You are an expert at summarizing long articles. Proceed to summarize the following text: 
in the present year 2000 we are celebrating the 100th birthday of quantum theory and the 75th birthday of quantum mechanics . thus it took only 25 years from the first inception of the new theory until its final consolidation . quantum field theory is almost as old as quantum mechanics . but the formulation of a fully consistent synthesis of the principles of quantum theory and classical relativisitic field theory has been a long and agonizing process and , as a matter of fact , has not yet come to a satisfactory end , in spite of many successes .    the best approximation to nature in the microscopic and relativistic regime of elementary particle physics which we presently have , the so called _ standard model _ , does not yet have the status of a mathematically consistent theory . it may be regarded as an efficient algorithm for the theoretical treatment of certain specific problems in high energy physics , such as the perturbative calculation of collision cross sections , the numerical analysis of particle spectra etc . yet nobody has been able so far to prove or disprove that the model complies with _ all _ fundamental principles of relativistic quantum physics . this somewhat embarassing situation is not so widely known . it is therefore gratifying that the clay mathematics institute has recently drawn attention to it by endowing a price of 1.000.000 $ for the mathematical consolidation of an important piece of the standard model , the yang  mills  theory . so the mathematical and conceptual problems of relativistic quantum field theory are in many respects a rewarding field of activity for mathematical physicists . there are two strategies to make further progress . either one tries to improve the existing mathematical methods for the treatment of models of physical interest . this is the approach of constructive quantum field theory @xcite . or one proceeds from a sufficiently rigid mathematical framework , consistent with all basic principles of quantum field theory , and aims at new conceptual insights and constructive ideas by structural analysis .    such a general framework which is useful for the solution of both , conceptual puzzles and constructive problems , is _ algebraic quantum field theory _ ( aqft ) , frequently also called _ local quantum physics_. it was invented by rudolf haag and daniel kastler @xcite and has proven to be consistent with the developments in elementary particle physics for several decades . it is the aim of this contribution to recall the physical ideas and mathematical structures underlying aqft and to outline some recent interesting results which reveal its flexibility for the treatment of a variety of problems . more systematic recent reviews of this approach can be found in @xcite . the relation between the conventional approach to quantum field theory , based on the lagrangian formalism , and algebraic quantum field theory may be compared with the concrete and abstract approaches to differential geometry . if one is dealing with concrete ( computational ) problems in geometry , it is natural to use coordinates , tensor fields , christoffel symbols etc , whereas in the general structural analysis one relies on intrinsic concepts such as the notions of manifold , fiber bundle , connection etc . both points of view have their virtues and full insight is only gained by combining them . the situation is similar in relativistic quantum field theory . in the concrete lagrangian approach , one specifies the field content of the theory acting on the given space time manifold @xmath0 as well as a corresponding lagrangian . the difficult step is `` quantization '' which is accomplished either by appealing to the correspondence principle ( canonical quantization ) or to general structural results according to which the problem can be reformulated in terms of classical statistical field theory ( euclidean approach ) . if everything has been said and done , one obtains the vacuum correlation functions of the fields . from them one can reconstruct a hilbert space on which the observables , such as the stress energy tensor , the currents etc . act as operators . the lagrangian approach is well suited for the computational treatment of concrete models , but it is not intrinsic . for different lagrangians with different field content may describe the same physics . this phenomenon of `` quantum equivalence '' of classical field theories has been observed in many examples @xcite . one may therefore ask whether there is a more intrinsic way of describing relativistic quantum field theories .    