1
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And, expressed in numbers, 1 over 137, i.e. slightly less than one percent,
2
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the magnetic moment of the electron can deviate from the Dirac value.
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This deviation has now been calculated using quantum electrodynamics with an accuracy of one in one million,
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i.e. accurate to six decimal places, by going right up to the square of Sommerfeld’s fine structure constant.
5
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The latest measurements, which were carried out by Bloch and his colleagues,
6
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have indeed also confirmed this value to an accuracy of one millionth.
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This means, all six decimal places of this number are confirmed by the experiment,
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the last decimal place then has some experimental uncertainty.
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One is therefore justified in saying that quantum electrodynamics has proved to be as good as one could hope for.
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There are certain aspects of quantum electrodynamics which have not yet been completely confirmed by experiment,
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although they also represent a deviation from classical electrodynamics and would therefore be interesting.
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According to quantum electrodynamics an interaction between light quanta should exist, for example.
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It should be possible for one beam of light to deflect or scatter another one,
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but this effect is so difficult to measure that no one has yet succeeded in doing it.
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There should therefore be no doubt that the predictions of quantum electrodynamics also accurately apply here as well.
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In this respect extraordinary progress has really been achieved during the past ten years, and it is possible to say
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that quantum electrodynamics is in an almost complete form, it is difficult to think that there is anything about this theory
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which can still be improved.
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Nevertheless, it is known that if this quantum electrodynamics is applied to the collision of very high-energy particles, i.e. if one wanted
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to pose the question, what happens if two electrons with enormous energy, let us say one hundred billion electronvolts, hit each other,
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then this quantum electrodynamics would certainly provide a wrong answer.
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The reason is that in reality heavy elementary particles are created, mesons and so on, while according to quantum electrodynamics it is
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precisely this which would not happen.
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This is not a surprise, however, the reason for this is that for such processes with very high-energy transfer it is no longer possible to
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separate the electron-light quantum field from other elementary particles, i.e. the form system must necessarily fail here.
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The fact that this is the validity limit for electrodynamics can also be seen from a quite different example.
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There are of course also different particles, which are not so many times lighter than the protons,
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but still slightly lighter than the protons, I mean the mesons, the pi-mesons.
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They are around seven times lighter than the protons.
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It would be possible to imagine that these particles could also be split from the protons, by saying:
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I consider only processes where it is not possible to create protons anew.
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So for such processes I do not need to take the heavier elementary particles into account, it is sufficient to describe the behaviour of the
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light elementary particles, i.e. the mesons among themselves.
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Now here in meson physics the difference in mass to the heavier elementary particles is much less than in electron physics.
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In addition, the coupling constants, about whose significance I want to say something later, are much larger,
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one should therefore not expect at all that the separation works here and in fact it does not work.
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There have been many attempts during recent years to mathematically formulate meson theory,
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but no satisfactory agreement with experience has been achieved here.
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The reason is that it is obviously not possible to separate the mesons from the field of the other heavy particles.
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This means that one will possibly be forced, if one works in meson physics,
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to have to describe the complete field of the elementary particles.
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And this of course is much more difficult, it is then not possible to separate the problems,
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the totality of the elementary particles must be described in one go, so to speak.
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Now, when one attempts to do this and this of course has already been tried and Mr. Yukawa has already talked here about a special form of
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such an attempt, in this attempt there are very peculiar difficulties,
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which have also occurred already in quantum electrodynamics in a slightly different form.
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I would now like to say a few words about these difficulties.
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What happened in pure mathematical terms was that initially the field equations were written, as they were familiar from classical theory,
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Maxwell’s theory, for example, the quantum conditions were then applied to these field equations and it was assumed that something
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experimentally sensible had to result from this.
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In reality it turned out that the equations do not converge, that something infinite results, i.e. the results become useless.
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This was already the case with quantum electrodynamics, and there this could only be avoided by using a certain trick,
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but I do not want to go into more detail about this here.
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Why do we initially get these infinities when we quantise field equations?
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At the very beginning, people probably thought that this was simply down to a clumsy mathematical formulation of the equation, but meanwhile
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it has turned out that there is a principal difficulty, which one needs to analyse in detail, before one can hope to come to a solution.
