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Ladies and Gentlemen,
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Over the last years, a great deal of new experimental material has been collected about elementary particles.
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Along with the long-known building blocks of matter: electrons, protons and neutrons,
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which we have already known about for more than a quarter or half a century,
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today we are familiar with a host of similar structures, we can say around 25 new, different types of elementary particles
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which in many of their properties appear similar to the elementary building blocks of matter,
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but which are generally capable of existing for only a very short time,
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and which frequently decompose radioactively already after, shall we say, a hundred-millionth of a second.
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This wealth of experimental material about elementary particles presents theoretical physicists with the task of investigating
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and finally formulating in mathematical terms the natural laws that determine the structure,
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in other words, the existence, and properties, of elementary particles.
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I would first like, in my lecture today,
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to deliver a critical analysis of the developments in this field during the last approximately two and a half decades,
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and in the second part of my lecture I will then address an attempt
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that our institute in Göttingen has made over recent years to clarify such phenomena relating to elementary particles.
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I would like to come straight to the point here and say that,
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with this very complicated field, we are not concerned for now with finding definitive solutions,
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but that at best we can locate the place where such solutions should be sought.
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And this is to be the task of the second part.
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But let us first consider the historical development:
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already from a very early stage, from around the time when quantum mechanics was completed, towards the end of the 1920s,
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it was very clear that we could only arrive at an understanding of elementary particles
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if quantum theory could be combined with the theory of wave equations,
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in other words, if one can quantise so-called field equations.
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The fact that this is so can actually already be derived from Einstein's works, from the early period of quantum theory,
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which showed that the application of quantum theory, for example to the Maxwell electromagnetic waves area,
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results in the existence of light quanta.
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Thus, this example showed that the application of quantum conditions to a field theory
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explains the existence of the related particles.
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So, the path ahead seemed clear; quantum theory needed to be applied to wave equations,
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and specifically to the various wave equations that were known at that time,
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with the hope of gradually moving forward to an understanding of elementary particles.
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At the start, this path was also pursued without – or so it seemed – major difficulties.
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As in the first works – I cite for example work done by Jordan, Klein and Wigner,
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and then by Pauli and Jordan, and by Dirac.
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In these works, the application of quantum theory to field equations
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appeared to be a clear mathematical process that initially also appeared to lead to reasonable results.
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But then quite rapidly, early in the 1930s, major difficulties emerged.
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Weisskopf was able to prove that, when consistently applying this quantisation rule to waves, infinite self-energies,
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namely infinite electron masses, emerged, in other words, mathematics diverges here,
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and fails to deliver any meaningful results at all.
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Despite this very major difficulty, a very large number of rational physics results were gained from these theories.
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If, for example, one structured the theory as a perturbation theory,
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and then discontinued this development at a sufficiently early stage,
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that is, the calculation was so to speak consciously made imprecise, then rational physics results emerged.
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I am thinking of Dirac's theory of radiation, theory of resonance lines, dispersion and so on.
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Especially with the application of quantum theory to electromagnetism, lots of good results were achieved
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that have stood the test of experience, while at the same time it was clear
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that consistent application of these mathematics would lead to nonsense, namely to infinitely large equations,
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in other words, to diverging equations.
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And at the start, it was naturally assumed that these divergences, these mathematical infinities,
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were somehow due to clumsiness on the part of the calculation.
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For instance, it was hoped that a better perturbation theory
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or a complete move away from perturbation theory could result in convergent, that is, to rational results.
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In fact, however, during the 25 years that have since elapsed,
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no real progress has been achieved as far as this point is concerned,
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and we now have every reason to assume that we are dealing here with a very fundamental difficulty
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that can be solved only by fundamentally new methods, in other words,
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by moving away from the fundamental preconditions of physics that have applied to date.
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The fact that we are dealing here with a fundamental problem is something
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that we can perhaps now clarify in the following manner.
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We can say that a conflict arises here between quantum mechanics,
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whose characteristic feature can perhaps be regarded as the uncertainty principle,
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and the space-time structure of the special theory of relativity about
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which we heard yesterday in the lecture given by Mr. von Laue.
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And we are concerned here with the following conflict: the uncertainty principle of quantum mechanics states
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that if one wishes to very precisely determine the location of a particle or a system,
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you have to accept a major uncertainty in terms of determining the impulse,
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or if you wish to precisely determine the point in time,
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you have to accept a very major uncertainty in terms of energy, and vice versa.
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In the space-time structure of relativity theory, however,
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the most important postulate is the idea that all effects are propagated at only the speed of light.
