Localization and the interface between quantum mechanics, quantum field theory and quantum gravity II (The search of the interface between QFT and QG)

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 I (The searc h of the in terface b et w een QFT and QG) dedicated to the memory o f Rob Clifton to be pu b lished in ”Studies in History and Philosophy of Physics” 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, G erman y Octob er 24 , 2018 Con ten ts 1 In tro duction to the s econd part 2 2 The split inclusio n 3 3 Lo calization-induced v ac uum p olarization, pres en t view and history 7 4 The elusive concept of lo calization en tropy 11 5 Holography on to horizons, BM S symm etry enhanceme n t 15 6 The lo cal co v ariance principle 24 7 Resum´ e, mi s cellaneous comments and outlo ok 26 Abstract The main topics of th is second p art of a tw o-part essay are some conse- quences of the phenomenon of vacuum p olarization as the most imp ortant 1 physical manif estation of mo dular lo c alization. Besides p hilosophically un- exp ected conseq u ences, it has led to a new constructive ”outside-inw ards approac h” in which the p ointlik e ﬁelds and th e compactly lo calized op- erator algebras which they generate only app ear from in tersecting m uch simpler algebras lo calized in noncompact w edge regions whose generators hav e ext remely mild almost free ﬁeld b ehavior. Another consequence of v acuum p olarization presented in this essa y is the lo calization entropy near a causal horizon which follow s a logarithmi- cally mo diﬁed area la w in whic h a dimensionless area (th e area divided by the square of d R where dR is the th ic kness of a ligh t sheet) app ears. There are arguments that this logarithmically mo diﬁed area law corresp onds to the vol ume law of th e standard heat b ath thermal b ehavio r. W e also ex- plain the symmetry enhancing eﬀect of holographic pro jections onto the causal horizon of a region and show th at t he resulting inﬁnite dimensional symmetry groups contain the Bondi-Metzner-Sachs group. When the ﬁrst versi on of t his p ap er was submitted to hep -th it w as immediately remov ed by the mo derator and placed on phys. gen without any p ossibilit y to cross list, even though its con tent is foundational QFT. With the interven tion of a member of the advisory comitee at least the cross-listing hop efully seems no w to b e p ossible. 1 In tro duction to the second part Whereas the ﬁr st pa r t [1] pres en ted the interface betw een (r elativistic) Q M and QFT, this second part fo cus s es on the interface o f QFT with g r avit y , or mor e precisely on what hitherto was pr e s umed to deﬁne this in terface 1 . As a re s ult o f the very diﬀer en t na tur e of the lo calization concept o f QFT, a bipartite pa rti- tion in to a subalgebra of a c ausally closed r egion and its ca us al disjoint do es no t tensor facto rize; the sha rp lo ca lization ge ne r ates inﬁnitely la rge v acuum pola r - ization whic h destroys the quantum mechanical entanglemen t co ncept. This is how ever not the end of the sto ry , sinc e in QFTs with reaso nable phase space de- grees o f free dom one can enforc e a tensor fa ctorization via the split co nstruction which re-crea tes so me but not all asp ects of quantum mechanical entanglemen t. The next sectio n pr esents this impo rtant ” splitting” idea. Althoug h there are mathematical examples o f QFT which violate the prer equisites for splitting, a physically motiv ated pha se space densit y of QFT exclude such cases. In pa r - ticular the existence of the co mpleteness pro per ty of as ymptotic incoming or outgoing particles in theories with a ﬁnite n umber of particle sp ecies imply the splitting pr o pe rty . The third section addr esses the most imp ortant ph ysica l im- plication of lo calizatio n, namely vacuum p olarization and prepar es the ground for the presentation of lo calization e n tro p y in s ection 4. It is shown that if in an interacting theor y one ” bangs” with a lo ca l op erato r A on to the v acuum, the so -obtained lo cal v acuum exc ita tion s ta te A Ω has inﬁnitely man y par ti- cle/antiparticle comp onents whose a nalytic c o nt inuation determine all formfac- 1 An example is the concept of entrop y associated with horizons, whic h acco rdi ng to our results in this essa y can b e f ormulat ed and understo od in the s tandard setting of LQP . 2 tors of A throug h the crossing relations. Section 5 expla ins the mathemati- cal/conceptual meaning of holo graphy o nto horizons a nd shows how the los s in information and spa cetime symmetry can b e r econciled with a huge in con- formal symmetry: the ho lographic pro jection a dmits inﬁnite dimensional sym- metry groups which co n tains in par ticular the classical Bondi-Metzner -Sachs group [22]. T he increa se of conformal symmetry on the ho rizon do es not help in inv erting the holo graphic pro jectio n back tow ards the reconstruc tio n of the bulk. The per haps greatest progress has b een the ada ptation o f the Eins tein loca l cov ariance principle to the quantum r ealm [39] which is brieﬂy sk etched in sec- tion 6 . This go es a lo ng way towards the ” ba ckground independence” which is the would-b e ”Holy Grail” of the still elusive QG. The mo dular theory of whic h relev ants parts for the 2-par t paper w ere pre- sented in part I, plays an imp ortant r ole in connecting the entropy asp ects of the heat-bath situation with those o f lo calization. According to o ur b est knowledge the connec ting formulae in section 4 are new. 2 The split inc lusion There is one prop erty of LQP which is indisp ensa ble for understanding how the quantum mechanical tensor factorization ca n b e reconciled with mo dular lo calization: the split pr op ert y . Deﬁnition: Two monads A , B ar e in a split p osition if the inclusion of monads A ⊂ B ′ admits an interme diate t yp e I factor N such that A ⊂ N ⊂ B ′ Split inclusions are very diﬀerent from mo dular inclusions or inclusio ns with conditional exp ectations. Their main prop erty is the existence an N -dep endent unitarily implemen ted isomorphism of the A , B g e nerated op era tor algebra into the tensor product algebra A ∨ B → A ⊗ B ⊂ N ⊗ N ′ = B ( H ) (1) The prerequisite for this fac torization in the LQP context is that the monads commute, but it is well-known that lo cal comm utativity is no t suﬃcient, the counterexample be ing tw o do uble cones which touch each other a t a spacelike bo undary [2]. But as so on as o ne lo ca lization region is spacelike separated from the other by a (ar bitrary small) spacelike s ecurity distance, the interaction-free net sa tis ﬁe s the split prop erty under very ge ne r al conditions. In [3] the relev ant ph ysic al prop erty was ident iﬁed in form of a pha se s pa ce prop erty (part I, sec- tion 4 ). Unlike QM, the num b er of deg rees of free do m in a ﬁnite phase space volume in QFT is not ﬁnite, but its inﬁnity is in some sense mild; it is a nucle ar set for free theories and this n uclearity requirement 2 is then po stulated for in- teracting theories [2]. The physical r eason b e hind this nuclearity requirement is that it allows to show the exis tence of temper ature sta tes once one knows that a QFT exists in the v acuum repres ent atio n. E ven more imp o rtant its v a- lidit y preven ts the vio lation of the causal shadow prop erty which states that 2 A set of ve ctors is nuclear if it is con tained i n the range of a trace class op erator. 3 the degrees of freedom in the ca usal s hadow of a spacetime region are the same as thos e in the origina l r egion: A ( O ”) = A ( O ) which is the algebra ic analog of hyper bo lic Cauch y propa gation. All these pro per ties a r e formally true in a Lagra ng ian setting but they cons titute a physical safety k it for mov ements o ut- side the standard quantization para llelism to classica l ﬁeld theor y as holo graphy onto horizons, AdS-CFT corresp ondences etc. The split prop er ty for tw o s ecurely c a usally separated a lgebras ha s a nice ph ysic al interpretation. Let A = A ( O ) , B ′ = A ( ˇ O ) , O ⊂ ˇ O . Since N contains A and is contained in B ′ (but witho ut car rying the assignment of a sha rp lo cal- ization b etw een O and ˇ O ), one ma y ima gine N as an algebra which shares the sharp loc alization with A ( O ) in O , but its lo c alization in the ”co llar” b etw een O a nd ˇ O is ”fuzzy” i.e. the colla r subalge bra is like a ”haze” which do es not really o cc upy the co lla r region. This is prec isely the regio n which is conceded to the v acuum p olar iz a tion cloud in or der to sprea d and thus avoid the inﬁnite compressio n into the surface of a sharply lo ca liz ed monad. If we take a se q uence of N ′ s which appro ach the monad A , the v a cuum po larization clouds b ecome inﬁnitely la rge in such a wa y that no dir ect deﬁnition of e.g. their ener gy o r ent ro py conten t is po ssible. The inclusion o f the tensor algebr a o f monads into a type I tensor pro duct (1) lo oks at ﬁrst sig h t like a d ´ ej` a vu of QM tensor factor ization, but there ar e int ere sting and impor tant diﬀerences. In Q M the tensor factor ization obtained from the Born lo calization pro jector and its complement is automatic since the v acuum of Q M (or the g round state of a quantum mechanical zero temp era ture ﬁnite density system) tensor factorizes. In QFT the v acuum does no t tensor factorize a t all, but there are other sta tes the so- c a lled ”split v a cuum” sta tes in the Hilber t space which em ulate a tensor-facto rizing v acuum in the se nse that exp ectation v alues of op erator s in A ( O ) ∨ A ( ˇ O ′ ) factor ize in the split v acuum h 0 split | AB | 0 split i = h 0 | A | 0 i h 0 | B | 0 i , A ∈ A ( O ) , B ∈ A ( ˇ O ′ ) (2) but there remains a huge conceptual diﬀerence to the quantum mechanical Bo rn factorization of the ”nothing” s ta te. The s plitting pr o cess requir es the supply of energy since the split v acuum has inﬁnite v acuum po larization (with ﬁnite mean energy) in the co llar region which is spacelike to O ∨ ˇ O ′ . The physical states of QFT a re by deﬁnition the sta tes with ar bitrary la rge but ﬁnite e nergy . Their massive par ticle conten t is ﬁnite but they may co n tain (as it is the c a se in QED) inﬁnitely man y zero mass particles. In contradistiction to QM thes e s ta tes only tensor-facto r ize after a spatial split, in which ca se the reduced v acuum and all ﬁnite energy states b ecome thermal Gibbs state with re spe c t to a split-related Hamiltonian. Without the split the A ( O )-reduced v acuum state is a sing ular 3 thermal KMS state. The problem o f ph ysical realizability has no t been given muc h attention in foundational discussio ns of QM. In QFT this issue is mor e serious since the situations a re muc h more co un ter- intuitiv e, as was shown b efore with the particle b ehind the moo n a rgument for the global v acuum. This proper ty is 3 A singular KMS state denotes a KMS state which is not a Gibbs state. 4 absent in a split v acuum state; the split deﬁnes a barrier , but it is unclear how such split states ca n be prepared a nd monitored. Most fo unda tional prop erties o f QM, a s violation of Bell’s inequalities, the Schroedinger cat prop erty a nd many other stro ng deviations from classica l rea l- it y can b e exp erimentally veriﬁed. This is generally not poss ible for the v acuum po larization caused prop e r ties whic h result from modular lo caliz a tion simply bec ause macros c opic manifestatio ns are to o small. A typical example is the Unruh eﬀect i.e. the ther ma l manifestation of a uniformly accele rated particle counter in the global v acuum, where the tempera ture created by an accelera tion of 1 m/ sec is 10 − 19 K to o s ma ll for ever b eing registered. But for the per ception of the reality which underlies LQP the diﬃculty in registering such eﬀects do es not diminish their conceptual impor tance. The characteriza tion of the restriction o f the global v acuum to a lo cal algebra in ter ms of a ther mal state for a mo dular Hamiltonian holds, independent of whether the lo cal algebra is a sharply lo c alized mo nad A ( O ) or a type I factor N contained in a lar ger sharply lo calized algebr a ˇ O , as in the ab ov e splitting construction. The only diﬀerence is that the in the sec o nd case the KMS state is also a Gibbs state i.e. the Hamiltonia n on N has a discr e te sp ectrum (in c ase ˇ O is compa c t). This therma l reinterpretation of r educed states do es not o nly hold for the v a cuum, but applies to all s tates which are of physical r elev anc e in particle physics i.e. to all ﬁnite energy states for which the Re e h- Schlieder theorem applies . Since KMS states on type I factors ar e Gibbs states, there exists a dens it y matrix. The r efore these Gibbs state can hav e a ﬁnite energy and ent ro py conten t which for monads is imp ossible. But a monad may be approximated by a sequence of type I factors in complete analogy to the ther mo dynamic limit. In fact the thermo dynamic limit is the only place where a monad algebr a a ppea rs in a QM se tting; an indica tio n that this limit is accompa nied by a qua litative change is the fact that one lo o ses the density matrix nature o f the Gibbs state which changes to a mor e singula r KMS sta te which simply do es not exis t o n quantum mechanical type I algebra s. A related fa ct is the breakdown of the tensor facto rization into physical degr ees of freedom a nd their ”shadow world” in the thermo dyna mic limit. This implies that the ”Thermo ﬁeld formalism” is only a pplica ble for the b oxed QFT (type I), it loo s es its meaning in the thermo dynamic limit when the sta te b ecomes a singular KMS state and the algebra turns in to a monad. The structural diﬀer e nc e can b e traced ba ck to the nature of mo dular Ha mil- tonians. Whereas for a mona d the mo dular Hamiltonian has contin uous sp ec- trum 4 and hence an ill-deﬁned (inﬁnite) v alue of energy and entrop y , this is not the cas e for the N -associa ted density matrix constructed fro m the split situa- tion. So the wa y out is ob vio us, one m ust imitate the thermo dyna mic limit by constructing a s equence o f t yp e I factor s (a ” funnel” N i ⊃ A ( O ) by tightening the split) whic h co nv erge from the outside tow ar ds the monad; equiv a le ntly one 4 A typ ical example is the thermodynamic V → ∞ limit in whic h the discrete spectrum of the Hamiltonian in a b ox turns into a contin uous spectrum and the H amiltonian b ecomes the modular Hami ltonian on a monad. Gibbs state. 