v arian t lo calization In this section it will b e shown that the diﬀerence b etw een QM and LQP in terms of their lo c a lization res ult in a surpr ising distinction in their notion of entan- glement. W e will contin ue to us e the w ord Born lo c aliza tion for the proba bilit y density o f the x-spa ce Sc hro e dinger wa ve function p ( x ) = | ψ ( x ) | 2 ; where a s its adaptation to the inv a r iant inner pro duct of r elativistic wa ve functions which was done by Newton and Wigner [11] and will refer r ed to as BNW lo caliza tion. Being a bona ﬁde probability density , one ma y c har acterize the BNW lo ca l- ization in a spatia l reg ion R ∈ R 3 at a g iven time in terms of a lo caliz a tion pro jector P ( R ) which app ears in the s pec tral decomp ositio n of the selfadjoint po sition oper ator. The standa rd version of QM and the v arious se ttings of mea- surement theory rely heavily on these pro jectors; witho ut BNW lo c a lization and the ensuing pro jecto rs it would b e imp ossible to formulate the co nceptual basis for the time-dependent scatter ing theory of QM and Q FT. The BNW p osition opera tor a nd its family of spa tial region-dep endent pro - jectors P ( R ) is no t cov ariant under Lo rentz b o osts. F or Wigner , to whom mo d- ular loca lization was not av ailable, this fra me dependence rais e d doubts about the conceptual soundness of QFT. Apparently the e x istence o f c o mpletely co- v ariant correla tion functions in renormalize d p erturbatio n theor y did no t sa tisfy him, he wan ted a n under standing fro m ﬁrst principles and not as an outgrowth 19 of some fo rmalism. The lack of co v ariance of BNW loca lization in ﬁnite time pro pagation leads to frame-dependence and superluminal eﬀects, which is why the terminolog y ”relativistic QM” has to be taken with a grain o f salt. How ever, as a lready emphasized, in the asymptotic limit o f large timelike separa tion a s required in scattering theo ry , the cov a r iance, frame- indepe ndence and causal relations are recov ered. As shown in section 3 o ne obtains a Poincar´ e-inv ariant unitary Møller op erator a nd S-ma trix whose DPI construction within an interacting n-pa rticle Wigner repr esentation of the Poincar´ e group which also guar anties the v alidity of all the macro - causality requirements (s pa celike clus tering, absence of timelik e precursor s, causal res cattering) which can b e formulated in a particle s etting i.e. without taking reco ur se to in terp ola ting lo cal ﬁelds. E ven though the lo caliza- tions of the individual pa rticles a re fr ame-dep endent, the asymptotic relatio n betw een BNW-lo calized even ts is given in terms o f the geometrically asso ciated c ovaria nt on-s hel l momenta or 4-velo cities which desc r ib e the as y mptotic mov e- men t of the c.m. o f wa ve pac kets. In fact al l observations on p articles always involve BNW lo c aliza tion measurements. The situatio n of pr opagation of DPI is s imilar to that of pro pagation of acoustic wa ves in an elastic medium; a lthough in neither case there is a limiting velocity , there e x ists a maxima l ”eﬀective” velocity , for DPI this is c a nd in the acoustic case this is the velocity o f sound in the particula r medium. In compar ing QM with QFT it is often co nvenien t in discussions ab out conceptual issues to rephrase the conten t o f (nonrelativis tic) Q M in ter ms of op erator alg ebras and sta tes (in the sense of positive exp ectation functional on op erator alg ebras); in this way one a lso achiev es more similarity with the formalism of QFT and develops a greater aw areness for gen uine co nceptual antinomies. In this F o ck space se tting the ba sic quantum mechanical op erators are the creation/annihilation op er ators a # ( x ) with [ a ( x ) , a ∗ ( y )] gr ad = δ ( x − y ) (8) where for F ermions the graded commutator stands for the anticomm utator . In the QFT setting it is not fo rbidden to work with s uch op era tors (the F ourier transforms of the Wigner c r eation/a nnihila tion op er a tors), except tha t it b e- comes nearly imp ossible to keep track of cov ariance and ex pr ess lo cal obser v ables in terms of them 19 . The g round sta te for T= 0 zer o matter density sta tes is annihilated by a ( x ) , whereas for ﬁnite density one enco unt ers a state in which the levels are o ccu- pied up to the F ermi surface in case of F ermions, a nd co n tains a Bo s e-Einstein condensate g roundstate in case o f Bo s ons. In QFT the identiﬁcation of pure s ta tes with state- vectors o f a Hilb ert space has no in trinsic mea ning a nd often cannot be ma int ained in conc r ete s ituations. 19 In f act lo cal observ ables would app ear nonlocal. The incorrect use of these op erators led Irving Segal to the conclusion that l ocal observ able subalgebras in QFT are quan tum mec hanical t yp e I factors a claim which he withdrew after becoming aw are of the results by Araki [49] who show ed that they are of t yp e II I (later r eﬁned to the unique ”h yp erﬁnite type II I 1 ”). 20 F or the re ason of facilitating the co mparison with QM we us e the uniﬁed F ock space setting instead of the Sc hro edinge r form ulation. Although DPI is formu- lated in F ock space, ther e is no useful s e cond quantized forma lis m (8). The global algebra which contains all observ ables indep endent o f their lo- calization is the alg ebra B ( H ) o f all b ounded op era tors in Hilb ert space. Phys- ically imp ortant un b ounded op erators ar e not members but ra ther hav e the mathematical status o f being aﬃliated with B ( H ) and its subalgebra s ; this bo okkeeping makes it p os s ible to apply p ow erful theor ems from the theory of op erator algebra s (whereas un b ounded op erator s a re treated on a case to cas e basis). B ( H ) is the corr e ct globa l description whenever the physical system under discussion arises as the weak closure o f a ground state represe ntation o f an irreducible system of op erator s 20 be it QM or LQP . Accor ding to the clas- siﬁcation of operato r algebr a s, B ( H ) and all its m ultiples are of Murr ay von Neumann type I ∞ whose characteristic proper ty is the existence of minimal pro jectors ; in the irre ducible case these are the one-dimensional pro jector s b e- longing to mea surements which cannot b e reﬁned . There are prominent physical states which lead to diﬀeren t glo bal situations as e.g. thermal KMS s tates, but for the time b eing our interest is in ground states. The structural diﬀerences be t ween QM and LQP emerge as so on as one deﬁnes a physical substructure on the basis o f lo caliza tion. It is well k nown that a dissection of space into nonoverlapping spatia l r egions i.e. R 3 = ∪ i R i implies via Born lo calizatio n a tenso r factor ization of B ( H ) and H B ( H ) = O i B ( H ( R i )) (9) H = O i H ( R i ) , P ( R i ) H = H ( R i ) ˜ X op = Z a ∗ ( ~ x ) ~ xa ( ~ x ) d 3 x = Z ~ xdP ( ~ x ) (10) where the third line contains the deﬁnition o f the p o sition opera tor and its sp ectral decomp osition in the b oso nic F o ck s pace. Hence there is or tho gonality betw een subspaces b elong ing to loca lizations in nonov erlapping reg ions (orthog - onal Bo rn pro jectors) and one may talk ab out states which ar e pure in H ( R i ) . As well kno wn from the discussion of entanglement, a pure state in the global algebra B ( H ) may not be of the sp e cial tenso r pro duct form but rather b e a sup e rp osition of factorizing states; the Schmidt deco mpo sition is a metho d to achiev e this with an intrinsically determined bas is in the cas e of a bipartite tensor factoriza tion. States which are not tensor pro ducts, but r a ther sup erp o s itions of such, are called e ntangled; their reduced density matrix o bta ined by averaging ov er the environmen t of R i describ es a mixed state on B ( H ( R i )). This is the standard formulation of Q M in whic h pur e states are vectors a nd mixed states ar e dens ity matrices. 20 The closure in a thermal equilibri um state associated with a con tinuous sp ectrum Hamil - tonian leads to a unitarily inequiv alen t (t yp e II I) operator algebra without minimal pro jectors. 21 Although this quantum mechanical ent ang lement can b e related to the no- tion of entrop y , it is an entrop y in the sense of information the ory a nd not in the t hermal sense . One cannot crea te a ph ysica l temp erature as a quantita- tive measur e of the degree of q uantum mechanical entanglemen t in this wa y . which results from BNW-r estricting pure globa l states to a ﬁnite region and its outside environmen t. In particular the g round state alwa ys facto rizes, a spa - tial tensor factorization never causes v acuum p o la rization and ent ang lement in QM setting. The net s tructure of B ( H ) in terms the suba lgebras B ( H ( R i )) is of a kinematical kind; a lthough the reduced sta te may be impure, there is no B ( H ( R ) r e duced Hamiltonian r elative to which an impure state in QM beco mes a KMS s tate. Here Q M s tands for any QT without a maximal pro pa gation sp eed i.e. one which lacks causa l pr opaga tio n a nd v acuum p olar ization. The LQP coun terpar t of the Born-lo ca lized subalg ebras a t a ﬁxed time are the obser v a ble a lgebras A ( O ) for spa cetime double co ne r egions O obtained fr om spatial r egions R by caus al completion O = R ′′ (causal complement taken t wice); they for m what is called in the terminolo g y of LQ P a lo c al net {A ( O ) } O ⊂ M of op erator a lgebras indexed by r egions in Minko wski spacetime ∪O = M which is sub ject to the natural and obvious requir e ments o f isotony ( A ( O 1 ) ⊂ A ( O 2 ) if O 1 ⊂ O 2 ) and ca us al lo ca lity , i.e. the a lgebras co mm ute for space like separa ted regions. The c onnection with the sta nda rd formulation of Q FT in terms o f point- like ﬁelds is that smea red ﬁelds Φ( f ) = R Φ( x ) f ( x ) d 4 x with supp f ⊂ O under reasona ble gener al conditions g enerate loc a l algebras. P ointlik e ﬁelds, which by themselv es are to o singular to be op erator s (ev en if a dmitting unbound- edness), hav e a w ell-deﬁned mathematical meaning as op er a tor-v alued distri- butions brieﬂy referr ed to as generato rs o f algebras . The singular natur e of generating ﬁelds is therefore no t a pathological a sp ect leading to inescapa ble ultraviolet catas trophes, but rather a natur al attribute of pass ing from classica l to quantum ﬁelds. The real cumberso me as pec t is not their singular b ehavior but their mult i- tude; there are m yria ds of ﬁelds whic h genera te the same net of lo cal ope r ator algebras and int er po late the same par ticles wherea s in cla s sical ﬁeld theory they could b e distinguished by classical ﬁeld measurements. In this sense gener ating ﬁelds play a similar role in LQ P as co or dinates in mo dern diﬀerential geometr y i.e. they co ordinatize the net of spacetime indexed op erator algebras and only the latter has an in trinsic meaning ; in particular the particles and their collisio n theory can be o btained from the lo cal net without being forced to distinguish individual o p e rators within a lo cal alg ebra. But as the use of pa rticular co ordina tes often facilitates ge o metrical ca lc ulations, the use of pa r ticular ﬁelds, with e.g. the one with the low est shor t- distance dimen- sion within the inﬁnite charge equiv alence cla ss of ﬁelds, can grea tly simplify 21 calculations in QFT. Therefor e it is a problem of practica l imp or tance to con- struct a co v ariant basis of lo cally cov ariant p o intlik e ﬁelds of an equiv alence 21 The ﬁeld whic h is ”basic” in the sense of a Lagrangian ﬁeld in a Lagrangian approac h is generally simpler to deal with than composi tes of that ﬁelds (the Mass i ve Thirring ﬁeld i s simpler than the Sine-Gordon ﬁeld which maybe deri v ed from it). 22 class. F or mas sive free ﬁelds and for ma s sless free ﬁelds of ﬁnite helicit y suc h a basis is esp ecially simple; the ”Wick-basis” o f c omp o site ﬁelds still follows in part the lo g ic of classica l comp osites (apart from the deﬁnition o f the double dot : :). This remains so ev en in the presence of in teractions in whic h case the Wic k-or dering gets replaced b y the technically more demanding ”nor mal ordering” [50]. F or free ﬁelds in cur ved spacetime (CST) and the deﬁnition o f their comp os ites it is imp or tant to require the lo c al c ovari ant tr ansformation b ehavi or under lo ca l is o metries [51]. The co nceptual fr amework of QFT in CST in the presence of inter actions has also b een largely understo o d [52]. W e no w r eturn to the main question na mely: what changes if w e pa ss from the BNW lo calization o f Q M/DPI to the ca us al lo calizatio n of L QP? The crucial prop erty is that a lo calized alg ebra A ( O ) ⊂ B ( H ) together with its commutan t A ( O ) ′ (whic h under v er y g e ne r al conditions 22 is equal to the alg ebra of the causal dis jo in t of O i.e. A ( O ) ′ = A ( O ′ )) are tw o von Neumann factor alg ebras i.e. B ( H ) = A ( O ) ∨ A ( O ) ′ , A ( O ) ∩ A ( O ) ′ = C 1 (11) In contrast to the Q M algebra s the lo cal factor algebra s are not o f type I and B ( H ) do es not ten sor-factorize in terms of them, in fact they ca nnot even b e embedded into a B ( H 1 ) ⊗ B ( H 2 ) tensor pro duct. The pr ize to pay for ignoring this imp ortant fact a nd imp osing wro ng structure s is the app ear ance o f spurious ultr aviolet diver genc es, the typical way of a QFT mo del to resist enforcing an incompatible structur e on it. On the p ositive side, as will b e explained in the se cond part of this essay , without this sig niﬁcant c hang e in the nature of algebras ther e would b e no hologra phy onto causal horizo ns a nd the resulting h uge symmetry enhancement to inﬁnite- dimens io nal (BMS) groups , and of course there would b e no thermal behavior caused by lo calization and a fortiori no area -prop or tional lo calizatio n ent ro py . The situation in LQ P is radically diﬀerent from that of e ntanglement and pure versus mixed states in QM since lo cal alge br as a s A ( O ) have no pur e states at al l ; so the dichotom y b etw een pure a nd mixed states breaks down and the kind of entanglemen t caused by ﬁeld theoretic lo calization is muc h more vio le nt then that co ming from BNW-lo caliza tion 23 , in the terminology of Ruetsche [3 ] these s tates are instrinsic al ly mixe d. This implies that the standar d pure - mixed dichotom y do es not e x tend b eyond QM i.e. such intrinsically mixed states do not exist in a ny natural wa y on B ( H ) . A t the moment in whic h they come int o being as e.g. ther mo dynamic limit s tates in the inﬁnite volume limit, the algebra has cea sed to b e of the qua nt um mechanical B (H) type a nd b ecome a t yp e I I I ope rator algebra [9 6]. The thermodyna mic limit co nstruction at ﬁnite 22 In fact this duality relation can alwa ys be ac hieve d b y a process of maximalization (Haag dualization) which increases the degrees of freedom i nsi de O . A p edagogical illustration based on the ”generalized free ﬁeld” can b e found in [ 53]. 23 By introducing in addtion to free ﬁelds A ( x ) whic h are co v ariant F our i er transf orms also nonco v ariant F ourier transforms a ( ~ x , t ) , a ∗ ( ~ x, t ) one can expli ci tly that the latter ar e relativ ely nonlocal. 