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Thoughts on the synthesis of quantum physics and general relativity and the rôle of space-time
Nuclear PhysicsB (Proc. Suppl.) 18B (1990) 135-140
135
North-Holland
T h o u g h t s on the Synthesis o f Q u a n t u m Physics and General Relativity and the R61e o f Space-Time Rudolf Hang II. Institut ffir Theoretische Physik, Universitgt Hamburg
Abstract. Following a review of the frame of local quantum physics some remarks concerning the incorporation of principles of general relativity are made. The possibility of developing a theory of events by accepting irreversibility on a fundamental level is suggested.
1. Generalities This meeting is in honor of Raymond Stora. and I find it very appropriate that the proposed subject is "theoretical physics", not "mathematical physics" or "recent results in elementary particle physics". This allows a wider perspective, starting from the recognition that theoretical physics has always had three essential ingredients which, for want of a better short expression, may be called philosophy, mathematics, phenomenology, and none can be ignored for a longer period without detrimental consequences. By philosophy I mean the formation of concepts, resulting in a language, abstracted from qualitative features of our experience and created by intuition: Space. time, matter, forces, particles, fields, temperature, causality, complementarity, Lagrangean, Smatrix, entropy, gauge invariance ..., together with the surrounding syntax of principles and recipes, partly fitting together, others not fully integrated or even contradictory. To become a theory, the conceptual structure must have a counterpart in a mathematical structure, and it must be equipped with a "physical interpretation" prox~ding the bridge to observed phenomena. The theory may be completely rigid or it may be flexible, needing further specification which can be supplied by new experimental evidence. If the degree of flexibility of the scheme is very large one may prefer to call it a frame rather than a theory. Classical mechanics is such a frame as long as the types of forces are not specified. I shall sketch in the next section the frame which I consider the best one existing at present. It may
be called local quantum physics because it results in a natural way by the fusion of the (classical} principle of locality at the level of special relativity with the concepts and mathematical structure of quant u m physics. It is believed that quantum field theories like QED, QCD, the standard model are exampies of specific theories within this frame. Possibly other models; not constructed from fields, may be contained. There remain some dark spots, and, after sixty years of the agonizing history, of quantum field theory, it is still conceivable that these dark spots veil a fundamental internal incompatibility. On the positive side on can say that we do have an "almost theory" which, as far as it can be handled, is in agreement with all known facts in experimental high energy physics, and we have a frame which is flexible enough to adjust to new experimental results. In fact, it is hardly conceivable that any experimental result in high energy physics found in the next decades could be incompatible with this frame. Nevertheless there is a wide spread feeling among theorists that a radical change of the paradigma is imminent. I share this feeling but, trying to rationalize it, its roots appear to be "theory internal ~. i.e. not forced by recently discovered phenomena but by a long standing dissatisfaction with our acceptance of space-time as primarily given, with the difficulty of incorporating the tenets of general relativity in the frame, with uneasiness about the depth of our understanding of quantum physics, and. last but not least, with the infiltration of modern mathematics such as algebraic topology into the consciousness of theoretical physicists.
0920-5632/91/$3.50 © Elsevier Science Publishers B.V. (North-Holland)
R. Haag/ Synthesis of quantum physics and general reMth:'ity
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2. Local q u a n t u m p h y s i c s We accept space-time as given and adequately represented by 4-dimensional Minkowski space, denoted by Ad. and equipped with its causal (and metric) structure which is unaffected by the physical state. W'e assume that we can place detectors into spacetime and interpret their response ideally as an event confined to a specified, finitely extended open region of .A/I. From the standard formalism of quantum physics we take it that such a detector should be mathematically represented by an element of an involutive algebra (*-algebra), that the mathematical representers of general observables can be algebraically constructed from these simple ones, and that compatibility of observables corresponds to the commutativity of their representers. Furthermore, we assume that detectors operating in causally disjoint regions of space-time are compatible. We observe the following parallelism between regions in Minkowski space and algebras of bounded operators acting on a Hilbert space. Let (9 be an open region. Its causal complement O' shall be the set of all points in A4 which are spacelike to all points of O. The causal completion of O is defined as the set O", the causal complement of the causal complement of O. One finds that the set K of causally complete regions in .M is an orthocomplemented la.tt~ce. With Ki E 1C we have K'¢_ K :
relativity is given by a m a p from. the causally complete regions in .£4 into the set of yon Neumann subalgebras of B(7-/) K E ~ ---~ n ( K ) c
We call the collection {R(K)}, K E /C a net of local algebras. The vectors in 7-I represent in the familiar way physical states (expectation functionals for the local observables). Some further properties of the net, restricting the scheme, are needed to ensure that we get a physically reasonable theory. Since there is no point here in getting involved in details I list the ones which are essentially used in a form which is not optimal. i) There is a unitary representation of the translation group
where a denotes a translation 4-vector. has
One
V ( o ) R ( K ) V - l ( o ) = R(K + a) and requires that the simultaneous spectrum of the generators Pg is confined to the forward cone ("positive energy requirement" ). ii) There is a translation invariant ground state, the vacuum. Its state vector Q satisfies
K"=K,
PvQ =0.