5 may a pproximate fr om the inside. In the next s e c tion the split limit, in which a double-co ne lo calized mo na d is approximated by a sequence of type I ∞ factors N i , will b e pres ent ed. In this case the mo dula r g roup of ( N i , Ω) leads to a Gibbs situatio n i.e. the restriction of the v acuum to the a lgebra N i is a Gibbs state at the sa me mo dular tem- per ature as the that a sso ciated with the restrictio n to the monad. The main distinction to the standard heat bath situation is that Ha milto nia ns which re- sult fro m res tricting the v acuum to an N i obtained from the split constr uction is constructed ac cording to very diﬀerent principles fro m tha t o f Gibbs s tates in a ﬁnite qua ntization b ox with the help of the Hamiltonian which descr ibes the time developmen t in a quantum mec hanica l inertial frame. Nevertheless for t wo-dimensional chiral theories there exists a rigoro us relation betw een the tw o kinds of therma l behavior: the inverse Unr u h eﬀe ct . The lo caliz a tion-caused thermal manifestation in a chiral theory is related by a conforma l trnaforma- tion to the thermo dynamic limit o f a one-dimensio nal glo bal heat bath iner- tial system [2 1][19]. There are pre s ent ly no reliable computational techniques for dealing with mo dular Hamiltonians and their split a pproximands, although there is no la ck of mathematica l precision in deﬁning these ob jects. In the next section we will try to overcome this situation by inv oking geometr ical arg umen ts leading to a lo calization of the v acuum polar ization cloud inside a ligh t- she et . The split prop erty do es not hold in all axiomatic mo dels of LQP but ther e are rather go o d arguments that it is v alid in ”physical” mo dels which shar e a ph ysica lly relev ant prop erty with Lagr angian mo dels. What is important is no t the Lagrang ian q uant iza tion but rather the causa l timelike propagatio n asp ect which historically has b een r eferred to as the time-slice prop erty [4] or the c a usal s hadow prop er t y . This prop erty is violated in theo ries with to o many phase spac e degrees o f freedom. The the study of what co nstitutes a physical phase space densit y started in the 60 s [5] when for the ﬁrst time it was sho wn that the implementation of the relativistic causa lity pr inciple requires a bigger cardinality of phase space degree of fr eedoms than in QM. Whereas the n umber o f degrees of free do m per unit cell in phase spa ce is ﬁnite, these authors found the cardinality in QFT is inﬁnite in the sense of being a compact set. This result was later sharp ened to nu cl ear < compact [6] and v arious importa n t prop e rties w ere shown to b e consequences , among them the exis tence of ther mal sta tes for all temper atures and the splitting prop erty [2 0]. The implies the abs ence of an un want ed Hagedo rn tempera ture as it o ccurs in inﬁnite comp onent QFT with to o many degr ees of freedom as string theo ries. It is easy to in ven t QFTs which violate this b ounds in the sense of ha ving to o many deg rees o f freedo m, the generalized free ﬁelds with suitably incr easing Kall´ en-Lehmann dens it y are the simplest ex amples. Unfortunately the k nowledge abo ut these concepts go t los t, o therwise how can one explain that a worldwide communit y works on a narrow sub ject as the AdS 5 -CFT 4 corres p ondence with mo re than 6 0 00 contributions witho ut b e- coming aw ar e that it is structurally impo ssible to hav e theories with physical degrees of freedom on b oth sides o f this corr esp ondence? The naive exp ectation 6 (back ed up b y unconvincing ca lculations outside any mathematical control) in these pap ers are in co n tra diction with established facts 5 . O ne can safely assume that this knowledge has b een lost and even worse, that those few correct pap ers which have b een published on this co rresp ondence [2 6][27][8] are be yond co n- ceptual rea ch o f the t ypical particle ph ysicis t with a co n temp ora ry (p os t string theory) q ua nt um ﬁeld theo retical ba ckground. If one lo o ks at the thousa nds of publication on this matter o ne cannot es cap e the conclusion that the deep conceptual knowledge of the past has b een replac e d by a more metaphoric mo de of thinking in the pursuit of a n ultra-r e ductio nist theory of everything (TOE). In the a bove form the po s itioning of monads aims at characterizing LQP in Minko wski spacetime. This b egs the question whether there is a gener alization to curved s pacetime (CST). A very sp ecial explor atory attempt in this direction would b e to inv estigate the Diﬀ(S 1 ) symmetries b eyond the Mo ebius gro up in chiral theor ies for their poss ible modula r origin in terms of p ositioning mo na ds relative to refer ence s tates whic h are diﬀerent fr o m the v acuum. Since the extende d c hiral theories which r esult from null-surface ho lo graphy (and no t fr om chiral pro jectio ns o f a tw o-dimens io nal confor mal QFT) seem to hav e great constructive p otential, this question could be of pra ctical interest. I exp ect that b y pursuing the algebr aization of QFT in CST accor ding to the p ositioning of monads viewp oint, one will learn imp ortant lesso ns ab out the still unknown QFT/QG in terfac e . It would b e of gr eat in teres t to under stand whether the isometric isomorphisms r elated to the local co v ariance principle (see las t se c tio n) have mo dular ro ots similar to symmetries of the v acuum in Minko wski QFT. A conser v ative approach whic h explor es unknown asp ects of QFT while sta ying ﬁrmly r o oted in known principles seems to be the most promising path for pushing the b orderline of QFT in CST further to wards the still unknown QFT-QG in terface. 3 Lo calization-induced v acuum p olarization, presen t view and history The phenomenon of v acuum p olar ization has been the p oint of departure of many metaphors of whic h the v acuum in QFT as a ”steaming br oil” is perha ps the b est known b ecause it o cca sionally even entered textb o oks. In order to sup- po rt this image the app eal to a shor t time vio lation of the energ y conser v ation allow ed b y the uncer taint y rela tion was made. A less metaphor ic view co mes from lo cally ”banging” on the v acuum i.e. applying a compactly lo ca lized op er- ator to the v acuum state. Suc h a state is characterized by its n-par ticle matrix elements for all n a nd these n- pa rticle v acuum p olariza tion compo nen ts are in turn sp ecial b oundar y v alues of a n analytic n- particle master function whose 5 The shar i ng of the SO(4,2) conformal symmetry preven ts a dilution of degrees of f reedom in passing from the higher- to the l o wer- dimensional theory . Starting f rom a standard confor- mal QFT (the sup ersymmetric N=4 Y ang-Mills theory is a potential candidat e) the degrees of freedom transplant ed to the higher di mensional AdS spacetime are not suﬃcient to ”ﬁll” the AdS bulk and remain ho vering at the boundary . 7 diﬀerent o ut-in pa rticle distributions obtained from the v a c uum po la rization comp onent by cross ing are the formfactors of A. A Ω ≃ { h 0 | A | p 1 , ...p n i in } n (3) cr os sin g → { out h− ¯ p k +1 , ... − ¯ p n | A | p 1 , ...p k i in } n where the neg ative mass s hell momenta − ¯ p denotes the a nalytic contin uation which is part o f the cr ossing pr o cess and the ba r is the reminder that the pa rticle is an antiparticle to the orig inal par ticle (this can b e omitted in case of self- conjugate par ticles). In case of A ∈ A ( O ) being the identit y op era tor there is no ”bang ing ” onto the v acuum; in that c a se we are dealing with the S-matrix for which many ma trix-elements v anish as a r e sult o f the energ y momentum conserv ation betw een the particles without the feeding in from the localize d op erator A. In particular the S-matrix is free of v acuum polar ization clouds, since the iden tity leav es the v acuum in v a riant and do e s not bang . The crossing prop erty is o ne of the deep est characteristics linking ﬁelds and particles. It had b een known for a long time that ther e are no compa ctly lo calized op era to rs which applied to the v acuum generate a one - particle state without a n admixture of a v acuum polar ization cloud except if the theor y is that o f a free ﬁeld 6 . One- particle op erato rs exist as in- or out- op er ators (or unitary transforms thereof ) in the full alg ebra B ( H ) , but they have a v ery nonlo cal relation with r esp ect to the lo ca lized op e r ators. The wedge r egion is a bo rderline lo calizatio n in that there exist wedge-loc a lized o per ators whic h even in the pr esenc e of inter actions cr e ate p olarization-fr e e one-p article (and also m ultipar ticle incoming) states . These op er ators are not identical to the incoming creation/a nnihilation op erato r s but they shar e the sa me Reeh-Schlieder domain for w edge lo calization. The w edge restricted v acuum state is a KMS state and the KMS pr op e rty is int imately rela ted to the one-step cross ing [44]. A pro of of cros s ing fro m these lo caliza tio n pr op erties would g o be yond the more mo dest aims of this pap er. Whereas in QM, re la tivistic or not, one has great lib erty in ma nipulating int era ction p o tent ials without leaving the setting o f QM so that almos t any prescib ed o utcome can b e accommo dated, this is not the case in Q FT. There the lo cality pr inciple is very restr ictive and this tight ness even show up in theorems ab out the S-matrix as the Aks theorem saying that in a 4 -dimensional QFT nontrivial elastic scatter ing is not p ossible without the presence of inelastic comp onents [11]. F or the formfactors the statement (3) has a more p opular stronger fo r m which, although probably pr ov able, has a c c ording to my bes t knowledge presently the status o f a fact b eing supp orted b y exp erience. This is the appar ent v alidity of a k ind of benevolent Murphy’s law: al l c ouplings of lo c al op er ators to other channels (in the c ase of formfactors multip article channels) which ar e n ot forbid den by sup ersele ction rules actual ly do o c cur. Of course one needs to ” bang o n to the v ac uum” , there is no ”b oiling soup” in a an iner tia l frame without heating the ”v acuum stove”. 6 Within the Wight man setting this has b een kno wn as the Jost-Schroer theorem [9]. A stronger form w as r ecen tly prov en in the algebraic setting [ 10]. 8 The formfactor asp ect of a lo cal op erator is p erhaps the b est QFT illustr ation of Murph y’s law to particle ph ysics . This tight coupling o f channels thro ugh the realization of the loca lit y principle is bo th a blessing a nd a curse. It attributes a holistic s tr ucture to QFT whic h on the one hand aggr av a tes the strategy to divide the diﬃcult problem of (nonperturbative) model construction in to easier pie c es, but on the other ha nd is the main reaso n why this theory is muc h more fundamen tal than Q M. In particular it do es not supp ort the pres en tly fashionable idea of ”eﬀective” QFT in which the holistic asp ect is larg e ly ig nored and which is eulogize d when it do es give a wan ted result and disre garded if it do es not. So me interesting and p er tinent rema rks ab out the imp ortance o f the holistic p oint of view in c o nnection with the pro blem of the energy density of the cos mological reference state can b e found in [12] V acuum po larization as a concomitant phenomenon of QFT w as discovered a long time b efore the r ole of locality to ok the center stage. It is interesting to reformulate Heisenberg’s ﬁrst obser v ation in a more mo dern context by deﬁning partial c har ges by limiting the charged reg ion with the help of smo oth test func- tion. In Heisenberg’s more forma l setting the pa rtial charge of a fr ee conser ved current in a s pa tial volume V is de ﬁned as Q V = Z V j 0 ( x, t ) d 3 x (4) j µ ( x, t ) =: φ ∗ ( x, t ) ↔ ∂ µ φ ( x, t ) : Int ro ducing a mo men tum spa ce cutoﬀ, the no r m of Q V | 0 i turns out to diverge quadratically which together w ith the dimensio nlessness o f Q is tied to the area prop ortiona lity . Hence already o n the ba sis of a crude dimensional reas o ning one ﬁnds an area prop or tionality of v acuum p olarizatio n. The cutoﬀ was the prize to pay for ignor ing the singular na ture of the cur rent which is really no t an op era to r but ra ther an o per ator-v alued distribution. The moder n remedy is to take ca re of the divergence by treating the sing ula r current as a n op era tor-v a lued distributio n. Such calculations hav e b een done in the 60s [13][14] b y using spacetime test functions which r egularize the delta function at coa lescing times and are equa l to one inside the ball with radius R and fall oﬀ to zero smo othly betw een R and R +∆ R. Using the co nserv a tion law of the current one ca n then show [13] that the action of the regularized par tial charge on the v acuum is compressed to the shell ( R, R + ∆ R ) and diverges quadratically with ∆ R → 0 i.e. As exp ected, the v ac uum ﬂuctuations v anish weakly a s R → ∞ (even str o ngly by enla rging the time smearing supp ort of g together with R [15]) i.e. the limit c o nv erges indep endent of the s p ecia l test function weakly 7 to the global charge oper ator lim R →∞ Z f R, ∆ R ( ~ x ) g ( t ) j 0 ( x, t ) d 4 x = Q (5) 7 Although the norm diverges, the inner pro duct of Q R | 0 i with localized states conv erges to zero in compliance with the zero c harge of the v acuum. 