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Ladies and Gentlemen,
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Mr. Menke just pointed out that I have already talked very often about fundamental particles here;
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in apologising for the fact that it is always the same topic I can say only that there is hardly any field of modern physics
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which is undergoing such rapid progress as the physics of fundamental particles.
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New experimental facts are being discovered almost every year, new theoretical ideas published
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and even since the last talk I held here 3 years ago, 10 new fundamental particles have been discovered.
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If one talks about a unified field theory of fundamental particles, or if one strives to produce one,
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one starts roughly from the following consideration: we know from the experiments during the last decades
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that fundamental particles can be converted into each other.
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The fundamental particles are therefore not, as was assumed in the past, the unchangeable eternal building blocks,
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the smallest building blocks of matter, instead they can be converted into each other.
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And, this is especially important, all fundamental particles can be converted into all others.
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By this I mean that if we take any two fundamental particles and fire them at each other with very high energy,
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new particles are formed and, in principle, particles of any type can be produced.
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The simplest description for this fact would be to say there is actually only one uniform matter,
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it could also equally well be called energy, and this matter or energy can exist in many different forms,
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which are called fundamental particles or just particles.
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There is no basic difference between fundamental and composite particles.
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This can also be expressed by saying: energy becomes matter by converting into the form of fundamental particles.
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If one starts from this general idea, it is also very natural to assume that there must be a law of nature
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from which the different fundamental particles with all their properties follow.
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I would like to talk today about a specific attempt at such a unified field theory of fundamental particles,
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which could maybe be characterised in the simplest way by starting with the equation which is to form the basis of such a theory.
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Could I please have the first slide? Here you see an attempt at an equation of matter.
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I only want to say a few words about the letters, and later not talk as much about mathematics as you may fear,
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but the quantity psi here on top represents matter so to speak, XYZ represents space-time
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and the gammas are matrices which Dirac introduced in connection with the representation,
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the mathematical representation of the Lorentz group, i.e. the properties of space-time, which we know from the theory of relativity.
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Maybe I should also say that re-writing this equation, which is here at the bottom, now appears to be useful in several ways,
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it illustrates the symmetry properties of the equation in a slightly simpler way still than the top equation.
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The specialist who deals with such things sees that this equation is invariant with respect to the Lorentz group;
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furthermore it says that this quantity psi, i.e. if one now represents matter, that it is a spinor,
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as the mathematicians say, in the space of Dirac’s spin variables as well as in the space of the so-called isospin,
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this will also be discussed later, and one can therefore see that both these group properties are contained in the equation.
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But this special form is not important at the moment, I want only to emphasize right at the start
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that it is not a particularly extravagant and unusual claim to believe that the whole complex spectrum of fundamental particles
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with all their properties results from an equation which looks so simple.
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After all, we are already very familiar with similar conditions from the theory of the atomic shell,
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which was the focus of interest around 40 years ago.
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Remember the complex optical spectrum of an iron atom with its many hundred lines of different intensities,
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different wavelengths, for example.
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And it took a very long while until it was possible to assign and explain this spectrum at all.
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Nevertheless, it is now known from Bohr’s theory of the atom and its mathematical specifications
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and quantum and wave mechanics that it is possible to write down a Schrödinger equation which looks simple,
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for the iron atoms, from which all these spectral lines with their intensities and other properties then follow.
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Now, writing down such an equation achieves very little, and when I talked about this 3 or 4 years ago,
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Pauli was justified in saying that this was only an empty frame in which the picture still needed to be painted.
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Because it was only then that the mathematical analysis started,
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one now needs to apply mathematical methods to try to determine what the statements of this equation really are.
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Whether the eigenvalues of the equation really make it possible
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to represent the empirically observed fundamental particles, for example.
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Two points of the theory which is characterised by this equation now extend very characteristically over and above
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the earlier quantum mechanics and earlier attempts at quantum field theory.
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Firstly, the mathematical axiomatic theory is different.
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We were forced to assume an extension of the basics such that the state space in which quantum mechanics operates does not have,
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as the mathematicians say, a definite metric, but an indefinite metric,
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and that the concept of probability thus initially becomes a problem.
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Secondly, it has turned out to be necessary to assume in this theory that the fundamental state,
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which in the earlier physics was considered simply as nothing, as the vacuum, as empty space,
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that this does not have the full symmetry of the equation, that it has physical properties,
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and must therefore really not be called vacuum, but world.
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And these two very radical changes have naturally caused many discussions,
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and the decisive progress has been achieved in these two points in particular in recent years.
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I would therefore like to divide up the topic of my report here into three parts.