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That, so to speak, a clear barrier exists between those events over
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which one can exert no influence as far as the future is concerned,
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and those events over which it is certain that one can no longer exert an influence,
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precisely because a light wave can no longer reach the related event point.
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Or vice versa in the past: about certain events, we can generally know something
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because a light wave could still reach us from this point.
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About others it is quite certain that we cannot learn anything
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because in this case the light wave could no longer reach us.
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In other words, the distinction in this manner between future and present, or between past and present,
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is absolutely clear in the theory of relativity.
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And this was already expressed 50 years ago when the theory of relativity was being developed,
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when it was said: in the theory of relativity, we cannot assume any actions at a distance ,
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we can allow only for so-called locality, in other words, an effect from a point to a neighbouring point,
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so that this type of effect is propagated at the speed of light.
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Now, it is precisely such an absolutely clear distinction between the present and the future, for example,
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or between the present and the past, that necessitates,
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according to the uncertainty principle of quantum theory, an infinite lack of clarity of the impulse or the energy,
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resulting in infinite impulses and energies.
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So this without any doubt correctly describes the root of all these problems.
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Of course, such a simple argument is insufficient to prove that these things cannot really be sorted out.
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But the fruitless efforts of the last 25 years have made it very likely
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that it is impossible to eliminate the contradiction that exists here by, so to speak, a mathematical sleight of hand.
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So this difficulty has simply continued to exist for the time being,
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although the approximation methods of perturbation theory have nevertheless allowed many good results
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to be derived from this quantum field theory.
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I want to outline just a few important steps from the subsequent development.
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For example, this quantum theory of waves was applied to beta decay, which was the Fermi Theory of Beta Decay,
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thereby providing a good explanation of the distribution of energy,
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the relationship between energy and lifespan in the case of beta decay.
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This theory of beta decay then also provided a very important distinction
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for the subsequent period between two different types of interactions.
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And since I wish to return to this point later,
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I would like to briefly characterise these two types of interaction for the time being, and simply introduce a name for now.
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I want to refer to one interaction as being of the first type, and to the other interaction as being of the second type.
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The first type of interaction is characterised by the fact that when several elementary particles,
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or, let us say, two elementary particles, collide,
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so that the interaction either declines with the growing energy of these particles,
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or remains constant, or at least doesn't increase.
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This also means, at the same time, that the interaction varies with these particles' wavelength so that,
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given decreasing wavelength of the related waves, it then either remains constant or diminishes.
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The interaction of the second type has the opposite property,
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meaning, if the energy of the interacting particles increases, the interaction also grows.
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And this initially has the following important consequence in terms of physics:
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assume that, in the case of small energies, such interaction is also small.
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This then means that within the mathematical formalism of quantum theory, for example,
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in the case of the collision or deflection of particles, a new particle, for example a new light quantum,
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is always generated, as one particle occurs in the first approximation in perturbation theory,
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two in the second, three in the third approximation.
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As the approximations now converge in a good way, however, almost always only one particle is emitted.
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And this would also be the case however high the energies were in the colliding particles,
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because in the case of interactions of the first type, the interaction does not increase with growing energy.
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Only individual particles could then be generated in the emission.
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But if the interaction increases with growing energy, then it is quite certain that, given high energy,
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we will eventually reach a point where the interaction has become so great that a perturbation calculation no longer converges,
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meaning that the first and the second, the third and the hundredth approximation will all be approximately equally great.
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And this means that it is also equally likely that as many as 20 particles are generated at once, and not just one.
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This depends on the energetic conditions of course.
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So the interaction of the second type results in multiple particle generation.
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Meanwhile, a number of experiments have shown that these interactions of the second type do occur in nature,
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for example when generating so-called pions from the collision between nucleons, between nuclear particles.
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Now I would like to mention a further, slightly different type of result.
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In other words, this differentiation between the two interactions, this was still before the war,
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whereas the one that I am about to talk about now came somewhat later.
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One had, as I have already said, the impression
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that this fundamental contradiction between relativity theory and quantum theory
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could be bridged only through modifications to the theoretical fundamentals.
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This naturally gave rise to the question: well, what will then remain at all of the previous theory if we modify the fundamentals?
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And then it became clear that certain mathematical quantities do exist that one will always need to utilise
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in order to describe the experiments, and which consequently, so to speak, form a stable component of existing theory.
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Physicists refer to this mathematical structure as the S-matrix or the scattering matrix,
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and this quantity can be explained as follows: when elementary particles collide,
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each collision potentially generates new particles, and the collided particles are deflected and so on.
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And it is probably very difficult to describe in detail what occurs during each collision.
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It is quite certain, however, that one will have to describe which particles enter, and which particles exit, the collision.
