TRANSCRIPTION OF THE TALK TO THE AKRON PHYSICS CLUB, FEB. 28, 2011.

[The actual transcription is copied without brackets. Comments that I have added appear, as this one, inside of the marks.]

[The double dash '--' is used to delimit a slide title in the set of corresponding slide images.]

[--The Pauli Exclusion principle--]

This talk is based on a talk I gave in Italy in the fall of 2008. There was a conference on the Pauli exclusion principle there, attended by people from all over the world. I heard at least five languages spoken and there were several other people, from India, Africa, or other countries, who did not say a word in their native language during the conference. It’s a group of people that have gotten together recently over the last ten years, to try to understand. Progress has been slow and difficult. Now, what I'm going to do is. . .

This is an

--Outline--

of the talk.

[The Pauli exclusion principle has its origins in the earliest days of quantum mechanics.] The Pauli principle, of course, has probably the highest notoriety to comprehensibility ratio of any of the ideas in physics. It is important in all areas of physics and no one really knows why it is like that. [It has been an intransigent problem of theoretical physics and remains enigmatic today.] [The talk has five parts.] So, I am going to talk about, first, a short history [following the book by Duck and Sudarshan] and after the short history I am going to talk a little bit about some of my early experiences with the Pauli principle and how I became at all interested.

Then I am going to talk about spinor space, which is a theoretical construction that I have written a number of papers about. It is [a new approach to problems in physics and] the way I work on these things--a brief discussion of that. Then I will talk about how that relates to the Pauli principle. Then, if there is time, I want to mention a few ongoing questions that I am thinking about now.

Now, the beginning goes back to this,

--Mendeleev’s periodic table--

When I was young, this is the periodic table I remember, [from] when I was probably ten years old. It must have been an original reprint of the article, stuffed in back of an [aging] mathematical tables volume. I kept looking at this and I started learning chemistry from this particular version of the periodic table. [It was a predictive tool for Mendeleev.] We do not think of sodium and copper and silver as being related, but on this table they are, and in substance they are in the real world. The modern table that we see now came out, at least from my point of view, ten or twenty years later. [It established a useful predictive pattern for the properties of the elements and allowed the chemists to fill in the empty squares.] And so, the purpose of putting this table up here is that this was the phenomenology that generated the Pauli principle. Bohr believed that this array off data could be explained by quantum mechanics but he could not do it himself. So Stoner and Pauli started to study it and eventually came up with the modern set of atomic states that account for the positions of the electrons in the system of atoms and it was based on the idea that there is only one electron in each state. And that business of putting exactly one electron in each state became the Pauli principle.

[--Quantum orbitals--]

Actually, somewhat later, Schrödinger developed quantum orbitals and these added more structure to the understanding of Mendeleev's table. [The orbitals confirmed Pauli's interpretation of the phenomenology. This gave credence to the positioning of electrons and the properties of the elements in chemical bonding. The orbitals are filled with one electron pair each; others must find a new space.] And of course Goudsmith and Uhlenbeck discovered that spin, and that completed the discussion of the quantum numbers. [There are four quantum numbers that justify Mendeleev's classification. The modern version uses the values: principle -- n, orbital -- l, magnetic -- m, and spin -- s. The Pauli principle became the quantification of an electron in space.]

The next problem, the next big step had to do with

--Identical particles--

and it comes from spectroscopy. I want to take a little bit of time to explain it because we don't really even think about this now. [This advance in understanding was made by Dirac, but also simultaneously by Heisenberg, Wigner, and Jordan.]

But let's suppose that we have a two electron transition. Say helium because that is the easiest one. I'll just pick arbitrary states, I have not tried to drawn the diagram to scale or anything. You have two states on the left, it goes to the right and you have two new states. Now that transition, presumably, could go in two different ways, the upper electron on the left could go to the lower electron on the right, and the middle electron could go across, or . . . they could switch. . . . well they could switch . . . and so on. There are four possibilities, so you should get four lines. And, in fact, the transition only has one line. And that was the problem. You see, we can think about the two electrons as being separate, but God thinks that they are the same. He doesn't know the difference between the electrons, and so there is only one transition with two electrons. This bit of phenomenology was studied by Dirac.

[Atomic spectroscopy showed that a two electron transition had only one spectral line, not four as one would naively expect. This implied that 'God' did not know the difference between electrons. The mathematics had to follow the phenomenology.]

And, if you start out with a two electron state and you go to two other electrons, you have some kind of a probability. But the probability cannot depend on the order of the electrons.

