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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
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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
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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:
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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
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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
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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
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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
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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
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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
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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
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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.
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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:
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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
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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