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ESI The Erwin Schrodinger International Institute for Mathematical Physics Boltzmanngasse 9 A-1090 Wien, Austria Ob jects, Events and Localization Rudolf Haag Vienna, Preprint ESI 541 (1998) Supported by Federal Ministry of Science and Transport, Austria Available via http://www.esi.ac.at March 20, 1998 Objects, Events and Localization Rudolf Haagy 1 General remarks The words in the title have an intuitive meaning but not a precise one in existing theory. The two relevant aspects of Quantum Theory which pose some problems for these notions are 1) Entanglement 2) Coherence. Both arise from the superposition principle and both limit the possibility of subdividing the universe into individual parts. Entanglement means that the properties of a combined system of several objects are not (in general) describable in terms of properties of the individual objects. This manifests itself in the type of correlations found in the joint probability distributions of events in which dierent of these objects are involved. Coherence, on the other hand, has a similar eect on the separation of events. Parallel to the situation with material objects, there is some atomicity, some discreteness in the event structure. Niels Bohr speaks of this as the irreducibility of a \Quantum Process". Yet again, the division of a complex event into individual subevents is not sharply denable as long as there is a chance of obtaining an interference between the assumed subevents. Thus, if we insist on absolute precision and believe in the ultimate and unrestrained validity of the general formalism of Quantum Theory we may not be able to subdivide the universe into individual things, single out individual objects, individual quantum processes, individual facts. On the other hand we know that we can do physics and physics depends on the possibility of such subdivisions. It accepts facts and tries to relate them. Let me say right here that I do not see that anything can be gained if Expanded version of lecture given at the Max-Born-Symposium on \Quantum Future", Przieka, September 1997 yMailing address: Waldschmidtstrasse 4b, D-83727 Schliersee-Neuhaus 1 one delegates the creation of facts to impressions in the mind, to consciousness of any sort. The reliability and accuracy of contents of our own consciousness is certainly far below the one aimed at in physical experiments. The reconciliation between the holistic aspect and the need for subdivision depends on idealizations. Idealizations are invoked whenever we try to apply a specic theory to a particular situation and they are even present in the formulation of the theory itself. The concepts and the language used in a theory are adapted to a certain range of experience and we have witnessed many radical changes in the past. Personally I believe that there is no \theory of everything". Even if one disagrees with this assessment one must concede that Quantum Physics as we know it today is not the ultimate wisdom. Let me elaborate somewhat. I have not yet mentioned space and time. In the well established, successful parts of Quantum Physics the space-time continuum is assumed as a pre-given arena in which the drama of physics plays. This continuum is by denition divisible into disjoint subsets. By the development of Quantum Field Theory in the past decades it has become clear that it is the divisibility of space-time which provides the key for the physical interpretation of the formalism. In fact, no other information about the meaning of the symbols is needed besides their assignment to regions in space-time [1]. This is true on the level of the standard use of the term \observable" involving the Bohr-Heisenberg cut. On the observer side of the cut we can consider detectors, about which we know nothing beyond their placement in space-time and their ability to signal some deviation from the vacuum. What they detect can be elaborated by monitoring experiments involving only the geometry of coincidence arrangements of such detectors, provided that the theory gives us a discrete set of possible types of stable, compact objects (particles) and provided we are in a situation in which the mean density of such particles is low (i.e. there is lots of vacuum around). | If we look at the structure of the theory itself then the situation is not so satisfactory. One assumes there an assignment of mathematical symbols to arbitrarily small space-time regions. For present day high energy physics it is important that this assignment is meaningful at least down to diameters of 10? cm and this limit reects only present day knowledge. On the other hand the objects that are described are (hopefully) the physical particles, where we do not distinguish between \elementary" and \composite". In a relativistic theory there is no sharp concept of localization for a particle. Its \position" becomes meaningless below its Compton wave length. This is, for electrons and hadrons 10? resp. 10? cm i.e. by orders of magnitude larger than the above mentioned distances. E. P. Wigner, who had pointed out this limitation in 1949, considered it as an indication that Quantum Field Theory is on the wrong track. In his typical mild sarcasm he liked to say: \There are those of us who believe that there are no points. What do you think?" Clearly the small lengths mentioned do not refer to the placement of any objects in spacetime. The experimental equipment used, to which the Bohr-Heisenberg cut applies, concerns the accelerator and the detectors, not the position of particles. The accuracy 17 10 2 13 in determining the relative position of detector signals is of the order of 10? cm. If distances of 10? cm have a meaning, and I believe they have, then they must be interpreted as intrinsic extensions of a collision event, smaller by orders of magnitude than the denable localization of the particles that produce it. In particular they do not relate to the measurement of some sharply localized observable. Neither one of the particles involved in the collision can be regarded as a measuring instrument for the other. Instead we infer that there are clearly separable irreducible events. This is one reason to dissociate the notion of \event" from that of a \measuring result" and to consider it as a primary concept. An individual event, just like an individual object (say a particle) is an idealization. It may depend on the prevailing circumstances and can be made absolutely precise in the present theory only as an asymptotic notion. But it seems to be a necessary concept for the understanding of the role of space-time. It is often said that Quantum Theory is eminently successful, that its predictions are veried in countless cases and that no phenomenon has been found which contradicts it. Yet, to this day, there remains some uneasiness about its status, some disagreement concerning its interpretation. This is not restricted to crackpots. Dierent camps of eminent scientists advance widely dierent opinions. Why? Of course there are deeply rooted metaphysical beliefs which cannot be proved or disproved with the methods of physics. But if we could avoid some misunderstandings of the meaning of words, of the terminology, some consensus on what we understand and where problems remain might be reached. For this it seems to me imperative to keep