23 temper ature gives also the corr ect hint to the nature o f in trinsica lly mixed states; they are typically ”singular ” KMS sta tes i.e. KMS states which although b eing the thermo dynamic limits of Gibbs state cannot themselves b e r epresented in the Gibbs form b ecause the KMS Hamiltonia n has co n tinuous sp ectr um. Unlik e Born loc a lization, ca usal lo ca liz ation is not r e lated to p osition oper - ators and pro jector s P ( R ); rather the op erato r algebras A ( O ) are of an entirely diﬀerent kind than those met in g round state (zero temp era ture) QM ; they are all iso morphic to one a bstract ob ject, the hyperﬁnite type I I I 1 von Neuma nn factor also referred to as the monad the unique factor behind Araki’s 19 6 3 dis- cov ery [49] . As will be seen later LQ P creates its wealth y mansio n from just this one kind of brick; all its structural richness co mes from pos itioning the bricks, there is no thing hidden in the structure o f one bricks. In a later s ection it will be explained how this emerges from modular lo calizatio n and a related op erator formalism. The situation do es not change if o ne ta kes for O a region R a t a ﬁxed time; as stated befor e, in a theory with ﬁnite pro pa gation sp eed o ne has A ( R ) = A ( D ( R )) , where D ( R ) is the dia mo nd shaped double cone subtended b y R (the c a usal shadow of R ) . Even if there are no po int like generator s and if the theory (as the result of the existence of an elementary length) only admits a macro scopically lo calized net o f algebr as (e.g. a net of non-triv ia l w edge- lo calized factor algebras A ( W ) with tr iv ial double cone int er section algebras A ( O ) = { c 1 } ) , the a lgebras would still not tens o r factoriz e B ( H ) 6 = A ( W ) ⊗ A ( W ′ ) . Hence the prop erties under discussion are not directly related to the presence of s ing ular gene r ating po intlik e/stringlike ﬁelds but ar e co nnected to the existence of well-deﬁned (sharp) causa l sha dows. T her e is a hidden singula r asp ect in the s harpness of the O -lo caliza tion whic h generates inﬁnitely la rge v acuum p olariz ation clouds o n the caus a l horizo n o f the lo caliza tion. In the last section a metho d (splitting) will b e presented which p ermits to deﬁne a split- distance dep endent, but otherwise intrinsically deﬁned ﬁnite thermal entropy . Most divergencies (but not all, since the divergence of lo ca lization entropy for v anishing splitting distance is an unav oidable consequence o f the pr inciples) in Q FT a re the result of conceptual err ors in the form ulation resulting from tacitly iden tifying Q FT with some sort of relativistic QM 24 and in this w ay ignoring the in trinsica lly singular nature of p o intlike lo calized ﬁelds. Often it is thought tha t the av oidance of lo cality in fav or of nonlo c al cov ar i- ant op erato r s eliminates the singula r s hort distance b ehavior. But this is not quite true as evide nce d by the Kallen-Lehmann representation of a cov ariant scalar ob ject h A ( x ) A ( y ) i = Z ∆ + ( x − y , κ 2 ) ρ ( κ 2 ) dκ 2 (12) which was prop osed precisely to show that even without demanding lo cality , but retaining o nly cov aria nce and the Hilb ert s pace str uctur e (p os itivit y), a certain 24 The corr ect treatment of p erturbation theory which takes in to accoun t the si ngular nature of pointlike quan tum ﬁelds may yield more free parameters than in the classi cal setting, but one i s never r equired to confront inﬁnities or cut-oﬀs. 24 singular b ehavior of cov ar iant o b jects is unav oidable. In the DP I scheme this was av oided, b eca use even tho ug h there are pa rticles at all times, there are no cov aria n t (tensors, spinors) ob jects a t ﬁnite times, the o nly cov ar iant q uantit y arises in the form of the in v ariant S- matrix in the t → ∞ limit. The next section shows that a separa tion b etw een cov a riance and lo calization in the pur suit of a less singular mor e nonlo cal theor y is a futile endeav our, at least as long as o ne do es not sub ject spacetime itself to a radical r evision. In the algebr aic formulation the cov ar ia nce req uir ement refers to the geom- etry of the lo calization region A ( O ) i.e. U ( a, Λ) A ( O ) U ( a, Λ) ∗ = A ( O a, Λ ) (13) whereas no a dditio nal require men t a b o ut the trans formation b ehavior under ﬁnite dimensiona l (tensor, spinor ) Lorentz representations (whic h would bring back the unboundedness and thus pr even t the use of p owerful theorems in op- erator a lgebras) is impo sed for the individual op era tors. The singular nature o f po int like ge ne r ators (if they exis t) is then a purely mathematical consequence. Using such singula r ob jects in p ointlik e interactions in the same wa y a s one uses op erator s in QM leads to self-inﬂicted divergence pro blems. W e have seen tha t althoug h QM and QFT can b e describ ed under a com- mon mathema tica l ro of of C ∗ -algebra s with a state functional, as so o n as one int ro duces the physically imp ortant lo calizatio n s tructure, signiﬁcant concep- tual diﬀerences app ear. Thes e diﬀerences show up in the presence of v acuum po larization in QFT as a result of causal lo ca liz ation and they tend to hav e dramatic consequences ; the most prominent ones will b e pr esented in this a nd the subseq uent sections, more will be con tained in the second ess ay . The net structure of the o bserv ables allows a lo c al c omp arison of states : t wo states are lo cally equal in a regio n O if and only if the exp ectation v alues of all op erator s in are the same in bo th s tates. Loc a l deviations fro m any s tate, in particular fro m the v acuum state, can be meas ured in this manner ; states which are equal on the ca usal complemen t A ( O ′ ) that are indistinguishable from the v acuum a re called lo caliza ble in A ( O ) (”str ictly lo c a lized states” in the se ns e of Lich t [54]) can b e deﬁned. Due to the unav oida ble correlatio ns in the v acuum state in r elativistic qua nt um theory (the Reeh-Schlieder prop erty [5]), the space H ( O ) obta ined by applying the o p e rators in A ( O ) to the v acuum is, for a ny op en re gion O , dense in the Hilb ert space and thus far fro m b eing o rthogona l to H ( O ′ ). This somewhat coun ter-intuitiv e fa ct is insepara bly linked with a structural diﬀerence b etw een the lo cal algebra s and the alg ebras encountered in non-rela tivistic quantum mec hanics (or the global a lg ebra of a quantum ﬁeld asso ciated with the entire Minkowski space- time) a s mentioned in connection with the breakdown of tenso r-factor iz ation (11 ). The res ult is a particular b enevolent form o f ”Murphy’s law” for interact- ing Q FT: everything which is not forbidden (by sup ersele ction rules) to c ouple, r e al ly do es c ouple . On the lev el of interacting particles this has b een termed nucle ar demo cr acy : an y particle whose sup ersele c ted charge is contained in the sp ectrum whic h r esults from fusing the charges in a cluster of par ticles can be 25 viewed as a b ound state of that cluster of particles. Nuclear demo crac y even strips a particle with a fundamental charge o f its individualit y since such an ob ject can be consider ed as bound of itself + an arbitra ry num b er o f parti- cles with non-fundamental charges. This r enders interacting Q FT conceptually m uch more attractive a nd fundament al than QM, but it a lso co n tributes to its computational co mplexit y i.e. the b enevolen t character o f Murphy’s LQP law unfortunately do es not necessarily ex tend to the co mputatio nal side, a t lea st if one limits oneself to the standard to ols of QT. The Reeh-Schlieder prop erty [5] (in mor e p opular but less precise termi- nology: the ”sta te-ﬁeld r elation”) is p erhaps the strong est realiza tion of Mur- ph y’s law since it sec ures the existence of a lo calization r egion dep endent dense subspace H ( O ) = A ( O )Ω ⊂ H whic h canno t b e asso ciated with a no nt riv ial pro jector. It also implies that the ex p ecta tion v alue o f a pro jection op era tor lo calized in a b ounded region c annot be in terpreted as the probabilit y of de- tecting a particle-like ob ject in tha t region, since it is necessar ily nonzero if acting o n the v a cuum state. The A ( O )-r educed ground s ta te is a K MS ther- mal state at a appropria tely normalize d (Ha wking) temp erature (more in par t II). The intrinsically deﬁned mo dular ” Hamiltonian” asso ciated via mo dula r op erator theory 25 to a ”standar d pair ” ( A ( O ) , Ω vac ) is a lwa ys av ailable in the mathematical sense but a llows a physical in terpre tation o nly in those ra re cases when there exists an inv ar iance g roup of O which is a subg roup of the space- time group leaving Ω vac inv ariant. W ell known cases a re the Lor entz b o ost for the wedge regio n in Mink owski spac e time (the Unruh eﬀect) and the gener a - tor of a double-cone pr eserving conforma l transformation in a conforma l theo ry and certain Killing symmetries in bla ck hole physics. Its general purp ose is to give a n intrinsic description of the A ( O )- reduced v a c uum state in terms of an KMS state o f an Hamiltoian ”mov ement” where we us ed brack ets in or der to highlight the fact that this is gener ally not a geometr ic mov ement but only an algebraic automor phism of A ( O ) (and simultaneously of A ( O ′ )) which res p ects the geometric b oundaries (the ca usal horizon) of O 26 . It is never the Hamilto- nian asso ciated with a globally iner tial reference frame as in case of heat bath thermal sys tems . There exis ts in fact a whole family of mo dular Hamiltonians since the op era- tors in A ( O ) natur a lly fulﬁll the KMS condition fo r any sta ndard pair ( A ( ˇ O ) , Ω vac ) for ˇ O ⊃ O : i.e. the diﬀerent mo dular Hamiltonians and the KMS states change with the causally closed world ˇ O of the observer. The surpris ing asp ect is that the causal localiza tion structure of one QFT leads to an inﬁnite s upply of diﬀeren t Hamiltonians without any change of in teractions . The c hange of the mo dular Hamiltonian K O via a change of the lo calization reg ion will le a d to a new Hamiltonia n whose a utomorphic movemen t maintains the new r egion but leav es (after some ” mo dular time”) the o ld regio n i.e. this is not a family of Hamiltonians on a quantum mechanical algebra. Of par ticular in terest is the res tr iction of a mo dula r a utomorphism to the horizo n of a caus a lly closed 25 The mo dular Hamitonian is the i nﬁnitesimal generator K mod of the mo dular group ∆ it ≡ e − itK mod . (see next tw o sections). 26 In fact it induces a geometric mov ement on the horizon. 26 region H or ( O ); ther e ar e go o d indications that this deﬁnes a diﬀeomor phism which b elongs to the inﬁnite dimens ional B o ndi-Metzner-Sachs subg r oup of a gigantic symmetry group o f holo graphic pro jection onto hor izons (see part I I o n hologra phy). The s ituation just descr ibe d is one of extreme ” virtuality”, i.e. there is gen- erally not even the p ossibility to view it in terms o f an Gedankenexp eriment of a non-inertial (acceler ated) O - conﬁned observer for w ho m the mo dular mov ement is an O - pres erving diﬀeomorphism; such pure algebra ic movemen ts without individual orbits are often called ”fuzzy”. Whenever the mo dular mo vemen t passes to a diﬀeomorphism one can at least en visag e a Geda nkenexperiment which keeps the observer o n an O -pr e s erving track b y appropriate acceler a tions. The only geometr ic c a se in Minkowski spacetime is the situation prop osed ﬁrs t by Unruh [46], when O is a wedge i.e. a r egion W which is b ounded by tw o int ers ecting lightf ro nt s which o nly share the 2-dim. edge of their intersection.. Conformal theories for which the obse r v ables live in the Dirac-W eyl compacti- ﬁcation f M of the Minko wski spacetime lead to mo dular diﬀeomo r phisms even for compa c t double cones D 27 . The most interesting and prominen t case comes abo ut when space time cur- v ature is cr e ating a black ho le. In case there ar e time-like Killing or bits and a n extension of the space time such that the black hole horizo n is a e vent horizon in the sense of dividing the extended manifold into a causally inside/outside with separate K illing movemen ts, o ne is in the classical Hawking-like situation. What one in a dditio ns needs for the quantum setting is the existence of a quantum state which is inv a r iant under the Killing g roup ac tion. In the case of the Sch warzschild black hole all these r equirements are fulﬁlled, the extension is the Sch warzschild-Krusk al extensio n a nd the inv ariant state is the Hartle-Hawking state Ω H − H . In this ca se ( A ( O S − K ) , Ω H − H ) is a standard pair and the mo dular movemen t is the K illing orbit which resp ects the black hole even t hor izon. Where as the c a usal horizons in the pr evious Minko wski spa cetime examples w as an extremely ”ﬂeeting” ob ject, a black hole even t horizon ha s an in trinsic metric-imprinted p os ition. Besides their a strophysical interests, black holes ar e therefor e of considera ble philoso phica l interest. The only future developmen t which could still enfo r ce a sig niﬁcant modiﬁca tion of the present concepts is the still unknown quantu m gr avity (more remarks on QG in par t I I). F or computations of thermal pr op erties, including thermal entropy , it does not matter w he ther the horizon is a ”ﬂeeting” observer-dep endent causa l lo - calization 28 horizon or a ﬁxed curv ature ge nerated blac k hole event horizon ; only its direct observ able signiﬁca nc e depends on the black hole ev ent horizo n. This leads to a pictur e ab out the LQP - QG (quantum gr avit y) in terface which is s omewhat diﬀeren t from that in mos t of the literatur e; we will return to these issues in connectio n with the presentation of the s plit pro p erty in part I I of the essay . 27 The r egion obtained by inte rsecting a f or wa rd lightcone with arbitrary ap ex with an bac kward li gh tcone; 28 The lo calization entrop y which dep ends on the ”split” size (see below) is ho wev er an imp ortan t prop ert y of the model, even if it not direct y exper iment ally accessible. 27 Causality in relativistic quantum ﬁeld theory is ma thematically expres s ed through lo cal commutativit y , i.e., mut ual commutativit y of the algebra s A ( O ) and A ( O ′ ). There is an intimate co nnec tion o f this prop erty with the p os sibil- it y of prepar ing states that exhibit no mutual co rrelations for a given pair of causally disjoint r e g ions. In fact, in a re c e n t pap er Buchholz a nd Summers [7] show that lo cal co mm utativity is a necessary condition for the existence of such uncorrela ted sta tes . Conv ersely , in combination with some further pro per ties related to deg rees of freedo m densities (split prop erty [57], existence of sca ling limits [5 8]), lo ca l commutativit y leads to a v ery satisfactory picture of statistic al indep endenc e and lo c al pr ep ar abilty of states in relativistic quantum ﬁeld theory . W e refer to [59][60] for thor ough discussions o f these matters and [55][15] for a brief r eview of s ome physical conseq uences. The last tw o pap ers a lso explain how the a bove men tioned concepts av oids spurio us problems r o oted in as sumptions that are in c o nﬂict with basic pr inciple s of rela tivistic qua nt um physics. In pa rticular it can b e shown how an alleged diﬃculty [8][9] with F ermi’s famous Gedanken- exp eriment [56], which F ermi pro po sed in order to s how that the velocity of light is a lso the limiting propaga tion velo c it y in quantum electr o dynamics, can be resolved b y taking [5 5] int o ac count the progr ess on the conceptual issues of causal lo caliz a tion and the gain in mathema tical rig or since the times of F er mi. After having dis cussed some signiﬁcant co nceptual diﬀerences betw een QM and LQP , o ne naturally asks for an a rgument why and in which way QM app ears as a nonrelativis tic limit o f LQ P . The s tandard kinematica l re a soning o f the text- bo oks is acceptable for fermionic/b osonic sys tems in the sense of ”F APP”, but has no t muc h streng th o n the c o nceptual level. T o see its weakness, imagine for a mo ment that w e would live in a 3-dim. world of anyons (a be lia n plek- tons, where plektons are Wigner particles with bra id gro up statistics). Such relativistic o b jects ar e by their very statistics so tightly interw ov en that there simply a re no compactly lo calized free ﬁelds which only crea te a lo calized an yon without a v acuum p o la rization clo ud admixture. In such a world no nonr ela- tivistic limit which ma int ains the s pin-statistic co nnection co uld lea d to QM, the limiting theory would rather r emain a nonr elativistic QFT . In o rder to av oid misunderstandings, its is not claimed here that the is sue o f nonr elativistic lim- its of any interacting r elativistic QFT is ma thematically understo o d 29 , ra ther the statement is that plektonic (bra id-group) commutation rela tions, relativis- tic or nonrelativistic, interacting or not, are inco mpatible with the structure o f (Sc hro edinger) QM. I n 4-dimensiona l spacetime ther e is no such obstacle ag ainst QM, simply b ecause it is not the F ermi/Bose statistics which causes v acuum p o- larization; to formulate it more pr ov o catively: there would be no Schroedinger QM without the existence of free r elativistic fermions /b osons. 