K , ~, A-~ - K, n h"~ = (K; v K ~ ) ' .
where t2 and n denote the ordinary set theoretic union and intersection. On the other hand let B(7-/) denote the set. of all bounded operators acting on the Hilbert space 7"/, and S any subset of B(7"/). The complement of S, defined as the commutant S' (the set of all bounded operators commuting with every element of S) is a yon N e u m a n n algebra. We may call S" the causal completion of S. The set of yon Neumann algebras A" is an orthocomplemented lattice. If Ri E d then
R:'=R,.
R1 v R2 --- (R1 U R2)". R, A R2 - R, n Rz = (R', v R~
(1)
respecting the operations v,/~ and '
K3 V K2 ~ (KI t..JK2)",
R; 6 A/';
A~
)'.
The preceding discussion suggests that a quantum theory incorporating the locality principle of special
iii) All R ( K ) are isomorphic as yon Neumann algebras to the (unique) hyp':Lfinite factor of type
III~ . REMARK 1. ~:e may claim that a map (1) with the properties listed suffices to define a theory including its physical interpretation (see remark 3). The hard part is to construct such a map. We have only the rudiments of a classification of possibilities (see remark 2). In spite of the simple appearance the requirements are very strong. They demand for instance the existence of a "hyperbolic equation of motion": Let {Ki} be a covering of a neighborhood of part of the space-like plane t = 0 and K be contained in the causal completion of u K i , then K is contained in the algebra generated by the Ki. REMARK 2. The comparison with conventional q u a n t u m field theory is the following. The vectors
R. Hang~Synthesis of quantum physics and genera/relativity of ?f describe t h e " v a c u u m sector" of the theory, i.e. states without a n y charges. The elements of R{K) are the gauge invariant quantities which can be cons t r u c t e d from the fields smeared out with test functions whose s u p p o r t lies in K . The net R contains nevertheless all i n f o r m a t i o n about the charge structure, essentially because from any charged state we can make an u n c h a r g e d one with almost the same local a p p e a r a n c e by adding a compensating charge sufficiently far away. To uncover the charge structure it is necessary to consider representations 7ri of the net R which are globally not unitarily equivalent to the vacuum sector though for each finite region K ( a n d also for infinite regions with a sufficiently large causal c o m p l e m e n t ) l r ( B ( K ) ) is u u i t a r i l y equivalent to R(K). The set of equivalence classes of such representations corresponds to the set of charge quant u m numbers. T h e y are related to the existence of non inner m o r p h i s m s of the net with certain localization properties. Quite a lot is known about the possibilites which m a y arise 1,2,a, a n d this m a y be regarded as a first step towards a classification of such theories. Physical consequences are the composition law of charges, charge conjugation, exchange sylrmxetry ( " s t a t i s t i c s " ) for identical charges, and the relation to gauge groups. REMARK 3. Concerning the physical interpretation we do not need to know which o p e r a t o r in B(Tf) corresponds to the observable measured by a particular device, c o n s t r u c t e d according to a specific blue print. It suffices t h a t we can reduce questions to geometric configurations of detectors in Minkowski space (coincidence a r r a n g e m e n t s ) and that an individual detector can be characterized as projection operator annihilating the v a c u u m and belonging "essentially" to the algebra R I O ) . where C0 is the macroscopic region in space-time where it operates. (Footnote: the two characterizing properties can not be precisely satisfied simultaneously but can be adequately app r o x i m a t e d . ) Tlfis allows the definition of the notion of "particle". A single particle state is a state which can yield a signal in a single detector but not in a coincidence arrangement of two detectors placed sufficiently far a p a r t at any time. One can then express collision cross sections of particles in terms of (limits of) v a c u u m expectation values of products of localized detectors. No reference to any correspondence between particle types and fields is needed. The above definition of "particle" is more general t h a n W i g n e r ' s definition associating a particle with an irreducible representation of the covering group of
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t h e P o i n c a r $ group. The two definitions are equivalent in the case of a massive theory whereas in a t h e o r y without mass gap t h e mass of a charged particle will not be in general a discrete eigenvalue of P~ ("infraparticle"). T h e r e remain three o u t s t a n d i n g problems in this frame. T h e first is to show t h a t there exist single particle states in the theory, and. related to this. that all states appearing in the representations or, can he characterized in terms of incoming particle configurations. This has been almost achieved in recent work of D. Buchholz 4. T h e crucial input is the hyperfiniteness of the local algebras. The second task is to p u s h on the classification theory, in p a r t i c u l a r to u n d e r s t a n d what is the counterpart of a minimally coupled gauge theory in this language. Here I a m optimistic t h a t essential progress can be made soon so t h a t specific theories within the frame can be characterized. The t h i r d is the existence problem of a nontrix4al theory. It is conceivable that due to the fact t h a t we still use sharply bounded regions there r e m a i n unsurmountable ultraviolet problems though this is disclaimed as unlikely according to the knowledge acquired in constructive f i d d theory, and renorrealization group arguments. There is. of course, a fourth task, nanlely to develop sufficiently powerful a p p r o x i m a t i o n methods so that a detailed comparison with experiments becomes possible. All this will take t i m e but the prognosis looks favorable. 3. T h e c h a l l e n g e o f g e n e r a l r e l a t i v i t y Classical general relativity still considers spacet i m e as a 4-dimensional manifold. Its causal structure and even its topology, are. however, not given a priori but depend on the physical situation {the state). In a corresponding quantum theory, we m a y still t r y to maintain the association of algebras to space-time regions in the small. Otherwise put. we have t h e task of defining a manifold with an event s t r u c t u r e over it. Like in manifold theory, one starts from coverings. An element of a covering is now a (slna]] t region of R 4, equipped with a net of algeb r a s for its subregions. In the limit of arbitrarily small size this may be called a germ of the theory. T h e r e are two distinct problems: 1) the characterization of germs. 2) the building up of the theory from local information, i.e. from the elements of a covering a n d transition o p e r a t o r s (generalizing the transition functions of the manifold case).