9 With other w ords in the limit of globa l charges the v acuum po larization drops out together with the test function dependence. The interesting question in the context of the present section is the ques tion of what is the R , ∆ R dependence when ∆ R → 0 . The answer dep ends on the dimension of spacetime and the leading divergence ca n b e calcula ted for free currents in massles s theor ie s. The simplest cas e is tha t of a chiral current which is lo calized on a ligh t ray h j ( x ) j ( x ′ ) i ≃ 1 ( x − x ′ + iε ) 2 (6) k Q ( g R, ∆ R )Ω k ˜ ln R ∆ R i.e. diﬀerent from QM the dimensionles s p artial ch ar ges diverges in Q FT, the ﬁrst ma nifestation of v a cuum po larization as ﬁrst observed by Heisenberg . In higher dimensio na l QFT the loga rithmic behavior is modiﬁed by powers in R ∆ R ; in par ticular in d= 1+3 one obtains an (log arithmically corr ected) area law k Q R, ∆ R Ω k ˜( R ∆ R ) 2 ln R ∆ R (7) The v acuum p olariza tion-caused ∆ R → 0 div er gencies of this partial charge op erator are preempted in a certain sense b y the behavior of the dimensionles s lo calization-entropy; ho wever despite similarities the computation of the latter is c onceptually mor e inv olved. The rea s on is that the en tropy is inherently nonlo cal in the sense that it cannot be obtained by a integrating a pointlike conserved curr ent (or a n y other op erator) but rather enco des a holis tic asp ect of an entire algebr a. Nevertheless the splitting pr op erty (for a descr iption of its history see [2]) is in a certain sense the algebraic ana log of the test- function smearing on individual ﬁeld oper ators. Entrop y in QM is an infor ma tion theore tica l concept whic h mea sures the degree o f entanglemen t. The standard situa tion is bipartite spa tia l sub division of a global system so that global pure states become entangled with respec t to the s ubdivis io n i.e. they can b e written a s a sup erp os ition of tensor pro duct states. The entrop y is than a num b er computed a ccording to the von Neumann deﬁnition from the reduced impure state which r esults in the s tandard wa y fro m av eraging ov er the oppos ite compo nen t in one of the tensor factors. The traditional quantum mechanical way to compute en tang lemen t ent ro py was applied to QFT o f a halfspace (a Rindler wedge in spacetime) for a s y s- tem of free ﬁelds in a inﬂuential 1984 pap er [1 6]. T he authors star ted from the assumption that the total Hilb ert space factorizes in that belo nging to the halfspace QFT and its opp osite. The calculatio n is ultraviolet divergen t and af- ter intro ducing a momentum spa ce cutoﬀ κ, the authors show ed that the cutoﬀ depe ndence is co ns istent with an ar ea b e havior. S/ A = C κ 2 (8) where in the conformal case C is a constant, κ is a momen tum space cutoﬀ a nd S/ A denotes the s ur face dens ity of entrop y . The metho d of co mputation is a gain 10 the in tegr ation ov er the degrees of freedom of the complement reg io n and the extraction of the entrop y fr om the r esulting reduced density matrix state whose degree of impurit y e nco des the measure of the inside/o utside entanglemen t. This calcula tion should b e seen in analog y with Heisenberg’s mo mentum space cut-oﬀ calcula tion of v acuum pola rization in the pa rtial charge (4). In bo th cases the starting formula is morally correct but factually wrong. Neither is the pa rtial c har g e inside a regio n deﬁned b y a volume integral nor do (as we know from discussio ns in prev io us sections) glo bal states in QFT p ermit an inside/outside factoriza tio n. These incorr ect assumptions create the divergen- cies which are then kept under the lid by the p opular emer gency kit of QFT: momentum spac e cutoﬀ. In b oth ca ses dimensional arguments lead to an area prop ortiona lity (with lo garithmic c orrections p ossibly es c aping the cons idera- tion). The main adv a nt ag e of the present spacetime approach v er sus a mo mentum space cutoﬀ argument is that the s plit prop erty teaches us that the v a cuum po larization cloud hov ers near the ho rizon in the split r egion characterized by the s heet s ize ∆ R . The divergence for ∆ R → 0 indicates in no wa y a conceptual inconsistency or shortcoming of Q FT which must must b e ov ercome with the help of quantum gravit y . With other w or ds the lo calization entropy is a no tion within a sp eciﬁed QFT, it do es not need a ny reference with r e spe c t to an ill- deﬁned nonlo cal ” cutoﬀ theory” 8 . There ar e simply some quantities whose shar p lo calization ca uses divergencies but whose global v alue is per fectly ﬁnite; for the global charge it is the ﬁnite v alue car ried by o ne or s everal particles and in the case of the entrop y its global v alue in the gr ound state v anishes. 4 The elusiv e concept of lo calization en trop y Let us ﬁrst apply the previo us ly presented s plit idea to a tw o- dimensional con- formal QFT in whic h case the double cone is a tw o -dimensional spa cetime reg ion consisting of the forward and backw ard causal shadow of a line of leng th L at t = 0 sitting inside larger cone obtained by augmenting the baseline o n b oth sides b y ∆ L. As a result of the as sumed co nfo r mal inv aria nce of the theo ry , the canonical split algebr a inherits this inv ariance and hence the en tro p y E nt of the canonical split alg ebra ca n only b e a function of the cr oss r atio of the 4 points characterizing the split inclusion E nt = − trρl nρ = f ( ( d − a ) ( c − b ) ( b − a ) ( d − c ) ) (9) wit h a < b < c < d = − L − ∆ L < − L < L < L + ∆ L where for conceptua l cla r ity we wrote the formula for gener ic p osition of 4 p oints. Our main int er est is to deter mine the leading behavior of f in the limit ∆ L → 0 (t wo pair s of p oints coa lesce) which is the analog of the thermo dynamic limit V → ∞ for heat bath thermal systems. 8 The concept of a theory with a cutoﬀ cannot ev en b e deﬁned in the pr esence of interactions, ev en if one limits the construct ion to the family of soluble factorizing models. 11 The as ymptotic estimate for ∆ L → 0 can b e carried out with a n algebra ic version of the r eplic a trick which use s the cyclic orbifold constructio n in [17]. First we write the entrop y in the form E nt = − d dn trρ n | n =1 , ρ ∈ M can ⊂ A ( L + ∆ L ) (10) Then one uses a gain the split pr op erty , this time to map the n-fold tensor pro d- uct of A ( L + ∆ L ) from the replica trick into the algebra of the compactiﬁed line ˙ R = S 1 with the help of the n th ro ot function n √ z . The part whic h is inv a riant under the cyc lic p ermutation o f the n tensor facto r s deﬁnes the algebra ic version [17] o f the replica trick. The transfo rmation prop er ties under Mo ebius gro up are now given in terms of the following subgroup of DiﬀS 1 written for mally as n s αz n + β ¯ β z n + ¯ α , L ′ ± n = 1 n L ± n , L ′ 0 = L 0 + n 2 − 1 24 n c (11) dim min = n 2 − 1 24 n c where the ﬁrst line is the natura l embedding o f the n-fold covering of Mo eb in diﬀ(S 1 ) and the corresp onding for m ula for the generators in terms of the Virasoro generator s. As a consequence the minimal L ′ 0 v alue (spin, ano malous dimension) is the one in the second line. With this additio na l information coming from r epresentation theory w e ar e able to determine at least the singular behavior of f for coalescing p oints b → a, d → c E nt sing = − l i m n → 1 d dn  ( d − a )( c − b ) ( b − a )( d − c )  n 2 − 1 24 n = c 12 l n ( d − a )( c − b ) ( b − a )( d − c ) (12) Since the function is o nly deﬁned at integer n, one needs to inv oke Car lson’s theorem. Apar t from the split setting the calculation follows the same steps as entrop y calculatio ns in condensed matter ph ysics [1 8 ] which is based on cer- tain ass umed prop erties of functional integrals which p er mit the avoidance of momentum space cutoﬀs. The r esulting leading con tribution to the entropy reads E nt sing = c 12 ln ( d − a )( c − b ) ( b − a )( d − c ) = c 12 l n L ( L + ∆ L ) (Λ L ) 2 (13) where c in typical cases is the Virasoro co nstant (which app e ars also in the chiral holo graphic lightray pro jection). This result was pr eviously [19] obtained by the ”inv erse Unruh eﬀect” for chiral theor ies which is a theorem stating that for a confor ma l QFT on a light- like line the K MS state obtaine d by res tr icting the v acuum to the algebr a of a n int erv al is unitarily equiv alent to a g lo bal hea t bath tempera ture state for a cer- tain (geometry -dep e nden t) v alue o f the temp eratur e. The chiral inv erse Unruh eﬀect inv olves a c hange o f length par ametrization; the length prop ortio na lit y 12 of the heat ba th entrop y (the well k nown volume factor) is transfor med into a log arithmic length measur e. The inv erse Unruh e ﬀect has only b een estab- lished in chiral QFT, but it p oints tow ards a question o f signiﬁcant conceptual and philo sophical impor tance: is there a str uctural relation betw een heat ba th and lo ca lization-caused thermal b e havior o r are do they represe nt t wo unrelated ph ysic al phenomena ? On exp ects the tw o monads to b ehave in the s ame way after reparametr izing in a wa y whic h accounts fo r the diﬀerent spa cetime a spe c ts of the tw o monads and their diﬀerent approximations b y t yp e I fac to rs. In the thermo dynamic case the monad is appr oached by type I algebras of Gibbs states o n systems in a box o f v olume V in the limit V → ∞ whereas approximations of the A ( O ) mo nade is done by the type I factors obtained from the split prop erty . since the v a cuum restricted to split type I factor s also turn out to b e thermal (with r esp ect to the mo dular Ha miltonian) one exp ects a universalit y in the tw o kinds of thermal b e havior. Ther e fore the rele v ant ques tion is: ca n the volume divergencies of the heat bath thermo dyna mic entrop y be set in rela tion to the ∆ R → 0 divergence in the area b ehavior (possible mo diﬁed by a logarithm) caused by v a cuum p olariza tion as in (7) ? And do all dimensionless lo c a lized ob jects ha ve the sa me leading div erg ence for ∆ R → 0 . Since QFT do es not know any frame-indep endent p osition op era tor (”eﬀective” substitutes have no place in conceptual a rguments) the question a rises whether QFT can o ﬀer a n analog to the Heisenber g uncertaint y r elation. A universal relation betw een the leading en tro py/energy incr ease with the shar pness o f loca lization is as clo s e as one could come. This ther ma l universality h yp othesis w ould suggest the following corres p on- dence b etw een the heat bath and the lo c alization entrop y (( k T ) n − 1 V n − 1 ) T =2 π ≃ R n − 2 (∆ R ) n − 2 l n R 2 (∆ R ) 2 (14) E nt ( h.b. ) T =2 π = E nt ( l oc ) (15) where the ﬁrst line express es the reparametrizatio n o f the dimensionless (n- 1)-volume facto r in terms of a dimensio nless lo g arithmically corr ected dimen- sionless area factor. Since lo calization therma lity is a phenomenon of modula r theory which do es not know an ything a bo ut kT, the dimensionless area is ob- tained from the thic kness of the light slic e which a ppea red already as a ∆ L in the loga rithmic divergence for n=2 , the case in w hich we pr esented an rig- orous pro of base d o n the c hira l inverse Unruh eﬀect. The ab ov e equa lit y for the entropies mea ns in particlar that the tw o matter dep endent ﬁnite constants in front o f the lea ding divergencies are the same if we use the sa me quantum matter for the heat ba th a nd the lo caliza tion situatio n. F or n > 2 the relation is conjectural but its vio lation would cause ser ious problems in our under s tanding of QFT. Naturally this corresp ondence can only be expected for the leading term in the thermo dynamic limit V → ∞ resp ectively in the ”funnel” limit ∆ R → 0 of decr easing split distance. 13 A mathematica l pro o f would amount to the calcula tio n o f the von Neumann ent ro py o f the density matrix ρ which res ults fr o m the re striction o f the v acuum to the split tensor fa c to r, a ta sk which go es b eyond the pr esent computational abilities in QFT. How ever it is po ssible to present some more details supp or ting details about the geometrica l asp ects of the situation which are clo sely related to the lea ding ∆ R → 0 b ehavior of the dimensio nless partia l c harg e ope r ators (7) caused b y the v a cuum p o larization cloud. Compared with the c hira l mo dels in the b eginning of this sec tion which c a n be controlled quite elegantly with the r eplica metho d, the question of higher dimensional lo calization ent r o py lo oks more inv olved. In ter ms o f inclusions and relative commu tants the funnel approximation to the double cone situation is describ ed in terms of the following split inclusion [20] A ( D ( R )) ⊂ N ⊂ A ( D ( R + ∆ R )) (16) A ( ring ) ≡ A ( D ( R )) ′ ∩ A ( D ( R + ∆ R )) , N = A ( D ( R )) ∨ J r ing A ( D ( R )) J r ing where N is the ca nonically ass o ciated type I alg ebra in terms of which there is tensor factor ization as in (1) a nd canonica l means that there is an explicit formula in ter ms of the double co ne alg ebra lo calized symmetrically a round the orgin with radius R and a larger one with ra dius R + ∆ R. The canonical for mu la for N is written in the third line whe r e J r ing is the mo dula r reﬂection for the ring algebra deﬁned in the second line. Note that this ring regio n is contained in a light she et betw een the tw o horizo ns of D ( R ) and D ( R + ∆ R ). The crucial geo metric input which leads to the desired result is the realiza tion that the r elev ant part for the ar ea-like b ehavior is the fact that the v acuum on N only con tributes in the r ing r egion since on D ( R ) it is indistinguisable from the old v acuum. The ring reg ion is pro po rtional to the a rea, and allowing for the previously esta blis hed logar ithmic behavior in lightlik e dir ection, one e nds up with [22] E nt ( D ( R )) ∆ R → 0 ≃ C ( n ) R n − 2 (∆ R ) n − 2 c 12 l n R ( R + ∆ R ) (∆ R ) 2 , C (0) = 1 (17) where the logarithm is the o nly singularity in c hira l conformal (n= 2) mod- els. The naive g eometrical ar gument would fav or the dimensionless area law inv o lving the ring size ∆ R w itho ut the log a rithm, whereas the pr esenc e of the lo garithm c an b e viewe d as r epr esenting a lightlike length factor which a ccording to the chiral inv erse Unruh eﬀect is mapp ed int o a lo garithmic divergence. In this wa y of counting there is a per fect match with the (n-1)-volume factor apart form the fact that one length factor has to be mapp ed into a log arithm. The result contradicts the p opular folklor e that QFT is incomparible with an area behavior, which is sometimes used delineate QFT in CST from QG. The presence o f the log arithm is imp ortant for our conjecture o f thermal universal- it y [22] which would ﬁnd its mos t p erfect ex pression in the existence of a yet hypothetical hig her dimensio nal inv ers e Unruh eﬀect (mo r e in the concluding remarks ); this remains an in teresting problem for future resea rch. 14 5 Holograph y on to horizons, BM S symmetry en- hancemen t The sp ecial role o f null-surfaces as c ausal b oundarie s, which de ﬁne places around which v acuum p olariz a tion clouds form, s uggests that there may b e more to ex- pec t if o ne only could ma ke QFT on a light-fr ont a