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I want to first talk relatively briefly about the mathematical axiomatic theory on which this equation is based,
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I want to then talk in some more detail about the asymmetry of the fundamental state and the applications
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which one can draw from this assumption, and the theory of the so-called ‘strange particle’ and electrodynamics in particular.
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And then in the third part I want to discuss the comparison of what the theory allows us to derive
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and what has been found experimentally in recent years.
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So first the mathematical axiomatic theory.
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I want to first give you some history.
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When Einstein formulated the special theory of relativity, he made a radical change with respect to Newton’s mechanics.
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You all know that he put forward a different proposition on space and time.
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This different proposition on space and time had only become possible by him making a different assumption on action.
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While Newton claimed that there are forces acting at a distance, so-called forces with action at a distance,
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Einstein allowed only so-called short-range actions, i.e. really only actions from point to point.
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And this is the only thing which made it possible for the relativistic structure of space and time to be compatible
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with the idea of causality which we have to assume here.
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In this respect, the quantum mechanics which was developed during the 1920s continues Newton’s mechanics.
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It also uses forces with actions at a distance.
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But as soon as the physicists moved from the theory of fundamental particles to the quantum field theory,
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they were forced to make the same radical step which Einstein made back then from Newton’s theory to the theory of relativity.
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They were forced to move from action at a distance to close-range actions, to point interactions.
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This then had the consequence, as was observed in the 1930s, that the mathematics which apparently results
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as a natural generalisation of quantum mechanics in quantum field theory, that this mathematics leads to contradictions,
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that the values which are calculated become infinite, and therefore no longer have any meaning.
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There has been a long search for ways of getting out of this difficulty, and the most interesting proposal
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in my view was made in 1942, i.e. precisely 20 years ago, by Dirac.
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Dirac explains that one can avoid the mathematical difficulties, at least as a first approximation,
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let us say, if the metric in the state space is made indefinite.
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In order to not stop with this mathematical term, I would like to immediately come to the objection to this idea
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which was then raised by Pauli one year later.
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Pauli stated that if one uses an indefinite metric in Hilbert space, that the probability one calculates
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from the theory may then possibly have negative values, and this is nonsense, of course,
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because there is no such thing as negative probability.
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The difficulty appeared initially insurmountable and this proposal was therefore not pursued for a longer time.
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But around 10 years later, in 1953, we then took up Dirac’s proposal again in Göttingen.
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We hoped to be able to modify it in the following way.
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We need the probability interpretation of quantum theory where we experiment, of course.
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So if we place a detector somewhere with which we count fundamental particles,
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then what we require from the theory is that it will state the probability that fundamental particles will arrive there.
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Of this kind and that kind, of this frequency and that frequency.
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But it is not so important that we can apply the probability concept inside the atomic nucleus as well,
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for example, because we cannot measure inside an atomic nucleus anyway.
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What we measure are only ever fundamental particles which have covered long distances, i.e. into our equipment.
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It was therefore hoped that it would be possible to modify this theory so that the concept of probability
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could be maintained for fundamental particles which are separated by a large distance,
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i.e. for the so-called asymptotic behaviour of the waves to be dealt with in quantum theory,
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while the concept of probability had to be dropped for the conditions on the very small scale.
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Well, this idea was then tested on a very interesting model, which had been developed by the Chinese physicist Lee.
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And this represents, if you like, a very much simplified mathematical version of quantum electrodynamics.
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Pauli and Källén were able to show that if one treats this model according to the methods of quantum electrodynamics,
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that one then automatically enters into the indefinite metric
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and that one initially even gets into the difficulty with the probability concept as a matter of principle.
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It was then successfully shown that under specific assumptions, which have to be made in addition there,
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it can be arranged that the asymptotic concept of probability does become proper again, it is only lost on the small scale.
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You can see therefore that this whole question is very complicated, but in recent years
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in particular a number of papers have been published which have investigated in more detail
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whether this separation between asymptotic and local behaviour can be carried out,
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and at least different models have been provided where this is actually possible.
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I think it is more important that one can also provide a theoretical reason why this separation
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into asymptotic and local behaviour is possible.
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If one is also interested in the conditions in the atomic nucleus, i.e.
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takes all possible experiments into account, then one also has to represent all the symmetry groups
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which are observed in nature, in particular the Lorentz group as well.
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Now it is known that this group is a non-compact group, in mathematical terms,
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and the indefinite metric is a natural representation for non-compact groups.
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If one is interested in what happens at large distances, however, i.e. with the fundamental particles
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if they are far away from their collision point, then this group is reduced to a compact group,
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because one only considers processes which belong to a specific energy and a specific momentum.
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And for the compact groups, as I said, the definite metric as a probability concept is a natural representation.