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R. Hang~Synthesis of quantum physics and general relativity
Some prelinfinary studies of both problems have been attempted 5.6. but these attempts do not carry far yet. Therefore I linfit myself to two remarks here. i) In a semiclassical treatment of gravitation the metric and afline connection can be related to scaling limits of the quantum state. This remark is useful in the discussion of Hawking radiation 5.7 it) In full q u a n t m n gravity we should not expect that the local algebras can be chosen as yon N e u m a n n algebras. There is no causal complement of a region if the metric fluctuates. Therefore the commutant of a local algebra should be trivial. Alternatively speaking, if the algebra of a region is represented by .~n operator algebra acting on a Hilbert space, then it must be weakly dense in B(7"/) and closed in a much finer topology. This is not as pathological as it looks at first sight. The same situation prevails in non-relativistic q u a n t u m field theory for regions of finite extension in space and time (due to the inlinfited propagation velocity of signals ). 4. S p a c e - t i m e as a d e r i v e d c o n c e p t
All existing (useful) theories start from space-time as a primary concept. It provides the arena for physics, or, in another picture, the vessel into which matter is put and events take place. General relativity recognizes that the vessel may be deformed by its contents but keeps it as a 4-dimensional continuum. W h y four dimensions? Why a continuum? Should one regard degrees of freedom associated with charges as coordinates of space on the same footing with the others (Klein-Kaluza type theories), or consider a bundle with a physically distinguished projection on a 4-dimensional base space? Generalizing the scheme of section 2. one may consider abstract nets: an algebra with a distinguished system of subalgebras partially ordered by inclusion. As U. Bannier 8 has shown, this defines under rather natural and mild conditions a ttausdorff space such that the distinguished subalgebras are associated to the open sets of this space. A conceptual criticism of the acceptance of space as a vessel for physics was originally directed by Leibniz against Newton's concept of absolute space, but remains pertin~!t ~_gain today. Leibniz maintains that space is just a system of relations. In particular, space without matter is meaningless. B.Russel :~
comments: "Although both physicists and philosophers tended more and more to take Leibniz's view rather than Newton's, the technique of mathematical physics continued to be Newtonian." "When we deny Newton's theory of absolute space, while continuing to use what we call points in mathematical physics, our procedure is only justified if there is a structural definition of point and of particular points in the theory." Classical field theory replaces Newton's absolute space by a continuous dynamical system, the "ether", and though the term "ether" has come into miscredit with the development of relativity theories we should remember that at the root of the locality principle lies the idea that space-time points (all of them) are the carriers of physical reality. Thus Einstein, always in quest of reality, advocated the rehabilitation of the term "ether" in 1920. Atomic physics and quantum theory changes this picture, and Leibniz's criticism becomes acute again. Of course we would be happy to reduce space and time to the r61e of ordering relations. But. ordering of what? This touches the problem of "reality" in q u a n t u m physics. In the generally accepted epistemological interpretation of quantum theory we must divide the universe in two parts: The "observed system" E and the observer with his equipment M. The singling out of a system E constitutes a (to some degree arbitrary) mental decision on the side of the observer but it was emphasized by Heisenberg that, while the cut can be shifted to some extend, it cannot be avoided. A q u a n t u m theory of the universe is a contradiction in itself. The task of the theory is to relate observed phenomena, and these are interactions between E and a~1. Without an observer the scheme is empty. But the term "observer" suggests a h u m a n being. At least the observer nmst be equipped with a m i n d and consciousness, and these are aspects not treated in physical theory. Thus if we rigorously foUow the accepted epistemology of quant u m theory we cannot talk about events in the "real world" (beyond consciousness): in particular, "history of the universe" belongs to metaphysics. I have drawn the picture in such harsh lines to bring out a discrepancy between the instinctive feeling of physicists (or at least a large number of them) and the professed epistemological position. One may say that this epistemology is just, the "orthodox" position, a minimal interpretation, avoiding everything which is not on save ground. This may be so but I