29 The argumen ts ab out the nonrelativistic limit og QFT ha ve remained metaphoric; how ev er the existence of exactly sol v ed interact ing 2-dim. QFTs raises now hopes that age old problem will b e b etter understoo d. 28 6 Mo d ular lo calization Previous ly it was mentioned on several o ccasio ns that the lo calization under- lying QFT c a n be fre ed from the co n tingencie s of ﬁeld co ordina tiza tions. T his is a chieved b y a physically as well mathematically impressive, but fo r histo ric and so ciolog ical rea sons little known theory . Its name ”mo dula r theo ry” is of mathematical or igin and refer s to a v a st g e neralization of the (uni)mo dularity encountered in the relation be tw een left/rig h t Haar measur e in group repre- sentation theory . In the middle 60 s the mathematician T omita pr esented a signiﬁcant generalization of this theor y to op era tor algebra s and in the subse- quent years this theory r eceived essential improv ements from T akesaki and later from Connes. A t the sa me time Haag, Hugenho ltz and Winnink published their work on statistical mechanics of o pe n systems [5]. When the ph ysicists and ma themati- cians met at a conference in Baton Roug e in 19 66, there was sur prise a bo ut the similarity o f conc e pts, followed by deep appreciation a b o ut the perfectio n with which these indep endent developmen ts supp or ted ea ch other [61]. Physicists not only ada pted mathema tica l terminolo gy , but mathematicians also to ok some o f their terminology from physicists as e.g. K MS states which r efer to Kubo , Mar- tin and Sch winger who int ro duced an analytic pr op erty of Gibbs states mer ely as a computational too l (in or der to av oid computing traces), Haag, Hugenholtz and Winnink re a lized that this pro pe rty (which they termed the KMS prop er t y) is the only asp ect which survives in the ther mo dynamic limit when the trace formulas lose s its meaning and must be replaced b y the analytic KMS b ounda ry condition. This turned o ut to be the right concept for for m ulating a nd s o lving problems directly in the setting of op en systems. In the present work the terminolog y is mainly used for thermal states of o p en systems which are not Gibbs states. They are t ypical for LQP , for example every mult ipar ticle state Ω particle of ﬁnite ener gy , including the v acuum, (i.e. ev ery ph ysical particle s tate) up on restriction to a lo ca l algebra A ( O ) bec o mes a KMS state with respec t to a ”mo dular Hamiltonian” whic h is canonically deter mined by ( A ( O ) , Ω particle ). Connes, in his path-breaking w ork on the classiﬁcation of von Neumann factors [6 3], made full use o f this hybrid ma th.-phys. terminology whic h dev el- op ed after Baton Rouge . Now adays one ca n meet mathematicians who use the KMS prop erty but do not know that this was a mere computatio nal too l by 3 ph ysic ists (Kub o. Mar tin and Sch winger ) to av oid calcula ting traces and that the co nceptual aspect was only realized la ter by Haa g Hugenholtz and Win- nink who gav e it its ﬁnal na me. One can hardly think o f any other co nﬂuence of mathematical and physical ide a s on such a pro found and at the sa me time equal and natura l le vel as in mo dula r theory ; even the Hilb ert space formalism of QM already existed for man y years b efor e quantum theorists b ecame aw are of its use. Abo ut 10 years after Baton Roug e, Bisognano and Wichm ann [17] discovered that a v a cuum state restr icted to a w edge- lo calized oper ator algebra A ( W ) in QFT deﬁnes a mo dular se tting in which the restricted v acuum b ecomes a 29 thermal KMS state with resp ect to the wedge-aﬃliated L-b o ost ”Hamiltonian”. This step mark s the b eginning of a very natur a l yet unexp ected relation b e t ween thermal and geo metric pr op erties, one which is totally characteristic fo r Q FT i.e. which is not shared by cla ssical theory no r by QM. Therma l asp ects of black holes were how ever disc ov ered indep endent o f this w or k, and the ﬁrst ph ysicist who saw the co nnection with modular theory was Ge o ﬀrey Sewell [18]. The theor y b ecomes more a ccessible for physicists if one introduces it ﬁrst in its mor e limited spatial- instea d of its full alge br aic- cont ext. Since as a foundational structure of LQP it merits more attent ion than it hitherto rec eived from the particle physics communit y , so me of its metho ds and achiev ements will be presented in the seq uel. It has b een realiz ed by Brunetti, Guido and Longo 30 [13] that there exists a natural lo caliza tion structure on the Wigner r epresentation s pace for any po sitive energy repres e ntation of the prop er Poincar´ e g roup. The sta rting point is a n irreducible r epresentation U 1 of the Poincar´ e´gr oup on a Hilb ert space H 1 that after ”se cond quantization” becomes the single-par ticle subspace of the Hilber t spa ce (Wigner- F o ck-space) H W F of the quan tum ﬁelds act 31 . In the bo sonic case the co ns truction then proc e eds according to the following steps [13][62][15]. One ﬁrst ﬁxes a r e fer ence wedge regio n, e.g. W 0 = { x ∈ R d , x d − 1 >   x 0   } and considers the one-pa rametric L-b o ost group (the hyperb olic r o tation b y χ in the x d − 1 − x 0 plane) w hich leaves W 0 inv ariant; one also needs the reﬂection j W 0 across the edge of the wedge (i.e. a lo ng the co or dinates x d − 1 − x 0 ). The j W 0 extended Wigner re pr esentation is then used to deﬁne t wo commuting wedge- aﬃliated op era tors δ it W 0 = u (0 , Λ W 0 ( χ = − 2 π t )) , j W 0 = u (0 , j W 0 ) (14) where attent ion should b e paid to the fact that in a p ositive energ y r epresen- tation an y operato r which inv erts time is necessa rily a ntilinear 32 . A unitary one- parametric strongly c ontin uous subg roup as δ it W 0 can b e written in terms of a selfadjoint generator K as δ it W 0 = e − itK W 0 and therefore pe rmits an ”a na- lytic contin uation” in t to an unbounded densely deﬁned po sitive o per ators δ s W 0 . With the help of this op erator o ne deﬁnes the unbounded antilinear op erato r which has the same dense domain as its ”r adial” part s W 0 = j W 0 δ 1 2 W 0 , j δ 1 2 j = δ − 1 2 (15) Whereas the unitary oper ator δ it W 0 commutes with the r eﬂection, the antiu- nitarity of the reﬂection c hanges the s ign in the analytic contin uation whic h 30 In a more limited conte xt and with less m athematical ri gor this was independent ly pro- posed in [14]. 31 The construction works for arbitrary posi tiv e energy represen tations, not only irr educible ones. 32 The w edge reﬂection j W 0 diﬀers from the TCP operator only by a π -rotation around the W 0 axis. 30 leads the commutation relation betw een δ and j in (15). This causes the inv o- lutivit y of the s- op erator on its do main, a s well as the identit y of its range with its doma in s 2 W 0 ⊂ 1 dom s = ran s Such op erators which ar e unb ounde d and yet involutive o n their domain are very un usua l; according to my best knowledge they only app ear in mo dula r theory and it is precisely these unusual pr op erties which a re ca pable to enco de geomet- ric lo caliza tio n pro pe rties into doma in pr op erties of abstr act quantum op era tors, a fantastic achiev ement completely unknown in Q M. The mo re genera l algebr aic context in whic h T o mita discovered mo dular theo r y will be men tioned later. The idemp otency means that the s-op era tor has ± 1 eig enspaces; since it is antilinear, the +space m ultiplied with i changes the sign and b ecomes the - space; hence it suﬃces to introduce a notation fo r just one eigenspace K ( W 0 ) = { domain of ∆ 1 2 W 0 , s W 0 ψ = ψ } (16) j W 0 K ( W 0 ) = K ( W ′ 0 ) = K ( W 0 ) ′ , dual i ty K ( W 0 ) + i K ( W 0 ) = H 1 , K ( W 0 ) ∩ i K ( W 0 ) = 0 It is imp or tant to b e aw are that, unlike QM, we ar e her e dealing with r eal (closed) subspaces K o f the co mplex one-particle Wigner repr e sentation space H 1 . An a lternative which av oids the use of re a l subspaces is to directly dea l with complex dense subspac e s H 1 ( W 0 ) = K ( W 0 ) + i K ( W 0 ) as in the third line. Int ro ducing the graph norm o f the dense space the complex subspace in the third line b ecomes a Hilb ert s pa ce in its own right. The second and third line require some e x planation. The upp er das h o n regions deno tes the causal dis jo in t (whic h is the opp osite wedge) whereas the das h o n real subspaces means the symplectic complement with respec t to the symplectic form I m ( · , · ) o n H 1 . The tw o proper ties in the third line are the deﬁning r elations of what is called the standar dness pr op erty o f a real subspace 33 ; any standard K space per mits to deﬁne an a bstract s- op erator s ( ψ + iϕ ) = ψ − iϕ (17) s = j δ 1 2 whose p ola r decomp osition (written in the second line) yields tw o mo dular o b- jects, a unitary mo dular gro up δ it and a a n tiunitary reﬂection which genera lly hav e howev er no geometric signiﬁcance. The do main of the T omita s -op erato r is the same as the domain o f δ 1 2 namely the real sum of the K s pace a nd its 33 According to the Reeh-Schlieder theorem a lo cal algebra A ( O ) in QFT i s in standard posi tion wi th respect to the v acuum i.e. it acts on the v acuum in a cyclic and separating manner. The spatial standardness, whic h follows directly from Wigner representat ion theory , is just the one-particle pro j ection of the Reeh-Sc hlieder proper ty . 31 imaginary mult iple. Note that this domain is determined solely in terms o f Wigner gr oup representation theo ry . It is ea sy to obtain a net o f K-spa ces by U ( a, Λ)-transforming the K-spa ce for the distinguis hed W 0 . A bit mor e tricky is the construction of s harp er lo calized subspaces via in tersections K ( O ) = \ W ⊃O K ( W ) (18) where O denotes a causally c o mplete smaller reg ion (noncompact spac e like co ne, compact double cone). Intersection may not be standar d, in fact they may b e zero in whic h case the theory a llows lo caliza tion in W (it a lwa ys do e s ) but not in O . Such a theo ry is still causal but not lo cal in the sense that its a sso ciated free ﬁe lds are po int like. One can show that the in tersection for spacelike co nes O = C for all pos itive energy is alwa ys sta ndard. A standar d subspace is uniqely aﬃliated with a T omita s-in volution (17). A t this p oint the important ques tion ar ises why , if these lo ca liz a tion sub- spaces are imp ortant for particle ph ysics they did not appea r already at the time of Wigner? After a ll, unlike the Wigner p ositio n op era tors, these spaces are frame indep endent (cov ariantly deﬁned) and for t wo causally separa ted re- gions O 1 and O 2 regions the simplectic inner pro duct v anishes I m ( ψ 1 , ψ 2 ) = 0 , ψ i ∈ H ( O i ) (19) [Φ( ψ 1 ) , Φ( ψ 2 )] = 0 Hence the symplectic inner pro duct of mo dular lo ca liz ed o ne-particle wa ve func- tions is nothing else than the fre e ﬁeld co mmutator function: the mo dular lo- calization preempts the algebr aic structure of free ﬁelds without having the use of any quant izatio n formalism. Na tur ally this would hav e b een of great interest to Wig ner, but the modular lo calizatio n concepts were only a v ailable more tha n half a cen tury later. Note that the rela tivistic DP I setting also starts from Wigner par ticles but it completely ignor e s the pres ence of this mo dular lo ca lization s tructure which, would anyhow not b e consistent with the DPI interactions. There are three classes of irre ducible p ositive energy repre s ent atio n, the family of mass ive represe ntations ( m > 0 , s ) with ha lf-int eg e r spin s and the family of ma ssless r epresentation whic h consists really o f tw o subfamilies with quite diﬀere nt prop erties na mely the (0 , h = half-integer) clas s, o ften called the neutrino-photon cla ss, a nd the r ather lar g e cla ss of (0 , κ > 0 ) inﬁnite helicity representations par a metrized by a contin uous-v alued Casimir in v ariant κ [15] . F or the ﬁrst tw o classes the K -space the standa rdness pr op erty a lso holds for double cone intersections O = D for a rbitrarily small D , but this is deﬁnitely not the ca s e for the inﬁnite helicity family for which the lo caliz ation spaces for com- pact spacetime r egions turn out to b e trivia l 34 . Passing from lo caliz ed subspace s 34 It i s qui te easy to prov e the standardness for spacelike cone lo calization (leading to singular stringlike generating ﬁelds) j ust from the p ositive energy prop ert y which is shared by all three families [13]. 32 K in the representation theo retical se tting to singular c ov ariant generating w av e functions (the ﬁrst quantized a nalogs of gene r ating ﬁelds) one can show that the D lo calizatio n lea ds to p ointlik e singular gener ators (state-v alued distributions) whereas the spacelike c o ne lo c alization C is as so ciated with semiinﬁnite space- like s tringlike s ing ular generator s [15]. Their second quantized coun terpa rts are po int like or s tr inglike cov ar iant ﬁelds . It is remar k able that one do es not need to introduce generators which are lo c a lized on hyp e rsurfaces (bra nes). Although the o bserv ation tha t the third Wigner repres ent atio n cla s s is not po int like g enerated was made man y decades ago, the statemen t tha t it is semi- inﬁnite string-g enerated a nd that this is the worst p os sible case of sta te lo caliza- tion (which needs the knowledge of mo dular theory) is of a mor e recent vin tag e [13][15]. But wha t is the physical signiﬁcance of mo dula r lo ca liz a tion of wa ve function which, diﬀerent fr om the pr o babilistic BNW lo calized states are obviously frame- independent and hence cannot b e used for descr ibing the dissipa tio n o f wa ve pack ets and the related sca tter ing theory? The answer is that they a re the pro jections of the dense subspaces (the Reeh-Schlieder domains) g e nerated by applying mo dula r extension of the localized subalgebra A ( O ) to the v acuum onto the one-particle space P 1 H ( O ) = K ( O ) + i K ( O ) ≡ H 1 ( O ) (20) H ( O ) = domS ( O ) , S ( O ) A Ω = A ∗ Ω , A ∈ A ( O ) In other words the one-pa rticle dense lo ca lization spaces a re pro jections of the Reeh-Schlieder space s 35 . There is a very subtle as pec t of mo dular lo caliza tion which one encounters in the second Wigner representation class of mas s less ﬁnite helicity representations (the photon, g raviton..class). Wherea s in the massive ca se all s pino rial ﬁelds Ψ ( A, ˙ B ) the re la tion of the physical spin s with the tw o s pino rial indices follows the naive a ngular momen tum comp ositio n rules [16]    A − ˙ B    ≤ s ≤    A + ˙ B    , m > 0 (21) s =    A − ˙ B    , m = 0 the second line contains the sig niﬁcantly reduced num b er o f spinoria l desc r ip- tions for zer o mass and ﬁnite helicity repres entations. What is going on here, why is there, in co n tra distinction to c la ssical ﬁeld theory no cov aria n t s=1 vector-po tential A µ or no g µν in case of s=2 ? Why are the admiss ible cov a r iant generator s of the Wigner repres ent atio n in this cas e limited to ﬁeld strengths (for s=2 the linear ized Riema nn tensor )? The short answer is that all these missing gener ators exist as stringlike co - v ariant ob jects, the above restrictio n in the massless cas e only r esults from the 35 At the time of the discov ery of the densit y of the spaces A ( O )Ω the mo dular theory w as not yet known. There is no c hange of conten t if one uses the same terminology for their modular extension domS ( O ) . 