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Ladies and Gentlemen,
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I would like to speak today about an attempt at a theory of elementary particles, a unified theory of elementary particles,
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and I would first like to say a few words perhaps about the physicists who worked on this.
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First I would like to mention the contribution made by my friend Pauli, who unfortunately passed away much too young,
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and who we all sorely miss here. Pauli had, following its discovery by the two Chinese physicists Lee and Yang,
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taken a renewed interest in the elementary particle he himself had predicted about 30 years earlier, namely the neutrino.
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And he had discovered a new symmetry property in the wave equation of the neutrino.
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Now, the importance of symmetry properties for the smallest components of matter is known not just to physicists,
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but even to philosophers, who can read it in Plato.
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Thus, the symmetry property is one of the most important quantities or one of the most important things
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that we can talk about in physics today, just as 2000 years ago.
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And Pauli had, as I mentioned, discovered a new symmetry property in the neutrino wave equation.
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At about the same time, or even the previous year, we were involved with non-linear spin theory in our group in Göttingen,
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a theory which was thought to be a model for a subsequent theory of elementary particles.
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Following the discovery by Lee and Yang, I had attempted to incorporate their thinking into this non-linear spin theory
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and had come upon an equation that struck me as particularly simple, actually simpler than the equation
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that I had previously investigated.
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And after having given a lecture in Geneva on this equation about two years ago, I visited Pauli in Zürich.
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And Pauli immediately discovered then that the new equation was also invariant with respect to his symmetry group.
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And with that arose the possibility now for the first time that this extraordinarily simple equation -
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or I would like to say, very simple equation in a certain sense - could account for all of the aspects of elementary particles.
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Pauli was enormously enthusiastic about this new potential at first, but then afterwards was very disappointed
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because further difficulties indeed became apparent that he could neither intially solve nor answer.
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We discussed this equation a great deal one year ago in Geneva and even more extensively in Italy, in Varenna,
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at the so-called summer school of Italian physicists, and were in complete agreement about all the details of the theory
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that had already been worked out.
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But Pauli judged the subsequent possibilities more pessimistically overall than I did.
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However, in the middle of the discussion, which we always conducted in letters, he unfortunately passed away in December.
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In the meantime, a lot of mathematical work has been done in conjunction with this equation,
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and I would also like to now mention at this point the names of my colleagues in Munich: They are Dr. Dürr, Dr. Mitter,
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Dr. Schlieder, and Dr. Yamazaki. When I lecture today about this field, or more correctly,
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about the new results in this field, then I would like to divide them as follows:
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I first would like to explain the fundamental thinking behind this theory from a very general point of view.
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However, I do not want explain the mathematical details at this point,
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which would be difficult to understand for such a broad audience, but would rather mention, in somewhat greater detail,
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the difficulties that still existed in this theory a year ago, and also state what answers to these difficulties
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we believe we are now able to give. And then I will go into the newer results that have been worked out by me and my colleagues,
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whom I have just mentioned, which will probably appear in detail in a German periodical a few days from now.
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Firstly, therefore: What are the basic ideas behind this theory?
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May I request the first image please, in order to clarify the problem?
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In this picture, which is a type thoroughly familiar to physicists, I would like to explain briefly
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what sort of problem is involved.
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Physicists research elementary particles by means of very energetic elementary particles
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that are either drawn from large machines or from cosmic radiation, and let these particles collide
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with other particles or with atomic nuclei. The particles split the atomic nucleus and thereby create new particles.
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I want to briefly explain what is roughly involved in the one case visible in the photographs here.
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Thus, for example, a proton from the left above collides with a proton in an atomic nucleus here.
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You can see that a large number of particles are ejected from this atomic nucleus.
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Most of them, namely the particularly wide black tracks, are protons that previously made up the atomic nucleus
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as its key components.
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However, there are also individual, narrower tracks, and you can notice for example a track moving perpendicularly downward -
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that is an elementary particle of a type that was discovered just about six to eight years ago, known as a tau meson.