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In other words, the asymptotic behaviour of the waves in infinity, namely both the incoming and outgoing waves,
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we need to present this in a secure mathematical form, and the mathematical measurement
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that performs this was referred to already in the earlier quantum theory as the scattering matrix.
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One could therefore study the mathematical properties of this scattering matrix, and from the relationships
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that resulted one could learn something about the behaviour of the quantities that really occurred in experiments,
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including, for example, the cross sections, the deflection of particles, the forces and so on.
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With this, one had a mathematical basis for a theory of which we still had no knowledge, insofar as we knew
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that this theory would include at least such a scattering matrix,
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so there must be a general mathematical formalism that allows one to derive this scattering matrix.
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And vice versa, the experimental physicist who studies the collision of elementary particles will, in the final analysis,
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be able to analyse his experiments in such a way
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that he writes down as the result the so-called matrix elements of this scattering matrix.
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Then, in the period after the war, during the first years after the war,
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the following discovery brought about important progress in this whole area:
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already before the war, Kramers for the first time proposed the idea that the mass and the charge,
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for instance of the electrons in quantum electrodynamics,
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are affected by the interaction between the matter field and the radiation field.
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And this means that if one characterises the interaction between matter field and light field in the equations,
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and if one originally utilises in this equation a mass m0 or a charge E0 for the electron,
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then these equations, if the entire calculation is performed, will generate another mass or another charge.
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In fact, it even became clear that, if you calculate correctly, this mass and charge then become infinite,
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in other words, actually meaningless in terms of physics.
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Kramers had once proposed the idea that this entire mathematical formalism should be reversed so that,
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so to speak, the final mass and final charge that emerge at the end be identified as constants in terms of physics,
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and that the quantities m0 and E0 that one starts with should be left indeterminate for the time being.
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Now this renormalisation program, as we refer to it, was taken up with the greatest success after the war by Bethe,
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and Bethe was able to show that, if you turn things around
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so that the infinities that had previously hampered any progress in the theory disappear for the time being,
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then you get convergent results, and also results that were extremely interesting in terms of physics.
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Meanwhile, the American physicists Lamb and Rutherford had namely shown
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that the old formulas for the fine structure of hydrogen lines were not exactly precise,
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and that characteristic deviations existed here between experiment and the earlier theory,
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in other words, between the Sommerfeld formula and the Dirac theory.
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Precisely these deviations could now be clarified through Bethe's renormalisation with the utmost accuracy,
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in fact with quite astounding precision, to a number of decimal points,
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the fine structure of the spectral lines could be calculated and found to be in accordance with the experiment.
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This was a very major success, because it showed
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that these quantum electrodynamics were actually much better than had originally been believed,
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or could have been hoped originally.
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And from this, from this really major success, a very far-flung hope now arose.
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The hope arose namely that this renormalisation process would now rid the theory of all of the infinities
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that had proved so disruptive to date.
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Roughly the following idea was proposed: one will be able to distinguish between two types of theory
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–between those where this renormalisation process feeds through to finite results, and those where this is not the case.
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The latter, however, were from the outset not taken into consideration for the elementary particles.
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And it then emerged that the differentiation between these two types of theory
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fell directly into line again with the distinction between the two types of interaction,
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in other words, what I referred to previously as interaction of the first and the second type.
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Theories with an interaction of the first type can be converted into convergent theories through renormalisation,
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theories of the second type, with interactions of the second type, cannot.
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So, for the time being, one simply shrugged off back then the fact
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that this interaction of the second type clearly occurs in nature.
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People were, nevertheless, not so sure about this, and formulated the hope that only theories of the first type could exist,
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that the theory of the first type reflects nature correctly, so to speak,
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and that this would remove the infinity and the mathematical problems.
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As a result of this development, the mathematical structure of this quantum field theory was investigated very precisely.
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I recall work done by Tomonaga, Schwinger, Feynman and many others.
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And we now have a much better overview of the structure of these entire theories than back then.
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But the result was that the hopes of which I've just spoken, that they were quite unjustified.
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And it is particularly thanks to the work of Pauli and Källén,
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who proved that this renormalisation process does not get us past the fundamental mathematical problems
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that we have known since 1930 in quantum field theory.
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It became clear namely in the works of Pauli and Källén, that if you conduct this renormalisation process,
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at least in a simple case where you can have a real overview of the mathematics,
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the renormalisation process then results in a deviation from previous quantum theory due to the fact
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that the interaction that stands in the Hamilton function has actually become imaginary.
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This then meant that the S-matrix, in other words, the scattering matrix, is not unitary,
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which in un-mathematical language means that one would be forced to introduce negative probabilities,
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and this is so to speak logically nonsense, and can't be done.