conceptua lly and mathemati- cally v alid co ncept. That this can b e indee d achiev ed is the re sult of holo graphy . Hologra ph y clariﬁes mo st of the problem whic h were ra is ed b y its predecesso r , the ”lightcone quant izatio n” and ex plains why the older metho d faile d. O ne of the reasons has to do with sho r t distance b ehavior since the na ive restriction o f ﬁelds to space- o r light-lik e submanifolds requir e the v alidity o f the ca nonical quantization for malism i.e. a shor t distance dimension not worse than sdd=1 , even though the there is no such restr ic tion o n the dimensio n of chiral ﬁelds living on a lightray . How ever the causal lo c alization pr inciple in its algebraic for m ulatio n p er- mits to attach to each region the algebr a o f its causal sha dow. F or null-surfaces the situation with resp ect to p ointlik e generators improv es. In that case the observ able algebras index e d b y regions on the lightfront are really pointlik e ﬁeld-generated and the ﬁeld generators are transversely extended c hiral observ- able ﬁelds C ( x, x ) where x denotes the lightlik e co ordina te o n the ligh tfront and x parametriz e s the n-2 dimensional transverse submanifold. The absence of tra nsverse v acuum p olar ization would sugg est to exp ect their commutation relations to be of the form [ C i ( x 1 , x 1 ) , C j ( x 2 , x 2 )] = δ ( x 1 − x 2 ) m X k =0 δ ( k ) ( x 1 − x 2 ) ˇ C k ( x 1 , x 1 ) (18) where the n umber m of opera tor contributions o n the right dep ends on the scale dimensions of the t wo o per ators on the left hand side. The tra nsverse delta function expr e sses the absence of transverse v acuum pola rization which is a rigoro us-mo del indep enden t result of the alg ebraic se tting [21]. As for standard chiral ﬁelds the scale dimensions a re unlimited (no r estriction to canonic ity as for equal time commutations) 9 . The ˇ C ( x, x ) denote x dep endent chiral ﬁelds of which has to kno w (as a cons equence of the absence of tra nsverse v acuum ﬂuctuations) only the pro duct str uctur e in x, x ′ at the sa me x which Suc h a commutation relation, with the exception o f d= 1+1 where there is no transverse depe ndence , can howev er not b e quite c o rrect for comp os ite ﬁelds as a simple free ﬁe ld calculation for : A 2 ( x ) : sho ws [23] in which case an ill-deﬁned square of a delta function a ppea rs (see b elow). This is p erhaps a reminder that o ne should not aim for the holo graphic pro jection of individual pointlik e ﬁelds in any literal sense, but ra ther seek po in tlike gene r ators of the holographically pro jected algebra ac c ording to: bulk ﬁelds → bulk lo cal algebr a s → hologra phic pro jection → constructio n of pointlik e g enerators . 9 There can be higher deri v atives i n the transverse direction but they ar e alwa ys ev en whereas the light -l i k e delta functions are o dd. 15 The modula r lo c a lization theor y plays a crucial role in the construction of a lo cal net on the lightfron t a nd its generating ﬁelds and for this reas o n one must start with o per ator algebras which is in a standard p osition with resp ect to the v acuum. Since the full lightfron t algebr a is identical to the glo bal algebr a o n Minko wski s pacetime, one must s tart with a subreg ion on the lig h tfront a nd the largest such regio n is half the lightfront who s e causal completion is the wedge (so that it can b e seen as the wedge´s causal (upp er) horizon H ( W ) 10 ) A ( W ) = A ( H ( W )) ⊂ B ( H ) (19) It is very imp orta n t to avoid to pro ject mo re into this e q uation than wha t is actually written: this equality refers o nly to the p osition of the tw o alg ebras within the full algebra B ( H ); it do es no t re fer to their lo cal subs tr ucture. The latter would b e v ery diﬀerent indeed; the lo ca l substructure consisting of the net of (a rbitrarily small) double cones ins ide A ( W ) and that on H ( W ) hav e no direct rela tion. The lo cal substructure on the horizon A ( H ( W )) is o btained by int er sect- ing diﬀere nt W algebra s which hav e their horizons on the sa me lightfront . In 4-dimensional Minkowski spacetime they are connected b y a 7-pa rametric sub- group of the 1 0-para metric Poincar´ e gro up cont aining : 5 transformatio ns which leav e W inv ariant (the b o ost, 1 lightlik e translatio n, 2 tr ansverse tra nslations, 1 transverse rota tio n) and 2 tr ansformations which change the edg e of W (the t wo ”translatio ns” in Wigner ’s Little Gr oup ). This 7-para metric subgr oup is precisely the inv aria nce g roup of the ligh tfront, but a s a consequence of the absence of transverse v acuum p olar ization of QFTs on null sur faces, the loss of symmetry is more than c ompe ns ated for by a gigantic symmetry gain leading to a n inﬁnite parametr ic symmetr y containing the Bondi-Metzner-Sa ch s gr oup. Although the net structure o f the bulk deter mines that on a lightfron t, the inv e rse is no t true, it is not p oss ible to c o nstruct the net structure of A ( W ) from that of A ( H ( W )) . The additional information beyond the intrinsic data of the A ( H ( W )) net which will secur e unique inv ersio n can hav e diﬀerent appea r - ance: Poincar ´ e transfor mations or characteristic propaga tion laws oﬀ H ( W ) or the rela tive p ositioning (forming a kind o f algebr aic GPS syst em ) of not more than three lightfron ts in diﬀeren t a ppr opriate relative p ositions. The loss of information, of phas e spa ce deg rees of freedom a nd of symmetries (those whic h transform out of the null surface) are all in terco nnected and related to the pro- jective nature of hologra ph y onto horizons a pro jection. The only k nown case of a bona ﬁde corresp ondence is the AdS n -CFT n − 1 isomorphism in which case the symmetr y groups are iden tical. F or free ﬁelds the construction can b e done explicitly . Since it is quite int ere sting and sheds some lig h t on why the holography works where the old lightcone quantization did not s ucceed, the re mainder of this section will b e used to present the free ﬁeld holography 11 . 10 This is the quantum version of causal propagation with charac teristic data on H ( W ) . A smaller region on LF does not cast a causal shado w. 11 I am indebted to Henning Rehren who informed me that simil ar idea can be traced back 16 The cr ucial pr op erty , which p ermits a dir ect holo graphic pro jection, is the mass shell representation o f a free scalar ﬁeld A ( x ) = 1 (2 π ) 3 2 Z ( e ipx a ∗ ( p ) d 3 p 2 p 0 + h.c. ) (20) With the help of this r epresentation one can dir ectly pas s to the lightfron t by using lightf r o nt adapted co o rdinates x ± = x 0 ± x 3 , x , in which the lightfron t limit x − = 0 can b e taken without caus ing a divergence in the p-integration. Using a p-pa rametrizatio n in terms of the wedge-related hyperb olic a ngle θ : p ± = p 0 + p 3 ≃ e ∓ θ , p the x − = 0 restr iction of A ( x ) A LF ( x + , x ) ≃ Z  e i ( p − ( θ ) x + + i p x a ∗ ( θ, p ) dθ d p + h.c.  (21)  ∂ x + A LF ( x + , x ) ∂ x ′ + A LF ( x ′ + , x ′ )  ≃ 1  x + − x ′ + + iε  2 · δ ( x − x ′ )  ∂ x + A LF ( x + , x ) , ∂ x ′ + A LF ( x ′ + , x ′ )  ≃ δ ′ ( x + − x ′ + ) δ ( x − x ′ ) The justiﬁcation for this formal manipulation uses the fact that the equiv alence class of test function which hav e the s ame r estriction ˜ f | H m to the mass hyper - bo loid of mass m is ma pped to a uniq ue test function f LF on the lig ht fro nt [25][21]. One easily veriﬁes the iden tity A ( f ) = A ( { f } ) = A LF ( f LF ) . But no te also that this identit y do es not mea n that the A LF generator can b e us ed to construct the lo calizatio n structure fro m that of a characteristic initial v alue problem which, concerning lo calization issues, is very diﬀerent from a Cauch y initial v alue problem. Even classically the lightfron t-bulk relation is prima rily one be t ween symplectic subspaces of the glo bal symplectic space of all clas sical wa ves, rather than relatio ns betw een individual solutions . The r estriction to transverse or lig ht like co mpa ct data does not improve the loc alization within the wedge, it only causes ”fuzziness” , i.e. lack of re- conv ertibility of an a lgebraic automo rphism to a geometr ic diﬀeomorphism. So algebraic hologr aphy from a wedge in the bulk to its horizo n is only inv ertible if one k nows the law of characteristic propa gation from the ho rizon in to the bulk; in the interaction free ca se this means the knowledge of the bulk mas s which was lost in the holographic pro jection. This law has no geometr ic pres ent atio n, i.e. the loca l substructure of a wedge algebra A ( W ) cannot b e ge o metrically enco ded int o A ( H ( W )), although the t wo global alg ebras ar e identical. This also applies to even t horizons in curved spac e time; is incompatible with the idea that the ful l information contained in the lo cal bulk s ubstructure o f a reg ion can b e loca lly enco ded into its horizon. In discussing the horizo n-bulk re lation it is eas y to overlook the fa ct that the representation o f the ligh tfront generators in terms of the Wigner creatio n to wo rk by Ka y and W ald from the 90s [24]. A pr esen tation of free ﬁeld holography f rom a more functional analytic point of view can b e found in [33] and references therein. 17 and annihilation op erator s a ( p ) , a ∗ ( p ) as in the ﬁrst line of (21) is not intrinsic, rather the in trinsic characterization of the theory is con tained in the structure of it cor r elation functions or its comm utation relation. The characteristic data in the in tera ction free case have lost all reference to a mass a nd unless one adds this information there will b e no uniq ue hologr aphic inversion. The uniqueness situation is in no way b etter in interacting theories . Even in the free ﬁeld sit- uation the mutual fuzzines s b etw een co mpact lo calized regions on H ( W ) and regions in the W -bulk o r the in verse situation r emains. In the spir it o f LQ P int rins ic ness, the reconstruction of the lo cal subs tructure A ( W ) requir es the knowledge of the a ction of the Poincare gr oup or that of the relative positions of several n ull-sur fa ces. It turns out that the enlargement o f group symme- try b eyond the 7-pa rametric subgroup a nd the incr ease of degrees o f freedom through the relative p ositioning are alternative wa ys for reconstructing the bulk from it holographic pro jection (more remar ks below). Not knowing anything ab out QG, it is diﬃcult to refute or supp ort the claim that ther e ar e holog raphic ”sc r eens” in QG which store al l bulk informatio n; but this is deﬁnitely not the situation in QFT o n causal horizons or on e vent horizons as they o ccur in QFT in CST. There are fewer degrees of freedom in a QFT o n LF than in the bulk QFT. More knowledge as that of the a ction of LF-changing Poincar´ e tra nsformations incr eases the cardinality of degrees of freedom. The us efulness of the hologra phic LF pro jection is inexorably linked to the thinning out of degr ees of freedom. The b est w ay to appreciate this happy circumstance is to lo ok at co r resp ondences fo r which this fails to b e true (see below). A ttempts at nonp erturbative constructions of QFT inevitably amo unt to sub div iding the problem in to s impler pieces which only addres s certain asp ects of the holistic QFT pro ject. Ho lo graphy on hor izons is a ra dical spacetime reorder ing of a given quantum matter subs trate. The latter may be a W eyl al- gebra (the more rigoro us formulation for (20) and (21)), a CAR alg ebra, or that describ ed in terms of a lo ca l interaction. In all cases the spacetime r eorga ni- zation acc ording to the LF ordering s tructure simpliﬁes certain ﬁeld aspects at the exp ense of particle asp ects whic h b ecome ma sked (hidden in the holog raphic inv e rsion for which more information is re quired). Behind this idea o f ”thinning” degrees of free do m a nd lo osing informatio n in hologra phic pro jection there is the concept of a natur al phase sp ac e density for a given spac e time dimension. Intuit ively sp eaking the idea b ehind this is to hold onto as ma n y prop erties a s p ossible from Lag rangian QFTs in situations outside the La grangia n qua ntization setting. As mentioned in sectio n 2 of the part I this notio n, introduced b y Haag and Swieca [5] in the 60s and later reﬁned to the n uclear it y r equirement by Buchholz a nd Wichmann [6] in the 80s, demands that the phase space densit y of degre e s of freedo m in QFT, which is compatible with mo dula r lo ca liz ation, is bigg er than the ﬁnite degrees of freedom p er phase space cell of QM; but the inﬁnite degrees of freedom als o should not go b eyond that of a nu cle ar set, since otherwise the causal propag ation, the existence ﬁnite temper ature statistical mechanics and the asy mptotic par ticle interpretation will be endangere d. 18 Such a situation arises in the AdS 5 -CFT 4 corres p ondence b eca use if one choses one side, say the one with the larg er s pacetime dimension, a s b eing of Lagra ng ian orig in (i.e. with a natur a l phase s pace density), the o ther side of a correp ondence is uniquely determined and the only thing one can do is to lo ok whether its degr e e s of freedom are na tural or not. The naive argument would s uggest that when one passes to a lower dimensiona l world one has to o many degre e s of freedom i.e. natura lit y is lost. In the opp osite direction o ne exp ects that the 5 -dimensional AdS theor y obtained from a na tural CFT 4 mo del is to o ”a nemic” the AdS theo ry coming fro m a normal CFT turns out to be ”anemic”. Both statements can be ma de precise and exempliﬁed b y explicit free ﬁeld calculations starting from either side [8]. Strangly enoug h, although notice d v ery pre c iesly by Rehren, who g av e a mathematical pr o of o f this corr esp o ndence 12 [26][27], this issue ha s not b een addressed by the Maldacena communit y [28] who ﬁrst formulated the conjecture ab out the corres po ndence in the context o f a conjecture co ncerning the relation of gr avit y on AdS 5 with a conformal s up er s ymmetric N=4 Y ang -Mills theor y in 4 dimensio ns b oth thoug ht of as theories with standard ph ysic al deg rees of freedom. There is a v a st comm unity with mo re than 6000 publica tions who tried to suppor t Maldacena’s conjecture, but scientiﬁc truth a re not decided according to the size of glo balized communities. In fact such co mmunities follow completely diﬀerent pattern 13 than a critical discour se b etw een individuals o r small groups of individuals collab orating on one sub ject. T he conjecture has so on its 20 th anniversary with no tangible res ult but an ever lar ger n umber o f publications with increasingly outr a gious claims. The fact is, a nd every particle physicist of a suﬃcient age will conﬁrm this, that several dec a des of communit y building aro und the idea of a ”theor y o f ev- erything” a nd extr e mely bad leadership has cr eated a n exp ectation of s alv atio n in which the level o f knowledge falls far back b ehind wha t is needed for resea rch at the fron tiers of QFT. Although no t so obvious, the degrees o f freedom ar g ument can also b e ap- plied to brane physics; ag ainst naive intuition br anes co n tains the same car - dinality of deg r ees o f free do m a s the bulk[44] a nd he nc e the s a me arguments ab out spacetime dimenion compatible naturalness applies. W e hav e seen that the hologra ph y of bulk matter on W to the horizon H ( W ) is not a corr esp ondence but a pro jection. So it is clear that the loss of information or the reduction of degr ees of freedom for the preser v ation of naturalness is a priviledge of holo graphy on null-surfaces. This explains wh y hologra ph y onto hor izons is extremely useful. F o r the ca s e at hand, namely the bulk- and lightfront - genera tors, this pro jectiv e natur e o f holography a sserts itself via the fact one cannot r econstruct the bulk from the space o f H ( W ) . But 12 Something which is ill in the ph ysical-conceptual setting, maybe perfect on the mathe- matical side. 13 In fact for the ﬁrst time in the his tory of particle theory there i s a deep sc hism b etw een a ma jor it y who has b een rais ed in the shadow of a theory of eve rything and a scholarly mi nority with pr ofound kno wledge of QFT who are in a ivory tow er against their own choice. Particle theory has en tered a deep crisis. 