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I may also perhaps mention: the photograph here comes from a balloon expedition that has been carried out jointly
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by English, Italian and our physicists from our institute in Sardinia, Italy.
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In this experiment, photographic plates are sent to high altitude and exposed there to the effects of cosmic radiation,
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and an image like this is obtained afterwards through microscopic examination of the plate.
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This elementary particle moving perpendicularly downwards, the tau meson, continues along in the photographic plate
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and at a subsequent location was photographed once again. You see this on the right.
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Again on the right, this particle enters once again from above and makes a pair of collisions at the point
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where its track becomes wider and displays a curvature, you can hardly see the collisions on the plate,
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and then finally comes to rest. After coming to rest, it decays into three more particles known as pi mesons,
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each of which decay again into a mu meson and an invisible neutrino.
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The mu meson in turn decays into an electron and two invisible neutrinos.
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You therefore can see an example here in which elementary particles are created through this kind of decay process,
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and that then decay radioactively and thereby mutate into other particles.
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This special image that has been selected here also shows yet another elementary particle whose discovery was somewhat later,
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I believe, about five years ago. That particle is known as a sigma hyperon, a particle heavier than a proton.
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It is moving horizontally to the right in the left-hand image.
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It is a wide, black track and at the end of this trace, the particle decays into a pi meson and a neutron.
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This image is only meant to illustrate the experimental facts that must be explained.
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We see in the experiments that new particles are created by sufficiently energetic collisions between elementary particles,
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and that these new particles in turn radioactively decay into other particles, and so on.
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I do not wish to explain the experimental methods in detail here that now that allow us to decide
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why the one particle is a pi meson and the other a tau meson, and so forth.
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Rather, I prefer to draw a qualitative conclusion from this image.
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I believe what we learn from such images is that we may not view elementary particles as indestructible, unchanging,
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ultimate constituents of matter. For we certainly see that the particles transmute into one another.
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Obviously the most correct way of speaking about these processes is to say:
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all these elementary particles are made of the same stuff, so to speak,
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and this stuff is nothing other than energy or matter ..., let us say, than energy.
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One can also express it perhaps so: the elementary particles are only various forms in which matter can manifest itself.
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Energy becomes matter, in that it assumes the form of an elementary particle.
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And if we interpret the elementary particles in this way - and based on current experiments,
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we can no longer be in any doubt that we are describing the events correctly -
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then the question immediately arises for theoretical physicists: Well, why do exactly these forms of matter exist in nature,
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as manifestations of matter? Why do the elementary particles have precisely those properties that we observe experimentally, etc.?
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That is, why does matter have to occur in just these kinds or in this excess of forms?
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Apropos, I would like to mention: There are many different elementary particles.
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We currently know of about 25 to 30 different kinds. Thus, 25 to 30 different forms which energy can take in becoming matter.
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When we endeavour to bring some theoretical order to all of these aspects,
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we obviously hope that all of the different forms that we have before us as elementary particles in the experiments,
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that they spring from one simple natural law.
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Meaning, there is just one fundamental natural law leading to just these elementary particles being formed and no others.
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This same unified natural law must then also stipulate the forces between the elementary particles,
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it must in fact allow us to actually derive all of the properties of the elementary particles.
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Experimentally, a large number of these kinds of images are actually available to us as material for such an examination,
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as you have just seen, and those of related experiments.
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That is, observations of the transmutation of particles, of the forces that they exert upon one another, the life spans, etc.
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Now, the basic premises of the theory that I want to talk about here, of the mathematical representation
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of these experimental events, would be roughly the following: Obviously there would be no sense starting from the view
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that the elementary particles are something given, and then introducing mathematical symbols for these elementary particles,
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which we then associate with a natural law.
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That would be unreasonable because certainly the elementary particles should not be in any way prerequisite to,
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but rather the consequence of natural law.
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We want to have the elementary particles with all their properties derive a priori from natural law,
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and for that reason we cannot insert them as something already given.
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We also cannot, as is often done in the conventional theory, introduce a new wave function or wave operator
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for every sort of elementary particle and then attempt to represent the complicated train of events.