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So theories of this type cannot be interpreted in terms of physics.
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Another way of expressing this problem – as set out in the work of Pauli and Källén – is
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that this type of renormalisation results in new states that Pauli und Källén refer to as "ghost-states",
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because they do not behave rationally vis-à-vis physics.
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And these states modify the metrics in the space of these states, in the so-called Hilbert space,
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in such a way that only the metrics become indefinite, and this again results in negative probabilities.
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So in other words: although the renormalisation process removed the infinities from the theory,
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it introduced negative probabilities, which is absurd in terms of physics, and is insufficient to interpret experience.
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The question as to whether this is now the case in all such renormalisation theories, such as in the simple Lee model,
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which Pauli and Källén dealt with, is still open.
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Here, it has not yet been possible to develop the mathematics far enough.
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But it is probably at least likely.
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Now the result that emerged from these mathematical investigations was actually also quite pleasing in terms of physics
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insofar as more recent experiments had meanwhile shown
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that these quantum electrodynamics proved incorrect in the case of large energies.
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Processes have meanwhile been observed where, for example, many gamma quanta arise from a single collision process.
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But if this is how quantum electrodynamics are, or are how they would be,
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in other words, with interactions of the first type alone, convergent mathematics and so on, then this shouldn't occur at all.
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So the fact that, at times, many gamma quanta are generated in collision processes,
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that is many light quanta are generated at once, already proves
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that quantum electrodynamics are not at all how one previously imagined.
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So, to this extent, the results of mathematical analysis and the results of the experiment fit well together,
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but only in the negative sense, whereby there is something a bit wrong with quantum electrodynamics.
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We can now ask: is this actually a very unsatisfactory situation, or is this a satisfactory situation of theoretical analysis?
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Now, if we assume that this hope was to be justified, could we get from there to a theory of elementary particles?
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And here we must actually reply straight away with a "No".
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Because then we would need to say something along these lines:
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in order to one day understand the masses of elementary particles, one would jot down – as the physicist says –
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something like a big Hamilton function, in other words, a big expression for interactions
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where one introduces a wave function for each type of elementary particle.
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And then you have some complicated interactions.
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And then you would prove that the theory can be renormalized only for quite particular masses of these elementary particles
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and for certain quantities of these interactions, thereby generating the entire masses of the elementary particles.
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So this is how they would be explained.
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Now we can see immediately that you can only reach this through a detour through a quite monstrous set of mathematics,
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because we already know about 25 different types of elementary particles.
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And what the mathematics would look like when 25 different wave functions are introduced,
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anyone who has previously had anything to do with such mathematics can imagine.
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So, this is clearly a hopeless start.
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And I would like for this reason …, yes, perhaps I should firstly also say the following:
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over the past years, particularly after these problems occurred in renormalisation,
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the increasing tendency has consequently been to be basically interested in only the scattering matrix,
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in other words, in only that which the experimental physicist directly delivers.
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And a number of interesting mathematical relations for this scattering matrix have been found,
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partly in connection with the causality requirement as it occurs in the theory of relativity,
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partly in connection with certain dispersion relations.
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But I don't wish to go into these details here.
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I would like to just say by way of criticism concerning this whole type of treatment of physics,
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that it roughly looks as if the following situation were to have existed in 1900: let us assume
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that although the principles of quantum mechanics were known in 1900,
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that nothing was known about the Bohr atomic model,
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and that nothing was known about the Coloumb forces between electrons and proteins and nuclei.
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Then one would have probably proceeded similarly to now, one would have introduced a wave function for the oxygen atom,
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another for the carbon atom, a third for the hydrogen atom and so on,
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in other words, constantly new wave functions for various atom types.
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One would have then been able to study the S-matrix, the scattering matrix for the collisions,
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thereby sorting out the gas kinetics.
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One would have been able to discuss the Ramsauer effect as a resonance phenomenon and so on.
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But it is quite clear that all such efforts would not have brought us any closer to the crux of the problem at all.
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Because the actual problem was to understand the structure of these atoms
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or, as we now know, to separate them into a statement about electrons and nucleus.
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So in other words: in order to break through to the crux of the problem, it is clear that something else needs to be done,
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which brings me on to the second part of my lecture.
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I would first like to discuss some principles according to which, as I believe, we now need to proceed.
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First, it seems to me quite certain
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that in such a theory one cannot start by introducing wave functions as primary quantities for any of these elementary particles.
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So, for example, the wave function for protons or neutrinos or electrons or the like.
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Because in this real theory of elementary particles,
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the particles should emerge as the solution of a system of equations, meaning, you cannot plug them in here.