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So it can be understood that the attempted division into asymptotic regions and non-asymptotic regions is reasonable here,
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and it may be possible to describe how the physicists view the problem at the moment in the following way.
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It can be said that there is firstly a conservative line of thought, maybe mainly represented by the American Whiteman,
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here in Germany by Nehmann, Simancik and others,
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which now attempts to connect this axiomatic theory as closely as possible with quantum mechanics.
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They try to manage with a state space of definite metric, assume the existence of a field operator
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and in order to represent causality they then say that the field operators should commutate and anti-commutate
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at points with space-like separation.
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A number of interesting results have been derived from this axiomatic theory, which also really fit in with experience
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and which obviously represent a part of reality correctly.
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On the other hand, it has not yet been possible to provide any mathematical model which satisfies all these axioms
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and contains a non-trivial interaction, i.e. it is doubtful whether the axiomatic theory,
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which is very narrow at this point, can be fulfilled.
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There is therefore a different line of thought which advocates a slightly more radical,
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opposite extreme and which was championed last year by Chew in America, for example.
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Chew gave a talk at the conference in La Jolla a year ago, which he called his “Declarations of independence”,
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where he said we don’t need the complete field theory, it would suffice to produce a mathematics
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which describes only this asymptotic behaviour of the particles.
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Now, to describe the mathematical tool, this asymptotic, is the so-called scattering matrix or S matrix
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and Chew said it should be sufficient to make a number of reasonable mathematical demands for this S matrix,
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e.g. in order to represent causality one had to demand that specific analytical properties existed
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and then one would not have to take care of the question of the metric or the particles at all.
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This is a very extreme point of view with the other line of thought, and maybe now I will ask for the second slide.
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I would like to briefly compare the different steps of the axiomatic theory which are possible here.
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The widest frame, which was championed by Chew last year, for example, is this axiomatic theory, which is here at the top.
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This assumes the existence of a scattering matrix, an S matrix, which must of course have the necessary group properties,
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symmetry properties, in order to represent nature correctly, and so that the demands of the relativistic,
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i.e. Einstein’s causality, are also maintained, this matrix must have specific analytical properties.
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This theory is therefore also very strongly based on the so-called dispersion relation, which is very well proven experimentally.
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It is possible to now narrow down this very wide framework, make it narrower,
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by adding something, by saying not only should this apply, but also requiring the existence of a local field operator,
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which in order to represent causality commutates and anti-commutates for space-like states
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and an appropriate Hilbert space should exist.
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At this point nothing is said about the metric in this Hilbert space, which can be definite or indefinite.
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And finally there is a third, even narrower axiomatic theory, in which both demands are made, both postulates,
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but a third demand is added, that the metric of the Hilbert space must be definite and that, in addition,
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the asymptotic states should be sufficient to open up this complete Hilbert space.
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This third axiomatic theory is therefore the narrowest, this is the one which I have called the conservative one,
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and that is the widest one, while the one in the centre is precisely the axiomatic theory
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which is the basis of this unified field theory, the subject of my talk here today.
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Now, at a Solvay conference in Brussels last autumn we also talked a lot about this axiomatic theory
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and provided this abstract, slightly mathematical issue with a physical aspect as well,
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which although it strictly is not part of it, makes the conditions very clear.
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If one assumes that the narrowest, the third axiomatic theory is correct,
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then one is actually forced to differentiate between fundamental particles and composite particles.
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Or let us say between real fundamental particles and only semi-fundamental particles.
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And the real fundamental particles are characterised by the fact that they have an infinitely hard and infinitely small,
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i.e. point nucleus in their centre, while the composite particles do not have this point nucleus.
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If one believes this last axiomatic theory, one must really differentiate between fundamental particles
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and those which are not fundamental.
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If one of the first two axiomatic theories is the basis, however,
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then it is very obvious to say there exists no fundamental particle at all which has such a hard, point nucleus in the centre.
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This question must be answered experimentally as a matter of principle,
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and the experimental question has a close relationship with what we have heard here at this meeting from Mr. Hofstadter.
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In Mr. Hofstadter’s experiments attempts are made to precisely measure the density distribution of a fundamental particle,
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the nucleon, for example.
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And it is possible to decide in principle whether the density distribution is such that there is an infinitely hard,
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point nucleus in the centre, which is surrounded by a cloud of matter having a diameter of around 10 E-13 cm,
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or whether the fundamental particle is only a cloud without a nucleus.
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In order to decide such a question, it is necessary to investigate collisions of extraordinarily high energy, however.