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No doubt, you obtain a mathematical description of individual processes with this kind of representation,
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but will hardly be able to encompass all the interrelationships.
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We will therefore have to assume that we represent matter in an arbitrary form.
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We must therefore introduce some kind of mathematical symbol for matter and from that say
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the theory simply begins with accepting that something like matter exists,
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and for this I may introduce a mathematical quantity representing matter.
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And since matter is additionally in space, is in space and time, this mathematical quantity
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that represents matter must therefore also somehow be related to space and time.
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Perhaps I may write this briefly on the blackboard.
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We can thus say: The initial prerequisite is the existence of matter, and it follows that we can introduce a quantity,
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I will call it Psi of X, with Psi so to speak standing for matter and X for space and time.
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And mathematically one will, according to what we know about quantum field theory -
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and we know that these statistical laws are evidently just the right description of nature,
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and mathematically one will say that it must be a quantum field operator, and indeed -
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I want to briefly write in addition to this - that it must be a spinor operator.
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We mean the following by this: First, we understand a spinor to be a quantity having two components.
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The mathematics of spinors was studied very early on by Pauli and introduced into physics.
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It follows from an experimental fact that we need these sorts of spinors.
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We know there are many elementary particles that have what is known as spin one-half, that is,
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their moment or angular momentum is half of Planck's constant. To represent these kinds of elementary particles, you need spinors.
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And for that reason, the original field dimension for matter must also be a spinor,
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for otherwise we could not represent these spin one-half particles.
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That it is an operator ..., well, the term operator is naturally comprehensible only to physicists and mathematicians,
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but we can perhaps explain that this quantity Psi in mathematics, as it were, simply makes matter out of nothing.
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What I mean is, in the mathematical representation we need to somehow move from nothing, that is,
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from a vacuum to a state in which matter is present. And this transition is accomplished simply through the operator Psi.
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Now, not much is accomplished apparently though this general statement, but you will see that we, through very few steps,
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indeed arrive at very definite predictions through mathematics.
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The next premise that we will insert into this theory is simply: Something like natural laws must exist.
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And a natural law means that we are able to predict something about the future state of the world from the present one,
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or about the past state of the world as well. Formulated mathematically, that means this spinor operator Psi must be sufficient
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to describe any possible state of the world during a very short time interval - now - that is,
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we say between now and a moment later, over a very short time interval.
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For we must be able to predict something about the current state.
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And if you make this assumption, it means in mathematical language that you therefore
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can produce all existing vectors of a Hilbert space of quantum states by applying the operator Psi
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over the time interval between t and delta t from the vacuum.
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Now, this mathematical formulation says something of course only to the specialists in the field.
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Therefore, if we require this prerequisite, then something very important immediately follows, namely,
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that there must be a valid differential equation as a function of time for this spinor operator.
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I will represent it in this way: We know that natural law must exist.
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Consequently, there must be a differential equation for Psi in time, Psi as a function of t.
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Now we can go one step further. We know, of course, that interactions exist in nature, that forces exist.
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All of nature and the entire interplay of events in nature are certainly based upon forces being able to affect particles,
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and that it is through just this effect of forces upon masses that the dynamics of the world are possible.
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Therefore, there must be interaction.
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And from the interaction an important property of this differential equation follows: It must be a non-linear equation,
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for interaction is actually only ever described by non-linear terms in mathematics.
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There are a few exceptions to this rule, but I do not want to discuss them here.
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Therefore, on the left I want to write: Interaction and consequently a non-linear equation.
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Finally, we need one further very important and general premise.
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And this premise can, if you use the word carefully enough, be labelled with the word causality.
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Here I must make a proviso, however: You know recently from modern atomic physics
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that causality is no longer valid in a certain sense.
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The recent interpretation of Psi waves as probability waves, for example, comes from Born.
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This movement in modern physics is not touched on in the theory discussed here.
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All mathematical processes are actually meant as predictions about the probability that something happens.
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However, the word causality has yet another aspect.
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What we mean with the word causality is that the effect cannot be earlier than the cause.