several regimes which frequently enter in discussions strictly apart: 4 17 a) Nature. \The laws of nature". b) Knowledge about nature by an individual human being. c) Collective knowledge of the human species, ultimately laid down in books. Knowledge refers to the mind and I restrict it here to the human mind. This does, of course not mean that I deny faculties of knowledge to an animal but this is beside the point. Dogs and horses have not contributed to the physics literature and cats entered only in a passive role. If in discussions of Quantum Theory we use extrapolations from our human experience of mental faculties, introducing for instance such notions as universal consciousness, abstract information : : : then we enter into the realm of metaphysics. There may be some suggestive truth value in it like in a lyrical poem but this lies outside the range of competence of physics. One can say: the Schrodinger wave function (or density matrix) represents information. True enough. But information of whom and about what? It is information of a team of human observers about past facts which are relevant for the occurrence of some facts in the future. I use the word fact in a realistic sense. This may be challenged on two dierent grounds. Philosophically by disclaiming the existence of facts independent 3 of consciousness; or from Quantum Theory by referring to the diculties arising from the superposition principle as mentioned at the beginning. The latter concern the problem of divisibility of nature, the title of Heisenberg's last book: \The part and the Whole". The philosophical question has no bearing on what is actually done and achieved in physics. The physicist proceeds from the assumption that he is confronted with a great unknown, called nature, which is something beyond and apart from his knowledge. Nature is relentless in rejecting his ideas by the emergence of unexpected facts about whose reality he cannot reasonably oer any doubts. He proceeds and must proceed on the basis of an \as if" realism. His empirical basis rests on the belief in facts and the subject of his search is the relation between facts. Let me return to the three regimes mentioned above. I want to assert: A. The subject of physics is \nature" and, whatever this means precisely, it is something beyond and apart from human knowledge. B. Individual knowledge is gained by observation, typically by an experiment. In describing the set up and the result of an experiment we are bound by limits emphasized by Bohr: \We must be able to tell our friends what we have done and what we have learned." Bohr concludes from this that in the description of both the arrangement and the result we are bound to use \the language of classical physics". I consider this warning as extremely important but feel that the word \classical" is unfortunate and gave leeway to many futile discussions. It is worthwhile to look more closely at what is involved. In an experiment we control the placement and motion of some specied material bodies in space and we note nal facts which are described as some conspicuous phenomena localized somewhere in space-time. Since not all details are relevant it is useful to introduce some abstractions. We omit for instance the description of the electric power plant of the details of wires etc. and replace this by saying that we have arranged a specic electromagnetic eld in the experimental areas. This eld is computed using classical Maxwell theory. But Prof. Haroche may also say that he sends hydrogen atoms in a state with principal quantum number 54 through a cavity though this is not part of the language of classical physics. What is important is that we are forced to use a realistic language which may use the fruits of work of past generations of physicists but which amounts, in the last resort, to a description of the placement of material bodies in space-time. Note that this is precisely the information needed to infer ne details of nature from a theory in the frame of Local Quantum Physics under some provisos. The most important ones are that we live in surroundings in which material bodies are concentrated to occupy a small part of available space with \lots of vacuum" around and secondly that we can guarantee stability in the repeated use of the same equipment in several experiments. Without 4 these lucky circumstances our ability of acquiring individual knowledge would be in poor shape. C. Collective knowledge is more than all the pieces of individual knowledge gained in experiments. One prime reason for this has again been stated by Niels Bohr. He called it the principle of complementarity. Each experiment can focus only on one aspect of nature and thereby veils another. An (admittedly rather poor) analogy may be the comparison of nature with a topologically non trivial manifold and of an experiment with the establishment of a chart for a part of it. Bohr saw in complementarity a generalization of the relativity principles. The ndings of an observer do not only depend on his position and motion but also on his choice of the aspect he wants to study. Collective knowledge implies the development of a theory uniting all individual knowledge gained in experiments in a coherent picture. This is a discontinuous step, involving the creation of concepts which have no longer any direct relation to experimental procedure. In other words it involves free creations of the mind. The judgment on it depends primarily on the ratio of output to the adjustable input assumptions. In the case of nonrelativistic Quantum Mechanics including interaction with photons this is overwhelming. The input are a few constants (previously determined by experiments), a well dened, mathematically natural scheme, the hypothesis of spin and the Pauli principle. The harvest is an enormous wealth of consequences ranging from nest details in atomic physics to the structure of bulk matter. In relativistic Quantum Field Theory the balance is still good. There are many veried consequences, some even spectacular. There remain also unresolved problems: the trustworthiness of approximations is often unclear, there are many adjustable parameters and there are questions about the consistency of the mathematical scheme. Still there are enough reasons to believe that Quantum Field Theory is the natural extension of Quantum Mechanics encompassing a much wider range of phenomena. Some aspects of the formulation of Quantum Theory as xed around 1930 have evoked criticism and prompted many eorts towards a \better understanding", ranging from attempts to supplement or modify the interpretation to the search for a deeper theory from which Quantum Theory would appear as a semiphenomenological approximation. One of these aspects is \indeterminism": optimal knowledge of past history does not enable us to predict future events with certainty but only the probability of dierent possibilities. Of course it is futile to dispute whether the ultimate laws of nature should be deterministic. This is a matter of personal metaphysical belief. My own personal belief is that it would be a nightmare to think that we live in a world governed by 5 inescapable deterministic laws. What can be discussed, however, are the virtues and vices of specic proposals for a deterministic alternative theory. Here none of the existing proposals appears to me