a fully satisfactory answer to this question is provided by aqft . in this approach the basic objects are the algebras generated by the observables localized in given space time regions ; fields are not mentioned in this setting and are regarded as a kind of coordinates of the algebras . the passage from the field theoretic setting to the algebraic one requires the following steps :    * determine the set @xmath1 of observables of the underlying theory for each space  time point @xmath2 * construct for each relatively compact space time region @xmath3 a corresponding _ local algebra of observables _ , @xmath4 i.e.  the von neumann algebra ( double commutant ) generated by the respective observables in the underlying hilbert space .    in view of the fact that the observables @xmath5 are only defined in the sense of sesquilinear forms ( or as operator valued distributions ) , the latter step is somewhat subtle . but it has been shown to be meaningful in the models which have been constructed so far @xcite and also in the general wightman setting of quantum field theory under some very general conditions @xcite . the resulting structure is an assignment of algebras to spacetime regions , @xmath6 which is called a _ local net _ in view of its order preserving properties . any such net inherits some fundamental properties from the underlying field theory . these are @xcite    * locality : operators localized in causally disjoint regions commute * covariance : the space  time symmetries act by automorphisms on the net * stability : there exist distinguished states ( expectation functionals ) on the net , describing stable elementary systems such as the vacuum .    whereas the first two points are well understood and have an obvious physical interpretation , the mathematical characterization of elementary states on arbitrary space  time manifolds is a more difficult issue which is still under discussion , cf . @xcite and references quoted there . for the class of maximally symmetric spaces there are no such problems , however .    according to the deep insights of haag and kastler , the full physical information of a theory is already contained in the net structure , i.e.  the respective map from space  time regions to algebras . phrased differently , equivalent quantum field theories can be identified by the fact that they generate isomorphic local nets . this assertion may be somewhat surprising at first sight since the passage from the observables , which normally have a specific physical interpretation , to the local algebras seems to be a rather forgetful operation . that no information is lost in this step has been confirmed by now by numerous results @xcite . a direct way of seeing this has been established by fredenhagen and hertel @xcite who proved for the class of minkowski space theories that the set of basic observables can be recovered from the local net by the formula @xmath7 the somewhat tricky point in this reconstruction is the need to proceed from the algebras @xmath8 of bounded operators to unbounded sesquilinear forms . it is accomplished by completing these algebras in a suitable locally convex topology , indicated by the bar . it should be noted , however , that the algebraic framework is in some respect more flexible than the field theoretic setting . for the local algebras may also accommodate extended objects , such as wilson loops or finite mandelstam strings , which are not built from point like observable fields . aqft is thus compatible with the structures found in the quantum field theories of present physical interest and complements them by putting emphasis on their intrinsic features . it may thus be regarded as a minimal setting for the description of the systems appearing in high energy physics . we discuss in the following some issues where the virtues of this approach become manifest . guided by insights gained from algebraic quantum field theory , brunetti and fredenhagen @xcite have recently established a perturbative construction of local nets on arbitrary globally hyperbolic space times @xmath0 . they propose to treat this problem in two steps : first one constructs the nets of local algebras in some convenient hilbert space representation , thereby fixing the theory . this step requires control of the notorious ultraviolet divergences ( renormalization ) and can be handled by configuration space methods and microlocal techniques . infrared problems , related to the so  called adiabatic or infinite volume limit , do not appear in this construction . in a second step one may then turn to the determination of the states of physical interest on this net and to their analysis . this requires the passage to new hilbert space representations of the algebraic structures and may thus be regarded as a problem in the representation theory of local nets .    