[--Symmetry types--]

That is, you can define the [particular] state either way and the electrons do not know that they are different. So you have two possibilities and because you switched them twice; it has to go back to what it was before. You can have either |mn> = + |nm> or |mn> = - |nm>. And over a period of time, later on, it was discovered that the plus sign is for Bose [-Einstein] statistics and the minus sign is now what we call Fermi-Dirac statistics. If you have two of those going on at the same time you can combine them and if you start with two Fermi-Dirac particles and combine them, then there is no sign change and you end up with a Bose system. For this reason, people consider the Fermi-Dirac system to be more fundamental. You can generate Bose statistics from Fermi-Dirac statistics, but not the other way around. [Because the multi-electron 'transition probabilities' do not depend on the order used to label the electrons, the exchange of any two particles in the transition element must either leave the matrix element unchanged or changed by a sign. [In time, it was observed that these properties were associated with a particular value of the spin. Integer multiples of the spin give the Bose-Einstein statistics and the half integer values give the Fermi-Dirac statistics.]

There are some applications. The standard application for

--Bose-Einstein statistics—

is black body radiation. Here we use what is called the anti-commutator, a bit of algebra that I am sure you all have seen. It is very much a bookkeeping device. People use it, it works, but no one understands why it works or where it comes from--the formulas. And so to understand Bose-Einstein statistics, you would have to understand where this comes from. Now, it is generally known, but not often mentioned, that you can consider photons [and their quantum properties] as coming from the particles in the wall of the cavity, when you do black body radiation. In fact all of the statistics of the photons can be accounted for by the properties of the particles that emit or absorb them. They [the photons] would not exist in any other case. This provides some evidence that maybe there is another way to do it. Pauli thought that probably this formalism was an interim formalism. It has not turned out that way, and the question is still open. [Photons, 'particles of light' have integer spin equal to one and must be symmetric under interchange. The statistics of black body radiation are the result.]

--Fermi-Dirac statistics—

appear when you have spin one-half particles, they are seen in the statistical mechanics of gases. It is a fairly rare observation, it is not common like photons are. If you have atoms of spin 1/2 and you cool them down enough, you get to see a difference in heat capacity. The Fermi-Dirac statistics do not have a classical analysis. They have always remained somewhat enigmatic. No one knows why they are like that at all. You can get rid of the Bose statistics by assuming that they are classical. But the Fermi statistics are really tough. [These particles are anti-symmetric under interchange. The raising and lowering operators associated with each symmetry type are used in quantum field theory to describe systems of multiple particles. This formalism is well-established but tells little about the fundamental mechanisms involved. It is a bookkeeping tool rather than a physical explanation. It is known that the photon statistics can be derived from the properties of the particles that emit or absorb them. The photon field, in this sense, is basically classical and has quantum properties that come from the sources. The Fermi statistics, commonly seen for electrons, do not have an accepted classical description. The electron behavior has remained enigmatic, although, apparently well described by the operator formalism.]

The next big step is the Dirac electron theory; I am not going into it [in

any detail].

--Dirac electron theory

It gave a more fundamental basis to the spin 1/2 particles, especially the electrons.

You have a wave function which is a four element column vector. They are complex numbers, and you have this wave equation which describes the evolution of that column vector. [There are four anti-symmetric matrices that are anti-symmetric under multiplication. The four matrices are used for the equation itself, but a fifth one also exists and appears in weak interaction theory. With the unit matrix, there are six in all.] That is sufficient for the present. And this Dirac electron theory predicts the atomic properties very well. It is a good theory of electrons.

[--Pauli's proof of exchange anti-symmetry--]

Pauli thus wanted to prove that only one electron could exist in any state. He observed it and derived it as a phenomenological statement. He wanted to show that it was really true. His ideas go back to the beginning of quantum mechanics, special relativity, and quantum field theory. His arguments show that the transformation properties of particles, when you group them together, have to be in certain forms. And if they are not, it does not work right. [It was a major goal of Pauli to prove that particles with the half-integer spin had to have Fermi statistics. His first result demonstrated the essential idea, which was based on the transformation properties of two or more Fermions under

special relativity. ]

[--Spin and statistics history--]

I am not going to try to go through this. As far as I am concerned, it is some of the most abstruse mathematics available. Pauli's paper is not too bad, but people have been studying it since then and this is the list of the history of these types of studies. [This result was the first of a long series of developments by a number of leading physicists. The modern arguments show that an odd half integer spin particle cannot be symmetrical under identical particle interchange and that an integer spin particle cannot be anti-symmetrical. The proofs depend on special relativity and do not go through if the complexities of modern gravitational theory are included.]