as attractive or even natural. This includes the recent developments of David Bohm's ideas on particle trajectories [2]. There one has to consider the wave function of a particle as a realistic property of an individual, governing the law which its trajectory must follow. This means that we cannot associate a single trajectory to a single particle without considering the wave function i.e. a family of other possible trajectories. The essential problem, namely the relation of the wave function to the occurrence of observed discrete events is not touched. This will become even more painful if one considers relativistic situations with particle creation. The use of the notion of probability as an intrinsic, irreducible aspect of the laws of nature needs some consideration. Our perception of nature shows us a history in which each event is unique. We must then regard probability as a quantiable attribute for each individual situation. K. Popper suggested (1956) to use the term propensity instead of probability to distinguish this mental picture from the customary use of probabilities associated with ensembles and large numbers [3]. In a recent paper entitled \The Ithaca Interpretation of Quantum Mechanics" N. D. Mermin based his analysis of quantum mechanical statements on such an understanding of the probabilities used in Quantum Mechanics [4]. He called it \objective probability". Although it is somewhat tedious to keep using the unfamiliar term propensity instead of probability I decided to make this eort whenever speaking of not yet realized possibilities in individual cases. It is clear that a propensity cannot be veried. Suppose I am faced with a choice between twenty dierent paths I could follow and I evaluate according to my knowledge the chance of getting killed if I take path Nr. 5. Then it does not help me that this path has the highest propensity assignment for survival among all others. I may get killed anyway. Only the life insurance company dealing with many similar situations can verify the correctness of my evaluation. The test of a propensity assignment needs our ability to create many \equivalent" situations, to form an ensemble in which the relative frequency of occurrence of some phenomenon can be observed. This needs human action and the human mind. Propensity itself is independent of this but it is unobservable. The use of concepts in the theory which are not directly amenable to observation is neither forbidden nor unusual. It seems unavoidable. The second feature in the quantum mechanical formalism which has caused much search for better understanding is the superposition principle. On the mathematical side it is simple enough. There is an underlying linear space over the eld of complex numbers, called \state space". Linear combinations of state vectors give other possible state vectors. One might expect that there is a simple counterpart of this basic mathematical structure on the physical side. This is not so. We do not have a general prescription for preparing experimentally a superposition of given states. The special cases in which we know how to do this concern primarily superpositions of 6 states which can be transformed into each other by means of external elds and the coherent states of photons (superpositions of states with dierent number of photons). Are there serious reasons to believe in a limitation of the general validity of the superposition principle? We know some limitations called \superselection rules". Strict superselection rules forbid (coherent) linear combinations of states which dier in some charge quantum number (electric, baryonic, leptonic, : : :). Since besides the photon there are no stable structures which do not carry some sort of charge quantum number this implies a strong limitation, in particular if we note that for practical purposes this extends to states with dierent localization of the charges even if their total charge quantum numbers are the same. Other superselection rules \for practical purposes" apply to macroscopic variables. They occur in the thermodynamic limit, in the classical limit and in many discussions of \decoherence" (see e.g. [5]). As mentioned at the beginning we need decoherence for the denition of events and disentanglement for the denition of objects. For both we need (basic or eective) limitations of the superposition principle. To fully evaluate the scope of such limitations we need the development of a self-consistent theory of measurement. By this I mean that we cannot claim that a self-adjoint operator in Hilbert space corresponds to an \observable". We have to respect the restrictions arising from the fact that any experimental arrangement has to use objects existing in nature, and is subject to interactions existing in nature, both (hopefully) described by the theory itself. In other words we cannot use Maxwell demons in a Gedankenexperiment. The third controversial feature in the standard formulation is the special role of the observer. For me this is the pivotal point. The \observer" is the only instance where decisions are taken, where facts are established. In the interpretation advocated by some eminent scientists, especially by London and Bauer, von Neumann, Wigner, the \observer" means ultimately the realm of consciousness, a realm beyond the range of physical arguments. In the so called \Copenhagen Interpretation" the observer may be regarded as just one side of a largely arbitrary cut between two parts of the world: on the one side there is the part of the physical world which one chooses to single out for study. On the other side there is the observer with his instruments which must be described in realistic language and to whom quantum laws cannot be applied. The cut may be shifted but never eliminated as long as we keep the form of orthodox quantum theoretic reasoning. The observer is the Archimedean point in standard quantum theory and is protected from the application of the theory as an extraterritorial region. In other words, the Schrodinger wave function (and its generalizations \density matrix", the \state") is only half of the picture and the Schrodinger equation does not describe the occurrence of events. It describes possibilities, not facts. The element of decision is lacking and the observer is needed to supply it. The result of an experiment is in each individual case a decision between dierent possible alternatives. This decision belongs to the realm of nature i.e. it does not depend 7 on impressions in any human mind. In modern technology it may be registered by a computer print-out. The characteristic properties in which we are interested are not changed by looking at the document. In the early days of Quantum Mechanics (1927) Dirac called the result of an observation \a decision by nature". This terminology was rejected at that time by Heisenberg who wanted to attribute decision making to the observer . There are two sides to this question. The experimenter decides indeed which aspect he wants to study but he poses a question to nature to which he expects to receive an answer in the form of a measuring result in each individual case. In the run of the experiment the question is repeated and dierent results appear yielding ultimately some probability distribution for various possibilities. It is not in the power of the experimenter to inuence