19 the hologr aphic pr o jection is nevertheless very us eful be c ause it co n tains still a lot of informa tions ab out the bulk in a m uch simpler more a ccessible fashion. I t is this a sp ect of simpliﬁcation at the exp ense of information co mpleteness which makes ho lographic pr o jection that use ful. Of cour se this co uld als o happ en in the case of corresp ondences; even though a CFT 4 viewed fro m the unphysical AdS 5 description has lost its physical interpretation, certain mathematical as p ects may s till simplify . Historically the ”lightcone quantization” which prec e de d lightfront holog r a- ph y shar es with the la tter part of the motiv ation, namely the idea that by using lightlik e directions one can s implify certain asp ects of an interacting QFT. But as the terminolog y ”qua nt izatio n” re veals that this was mixed up with the erro- neous idea that in order to achiev e simpliﬁcation one needs a new quantization instead of a r adical space time r eordering of a given abstract a lg ebraic op era - tor s ubstrate whose Hilb ert s pace is a lwa ys ma in tained. As o ften such views ab out QFT results from an insuﬃcient appre ciation of the a utonomy of the causal lo cality principle by not s eparating it suﬃciently from the co ntingency of individual pointlike ﬁelds. F or mally mass shell repr esentations also exist for interacting ﬁelds. In fact they appea red shortly after the form ulation of LSZ scattering theory and they were in tro duced in a pape r by Gla ser, Lehmann and Zimmerma nn [29] and bec ame known under their short name of ”GLZ repr esentations”. They expres s the interacting Heisenberg ﬁeld as a pow er series in incoming (outgoing ) free ﬁelds. In case there is o nly one type of particles one ha s: A ( x ) = X 1 n ! Z · · · Z V m a ( p 1 , ...p n ) e i P p k x : A in ( p 1 ) ...A in ( p n ) : d 3 p 1 2 p 10 ... d 3 p 1 2 p 10 (22) A in ( p ) = a ∗ in ( p ) on V + m and a in ( p ) on V − m a ( p 1 , ...p n ) p i ∈ V + m = h Ω | A (0) | p 1 , ...p n i (23) where the integration extends ov er the fo rward and ba ckward mass shell V ± m ⊂ V m and the pro duct is Wick order ed. The co eﬃcient functions for all momenta on the forward ma s s shell V + m are the v a cuum p ola r ization comp onents of A and the v ario us formfactors (matrix elements b etw een in ”ket” and o ut ” bra” states). In the GLZ se tting the co eﬃcient functions arise as the mass s hell bo undary v alues o f F o urier-tra nsformed r etarded functions. The conv ergence status o f these series is unknown 14 . This mass- shell rep- resentation is inherently nonlo cal. Nevertheless o ne ma y hop e that it do es not only r e present a lo cal bulk ﬁeld but that its ligh t front res triction is a lso lo- cal. Superﬁcia lly there is no problem with placing the GLZ r e presentation o n the lightfron t. Ho wev er the applica tion to : A 2 ( x ) : the Wic k-order ed c o m- po site of the free ﬁeld shows that there is an obstructio n against a simple- minded p ointlik e formulation (18) since the Wick deco mpo sition o f : A 2 ( x ) : LF : A 2 ( x ′ ) : LF contains squares of transverse delta functions which, as the res ult of 14 In con trast to the p erturbativ e expansion which is known to div erge even in the Borel sense, the con verge nce status of GLZ had not b een settled. 20 having los t the energy momentum po sitivity in the transverse compo nent s, are incurably divergent. T his tr ansverse delta pr oblem is a bsent in the hologr aphic pro jection o f tw o- dimensional massive theories . The decisive prop erty is how- ever not whether genera ting ﬁelds on LF come from p oint wise manipulations on bulk ﬁelds, but rather whether a net o n LF can b e describ ed of genera ting ﬁelds. B ut since the concre te calc ula tions in terms of individual ﬁelds is more familiar one would lik e to hop e that there is a so lution to the transverse delta problem. The holography on horizons contains some not entirely unders to o d problems of spin and statistics. Only Bose ﬁelds with integer short dista nc e dimensions , as those a sso ciated with co ns erved cur r ents (conser ved charge curr en ts, the ener gy- momentum tensor), can hav e bo sonic hologr aphic pro jectio ns wher e as (b os o nic or fermionic) bulk ﬁeld with ano ma lous shor t distance dimensions pass to plek- tonic lig h tfront ﬁelds for which the anomalo us dimension, the anomalous spin and their braid gr oup statistics a re int er connected via the chiral spin&statistics theorem [10]. This change of the statistics in passing from bo s ons/fermions with ano ma lous dimensions to ligh tfront ﬁelds with any onic/plekto nic statistics is formally taken ca re of by the GLZ formula. These tra nsmu tation prop erties with resp ect to statistics are mo re co nve- nient ly studied in the simpler c o nt ext of absence of trans verse dimension i.e. in the hologr aphic pro jection of tw o-dimensiona l QFTs onto the light ray . In this case the afore men tioned obstruction is a bs en t. In particular for the facto r izing mo dels presented in the section on a lgebraic asp ects of mo dular theory , there are on-shell represe n tation of lo cal ﬁelds in terms of certa in wedge generating creation/a nnihilation o per ators, the Zamolo dchik ov-F a ddeev algebr a g enerators (see part I), which replace the incoming creation/ a nnihilation op erato rs in (22) and lead to a co eﬃcie nt functions which are identical to the crossing symmetric formfactors. The po in tlike ﬁelds in the ma ss shell r e pr esentation highlight so me inter- esting pro ble ms whose b etter understanding is impo r tant for autonomous no n- per turbative constructions of mo dels in QFT i.e. constructio ns which do not depe nd on Lagra ngian quant izatio n as those presented in part I. The more rig- orous alg ebraic metho d b y its v ery na ture (using relative co mm utants) only leads to b osonic hologra phic pro jections. This means that the extended chiral structure on the lightfron t only contains integral v alues in its short distance sp ectrum; i.e. the generating ﬁelds a re of the kind o f the chiral c o mpo nen ts of t wo-dimensional co nserved curr ent s and energy-mo men tum tensors. Hence only a small s ubalgebra of the bulk algebra 15 asso ciated with transverse extended currents, energy momentum tens o r etc. will hav e a b osonic holog raphic im- age; there w ould b e no anomalous dimension ﬁeld in the algebraic holographic pro jection. Apart fro m conse r ved currents whose charges must b e dimension- less, ﬁelds a re no t pr otected ag ainst car rying non- integer shor t distance sca le dimensions; such ﬁelds w ould not pass the algebraic metho d o f holo graphy . 15 Apart from conserved current s whose charges must be di mensionless, ﬁelds are not pro- tected against carrying non-in teger s hort distance scale dimensions. 21 Clearly some of these ideas, as important for the future development of QFT as they may app ear , are not yet mature in the sense of mathematical ph ysic s. Therefore it is go o d to know that there exists an excellent theoretical lab orator y to test such ideas in a b etter controlled mathematical setting, the t wo-dimensional factoriz ing mo dels a nd their this time b ona ﬁde (no trans verse extension) chiral holo graphic pr o jection. F ro m a pr e vious section on mo dular theory in part I o ne knows tha t these mo dels hav e rather simple on-shell w edge generator s Z ( x ) whic h still maintain a lo t of similarity with free ﬁelds. In that case Za molo dchik ov prop osed a co nsistency arg ument which led to interesting constructive conjectures ab out relatio ns be tw een facto rizing mo dels and their critical univ ers ality classe s repres ent ed in form of their conforma l shor t distance limits. F ro m a conc e ptual viewp oint the cr itica l co nformal limit leading to univer- sality classes is very diﬀerent from the holog raphic pro jection. The former is a diﬀerent theory whos e Hilbert space has to b e reconstructed fr om the massless correla tion function, whereas the latter keeps the or ig inal Hilb ert s pa ce a nd only repro cesse s the spacetime or dering of the origina l quantum s ubstrate. Assuming that one knows the chiral ﬁelds on the ligh tray as a p ow er series in term of the Zamolo dchik ov-F a ddeev op er ators 16 [30], o ne has a unique in version, i.e. the hologra phic pro jection b ecomes an isomorphis m. Calculations o n tw o mo dels [31 ], the Ising ﬁeld and the Sinh-Go rdon ﬁeld, hav e shown that the universality cla s s metho d and the holog raphic pro jection lead to identical results 17 . Whereas the anoma lous dimension of the Sinh- Gordon ﬁeld can only b e co mputed approximately in ter ms of doing the in- tegrals in the lo west terms in the mass shell contributions, the serie s for the Ising order ﬁeld can be summed exactly and y ie lds the ex pected num b er 1/16. This is highly sugges tive for re interpreting the more sp ecula tive Zamolo dchiko v wa y of relating factor izing mo dels with chiral mo dels in the c o nceptually clearer setting of holographic pro jections. The gain in modular gener ated symmetry is p erhaps the most in triguing asp ect of holo graphy . In g eneral the mo dula r theory for c a usally complete spacetime reg ions smaller tha n wedges lea ds to algebraic mo dular groups which cannot b e enco ded in to diﬀeomeo phisms of the underlying spacetime manifo ld; the gener ators of these gro ups are a t b est pseudo-diﬀer en tial op erators . Ho w- ever there ar e strong indications that their restriction to the horizon are always geometric. So it may b e useful to constr uct the bulk mo dular groups from those of their ho lo graphic pro jection. The constructive knowledge a bo ut chiral theories has very m uch pr o gressed [32] and it would b e nice to b e able to use that insight to construct massive bulk theo r ies with chiral models b eing the hologra phic input. Let us ﬁnally a ddr ess the sy mmetr y enhancement which leads to the inﬁ- nite Bondi-Metzner -Sachs symmetry gro up which these author s discov ered in 16 F rom the p oin t of view of c hir al mo dels such a representat ion is of course somewhat un usual. 17 The consistency of the holographic lightra y pr o jection with the critical limi t f or f actorizing models was v eriﬁed i n s bl ac kb oard discussion with Mi c hael Karowski. 22 asymptotically ﬂat solutions of cla ssical general r elativity . In the case o f the free ﬁeld it is not diﬃcult to see [2 2] that the absence of tra nsverse v acuum po larization leads to a slightly la rger symmetry than the transverse Euclidean group; the transverse delta functions p ermits a compactiﬁcation to the Riemann sphere on whose complex ζ , ¯ ζ coo rdinates ( ζ = x + iy ) the g roup SL(2,C) acts as a fr actional tra nsformation, just as the covering o f the Lorentz group. Restrict- ing the Diﬀ(S 1 ) group to the symmetr y group of the v acuum which is the ﬁnite parametric Mo ebius group; imp osing in addition the req uirement of the preser - v ation o f the p oint a t inﬁnit y in the lightlik e direction the gr oup is the a x+b translation dilation gr oup. By itself this w ould b e a tw o parametric group, but the fact that the tw o parameters can b e functions of ζ , ´ ζ makes jointly gener ated group an inﬁnite para meter group x → F Λ ( ζ , ¯ ζ )( x + b ()) ζ , ¯ ζ (24) ( ζ , ¯ ζ ) → U (Λ)( ζ , ¯ ζ ) , U (Λ ) ∈ S L (2 , C ) The g roup comp ositio n law F Λ ′ (Λ( z , ¯ z )) F Λ ( z , ¯ z ) = F Λ ′ Λ ( z , ¯ z ) requires the mul- tiplicative fa ctor to be of the form F Λ ( ζ , ¯ ζ ) = 1 + | ζ | 2 | aζ + b | 2 + | cζ + d | 2 (25) whereas the functions b ( ζ , ¯ ζ ) are from a function space which is the closure of C ∞ ( ζ , ¯ ζ ) functions on the Riemann sphere in some top ology . The somewhat unexp e cted prop erty is that the action of SL(2,C) o n the function spa ce co n tains (in its linear par t) the a copy of the semidirect pro duct action o f the Lo rentz group o n the translations i.e. the inﬁnite dimensio nal BMS group con tains the Poincare gro up. F or mor e informations esp ecially on the p osition of the Poincare inside the BMS gro up we r efer to a comprehensive pap er by Dappiaggi [33 ]. One expects this tra nsformation o n a classical Penrose double cone hor izon at inﬁnity since on such a ”sc r een” the Poincar´ e group acts naturally . But its app earance already on co mpa ct quantum double cones is at ﬁrst sight s omewhat astonishing although the split pr op erty yields a mathematica l explanation [22]. It is helpful to take notice that in addition to the thermal pr op erty of the v acuum reduced to one tensor factor as e xplained in the pr evious section, the split prop erty p er mits also to ” lo calize” g lobal s ymmetries whic h constitutes an a nalog of the classical Noe ther theor em [2]. This pure a lgebraic de r iv atio n do es no t require to deﬁne a conserved current with the help of the Lag rangian quantization, one even do es not hav e to know ho w to construct without being forced to p ostulate the existence of singular current opera tors as the quantum counterparts of the classical c o nserved Noether currents. This int rins ic (i.e. not rely ing on a quantization parallelis m) ” lo calization” o f globa l symmetries based on the split prop er t y als o applies to the lo calization changing Poincar´ e symmetry if one r estrict the g roup parameters to suﬃciently small v a lues so tha t the lo caliza tion of the transfo rmed op erator s stays inside the chosen lo calizatio n region [2]. 