I worked on some of these papers here, a couple papers by Streater and Wightman, and Jost. I studied them at Berkeley, in the seventies. The proofs have gotten more elegant, more refined, more precise, but they have offered no real understanding. I would have liked to have had a chance to try to disassemble some of these things, but someone did it before me.

[There are criticisms that can be made from the point of view of the 'spinor-space' theory that I will soon present. However, none of the comments that I could have make would be any more effective than this short question that appeared in the American Journal of Physics.]

--Neuenschwander's Question--

That is in a paper by this fellow, I think it is pronounced 'Neuenschwander'. Is that how you would say that? It's a famous paper, that is the whole paper right there. He quotes Feynman first, and says Feynman says "Why is it that particles with half integral spin are Fermi particles whose amplitudes add with a minus sign, whereas particles with integral spin are Bose particles whose amplitudes add with a positive sign. We apologize for the fact that we cannot give you an elementary explanation. An explanation has been worked out by Pauli from complicated arguments of quantum field theory and relativity. He has shown that the two must necessarily go together but we have not been able to find a way to reproduce his arguments on an elementary level. It probably means that we do not have a complete understanding of the fundamental principle involved. So Neuenschwander writes one sentence. He's saying this for one sentence: "Has anyone made any progress towards an elementary argument for the spin statistics theorem?" That's what he said. And it is a very polite way of saying "Do you folks know what you are doing?"

Q1: "This says question number seven. What is this set of questions?'

Oh, this is from the American Journal of Physics. They have a little section, at the beginning of each issue, in which people write in questions. You can ask your question and they publish the question and someone else writes in an answer. You can read them. And it has been going on for many years. And there are lots of question there. The journal normally publishes questions and answers which are not controversial. If you have a new idea or new question, [if it is adaptable to standard thinking,] it may get published. Its intended for teaching purposes, I think is really the orientation of the journal. This is really asking the people who provide the standard answers, "Do you really know what the standard answer . . ., is and what you are doing?" And he did it much better than I ever could have.

Q2: "And what year did this appear?"

Oh, that has been lost from the slide, it’s about '95, '99 or something like that, about ten or fifteen years ago. This question, a single question, inspired a book by Duck and Sudarshan on the spins statistics theorem, the Pauli principle and it also probably started the series of lectures--meetings that I gave my talk in. They are saying, "Let’s start over on this one, what is going to happen next?" Nobody understands it.

[--Molecular Spectroscopy--]

So, having said that, I would like to go to a discussion of some of my experiences with the Pauli principle. I got interested through molecular spectroscopy. It turns out that in diatomic molecules when they are rotating you can see effects of the Pauli principle in the spectrum. This is an example of what happens in say, oxygen. Oxygen has no nuclear spin. It is a spin zero particle, it is a Boson. And that means that two of these particles will have the same phase. Now if you put them in a ground state so that they have no angular momentum, they add because of the same phase and you end up with a final state. But if you put them in a state that has angular momentum one, then you have a wave function that has one cycle per rotation. Both particles have one cycle per rotation but they are shifted by 180 degrees. The peak of one wave function matches the bottom of the other and the peak of the second wave function matches the bottom of the first. So they add to zero. And what that means is that there is no solution; in fact there is no state. If you look at the oxygen molecule, half of the states are missing. And there was a nice fellow at the meeting, who did carbon dioxide, named Lien, and he could see the spectrum of carbon dioxide, you have a carbon in the middle, two oxygens rotating about the carbon. They do exactly the same thing; all of the odd states are missing leaving only even states. And then he went and looked with the laser, right exactly where these odd states should have been and he set a limit of 10 to the 13^th--the strength of existence of the state. And it is worth emphasizing that it is not a question of transition rate, the states do not exist and there has never been an exception observed. There is no evidence of the missing state whatsoever.

Now here is some early data. This is picture of a rotational spectrum by a fellow named

--Rasetti--,

--Italian fellow, that was published in 1929. It is a Raman spectrum. I am not going to go into how to do a Raman spectrum in 1929. It is not easy. In the case of oxygen here you have alternate lines missing, and you can tell if you calculate the moment of inertia that in fact half of them are gone, and in case here of nitrogen you can see there are some very faint lines in between the dark lines. This image has been through the original photographic plate and the processing of the publisher, then it was zeroxed; it was put on my computer; clipped out; run through gimp; and so on. It’s been through a lot. But very faintly you can see the lines there; they really are there on the original. And there is an alternation of intensity. In the nitrogen case, you have a spin of both nuclei, and so you get different amounts of symmetric and anti-symmetric state, causing the alternation of intensity. This fact is actually fairly important in the history of physics because it indicated that the spin of the nitrogen nucleus was one instead of one-half. People believed that nitrogen nucleus had seven electrons and fourteen protons, which would be an odd number of spin one half particles. The [observed] spin of one led to the detection of the neutron and eventually the neutrino. It was a very long time ago. [More recent data is clear; the effect is present exactly as proposed in the thirties. All current experiments (O2, H2, D2, N2 . . .) support the effect. There are no dissenting observations. ]

I also brought some more modern data along on molecules.