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The difficulty which occurs here is the unification of quantum theory and the theory of relativity.
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Or to be more precise, it is the difficulty of bringing the uncertainty relation of quantum theory into harmony
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with the space-time structure of the special theory of relativity.
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Now, what is this space-time structure?
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If one calls all those events which one can in principle still influence, future events, and all those which one can in principle at least
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get some information on, past events, then we would say from our daily experience that these two groups of events, which I have just called
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future and past, that they are only separated from each other by one infinitely short moment, which can be called the present moment.
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In Einstein’s special theory of relativity we learned that the structure of space and time is slightly more complicated,
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however, and in any case slightly different.
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If we take again this definition of future and past, which I stated just now, i.e. all events which one can in principle still influence are
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in the future, and all those from which I can in principle still get some information are past, then these groups of events are in reality
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separated by a finite time interval, whose size depends on the distance of the location where the events take place from the observer.
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And this whole space can also be called the present space.
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It is only important to state that this word present now refers to a finite time interval.
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This means, if we think of events at a distant star, let us say events on the star Sirius, then there is a space, a time interval of the
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order of a few years, between the group of the future and the group of the past events.
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And this time interval can be called the present space, i.e. any events which occur during this period on Sirius can be thought to be
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simultaneous with events happening here, and are in the strict sense of the theory of relativity
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also simultaneous from certain coordinate systems.
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Now, this special space-time structure of the theory of relativity made it necessary that the causality requirement can only be linked to
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the theory of relativity in a very special mathematical form.
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If one wants to maintain that the action always brings about the cause (sic), i.e. if one wants to state a consistent sequence of cause and
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action, this is no longer possible in the theory of relativity
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– in the sense in which it was possible in classical mechanics - with forces which act at a distance.
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Rather, this is then only possible by stating an action can only go from one point to another point in the immediate vicinity.
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All actions must of course propagate with the speed of light at most, i.e. not faster, and therefore those actions can be best described
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with differential equations of the type of the wave equation of light,
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i.e. with hyperbolic differential equations, which describe such an action from point to point, a so-called contiguous action,
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and where such actions then propagate with the speed of light, or with the speed of light at most.
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Since the special theory of relativity, one has therefore been assuming in a quite general way that all fundamental laws of physics had to
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be written as differential equations of the type of a relativistic wave equation.
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It is precisely this consequence of the theory of relativity which causes the problems with the uncertainty relations of quantum theory.
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In quantum theory it has turned out that it is not possible to know position and velocity of a particle,
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for example, simultaneously with unlimited precision.
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Either, the position can be determined very precisely, then the velocity is not known, very indeterminate,
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or the velocity can be determined with very high precision, then the position is known very inaccurately.
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In other words, therefore, a very precise localisation of any event results in a very large imprecision in terms of momentum and energy.
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If I demand such a contiguous action, however, i.e. if I say that from this point here actions can only be transferred into the immediate
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vicinity, and not to regions further away, this means of course an infinitely sharp localisation; this means that a contiguous action theory
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is a theory where the action from one point to another is localised with infinite sharpness.
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Such an infinitely sharp localisation corresponds to an infinite momentum, however, an infinite amount of energy: in this case energy and
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momentum must be infinitely undetermined, and these are precisely the infinities which in the mathematics of this quantum theory of the wave
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fields have always been causing problems.
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These infinities are thus by no means accidental things which could be avoided by more skilful mathematics, but they are caused directly by
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the space-time structure of the special theory of relativity and the uncertainty relations of quantum theory.
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It is possible to try and escape these consequences or this dilemma in some way or another.
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And one can point out that we - although we can see in the theory of relativity that we can describe the relativistic behaviour correctly
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with such differential equations – we are really not sure that it is imperative to describe nature with such differential equations.
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There could therefore be some kind of action at a distance at least over small distances, we need only to make sure that such action at a
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distance fits with the invariance requirements of the theory of relativity.
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If one proposes this programme, then the first question is, what if anything remains of our previous theory.
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What do we need to drop, what can we keep.
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We can then state: in any case it is obviously possible to observe what the mass of the elementary particles is in collisions and so on,
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with which speed they leave a collision, a collision process, how many particles leave with which speeds etc.