in minkowski space theories both problems are frequently treated simultaneously because of the possibility of characterizing the vacuum state directly by momentum space properties ( spectrum condition ) . but , as indicated above , this strategy does not work for arbitrary space  time manifolds . thus the brunetti fredenhagen approach is a very natural way of circumventing these difficulties . following @xcite , we outline this method by discussing the theory of a self - interacting scalar field on a given space - time @xmath0 . the perturbative construction of the net requires the following steps :    * consider the free scalar field @xmath9 on the space  time @xmath0 , @xmath10 = -i \ , \delta(x , y ) \ , 1,\ ] ] where @xmath11 denotes the dalembertian and @xmath12 the causal commutator function on @xmath0 , in a regular hilbert space representation induced by some hadamard state @xcite . * construct _ wick powers _ of the free field for @xmath13 , @xmath14    the existence of these operator  valued distributions was first established in @xcite by methods of microlocal analysis and shown in @xcite to be largely independent of the chosen regular hilbert space representation . these wick powers are the building blocks for the construction of interacting fields . * define the time ordered exponentials ( local s  operators ) @xmath15 where @xmath16 is any test function and @xmath17 is the volume form on @xmath0 . ( more generally , one considers such exponentials for finite sums of wick powers . ) the proof that these exponentials are meaningful expressions is the most difficult step in the construction . it has been established in @xcite in perturbation theory by defining @xmath18 as formal power series in @xmath16 . the coefficients in this series suffer from ambiguities due to short distance singularities which require renormalization . this problem is solved by generalizing methods of epstein and glaser ( causal perturbation theory ) . if @xmath19 , there does not appear a proliferation of these ambiguities with increasing order of perturbation theory ( renormalizability ) . * use bogolubov s formula @xmath20 to define , for given interaction density @xmath21 , local operators for the cutoff density @xmath22 .    up to this point the construction is akin to the treatment of minkowski space theories , although the technical details are more involved in view of the absence of space  time symmetries . the adiabatic limit of @xmath23 for @xmath24 may not exist , however , due to infrared problems caused by the _ ad hoc _ choice of a defining representation of the interacting theory . this difficulty can be circumvented by the following important observation @xcite . let @xmath3 and let @xmath25 . there exists a unitary operator @xmath26 such that @xmath27    in view of this result it is meaningful to define local algebras , setting for @xmath3 and any @xmath16 with @xmath28 @xmath29 according to the preceding proposition , these algebras are unique up to isomorphisms @xmath30 , which do not change the physical interpretation . one may thus proceed to an algebraic adiabatic limit by considering the algebras @xmath31 the inclusion ( net ) structure of this family of algebras is given by their natural embeddings and the algebraic operations of addition , multiplication as well as the * operation in @xmath32 are pointwise defined for each @xmath16 . in this way one arrives at a ( perturbative ) net @xmath33 for the given interaction . thus the algebraic point of view leads to a natural perturbative construction of nets of local algebras in any space time @xmath0 . similar methods have also been applied to the construction of local nets of observables in gauge theories in minkowski space @xcite , cf . also @xcite and references quoted there . we turn next to a conceptual problem in minkowski space quantum field theories , namely the asymptotic particle interpretation . in the gauge theories of physical interest , the basic fields are in general unphysical and not related to stable particles . one has therefore to develop methods to determine the particle features of these theories from the vacuum correlation functions of the local observables . the confinement problem in quantum chromodynamics and the infraparticle problem in quantum electrodynamics are two well known examples illustrating this issue . here the algebraic point of view led recently to some interesting progress . the following steps are necessary in the particle analysis of any theory :    * introduce some meaningful particle concept * establish methods to determine the particle content of the theory * analyze the properties of these particles * develop a scattering ( collision ) theory . one has a quite satisfactory understanding of these points in theories with short range forces ( mass gap ) . there the natural starting point is wigner s particle concept , according to which the possible states of a particle are described by vectors in some irreducible representation of the poincar group or its two  fold covering , respectively . it is well known , however , that this approach does not work , for example , in theories with electrically charged particles , cf . for example @xcite . so it seems desirable to develop a universal particle concept which applies also in those cases . such a more flexible particle concept was introduced in @xcite . it is based on the notion of _ particle weight _ which is a generalization of dirac s idea of an improper momentum eigenstate of a particle . there are two possibilities of looking at these improper states .    * traditional : improper momentum eigenstates are regarded as vector valued distributions , i.e.  maps from a space of wave functions into the physical hilbert space , @xmath34 it is anticipated in this approach that the improper states become normalizable by superposition ( interference effects ) . * alternative : improper states of _ fixed _ momentum are regarded as linear maps from a space @xmath35 of localizing operators into the physical hilbert space , @xmath36 here the improper states become normalizable by localization . it is important to notice that the second approach is more general than the first one . it is expected to be applicable even if the superposition principle fails for the improper states . this happens , for example , if the process of localization is inevitably accompanied by particle production ( such as in quantum electrodynamics , where infinite clouds of soft photons are produced ) .    in quantum mechanics , suitable localizing operators would be rapidly decreasing functions of the position operator ; but the notion of position operator is not meaningful in a field theoretic setting . nevertheless , localizing operators @xmath37 exist in aqft in abundance and are easily constructed . simple examples are all operators of the form @xmath38 here @xmath39 is any test function whose fourier transform vanishes in the forward light cone and @xmath40 , where @xmath41 are the unitaries inducing the space  time translations @xmath2 and @xmath42 is any local observable . the operators @xmath37 can be shown to annihilate all states ( disregarding vectors of arbitrarily small norm ) which do not describe excitations of the vacuum in some ( sufficiently large but finite ) space  time region @xmath43 . so , roughly speaking , they `` project '' onto states which differ from the vacuum in @xmath43 . in this sense they are localizing operators . it is technically important that these localizing operators form a left ideal @xmath44 in the c@xmath45algebra @xmath46 generated by all local observables . it is convenient to proceed from the improper states to corresponding positive , linear and non normalizable functionals on the domain @xmath47 , @xmath48 called particle weights . these functionals can be characterized in an intrinsic manner . in particular they are extremal and invariant under space time translations . as @xmath49 is a @xmath50algebra , one can recover by the gns reconstruction theorem the improper particle states from these particle weights .    after having introduced in aqft a general particle concept , one has to develop methods to determine the particle content of a theory ( described by particle weights ) . this is accomplished by analyzing the timelike asymptotic properties of the physical states in the vacuum sector @xmath51 of the theory . so let @xmath52 be any expectation functional induced by vectors in the subspace @xmath53 of the physical hilbert space @xmath54 . one then considers the functionals @xmath55 these expressions are mathematically meaningful as a consequence of locality and the shape of the energy momentum spectrum @xcite . moreover , the family of functionals @xmath56 is equibounded . so this family has limit points @xmath57 the functionals @xmath58 are , by their very construction , invariant under translations , but highly mixed . it is therefore natural to ask whether they can be decomposed into particle weights . an affirmative answer to this question was recently given by porrmann @xcite . more specifically , there holds the following statement . @xmath59 where each @xmath60 is a particle weight of momentum @xmath61 and `` internal index '' @xmath62 and @xmath63 is a measure depending on the initial state @xmath64 . ( the internal index @xmath62 describes the intrinsic features of a particle , such as its charge quantum numbers and spin . ) thus the preceding asymptotic construction and subsequent decomposition of functionals provides a general method to determine the stable particle content of any theory , including the charged particles ( which appear in the vacuum sector only as pairs of opposite