[--Modern observations--]

The electron data is pretty solid. Errors, the upper limit on the electron data is 10 to the minus 44 right now, maybe slightly more than that. The question for the electron is pretty much closed, 10 to the minus 44 is good enough for me. This is another spectrum showing missing lines. This is an oxygen band in air in the deep red. This was taken down at Kitt peak.

I was down there one day when they were doing this kind of work. These four guys walked into the room. I was minding my business in the back with my stardust experiments. All of a sudden, they all put on their sunglasses. They were all big enough that you would not want to meet them in a dark alley. I did not know whether it was the four blind mice or the four musketeers as they started rummaging around in the back and started pushing buttons on a panel that I never used and started making growling noises, clunks, and whizzes. They were starting up the telescope. It was a sixty inch telescope, and it collects that much sunlight. It focuses down into the room where you take a piece of that and do a spectrum. It is an absorption spectrum in air. So, these numbered groups, here numbered 1, 3, 5 are absorption lines from oxygen bands. They are split because of the electronic splitting of the ground state in oxygen. The even numbered lines are missing. Now you can tell if you look at the small lines. See, here are two small lines and here are two small lines. Those are in the place where you would think the missing lines would be but it turns out that they are oxygen-16 oxygen-18 lines, and you know because the oxygen-16 oxygen-18 molecule is slightly heavier, so it has a slightly higher moment of inertia; so the lines are spaced a little closer together. If you go up a little further here, you see that the inner lines shift a bit. And here you can see all four of them 1,2,3,4. There is no missed alternation of intensity in the oxygen-16 - oxygen-18 lines because the molecules are different, the atoms are different and they do not satisfy the Pauli exclusion principle. It only occurs in the stronger oxygen-16 lines where the nuclei are identical. And there are also oxygen-17 oxygen-16 lines here, even weaker, which showed also the isotope effect. And both sets of lines are there, there is a bunch of missing lines. Spectroscopists know that there are lots of extra lines in the spectrum, but if a line is missing they get very excited. That's the point that you would want to make here. This spectrum is not as good as Lien's data. It might set up a limit that may be 10-4 or 10-5, on this structure but it was not taken for that purpose. It was taken for just the atmospheric absorption. Part of this band was actually part of the Fraunhoffer spectrum, [taken] in 1814. And so it has been around for two hundred years. And if you are interested in the history of physics, you should go back and look at it because there is one-hundred and ten years from the observation of this band until quantum mechanics was invented and it took another twenty years until the analysis of the band was at its modern state; and we are still fussing about the Pauli principle. Maybe in a few more years we will really figure this out. [Tests of molecular spectra are still ongoing.]

[--Evolution of spinor space--]

Now, this is a short discussion of how I eventually got to where I am in spinor space right now. In `68, after looking at spectra not unlike what we just talked about, I decided to start studying spin one-half particles. It was a long time ago. And I had various ideas; one of them was called a gradient spinor which I will talk about later. I was impeded because I needed a geometrical theory of quantum mechanics. Spin is a very geometrical subject. So over the years, I kept working on it. We have talked about some of these things. I am sure that I have talked about them here. In 2002, I had a sufficiently convincing five dimensional theory that I could start doing spinor space work. So that is what I did.

Well, let us talk about spinor space a little bit

[--Basic spinor space--]

because that was the basis of the talk in Italy.

This is what it looks like. The idea is that if you define a set of complex coordinates and that these coordinates locally act like a spinor. It is a space that acts like spinors act in four-space. This is a gradient spinor.

You define spinors as gradients so that the Dirac spinor is a gradient in spinor space. There is a way to map this space into space-time. This equation tells you how to do that, but only infinitesimally, there is no large map. One

space is all stirred up as it is mapped to the other space.

[--Conformal waves in spinor space--]

And if you argue that the space satisfies the conditions of Riemannian geometry, you will found out that you have an equation like this, where you have this eight-dimensional D'Alembertian. That is the invariant equation in eight dimensional space. I use that little funny symbol for it, the octagon folded over on its sides. If you draw a straight octagon it looks like a circle, and does not work very well. And if you do appropriate chain rule manipulations, you can find out that within this equation is the Dirac equation in five dimensional form. And that means that the Dirac equations satisfy that eight dimensional D'Alembertian.