R. Hang~Synthesis of quantum physics and genera/relativity have not seen any alternative, complete and and consistent interpretation of quantum theory with one possible reservation. I have suppressed one important aspect, namely the distinction between "microscopic" and "macroscopic" physics. One may indeed argue that an observation does not need consciousness but only the transmutation to a macroscopitally recorded feature which is factual, irreversible due to the second law of thermodynamics and that this law can be derived from the quantum theoretical description of the microscopic world by going over to a "coarse grained" description as done in statistical mechanics. But can the second law of thermodynamics, the irreversibility of all processes in nature, be really derived from reversible fundamental laws? Certainly not without additional qualitative assumptions about initial conditions. In laboratory situations these are naturally set by the experimenter who purposely creates a "thermodynamically improbable" situation after arriving in his lab. On the cosmic scale or even in meteorology the experimenter has to be replaced by God or the "given state of the world", ultimately the "big bang". It appears to me unsatisfactory to anker the explanation of one of the most basic facts of experience, the "arrow of time", the difference between past and future, of fact and potentiality on such grounds. We should question ourselves why we believe that the fundamental laws cannot incorporate the arrow of time, why we think that irreversibility belongs to the domain of statistical mechanics and only arises by "coarse graining". I believe this is an undue prejudice. In an indeterministic theory we may accept as a fundamental feature also the spontaneous appearance of events, the irreversible transmutation from a potentiality to a fact. The notion of event replaces then the marked point in space-time, the notion of causal ties between events corresponds to the ordering rdations in Leibniz's understanding of the r61e of space. Some arguments in favour of such an approach together with indications of how it may be mathematically implemented and how it relates to existing theory in the regime of low density, high energy, are given in 10. Let. me emphasize here that the unity of space and time is essential in this picture. We have learned this as a lesson of relativity theory, but is is also suggested by the reality problem in quantum theory. A marked point in (3-dimensional) space would (ideally) be physically realized by the position of a paxtide at a particular time. Quantum mechanics tells us that no objective reality can be
139
attached to this concept. A "position measurement'of an electron dose not reveal a property which the dectron possessed prior to the measurement, but a property of the interaction process between the electron with a molecule in the detector. We axe not forbidden to attribute objective reality to the approximate position in space-time of this process. There is no "world line" associated with the particle but an approximate space-time region with an event. A particle is the simplest causal tie between two events. Let me close here and return to the cause of this colloquium, the 60 th birthday of Raymond Stora. The reason why I followed the invitation with great pleasure has something to do with the tribute given by Einstein to Planck, a tribute which I learned f~om a quotation given by Nussenzweig on the occasion of the 70th birthday of Guido Beck: "In the temple of science axe many man.ions, and various indeed are they that dwell therein and the motives that have led them thither. Many take to science out of a joyful sense of superior intellecttml power; science is their own special sport to which they look for vivid experience and the sath-faction of ambition; many others axe to be found in the temple who have offered the products of their brains on this altar for purely utilitarian purposes. Were an angel of the Lord to come and drive all the people belonging to these two categories out of the temple, the assemblage would be seriously depleted, but there would still be some men, of both present and past times, left inside... Our Planck is one of them, and that is why we love him." I feel that the spirit of this tribute must be remembered. Raymond is, for me. one of the not so frequent manifestations of this spirit in our times.
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R.
ltaag/Synthesis of quantum physics and genera/relativity
References
1. S.Doplicher, R.Haag and J.E.Roberts, Commun.Math.Phys. 15, 173 (1969): 23, 199 (1971); 35, 49 (1974) 2. D. Buchholz and K.Fredenhagen. Commun.Math.Phys. 84, 1 (1982) 3. S.Doplicher and J.E.Roberts, Annals of Math. 130.75 (1989); J.Functional Analysis 74, 96 (1987) 4. D.Buchholz, private communication 5. K.Fredenhagen and R.Haag, Commun.Math.Phys. 108, 91 (1987) 6. R.Haag, in "Quantum Theories and Geometry", M.Cahen and M.Flato, eds., Kluwer Acad.Publ., Doordrecht 1988 7. R.Haag, H.Narnhofer and U.Stein, Commun.Math.Phys. 94, 219 (1984) 8. U.Bannier, Thesis, Hamburg Univ. 1987 9. B.Russell, "Human Knowledge, its scope and limits", George Allen and Unwin Publ., London 1976 10. R.Haag, "Fundamental irreversibility and the concepts of events", Commun.Math.Phys., in print.