23 Using the notation of the do uble cone lo ca lization deﬁned in the prev ious section o ne obtains a representation o f the full Poincar´ e group o n the tensor factor N which for suﬃcien tly small parameters act on op er ators A ∈ A ( D ( R )) the sa me wa y a s the global symmetry . In the ring region or its light sheet prolonga tion D ( R ) \ D (( R + ∆ R )) , which constitutes the fuzzy lo calized part of N which surrounds its shar ply lo calized nucleus A ( D ( R )) , the Poincare group do es not act geometrically in a way which can b e enco ded in to a g eometric diﬀeomorphism; of course it never fails to b e an algebraic a utomorphism. Hence the s plit situation for a double cone crea tes an analog situatio n to a Penrose screen except that the Poincar´ e subgroup of the BMS gro up is an unphysical extension of the partial physical Poincar´ e gro up for parameter v alues for whic h the b ounda ry of the region of in teres t in A ( D ( R )) pa s ses int o the ∆ R split r ing - like or light-sheet region with ∆ R → 0 in the hologra phic limit in which light- she et → holo gr aphic scr e en . With o ther words the artifact a Poincar ´ e subg roup of the holog raphic BMS g roup is explained in terms o f the artifact o f a loca lized Poincar´ e s ymmetry resulting from the s plit constr uction. 6 The lo cal co v ariance principle Less than a decade ag o the holistic structure QFT in CST was signiﬁcant ly enriched b y the formulation o f the lo ca l cov ariance principle [3 9]. Preliminary studies in this direction beg an at the beg inning of the 90s with the realiza tion that even in the ca se of a free quantum ﬁeld the deﬁnition of a n energy- s tress tensor with prop erties similar to those of the clas sical expressio n whic h ent er s the right hand side of the Einstein Hilb ert equation is a very nontrivial mat- ter as so on as cur v ature enters [3 7]. One problem is tha t even in Minko wski QFT, where a unique deﬁnition in terms of the Wick-ordered expression of the classical form is av ailable, the energy density is no t b ounded b elow, since one can ﬁnd state vectors on which the energ y density T 00 ( x ) takes on ar bitr arily large negative v alues [34]. Whereas this r esult does not create serious prob- lems in standar d QFT, it causes pr oblems with the quantum coun terpa rt of certain stability theorems which follow from p ositivity inequalities for the clas- sical stress energ y tensor whic h enters o n the right hand side o f the Einstein Hilber t equatio n. It started a ﬂurry of inv estigatio ns which led to state-independent lo wer bo unds o f T 00 ( f ) for ﬁx e d test functions as well as ine q ualities o n subspaces of test functions [35]. These inequa lities which inv olve the free str ess-energ y tensor were then g eneralized to curved space time 18 . In the presence of curv ature the main problem is that the correct deﬁnition o f T µν ( x ) is not obvious since in a generic spacetime there is no v acuum like s ta te (which is distinguished b y its high s ymmetry) to which the op erator or der ing could refer; to pla y that po in t split game with a n ar bitrarily c hosen state will not pro duce a locally cov ariant energy stress tensor since sta tes (in co ntradistinction to o pe r ators) are inevitably globa l in tha t their dep endence on the spacetime metr ic is not 18 F or recen t publication with many references see [36]. 24 limited to the inﬁnitesimal sur rounding of a p oint (which would be required b y a lo cal co v ariance principle). A s tr ategy to obtain lo c a lly cov ar iant lo cal quantum ﬁeld pro duct for the energy-mo mentum tensor which is no t asso cia ted with a particular state w as given in 19 94 by W ald [37] in the setting o f free ﬁelds. His p ostulates gave rise to wha t is now adays referred to a s the lo c al c ovarianc e principle which is a very nontrivial implemen tation o f Einstein’s classical cov arianc e princ iple of GR to quantum matter in curved spac etime (after freeing the classical principle from the relics of its physically empt y co ordina te inv ariance interpretation). The re q uirements introduced by W ald deter mines the corr e ct ener gy-momentum tensor up to lo ca l curv ature terms (whose degr ee dep ends on the spin of the free ﬁelds). The metho d of W ald is so mewhat surpris ing since it do es not consists in taking the coincidence limit after subtracting from the po in t split expr ession the exp ectation v alue in one o f the sta tes of the theory . Rather one needs to subtract a ”Hadamard pa rametrix” [38] i.e. a function which dep ends o n a pair of co o rdinates and is deﬁned in geometric terms ; in the limit of coalescenc e it depe nds only o n the metric in a neighborho o d o f the po int of co alescence. Only then the globa l dep endence on the metric car r ied by states can b e eliminated in fa vor of a lo cal cov a riant dep endence on g µν ( x ) and its deriv atives. As a result the so-constructed str ess-energ y tensor at the point x dep ends o nly on the metric in an inﬁnitesimal neighborho o d of x . Already W a ld’s work co ntains the impo rtant messag e that in order to con- struct the co rrect tenso r it is not eno ug h to lo ok at one mo del o f a QFT in a particular curved spacetime background, but one is o bliged to lo ok simu lt ane- ously at al l diﬀer ent sp ac et ime or derings of abstr act quantu m matter (in W ald’s case the abstract W eyl algebra quantum matter) in order to b e a ble to correctly describ e the algebr aic structure of one particular mo del. The implementation of the lo cal co v a riance principle requires a strict se pa ration of the a lgebraic structure from states; settings o f QFT in whic h the tw o are mixed together as functional int eg r al approaches or other formulations in terms of exp ectation v alues are unsuitable. In fact it is not an ex aggera tion to think that without the dic hotomy b et ween spac e time indexed nets of o pe rator algebras and states inherent in algebra ic QFT, the formulation of QFT in CST would not reached the pres ent level of clar it y . In [39] the formulation of the lo cal cov ariance principle attained its pr e sent form. There are tw o diﬀerent but connected formulations, o ne working with nets of caus ally closed nets of space time- index ed op era tor algebr as and the o ther one in terms of p ointlik e cov ariant ﬁelds. The diﬀerence to the standard for mu latio n of a global algebra with its ca us ally c lo sed subalgebr as is that alg ebras which are ”living” o n isometric caus a lly closed parts of spacetime and a re in addition algebraic is omorph (ar e made from the same abstra ct matter substrate) are considered on equa l fo oting. The totality of observ ation whic h can b e made on isometric isomor phic subalgebras is identical and indep endent of diﬀerences which ma y show up in their surrounding. This go es a long wa y towards what is considered as the characterizing prop erty of QG: the ba ckground indep endence. Some resear chers of QG want to go one s tep beyond isomo rphy a nd lo ok for 25 equality in the spirit of gauge inv aria nce by integrating over gauge ﬁelds but a prop osal to implemen t this ides is still missing . The lo cal cov ar ia nce principle can also be expres s ed in ter ms o f p ointlik e cov ariant (under lo cal isometr ies) ﬁelds. In contradistinction to s ta ndard s pace- time sy mmetries in QFT (e.g. Poincar´ e s ymmetry) these symmetries do not come with a state whic h is left glo bally inv aria n t. They are like the Diﬀ(S 1 ) symmetry beyond the Mo ebius group of chiral conformal QFT on the circ le in which case there is also no state which is globa lly inv aria nt under diﬀeomorphism beyond the Moebius gro up. Recently these renormalizatio n ideas were a pplied to computations of back- reactions o f a scalar massive fre e quantum ﬁeld in a spatially ﬂat Rob ertson- W alker mo del [4 1]. As a substitute for a v ac uum state o ne uses a state of the Hadamard form since these s tates fulﬁll a the so-called microlo cal sp ectrum condition whic h em ulates the sp ectr um condition in Minko wski spacetime. The singular part of a Hadamard state is determined by the geometry of spacetime. The renormaliza tion requirements of W ald lea d to a an energy momen tum ten- sor with 2 free par a meters whic h ca n b e co n venien tly repr esented as functiona l deriv a tiv es with resp ect to the metr ic of the tw o quadr atic inv ariants which one can form from the Ricci tensor and its trace. In [40] the resulting background equations were analyze d in the s impler co nformal limit and it was found that the qua nt um bac krea ction stabilizes solutions i.e. accomplishes a task whic h usually is a s crib ed to the phenomeno logical cos mo logical constant. Without the simplifying assumption the linear dep endence on a free reno rmalization pa- rameter guar anties that any measured v alue can be ﬁtted to this backreaction computation. The principles o f QFT canno t de ter mine renormaliza tio n pa ram- eters. Hence fr om a QFT p oint of view there is no co s mological problem whic h places Q FT in contradiction with astrophysical observ ations. A consistency chec k would only be p ossible if there ar e other measurable astrophysical quan- tities which fa ll int o the setting of quantum bac kre a ction o n spatially ﬂat R W cosmolog ies. Last not least the requirement of the lo ca l cov ar iance principle to co nsider a given quan tum matter s ubstrate simultaneously in al l CST helps to ma in- tain some asp ects of par ticles, whos e Wigner characteriz ation only applies to Minko wski spacetime. Since the latter is included in the cov ariance deﬁnition it is suﬃcient to ﬁnd an region o f the g iven CST which is isometric to a Mink owski space region in order to secure ob jects which b ehave in a certain limited space- time reg ion as par ticles (see la st sec tio n). 7 Resum ´ e, miscellaneous commen ts and ou tlo ok F or a lo ng time the conce ptua l diﬀerences betw een relativistic quant um me- chanics 19 and QFT in which the maximalit y of propag ation is build into the 19 In r elativistic quan tum mec hanics (DPI of part I) the velocity of ligh t is not a maximal propagation ov er ﬁnite distances but rather a l i miting velocity f or the leading asymptotic 26 algebraic causa lit y structur e were not suﬃciently appreciated. Even in co n- tempo rary a rticles one ﬁnds the terminolog y ”rela tivistic QM” ins tead o f QFT. Perhaps one reason is that many p eo ple b elieve that relativis tic QM, as a sep- arate sub ject from QFT, does not really exist so that the somewhat sloppy terminology does not rea lly ma tter . But the exis tence of the DPI presented in part I shows that this is not correct; the direct particle in tera ction theory is a r elativistic theory of pa r ticles whic h fulﬁlls all requirement which one is able to implement using e xclusively prop erties of pa rticles. As men tioned in part I, even crea tion/annihilation proce sses of particles in scattering pro cess e s ca n b e describ ed in DPI by in tro ducing suita ble channel couplings ” b y hand”. What is how ever characteristic of int er acting QFT and has no place in DPI is the notion of interaction-caused inﬁnite v acuum p olariza tion. In pa rt I the fundamental diﬀerences were explained in ter ms of tw o fundamentally diﬀerent lo calization concepts. F or tunately these very diﬀerent lo calizations co alesce as ymptotically i.e. the quantum mechanical Bo rn-Newton-Wigner lo calizatio n b ecomes cov aria nt in the asymptotic limit of s cattering theory and its quantum mechanical pro bability concept p ermits to e x tract cro s s sections from scattering a mplitudes. So p erhaps it is b etter to de-emphasize the ”b ottle half-empty” vie w expressed in (see part I) the title Re eh-Schlie der defe ats Born-Newton-Wigner and take a mo re ha r- monic per sp ective by viewing R-S and BNW, as an asymptotic al ly harmonious p air. Any o ther result would have ca used a disaster in the relation b etw een particles a nd ﬁelds. DPI reaches its conceptua l limit if it comes to the notion of formfactor s. It is an interesting question whether LQP has any new mes sage for the main philosophical pr oblem of the 20 th cent ury p osed b y QT: the controv ersy betw een Bohr’s (and more gene r ally the Co penha gen) holistic view of quantum r e ality and Einstein’s insistenc e in indep endent elements of r e ality. I think it does . On the one hand it pus hes the ho listic p oint o f view to its ex treme as exempliﬁed in the v ario us wa ys it realizes an extr eme form of connectedness which we tried to highlight by calling it ” Mur ph y’s theor em” (” what can co uple doe s couple”) as illustr ated b y Reeh- Schlieder pr op erty , the a na lytic crossing connectio n of the diﬀerent for mfac tors of a lo ca l op erator with its v acuum p olar ization and in a muc h strong er form by the characterization of a LQP in terms of a ﬁnite nu mber of monads in a sp eciﬁc mo dular position. But on the other hand there is a lso the split prop er ty which creates a situation close to E instein’s view. If one interprets E ins tein’s maxim in an appropria te way , na mely as the preserv a- tion of the totality of all p ossible measurements in the pr esence o f uncon tro lled activities in a space like sepa rated la bo ratory instead o f exc luding the ho listic EPR situation, then there is no a n tag onism b etw een the t wo views. The rec- onciliation maybe diﬃcult fro m an intuitiv e v ie wpo in t, but the exis tence of a mathematical consistent pr esentation clear ly s hows that intuition is not alwa ys reliable a nd sometimes needs mathematical g uidance. con tribution of a wa ve funtion. In this r espect it is the relativistic coun terpart of the speed of sound in a nonrelativistic system of coupled oscillators. 