[--Geometrical interactions--]

Now there are presumably other things going on in eight dimensions. But you have to find them. So you consider other possible things, think about it a bit, and you test them against experiment, against other theory, against phenomenology. Maybe you can make more sense out of it, maybe you cannot.

This is sort of a schematic discussion of how you compare something in eight dimensions with something in four dimensions. You might relate the two together.

[--Local electron--]

And one of the important ideas is that you match plane waves, so that a plane wave in space time, if you add the mass, becomes a plane wave in five dimensions and eventually becomes a plane wave in eight dimensions when you add the spin, too. So you are still doing wave theory.

[--Spinor space I: simple particle--]

Now I have a short sequence here, which I added from the original talk, to try to give you a feeling for how spinor space really works. What I am going to do? And this is pretty simple diagram, I am going to shoot the particle in from the left and then there is going to be a region, symbolized by the box, where I am going to use the eight-dimensional equations of spinor space. Then the particle is going to come out the other side, and by looking at what happens to the particle. I can gain some knowledge about what happens in spinor space.

Q3: So is that box in our real three dimensional space?

Well, it is just a different way of thinking about the same particle.

Q4: That could be the track that we are seeing with the arrows.

Presumably the track goes . . .

Q5: Is that in three dimensional space?

This picture is drawn in three dimensional space. This is a picture drawn in three dimensional space.

Q6: OK, OK

That's not essential, but it may help.

[--Spinor space II: wave particle--]

If you try (first of all, you might try) to shoot a classical particle through. It turns out that the spinor space takes [only] Dirac particles. You can not use a classical a particle, you do not know what happens when you try to put a classical particle through here. [Classical particles must be divided up into quantum particles -- electrons, muons, tauons.] You have to use a wave particle. So over on the left, a wave particle comes in and a wave particle comes out. Then if you try to do the calculation in the spinor space, you still cannot do it. What happens is that on the left side there are lots of directions, but in the spinor space, there are more directions. And so, right when you get to the boundary of the spinor space, you have too many directions coming out for the number of directions going in.

[--Spinor space III: wave particle with spin--]

And it turns out, if you use information about the spin, then the motion of the spin into the direction in the spinor space. And when you get to the other side, you can separate the motion of the spin and the direction of the spin comes back out. This actually works for gravity, the gravitational field, you can watch the particle fall. It works for electromagnetism. The rotation of the spin comes out right if you put a magnetic field there. According to Dirac's theory, the electromagnetic field makes the spin precess. It's the . . .

Q7: Are you are saying that the box region is a region of some potential? The gravitational potential, or . . .

The box is the region where we are thinking about it in a different way.

Q8: But what is happening in the box?

In the box, we are using a different set of equations to describe the same particle as it goes through there.

Q9: The particle is free inside and outside the box, always free?

Actually, the transformation laws allow arbitrary electromagnetic and gravitational fields. So you can have a forced motion, inside and outside the box. The particle might be in an electric field or falling. The spin might be precessing. We can calculate that in space time, when it is in the box, the spin and the velocity combine to make one wave. That wave moves through spinor space and from the way that wave must move in spinor space, you can begin to understand how forces work in spinor space. It is an exploration of what happens to a particle inside a region where you would use spinor space for calculations. . . . [As an electron enters spinor space its new direction in eight dimensions depends on both its direction in space-time and its spin. It is conjectured that conformal distortions are sufficient to account for changes in motion. ]

[--Spinor space IV: pair production--]

So, if we have arbitrary electromagnetic fields, we can put the field of a nucleus in there. If you shoot in a gamma ray, you can make a pair. So by electromagnetism you get a positron and an electron out. [The particle may go in and out from the same side. This can be used to represent pair production. In five or eight dimensions, as distinct from field theories in space-time, the motion is smooth and connects the electron to the positron in the extra dimensions.]

[--Spin space V: weak interaction?--]

I decided, after thinking about this for a while, that I would investigate one more question. I would investigate what other interactions could exist in spinor space besides the gravity and the electromagnetism. And so, I did that. It looks like this, that if you carry it to an extreme, you can put an electron in, you always get a mass zero particle out with the spin pointing backwards. Presumably this is a neutrino in a spinor space, and it is not obvious at first why you would be able to do that. But when making the conversion to spinor space, you used the fifth Dirac matrix. And that Dirac matrix is precisely the matrix that is used to describe weak interactions. When you put that into the geometry you have the capability of deducing the weak interaction.