135
North-Holland
T h o u g h t s on the Synthesis o f Q u a n t u m Physics and General Relativity and the R61e o f Space-Time Rudolf Hang II. Institut ffir Theoretische Physik, Universitgt Hamburg
Abstract. Following a review of the frame of local quantum physics some remarks concerning the incorporation of principles of general relativity are made. The possibility of developing a theory of events by accepting irreversibility on a fundamental level is suggested.
1. Generalities This meeting is in honor of Raymond Stora. and I find it very appropriate that the proposed subject is "theoretical physics", not "mathematical physics" or "recent results in elementary particle physics". This allows a wider perspective, starting from the recognition that theoretical physics has always had three essential ingredients which, for want of a better short expression, may be called philosophy, mathematics, phenomenology, and none can be ignored for a longer period without detrimental consequences. By philosophy I mean the formation of concepts, resulting in a language, abstracted from qualitative features of our experience and created by intuition: Space. time, matter, forces, particles, fields, temperature, causality, complementarity, Lagrangean, Smatrix, entropy, gauge invariance ..., together with the surrounding syntax of principles and recipes, partly fitting together, others not fully integrated or even contradictory. To become a theory, the conceptual structure must have a counterpart in a mathematical structure, and it must be equipped with a "physical interpretation" prox~ding the bridge to observed phenomena. The theory may be completely rigid or it may be flexible, needing further specification which can be supplied by new experimental evidence. If the degree of flexibility of the scheme is very large one may prefer to call it a frame rather than a theory. Classical mechanics is such a frame as long as the types of forces are not specified. I shall sketch in the next section the frame which I consider the best one existing at present. It may
be called local quantum physics because it results in a natural way by the fusion of the (classical} principle of locality at the level of special relativity with the concepts and mathematical structure of quant u m physics. It is believed that quantum field theories like QED, QCD, the standard model are exampies of specific theories within this frame. Possibly other models; not constructed from fields, may be contained. There remain some dark spots, and, after sixty years of the agonizing history, of quantum field theory, it is still conceivable that these dark spots veil a fundamental internal incompatibility. On the positive side on can say that we do have an "almost theory" which, as far as it can be handled, is in agreement with all known facts in experimental high energy physics, and we have a frame which is flexible enough to adjust to new experimental results. In fact, it is hardly conceivable that any experimental result in high energy physics found in the next decades could be incompatible with this frame. Nevertheless there is a wide spread feeling among theorists that a radical change of the paradigma is imminent. I share this feeling but, trying to rationalize it, its roots appear to be "theory internal ~. i.e. not forced by recently discovered phenomena but by a long standing dissatisfaction with our acceptance of space-time as primarily given, with the difficulty of incorporating the tenets of general relativity in the frame, with uneasiness about the depth of our understanding of quantum physics, and. last but not least, with the infiltration of modern mathematics such as algebraic topology into the consciousness of theoretical physicists.
0920-5632/91/$3.50 © Elsevier Science Publishers B.V. (North-Holland)
R. Haag/ Synthesis of quantum physics and general reMth:'ity
136
2. Local q u a n t u m p h y s i c s We accept space-time as given and adequately represented by 4-dimensional Minkowski space, denoted by Ad. and equipped with its causal (and metric) structure which is unaffected by the physical state. W'e assume that we can place detectors into spacetime and interpret their response ideally as an event confined to a specified, finitely extended open region of .A/I. From the standard formalism of quantum physics we take it that such a detector should be mathematically represented by an element of an involutive algebra (*-algebra), that the mathematical representers of general observables can be algebraically constructed from these simple ones, and that compatibility of observables corresponds to the commutativity of their representers. Furthermore, we assume that detectors operating in causally disjoint regions of space-time are compatible. We observe the following parallelism between regions in Minkowski space and algebras of bounded operators acting on a Hilbert space. Let (9 be an open region. Its causal complement O' shall be the set of all points in A4 which are spacelike to all points of O. The causal completion of O is defined as the set O", the causal complement of the causal complement of O. One finds that the set K of causally complete regions in .M is an orthocomplemented la.tt~ce. With Ki E 1C we have K'¢_ K :
relativity is given by a m a p from. the causally complete regions in .£4 into the set of yon Neumann subalgebras of B(7-/) K E ~ ---~ n ( K ) c
We call the collection {R(K)}, K E /C a net of local algebras. The vectors in 7-I represent in the familiar way physical states (expectation functionals for the local observables). Some further properties of the net, restricting the scheme, are needed to ensure that we get a physically reasonable theory. Since there is no point here in getting involved in details I list the ones which are essentially used in a form which is not optimal. i) There is a unitary representation of the translation group
where a denotes a translation 4-vector. has
One
V ( o ) R ( K ) V - l ( o ) = R(K + a) and requires that the simultaneous spectrum of the generators Pg is confined to the forward cone ("positive energy requirement" ). ii) There is a translation invariant ground state, the vacuum. Its state vector Q satisfies
K"=K,
PvQ =0.