the answer in an individual case. | There are only three logically possible positions one may take. Either we accept that the appearance of an event is a decision by nature, a decision in which some measure of freedom is allowed due to an intrinsic indeterminacy of the laws of nature. Or, as Wigner proposed, one postpones the appearance of an event to the appearance of an impression in the mind. Or one believes in ultimately deterministic laws, veiled by uncontrollable hidden parameters. I adhere to the rst of these positions and have given some reasons why I consider the other two as not helpful though they cannot be excluded on logical grounds. Nor can any of these positions be excluded on the grounds of any known empirical nding. I feel that Dirac's early formulation catches some essential aspect but shall use the less poetic term \event", or more precisely \coarse event" instead. There remains the problem that according to the present status of the theory there is no absolutely sharp denition of an event, there are no absolutely precise facts. Thus one may have to decide on the desired precision in carving out an individual event. Typically it would be the maximal contrast which could arise in a future interference between two putative events under the prevailing circumstances in a macroscopic space-time region. This is some reminiscence of the Bohr-Heisenberg cut but one which depends in a quantiable way on the existing situation including the position of all material objects in the space-time region under consideration. To evaluate it we need a selfconsistent theory in which all relevant objects are included as parts of the physical system. One might argue that such a theory is impossible because of the epistemological argument of Bohr. We are forced to describe the set up and result of an experiment in realistic and necessarily coarse grained language and we want to obtain much ner features which are not describable in this language. Therefore we need the cut. This is certainly an indispensable starting point but it concerns the individual knowledge obtainable in an experiment. We have described how by some tortuous process one can 1 Bohr felt uneasy with the terminology but in later years he used it himself with apologies for the undue personication of nature it suggested. 1 8 advance from there to knowledge about ne features, provided we are granted the lucky circumstances in which we nd ourselves which allow the performance and analysis of such experiments. The synthesis of such ndings in a theory representing collective knowledge is something else. It involves free creations of the mind, concepts which are no longer directly related to experimental procedure. Thereby it allows extrapolations to situations beyond our lucky circumstances. Astrophysics and Cosmology are examples where we do not get far if we insist on the central importance of the BohrHeisenberg cut though the latter is important in gaining the empirical information on which the theory is based. 2 Proposal of a strategy After these preliminaries we can address the following interrelated questions: Can we replace the central role of the observer by some concept which is not extraterritorial? Can we improve our understanding of the relation between physical concepts and spacetime? In other words, what do we mean by localization? I want to suggest the following strategy. 1) Consider the notion of \event" as a primary concept (generalizing the phenomena called \result of observation"). Events are not tied to observation. 2) Consider \material objects", for instance particles, as \causal ties" (or \links") between events. 3) Attribute the property of localization to events, not to material objects. It is a localization in space-time. The idealized picture is a space-time point. Actually it is an approximately denable region in space-time whose extension and sharpness depends on the nature of the event. 4) A causal link, say a particle, represents a potentiality until it has fullled its mission i.e. until some target event is concluded. Its characteristic property, its \state" is a propensity assignment relevant for the occurrence of a subsequent event. Since there are several potential links emanating from one event there will be correlations between the subsequent events caused by them. The joint propensity assignment concerning the localization of subsequent events 1; : : : ; n linked to a source event is a function in n-fold conguration space-time, restricted by the causality conditions that the xk ? x are positive time-like whereas there is no further restriction for the position of the xk ? xj . In particular, correlations may extend into space-like directions. 5) An event is considered as real, marking the appearance of a new fact. The universe is regarded as an evolving history of facts; mathematically as an evolving graph or category whose points are the events and whose \arrows" are the causal links. Its 9 shifting boundary separates a past which is factual from a future which is open. Thus this picture must incorporate the \arrow of time" on the fundamental level. It relates to the arrows representing causal links. The realization of an event is irreversible. Bohr mentions the \essential irreversibility inherent in the very concept of observation" but attributes this to the needed amplication and thus implicitly to the realm of statistical mechanics. Here we replaced the concept of observation by the concept of event and, while amplication is needed for recognition, it is not considered as an essential prerequisite for an event. Comments. 1) The proposal of a \`theory of events" has been advanced twenty years ago by Henry Stapp [6]. Some of his axioms are very similar to the strategy outlined above and they preceded my attempts by a wide margin. But there are also some essential dierences in our views, in particular in the interpretation of the EPR-type phenomena, the meaning of \non local character of Quantum Mechanics" and the role of consciousness and mind. 2) C.F. von Weizsacker [7] has emphasized for many years that probability assignments in physics are future-directed, that the past is factual (we have documents) whereas the future is open and time enters essentially in this context. 3) In a series of papers [8] Ph. Blanchard and A. Jadczyk have described a formalism which generates real events in the interaction of an atomic object with a macroscopic measuring device. The latter is idealized as a classical system in the sense that it is described by a commutative algebra which may include some discrete variables. The scheme, called \event enhanced quantum theory", introduces irreversible decisions into the interaction process and yields a good phenomenological description of the quantum measurement process. It is related in spirit to points 1 and 5 of the strategy outlined above but allocates events only to the interaction between a small and a large system retaining the ad hoc distinction between a classical and a quantum part of the world. 