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And when we take this prediction together with the prediction from the theory of special relativity,
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that effects cannot propagate faster than the speed of light, that therefore space and time have an actual structure,
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which was discovered by Lorentz and Einstein fifty years ago, then the word causality in this special form
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therefore contains a very specific prediction about how the differential equation for this spinor operator must look.
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Namely, that it must be a Lorentz-invariant equation, that is,
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a differential equation that satisfies the theory of special relativity, and in addition,
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we can state a further property of this spinor operator that is extraordinarily important.
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Namely, that we are able to determine that for space-like intervals,
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these spinor operators must always be commutable or anticommutable with one another.
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For if this were not the case, then this simplest concept of causality,
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namely that the effect must always be later than the cause, and that effects can only be propagated at the speed of light,
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would not be satisfied. Therefore, I want to write here again: causality.
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And from causality comes a Lorentz-invariant differential equation,
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or invariant differential equation (if I may abbreviate it), and further,
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another prediction follows about the commutation relation (I also do want to write this down for the specialists in the field -
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it looks like this: X minus X^2 should be equal to 0, that is for space-like intervals greater than 0.
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And actually, the entire theory follows from what is now on the board. There are no further premises in it. The next thing ...
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Yes, I want to mention one more mathematical consequence, although I do not want to deal with the mathematics in detail today.
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The circumstance, that we are dealing with a non-linear differential equation,
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that in addition the commutation functions should vanish for space-like intervals - these two circumstances already establish
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how the commutation relations have to be for time-like intervals and in particular for when we are
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on what is called the light cone, meaning when we are observing the location where the effect propagates with the speed of light.
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It follows namely from the two premises - a non-linear equation and space-like commutation -
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that we do not have that kind of singularity on the light cone that we are familiar
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with from the usual linear differential equations, that is, what are called the Dirac delta functions,
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so that these Dirac delta functions must be replaced by another kind of singularity, indeed by a smaller singularity.
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That has immediate, very broad mathematical consequences.
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One must take seriously a suggestion Dirac made about 15 or no, 17 years ago.
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You must also contemplate an indefinite metric in the so-called Hilbert space of quantum-theoretical states.
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Perhaps I may include a few words about the history of this suggestion made by Dirac.
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As I mentioned, Dirac had suggested in January 1942 that one would have to resort to this indefinite metric
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in Hilbert space in order to avoid a mathematical singularity in quantum field theory.
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Then Pauli was able to determine a short time thereafter that this suggestion suffers from a serious objection.
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Namely, that then the quantities, which usually occur in the mathematical interpretation of quantum theory ...,
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in the physical interpretation of the probabilities in quantum theory, that these quantities can then become negative.
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Now, a negative probability makes no physical sense of course, and cannot be interpreted.
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For that reason, based on the investigation by Pauli, we thought that this indefinite metric
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could not be any sort of starting point for describing nature.
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Nevertheless, we once again took up this suggestion by Dirac about eight years ago in Göttingen
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and have been successful in showing special examples where Pauli's objection simply does not actually occur.
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Therefore, we now know at least that there are certain cases in which Pauli's objection does not necessarily arise,
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so that indeed the introduction of the indefinite metric does not represent a priori an insurmountable flaw for a theory.
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If you now take these premises as they are represented on the board seriously, then you can simply say:
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ok, let us try to formulate a natural law from them, and then we need to take a look at
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whether it can approximate the real world of elementary particles.
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We therefore have to take a non-linear differential equation that is Lorentz-invariant,
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that is valid for a spinor operator of Psi as a function of X,
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and the simplest possible equation of this kind is the equation that I want to show in the next image. That is the equation.
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It looks somewhat complicated as it appears here, the gamma mu and gamma 5 matrices introduced by Dirac occur in it.
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However, one can write the equation in other mathematical forms that then also demonstrate that it is
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in principle the simplest non-linear spinor equation that exists.
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And it is indeed the simplest because it has the greatest symmetry.
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And this prompts the question: Is this simplest spinor equation actually the correct equation?