K , ~, A-~ - K, n h"~ = (K; v K ~ ) ' .
where t2 and n denote the ordinary set theoretic union and intersection. On the other hand let B(7-/) denote the set. of all bounded operators acting on the Hilbert space 7"/, and S any subset of B(7"/). The complement of S, defined as the commutant S' (the set of all bounded operators commuting with every element of S) is a yon N e u m a n n algebra. We may call S" the causal completion of S. The set of yon Neumann algebras A" is an orthocomplemented lattice. If Ri E d then
R:'=R,.
R1 v R2 --- (R1 U R2)". R, A R2 - R, n Rz = (R', v R~
(1)
respecting the operations v,/~ and '
K3 V K2 ~ (KI t..JK2)",
R; 6 A/';
A~
)'.
The preceding discussion suggests that a quantum theory incorporating the locality principle of special
iii) All R ( K ) are isomorphic as yon Neumann algebras to the (unique) hyp':Lfinite factor of type
III~ . REMARK 1. ~:e may claim that a map (1) with the properties listed suffices to define a theory including its physical interpretation (see remark 3). The hard part is to construct such a map. We have only the rudiments of a classification of possibilities (see remark 2). In spite of the simple appearance the requirements are very strong. They demand for instance the existence of a "hyperbolic equation of motion": Let {Ki} be a covering of a neighborhood of part of the space-like plane t = 0 and K be contained in the causal completion of u K i , then K is contained in the algebra generated by the Ki. REMARK 2. The comparison with conventional q u a n t u m field theory is the following. The vectors
R. Hang~Synthesis of quantum physics and genera/relativity of ?f describe t h e " v a c u u m sector" of the theory, i.e. states without a n y charges. The elements of R{K) are the gauge invariant quantities which can be cons t r u c t e d from the fields smeared out with test functions whose s u p p o r t lies in K . The net R contains nevertheless all i n f o r m a t i o n about the charge structure, essentially because from any charged state we can make an u n c h a r g e d one with almost the same local a p p e a r a n c e by adding a compensating charge sufficiently far away. To uncover the charge structure it is necessary to consider representations 7ri of the net R which are globally not unitarily equivalent to the vacuum sector though for each finite region K ( a n d also for infinite regions with a sufficiently large causal c o m p l e m e n t ) l r ( B ( K ) ) is u u i t a r i l y equivalent to R(K). The set of equivalence classes of such representations corresponds to the set of charge quant u m numbers. T h e y are related to the existence of non inner m o r p h i s m s of the net with certain localization properties. Quite a lot is known about the possibilites which m a y arise 1,2,a, a n d this m a y be regarded as a first step towards a classification of such theories. Physical consequences are the composition law of charges, charge conjugation, exchange sylrmxetry ( " s t a t i s t i c s " ) for identical charges, and the relation to gauge groups. REMARK 3. Concerning the physical interpretation we do not need to know which o p e r a t o r in B(Tf) corresponds to the observable measured by a particular device, c o n s t r u c t e d according to a specific blue print. It suffices t h a t we can reduce questions to geometric configurations of detectors in Minkowski space (coincidence a r r a n g e m e n t s ) and that an individual detector can be characterized as projection operator annihilating the v a c u u m and belonging "essentially" to the algebra R I O ) . where C0 is the macroscopic region in space-time where it operates. (Footnote: the two characterizing properties can not be precisely satisfied simultaneously but can be adequately app r o x i m a t e d . ) Tlfis allows the definition of the notion of "particle". A single particle state is a state which can yield a signal in a single detector but not in a coincidence arrangement of two detectors placed sufficiently far a p a r t at any time. One can then express collision cross sections of particles in terms of (limits of) v a c u u m expectation values of products of localized detectors. No reference to any correspondence between particle types and fields is needed. The above definition of "particle" is more general t h a n W i g n e r ' s definition associating a particle with an irreducible representation of the covering group of
137
t h e P o i n c a r $ group. The two definitions are equivalent in the case of a massive theory whereas in a t h e o r y without mass gap t h e mass of a charged particle will not be in general a discrete eigenvalue of P~ ("infraparticle"). T h e r e remain three o u t s t a n d i n g problems in this frame. T h e first is to show t h a t there exist single particle states in the theory, and. related to this. that all states appearing in the representations or, can he characterized in terms of incoming particle configurations. This has been almost achieved in recent work of D. Buchholz 4. T h e crucial input is the hyperfiniteness of the local algebras. The second task is to p u s h on the classification theory, in p a r t i c u l a r to u n d e r s t a n d what is the counterpart of a minimally coupled gauge theory in this language. Here I a m optimistic t h a t essential progress can be made soon so t h a t specific theories within the frame can be characterized. The t h i r d is the existence problem of a nontrix4al theory. It is conceivable that due to the fact t h a t we still use sharply bounded regions there r e m a i n unsurmountable ultraviolet problems though this is disclaimed as unlikely according to the knowledge acquired in constructive f i d d theory, and renorrealization group arguments. There is. of course, a fourth task, nanlely to develop sufficiently powerful a p p r o x i m a t i o n methods so that a detailed comparison with experiments becomes possible. All this will take t i m e but the prognosis looks favorable. 3. T h e c h a l l e n g e o f g e n e r a l r e l a t i v i t y Classical general relativity still considers spacet i m e as a 4-dimensional manifold. Its causal structure and even its topology, are. however, not given a priori but depend on the physical situation {the state). In a corresponding quantum theory, we m a y still t r y to maintain the association of algebras to space-time regions in the small. Otherwise put. we have t h e task of defining a manifold with an event s t r u c t u r e over it. Like in manifold theory, one starts from coverings. An element of a covering is now a (slna]] t region of R 4, equipped with a net of algeb r a s for its subregions. In the limit of arbitrarily small size this may be called a germ of the theory. T h e r e are two distinct problems: 1) the characterization of germs. 2) the building up of the theory from local information, i.e. from the elements of a covering a n d transition o p e r a t o r s (generalizing the transition functions of the manifold case).