Q10: When you have a gamma-5 it is very conventional.

Gamma-5 only occurs in the coordinate transformation. So this is a free scalar particle in eight dimensions. When you map it back, it has a direction. When you map it back, these properties come back out. You can set it up so that the particle when it gets to the far side of the eight dimensional space, it gets mapped back and corresponds to a particle with no mass and reverse spin direction. So this is about the weak interaction as contained inside the notion of spinor space. This is exciting, but it set me back, because, wow, I had been studying general relativity, I had been studying electromagnetism and quantum mechanics, and I did not know anything about nuclear physics or particle physics. One of the things you have to do is every morning you evaluate what you did the day before. You have to ask, well is this right, or is it wrong? By the next morning, I needed to know everything about nuclear physics and particle physics, weak interactions and strong interactions. I could not do that, it takes a little longer, but I thought I would try anyway. [It became clear that there are many things going on in spinor space, many yet to be uncovered.] And so I got to work on it. The morning went on into the afternoon; the afternoon went on to a week and a month. After about a year and a half, I had read the book about Blatt and Weisskopf, Gottfried and Weisskopf, and books about neutrinos and stuff. I was learning a lot, but I was getting pretty tired of it.

[--Escape!--]

I saw this poster. It was a meeting announcement. Here it is, a meeting on the spin statistics connection and related dynamics. [It was a meeting that comes by every five years about the Pauli exclusion principle.] There you see Pauli and you see the sodium spectrum and it is in a nice place. I thought, that is were I started. [That is, I started by working on the question of spin and statistics.] Maybe I will go back and talk about it. [In any case I was quite tired of just reading.] I should be able to say something about the spin statistics connection. I have what is in some ways a good theory of electrons, perhaps the best theory of electrons available. It was to be in six months, seven months. So I wrote a simple abstract and sent it in. Then I got to work. It turned out that it was not too hard to do spin-statistics in spinor space. The biggest difficulty was that you have to check the epistemology. You need to go back and find out why everything that you decided could be right as compared to what everyone else thinks. Are their reasons for believing what they are believing better than your reasons for believing what you are believing? A lot of the early work on spin-statistics was done by a fellow named Pascual Jordan. You may have heard of him -- Wigner and Jordan papers. Many of them are not translated. So I spent four of the six months trying to read papers and to check whether the epistemology was right—whether I could really do what I had said.

Now, this is the discussion that I presented at the meeting, in its essence.

[--An identified pair--]

We start out here with an electron. Up here I have a space time diagram. The x direction is presumably all of space. I use this diagram to explain because there are some relativistic processes involved.

And what you do is imagine successively increasing electromagnetic fields, so that the electron, which is unaccelerated, will have an acceleration, and a deceleration. Or, in successively higher fields the curve increases. You can actually get to a pair production situation. You will have an electron, a positron and an electron. If you increase the forces, you can move [the pair production or pair annihilation event] them out of the vignette. And then you can move the positron out too, leaving the two electrons. Now those two electrons are connected to each other. The positron is not in the picture, but it is there. I call these an identified pair. And this equation, octagon psi [=0], follows the electron to the positron, [along the motion through the pair production and pair annihilation events] and back to the other electron. These two electrons are in the same space and have not just identical equations, but one equation. They both satisfy the same equation. You can see here the beginning of the Pauli principle. If this picture is drawn in spinor space, the motion in spinor space is a gradient motion, and so two electrons cannot cross. They will be disjoint in spinor space. The problem is to translate this prediction of electrons in spinor space into space-time.

Q11: It is a one body wave equation that has multi-particle interpretation, same as the Dirac equation--the same property.

Well this . . .

Q12: You have a different mathematics. It has a lot more in it because it is eight dimensional.

Yea,

Q13: Nevertheless, it has the same property as the Dirac equation.

If you were to look at this for the Dirac equation, I think that you would use one Dirac equation for the electron here and another one for here. At least if you use modern field theory. I do not usually see them turning around the corner.

Q14: No, no, one equation.

One equation--the property carries around the corner. That’s pretty much what you need, so if you have two electrons which are disjoint in space-time, they have to be disjoint in spinor space. The problem comes if you have a bond, where there are two electrons that are on top of each other. You want to show that they are disjoint in spinor space.

Q15: Disjoint as the product of two spinors?

There are no products. . . .it means that in this space they have no two points in common.

Q16: OK

What is the word they use for it?--the support.

Q17: OK

The supports are disjoint.