33 cov aria n tizatio n to p ointlik e generators. The full r a nge o f spinorial p oss ibilities (21) retur ns in ter ms of s tr ing lo caliz e d ﬁelds Ψ ( A, ˙ B ) ( x, e ) if s 6 =    A − ˙ B    . These generating free ﬁelds ar e cov ariant and ”string-lo cal” U (Λ)Ψ ( A, ˙ B ) ( x, e ) U ∗ (Λ) = D ( A, ˙ B ) (Λ − 1 )Ψ ( A, ˙ B ) (Λ x, Λ e ) (22) h Ψ ( A, ˙ B ) ( x, e ) , Ψ ( A ′ , ˙ B ′ ) ( x ′ , e ′ i ± = 0 , x + R + e > < x ′ + R + e ′ Here the unit vector e is the spacelike direction of the semiinﬁnite str ing and the last line expresses the spacelike fermio nic/ b osonic spacelike c o mmu tatio n. The bes t known illustration is the ( m = 0 , s = 1) v ectorp otential repre s entation; in this case it is well-known that although a genera ting p ointlik e ﬁeld s tr ength exists, there is no p ointlike v ectorp otential a cting in a Hilber t spa ce. According to (22) the mo dular lo calization appro ach oﬀers as a substitute a stringlike cov ariant vector p otential A µ ( x, e ) . In the case ( m = 0 , s = 2) the ”ﬁeld strength” is a fourth degree tenso r which has the symmetry pro per ties of the Riemann tensor (it is often r eferred to a s the line arize d Riemann tensor). In this case the string-lo calized potential is of the for m g µν ( x, e ) i.e. res emb les the metric tenso r o f genera l re la tivity . Some consequence s of this lo caliza tio n for a reformulation of gauge theory will be men tioned in sectio n 8. Even in ca s e of massive fr ee theories wher e the repres e n tation theor etical ap- proach of Wig ner does no t r e quire to go b eyond p ointlik e lo caliza tion, co v ariant stringlike lo calized ﬁelds exist. Their a ttractive pro p erty is that they improve the short distance b ehavior e.g. a ma ssive pointlik e vector-po tential o f sdd=2 passes to a string lo calized vector p o tent ial of sdd=1 . In this way the incr ease of the sdd of pointlik e ﬁelds with spin s can be traded aga inst string lo calized ﬁelds of spin indep endent dimens io n with sdd= 1 . This obse r v ation would suggest the po ssibility o f a n enormous potential enlarge ment o f p erturbatively accessible higher spin in teraction in the sense of p ower counting. A diﬀerent kind of space like string-lo ca lization arises in d=1+ 2 Wigner rep- resentations with anomalous spin [64]. The ama zing p ow er of the mo dular lo- calization approach is that it preempts the spin-statistics connection a lready in the one-par ticle setting, namely if s is the spin of the particle (which in d= 1+2 may tak e on any real v alue) then o ne ﬁnds for the connection of the symplectic complement with the causal complemen t the generalized dualit y r elation K ( O ′ ) = Z K ( O ) ′ (23) where the s quare of the twist op erator Z = e π is is easily s e e n (by the co nnection of Wigner r epresentation theor y with the tw o- po int function) to lead to the statistics phase = Z 2 [64]. The fact that one never has to go beyond string lo calization (and fact, apar t from s ≥ 1, never beyond p oint lo calizatio n) in orde r to obta in genera ting ﬁe lds for a Q FT is re ma rk able in view of the many attempts to introduce extended ob jects into QFT. 34 It is helpful to b e a gain reminded that modula r lo calization which go es with real subspaces (or dense complex subspaces), unlik e BNW lo caliza tion, cannot be connected with probabilities and pr o jectors. It is ra ther related to causal lo calization a s pe c ts ; the standar dness of the K-space for a compact r egion is nothing else then the one-par ticle version of the Reeh- Schlieder pr op erty . As will b e seen in the next sectio n mo dular lo ca lization is also an imp ortant too l in the non-p erturbative cons tr uction of interacting mo dels. 7 Algebraic asp ects of mo dular theory A ne t of rea l subspac es K ( O ) ⊂ H 1 for an ﬁnite spin (helicity) Wigner repre- sentation can b e ”seco nd quantized” 36 via the CCR (W eyl) r e sp e ctively CAR quantization functor ; in this w ay one obtains a cov ariant O -indexed net of von Neumann a lgebras A ( O ) a cting o n the b oso nic o r fermionic F o ck s pace H = F ock ( H 1 ) built over the one-particle Wigner s pa ce H 1 . F or in teger spin/helicity v alues the modular lo c alization in Wigner space implies the identiﬁcation of the symplectic complement with the ge ometric complement in the sense o f re l- ativistic causality , i.e. K ( O ) ′ = K ( O ′ ) (spatial Haag duality in H 1 ). The W eyl functor takes this spatial version of Haag duality into its algebra ic c ounterpart. One pro ce e ds as fo llows: for ea ch Wigner w ave function ϕ ∈ H 1 the as so ciated (unitary) W eyl oper ator is deﬁned as W ey l ( ϕ ) := expi { a ∗ ( ϕ ) + a ( ϕ ) } ∈ B ( H ) (24) A ( O ) := al g { W ey l ( ϕ ) | ϕ ∈ K ( O ) } ′′ , A ( O ) ′ = A ( O ′ ) where a ∗ ( ϕ ) and a ( ϕ ) are the us ual F o ck space c reation and a nnihila tion op- erators of a Wigner particle in the wa ve function ϕ . W e then deﬁne the von Neumann a lgebra corresp onding to the loca lization r egion O in ter ms o f the op erator algebr a generated by the functoria l image of the mo dular co nstructed lo calized s ubspace K ( O ) as in the second line. By the v on Neumann double commutan t theorem, our genera ted op erato r algebra is weakly closed by deﬁni- tion. The functorial relation b etw een r e a l subspa ces a nd von Neumann alg ebras via the W eyl functor preserves the causal lo calization structure and hence the spatial duality pass es to its algebr aic counterpart. The functor also commutes with the improvemen t of lo calization through int er s ections ∩ acco rding to K ( O ) = ∩ W ⊃ O K ( W ) , A ( O ) = ∩ W ⊃ O A ( W ) as expr essed in the commutin g diagram { K ( W ) } W − → { A ( W ) } W (25) ↓ ∩ ↓ ∩ K ( O ) − → A ( O ) 36 The terminology 2 nd quan tization is a misdemeanor s ince one is dealing with a rigorously deﬁned functor within QT which has little i n common wi th the artful use of that parall ellism to classical theory called ”quan tization”. In Edward Nelson’s words: (ﬁrst) quan tization is a myst ery , but second quant ization is a functor. 35 Here the vertical a rrows denote the tigh tening of localiz ation b y in tersection whereas the ho rizontal ones denote the a ction of the W eyl functor. This com- m uting diagram expr esses the functorial relation b etw een particles and ﬁelds in the absence of interactions. In the interacting c ase the loss of the diagram and the unsolved par ticle-ﬁeld problems ar e sy nonymous. It is also the r eason why , in contrast to QM, the e x istence problem o f int er acting QFTs e ven after more than 80 y ear s rema ins unsolved. The wedge regions c o ntin ue to pla y a dis- tinguished role in attempts to construct interacting models (for s ome mo dular successes in d=1+1 see b elow). The case of half-integer spin repres ent atio ns is analogo us [6 2], apa rt from the fact that there is a mismatch betw een the ca usal and symplectic co mplements which m ust be taken care of by a t wist op er ator Z and as a result one has to use the CAR functor instead of the W eyl functor . In ca se o f the larg e family of ir reducible ze ro ma ss inﬁnite spin r e presen- tations in which the lightlik e little gro up is faithfully represented, the ﬁnitely lo calized K-spa ces ar e trivial K ( O ) = { 0 } and the m ost t ightly lo c alize d nontriv- ial sp ac es ar e of the form K ( C ) for C an arbitrarily narrow sp ac elike c one . As a double cone co nt ra cts to its co r e which is a p o int, the cor e of a space like co ne is a c ovaria nt sp ac elike semiinﬁnite string . The ab ov e functorial construction works the same wa y for the Wigner inﬁnite spin repr esentation, except that in that cas e there ar e no nontrivial algebras which hav e a s maller lo caliza tio n tha n A ( C ) a nd there is no ﬁeld which is sharp er lo calized than a semiinﬁnite string. As stated befo re, stringlike genera to rs, which are also av ailable in the p o intlik e ca se, turn out to have a n improved short distance behavior which makes them preferable from the p oint of view of for mulating interactions within the p ow er co unt ing limit. They c a n b e constructed fr o m the unique Wigner r epresentation by so called intert winers b etw een the unique c anonical and the many p ossible cov ar i- ant (dotted-undotted spinoria l) r epresentations. The Euler-La grange asp ects plays no direct role in these construction since the ca usal asp ect of hyper b o lic diﬀerential propagation are fully ta ken c a re o f by modular lo caliz a tion and also bec ause most of the spinor ial higher spin repr e sentations (21) anyhow ca nnot b e characterized in terms of E uler -Lagr ange equatio ns. The mo dular lo ca lization is the mor e gener al metho d o f implementating ca usal pro pagation than that from hyperb olic equa tions of mo tions. A basis of lo cal cov aria nt ﬁeld co or dinatizations is then deﬁned b y Wick comp osites of the free ﬁelds. The c ase which deviates fur thest from cla ssical be- havior is the pur e s tringlike inﬁnite spin ca se which rela tes a c ontinuous family of free ﬁelds with one irr educible inﬁnite spin r epresentation. Its no n-classica l asp ects, in pa rticular the absence of a Lag rangia n, is the reaso n why the space - time descr iptio n in terms of semiinﬁnite string ﬁelds has b een discov ered only recently rather than at the time of Jo rdan’s ﬁe ld quantization or Wig ne r ’s rep- resentation theor etical appro ach. Using the sta nda rd notation Γ for the s econd quan tization functor whic h maps r eal lo calized (one-par ticle) subspac es into lo calized von Neumann alge- bras and extending this functor in a natural wa y to include the images o f the K ( O )-as s o ciated s, δ, j which are denoted by S, ∆ , J, one arrives at the T o mita 36 T akesaki theory of the interaction-free lo ca l alg ebra ( A ( O ) , Ω) in sta nda rd p o- sition 37 H F o ck = Γ( H 1 ) = e H 1 ,  e h , e k  = e ( h,k ) (26) ∆ = Γ( δ ) , J = Γ( j ) , S = Γ( s ) S A Ω = A ∗ Ω , A ∈ A ( O ) , S = J ∆ 1 2 With this we arrive at the core s ta temen t of the T omita-T a kesaki theorem which is a statement ab out the t wo mo dular ob jects ∆ it and J on the a lg ebra σ t ( A ( O )) ≡ ∆ it A ( O )∆ − it = A ( O ) (27) J A ( O ) J = A ( O ) ′ = A ( O ′ ) in words: the reﬂection J maps an algebra (in standar d p osition) int o its von Neumann commut ant and the unitar y g roup ∆ it deﬁnes an o ne-parametric automorphism-g roup σ t of the algebr a. In this form (but without the last geo- metric statement in volving the geometric al causal complement O ′ ) the theorem hold in complete mathematical g enerality for standa rd pairs ( A , Ω). The free ﬁelds and their Wick comp os ites a re ”co or dinatizing” singular gener ators of this O - indexed net of op erator alg e bras in the sense that the smeared ﬁelds A ( f ) with suppf ⊂ O ar e (un b ounded op erator s) aﬃlia ted with A ( O ) and in a ce r tain sense gener ate A ( O ) . In the ab ove seco nd qua nt izatio n context the origin of the T-T theor e m and its pro of is clea r: the s ymplectic disjoin t passes via the functorial o p eration to the o pe r ator algebr a commutan t (21) and the spa tia l one-pa rticle automor phism go es into its algebraic counterpart. The deﬁnition o f the T o mita inv olution S through its ac tio n on the dense set of states (gua rantied by the standar dness of A ) as S A Ω = A ∗ Ω a nd the action of the tw o mo dular o b jects ∆ , J (2 6) is part of the g e neral setting o f the mo dular T o mita-T akesaki theo ry of abstract op erator algebras in ”standard p osition”; standardness is the mathematical terminology for the physicists Reeh-Schlieder pr o p erty i.e. the existence 38 of a vector Ω ∈ H with resp ect to which the algebr a acts cyclic and has no ”a nnihila tors” o f Ω . Naturally the pro of o f the abstract T-T theorem in the general s etting of op erator a lgebras is more involv ed 39 . The domain of the unbounded T omita inv olution S turns o ut to b e ”k ine- matical” in the sense that the dense set which features in the Reeh- Schlieder theorem is deter mined in terms o f the repr e sentation of the co nnected par t of the Poincar´ e group i.e. the particle/ spin s pe c trum 40 . In other words the Reeh- 37 The functor Γ preserves the standardness i. e. maps the spatial one-particle standardness int o its algebraic coun terpart. 38 In QFT an y ﬁnite energy v ector (whic h of course includes the v acuum) has this prop erty as well as any nondegene rated KMS state. In the mathematical setting it is sho wn that standard vectors are ” δ − dense” in H . 39 The local algebras of QFT are (as a consequen ce of the split property) h yp er ﬁnite; f or suc h op erator algebras Longo has given an elegan t pr o of [66]. 40 F or a w edge W the domain of S W is determined in terms of the domain of the ”analytic con tinuat ion” ∆ 1 2 W of the wedge -asso ciated Lor en tz-b oost subgroup Λ W ( χ ) , and for subw edge lo cali zation regions O the dense domain is obtained in terms of int ersections of wedge domains. 37 Schlieder domains in an interacting theor y with asymptotic completeness ar e ident ical to those of the incoming or outgo ing free ﬁe ld theory . The impor tant prop er t y which render s this useful b eyond free ﬁelds as a new constructive to o l in the pres ence of int er a ctions, is that for ( A ( W ) , Ω) the antiunit ar y inv olution J dep ends on the interaction, whereas ∆ it contin ues to be uniquely ﬁxed by the re presentation of the Poincar´ e gro up i.e. by the particle conten t. In fact it has b een known for s ome [14] time that J is r elated with its free counterpart J 0 through the scattering ma trix J = J 0 S scat (28) This mo dular ro le o f the scattering matrix as a relative mo dular inv ar iant betw een an interacting theory and its free counterpart comes as a surpr is e. It is precisely this prop erty which op ens the w ay for an inv ers e scatter ing con- struction. If one only looks at the dense lo calizatio n of states which features in the Reeh-Schlieder theorem, o ne miss es the dynamics. T her e is presently no other wa y to inject dynamics than generating these states b y applying op erators from op era tor algebra s . The pro p er ties of J are es sentially determined b y the relation of loca lized op era tors A to their Hermitia n adjoints A ∗ 41 . The physically relev ant facts emerg ing from mo dular theory can be con- densed into the follo wing statements: • The domain of t he un b ounde d op er ators S ( O ) is ﬁxe d in terms of int er- se ctions of the we dge domains asso ciate d to S ( W ); in other wor ds it is determine d by t he p article c ontent alone and ther efor e of a kinematic al natur e. These dense domains change with O i.e. the dense set of lo c alize d states has a bund le structu r e. • The c omplex domains D omS ( O ) = K ( O ) + iK ( O ) de c omp ose into r e al subsp ac es K ( O ) = A ( O ) sa Ω . This de c omp osition c ontains dynamic al in- formation which in c ase O = W r e duc es to the S-matrix (28). Assu ming the validity of the cr ossing pr op erties for formfactors, the S-matrix ﬁxes A ( W ) u niquely [28]. The r emainder of this subse c tion co nt ains some comments ab out a remar k - able constructive success of these mo dular metho ds with re sp e ct to a particula r family o f interacting theor ies. F or this one needs s o me additional terminol- ogy . L e t us enlarge the algebra ic setting by admitting unbounded op erator s with Wight man domains which are a ﬃliated to A ( O ) and let us a gree to just talk abo ut ” O -lo c a lized op era tors” when we do not w ant to distinguish betw een bo unded and aﬃliated unbounded o pe r ators. W e call an O -lo ca lized o per ators a v acuum p olariza tio n f ree g enera tor (PFG) if applied to the v acuum it g en- erated a one-pa rticle state without a dmixture of a v acuum-p olar ization cloud. 