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R. Hang~Synthesis of quantum physics and general relativity
Some prelinfinary studies of both problems have been attempted 5.6. but these attempts do not carry far yet. Therefore I linfit myself to two remarks here. i) In a semiclassical treatment of gravitation the metric and afline connection can be related to scaling limits of the quantum state. This remark is useful in the discussion of Hawking radiation 5.7 it) In full q u a n t m n gravity we should not expect that the local algebras can be chosen as yon N e u m a n n algebras. There is no causal complement of a region if the metric fluctuates. Therefore the commutant of a local algebra should be trivial. Alternatively speaking, if the algebra of a region is represented by .~n operator algebra acting on a Hilbert space, then it must be weakly dense in B(7"/) and closed in a much finer topology. This is not as pathological as it looks at first sight. The same situation prevails in non-relativistic q u a n t u m field theory for regions of finite extension in space and time (due to the inlinfited propagation velocity of signals ). 4. S p a c e - t i m e as a d e r i v e d c o n c e p t
All existing (useful) theories start from space-time as a primary concept. It provides the arena for physics, or, in another picture, the vessel into which matter is put and events take place. General relativity recognizes that the vessel may be deformed by its contents but keeps it as a 4-dimensional continuum. W h y four dimensions? Why a continuum? Should one regard degrees of freedom associated with charges as coordinates of space on the same footing with the others (Klein-Kaluza type theories), or consider a bundle with a physically distinguished projection on a 4-dimensional base space? Generalizing the scheme of section 2. one may consider abstract nets: an algebra with a distinguished system of subalgebras partially ordered by inclusion. As U. Bannier 8 has shown, this defines under rather natural and mild conditions a ttausdorff space such that the distinguished subalgebras are associated to the open sets of this space. A conceptual criticism of the acceptance of space as a vessel for physics was originally directed by Leibniz against Newton's concept of absolute space, but remains pertin~!t ~_gain today. Leibniz maintains that space is just a system of relations. In particular, space without matter is meaningless. B.Russel :~
comments: "Although both physicists and philosophers tended more and more to take Leibniz's view rather than Newton's, the technique of mathematical physics continued to be Newtonian." "When we deny Newton's theory of absolute space, while continuing to use what we call points in mathematical physics, our procedure is only justified if there is a structural definition of point and of particular points in the theory." Classical field theory replaces Newton's absolute space by a continuous dynamical system, the "ether", and though the term "ether" has come into miscredit with the development of relativity theories we should remember that at the root of the locality principle lies the idea that space-time points (all of them) are the carriers of physical reality. Thus Einstein, always in quest of reality, advocated the rehabilitation of the term "ether" in 1920. Atomic physics and quantum theory changes this picture, and Leibniz's criticism becomes acute again. Of course we would be happy to reduce space and time to the r61e of ordering relations. But. ordering of what? This touches the problem of "reality" in q u a n t u m physics. In the generally accepted epistemological interpretation of quantum theory we must divide the universe in two parts: The "observed system" E and the observer with his equipment M. The singling out of a system E constitutes a (to some degree arbitrary) mental decision on the side of the observer but it was emphasized by Heisenberg that, while the cut can be shifted to some extend, it cannot be avoided. A q u a n t u m theory of the universe is a contradiction in itself. The task of the theory is to relate observed phenomena, and these are interactions between E and a~1. Without an observer the scheme is empty. But the term "observer" suggests a h u m a n being. At least the observer nmst be equipped with a m i n d and consciousness, and these are aspects not treated in physical theory. Thus if we rigorously foUow the accepted epistemology of quant u m theory we cannot talk about events in the "real world" (beyond consciousness): in particular, "history of the universe" belongs to metaphysics. I have drawn the picture in such harsh lines to bring out a discrepancy between the instinctive feeling of physicists (or at least a large number of them) and the professed epistemological position. One may say that this epistemology is just, the "orthodox" position, a minimal interpretation, avoiding everything which is not on save ground. This may be so but I