4) A central point in the above strategy is the shifting of the localization concept from objects (e.g. particles) to events. This allows a clear separation of the so called \nonlocal aspects of Quantum Theory" and the causal relations which are restricted by the geometry of Minkowski space. We are used to talk about the \position of an electron at some (arbitrarily chosen) time" but add that in Quantum Mechanics this should not be considered as a real property \unless it is measured". What is implied here is that \position" is not an attribute of the electron itself, it is an attribute of the interaction process of the electron with another object, say a photographic plate. To the \event" of ionization followed by a chain reaction in this plate corresponds a localization in space-time not only in 3-space. As another example one might consider the -decay of a nucleus. The -particle is described by an outgoing, essentially spherical wave. 10 This describes a propensity for a possible event here or there provided it meets a collision partner. It does not describe a matter distribution in space. The collision on the other hand can be regarded as a decision for an event in an approximately well dened space-time region. | We inherited from classical mechanics a too materialistic picture of objects, considering the occupation of some region of space at given time as the primary property of matter. If Quantum Mechanics is regarded as resulting from Classical Mechanics by a miraculous formal procedure called "quantization" in which the classical picture persists in some symbolic way but is blurred out so that it cannot be taken any longer at face value then we come to the resigning conclusion that we cannot assign any realistic attributes to an atomic object and this conclusion is illustrated in innumerable discussion on \wave-particle duality". Here I suggest that we should forget the classical picture (which retains of course its value in the limit of large objects) and consider a particle (or more generally an object) only as a messenger between two events. It is born as an individual in the source event, carries as its attribute the potentiality for a range of subsequent target events, quantitatively described by a propensity assignment (a \quantum state"). The events on the other hand have localization as one of their attributes and provide the bridge to space-time. Ultimately one might say that the signicance of the notion of space-time reduces to the description of causal relations between events. Loosely speaking we replace the waveparticle duality by a dualistic structure of objects-events in the theory. In relativistic theory the evidence against attaching the localization concept to objects becomes even stronger. While in non-relativistic theory we can at least obtain wave functions which are strictly localized in a space region at some time the concept of localized states in relativistic theory is useful only in a qualitative sense. There the basic assumption reserves the notion of strict and arbitrarily sharp localization for \observables" and there is evidence for the physical relevance of this assumption. But the operational realization of an \observable" needs objects and therefore it appears more natural to shift the localization concept further and apply it to events. Unfortunately this last shift is not just a question of reinterpretation but requires a modication of the structure of the theory. 3 Discussion of some phenomena It should be clear that the aim of the strategy described is not primarily to propose another language for an existing theory but an extension and modication of the theory which strives to incorporate in a selfconsistent way two areas which are excluded in standard theory from the application of quantum laws: the emergence of coarse events and the role of space-time as relating events instead of assuming a classical space-time continuum in which physical processes are embedded. There arise the questions: 11 a) Can a scheme developed along the lines of the proposed strategy agree with all the experimental evidence which is handled with such outstanding success by existing branches of the theory? b) Where can we look for eects in which the dierence between this scheme and standard theory should be seen? The answer to either question involves, of course, an enormous amount of detailed work. However a few general points are visible from the following examples. 3.1 EPR-type phenomena and the division problem for objects Let us consider the example rst proposed by D. Bohm which led J. Bell to his famous inequality [9]. An instable particle of spin 0 decays into two particles with spin (event 0). On each of these particles the spin orientation is measured by a SternGerlach device. The orientation of the two Stern-Gerlach magnets (unit vectors e; f ) is set at some time (events 1 and 2). The subsequent possible measuring results can be either + or ? (events 3 and 4 for each of the particles). One determines the joint probability We;f (a; b) where a and b stand for + or ?. Here there is no diculty in clearly dening the ve relevant events and drawing the graph of causal links. In the gure ; represent the particles, ; the Stern-Gerlach devices. 1 2 3 γ 1 4 α β 0 δ 2 It indicates that the relative space-time placement of events 1, 2 and 0 or events 3 and 4 should be irrelevant whereas event 3 should be in the forward light cone from both 1 and 0, similarly 4 from both 2 and 0. If the separation between 1 and 2 or between 3 and 4 would be time-like then it would, of course, be possible that additional causal links, not incorporated in the gure could exist (signals from one region to the other). In the set up of the experiment such additional lines are excluded. The mere existence of correlations in We;f and their persistence for large separations is not strange and does not suggest any non-local aspect of the theory. For instance, if one replaces spin by electric charge and starts from an uncharged particle decaying into two oppositely charged ones then nobody will be surprised by the fact that a measurement of the charge of one particle tells us immediately what the charge of the other is, no matter how far 12 away this may be. The strange feature of the correlation tables here, which appears to violate \common sense", is that it casts doubts on the \reality" of the individual particles. We would be tempted to assume that each of the particles, once clearly separated, is in some \state", say resp. . The statistical ensemble considered will then be described by some probability measure d( ; ) which may have correlations between and . There should be some probability pe;a( ) for result \a" by the device in the setting e in case the state is and similarly a probability pf ;b( ). This would be the case if we can assign a quantum state to each link. The joint probability should then be of the form 1 2 1 1 2 2 1 1 2 Z We;f (a; b) = pe;a( )pf ;b ( )d( ; ) : 1 2 1 2 (1) Note that if, in addition, we wanted to assume that ; are \hidden variables" which allow a deterministic description then the functions pa; pb would be restricted to the values 0 or 1. But we shall not assume this. It can be shown now that the dependence of We;f (a; b) on e and f cannot be expressed in the form (1) for any choice of the p() and . I shall not reproduce the argument but refer to the original paper by Clauser et al. [10] and the presentation in the book by Omnes [5]. The conclusion is that the correlation cannot be understood as a correlation between states of the particles and . The 2-particle state produced in event 0 cannot be broken up in the intermediate region in spite of the