41 According to a theorem of Alain Connes [63] the existence of operator algebras in stan- dard p osition can b e inferred i f the real s ubspace K perm it a decompositions into a natural posi tive cone and its opp osite with certain facial properties of p ositive sub cones. Although this construction has b een highly useful in Connes classiﬁcation of von Neumann factors, it has not yet b een p ossible to relate this to ph ysical concepts. 38 The following three theor ems ha ve turned out to be useful in a constructive approach based on mo dular theory . Theorem ([29]): The ex ist enc e of an O -lo c alize d PF G for a c ausal ly c om- plete subwe dge r e gion O ⊂ W implies the absenc e of inter actions i.e. the gener- ating ﬁelds ar e ( a slight generaliz a tion [2 9] o f the Jost-Schro er theorem (refer red to in [65][67]) which still used the existence of p ointlik e cov a riant ﬁelds). Theorem ( [2 9] ) : Mo dular the ory for we dge algebr as insur es the existenc e of we dge-lo c alize d PFGs. Henc e the we dge r e gion p ermits the b est c ompr omise b etwe en int er acting ﬁelds and one-p article states 42 . Theorem ( [29] ) : We dge lo c ali ze d PFGs with go o d (Wightman-like) domain pr op erties (”t emp er ate” PFGs) le ad to the absenc e of p article cr e ation (pur e elasic S scat ) which in turn is only p ossible in d=1+1 and le ads to the factorizing mo dels (which hitherto wer e studie d in the setting of the b o otstr ap-formfactor pr o gr am [68]). The c omp act lo c alize d inter acting sub algebr as A ( O ) have no PF Gs and p ossess t he ful l inter action-induc e d vacuum p olariza tion clouds. Some additional comments w ill b e helpful. The ﬁr st theore m gives an in- trinsic (not dep endent on any Lag rangia n o r other extr aneous pr op erties) lo ca l deﬁnition of the presence of interaction, even though it is not capable to dif- ferentiate b etw een diﬀer ent kind of in teractions (whic h would b e reﬂected in the sha pe s of interaction-induced po larization clo uds). The o ther tw o theo rems suggest that the knowledge of the wedge algebr a A ( W ) ⊂ B ( H ) may ser ve as a useful sta rting p o in t for clas s ifying a nd constructing mo dels of LQP in a completely in trinsic fashion. Knowing genera ting o per ators of A ( W ) including their tr ansformation prop e rties under the P oinca r´ e gr oup is certainly suﬃcien t and constitutes the most practical wa y for g etting the co nstruction started (for additional infor mations see later section). All w edge a lgebras p os s ess aﬃlia ted PFGs but only in case they come with reasona ble domain prop erties (”temper ate”) they can pre s ent ly b e used in com- putations. This requirement only lea ves mo dels in d=1 +1 which in addition m ust b e factoriz ing (int egr able); in fact the modula r theory used in e s tablishing these connections shows that ther e is a deep connectio n be tw een integrability in QFT and v acuum p olar iz ation prop erties [29]. T emper ate PF Gs whic h generate wedge alg e bra for facto rizing mo dels hav e a ra ther simple algebraic structure. They are of the form (in the absence of bo undstates) Z ( x ) = Z  ˜ Z ( θ ) e − ipx + h .c.  dp 2 p 0 (29) where in the simplest case ˜ Z ( θ ) , ˜ Z ∗ ( θ ) are one-c omp onent ob jects 43 which ob ey the Zamolo dchiko v-F addeev comm utation relations [2 8]. In this wa y the formal Z-F device which enco ded the tw o-particle S-matrix into the commutation struc- ture o f the Z- F algebra r eceives a pr ofound s pacetime in terpr etation. Like fr ee 42 It is the smallest causally closed region (its localization representing a ﬁeld aspect) whic h con tains one-particle creators. 43 This case leads to the Sinh-Gordon theory and related models. 39 ﬁelds thes e wedge ﬁelds are o n ma ss shell, but their Z-F co mmutation relatio ns renders them non-lo cal, more precisely wedge-lo c a l [2 8]. The simplicity of the wedge generator s in factorizing models is in stark contrast to the richness of co mpa ctly lo calized op er ators e.g. of op er ators a f- ﬁliated to a s pacetime do uble cone D which aris e s as a relative commutan t A ( D ) = A ( W a ) ′ ∩ A ( W ). The w edge algebra A ( W ) has simple ge nerators a nd the full space of fo r mal o p er ators a ﬃliated with A ( W ) ha s the form of an inﬁnite series in the Z-F o per ators with co eﬃcient functions a ( θ 1 , ...θ n ) with analyticity prop erties in a θ -strip A ( x ) = X 1 n ! Z ∂ S (0 ,π ) dθ 1 ... Z ∂ S (0 ,π ) dθ n e − ix P p ( θ i ) a ( θ 1 , ...θ n ) : ˜ Z ( θ 1 ) ... ˜ Z ( θ 1 ) : (30) where for the pur po se o f a co mpact nota tion we view the cr e a tion part ˜ Z ∗ ( θ ) as ˜ Z ( θ + iπ ) i.e. as coming from the uppe r pa rt o f the str ip S (0 , π ) 44 . The r e- quirement that the series (30) co mmu tes with the tra nslated genera tor A ( f a ) ≡ U ( a ) A ( f ) U ∗ ( a ) aﬃliated with A ( W a ) deﬁnes formally a subspace of oper a tors aﬃliated with A ( D ) = A ( W a ) ′ ∩ A ( W ) . As a result of the s implicit y o f the ˜ Z gener ators one can c hara cterize these subspaces in ter ms of a nalytic prop erties o f the co eﬃcient functions a ( θ 1 , ...θ n ) . The latter a re r elated to the formfactor s of A which are the matrix elements of A b et ween ”ket” in and ”br a” out pa rticle states. The co eﬃcient functions in (30) o b ey the crossing prop erty . In this wa y the computational r ules of the b o otstrap- formfactor prog r am [6 8] ar e explained in terms of a n algebr aic construction [14]. This is s imilar to the old Gla ser-Lehmann- Z immermann representation for the interacting Heisenberg ﬁeld [69] in ter ms of incoming free ﬁeld. Their use has the disadv an tag e that the co eﬃcient functions ar e not related by the crossing prop erty to one a na lytic master function. The co nv ergence o f b oth series has remaind an op e n problem. So unlike the p erturbative series r esulting from renormaliz e d p erturbation theory which ha ve b een shown to diverge even in mo dels with optimal shor t dista nce behavior (even Bo rel resummability do es not help), the status of the GLZ and formfactor series remains unresolved. The main pr op erty one has to establish, if one’s aim is to s ecure the exis tence of a Q FT with lo ca l observ ables, is the standardness o f the double cone intersec- tion A ( D ) = ∩ W ⊃D A ( W ) . Ba sed on nuclearity prop erties of degr ees of freedom in phase space (discovered by Buchholz and Wichmann [7 3]), Lechner has es- tablished the standardness o f these in ters ections and in this way demons trated the nontriviality of the model as a localize d Q FT [30][74]. F or the ﬁrst time in the history o f QFT one now has a constructio n metho d which g o es b eyond the Hamiltonian- and mea sure-theor e tical appr o ach o f the 6 0s [7 5]. The o ld approach could only deal with sup errenor malizable mo dels i.e . models whose basic ﬁelds did not hav e a short distance dimensio n b eyond that of a free ﬁeld. 44 The notation is suggested b y the the strip analyticity coming f rom wedge lo calization. Of course only certain matrix elements and exp ectation v alues, but not ﬁeld oper ators or their F ourier transforms, can be analytic; therefore the notation is symbolic. 40 The factoriz ing mode ls form a n in teres ting theoretica l lab o rator y where problems, which accompanied QFT almost since its birth, res ur face in a co m- pletely new lig h t. The very existence of these theories, who se ﬁelds ha ve anoma- lous trans-cano nic a l s hort distance dimensions with interaction-dep e nden t strengths, shows that there is nothing intrinsically threatening ab out sing ular shor t dis- tance b ehavior. Whereas in reno rmalized per turbation theo ry the power co unt - ing rule only p ermits loga rithmic cor rections to the canonical (free ﬁeld) dimen- sions, the co ns truction of factor izing mo dels starting fro m wedge alg ebras and their Z gene r ators allow ar bitrary high p ow er s . That many problems of QFT are no t int r ins ic but rather caused b y a pa rticular metho d of quantization had already been sus p ected by the pr otagonis t of Q FT Pascual J ordan who, as far back as 1929 , pleaded for a formulation ”without (classic) cr utches” [76]. The ab ov e construction of factorizing mo de ls which do es not use a ny of the quanti- zation sc hemes and in which the mo de l do es not e ven c o me with a Lagrang ian name may b e considered at the ﬁrst realiza tion of Jordan’s plea at whic h he arrived o n purely philosophica lly gro unds. The sig niﬁcant co nc e ptua l distance be tw een QM a nd LQP beg s the question in what sense the statement that QM is a nonrela tivistic limit of LQP should be understo o d. By this we do no t mean a formal manipulation in a Lagrangian o r functional integral repr esentation, but an ar gument which starts fr o m the corre- lation functions or op era tor algebr as of an interacting L Q P a nd ex plains in what wa y an interacting QFT loo ses its modular lo calizatio n + v acuum p olar ization and mov es in to the conce ptua l setting o f Q M. This is far from evident since in certain cases as that o f 3-dimensional plektonic statistics the nonrelativistic limit retains the v acuum p olar ization, which is necessar y to s ustain the braid group statistics and thus b ecomes a nonrelativistic Q FT instead o f QM. Apparently such arguments do not yet exist. One attempt in this direction could consis t in starting from the known formfactors o f a factor izing mo del (as e.g. the Sinh-Gordo n mo del) a nd s tudy the simpliﬁca tio ns for small ra pidit y θ. An insight of this kind would constitute an essential improv ement of our understanding o f the Q M-QFT interface. Since mo dular theory con tinues to play a n importa nt role in the remaining section as well a s part I I, so me care is req uired in avoiding po ten tial misunder- standings. It is cr ucial to b e a ware o f the fa ct that by restr ic ting the glo bal v acuum state to, a say double cone algebr a A ( D ) whereup on it b ecomes a ther- mal KMS state, there is no change in the v alues of the global v acuum exp ectation v alues (Ω vac , A Ω vac ) = (Ω mo d ,β , A Ω mo d ,β ) , A ∈ A ( D ) (31) where for the standa rd nor malization of the mo dular Hamiltonian 45 β = − 1 . This notation on the rig h t ha nd side mea ns that the v acuum exp ectation v alues, if restr icted to A ∈ A ( D ) , fulﬁll an additional pro p e r ty (which without the restriction to the lo cal algebra would not hold), namely the K MS re lation (Ω mo d ,β , AB Ω mo d ,β ) =  Ω mo d ,β , B ∆ A ( O ) A Ω mo d ,β  (32) 45 The modul ar Hami ltonian lead to fuzzy motions wi thin A ( O ) except in case of O = W when the mo dular Hamiltonian is iden tical to the bo ost generator. 41 A t this po int o ne may wonder how a global v acuum state can turn in to a ther- mal state on a smaller a lgebra without any therma l exchange taking pla ce. The answer is that the in terms of ( A ( D ) , Ω vac ) ca nonically deﬁned mo dular Hamil- tonian K mo d with ∆ = e − K mod is very diﬀerent from the original transla tive Hamiltonian H tr whose low est ener gy eigenstate deﬁnes the v acuum, wher e as K mo d is the generato r of a mo dula r a utomorphism of A ( D ) which in the geo - metric terminolog y preferred by ph ysicis ts (even when it b ecomes ina ppropriate) describ es a ”fuzzy” motion inside D . The mo dula r automor phism is actually deﬁned on the g lo bal algebra B ( H ) where it acts in s uch a wa y tha t A ( D ) and A ( D ) ′ = A ( D ′ ) ar e automo rphically mapp ed into themselves. T he state vector Ω vac ∈ H is a zero eig env alue o f K mo d which sits in the middle of a symmetric tw o-sided sp ectrum. What has changed through the pro c e s s of restr iction is not the state but rather the way of lo o king at it: H mo d describ es the dynamics of a n ”obse rver” co nﬁned to D whereas H tr has o bviously no intrinsic meaning in a world restricted to D . In fact it tur ns out that the fuzzy a utomorphism b eco mes geo metric near the causal horizon of the re g ion O (see second part) The thermal asp ect of mo dular theor y r efers to the mo dular Hamiltonian; it do es not mean that one is cre a ting heat with r esp ect to the usua l inertial frame Hamiltonian; its ener gy conser v ation is a lwa ys maintained and observer-relev ant heat is never ge nerated as long as the observer’s s ystem r emains inertia l. Alr eady in this context of inertial obs erver in the ground state and a mo dular o bserver for whom this state b ecomes thermal, the attentiv e reader may corr ectly pr esume a n anticipation of the thermal manifestations o f black holes a s lo calized restric tions of a la rger sys tem (the Kr usk al extensio n of the Sch wartzsc hild black hole). Going back to the Unruh Gedankenexperiment featuring a non-inertial ob- server which in or der to follow the path of the modula r Hamiltonian of a Rindler wedge W must b e uniformely accele r ated in some spatial direction, the standar d question is the thermal asp ect of the W-reduced v acuum rea l or is it a math- ematical a sp ect car r ied to o far ? The Unruh eﬀect claims that this is really what the non-inertial observer mea sures in his taken along counter. Although the eﬀect is so tiny that it will pro bably never b e o bserved, the exis tence of the thermal radiation is a inescapable consequence of o ur most success ful theories. One is accus tomed to all kind of fo rces in noniner tial sy stems but where do es the nonzer o thermal r adiation density come fro m? In order to create a causal horizon the observer must b e uniformely ac- celerated whic h require s feeding energy into the system. In other w ords the realization of the inno cent lo oking restr iction in lo ca lization requires an enor - mous ene r gy expenditure thus revealing in o ne exa mple what is b ehind the ph ysic s of the har mless sounding word ”restrictio n”. Only when the mo dula r Hamiltonian des c rib es a mov ement which corr esp onds to a diﬀeomo r phism of spacetime is there a chance to think in terms o f an Unruh kind o f Gedanken- exp eriment. As was expla ine d before the mo dular situation is mor e physical in black hole situations wher e the p osition of even t ho rizons is ﬁxed by the metric independent o f what a n obser ver do es . This is underlined by the earlier men- tioned existence o f a pure sta te on the K rusk al extension of the Sch warzschild 42 solution (the Hartle-Hawking state); this state has the p ositio n of the e ven t horizon worked in and does not need a ny observer for its deﬁnition. Restricted to the region outside of the bla ck hole the mo dular automor phism des crib es the timelik e Killing mov ement which is as clo s e a s one can come to an iner tial path. T he co rrep onding Killing Hamiltonia n is the clos e st a na log of the inertial Hamiltonian in Mink owski spacetime. There remains the question to wha t ex tent qua nt um physics in a n Unruh frame is diﬀerent from that in an inertial frame. There are no particles (in the sense of Wigner) since the v acuum b e haves like a thermal densitiy in whic h counter exp eriments only p ermit the measur ement of r a diation densities as in standard ther mal radia tion or cosmic micr ow av e background ra diation. In fact it is quite straightf or ward to show the LSZ sc attering limit do es not exist in the Unruh b o ost time , a fact which is related to the tw o-sided sp ectrum of the mo dular Hamiltonian with resp ect to the W-reduced r educed gr ound state of the or iginal inertial system. T o wit, the g lo bal zero tempe rature Wigner -F o ck space can b e used a lso after the w edge restr iction, but the globa l n-particle states lo ose their intrinsic physical meaning. Apar t from the mo dula r asp ects the pro blems of the Unruh eﬀect hav e b een tr eated by ma ny author s including authors from the foundational c o mmu nity [44][45]. In fa c t there is a c ontinu ous family of mo dular ”Hamiltonians” which a r e the generator s the mo dular unitar ie s for s equences of included r egions. The mo dular Hamiltonian o f the larger re g ion will sprea d the smalle r lo c a lized a lgebra into the lar ger region. Besides the thermal des cription of restricted states there is one other macro- scopic manifesta tio n of v acuum po larization which has ca used unbelieving ama ze- men t in philosophica l circles namely the cyclicit y o f the v acuum (the Reeh- Schlieder prop erty) with resp ect to algebra s lo caliz e d in ar bitrarily small space- time region o r in its mo r e metaphoric presentation the idea that by doing some- thing in a small ea rthly la bo ratory for a n arbitr ary s ma ll fractio n of time one can approximate any s ta te ”b ehind the mo on” with ar bitrary precision by (howev er with ever increasing exp enditure in ener gy [5]). Both consequences of v acuum p olar ization a nd as such interconnected, they 46 are manifestations o f a n holistic b ehavior which in this extreme for m is a bsent in QM. Instea d of the division b etw een a n ob ject to be measured, the measuring apparatus and the e n vir onment, without which the mo dern quantum mechanical measurement theory cannot b e formulated, in LQP such a se pa ration is called int o question. By restricting the v acuum to the inside, one already s p eciﬁes the v acuum p olar ization driven dynamic on the causal disjoint. In the ”s ta te behind the mo on a rgument” the diﬃculty in a system- e n vir onment dichotomy is even more palpa ble. This is indeed an extremely surprising feature which go es consider ably be- yond the k inematical change caused by entanglemen t a s the result o f the qua n- tum mech anic a l divis ion in to mea sured system and en vir onment. It is this de- pendenc e of the reduced v acuum sta te on the lo calization r egion inside which it 46 Sometimes used as a metaphor for the Reeh-Schlieder prop erty . 