It's easy if the spins are aligned

[--Parallel electrons--]

because the electrons do not overlap in space-time and do not overlap in spinor space. The idea here is to prove that one spinor space works for all particles. There we have more particles no matter what. [Electrons with parallel spins can always be separated in space time and therefore also in spinor space. Conventional theory indicates that the wave functions are disjoint.]

--Anti-parallel electrons

In this case, we have the particles anti-aligned. They are on top of each other in space-time. How do you show that they are really not on top of each other in spinor space? You do the following argument. You say that those two electrons, in the helium atom, can be separated by an ionization pulse. So now we have two electrons separated in space-time. These are separated in eight dimensions, but the gradient flow does not allow them to be combined. If you integrate backwards in eight dimensions you can not put two electrons, with opposite spins exactly on top of each other because the electromagnetic field does not force them to be on top of each other. It is always a gradient flow. What that says is that, the particles, you can put as many electrons in the eight dimensional space as you want and they never lie on top of each other. It is a little bit easier to see that when you go actually watch a collision. I always thought it that was kind of instructive. [For anti-parallel spins the argument is more complicated. The electrons can always be separated in space-time and can then be made to separate in spinor space. Since the trajectories of the gradient flow cannot intersect, they must always be, and must always have been separated.]

[--Spinor wave propagation--]

Here, what I am doing is I have two electrons, electron 1 and electron 2. Imagine that they are coming at us out of the plane of the paper. They are slightly pointed towards each other, so they slowly move together. At some point they are going to hit. And right at that instant when they hit each other, we want to do the Pauli exclusion principle. We want to show that it works right for the Pauli exclusion principle, because that is where it matters.

[--Boundary development--]

If you go look at the numerical techniques for the calculation of this problem, at the instant of contact, you take this wave function [, 1] and subtract it from this wave function [, 2] to get this wave function [, on the right] and you take this wave function [, 2] and subtract it from that wave function [, 1] to get the new wave function on the left. That is how you force an anti-symmetric boundary condition numerically.

[--Wave function anti-symmetrization--]

It is a little easier to see if you draw this kind of a scheme here where I have picked two different looking electrons. When these two particles hit, this one, with a soft boundary condition subtracts from that one to give a decreasing boundary condition which is a little easier than that one. But this one is tougher than that one so it subtracts more from this and makes a negative boundary condition. It is simple enough, and the distortions here propagate through the electrons to the far side.

But it [, this behavior] is exactly predicted by spinor space because this equation says that the boundary condition is continuous and this one says that when you exchange the two particles, there is a sign change. The anti-symmetry is enforced easily by the fact that the two particles satisfy those equations.

Now, in spinor space, since the particles remain disjoint except for the particular point of contact, it is much simpler to follow them. You only have to worry about the anti-symmetrization. So that is the idea. It has been out for two years, I have not had any complaints, and I do not know what is wrong with it if there is something wrong with it. As far as I know it works. It is a new idea still.

[--Multiple electrons in spinor space--]

If you imagine, say, a bunch of electrons, here if you have these electrons in black coming out towards us; they might hit each other and we end up with these anti-symmetric conditions right where they each meet. But they all still satisfy the same wave equation, just like they did before; it is just that they need the anti-symmetrization carries on . . . in that dimensionality.

The Italians liked my bowl of

--Spaghetti--

the electrons are swimming around in there; they do not run into each other, but it is eight dimensional spaghetti.

Well alright, these are ideas that I am thinking about to go on from spinor space.

[--Orbitals in spinor space--]

Presumably, an orbital in space-time will have an orbital in eight dimensional space, except that it will only correspond to one direction of the spin. And so there is an issue of how to calculate what that would look like if you saw a hydrogen atom in spinor space. If you have a core, a core solution might have electrons next to each other, but might not stay in a simple arrangement. I think they look a lot like the regular ones, but not the same.

Here is another problem that I have been thinking about.

[--Calculational complexity--]

Actually, this was a flash, about two weeks before the meeting. That is always very awkward because you have this great idea and you do not know whether it is right or not. If it is right and you do not say anything, you loose it, when someone else will pick up on it. But if it is wrong and you say something, you get into all kinds of trouble.

It has to do with the calculation of wave functions when you have more than one electron. Here is have a [way] . . . we are going to numerically calculate a wave function, just so you can keep tract of the number of parameters. I have a box here. I am going to suppose that there is an electron wave function in here. And I am going to keep tract of it by just recording the value at the points. In space time, a three dimensional array of quantities is sufficient [for a stationary state]. So the total number of points is the cube of the value on the side. In spinor space, it is a little harder; I am guessing conservatively that it could be the fourth power. It really doesn't matter, it turns out that the argument goes through anyway.