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It seems to be the case that, if one has a point nucleus, then elastic collisions still occur even
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at the highest energies possible with a considerable probability, whereas, if there is only a cloud,
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the elastic collisions become as rare as one wants at sufficiently high energies.
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I myself am convinced that fundamental particles with such a hard nucleus do not exist,
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but the question cannot yet be decided experimentally, because sufficiently high energies are not yet available.
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But maybe collision processes with the large machines in Geneva and Brookhaven
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will bring us closer to deciding this question in the future.
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And with this I would like to finish this discussion of the mathematical axiomatic theory
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and now come to a more physical part, the second part, i.e. to the question of the asymmetry of the fundamental state.
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The field theory, which I am talking about here, was forced to assume this asymmetry of the fundamental state
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for the following reason.
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There is a property in fundamental particles which is called isospin.
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This is a property which was introduced empirically 30 whole years ago.
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Mathematically it is a property, similar to the angular momentum,
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but graphically means the difference between a proton and a neutron.
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Two particles which have nearly the same mass, but one of them is charged and the other neutral.
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It was determined a long time ago, i.e. 30 years ago, that the interaction between fundamental particles
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is at least approximately symmetrical with respect to rotations in this space.
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If one therefore converts a neutron into a proton, the forces do not change in the first approximation.
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And therefore the fundamental equation, which you saw just now, is invariant in respect of such transformations.
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But this invariance does not apply strictly in reality.
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The electric charge already breaks this invariance, which you can simply see from the fact
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that a proton is charged and a neutron is not charged.
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One therefore has to draw the strange conclusion that there are symmetries in nature which only apply approximately.
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And the natural interpretation for this is that one says the fundamental state of the world itself is therefore not symmetrical,
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but the world has a large isospin.
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Now this means empirically that, for example, the number of neutrons and the number of protons
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in the world are very different and this is indeed the case.
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This asymmetry is what makes it possible to understand that a proton and a neutron have a slightly different mass,
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because the proton is then a particle whose isospin is parallel to that of the world, while the neutron’s isospin
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is antiparallel to the world.
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This forced the scientists to assume the asymmetry.
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Then there was something else: if the fundamental equation is initially taken as it is written here,
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one would expect that a particle which has half-integer Dirac’s spin also has half-integer isospin,
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or, if it has an integer Dirac’s spin, then it also has integer isospin.
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But in reality there are so-called strange particles where this is just not the case.
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In order to explain those, it is therefore necessary to again take into account the asymmetry of the fundamental state.
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This had been a slightly audacious assumption, but it has meanwhile fortunately turned out that precisely
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the same conditions apply in other fields of physics, and we have learned a lot from the development
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of the theory of superconductivity, on which Mr. Bardeen gave a talk here.
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It turned out that the conditions are the same in superconductivity, i.e. the fundamental state does not have the full symmetry
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of the equations, but is degenerate, has a lower symmetry, does not possess the so-called gauge symmetry.
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And then Bogoliubov correctly pointed out that basically such conditions had been known earlier at many places
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and had simply been no longer thought of.
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It has already been assumed in the theory of ferromagnetism that the fundamental state of a ferromagnet
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has a magnetic moment, i.e. a direction, although the equation from which one starts has rotational symmetry.
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In the crystal formation, in the theory of superfluidity, we have the same conditions everywhere.
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Nambu in particular showed that even purely mathematically there are many similarities between
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the theory of fundamental particles and the theory of superconductivity.
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The scientists gained a kind of mathematical practice ground in the theory of superconductivity and the theory of ferromagnetism.
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We know that there are no fundamental mathematical difficulties in these two areas,
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we are on completely solid ground there and can now look how far the mathematical assumptions
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which we made in the theory of fundamental particles also apply here.
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And it did indeed turn out that very many features which had been assumed in the theory of fundamental particles
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could now really be proved here mathematically,
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i.e. the conditions can really be as they had been assumed to be in the theory of fundamental particles.
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The following was also very encouraging: in the theory of fundamental particles we were forced to use an approximation method
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for the numerical calculations, and this was very problematic .
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But it was the only one which could be applied at all to these types of field theory.
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It was possible to use the very same approximation method in the theory of superconductivity in order
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to see whether it provided good results there, and it turned out that it provided good results even there,
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that it is practically as good as the exact calculation.
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In this respect the theory of superconductivity provided a great deal of assistance for the theory
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of the asymmetric fundamental state for the fundamental particles.
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Now I want to discuss more specifically the theory of the strange particles which has been developed by Dürr and myself meanwhile.
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In order to understand the strange particles in their slightly weird property
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Wentzel once introduced a term which he called spurion.