43 is tested with lo calized algebr as which raises doubts ab out what are r eally no n- ﬂeeting p ers is ten t prop erties o f a material subs ta nce. The monad descriptio n in the next section strengthens this little holistic a sp ect o f LQP . As we have seen the thermal asp ects o f mo dula r lo caliza tion are very r ich from an epistemic viewp oint. The ontological conten t of these obse r v ations on the o ther ha nd is quite w eak; it is only when the (imagined) causal lo caliza tio n horizons pas ses from a Gedanken ob jects to a (real) even t horizo ns through the curv ature o f spacetime, that the ﬂeeting as pec t of ca usal horizons of observers pass to a n intrinsic o nt olo gical prop erty of spacetime in the case o f bla ck holes. But even if one ’s main interest is to do black hole physics, it is wise to avoid a presently popular ”shut up and compute” attitude a nd to under stand the conceptual basis in LQ P of the thermal aspect of loca lization a nd the pecu- liar thermal entanglemen t which contrasts the information-theo r etical qua n tum mechanical en tanglement. Igno ring these co nceptual a sp ects one ma y easily be dr awn into a fruitles s and pro tractive a rguments as it ha ppened (a nd s till happ ens) with the entropy/information loss issue. Up to now the terminology ”lo calization” was used bo th for s tates and for subalgebra s. In the absence of in teractio ns they are syno nymous; this is b ecause free ﬁelds are uniquely deter mined by positive energy r e presentations o f the Poincar´ e, in fact the gener ators of cov ariant wa ve functions pass dir ectly to generating ﬁelds. A repres e ntation which has no inﬁnite spin comp o nents is alwa ys po int like genera ted. This applies in pa rticular to string theory which is a misnomer for inﬁnite comp onent ﬁeld theory [31]. Suc h a clos e r elation b etw een algebraic and state lo caliz ation breaks down in the presence of interactions. It is p erfectly co nceiv able to hav e a theory with ” top ologica l charges” [5] which by deﬁnition cannot be descr ibed by compactly lo calizable op era tors but need spacelike string lo calizable g enerating ﬁelds. In that cas e the neutral observ able algebra ha s the usual compact lo ca lizability wher eas the charge-carr ying part of the total a lgebra ma y need semiinﬁnite string generator s for its description [70]. The fa ct that this p ossibility could even o ccur in massive QCD like theorie s makes it v ery in teresting, but unfortunately ther e is no illustrative example. The pr o blem o f lo ca lization is of pivotal relev ance for Q FT. But nowhere is glory a nd failure so interlink ed as in this issue. The misunder standings range from the comparatively harmless confusio n b etw een the BNW loc alization of states and the mo dula r lo calizatio n o f o bserv ables to the very serious misun- derstanding of string theory . Besides these g rav e consequences the inn umera ble confusions ab out particles, frame-indep endent lo calization of states and a lg e- bras have b een a harmless nuisance since they w ere comitted by individuals and not b y globalized commu nities. As the ex ample of the misinterpreted F er mi Gedankenexperiment show ed, such mistakes can b e corrected. In particular the 10 dimensional co v ariant inﬁnite comp onent unitar y su- per string representation of the Poincar´ e group coming from the quantization of the bilinearized Nambu-Goto Lag rangian is according to the befo re men tioned theorem (for re presentations which do not c o ntain Wigner’s inﬁnite spin r epre- sentation) a po intlike lo c a lized ob ject, and this also a pplies to its predecesso r, the dual reso nance mo del. F or a more de ta iled presentation of these p oints 44 see [31]. Every explicit co mputation of the (graded) commut ato r of tw o string ﬁelds car r ied out by s tring theor ists has co nﬁr med the inﬁnite comp onent p oint- like nature [71][72], but there is a s trange ide o logical spirit whic h pe r v ades the string communit y which prev ents them from saying clearly what they r e ally compute. Reading the tw o cited pap ers is a s trange e xpe rience beca use it shows that c orrect co mputations in times of a dominating metaphorica l idea are no guarant y for a corr ect interpretation. The a uthors c o me up with a ll kinds of metaphoric ideas (including that of a str ing of which one o nly sees a p oint) in order to av oid having to say ”inﬁnite comp onent p ointlike ﬁeld” which would place them outside their comm unity . An y philosophically motiv ated historian who wants to understand the Z e it- geist which led to string theor y a nd its v arious revolutions in the s e rvice of a theory of ev erything, should ﬁnd these (computatio nally correct but co ncep- tually str ange) pap ers a r ich s o urce of infor mation. Less than 7 decades after Bohr and Heisenberg remov ed the metapho r ic ar guments o f the o ld q uantum theory b y in tro ducing the concept o f o bserv ables, the discour s e within the string theory communit y is try ing to re- int ro duce metaphor ic arguments into the rela- tivistic particle disco urse. Sur ely one do es not want to miss the kind o f fruitful transient meta pho rs which at the end led to v a lua ble insights, but what is a r ea- sonable a ttitude with resp ect to a n o bviously incorr ect metaphor which hovers ov er particle theory ever since its b eginnings in the 7 0s? 8 String-lo calization and gauge theory Zero mass ﬁelds of ﬁnite helicity play a crucial ro le in ga uge theory . Wherea s in classical g a uge theor y a p ointlik e ma ssless vectorp otential is a p erfectly accept- able concept, the situation changes in QT as a co nsequence of the Hilb e rt space po sitivity , which for massless unitary r epresentations leads to the lo ss of ma ny spinorial rea liz ations (as expres sed in the second line of (21 )), in par ticular to that o f the vector-p otential without whic h it is hardly p oss ible to formulate p er- turbative QED. The tra ditio nal wa y to deal w ith this situatio n ha s b een to allow vector-po tentials to liv e in an indeﬁnite metric space a nd to add ghost degre es of freedom in intermediate calculations in such a way that the physical ob jects in form o f the lo ca l observ ables in a Hilber t space c o alesce with the ga uge o r BRST inv aria n t o b jects under a suitably deﬁned gauge or BRST gr oup action. The ghost degree s of freedom ar e like catalyzers in chemistry; they are not there in the initial set up of the pr oblem and they hav e gone at the end, but without there presence the renor malized perturba tive p ointlik e ﬁeld formalism would not work for zero mass spin s ≥ 1 quantum matter. At the bo ttom of the problem is a clash b etw een mo dula r lo ca lization and the Hilb ert space s tr ucture: although the ( m = 0 , s ≥ 1) represe ntations are pointlik e g enerated (21) there are only ﬁeld-strength but no po tential-t yp e generato rs with    A − ˙ B    > s. Ga uge theory tr ies to reso lve this clash by using a ca talyzer which vio lates the Hilb ert space setting of QT. 45 Despite the undenia ble success o f this kind o f quantum ada ptation of the p er- turbative gauge setting, there ar e tw o arg ument s agains t co nsidering the present formulation as the end of the story . One is of a mor e philosophical kind and the other p oints tow ards a serio us limita tion o f the gaug e formalism. F rom a philo- sophical p oint o f view this setting violates the maxim of Bo hr and Heisenberg that one should always lo o k for a formulation in which the co mputational steps (and no t only the ﬁnal result) can b e for m ulated in ter ms o f o bs erv ables. Mor e tangible is the ob jection that the existing gauge for malism a ims only a t lo c al observ ables. There are interacting generators of ph ysica l ob jects whic h do not admit pointlik e gener a tors but whose sharp est pos sible loc alization is semiinﬁ- nite string like; the mo s t pro minent ones are electr ic charge-car rying o p e rators [73]. Their construction is not par t of the sta ndard p ertur ba tive fo rmalism but they have to b e deﬁned ”by hand”. The b est loc alization for a c harg ed generating ﬁeld is tha t of a semiinﬁ- nite Dirac-J ordan-Ma ndels tam string characterized formal ly by the well-kno wn expression Ψ( x, e ) = ” ψ ( x ) e R ∞ 0 ie el A µ ( x + λe ) dλ ” (33) Using a v er s ion o f pertur ba tion theory whic h was esp ecia lly designed to incor- po rate this for mal DJM express ion into the n th order r enormaliza tion setting, Steinmann [77] s uc c e eded to attribute a renormalized perturba tive meaning to this fo rmal ex pression. Connected with this nonlo cality asp ect is the subtle rela- tion o f electr ically charged ﬁelds to charged particle s which shows up in infrared divergencies of on mass s hell ob jects. In addition a charged particle, e ven after a long time of ha ving left the scattering r egion, will never co mpletely esca pe the reg io n of inﬂuence of infr ared real (not virtual!) photo ns whose ener g y is below the (arbitra r ily small but no nv anishing) re gistering resolution and which therefore r emain ” invisible” (in the sense of unreg istered). This makes c har ge particles ” infr a particles” i.e. ob jects whose scattering theory do es not lead to scattering amplitudes but only to inclusive cross sectio ns. The infrared divergence problems in QED, ﬁrst studied in a s impler mo del by Blo ch and Nordsiek, whose phenomenologica l remedy required to trade scat- tering a mplitudes with inclusive cr o ss s ection [78], turned out to have a very profound conceptual explanation: the Hilbert space of Q ED do es not c ontain an irreducible r epresentation with a sha rp mass, less so can it be written in terms of antisymmetric tensor pro ducts of such s tates; r ather the electro n tw o-p oint function starts with a cut at m e which dep ends on e . F or this to o ccur the pres- ence of zer o mass particles is necessary but no t s uﬃcie nt. Their coupling for low energ ies must also b e suﬃciently strong, a requir ement which is fulﬁlled for the minimal coupling o f photons in QED but not for renor malizable couplings of ( m = 0 , s = 0 , 1 / 2) (e.g. not for the π - N coupling with mas s less pions). Also the converse holds, if the theory allows for one- particle sta tes in the sense that the theory has a mass-s hell than even if this mass shell is no t s eparated fro m the co n tinuum b y a ga p) the theory p o s sess a standard (LSZ) scattering theory [80]. F or global gauge symmetries, the idea that the loca l observ ables in their 46 v acuum r e pr esentation determines all charged r epresentation and, by suitably combining them, lead to the physical charged ﬁelds, was one of the mos t se mina l conceptual conques ts in lo cal q uantum physics [5]. The sup erselected charge- carrying ﬁelds are in this wa y (up to some co nv ent ions ) uniquely determined in terms o f the v a cuum r epresentation of the lo cal obs e r v ables. In d ≥ 4 these ﬁe lds are Bos e/F ermi ﬁelds which ac t irreducibly in a Hilb ert space which co n tains all supersele cted s e ctors assoc iated to the system of lo cal observ ables. The y transform according to a compact internal symmetry group whose existence is preempted by the incons pic uo us presence o f a co py of the dual o f a gr oup within the net of lo cal observ ables; the latter in turn is the is the ﬁxpo int subalg ebra of the ﬁeld algebra under the actio n of the internal s ymmetry (always global). Each compact gro up (with the exception of supe r symmetry) can appea r as the internal sy mmetry of a Q FT. With this str uctural ins ight a lo ng path of the somewhat mysterious 47 concept o f internal symmetries , which begun with Heisenberg’s S U (2) iso spin in nuclear physics, came to a b eautiful conclus ion. Global symmetry groups are a to ol by whic h the quantum locality principle arrang es the v arious lo cal super selection rules of a given observ able a lgebra belo nging to diﬀeren t sup erselection charges into a ﬁeld algebr a . What started so mysteriously with Heisenberg’s is ospin in the 3 0s, ended brilliantly with the DHR sup ersele ction theory and its lo ca l presentation in ter ms of sup erselectio n charge-carrying lo cal ﬁelds; showing again what the quantum causa l lo caliza tion principle is capa ble to achiev e but a lso how conceptually demanding it is to ﬁnd that path whic h reduces an o bserved prop er t y to its conceptual r o ots. In d= 3 ,2, the commutation relations may be plektonic or solitonic , meaning that the ﬁelds ob ey bra id gro up or so liton commutation relations which requir e semiinﬁnite s tringlike lo calization and lea d to a gener alized spin&statis tics theo - rem [64] and to a situa tio n in which the int er nal and spac etime symmetries allow no clear cut se pa ration among them. But as b efor e , the net of lo cal obser v a bles determines mo dulo s o me conven tions its full ﬁeld algebra which inc o rp orates the sup e rselected charge-ca rrying ﬁelds. In b oth cases , the low- and higher- dimen- sional ca s e, there is no b etter characterization for the inv erse pro blem neutr al observables → char ge-c arrying ﬁelds than the metaphor whic h Mark K ac used in c o nnection with an a coustic inv erse problem: ”how to hear the shap e of a drum”. It is a natural question to as k whether this reconstructio n pe r mits a more concrete formulation in the form of reconstr ucting bilo cals by breaking up lo c al expressions as the ele c tric current ¯ ψ ( x ) γ µ ψ ( x ) in an analo gous manner as was done in the 6 0s in o rder to rec onstruct bilo cals A ( x ) A ( y ) from Wick-ordered lo cals : A 2 ( x ) : via a ligh tlike limiting pro cess [81]. In the case that the lo ca l op erator s ar e asso cia ted with a lo cal gauge theory as Q E D, one ex p ects bilo - cals with ”gauge- bridges” be tw een the tw o p oints. The partial results on this problem are sca rce but encourag ing [8 2]. It would be a ma jor progres s in gauge theory if electrically charged bilo cals including gaug e br idges could b e obtained 47 This concept, whic h is cent ral to local quant um ph ysics, do es not exist in classical ph ysics; but as any quan tum ob ject it can be ”red bac k” therein (see also classical Grassmann v ariables from r eading back F ermions). 