27 The particle-ﬁeld r elation in the presence of int er actions is one of the s ubtlest asp ect of relativistic lo cal quantum physics; there has never b een a n y closure on this issue, nor would anybo dy who has a detailed knowledge ab out this sub ject exp ect one in the near future. Nob o dy at a high energy lab or atory ha s ever di- rectly measured a hadr onic quantum ﬁeld 20 . Even though all o ur int uition and the formulation of pr inciples enters the theory through lo ca l ﬁelds and space- time indexed algebra s of observ ables genera ted by them, quantum ﬁelds remain hidden to direct observ ations. What one really measures are either particles ent er ing a nd lea ving an interaction pr o cess, or thermal radiation densities and their ﬂuctuations a s in the micr ow av e background ra dia tion. Quantum ﬁelds or lo c a l observ able algebras a re the carriers o f the causa l lo cality principle 21 but, diﬀeren t from classica l relativistic ﬁelds whic h propag ate with a maxim um velocity , they hav e themselves no ontological status. The pro tagonists o f LSZ scattering theo ry coined a very appropr ia te w or d for this state of aﬀair s, they called ﬁe lds in pa rticle physics ”interpo lating”. In general there will b e inﬁnitely many in terp ola ting ﬁelds which int er po late the same particle. But there a re in- dications based on the use o f the cro ssing pr o pe r ty , that the in verse sca ttering problem has a unique solution with resp ect to the s ystem o f lo cal algebras [42] without any guar ant y for its existence. Besides the standar d Wig ner pa rticle setting whose connec tio n with ﬁelds is channeled through the (LSZ, Haag-Ruelle) scattering theory there are charged (infra)particles whose scattering theory in terms of inclus ive cr oss sections e x - ists in the for m o f computational recip es without conceptual backup. These particle-like o b jects corres p ond to ch ar ged ﬁelds in QED whic h o nly exist as semiinﬁnite strings i.e. are nonlo ca l (in the standar d use of this word where everything which is not p ointlik e genera ted is ca lled no nlo cal. It would b e naive to exp ect tha t the situation with resp ect to the necessity of introducing phys- ical nonlo cal observ a ble s decrea ses in pass ing from ab elia n gauge theo ries to Y ang -Mills theories . But it is pr e cisely the idea o f an equiv ale nc e class of in terp olating lo cal ﬁelds in their pro per t y of interpola ting the s ame particle which le d to the p ow- erful observed pro per ties as e.g. Kra mers-Kro nig dispe rsion rela tions which a particle-base d appr o ach as DPI c an not deliver. The exp erimental v er iﬁcation of a dispe r sion relation cannot select or rule out a particular Lagr angian mo del of ha dr onic interactions but ra ther is a test for the v alidity of the causal lo cal- ization principle. The par ticle based view is certainly thrown in to dis array when one studies QFT in no n inertial frames (e.g. the Rindler frame of the Unruh eﬀect) or in CST. According to the b est o f m y knowledge there e x ists no time-dep endent LSZ scattering in a (ﬂat) Rindler w or ld; although the glo bal and the w edge - lo calized QFT live in the same Hilb ert space, the g lobal par ticle states ca rry no intrinsic 20 There are how ev er certain distinguished composite ﬁelds, in particular the quantum analogs of Noether currents, whose form factors are used in the analysis of scattering data for certain deep inelastic processes. 21 In the nonin teracting case cov ered by Wigner’s representation theory this viewp oint has led to the understanding of string-lo calized generato rs of ”inﬁnite spin” representa tions [7]. 28 ph ysic al infor mation with res pect to the w edge-lo c a lized theory . Rob ert W ald, a leading resear cher on QFT in CST, has recently prop osed [38] to aba ndo n the pa rticle co ncept a ltogether and work under the h yp othesis that ﬁelds are directly mea surable. But measur a bilit y r equires a certain a mount of stability and individuality; quantum ﬁelds are ﬂeeting o b jects of which there ar e alwa ys inﬁnitely ma n y for which, in contradistinction to classical ﬁelds , there seems to exist no mea surable prop erty which allows to disting uis h the member s in an equiv ale nce class of ﬁelds which car ry the same charge. It is ha rd to b elieve how W a ld’s advise of abandoning particle s co uld work. Perhaps, as indicated at the end o f the previo us section, the lo cal cov ar iance la w lea ds to an argument why certain pa rticle manifestations in Minko wski spacetime can b e trans fered to ﬁnite regions in CST. In reco gnition o f this lack of observ ational distinctness for ﬁelds, the alge- braic approach to Q FT has placed sp ac etime-indexe d op er ator algebr as into the center stage. In such a setting the increase of kno wledge ab out a lo calized op- erator algebra takes place through a tightening in lo caliza tion and not via the increase in precision in measuring prop erties of an individual op er ator 22 . This ﬁts very nicely with scattering theory b ecause the in/o ut ﬁelds res ulting from diﬀerent op erato rs in the same algebra A ( O ) are ident ica l; their diﬀerences bec ome abso rb ed in to normalization facto rs [43] and it is at best the system of op era tor a lgebras which is deter mined by inverse sca ttering and never the individual ﬁelds. Although witho ut the notion of particles and scatter ing theory the physical world of QFT in CST would b e quite a bit p o or e r, it is by no means void o f all exp erimentally ac cessible asp ects. Even if one has no clear idea on the nature of the co smologica l r eference s ta te o f our universe (the CST replacement for the v acuum), one can study mo dels and compare the thermal aspects of the exp ectation v alues of the ener gy-stress tensor in the cosmic reference state with data from the cosmic bac kgr ound radiation [40], for this one do es not need the v acuum s tate a nd particle sta tes as they follow from Poincar´ e symmetry . A class of ob jects b etw een particles (o n-shell) and ﬁelds (oﬀ-shell) which are ideally suited for the study o f v acuum pola r ization are the formfactors , i.e. matrix-e le men ts of loca l o per ators b etw een br a o ut- a nd ket in- particle states. The s pec ia l matrix-elements with v a cuum on one side and all pa rticles on the o ther side characterize the v a cuum po larization of the lo cal ”ba ng ” on the v acuum A Ω , A ∈ A ( O ) . The g eneral formfac to r r esults from the v a cuum po larization co mpo nent b y a particula r on- shell ana lytic contin uation pro cess known as the cr ossing pr op erty. The latter is one of the most subtle prop erty in the particle-ﬁeld relation [4 4], its comprehension g o es signiﬁca nt ly b eyond that o f time-dependent scattering theor y . As the Unruh eﬀect, it uses KMS prop erties o f the wedge lo calized alg ebra 23 , the subtle p oint being that one needs to cons truct very sp ecia l wedge lo calized o pe r ators which applied to the v acuum 22 An except ion ar e those lo calized individual operators which result from the ”split lo cal- ization” of global symmetries (the b efore ment ioned quan tum No ether current s). 23 It testiﬁes to the conceptual depth of mo dular lo calization that it places such diverse lo oking i ssues as the Unruh eﬀect and the crossing property under one ro of. 29 generate pa r ticle states without admixture of v a cuum p olariz ation cloud [44]. The histor y of the cro ssing prop erty is a lso a pr ime exa mple of the disa strous consequences of a several dec ades lasting misunderstanding of a central co nce pt of QFT [44] in the absence of a pro fo und criticism. An exa mple of a deep a n tag o nism of QFT with resp ect to QM which is usually not p erceived as such comes from the ex ploration of mo dular lo caliza- tion. Whereas the lo calization of states is basically a k inematical notio n, its algebraic version inco rp orates most, if not a ll dynamics . The crucia l prop erty is the mo nad (hyper ﬁnite t yp e II I 1 factor) nature of the loca l a lgebras. In QM all Born-lo c a lized subalgebras a r e of the same t yp e as the g lo bal algebr a, namely type I ∞ factor B ( H ) , H ⊂ H glob . A monad in Q M only app ears at ﬁnite temper ature in the thermo dynamic limit. There is har dly an y textb o ok which emphasises the ra dically diﬀerent alg ebraic prop erties (see howev er [4 5]) from those of its ”b oxed” Gibbs state approximands 24 . In QFT as opp osed to QM, it is the monad structure which is the norma l situation and the q uantu m mechanical t yp e I ∞ prop erty which is the exception; the latter can only be constructed by ”splitting” a lo cal alge br a fr om its causal disjoint and in this wa y creating a fuzzy ”halo” in which the v acuum p ola rization can settle down to a (halo-dep endent ) temperate behavior leading to a ﬁnite (halo-dep endent) entrop y . So the region for the calculation of e ntropy is not the horizon itself but rather a light-sheet surro unding the horizo n of the lo caliza tion region. Hence the divergence of lo calizatio n entrop y in the limit of v anishing sheet size ∆ R → 0 is not a n indicatio n that QG must interv ene in order to rescue QFT from high energ y inconsistencies [16], but rather that the assumption of tensor factorization, whic h is the prer equisite of a bipartite entanglemen t situation, was not quite corr ect; the to ta l algebr a B ( H ) = A ∨ A ′ cannot b e written a s a tenso r pro duct e ven though A ′ is the comm utant of A . The split construction enforces the tenso r product situation but it brings a new parameter into the fray , the split size ∆ R. T he conceptual situation calls for great ca re in using standar d notions o f quantum information theory from QM in quantum ﬁeld theoretica l situatio ns in which thermal a sp ects of entanglemen t (and not the information theoretical) are domina n t. In particular the discus- sions ab out information los s in black hole physics s eems to have b een ca rried out without muc h apprecia tion for the ﬁeld theo retical subtleties address ed in this essay . Although the ter mino logy ”entanglemen t” strictly sp eaking do es not apply to a bipartite separation with sharp causa l b oundaries in Q FT, the liter- ature on entanglemen t unfortunately do es not s eem to diﬀerentiate b etw een the QM and the QFT cas e. There is of cours e the problem o f resp ecting a histor- ically accepted terminolo gy when its litera l meaning contradicts mathematica l facts [46]. Another issue presented in pa r t I is the question to wha t extend o ne needs 24 The thermoﬁeld formali sm of doubling of degrees of f reedom holds for the ﬁnite box and corresponds to the tensor factorization b etw een the boxed algebra and its commutan t. B ut b y not noting that this factorization br eaks do wn in the thermo dynamic l imit the aﬁcionados of thermoﬁeld theory miss an interesting ch ance of b ecoming aw are of a deep conceptual problem. 30 to go b eyond pointlik e ge ne r ators. W e rev ie wed the Wigner repres ent atio n theory in the modular localizatio n setting and reminded the reader that the only class which needs string like genera ting cov ar iant wav e function is Wig ne r ’s inﬁnite spin class whic h after a more tha n 60 y ear o dyssey , thanks to modular lo calization, ﬁna lly re a ched its ﬁnal p ositio n with resp ect to lo caliz ation. The algebraic notion of string like genera tor o f an algebra is how ever mor e restrictive in the sens e that it is describ e d by a n ” indecomp o aible” s tring-like lo calized ﬁeld Φ( x, e ) (with e the spacelike direction of the semiinﬁnite str ing a nd x its start) i.e. one which cannot b e res olved in terms of a p ointlik e ﬁeld s mea red along x + R + e. The a pplication of such an alg e braic string to the v acuum sta te is how ever dec ompo sable as a sta te in to ir reducible repr e sent atio ns of the Poincar´ e group and unless there ar e inﬁnite spin comp onents, the lo calization o f the state is p ointlik e even though the alg ebraic ob ject was an indeco mpo sable s tring 25 . The only illustratio n for s uc h an algebraic string, ment ione d in the ﬁr st par t, is the Dirac-Jo rdan-Mandelstam string. The Buchholz-F redenha g en setting [47] oﬀers ro om for pur e massive strings as one wants them in QCD, but since these strings do not leave any traces in p er tur bation theory , they remain b eyond what one is able to control with existing metho ds. W e also stressed that the widening of the se tting of lo caliza tion achieved through the mo dular formalism p oses new questions in volving massive vectormesons whose resolution could b e r elev ant for the interpretation of forthcoming LHC exp eriments. Lo calization is the ov err iding pr inciple of LQP , in fact it is the only principle and therefore the main and often diﬃcult tas k in the conceptional conquest of sp eciﬁc eﬀects and mechanisms in Q FT co nsists in ﬁguring out how a nd under what conditions they can b e derived from lo c alization. In pheno mena ﬁrs t observed in a Lagrang ia n qua ntization setting as e.g. Q ED infrared pro pe rties, sp ontaneous symmetry-br eaking a la Go ldstone, the Sc hwinger-Higgs screening mechanism or the cros s ing pr op erty most of the rich conc e ptua l-mathematical understanding came from the pur suit of this goal, and if there has not y et bee n a p e r fect understanding, it only means the the subtle connection of these phenomena with lo calization has not yet b een completely unraveled. The stor y of the Wigner inﬁnite spin r e pr esentation class shows that even for kinematical problems the unders tanding o f their lo caliz ation a sp e cts so metimes take mor e than half a century (see par t I). Even wher e it is leas t expected, namely in case of the mysterious quan tum concept of internal symmetries is a particular mo de o f r ealization of the lo ca lit y principle. The DHR theory show ed how the po ssible sup erselected repr esenta- tions of an obser v able algebra 26 and the ensuing g roup theor y which ties the diﬀerent sectors into one ” ﬁeld repr esentation” (o n which it acts in s uch a wa y that the lo cal observ able alg ebra reemer ges as the ﬁx p oint subalgebr a under 25 The s tate l ocalization structure i s exclusively determind by the represent ation theo ry of the Poincar ´ e group whereas the pr oblem of i rreducible algebraic generators in int eracting theories depends on the dynamics. 