wide separation of the events. Other models have been presented in which the oense against \common sense" appears even more glaring [11]. The basic aspect is not changed. We see correlations between events which cannot be reduced to correlations between states of subobjects involved in the events. This situation emphasizes the points 2; 3; 4 of the proposed strategy. The pattern of an individual history is xed only after the realization of the events; the propensities attached to the (not yet realized) causal links have no independent localization properties. The statement that we are dealing with two distinct particles means that there will be precisely two subsequent events which can be causally linked to event 0. The nature of these events does not only depend on these potential links but on the reaction partners they meet (here the orientation e; f of the Stern-Gerlach devices). The source event 0 denes a propensity contribution for the growth of the pattern which cannot be broken up into parts referring to the separate branches. Still, if we choose to look only at one branch of the subsequent pattern and focus only on one of the events, disregarding coincidence with the other, then the propensity for this next event is independent of whatever happens in the other branch. This important fact is a consequence of the commutativity of observables in space-like separated regions and allows a propensity assignment for any event caused by one particle (a \state of the particle") though such states do not suce to describe correlations. This indicates that the \non-local aspects" do not concern causal relations between events but correlations 1 13 2 in the propensity for the joint appearance of events, as in the simple example where the spin was replaced by the charge. Though it is not important for our discussion here it should be noted that entanglement can be studied in more general cases, namely whenever the state space of the system can be decomposed into a tensor product. This may be the case for a single object with dierent \degrees of freedom" such as the center of mass motion and the spin of a single particle. 3.2 Coherence and the division of events Consider the following situation. There are initially two electrons and one photon in some space-time region, suciently isolated from everything else. There may be a sequence of two events where the rst one is an elastic scattering of one electron and the photon. Later on one of the reaction products collides (again elastically) with the remaining electron. There are two possibilities depending on which of the two outgoing particles from the rst process is the partner in the second collision. We may either have electron-photon or electron-electron scattering in the second process. In both cases we have at the end again two electrons and one photon. The quantum theoretic calculation yields amplitudes for each of these histories and we have to add them to get the amplitude for the total process. If in squaring it the interference term is appreciable then we cannot claim that the total process can be decomposed into a sequence of two events with a choice between two alternative histories. The size of the interference term depends on the initial state of the 3-particle system. If the distance between the expected collision centers (see next section) is large then the interference term is washed out. In operational terms: no contrast can be demonstrated in any interference experiment whose instruments are bought to interact with the outgoing particles suciently much later when the total process can be regarded as concluded. In that case we can speak of two alternative histories consisting of a sequence of two events. In the other case the total process must be regarded as an irreducible 3-particle collision. (a) (b) 14 3.3 Localization in a two-particle collision Consider a situation where we have, naively speaking, only particles like electrons, atoms, molecules separated on the average by distances large compared to their intrinsic extension, to the range of interaction forces and to the parameters (2) d = (t=m) 21 ; where t is the mean time between collisions, m the mass (using natural units with h = 1, c = 1, so m? is the Compton wave length). We shall see that d can be interpreted as the essential extension in space-time of the individual collision process according to standard quantum theory. For simplicity we do not consider here the presence of large chunks of matter or external elds. We also exclude here the consideration of photons. For them a dierent discussion is needed because for them d of (1) is innite but for the process of a photon interaction with a massive particle the result (21) persists. We shall use relativistic kinematics with a view to high energy physics but also because the formulas become more transparent. The relevant processes will be collisions with two incoming particles and possibly several outgoing ones. Using Wigner's denition of a particle species as corresponding to an irreducible representation of the Poincare group the species is characterized by a sharp value of the mass and the spin. To this we have to add charge quantum numbers. In a semi-phenomenological description this is taken care of by considering instead of the Poincare group P the direct product of P with a global gauge group G . We shall, however, suppress charge and spin indices. A pure state of a particle may then be described by a wave function ' of the 4-momentum p with the constraint p = m and the normalization 1 2 2 Z h'; 'i = j'j d(p) with d(p) = (p ? m )d p : 2 2 2 (3) 4 Notation: the translation by the 4-vector a is represented by the unitary operator U (a) = eiPa . P = P is the \energy operator", P k = ?Pk (k = 1; 2; 3) the spatial momentum. Thus, a particle which is localized at time x around the space point x with an extension d of the localization region is described by a wave function '(p) = (p)eipx; x = (x ; x) (4) where is a smooth, slowly varying function (changing little in a momentum range of order d? ). We take care of the constraint to the mass shell by the -function in (3), not in ' itself. The probability amplitude for the momentum conguration in an outgoing nparticle channel may be written as 'out (p0 ; : : :; p0n ) = T (p0k ; p ; p ) (p0k ? p ? p )' (p )' (p )d(p )d(p ) (5) 0 0 0 0 1 Z 1 1 2 4 1 15 2 1 1 2 2 1 2 where T is expected to be calculable from the theory. There are some important qualitative features. Apart from very special subsets in the conguration space of momenta, T is a smooth and slowly varying function of all its arguments. The exceptional congurations are thresholds and sharp resonances. If p is the momentum change for which T changes signicantly then r = p? describes the extension of (one contribution to) the unsharpness of the position of the event. In the case of a resonance this is large for the total process due to the lifetime of an intermediate unstable particle. The process can then be broken up into the formation and subsequent decay of instable particles. In the case of a threshold it is large because of the almost zero velocity of some outgoing particle. We shall not discuss here the complications arising for such special congurations of the incoming momenta but limit ourselves to the generic case in which r may be interpreted as the range of the