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This spurion was a slightly weird fundamental particle, i.e. it was not a particle at all,
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e.g. it had neither energy nor momentum, nor a position, it had only an isospin and a parity.
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The only illustrative comparison I can provide is the cat which appears in the English tale of Alice in Wonderland,
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although this is a bit mystical: it talks about a cat which disappears into a mirror, and first the tail disappears,
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and then the body, and then the head of the cat, and only the sneering grin of the cat remains in the room.
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Now the grinning of the cat is the isospin of the spurions, so to speak.
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Now this idea, which Wentzel made purely phenomenologically, was the result of the mathematics of the field theory,
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which I am talking about, of its own accord, i.e. it turned out that this mathematics invents the spurions of its own accord,
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i.e. this property of the degenerate vacuum so that all assumptions that Wentzel made with his term spurion
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can be found again in the mathematics and vice versa; this mathematics could now be tested again
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in the theory of ferromagnetism or superconductivity.
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Maybe the simplest comparison is the one from the theory of ferromagnetism, where we therefore then need to replace
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the isospin with the spin.
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One can therefore also imagine the following in a ferromagnet: let us assume an excited electron,
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which borrows a spin 1/2 from the total magnetic moment of the ferromagnet
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so that the spin of the electron is no longer 1/2, but 1.
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It is therefore quite possible by attaching a spin wave to the electron in this way,
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that in a ferromagnet electrons orbit which do not have this spin 1/2 as they should, but spin 1.
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I do not really know whether such ferromagnets have actually been produced,
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I do not believe that it has been observed, but in principle nothing prevents this being possible.
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Calculations have been performed with the help of this spurion term and I want to provide a few more equations,
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but then quickly move on.
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Could I please have the next slide.
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Now, if one takes this idea of the spurions seriously, then one must, in order to describe a strange particle,
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expand Dirac’s equation of the nucleon, which would include only this term, gamma mu p mu and by this term,
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the i kappa, one must and can expand it by precisely two terms.
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Other terms are not possible due to the symmetry.
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And this one term provides so-to-speak an interaction of the parity of the spurion, like the parity of the nucleon,
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and this term provides an interaction between the isospin of the spurion and the isospin of the nucleon.
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If this equation, which contains only two constants which still need to be determined, is solved,
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then one obtains such an equation for the energy, i.e. the mass of the spurion.
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This still depends, as I said, on the two constants alpha and eta, which have to be determined with the aid of other requirements.
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I would like to ask for the next slide.
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If one calculates masses with this equation, then one can see that the following happens:
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as long as the two interaction constants alpha and eta are zero, for example,
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we have here the eigenvalue, i.e. the mass of the nucleon.
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If we then take only the parity interaction, the nucleon, this state splits up,
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this nucleon state which slightly changes in its mass here at the bottom remains, and there is only one higher state,
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which can be interpreted descriptively, i.e.
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as nucleon + spurion, but this state has a four-fold degeneration, and if we then also take into account the eta interaction,
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i.e. the isospin interaction, then this state splits up into four states.
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We should then expect that there exists a strange particle above the nucleon, a hyperon.
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This here is to have isospin zero, this here isospin 1, above this there should now again be an excited particle
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of isospin 1 and again of isospin zero.
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The two particles at the bottom are experimentally known, this here is the so-called Lambda hyperon and Sigma hyperon.
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At the time, the two top states were not yet known.
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But it is possible that they have now been found, I will talk about the experiments later.
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If one were to calculate this more accurately than was done in these initial equations,
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these levels could of course be shifted slightly with respect to each other,
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because the constants alpha and eta can again depend on the energy themselves,
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but in a first approximation the split looks like this.
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I should possibly say here that it has been possible to determine the constant alpha correctly
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in the numerical calculation according to the Tamm-Dankoff method, albeit in approximation.
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The second constant eta, which is very much smaller, could not be determined reliably, however,
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because the second constant changes very strongly even if there is only a very small error in the first constant.
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It depends very sensitively on the first one so that we can determine this constant at best with an inaccuracy of factor 4,
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and then we proceeded in practice such that we have preferred to take the experimental value, i.e. the value which best represents the experiments, but I must say that even if the constant were larger by a factor 2 or so,
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the picture would not be fundamentally different.
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The next slide, please.
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The same calculation was then made for the so-called bosons, these are therefore particles such as the Pi meson etc.
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and for them there are two states without the interactions of the spurions.
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This one here is an isotriplet, this is a Pi meson, the second state of the isosinglet,
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it had not yet been found when the theory was developed.
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It seems to have been found now, and this state seems to be the so-called Eta meson.
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If eta is still zero this state splits up into two, into three and others also into three,
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and if this constant eta is then taken into account, the ......