47 from lo cal currents by suc h a lightlik e splitting, so that formally the str ing like DJM ch ar ge generating ﬁeld (33) app ea rs in the limit of dumping one charge a t inﬁnit y . The pr oblem of po ssible pr esence o f interacting no nlo cal gener ating ﬁelds in the physical Hilb ert space b eco mes mor e s erious in theories involving vec- torﬁelds coupled a mo ng themselves. Wherea s one b elieves to hav e a physical understanding o f the lo cal (= ga uge inv ariant) comp os ites (whose p erturba tion expansion in terms of in v ariant corr elation functions ha s incurable infrared di- vergencies 48 ), there is no convincing idea ab out the conceptual status of the degrees of freedo m whic h ar e the analo gs of the charged ﬁelds in QED. F or many decades w e hav e been exp os e d to suc h ev o cating metaphoric words as quark- a nd gluon- conﬁnement. Wherea s such ideas are quite natural in Q M where they point to enclos ing quan tum matter in a po ten tial v ault, QFT has no mechanism of hiding deg rees of free dom by lo calizing them. The only mecha- nism thro ugh which deg rees o f freedom ma y esca pe observ ations in a theory in which lo ca lization is the dominating physical principle is a we akening of lo c al- ization i.e. the o ppo site of a qua ntu m mechanical v ault 49 . The delo ca lization of electrically charged particles due to surrounding photo n clo uds in QED is obviously no t suﬃcien t. Wha t one needs is the understanding o f a situation in which the g luon plays a double role: tha t of a charge carr ier and that of the pho to ns hov er ing around it. Contrary to the formal DJM expres sion for charged ﬁelds, there is little chance that the formal spacetime structure of s uch a situation can b e guessed by a nalogies. After this long informative detour let use return to the lo calization-QT cla sh one confronts in pas sing from cla s sical vectorpo ten tials to their quan tum co un- terpart and ask whether instead o f the indeﬁnite metric ” catalyzer ” we could hav e chosen o ne which sacriﬁes the p ointlik e lo ca liz a tion (for whic h there is no ph ysic al supp ort anyho w in view of what has b een sa id ab out charged ﬁelds) a nd instead works with what we obtained from combining the Wigner repres entation theory with mo dula r lo caliz ation (section 6,7)? If we s ucceed to set up a reno r malized p er tur bation theor y in terms of string- lo calized p otentials and explain the extreme delo calization of charges in terms of string-lo c alized vector potentials delo calizing massive complex matter ﬁelds through their QED interaction in a per sistent wa y , we would not need to talk ab out formalism- preserving ”catalyzers ” to start with. Instead we w ould have an alternative formulation in which the gauge inv aria nt lo cals ar e identical to the p ointlik e generated charge neutral subalgebra of an algebra whic h con tains all string - lo calized charged ﬁelds. So using string -lo calized p otentials A µ ( x, e ) from the start we would under- stand that the ghosts in Gupta-Bleuler and BRST w er e the prize to pay for insisting in setting up per turbation theory with only p ointlik e ﬁelds ag ainst our better structural knowledge that charged o b jects ar e nece ssarily noncompact lo- 48 Only if p erturbation theory is formulated in a pure algebraic setting and the problem of states is treated as a second step there is a chance to control the inf rared divergencies. 49 The use of lattice theory has also its li mitations; for example there is no l attice description of i nfraparticles. 48 calized and that point-lo calized vectorpotentials not only go against the Hilb ert space se tting, but also leav e the orig in of the string- lo calized quantum Maxwell charges in terms of prop erties of the interaction density shrouded in mystery . It is imp ortant no t to b e misundersto o d o n this p oint; we are not saying the the ga ug e theory setting is incorrect, it remains correc t for all p ointlike g a uge inv ariant ﬁelds which are compo sites of ﬁeld s trength and charge neutral matter ﬁelds but it fails for charge-carry ing ﬁelds. One could think that one would also be able to compute no nlo cal gauge inv ariants, whic h in the setting of the BRST formalis m would amount to deter- mine inv ariants under the nonlinear acting BRST transfo rmation; but this an impo ssible task. This should b e no sur prise since in this step physical no nlo cals hav e to ”p op out” o f unphysical ”po int like” genera ted ob jects. The par enthe- sis indicate that p ointlik e is not mea nt in a physical but only a formal sense. But for the p ointlik e generated subalg ebra of c harg e neutrals the gauge a p- proach is eﬃcient (p erha ps apart from Y ang-Mills theories) and based on a well-studied renormalization formalism but, since diﬀerent from the classica l int era ction which is p erfectly c o nsistent with the principle of c la ssical ﬁeld the- ory , the quant ized version le aves the Hilb ert space one needs the q ua nt um gaug e formalism to recov er it. All this ca n b e av oide d in a setting whe r e one couples string-lo ca lized p o- ten tials A ( x, e ); there are no g ho sts and therefore there is no need for a gauge formalism, all ob jects have an intrinsic string loca lization and the p o intlik e lo- calized form a subset. Descr ibing the so co nstructed theory in terms of ﬁeld strengths instead o f potentials the only stringlike ge nerators a re the charged ﬁelds. As w as explained in the sec tio n o n mo dular loca lization (21) one can ﬁnd intert winers from the Wigner repres entation to all spino r repres ent atio ns if one admits string-lo calized p otentials. In particular the one with the s tring- lo calized ﬁeld of low est Lorentz spin ( A µ ( x, e ) for helicit y h= 1, g µν ( x, e ) for h=2,..) has the low est shor t distance dimension namely sdd=1 and hence the optimal b ehavior from the viewp oint of renormaliz a tion theor y . It is quite s urprising that there are string-lo ca lized potentials for a rbitrary spin s with scale dimension=1 which is the p ow er coun ting prerequisite for en- countering renorma lizable interactions. So by avoiding to impo se the unphysical restriction to p ointlike interactions one also enlarg e s the scop e of renor maliz- ability . A formalism with this enor mo us r ange cannot b e exp ected ca nnot b e exp ected to fall into o ne’s lap, in fact it is prese ntly still in its infancy , it po ses completely new and lar g ely unsolved problems. But b efore commenting on this new task , it is helpful to delinea te what one expe c ts of such an alternative approach. Super ﬁcially the use o f such string - lo calized ﬁelds seem to be indistinguish- able from the axial gauge 50 ; in b oth ca ses the conditions ∂ µ A µ ( x, e ) = 0 = 50 The axial gauge i s the only one which (after adjusting the Lorentz-transformation prop erty of e ) is cov ar i an t and Hil bert-s pace compatible. So i t neve r was a gauge in the sense of sacriﬁcing the Hi l ber t space structure. A r enormal i zed perturbation theory was not p ossible because its serious infr ared problems were not understoo d as indicating string- instead of point-locali zation. 49 e µ A µ ( x, e ) ar e ob eyed. In the axial gauge int erpr etation the e is a gaug e pa- rameter and do es not participate in Lor entz transformatio ns, wher eas in case of string-lo ca lized ﬁeld the spac elike unit v ector transfo rms a s a string dir ection or, what is the same, as a p oint in a 3-dimensiona l de Sitter spacetime. The a xial gauge failed as a p er turbative computational to ol in the s ta ndard setting a s a result o f its incurable infrar ed divergence proble ms . In a w ay the string-lo caliz ed approach explains this as a conse q uence of qua nt um ﬂuctua tio ns b oth in x and e which makes it necessary to use testfunction smearing in x and e and to discuss coalescing p oint limits with the s a me ca re as for compo site ﬁelds . The guiding idea is that the use of str ing dep endent p o ten tials delo ca lizes the charged ﬁeld automatically so that there is no necessity to use ad ho c formulas as (33) and to engag e in the diﬃcult task to construct their renormalize d co un terpa rt. Whereas the standar d renor malization formalism for p ointlik e ﬁelds admits many diﬀerent v aria tio ns o f whic h the Epstein Glaser scheme is the one which uses causal lo c ality most he avily 51 But for the string-lo calized appro ach which admits no Lagra ngian form ulation, the Epstein- Glaser [83] is the only one. I n the p ointlike case the knowledge of the n th order determines the n + 1 order up to a term on the total diagonal which limits the free do m to the addition of po int like comp osites. If the power counting requirement for r enormaliza bility is no t only necessary but also suﬃcient one would have a n enormous enrichmen t of renor malizable int era ctions genera lizing ga uge theories or r a ther its string- lo calized reformula- tion. So this would op en a v ast new area o f research. The in teresting question is then whether not only interations o f zer o mass higher spin pa rticles with low spin mas sive matter but also massless higher s pin par ticles can in teract a mong themselves. F or s=1 the only known mo dels are the Y ang-Mills theories; one exp ects fo r s=2 mo dels whose Riema nn tenso r -like p ointlik e ﬁeld str engths ar e nonlinear expressions in string -like g µν ( x, e ) tensor p otentials. In b oth cases the exis tence of pointlik e compo sites of stringlike ﬁelds is the guiding princ iple and not the gro up theor etical struc tur e of the in tera ction in terms of stringlike po tent ials . Up to now the issue was how to couple massles s higher spin s ≥ 1 to low spin massive matter. F or massive higher spin there is no re pr esentation theo- retical argument to intro duce string-lo calize d p otentials since all the cov aria n t po ssibilities a r e (21) are realized. There is how ever the p ow er-co unt ing require - men t which go es ag ainst the use of the A µ ( x ) with ∂ µ A µ ( x ) = 0 since its s ho rt distance dimension is 2 instead o f one a s a res ult of the a dditional degree of free- dom w hich disting uishes the mas sive cas e from its massles s counterpart. Hence the pointlik e vectorﬁeld has the same shor t distance dimension as its ”ﬁeld strength” and therefor e falls o utside the power co un ting limit. There exists how ever a mas sive s tr ing-lo calized p otential A ( m ) µ ( x, e ) of s ddim. = 1 asso ci- ated with this ﬁeld streng th which for m → 0 passes to the zer o mass p otential. Its use certainly complies with the p ow er co unt ing requir e men t, but does its use 51 The other formulations based on Lagrangians and Eucli dean cont inuations use the close relations b etw een the classical ﬁelds and their Euclidean coun terparts. 50 lead to a n a cceptable physical theo ry? By acceptable w e mean a theo ry in which the zero order str ing lo caliza- tion do es no t s pr ead ” all over the place” i.e. in which there exist still p oint- like generated subalgebras as those generated by charge-neutr al ﬁelds in QED. F ortunately there is an additional mechanism, which acco rding to o ur pr esent knowledge seems to secur e pr ecisely this picture of the existence of lo cal o bserv- ables: The Sch winger-Higg s scre ening mechanism. It is the QFT analog o f the Debeye s c r eening in QM. The latter describ es the trans itio n from a long range Coulomb sy stem to a o ne in whic h the eﬀectiv e action falls oﬀ like a Y uk awa p o - ten tial (the comp ensating eﬀects of oppo site c har ges. In Q FT screening w ould be a muc h more violent mechanism be c ause b eca use the a nalog of ra nge of forces is the spacetime lo calizatio n of gener ating ﬁelds 52 . It was Sch winger’s idea that something like this could o ccur in ac tua l (spinor ) QED and render the who le theo ry mas s ive, but since he could no t ﬁnd a pertur - bative argument, he inv ented the Sch winger mo del (massles s QED in d=1+1 ) which only leaves the screened phase and appr oaches the ch ar ged Jordan mo del in the s hort dista nce limit [8 8]. The co nt r ibutio n o f Higgs consis ts in a more int ere sting mo del which allows a perturbative version of scre e ning ; the Higgs mo del in its origina l form is nothing but scr eened scalar QE D. T he scre e ning mechanism formulated o n scalar QED, using the string-lo c alized vectorp oten- tial ma ps this mo del to one with a massive stringlike vectorpotential and a rea l scalar ﬁeld, so that ha lf the degrees of freedom of the charged co mplex ﬁeld was used to con vert the ma ssless photon into a massive vectormeson. This pro c ess has nothing to do with what in the literature is c a lled sp ontaneous symmetry breaking (Goldstone). Certainly s creening leads to a symmetry re ductio n since the Maxwell charge is zer o and hence its super selection rule has b een los t. The nonv anishing lo cal exp ectation o f Φ is part of the pr escription a nd o f no intrin- sic signiﬁcance ; the intrinsic meaning is r elated to the co nserved c ur rents: the divergence of the charge in the case o f sp ontaneous symmetry-br e aking (a s the result of the existence of a Goldstone Boso n) and the v anishing of the charge in the cas e o f scr eening. A more detailed description can be found in [89]. The impo rtant question which remained unanswered in the 70s is whether this screening mec hanism is a p e culiar illustration for ho w an int er a cting mas- sive vectormeson can b e part of a p ointlik e lo cal QFT o r whether this is a spe c ial case o f a more genera l intrinsic mechanism which sta tes that in order to main- tain loc a lity in tera cting massive hig her spin particles m ust b e accompanied b y low er spin ob jects? Diﬀerent spins have b een linked together by the inv ention of s uper symmetry , but it w ould b e mo re natural to understand this a s a conse- quence of the lo cality principle. An supp orting argument was g iven within the BRST setting [84]: if one star ts with a massive vectormeson, the Higgs meson (but now with v anishing v acuum exp ecta tio n) has to b e intro duced for main- taining consistency of the BRST for malism. Only by removing the non-intrinsic 52 A mathematical theorem which explains the connection b et ween the gain of analyticit y in 3-poi n t functions indicating a gain in l ocali zation (screening) and the conv ersion of photons int o massi v e vecto rm esons can b e found in [40]. Swieca used the screening terminology in most of the publications but it s eems that this got lost during the 70s. 51 BRST formalism and using instea d the stringlike sdd=1 vector-p otentials one can hop e to under stand the c rucial role of lo ca lity in a conjectur ed lower spin c omp anion mechanism b ehind the Higgs issue. 