26 This example (the Doplicher-Haag Rob erts sup erselection theory [2]) is particularil y suit- able since nobo dy would expect group theory to emerge from classifyi ng i nequiv alent local represen tations of the observ able algebra; at no p oint is group theory vi sible. 31 the action of a c ompact gro up) is uniquely contained in the str uc tur e of the lo ca l observ ables. On the other hand the monad picture shows that even spacetime symmetries can b e enco ded in the abstract mo dular p ositioning within a s ha red Hilber t spac e . A brief explanation o f this last r e mark is as follows (see part I). A QFT in the a lgebraic setting is a net of spacetime-indexed algebra s. Hence it comes as somewhat of a s urprise that one can do with less; to get a concrete QFT going, o ne only needs a ﬁnite num b er of monads in a sp ecial r elative modular po sition. The reaso n why I used a whole section in pa rt I of may essay for a description of this prop erty (which up to now has remained without pra ctical use), is that I ﬁnd this very exciting from a philosophica l po int of view. It is the a lmost literal adaptation of Leibniz’s idea of what constitutes reality to the setting o f lo ca l quantum physics. A monad in isolation is no t muc h more than a p oint in geometr y , be sides the absence of pur e and mixed sta tes and the s tatement ab out what kind of states it admits instead it, is a n ob ject without prop erties . Surpris ingly its is not even necessar y to req uire that the algebras are mo nads, their modula r theory together with the p ositioning deﬁned in terms of modular inclusions or mo dular in ters ections [48] alo ne for ces the factors to b e of hyperﬁnite type I I I 1 , no other facto r algebr as can b e br ought int o that particular mo dular p osition. The structural r ichness of QFT re s ult solely fro m the relatio n betw een these monads; this includes not o nly the lo cal net o f qua nt um matter but also its internal as well as spacetime symmetries. There ar e other equiv alent wa ys to characterize a QFT in terms of the mo dular data of its lo ca l subalgebras [51]. F rom a practical p oint of view it turns out to be more useful to know the a ction of the Poincare group on the g enerator s on one ﬁxed wedge which is e q uiv alent to the p ositioning p oint of view, indee d this was the appro a ch b y which the e x istence of certain tw o -dimensional factorizing mo dels was established (sectio n 5 of part I). Perhaps the most pr ofound co nceptual-philosophica l con tra st b et ween QM and QFT ﬁnds its expression in these mo dula r enco ding. As mentioned in the last section in pa rt I, there ha ve b een other ideas which are desig ne d to highlight an underlying relationa l nature of QT; in particular Mermin’s view of QM [5 0] in terms of its c orr elations . It is diﬃcult to ﬁnd a mathematical backup which is as crystal clear as that in terms of mo dular p ositioning of monads. Mermin expressed his relational po in t o f v iew b y the following a p o diction: c orr elations have physic al r e ality, that what they c orr elate do es not. The LQP analog of this dictum would b e: re lative mo dular p ositions in Hilb ert sp ac e have physic al r e ality, the su bstr ate 27 which is b eing p ositione d do es not. The presentation of QFT in terms of po sitioning monads is v ery sp eciﬁc of LOP i.e. it has no analo g in QM i.e. Mermin’s rela tional view is no t a special case o f p ositio ning in LQ P . It has the additional adv antage that b e yond the 27 Mo dular p ositioning i s the m ost radical form of relationalism si nce the lo cal quantum matter ari ses together with in ternal and spacetime symmetries. In other w ords the concrete spacetime ordering is preempted i n the abstract modular p ositioning of the monads in the joint Hilb ert space. 32 metaphor ther e are har d mathematical fac ts . Of course it would be a serious limitation if this philosophical viewp oint is restricted to the c hara cterization o f Minko wski spacetime QFT. Despite all the progre s s with QFT in CST in relation with the formulation of the lo ca l cov ari- ance principle, it is too ear ly for such questions inv olv ing mo dular theory . As a preliminary test one co uld ask whether the diﬀeomo r phisms in Diﬀ(S 1 ) b eyond the Mo ebius transfor mations in a chiral theory , which as well-known do not leave the v acuum state in v ariant, hav e their o rigin in mo dular theory . It is clear that in order to a chiev e this one has to b e mor e ﬂexible with sta tes , i.e. using also other than the v acuum sta te and not ins is ting in a mo dula r a utomorphism of being globally geometric. In this connection it is very encourag ing that recent ly the idea of lo cal cov ariance found a satisfactory expr ession; witho ut having a precise description of this cr ucial principle, there would no t b e muc h c hance to make headw ay with mo dular loca lization metho ds in the CST setting of QFT. The res triction o f g lobally pure state (v acuum, par ticle states) to causally lo- calized subalg e bras A ( O ) leads to thermal KMS states a sso ciated with the mod- ular Hamiltonian as so ciated to ( A ( O ) , Ω). Mo dular Hamiltonians give ra rely rise to geometr ic mov ements (diﬀeomorphisms). Althoug h outside conformal QFT there is no compact lo ca lization region in Minko wski spacetime which leads to a fully ge ometric mo dular theory , there a re rather convincing a rgu- men ts that the mo dular auto mo rphism b ecomes geometric a t the horiz o n of the loca lization region. The reason is that the holog raphic pro jection onto the horizon is a (transverse extended) c hiral theor y . In case of timelike Killing symmetries in CST there may even exist an ex- tensions of the spacetime and a state on it such that the mo dular gro up of its restriction is iden tical to the Killing gro up 28 . One of the most mysterious aspec ts of lo calization-c aused ther mal behavior is its poss ible connection to ordinary i.e. heat bath thermality . The pr oblem is often referred to a s the inverse Unruh eﬀect b ecause one wan ts to know whether there exists a heat bath thermal system which can be viewed a s arising thr ough restricting the v acuum on an extended system i.e. in the spirit of an Unruh eﬀect. T his is indeed p o ssible for chiral theor ies and the connection b etw een the tw o systems is a conformal transformation which maps the standard volume (here length) factor in to the loga rithm of the splitting length. Although in higher dimension there is no rigor ous argument which relates the tw o kinds of thermal b ehavior, there is an educated guess. Apart from the case of chiral theories on the lightray , where the v a lidit y of the inverse Unruh eﬀe ct p ermits to tr a nsform the heat bath volume (here length) law into a logar ithmic divergence in ∆ R → 0 , a relation b et ween heat ba th- and lo ca lization-caused entrop y is unknown. F o r hig her dimensional lo c alization ent ro py the split prop erty suggests to consider the en tropy of a lig ht sheet whose thickness is the split distance ∆ R. By gener alizing the lo c alization entropy of chiral theories and by relying on the v acuum p olariz a tion div erg e nc e of a 28 The standard example is the Har tle-Ha wking state on the Kr usk al extension restricted to the region outside a blac k hole. 33 dimensionless charge we presented a formula for the leading divergence of the lo calization entrop y and gav e strong arg umen ts in fav or of a universalit y b etw een heat bath and lo caliza tion-caused en tropy . F rom an a lgebraic viewpo in t one parametrizes the approximation of a monad b y type I algebras in tw o diﬀerent wa y; in the thermo dynamic limit b y a L 3 prop ortiona l s e quence and in the funnel limit one o f the length factors was replace s by a log so that eﬀectively one obtains a L 2 (area) prop ortionality . The mo nade itse lf is s tructureless a nd the parametriz a tion only app ear s in the context if its use inside a physical theo r y . A ∆ R independent area law a s that of Be kenstein, in which the light s heet width ∆ R is replaced by the Pla nc k length, is not co mpatible with the lo caliza- tion structure of QFT which require s the quadr atic incr ease. There is a tight connection b et ween modular lo ca lization and the phasespace densit y of states. Whereas the pha se space density in QM is ﬁnite, that of QFT is n ucle ar. If one interprets the Bekenstein a rea law as coming from a future quantum theo ry of gravit y (QG) without s tandard quantum matter, the alge br aic structure of such a theo r y must b e that of a v ery low phasespace density , as e.g. an un- known QM or a combinatorial algebr a with trace sta tes. On the other hand the lo calization en tropy from QFT is the precis e entropical counterpart within the thermal setting of the Hawking radiatio n which do es not need any app eal to a yet unknown QG. Hence their remains a basic clash b etw een Ha wking ra diation, which Hawking derived from Q FT lo calized outside a black hole ho rizon, and the Bekenstein for mu la whic h w as inferred from in terpr eting a certain clas sical area formula. In the a rticle we als o presented exa mples of unnatural QFT in which the phase spac e densit y is to o hig h or to o low. Among the unph ysica l consequences are: the ex is tence of a Hagedor n temp era ture or the a bsence of an y therma l state, as well a s s e r ious problems with causal propag ation. In pa r ticular they cannot o ccur in causally propag ating situations as formally describ ed b y La- grangia n quant izatio n. The algebraic appro ach to QFT fro m its very beg inning [4][5] tried to isolate them. Their o nly use has b een to exemplify those unphys- ical prop erties whic h a no n- Lagrang ian approa ch m ust a void and understand those prop erties which one must r equire to ex c lude pa tho logies. They o ccur in inﬁnite compo nen t QFT as string theo ries 29 . One als o meets them in the AdS n -CFT n − 1 corres p ondence , an e x plicit illustra tion is provided b y taking a free ma ssive AdS ﬁeld which on the conformal side yields a g eneralized confor - mal ﬁeld with the mentioned pa tho logies. Unfor tunately the old insig ht s into what constitutes a natural QFT outside the La grangia n protection hav e b een lost o n pro tagonists o f the sup ersymmetric N=4 Y ang Mills – super gravitation AdS mo del. The attempt to remind them of the problems in their conjectures has rema ined without av a il [8]. The Bekenstein ther mo dynamical interpretation of a c e r tain quantit y in the setting of classical gr avit y rais e s the question whether it is not p ossible to inv ert this connec tio n i.e. to supplement the ther mo dynamical setting by reas o nable 29 Con trary to its name, the result of the canon ical quantization of the Nam bu-Goto La- grangian is not string-lo calized but represen ts a p oint-localized dynamic al inﬁnite c omp onent QFT [44][1]. 34 assumptions of a general geometric nature, so that the E ins tein Hilber t eq ua- tions are a consequence of the fundamen tal laws o f ther mo dynamics. Mo dular theory alr e a dy rela tes thermal b ehavior with lo caliza tion, hence a relation o f fundamen tal laws of thermo dyna mics with gravity is not as unex p ected as it lo oks at ﬁrst sight. The r eader is r e ferred to s ome very in teres ting o bserv a tions by Ja cobson [49]. Whereas there is hardly any doubt tha t a pa rt from problems of improved formulations the QM- Q FT interface ha d r eached its conceptual ﬁna l p os ition this is not the ca se with the interface b etw een QFT in C ST and QG. Up to recently the ge neral b elief was that the background indep endence a nd the en- tropical a rea pr op ortionality a re marking this in terface . But in b oth cases this had to be amended. On the one hand the new loca l cov ariance principle shows that loca l cov ar iance implies at least the unitary quantum equiv a le nc e o f QFT in spacetime regions which are isometric 30 which is a big s tep in the directio n of background indep e ndence. And if one ignores the logar ithmic factor the ar ea prop ortiona lity by itself cannot b e c hara cteristic for QG. Returning to the main p oint in pa r t I; there ar e hardly t wo concepts which are that diﬀere nt tha n re la tivistic QM and relativistic QFT. In textbo ok s this is consistently overlooked pro bably as a result of b elieving that beca use they share the Lag rangian quantization fo r malism and ℏ the only diﬀerence is taken care of by adding the word ”rela tivistic”. In b oth parts of this work we explained the diﬀere nc e in terms of the diﬀerent lo ca lization co ncepts which in turn is int imately r elated to the diﬀerence in the cardinalit y of phase spa ce degrees of freedom (ﬁnite in Q M, ” n uclea r” in QFT [4 4]). The ignoratio n or misun- derstandings o f these diﬀerences has b een the cause of a ma jo r derailment of particle physics [44]: str ing theory and the Maldacena co njecture. But modula r lo calization also led to the ﬁrs t existence pr o o f fo r certain in tera cting ﬁeld theo- ries (factor izing mo dels) after 80 y ear s of QFT, and it pro mises to revolutionize gauge theo ries [52]. In addition it gener ates the concepts which ar e neces sary t wo ”s plit” causally separ a ted reg ions so that the no tion of entanglement can also be used in QFT where it has thermal conseq ue nce s (lo ca lization-caus e d thermal b ehavior). Since the crisis of par ticle ph ysics or iginated from confusing the holistic asp ects of mo dula r lo caliza tion in QFT [4 4][1][53] in the aftermath o f S-matrix theory in the 60’s and ha s solidiﬁed ev er since, crea ting a quite misleading int uition even in pre s ent day QFT, the only wa y out is to correct this incorrect understanding of the most imp orta nt concept whic h constitutes the essence of QFT. 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Localization and the interface between quantum mechanics, quantum field theory and quantum gravity II (The search of the interface between QFT and QG)

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