interaction. Let us use in (5) the identity 1 4 (q) = (2)?4 Z eiqxd x (6) 4 and perform the integration over p ; p rst for xed x. Here we may replace the arguments p ; p in T by mean values pr (possibly depending on x). In most experimental situations one can argue that the momenta of the incoming particles are so sharply dened that the change of T within the margin of momentum uncertainty can be ignored. In this case the pr can be considered as being experimentally prescribed. Let us envisage here a somewhat dierent situation in which the incoming particles originate from previous interaction processes localized respectively in small regions around the (space-time) points xr with some extension d. Thus we shall use the form (4) for the incoming wave functions. When the components of x ? xr are large compared to d the integrals over p and p can be evaluated by the method of stationary phase, where in the slowly varying factors we can replace pr by pr with 1 1 2 2 1 2 q pr = mr(x ? xr )r? ; r = (x ? xr ) (the proper time distance): 1 2 (7) In wave mechanics this is intuitively understood by noting that a wave emitted from a source of small extension looks like a plane wave with momentum (7) at a point far from the source. pr is the classical momentum of a particle with mass mr in passing from xr to x. This brings (5) into the form (apart from numerical constants of order 1) 'out(p0 : : :p0n ) = T (p0k ; p ; p ) (p ) (p )(m m ) 21 ( )? = ei ptotx?m1 1?m2 2 d x (8) with p0 = p0k : Z 1 1 2 1 1 2 2 1 X tot 16 2 1 2 3 2 ( 0 ) 4 The form (8) may be regarded as an expression of Huygens' principle. 'out is a superposition of contributions originating from dierent space-time points. pr and r depend on x according to (7). The method of stationary phase. Consider the integral Z I = (q )ei(q)dn q (9) where is a smooth, slowly varying function and the phase factor ei oscillates rapidly except in the neighborhood of a point q^ where is stationary @ / @qr (^ q) = 0 : (10) The essential contribution to I comes from the neighborhood of q^. There we have @ 2 1 K rs = (^q ) : (11) (q ) = (^ q ) + K rs (qr ? q^r )(qs ? q^s ) ; 2 @q @q r s We get = (^q )ei(q^) j det K j? 2 (2i) 2 n : (12) R 1 The last factor comes from the integral e 2 iK rs r s dn . The extension of the relevant neighborhood of q^ in the direction of an eigenvector of K to eigenvalue i is given by 1 I q1 = ji j? 2 1 1 (13) : The essential criterion for the reliability of the approximation (12) is that the relative change of in the range of q is small compared to one. Again, in the x-integration of (8) we have a rapidly changing phase factor. So the method of stationary phase can be used again to evaluate the integral. The total phase is = p0 x ? m ? m (14) @ = p0 ? m ? (x ? x ) ? m ? (x ? x ) : (15) @x For xed p0 the essential contribution comes from the point x where 1 1 tot tot 1 1 1 1 2 2 2 2 1 2 tot @=@x = 0 : (16) This equation, which may be regarded as the expression of momentum conservation, can be solved for x. Let us denote the resulting point by x^. 17 To complete the evaluation of the integral (8) we expand around the point x^: (17) = + 12 N + : : : = (^x) ; N = @ =@x@x (^x) ; = (x ? x^) ; = (x ? x^) : (18) The usefulness and reliability of the approximation method depends on the size of N . If it is large compared to the r? we can stop in (17) with the quadratic term. N determines the extension and shape of the space-time region around the point x^ which contributes signicantly to the integral. This consists of the vectors for which 0 2 0 2 One has N = jN j < 1 : (19) r? mr (g ? pr;pr;=mr ) : (20) X 1 r=1;2 >From (19) and (20) one nds that the extension in the transversal direction (orthogonal to both pr ) is d = (m = + m = )? 12 (21) and in the longitudinal and temporal direction one gets in the extreme relativistic limit tr 1 1 2 2 d0 = q? (1=m + 1=m )? 12 1 1 1 (22) 2 2 where q is the center of mass momentum and q mr is assumed. We get then 'out (p0 : : : p0n ) = T (p0k ; p p ) (p ) (p )(m m ) 12 ( )? = j det N j? 12 ei0 : (23) 1 2 1 1 1 2 2 1 2 1 2 3 2 Here the pi; i; N ; depend on x^ which in turn is a function of p0 according to (15), (16). Before proceeding further we should understand the origin for the expressions (21), (22) describing the extension of an event. If the proper times r between events is large then the uncertainty product xp in each of the incoming links is large compared to 1. This means that for each particle there is a correlation between its (putative) momentum and position, expressed by a local momentum p(x). Even if the overlap of the wave functions of the incident particles is large the interference due to the phases is constructive only in a small region (this eect corresponds to the applicability of the method of stationary phase). The correlation between position and momenta of the individual particles is transformed into a correlation between the total nal momentum and the position of the collision center and the sharpness of this is such that with respect to p0 and x we have a minimal wave packet. The total collision amplitude 0 tot tot 18 is still a coherent sum of such minimal wave packets but since we cannot measure x and do not measure p0 with higher precision than the momentum uncertainty of the incoming particles this is for future events equivalent to a mixture of such minimal wave packets. The extension d given by standard quantum theory increases with the square root of i.e. it becomes large if the incident particles have a very small momentum uncertainty, very large correlation length. If we believe that the event should have an intrinsic localization then d should not increase indenitely with but have a bound. Unfortunately no existing experimental studies give us any information on this since it is extremely dicult to control the initial momenta and measure independently the total nal momentum with such a high precision. The probability for a process leading to a nal n-particle channel is given by W = 'out d(p0 ) : : : d(p0n ) : (24) tot Z 2 1 We split the integration over all momenta into an integration over the 4 components of the total momentum and (3n ? 4) remaining variables which we denote collectively by L, writing d(p0k ) = d p0 d(L) : (25) Since the total nal momentum p0 is coupled to the position x^ of the collision center the integration over p0 can be replaced by an integration over x (writing now x instead of x^). The functional determinant j@p0 =@xj is just det N . So we get according to (23) 4 tot tot tot tot Z W = w(x)d x ; w(x) = ; r = jr (pr )j mrr? : 4 1 2 2 3 (26) (27) The dimensionless quantity = (2) Z 2 jT j d(L) 2 (28) is a Lorentz invariant function of the initial momenta which in turn are determined by x and the position of the previous events according to (7). One has Z w(x) = w(x; L)d(L) with (29) w(x; L) = (2) jT j : (30) r dened in (27) depends on x and on the previous collision from which the link r originated and may be interpreted as the contribution of that link to the propensity 2 19 2 1 2 of the event at x, whereas w(x; L) gives the joint propensity, relevant for future events caused by the event at x. It is important to note that pr is not determined by the previous event alone but by the position of both the source event and the target event, the latter being the point x. As emphasized above, the causal tie becomes \real" only after both the source and the target events have been realized. Thus within the limits of the above approximations the predictive reasoning agrees with the proposed scheme and runs as follows: any pair of unsaturated links, remaining from the pattern of earlier events, has the possibility of joining to give birth to an event at a later point x with the propensity w(x) determined by the earlier events, thus completing these links and creating new potential links with a propensity function w(x; L) for the momentum conguration L which describes (contributions to) correlations in the development of future patterns of events. The step from the complex probability amplitudes (Huygens' principle) to positive probability functions is the step from the S-matrix to cross sections or rather to a transition probability per unit volume in space-time. In the standard discussion of this step one considers in (5) T as a function of p ; p and L (replacing p0 by p + p ). Assuming then that the wave functions 'r (p) are sharply peaked at the values pr one replaces pr in T by pr . Using the representation (6) of the -function one gets 'out = (2)? T eiptotx (x) (x)d x (31) where (x) = (2)? = '(p)eipxd(p) (32) denotes the covariant wave function in position space. Squaring and integrating over p0 gives (33) j'out(L; p ; p )j = (2) jT j j (x)j j (x)j d x : This agrees with the transition probability per unit space-time volume as given by (30) with r replaced by j rj . Conventionally one writes = F ; F = m m [(u u ) ? 1] 12 (34) where is the cross section and ui are the 4-velocities of the incident particles. The factor F introduces the relative velocity of the incoming particles in the relativistically invariant way suggested by C. Mller. Our computation above just shows that one can dispense with the assumption that the 'r are sharply peaked; they could, for instance be spherical waves emitted from the neighborhood of some points xr , provided these points are far separated. The only dierence is that then the pr and thereby also T become functions of x and that we see that the \event" has an extension given by (21), (22) which is not due to a limitation in the overlap of and in position space but to an interference eect establishing a relation between p0 and x. 1 Z 2 tot 0 1 1 4 2 Z 3 2 tot Z 1 2 2 2 2 2 1 2 1 1 2 tot 20 2 1 2 2 2 2 4 1 2 4 Concluding remarks Any meaningful physical statement involves some subdivision of nature, some distinction of individual elements. One way of subdividing is provided by the notion of spacetime. In classical eld theory this suces since the physical quantities are thought to be associated to the space-time points. In Quantum Theory this is not so. There is some atomicity concerning on the one hand stable structures (material objects), on the other hand irreducible facts (events). In the strategy proposed in section 2 I argued that the relation to space-time is established by events whereas material objects are considered as causal links between events without independent localization properties. Furthermore the feature of basic indeterminism in Quantum Theory demands a distinction between facts and propensities and this distinction relates to a distinction between past and future within the theory. We consider an evolving history of the pattern of events and causal links with an ever shifting boundary separating it from an open future. The \nonlocal features" of Quantum Theory refer to propensities for correlations between future events which are linked to a common source event. The geometry of Minkowski space governs the causal relations between events. I called this a strategy because it replaces the concept of observable which is basic in Quantum Theory by the concept of event. It is not a theory. It lacks a general denition of an event in terms of basic symbols of the theory. Our discussion was restricted to the special case of a low density situation where we may recognize particles as objects and events as collision processes. In this regime the discussion reduces to an S-matrix theory and I remember well a remark by Heisenberg (1956): \the S-matrix (and particles) are the roof of the theory, not its foundation". Still the strategy suggests a faint chance of nding an eect which distinguishes the proposed scheme from standard theory by a very precise study of localization and momentum transfer as mentioned in section 3.3. In general situations e.g. inside a liquid it is unclear how to single out individual events or, for that matter, individual objects. The so called constituents, electrons, protons, neutrons have no individuality. Yet the naive picture of matter being built up from such constituents has not only been extremely useful but expresses some feature in nature we do not fully understand. Consider experiments in high energy physics where an accelerated particle hits a target of bulk matter. Depending on the process of interest (in particular the primary energy and the reaction channel) the process is idealized as a collision with an eective partner, for instance with a single proton in the bulk matter. The ability to talk about an eective collision partner is explained in the constituent picture by saying that the binding energy of the constituents is small compared to the energy transfer in the reaction. On a more sophisticated level we see that the nature of the constituents is changing over many orders of magnitude as the energy transfer changes. We may have to take a whole crystal, a molecule, an atom, a nucleon or even a parton. It seems that it is the type of event which changes and this 21 is regarded as a manifestation of various constituents of matter. There is some granular structure. In the constituent picture it is encoded in the internal wave function of the bulk matter. But it manifests itself in the discreteness, the separability of various types of events irrespective of the lack of individuality of constituents and their changing nature. This seems to me a qualitative feature supporting the idea of the essential importance of the concept of event. Acknowledgement I have proted much from many discussions with Prof. B.-G. Englert and also want to thank the Erwin Schrodinger Institute in Vienna for the opportunity to clarify my ideas during a most stimulating workshop in the last week of September 1997. 5 References [1] R. Haag, \Local Quantum Physics", second edition, Springer Verlag, 1996. [2] D. Durr, S. Goldstein and N. Zanghi, J. Stat. Phys. 67, 843 (1992). [3] K. Popper, Conf. Proceedings Bristol 1956. [4] N. D. Mermin, \The Ithaca Interpretation of Quantum Mechanics", Preprint 1996. [5] R. Omnes, \The Interpretation of Quantum Mechanics", Princeton University Press 1994. [6] H. P. Stapp, \Theory of Reality", Found. of Phys. 7, 313 (1977); \Whiteheadian Approach to Quantum Theory and Generalized Bell's Theorem", Found. of Phys. 9, 1 (1979). [7] C. F. von Weizsacker, \Probability and Quantum Mechanics", Br. J. Phil. Sci. 24, 321 (1973). [8] Ph. Blanchard and A. Jadczyk, \Events and piecewise deterministic dynamics in event enhanced quantum theory", Phys. Lett. A 203, 260 (1995). [9] J. Bell, \On the Einstein-Podolsky-Rosen Paradox", Physics 1, 195 (1964). [10] J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526 (1974). [11] N. D. Mermin, \Quantum Mysteries Rened", Am. J. Phys. 62, 880 (1994). 22