charge , but asymptotically give rise to particle weights contributing to the mixtures @xmath58 ) . this result is a substantial generalization of work of araki and haag for massive theories with a complete particle interpretation @xcite . no _ a priori _ input about particles is necessary for its derivation ; this shows that the concept of particle weight is sufficient for the description of the asymptotic particle features of any theory . the next step in the analysis is the determination of the possible properties of particle weights . to this end one first constructs for each particle weight the corresponding sector of the physical hilbert space , @xmath65 one can then establish the following general results @xcite . * mass : the energy  momentum @xmath61 of the underlying improper state @xmath66 and therefore its mass @xmath67 can be sharply defined in the sector @xmath68 by the formula @xmath69 where @xmath70 is the unitary representation of the translations on @xmath68 ( which can be shown to exist ) . * spin : if , for given @xmath61 , there is a _ finite _ multiplet of particle weights , there exists a unitary representation @xmath71 of the ( covering group of the ) stability group @xmath72 of @xmath61 such that for @xmath73 @xmath74 where @xmath75 are matrix representations of @xmath72 and @xmath76 . thus if @xmath77 , the particle weights can have spin @xmath78 , in accordance with the results found by wigner . however , in contrast to the case of wigner particles , there may appear for particle weights with mass @xmath79 representations with arbitrary helicity . * coherence : let @xmath80 be the decomposition of @xmath68 with respect to the spatial momentum and consider the restrictions @xmath81 of the generator @xmath82 of the time translations to the respective subspaces . for the point spectrum of these restrictions there appear the possibilities @xmath83    the former case corresponds to the familiar situation of wigner particles , where the improper states @xmath84 are affiliated with the same sector and thus can coherently be superimposed . in the latter case , where @xmath85 has purely continuous spectrum for @xmath86 , the superposition principle fails for particle weights of different momenta . this situation is expected to prevail in theories with long range forces .    in view of the latter result it seems a rewarding experimental challenge to test the status of the superposition principle for electrically charged particle weights . the theory predicts that there ought to be a substantial difference in the asymptotic coherence properties of electrically neutral and charged particles .    in a final step one has to establish a collision theory for particles described by weights . in general one may not expect that a scattering matrix exists for these entities , but it is always possible to define and compute collision cross sections . a general method to this effect has been outlined in @xcite . we conclude this section with the remark that , in contrast to the case of many particle quantum mechanics , the completeness of the particle interpretation in quantum field theory is still an open problem , even in the case of short range forces . a survey of the state of the art can be found in @xcite . triggered by developments in string theory , known under the catchwords of holography or ads / cft correspondence ( cf . @xcite and references quoted there ) , there has recently emerged interest in the relation between quantum field theories on different space time manifolds . one speaks of _ holography _ if there is a correspondence between a quantum field theory on a space time @xmath87 and a theory on its boundary @xmath88 . in other words , given the theory on @xmath87 , one can unambiguously determine the theory on @xmath89 and _ vice versa_. a related notion is _ transplantation _ , where such a correspondence exists between theories on space times @xmath87 , @xmath89 of equal dimension . in the conventional field theoretic setting a satisfactory understanding of these issues seems impossible . it is clear from the outset that one can not identify point fields on @xmath87 and @xmath89 in a meaningful way , @xmath90 with regard to holography , the best one can do is to proceed from a field theory on @xmath87 by restriction to a corresponding theory on @xmath89 , cf .  @xcite . but it is in general impossible to recover from the latter data the original fields . similar problems hamper also the idea of transplanting fields . here the algebraic point of view provides a solution : bearing in mind that a theory is fixed by the underlying net , one realizes that it is sufficient ( and frequently possible ) to identify local algebras for certain sufficiently rich families of regions @xmath91 , @xmath92 , @xmath93 thereby establishing a rigid link between the two theories in question . this insight was first used by rehren in his analysis of the issue of holography @xcite . similar ideas were applied to the problem of transplantation in @xcite . we outline in the following the underlying simple geometrical facts . holography : + the simplest example , where the idea of holography can be illustrated , is the correspondence between quantum field theories on anti  de sitter space ( ads ) and on its minkowskian boundary . in proper coordinates , ads can be envisaged as a full cylinder whose tube like boundary is conformal minkowski space .    as indicated above , one has to identify suitable regions in the given space times in order to establish a correspondence between the respective theories . for the case at hand , rehren proposed to consider a family of causally complete wedge  shaped regions @xmath94 . their intersections with the boundary of ads are diamond shaped regions , @xmath95 , cf . figure 1 . it is crucial that with this choice there exists a bijection @xmath96 between these regions which is    * causal : @xmath97 , where the prime indicates causal complementation * symmetric : @xmath98 for @xmath99 , where iso and conf indicate the isometry and conformal group of the respective spaces * order preserving : @xmath100 if @xmath101 .    after these geometrical preparations it is straightforward to establish the desired correspondence between nets on ads and @xmath102ads as well as the corresponding unitary representations of the respective symmetry groups and the vacuum states , setting    * @xmath103  for  @xmath104 * @xmath105 * @xmath106 .    starting from a local , covariant and stable net on ads one obtains in this way a local , conformally covariant and stable net on minkowski space and _ vice versa _ ( by reading the defining relations from right to left ) . so one arrives at @xcite    there is a one  to one correspondence between local , covariant and stable qft s on ads and on @xmath102ads = conformal minkowski space . it seems plausible that the general idea of holography can also be applied to other space  time manifolds with suitable boundaries . + transplantation : + the possibility of identifying quantum field theories on space  times of equal dimension ( transplantation ) has been exemplified in @xcite for a special class of robertson  walker space  times ( rw ) . in this approach one makes use of the fact that these space  times can be conformally embedded into de sitter space ( ds ) . given this embedding , one chooses a certain specific family @xmath107 of diamond shaped regions in ds and , taking the intersection with rw of those regions @xmath108 whose edge lies completely in rw , one obtains a corresponding family of regions @xmath109 in rw , @xmath110 this construction is indicated in figure 2 . we mention as an aside that the respective regions can also be characterized in an intrinsic coordinate independent manner . given these regions , there exists a unique bijection @xmath111 : @xmath112 with the property of being    * causal :   @xmath113 * symmetric : @xmath114 , @xmath115    here @xmath116 denotes the induced action of the elements @xmath16 of the isometry group of de sitter space , iso ds , on the family of regions @xmath109 . this action can in general not be described by a point transformation on rw . but if @xmath117 one has @xmath118 .    on the basis of these geometrical facts it is straightforward to establish a one  to one correspondence between theories on ds and rw , setting    * @xmath119  iff @xmath120 * @xmath121 ,  @xmath122    an immediate consequence of the geometrical properties of @xmath111 is @xcite    each local , covariant _ ds_theory can be mapped to a local , covariant ( ultra symmetric ) _ rw_theory and _ vice versa_.    here the term `` ultra symmetric '' means that the resulting rw theory exhibits , besides the familiar space time isometries , an aditional geometrical symmetry which is _ not _ induced by point transformations . this situation differs from the ads / cft correspondence , where the minkowskian theory on the boundary of ads is conformally invariant . an intriguing recent result in algebraic quantum field theory is the insight that local nets can be constructed from a few local algebras in suitable `` relative positions '' @xcite . it is of interest in this context that the local von neumann algebras are universal ( model independent ) objects : they are generically to the _ unique _ hyperfinite type iii@xmath123 factor @xcite . so the starting point of this novel construction is a concrete and well  studied algebra , denoted by @xmath0 in the following ; the second ingredient is a standard ( cyclic and separating ) vector @xmath124 for @xmath0 in the underlying hilbert space . given