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We can say quite in general that the asymptotic behaviour of the waves or the particles in large space-time intervals can obviously still be
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observed, so this must also be a component of a future theory.
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It has also been ascertained that this asymptotic behaviour really can be represented by specific mathematical structures, which the
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physicist calls the S-matrix, and that if one drops all other characteristics of matter, that with the aid of this S-matrix a proper
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mathematical description really can be obtained for the things that happen during any collision processes.
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It is therefore possible to avoid all the infinities and to obtain a theory with only reasonable mathematics if one drops the need for a
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description of the local process, and is instead only interested in the asymptotic conditions.
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Now, if one does that, one maybe foregoes too much, however, because with this the complete causal sequence of events is destroyed.
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Now I do not mean those deviations from causality which already originate from quantum theory anyway.
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You know that quantum theory causes a certain statistical character of the atomic events.
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But this is not what is meant here, all that is meant is this character of the theory of relativity, that from an action,
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from any point actions can only spread to other points with the speed of light.
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This fact would then be destroyed, and this would result in such a radical intervention into the structure of the theory
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that all the time relationships would be in disarray.
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And it would no longer be possible to state in which temporal sequence two events have taken place.
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There is then the danger of obtaining very large deviations from this concept of relativistic causality,
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and we now know on the other hand that such large deviations do not occur.
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In recent years there have thus been attempts to write down theories which, although they have such deviations, allow only deviations from
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the causality principle of relativity theory in very small dimensions, i.e. in dimension of the order of magnitude of an atomic nucleus.
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The non-local theory which Mr. Yukawa has talked about here, for example, is one of this group of theories.
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Such non-local theories have therefore been written down and attempts have been made to come to a converging mathematics in this way,
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i.e. to a mathematical scheme which is sensible, and also corresponds fairly well to the causal events which we observe in nature.
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We do not yet know whether this programme can be carried out to its completion.
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It is certain that one can remove certain difficulties which exist in these local theories, i.e. in theories with contiguous interactions.
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But whether the non-local theories can be carried out in such a way, that on the one hand full mathematical convergence is guaranteed, and
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on the other deviations from relativistic causality occur only in very small space-time dimensions, this is something we do not yet know,
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and this must come out in the coming years.
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The success of this extension of the theory is not quite guaranteed, at any rate,
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and neither is it known whether this extension of the theory is really necessary.
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It is maybe possible to already state that it would be desirable if this deviation from the local, contiguous action character of the
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equation would only enter via quantum theory, because one does not have the impression that in the region
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where quantum theory does not play a role, i.e. for large dimensions,
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that in this case one should deviate somehow from the usual differential equations of the theory of relativity.
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Now this question of whether the future theory will be a pure contiguous action theory or a non-local theory,
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of the Yukawa type, for example, this is of course not only a mathematical problem, but in particular also an experimental problem.
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Which experiments for example could provide the information on which of these two alternatives should be selected?
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Now, according to quantum theory, as I said a while ago, a very precise localisation, a very sharp fixing of the position,
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corresponds to a very large uncertainty of the momentum or the energy.
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It is therefore only possible to get information on this issue of local or non-local theory
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through the very high-energy collision processes between different elementary particles.
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Nowadays in cosmic radiation we can observe collisions between particles of energies
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of the order of magnitude of one thousand billion electronvolts or more.
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And it is precisely these collisions which teach us something about the question of whether we are dealing with a causal,
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with a local or non-local theory.
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I would like to compare three alternatives here and briefly state how they would become apparent in the experiment.
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One possible alternative would be that it was a strict contiguous action theory, a contiguous action theory
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with so-called small interaction, i.e. with an interaction which causes only a small coupling between different fields.
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If this were the case, we would expect that in the collision of two elementary particles overall only one new elementary particle would ever
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be re-radiated, one or in the worst case only very few, but that this one elementary particle can generally be re-radiated
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with very high energy, that around half of the total energy, for example, can be carried away in this process.
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Now, the experiments certainly do not look like that.
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It does not really look like this would actually be the case in nature.
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We could also assume secondly that the theory is a non-local theory, i.e. a theory where the elementary particles are smeared out,
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so to speak, over a region of let us say the order of magnitude of 10-13 cm.