R. Hang~Synthesis of quantum physics and genera/relativity have not seen any alternative, complete and and consistent interpretation of quantum theory with one possible reservation. I have suppressed one important aspect, namely the distinction between "microscopic" and "macroscopic" physics. One may indeed argue that an observation does not need consciousness but only the transmutation to a macroscopitally recorded feature which is factual, irreversible due to the second law of thermodynamics and that this law can be derived from the quantum theoretical description of the microscopic world by going over to a "coarse grained" description as done in statistical mechanics. But can the second law of thermodynamics, the irreversibility of all processes in nature, be really derived from reversible fundamental laws? Certainly not without additional qualitative assumptions about initial conditions. In laboratory situations these are naturally set by the experimenter who purposely creates a "thermodynamically improbable" situation after arriving in his lab. On the cosmic scale or even in meteorology the experimenter has to be replaced by God or the "given state of the world", ultimately the "big bang". It appears to me unsatisfactory to anker the explanation of one of the most basic facts of experience, the "arrow of time", the difference between past and future, of fact and potentiality on such grounds. We should question ourselves why we believe that the fundamental laws cannot incorporate the arrow of time, why we think that irreversibility belongs to the domain of statistical mechanics and only arises by "coarse graining". I believe this is an undue prejudice. In an indeterministic theory we may accept as a fundamental feature also the spontaneous appearance of events, the irreversible transmutation from a potentiality to a fact. The notion of event replaces then the marked point in space-time, the notion of causal ties between events corresponds to the ordering rdations in Leibniz's understanding of the r61e of space. Some arguments in favour of such an approach together with indications of how it may be mathematically implemented and how it relates to existing theory in the regime of low density, high energy, are given in 10. Let. me emphasize here that the unity of space and time is essential in this picture. We have learned this as a lesson of relativity theory, but is is also suggested by the reality problem in quantum theory. A marked point in (3-dimensional) space would (ideally) be physically realized by the position of a paxtide at a particular time. Quantum mechanics tells us that no objective reality can be
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attached to this concept. A "position measurement'of an electron dose not reveal a property which the dectron possessed prior to the measurement, but a property of the interaction process between the electron with a molecule in the detector. We axe not forbidden to attribute objective reality to the approximate position in space-time of this process. There is no "world line" associated with the particle but an approximate space-time region with an event. A particle is the simplest causal tie between two events. Let me close here and return to the cause of this colloquium, the 60 th birthday of Raymond Stora. The reason why I followed the invitation with great pleasure has something to do with the tribute given by Einstein to Planck, a tribute which I learned f~om a quotation given by Nussenzweig on the occasion of the 70th birthday of Guido Beck: "In the temple of science axe many man.ions, and various indeed are they that dwell therein and the motives that have led them thither. Many take to science out of a joyful sense of superior intellecttml power; science is their own special sport to which they look for vivid experience and the sath-faction of ambition; many others axe to be found in the temple who have offered the products of their brains on this altar for purely utilitarian purposes. Were an angel of the Lord to come and drive all the people belonging to these two categories out of the temple, the assemblage would be seriously depleted, but there would still be some men, of both present and past times, left inside... Our Planck is one of them, and that is why we love him." I feel that the spirit of this tribute must be remembered. Raymond is, for me. one of the not so frequent manifestations of this spirit in our times.
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References
1. S.Doplicher, R.Haag and J.E.Roberts, Commun.Math.Phys. 15, 173 (1969): 23, 199 (1971); 35, 49 (1974) 2. D. Buchholz and K.Fredenhagen. Commun.Math.Phys. 84, 1 (1982) 3. S.Doplicher and J.E.Roberts, Annals of Math. 130.75 (1989); J.Functional Analysis 74, 96 (1987) 4. D.Buchholz, private communication 5. K.Fredenhagen and R.Haag, Commun.Math.Phys. 108, 91 (1987) 6. R.Haag, in "Quantum Theories and Geometry", M.Cahen and M.Flato, eds., Kluwer Acad.Publ., Doordrecht 1988 7. R.Haag, H.Narnhofer and U.Stein, Commun.Math.Phys. 94, 219 (1984) 8. U.Bannier, Thesis, Hamburg Univ. 1987 9. B.Russell, "Human Knowledge, its scope and limits", George Allen and Unwin Publ., London 1976 10. R.Haag, "Fundamental irreversibility and the concepts of events", Commun.Math.Phys., in print.