these quantities , one can consistently define a conjugation @xmath125 it is an anti  linear closable operator whose closure is denoted by the same symbol and whose polar decomposition has the form @xmath126 here @xmath127 is an anti  unitary operator , called _ modular conjugation _ , and the positive selfadjoint operator @xmath128 is the _ modular operator_. the corresponding unitary group @xmath129 is called _ modular group_. irrespective of the choice of @xmath0 and @xmath124 within the above limitations , there hold the following basic relations established by tomita and takesaki :    * @xmath130 * @xmath131    based on these facts , wiesbrock introduced in @xcite the notion of _ half  sided modular inclusion _ of a von neumann algebra @xmath132 , where @xmath133 likewise has @xmath124 as standard vector , by posing the condition @xmath134 the unitary group @xmath135 , obtained by the trotter product formula @xmath136 then has the following properties @xcite . let @xmath132 be a half  sided modular inclusion . the corresponding modular groups generate a unitary representation of @xmath137 such that    * @xmath138 * @xmath139 * the spectrum of the generator of @xmath135 is contained in @xmath140 * @xmath141 . an analogous result holds if @xmath140 is replaced by @xmath142 in @xmath143 . another useful concept , characterizing the relative position of von neumann algebras , is the notion of _ modular intersection _ @xcite : two von neumann algebras @xmath0 , @xmath133 are said to have modular intersection if @xmath144 is half  sided modular in @xmath0 and @xmath133 , respectively . similarly to the situation discussed in the preceding proposition , the modular groups can be shown to generate a unitary representations of a lie group in the latter case as well . these general mathematical facts have immediate applications in algebraic quantum field theory . local nets on @xmath145  @xcite + any modular inclusion fixes a local , covariant and stable net of local algebras on the light ray @xmath145 . one first assigns algebras to half lines , setting @xmath146 the algebras corresponding to arbitrary intervals @xmath147 \subset \rr$ ] are given by @xmath148 it then follows from the preceding proposition and the basic relations of tomita  takesaki  theory that the net @xmath149 on @xmath145 is local ( operators localized in disjoint intervals commute ) , @xmath150covariant and stable ( @xmath124 being a ground state for @xmath135 ) . thus , given two algebras in suitable relative position , one can construct a full chiral quantum field theory . ( in a similar way one sees that any modular intersection fixes a conformally invariant quantum field theory on the compactified light ray s@xmath151 . ) local nets on @xmath152  @xcite + in order to obtain in a similar manner local nets on higher dimensional space  times , one has to proceed from a larger set of algebras in appropriate relative positions . in the case of two  dimensional minkowski space , one starts from three algebras , forming two half  sided modular inclusions @xmath153 . by the preceding proposition one then has two unitary groups @xmath154 , where @xmath155 are interpreted as light cone coordinates of @xmath156 . these groups are assumed to commute , @xmath157 in this way one obtains a unitary positive energy representation @xmath135 of the translations @xmath152 on which the modular group of @xmath0 acts like a lorentz transformation , @xmath158 one then can proceed as in the preceding case and define algebras for wedge shaped regions of the form @xmath159 , setting @xmath160 the algebras associated with diamonds @xmath161 , cf . figure 3 , are obtained by setting @xmath162 in this way one arrives at a local , poincar  covariant net on @xmath152 where @xmath124 describes the vacuum vector . local nets on @xmath163 , @xmath164  @xcite + the construction of local nets from a few algebras was recently extended to three and four  dimensional minkowski space in @xcite . the crucial and difficult step in this approach is the formulation of conditions which guarantee that the modular groups affiliated with the algebras and the underlying vector @xmath124 generate representations of the poincar group @xmath165 . any family of algebras @xmath166 in suitable modular positions fixes a local , @xmath165covariant net on @xmath163 with vacuum state @xmath124 . we refrain from giving here the precise conditions on the algebras and only note that , in analogy to the cases discussed before , the corresponding modular groups generate representation of @xmath165 with positive energy . the algebras @xmath167 can consistently be assigned to wedge regions in @xmath163 and the local algebras associated with diamonds are defined by taking intersections . the converse of this statement is a well  known