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And correct would mean: From this equation, one can derive the elementary particles with all of their properties.
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And the equation's simplicity is matched by the difficulty in treating it mathematically.
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For this reason, a great deal of mathematical work had to be done at this point
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before we could give a few answers to the questions I have just posed.
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Now, the most important empirical data about elementary particles that we know of
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are the data concerning the symmetry properties of elementary particles.
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In this regard, I was reminded a short time ago once again of the ancient Greek philosophers;
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symmetry is always the principal feature, so to speak, of a physical structure.
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And thus the symmetry properties are the most important properties.
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Now I would like to request the image with the large table.
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This table is meant to provide you with a brief overview of the results of experiments.
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On the left side of the table shown vertically are the masses of the elementary particles and you see the entire list,
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so to speak, of the elementary particles in the form of points entered along this table.
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The lightest of the elementary particles are those with rest mass 0, those are the light quanta,
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they shown at the bottom as points. Also very light elementary particles are the electrons and positrons.
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Then come particles that are about 200 times heavier, what are called the mu mesons, and then above them the pi mesons,
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which are about 270 times heavier than electrons.
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The pi mesons are most closely associated with the forces that hold the atomic nucleus together,
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they are the particles first predicted by Yukawa in conjunction with the nuclear forces.
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They were observed as particles actually occurring in nature by Powell in England.
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Then above them come the K mesons, which are about 700 times heavier than ..., no, 960 times heavier than electrons, then the ...
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a short while ago for example you saw such a K meson in the images, that is the tau meson, it is a special sort of K meson.
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Then come the protons and neutrons, they are the longstanding components of the atomic nuclei,
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and finally further up the hyperons.
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Now, this entire menagerie of elementary particles, that is, this quite complicated train of events,
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can be systematically arranged, in that you can introduce quantum numbers for the elementary particles
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that simultaneously represent their symmetry properties.
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One can state in a somewhat simplified fashion: For every symmetry property of an elementary particle,
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there exists a quantum number, or perhaps more correctly the other way around: every elementary particle is characterised
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by an entire series of quantum numbers, and if you know these quantum numbers of the elementary particle,
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then you know what symmetry properties the particle has.
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Simultaneously, these quantum numbers are also an expression of the conservation laws that apply to the elementary particles.
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Therefore, the quantum numbers customarily play a role similar to the spin.
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We say that the spin is an integer multiple of, or a half-integer multiple of Planck's constant.
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And the total spin is conserved for any particle transmutation processes or creation processes.
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The quantum numbers that we have empirically found are entered here in the table on the right.
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I must mention at this point that this kind of representation originates from the theory.
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It comes from work which Pauli and I jointly wrote back then.
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But it is basically just a representation of the facts discovered previously by the experimental physicists.
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In the first column on the left are the elementary particles with the labels that are customary for physicists.
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On the right as the first quantum number is the electric charge, labelled as Q.
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That is a longstanding property of elementary particles, of course.
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Then comes a quantum number called the isotopic spin, or isospin.
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This property of elementary particles was discovered about 30 years ago.
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The isotopic spin has turned out to be a very important symmetry property,
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because it still poses certain riddles for physicists, since it is not a complete symmetry property.
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That is, the conservation laws that result from this symmetry property are only approximately valid,
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and that means that symmetry in nature is also only approximately valid.
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Then come the additional quantum numbers that are labelled here with L, which I want to say something about later.
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Then comes a quantum number N, that is what is known as the baryon charge or baryon number.
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This is for protons and neutrons 1, for example, but 0 for electrons or mu mesons.
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Then comes a quantum number known as IN one-half, which has a simple relationship to the baryon number.
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You can therefore say that the quantum number IN and LN also mean the baryon number and lepton number.
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We therefore arrive at a table of quantum numbers like this empirically.
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A few are left out, for example the spin is not listed here.
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And for physicists the question now arises, also for theoreticians: can one explain all these quantum numbers
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through the symmetry properties of this wave equation for matter that we have provisionally written down?
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To answer the question, one has to examine what symmetry property the equation itself hat, in other words,
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under what transformations is the equation invariant.