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In other words: if the wave function can be written out at all in this theory, which is naturally the question,
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but if this is the case, then it can only be as a wave function for matter,
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whereby it is entirely undecided whether this matter manifests later as a proton, meson, electron etc.
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It must therefore relate to a wave function and a wave equation simply for matter,
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and not remain for a particular type of elementary particle.
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Secondly: the mass of the elementary particles must be a consequence of the interaction.
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In any case, it must be closely connected with the interaction.
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Thus, it certainly makes no sense to write a linear-character equation
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that expresses absolutely no interaction.
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Because mass is a consequence of the interaction.
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So, it will be most correct if we write a formula in which there is only interaction,
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and where we can hope that the mass arises as a consequence of the interaction.
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In other words, we will have to write a nonlinear wave equation for the matter that looks as simple as possible.
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As simple as possible – for now, one has no real argument for this in terms of physics.
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We can only say that, it has always been the case, in the final analysis,
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that in physics to date, the basic equations were simple.
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Why nature has arranged things in this way is something that perhaps no physicist has ever fathomed out,
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but the entire science of physics is based on the hope or conviction that this is possible in the end.
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And now to turn to a solution for this type, we have, in Göttingen, investigated a particular equation
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which, although it is certainly simpler than the equation that leads to the real system of elementary particles,
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should already contain many elements of the real theory of elementary particles – this is the hope in this case.
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Now, matter …, the wave function of matter that is utilised here, it is the so-called spinor type,
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meaning it belongs to the half-integral spin,
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and this is required because nature includes particles with half-integral spins and whole integer spins.
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If one wishes to explain both particle types from one equation, one consequently needs to start with a half-integral spin.
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So, it must be an equation for a spinor wave function.
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And perhaps the simplest equation of this type is what I have written here on the blackboard.
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This equation is of the familiar Dirac type: Gamma^My * d Psi / d x^My, this would be so to speak the Dirac equation,
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and then comes a simple, nonlinear term.
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Now, this equation as it stands here would initially be, as we would say, a classic wave equation,
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and it only becomes a quantum theory equation through the process of quantising, in other words,
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through the introduction of commutation relations.
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And this is where the actual difficulties start: for as I have already said,
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when we quantise according to the normal rules of the game,
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then this is where the infinities arise, or ghost-states, or other mathematical contradictions.
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So, clearly, if we wish to lend such an equation any type of quantum theory meaning at all,
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we need to modify the commutation relations, and do something that isn't done in normal quantum theory.
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And here we have been guided by the following point of view: this commutation relation is first written one line below,
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perhaps not readable for all, but I can say in words what its essential aspect is.
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This commutation relation, namely the commutation between the wave functions at the two points – x and x' –
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if the causality requirement of the special theory of relativity is to be satisfied,
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this commutation relation must have the property that it disappears for space-type distances between the points x und x'.
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Because no effects should be propagated here.
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It should differ from zero for time-type distances, and now the question arises as to what should be on the light cone itself.
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And this is where we immediately encounter difficulties.
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If we are applying normal quantum mechanics, in other words, always observing only the transitions,
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for instance from the vacuum until the creation of a particle and back again,
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then one can prove that this commutation function becomes infinite on the light cone in such a way
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that the integral above it becomes finite.
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In other words, as physicists say, a Dirac delta function arises on the light cone.
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And in the case of this type of nonlinear theory,
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this consequence of quantum theory already results in contradictions and difficulties.
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And for the following reason: this commutation function, whose general character I have already depicted,
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should actually behave similarly, for reasons I cannot go into here,
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to a solution for the wave equation, the classic wave equation, which assumes a singular point.
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Because if I start at some point with a singularity, then according to the classic wave equation
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and according to the theory of relativity, effects arise only in the future and past cones,
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while the wave function is zero for space-type distances.
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So the commutation function should actually behave for reasons, which I cannot cover in greater detail here,
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similarly – in other words, in correspondence with – this propagation function.
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We often express this with simple words: "The commutator should be identical to the propagator".
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Propagator refers to this propagation function of the classic wave equation, and commutator refers to the commutation function.
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Now, if in the case of the nonlinear equation, one requires that these two should harmonise,
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and I believe that one should require this conclusion,
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then it is impossible to harmonise this through the commutation relations …
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with the commutation relation of normal quantum mechanics,
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then namely the Dirac delta functions should not occur on the light cone,
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and one is then prevented from doing anything except, in addition to the states that exist in terms of physics,
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introducing non-physics states, in other words, to a certain extent ghost-states,
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which ensure that this delta function does not occur on the light cone.