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And in this case it does not really matter whether we assume that the interaction is very large or very small.
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It will then be assumed that in any case only relatively little energy can be transferred.
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It is however possible, depending on how things are, how the interaction is, that either a large number of elementary particles
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are created or few, but in any case it should never happen, or happen only very rarely,
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that very high energy is transferred to all elementary particles, because this smearing out over a certain finite space,
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this means of course a decrease of the probability for large energy and momentum transfer.
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But this is not how it looks so far, although possibly this case is not closed yet,
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only relatively little experimental material is available at present.
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And finally, one could think thirdly that the theory is nevertheless a contiguous action theory, but one with very strong interaction.
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One would then expect that in the case of the collision of two elementary particles a large number of elementary particles were created,
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but that these elementary particles are created relatively often with low energy and only relatively seldom with higher energy.
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In Göttingen some time ago I attempted to follow up this possibility and to describe this release of elementary particles in the collision
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of two high-energy particles in a similar way to the description of blast waves from the location of an explosion,
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i.e. as nonlinear wave equation, but nevertheless as a proper contiguous action equation.
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And here one obtains results which seem to fit quite reasonably with experience.
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I say seem to fit, because the files on this case are certainly not yet closed either,
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relatively little experimental material is available so far.
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These very high-energy collision processes are relatively rare, and when Mr. Powell recently showed you pictures of those collision processes,
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then you must not conclude that one finds hundreds of those pictures every day, but these have been exhibition items, so to speak,
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which have been presented to us because they are particularly beautiful.
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These collision processes must therefore be investigated in a much larger number, and also analysed, only then we will obtain complete
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clarity on the type of interaction of the elementary particles.
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I should perhaps mention here that other experiments have been done to expand the mathematical options, Snyder for example has attempted to
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introduce a quantisation of space and time, and Mr. Dirac has talked here about a way of extending relativistic quantum mechanics
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by assuming an absolute time variable.
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So at many, many different places attempts are being made to expand the mathematical forms in such a way
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that they finally match the experience, but these attempts are not yet complete.
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And now I come to the finish.
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I have described for you the difficulties and ways of getting out of the difficulties, and I would like to state at the end,
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what we can do in concrete terms, and what is de facto now being done in the world in order to solve these difficulties.
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Now, in a way it is essentially a problem for experimental physicists.
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We have to investigate these high-energy collision processes even better, we need to know which elementary particles exist.
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We particularly need to get to know their interactions, for example selection rules, which elementary particles cannot convert into
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different ones, and we need to draw conclusions from this about the symmetry relationships of the equation of matter,
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which we cannot yet write down, but which will certainly exist at a later stage.
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From the purely practical point of view, there are two options for this: the balloon flights,
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which Mr. Powell has talked about, and then large machines, and maybe I should mention in this context that just yesterday a European convention was
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signed in Paris, according to which the participating countries, including West Germany,
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decided to establish a joint atomic physics institute in Geneva,
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whose heart will be a very large machine which will make it possible to accelerate elementary particles to around 30 billion electronvolts.
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When this machine has been built, then it will surely provide us with incredibly valuable findings on the behaviour of elementary particles.
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As long as this machine does not exist, we will be dependent on cosmic radiation,
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or on the smaller machines which are already in operation in America.
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And in addition to all this experimental work, the theoretical scientists can also do something, of course.
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They can look for mathematical structures which at least represent qualitatively what we observe.
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Qualitatively we have a very good idea what happens in the physics of the elementary particles.
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We know there are quite a number of stable and some unstable particles, we know they can change into each other.
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Therefore it should be possible to at least describe mathematical schemata which represent this behaviour in a qualitative way.
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Nobody has succeeded so far in writing such schemata which make mathematical sense
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and at the same time correctly describe the qualitative behaviour.
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There is therefore a problem, and one can hope that it will be solved during the next few years, but which is not yet solved.
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But even if this solution is difficult, this certainly does not mean some kind of pessimism, it is still possible to assume that the future
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physics of elementary particles will look quite simple, only getting there is difficult; and it is very difficult to find these very special
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mathematical structures which will govern the physics of the elementary particles in the future.
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And with this I would like to finish.