theorem by bisognano and wichmann @xcite , cf . also @xcite . these intriguing results should admit a generalization to other space  time manifolds , cf . also @xcite for a related approach . moreover , they seem to be of relevance for the classification of local nets and are possibly a step towards a novel , completely algebraic approach to the construction of local nets . the preceding account of recent results in aqft illustrates the role of this approach in relativistic quantum field theory : it is a concise framework which is suitable for the development of new constructive schemes , the mathematical implementation of physical concepts and ideas , the elaboration of general computational methods and the clarification of the relation between different theories as well as their structural analysis and classification . so this framework complements the more concrete approaches to relativistic quantum field theory and thereby contributes to the understanding and mathematical consolidation of this important area of mathematical physics . 99 a.  jaffe , constructive quantum field theory , p.  111 in : _ mathematical physics 2000 _ , a.  fokas et al . eds . , imperial college press 2000 r.  haag and d.  kastler , ( 1964 ) 848 d.  buchholz and r.  haag , ( 2000 ) 3674 d.  buchholz , current trends in axiomatic field theory , preprint hep - th/9811233 s.  coleman , phys . * d11 * ( 1975 ) 2088 m.  lscher , nucl . * b326 * ( 1989 ) 557 k.  intriligator and n.  seiberg , nucl . b , proc . * 45bc * ( 1996 ) 1 w.  driessler and j.  frhlich , ann . henri poincar * 27 * ( 1977 ) 221 h .- j . borchers and j.  yngvason , ( 1992 ) 15 r.  haag , _ local quantum physics _ , 2nd revised ed . , springer ( 1996 ) r.m . wald , _ quantum field theory in curved spacetime and black hole thermodynamics _ , chicago university press 1994 r.  brunetti , k.  fredenhagen and m.  khler , ( 1996 ) 633 d.  buchholz , o.  dreyer , m.  florig and s.j . summers , ( 2000 ) 475 c.j . fewster , class .  quantum grav . * 17 * ( 2000 ) 1897 h.  sahlmann and r.  verch , preprint math - ph/0002021 k.  fredenhagen and j.  hertel , ( 1981 ) 555 r.  brunetti and k.  fredenhagen , ( 2000 ) 623 m.  dtsch and k.  fredenhagen , ( 1999 ) 71 g.  scharf , _ finite quantum electrodynamics : the causal approach _ , 2nd ed . , springer 1995 d.  buchholz , phys . lett .   * b174 * ( 1986 ) 331 d.  buchholz , m.  porrmann and u.  stein , phys . * b267 * ( 1991 ) 377 d.  buchholz , on the manifestations of particles , p. 177 in : _ mathematical physics towards the 21st century _ , r.n . sen and a.  gersten eds . , ben gurion university of the negev press 1994 d.  buchholz , ( 1990 ) 631 m.  porrmann , the concept of particle weights in local quantum field theory , phd  thesis , universitt gttingen 2000 h.  araki and r.  haag , ( 1967 ) 77 d.  buchholz and m.  porrmann , ann . henri poincar * 52 * ( 1990 ) 237 d.  iagolnitzer , _ scattering in quantum field theories : the axiomatic and constructive approaches _ , princeton university press 1993 e.  witten , adv . * 2 * ( 1998 ) 253 m.  bertola , j.  bros , u.  moschella and r. schaeffer , ads / cft correspondence for n point functions , preprint hep - th/9908140 k .- h . rehren , ann . henri poincar * 1 * ( 2000 ) 607 d.  buchholz , j.  mund and s.j . summers , transplantation of local nets and geometric modular action on robertson  walker space  times , preprint hep - th/0011237 h.w . wiesbrock , ( 1993 ) 83 ;   erratum : ( 1997 ) 683 h.w . wiesbrock , ( 1997 ) 203 h.w . wiesbrock , ( 1998 ) 269 r.  khler and h.w . wiesbrock , modular theory and the reconstruction of four  dimensional quantum field theories , preprint , to appear in d.  buchholz , c.  dantoni and k.  fredenhagen , ( 1987 ) 123 j.j . bisognano and e.h . wichmann , ( 1975 ) 985 ; ( 1976 ) 303 h.j . borchers , ( 2000 ) 3604
algebraic quantum field theory is an approach to relativistic quantum physics , notably the theory of elementary particles , which complements other modern developments in this field . 
 it is particularly powerful for structural analysis but has also proven to be useful in the rigorous treatment of models . in this contribution a non  technical survey is given with emphasis on interesting recent developments and future perspectives . 
 topics covered are the relation between the algebraic approach and conventional quantum field theory , its significance for the resolution of conceptual problems ( such as the revision of the particle concept ) and its role in the characterization and possibly also construction of quantum field theories with the help of modular theory . 
 the algebraic approach has also shed new light on the treatment of quantum field theories on curved spacetime and made contact with recent developments in string theory ( algebraic holography ) .