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And as I had said already, the equation must be invariant under the Lorentz transformation.
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That is necessary so that the structure of space and time, which we are familiar with from Einstein, is correctly represented.
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The equation actually is invariant under the entire inhomogeneous Lorentz group as well,
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and the usual conservation laws for energy, momentum, angular momentum, and motion of centre of mass, etc.
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then follow from this invariance.
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However, we have not represented so far anywhere near all of the symmetry properties and quantum numbers that are shown here.
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One very important additional quantum number is the isotopic spin, which was discovered about 30 years ago.
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And I said previously, that Pauli had established a new symmetry property in the wave equation for the neutrino,
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precisely the transformation he discovered, and Gürsey then associated this group with isotopic spin
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based on work by his predecessor Schremp, whom I would like to mention here.
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[Perhaps I could show the next image.] Here are these Pauli transformation formulas.
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The equation is also invariant for these transformations discovered by Pauli, and Gürsey has just been able
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to show that these transformations allow the isotopic spin quantum number, or isospin to be represented.
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Thus, some of the quantum numbers that you have seen presented empirically in this image have been explained.
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The lower group, I still have to mention for the sake of historical accuracy,
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the lower transformation had first been discovered by Touschek, and Touschek had already announced
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that one may be able to associate them with the baryon number.
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And once one has recognised all of these transformations as genuine transformation properties of the equation,
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then it follows that you really can explain some of the quantum numbers shown in the table.
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However, two quantum numbers that caused difficulties in the first place still remain.
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And the difficulties of which I spoke a short time ago, which still remained in the theory a year ago,
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had to do with the existence of precisely these two quantum numbers.
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I now therefore come to this question of the difficulties.
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The quantum number table that you have just seen still contains two quantum numbers named LQ and LN there.
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And at least one quantum number of the two must be able to take on all values from minus infinity to plus infinity.
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And this means that there still must be one additional transformation property of the equation, if the equation is correct,
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that is of the same kind as being able to, let us say, revolve about an axis, that is, a continuous scalar transformation group.
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Besides this, there is still the difference, so to speak, between LN an LQ.
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That is a quantum number which has been named the strangeness of the particle,
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one which characterises the K mesons in contrast to the Pi mesons, for instance.
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This strangeness is a quantum number that does not have to take on all values from minus infinity to plus infinity,
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it is apparently only capable of taking on a couple of values, that is, it can assume the values of 0, 1, or 2, plus, minus.
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It would be sufficient for this quantum number to have discrete transformation groups still.
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And indeed, the equation also remains invariant for one additional single-parameter group however,
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which is referred to as the scalar transformation.
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If I may write on the board: Psi of X and L (L is the quantity that occurs in the equation) can transform into Eta
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to the three/halves power times Psi of X Eta and L Eta.
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In ordinary language: the entire world, so to speak, can expand or contract,
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and we would not notice this if the entire transformation took place similarly.
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That a scalar transformation is conceivable was also recognised very early on, reported to me during a visit in Yugoslavia
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that the Yugoslavian philosopher Boškovic already had the idea of this transformation in the 18th century.
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And this transformation is actually contained in this equation as well and it appears to be sufficient
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to explain at least one of these two quantum numbers.
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And for the explanation of so-called strangeness - the discrete transformation properties of the equation,
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of which there are several, will probably suffice.
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To this extent, we can therefore give a thoroughly satisfactory answer now to the first difficulty.
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A second, still-greater difficulty consisted of particles, indeed strange particles,
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existing that have isotopic spins in multiples of one-half and simultaneously an integer, regular spin.
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Or put the other way, an integer isospin and a half-integer regular spin.
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That was incomprehensible in such a theory at first, because the operator Psi always creates a half-integer spin, so to speak.
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If it therefore is applied an odd number of times, it creates a half-integer number for both kinds of spins,
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and if it is applied an even number of times, it creates an integer number.
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Now, this problem is associated with another very peculiar problem, which I have also already touched on earlier,
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that indeed certain ...