9 Building LQP via p ositi oning of monad s in a Hilb ert space W e hav e seen that mo dular lo c a lization of states and alg ebras is an intrinsic i.e. ﬁe ld- co ordinatiza tion-indep endent wa y to formulate the kind of lo ca lization which is characteristic for QFT. It is deeply satisfying that it also leads to a new constructive view o f QFT. Deﬁnition (Wiesbro ck [90]) : An inclusion of standar d op er ator algebr as ( A ⊂ B , Ω) is ”m o dular” if ( A , Ω) and ( B , Ω ) ar e standar d and ∆ it B acts like a c ompr ession on A i.e. Ad ∆ it B A ⊂ A . A mo dular inclusion is said to b e standar d if in addition the r elative c ommutant ( A ′ ∩ B , Ω ) is standar d. If t his holds for t < 0 one sp e aks ab out a -mo dular inclusion. The study of inclusions of op erator a lgebras has been an ar ea o f considera ble mathematical int ere st. Particle physics uses three diﬀerent kind o f inclusions ; bes ides the modular inclusions, whic h play the principa l role in this section, there ar e split inclusions a nd inclusions with c onditional ex p ecta tions (or using the name of their protagonist, V aughn Jones inclusions). Split inclusions play an imp ortant role in structura l inv estigatio n and ar e indisp ensable in the study of thermal a sp ects of lo calization, nota bly lo calization entropy (see second par t). Inclusions with conditional exp ectations r esult from reformulating the DHR theory of sup er selection secto rs which in its original for mu latio n us es the setting of lo calized endomorphisms of o bserv able alg e bras [5 ]. Inclusions A ⊂ B with conditional e x pe c ta tion E ( B ) cannot b e mo dular and the pre c ise under standing o f the r eason discloses interesting insig hts. According to a theorem o f T akesaki [85] the existence of a conditional exp ectation is tanta- mount to the mo dular g roup of the smaller alg e bra b eing equal to the restr ic tion of that o f the big g er. Hence the natural generalization of this situation is that the group Ad ∆ it B of the larger algebra ac ts on A for either t < 0 or for t > 0 as a b o na ﬁde compress io n (endomorphism) whic h precludes the existence of a conditional ex pec ta tion. In tuitively sp eaking mo dula r inclusions a re ” to o deep” to allow co nditional exp ectations. Cont inuing this line of speculative reas oning one would exp ects that inasm uch as ” ﬂa t” inclusio ns with co nditional ex pe c - tations are r elated to inner sy mmetr ies, ”deep” inclusions o f the mo dular kind should lead to spacetime symmetries. This r o ugh guess turns out to b e correct. The main aim o f mo dular inclu- sions is rea lly tw ofold, on the one hand to gener ate sp ac etime symmetry whic h than a cts on the o riginal alg ebras and creates a net of sp ac etime indexe d al- gebr as which are cov ar iant under these symmetries. F or the ab ove mo dular inclusion of t wo a lgebras this is done as follows: from the tw o mo dular groups ∆ it B , ∆ it A one can form a unitary gro up U ( a ) which together with the mo du- 52 lar unitary g r oup of the sma ller algebra ∆ it B leads to the commut atio n rela- tion ∆ it B U ( a ) = U ( e − 2 π t a )∆ it B which ch ar acterizes the 2- parametric tra nslation- dilation (Anosov) group. O ne also obtains a system o f lo cal algebras by applying these sy mmetr ies to the r elative comm utant A ′ ∩ B . F r om these r elative com- m utants one may form a new alg ebra C C ≡ [ t Ad ∆ it B ( A ′ ∩ B ) (34) In general C ⊂ B and we are in a s ituation of a nontrivial inclusion to which the T akesaki theo rem is applica ble (the modular group of C is the r e striction of that of B ) which leads to a conditional exp ecta tio n E : B → C ; C may also b e triv ia l. The most interesting situatio n aris es if the modular inclusion is standar d i.e. all three algebr as A , B , A ′ ∩ B ar e standard with resp ect to Ω; in that case we arrive at a chiral QFT. Theorem : (Guido,Longo and Wiesbro ck [86]) St andar d mo dular inclusions ar e in one-to-one c orr esp ondenc e with str ongly additive chir al LQP. Here chiral L Q P is a net of lo cal a lg ebras indexed b y the interv als on a line with a Mo ebius-inv ar iant v acuum vector and stro ngly addi tive refers to the fact that the remov al of a p o int from an interv a l do es not “damag e” the algebr a i.e. the von Neumann alge br a gener ated by the t wo pieces is still the orig inal a lgebra. One can show v ia a dualization pro cess that there is a unique asso cia tion of a chiral net on S 1 = ˙ R to a stro ngly additive net on R . Although in our deﬁnition of mo dular inclusion we hav e not said a n ything abo ut the na ture of the von Neumann a lg ebras, it tur ns out that the very requirement of the inclusio n b eing mo dular for ces bo th alg ebras to b e hyperﬁnite t yp e I I I 1 algebras . The clos eness to Leibniz’s idea a b out (physical) rea lity o f orig inating from relations betw een monads (with each mona d in is olation being void of individ- ual attributes) mo re than justiﬁes our choice of name; b es ides that ”mo na d” is m uch shorter than the s omewhat long winded mathematical terminology ”hy- per ﬁnite type II I 1 Murray-von Neumann fa c to r alg ebra”. The nice a sp ect of chiral mo dels is tha t one ca n pa ss betw een the o per ator algebra fo r mulation and the construction with p ointlik e ﬁelds without ha ving to make additional techn ica l assumptions 53 . Another interesting constructive aspect is that the op erator - algebra ic s etting p ermits to establish the existence of algebra ic nets in the sense of LQP for all c < 1 representations of the energy-mo mentum tensor algebra. This is muc h more than the vertex algebr a approa ch is able to do since that formal p ow er series approa ch is blind against the dense domains which change with the lo calizatio n regions. The idea of placing the monad int o modular pos itions within a common Hilber t space may b e g eneralized to mo r e tha n t wo copies. F or this purpo se it is conv enient to deﬁne the c o ncept of a mo dular interse ction in ter ms o f mo dula r inclusion. 53 The group theoretic arguments which go into that theorem [87] seem to b e a v ailable also for higher dimensional conformal QFT. 53 Deﬁnition ( Wiesbro ck [90] ) : Consider two monads A and B p ositione d in such a way that their interse ction A ∩ B to gether with A and B ar e in standar d p osition with r esp e ct to the ve ctor Ω ∈ H . Assume furthermor e ( A ∩ B ⊂A ) and ( A ∩ B ⊂ B ) ar e ± mi (35) J A lim t →∓ ∆ it A ∆ − it B J A = lim t →∓ ∆ it B ∆ − it A then ( A, B , Ω ) is said to have the ± mo dular interse ction pr op erty ( ± m i) . It ca n b e shown that this pr op erty is stable under taking commutan ts i.e. if ( A , B , Ω) ± mi then ( A ′ , B ′ , Ω) is ∓ mi. The minimal num ber o f monads needed to characterize a 2+ 1 dimensional QFT thro ugh their mo dular p ositioning in a joint Hilb e rt spa ce is three. The relev ant theorem is a s follows Theorem : (Wiesbrock [9 1]) L et A 12 , A 13 and A 23 b e t hr e e monads 54 which have the s tandar dness pr op erty with r esp e ct t o Ω ∈ H . A ssume furthermor e that ( A 12 , A 13 , Ω) is − mi (36) ( A 23 , A 13 , Ω) is + mi ( A 23 , A ′ 12 , Ω) is − mi then the mo dular gr oups ∆ it 12 , ∆ it 13 and ∆ it 23 gener ate the L or entz gr oup S O (2 , 1) . Extending this setting by placing an a dditional monad B into a suitable po sition with r esp ect to the A ik of the theor em, one arr ives at the Poincar´ e gr oup P (2 , 1) [92 ] . The action of this Poincar´ e gro up on the four monads g enerates a spacetime indexed net i.e. a LQP mo del and all LQ P have a mo nad pr esentation. T o a rrive at d=3+1 LQP one needs 6 monads [93]. The num b er of monads increases with the spacetime dimensions. Whereas in low space time dimensio ns the algebr aic p ositioning is natural within the lo gic o f mo dular inclusio ns, in higher dimensio ns it is pres ently necessa ry to take some additional guidance from geometry , since the n umber of p os sible modula r arrangements for more than 3 monads increases . There is an approa ch with similar aims of characterizing a QFT b y its mo dular data by Buc hholz and Summers [94]. Instead of the mo dular g roups these a uthors use the mo dular reﬂectio ns J. F or o ur pur p o se of characterizing lo cal quantum ph ysic s in terms of p ositio ning of monads the approach pr op osed b y Wiesbro ck bas e d on mo dular inclusions and intersections is more co nvenien t. Its o r gin da tes back to the o bserv ation that the Mo ebius group ca n be extra c ted from the mo dular gro ups of the quar ter circle a lg ebras [95]. W e hav e presented these mathematica l res ults and used a terminolog y in such a wa y that the relation to Leibniz philosophical view ab out relational reality is visible. 54 As i n the case of a mo dular inclusion, the monad property i s a consequence of the mo dular setting. But for the presen tation it i s more con venien t and elegan t to talk ab out monads f rom the start. 54 This is not the plac e to give a comprehe ns ive account, but only an a ttempt to direct the a tten tion o f the reader to this (in my view) star tling conceptual developmen t in the heart of QFT, a theory which despite its almo st 9 0 years of existence is still far fr o m its closure.. Besides the radically diﬀer ent conceptual-philoso phical outlo ok on what con- stitutes QFT, the mo dular setting oﬀers new metho ds of constr uction. F or that purp ose it is how ever more co nv enient to star t fro m one monad A ⊂ B ( H ) and assume that one knows the ac tio n of the Poincar´ e g roup via unitaries U ( a, Λ) on A . If one interprets the mo nad A as a wedge algebr a A = A ( W ) than the Poincar´ e action g enerates a net of wedge algebra s {A ( W ) } W ∈W . A QFT is suppo sed to hav e lo cal obs erv ables and hence if the double cone intersections 55 A ( D ) turn o ut to b e trivial (multiples of the identit y a lgebra), the net of wedge algebras do es not leads to a Q FT. This is exp ected to b e the alge br aic coun- terpart of a L a grang ia n which do e s not hav e a hav e a corr e sp o nding Q FT. If how ever these in ters ections are nontrivial, we would hav e a rig orous existence pro of; the existence of a generating ﬁeld fo r those double c o ne a lgebras is then merely a tech nical problem. There are of co urse tw o obvious sticking po ints: (1) to ﬁnd the action of the Poincar ´ e on A ( W 0 ) and (2) a metho d which establishes the no n-triviality of intersections of wedge alg e bras and lea ds to for mulas for their gener ating elements. As was explaine d in the pr evious section, bo th problems hav e b een solved within a class of factorizing mo dels [3 0]. Nothing is known a bo ut how to address these tw o p oints in the more gene r al setting i.e. when the temp ered PFG are not av aila ble. The monad setting has o nly b een formulated for Poincar´ e- c ov ariant QFT. A e xtension to lo cally cov aria nt QFT in CST is ex pe c ted to present a new path to wards the still elus ive Quantum Gr avit y . It is tempting to think of the diﬀeomorphisms of A QFT in CST to b e of modular o rigin. A pa rticularly sim- ple illustration is D if f ( S 1 ) , the diﬀeomorphism gr oup of chiral theories on a circle. It is well known that the v acuum is only inv ariant under the Mo ebius subgroup and ther e ar e no s ta tes which a re inv aria nt under higher diﬀeomor- phisms. The candidates for the higher mo dula r gr oups are the diﬀeomorphisms which ﬁx more than tw o p oints which can b e obtained from a covering con- struction inv olving r o ots o f fractiona l M¨ obius trans formations. The r esulting m ulti-interv al constructio n sugg ests to lo ok for the mo dular gr o up of a multi- int erv al; the problem is to ﬁnd the appr opriate states which lead to a geo metric mo dular gr oup. This problem was solved very recently by Longo, Kaw ahigashi and Rehr en [9 9]. The interesting asp ect of their so lution (in agree ment with the absence of eige ns tates of hig her diﬀeomorphisms) is that the res ulting mo dular groups are only partially geometric i.e. ge o metric only inside the multi-in terv al. This is of course what o ne exp ects in the case of iso metries in CST. Another interesting pr oblem which is on the v erg e o f being solved is the existence o f a hig her (m=0 ,s > 1) q uantum Aharonov-Bohm e ﬀect. The quantum 55 Double cones are the t ypical causally complete compact r egions which can be obta ined b y intersecting wedges. 55 A-B eﬀect in the setting of A QFT is the s tatement that the electro magnetic free quantum ﬁeld shows a violation of Haag duality [1 00] for a non simply connected toroidal spac e time region T A ( T ) ⊂ A ( T ′ ) ′ (37) whereas for simply co nnected r egions the equality (Haa g duality) holds . F or higher spin massive ﬁelds Haa g duality holds for an y r egion. The A-B inter- pretation is that that the rig ht hand side contains observ ables which ca nnot b e constructed from ﬁeld strengths in the torus. This violation of Haag dualit y has b een sho wn in an old unpublished work befo re the mo dular metho ds b e- came av a ilable. A mo dular approach to this pro blem yields mo re than just the violation o f Haag duality , one also can compute a mo dular group a nd there is a c lose relation to the previo us 4 -ﬁx p o in t problem. What makes this problem so fascina ting is the fact that it has a nontrivial extension to zero mas s s > 1 in which case higher genus A-B ﬂuxes result. So it places s= 1 gauge theory a nd the higher spin extensions on the same A-B fo oting. Finally we should mention one unsolved long- lasting issue of mo dular the- ory: the mo dular gro up of the free massive double co ne alg e bra (with resp ect to the v acuum) which is known to act ”fuzzy” (non-geometr ic) and has b een conjectured to hav e a Hamiltonian which acts as a pseudo- diﬀerential op er ator instead of a diﬀerential oper ator [101]. There are rather c o nvincing arguments that the hologr aphic pro jection o f such a situa tion lea ds to a geo metric mo dular mov ement o n the horizon [10 2]. This s uggested the idea that if one knew a for - m ula for the propaga tion of characteristic ma ssive data on the hor izon into the inside o f the double cone, the fuzzy ac tio n may simply come ab out by applying this for mu la to the geo metric gr oup on the ho rizon. Such a formula ha s r ecently app eared in [103] A m ( x ) = − 2 i Z LF dy + d 2 y ⊥ ∆ m ( x − y ) | y − =0 A ( y + , y ⊥ ) (38) where A ( y + , y ⊥ ) is the tr ansverse extended chiral hologr aphic pro jection of the massive bulk ﬁeld A m ( x ) . A sc a le tra nfo r mation on A ( y + , y ⊥ ) a cts on y + and since y − = 0 w e c an apply the in verse scale transformation to y − without changing anything. By renaming v ariables we can ma intain the orig inal unsca led v ariable in A if we replace the y in ∆ m by y + → λ − 1 y + , y − → λy − . Using the Lorentz inv aria nce of ∆ m we may s hift this transformation to the x . So the upshot is that the dilation on H ( W ) pa sses to the Lorentz b o ost on the bulk W . Let us now see how the ”fuzzyness ” develops in the case of a double cone. F or simplicity w e sta y in d = 1 + 1 and chose a double cone symmetric around the o rigin a s in [5]. Then the low er ma nt le of the cone with ap ex ( − 1 , 0) is a Horizon whose causal sha dow co vers the double cone. Every s ignal which ent er ed the double cone must have entered through the mantle. In this case the 56 propaga tion from the tw o pieces of the ma nt le leads to the sum A m ( x + , x − ) = − 2 i Z +1 − 1 dy + ∆ m ( x − y ) | y − =0 A ( y + )+ (39) + − 2 i Z LF dy − ∆ m ( x − y ) | y + =0 A ( y − ) Now the mo dular group on the Horizon is fractio nal namely the ” dila tion” which leav es the ﬁxed p oints y ± = − 1 , +1 inv aria n t instead of 0 , ∞ as in the ﬁrst ca s e. The mo dular group on b oth parts o f the hor izon is x ± ( s ) = (1 + x ± ) − e − s (1 − x ± ) (1 + x ± ) + e − s (1 − x ± ) (40) Diﬀerent fro m the previous case one ca nnot transfer this fractiona l change from the y to the x. This time ther e is no lo cal transfor mation, rather the a ction on A m ( x ) is fuzzy but stays ins ide the do uble co ne . It is how ever not pur ely alge - braic since it was o btained by combining the g eometric group o n the Ho rizon with the causal propa gation whose r e verberatio n asp ect causes the fuzzyness. It can b e shown that as in the case of a wedge the in teractio n do es not change any- thing, it is a lwa ys this semi-ge o metric modula r gro up. A mor e general discussio n including a calculation of the pseudo diﬀerential g enerators of these mo dular ac- tions will be contained in a forthcoming paper b y Brunetti and Mor etti. So it lo oks that there is some new movemen t o n this long-lasting issue. References [1] R. Clifton and H. 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Localization and the interface between quantum mechanics, quantum field theory and quantum gravity I (The two antagonistic localizations and their asymptotic compatibility)

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