[--Calculational complexity in Fock space--]

Now if you do it in regular space, you have to do what is called a Fock anti-symmetrization. It is because you have to satisfy the conditions of the Pauli principle. That means that the number of points increases. For each point of the first electron, you have to allow for every possible position in the other space [of the second electron]. The number of points in your computer program is k cubed times k cubed. And, in fact for n electrons you find out that the number of points is k cubed to the power n. This is well known; it has always been a problem when you are trying to calculate things in electron space. If you increase the number of electrons, it gets bad very fast. It is what mathematicians call an NP problem. It gets bad faster than any polynomial.

[--Separate electrons in spinor space--]

Now in spinor space, it appears to be much better. Because when you put the two electrons in there, they are beside each other, and they are already anti-symmetrized by the position.

[--Calculational difficulty spinor space--]

The anti-symmetrization occurs as a characteristic of the two wave functions as they are placed against each other. In a practical problem, this boundary may not be a simple plane as I have drawn. It could be, oh, a circle or a more complicated thing. But the number of points is essentially the same. It only takes so many points to describe the electron. That is the idea anyway. You get a dependence on small n of proportionality or at worst n squared, or n log of n perhaps.

--Comparison

These two situations flashed before my eyes in the form of this graph--about two weeks before the meeting. And that is why I was having trouble trying to decide what to do. Because you can see here this curve, with the number of calculations necessary, the complexity of the calculation, goes up pretty fast. This is log-linear paper that has 120 cycles. You have probably never seen such a piece of log linear paper. This, right here corresponds at 20, to calcium. A lot of important atoms of chemistry are in this range. The ratio here is 10 to the 110. The question is not whether you can do anything this way. You cannot do anything with that. It gets bad too fast. The question is whether there is anything here or not [in spinor space]. Is there a way to calculate, if you get up to this curve. You might be able to do calculations in spinor space of systems with large numbers of electrons.

There are not very many points on this curve. I think that there is a pretty good point here at two for helium, but one does not count. I don't know if there are any more calculations for this either. I am pretty sure that there is nothing for beryllium. We are not even sure of the relative heights of these graphs. There are no points on this curve [I.e. in spinor space], no calculations. This is an open question. Where does the line lie? When is n or n log n possible? It may not happen, but if it does, it is going to matter. That means that there would be large calculations of large electron systems that we cannot do now.

What else do I have here? Here is another problem that I have been thinking about.

[--Alpha--]

When you form an electron pair like this, you start electromagnetic interactions, and gravitational interactions between the electrons. Pauli said that there is no solution of the Pauli exclusion principle until you understand here the interaction comes from. I have not encountered any difficulty with that, but the question is still open. What happens to this interaction in this kind of space? What do they look like in spinor space?

This is an exciting question as well.

[--Quarks--]

And then we have the question of nuclei. If you send a nucleus into spinor space, what happens? The nucleus cannot go through it because it is not a Dirac particle. It has to be a Dirac particle. But we know that the nuclei satisfy the Pauli principle exactly. So it could be that the nuclei are made up of components that satisfy spinor space. And so the open question is "Are there quark wave functions in spinor space that make any sense?" So you might have a nucleus that goes into spinor space, you divide it up into three quarks; the quarks go through the spinor space, come out the other side and recombine into the nucleus. Can you make any sense out of that?

This last one is sort of a periodic table of elementary fundamental quarks and leptons.

[--Extra exclusion--]

This is the latest periodic table. The idea here is that these interactions are universal. I put the W up there because I was thinking of weak interactions. The electromagnetic interactions have a certain universality, the gravitational interactions have a universality, and it suggests that these particles exist in the same space. Whether you know how to do that or not, there are some experiments that you can do. If you really believe that the exclusion principle is caused by the currents that cannot cross. And you should be able to, in principle at least, to detect an exclusion principle between dissimilar fermions. In the past, [tests of the] the failures of the exclusion principle have always been [tests of] failures to see an exclusion between two electrons. But it could be that we are missing the point. That, [i.e. exclusions between electrons] always happens but that sometimes you can see exclusion because of volume lost, because of the failure of the muon to get into a nucleus. This is a hard project; you have to go through a lot of data; you have to go through a lot of different theories to try to understand if, in fact, there might be a shift in the spectrum of muonium due to extra exclusion.

That is all I know about the Pauli principle. I hope that you have found it as exciting as I have.

I will say that it is always nice to come and talk to the physics club.

Q18: Can you give us any kind of intuitive description of spinor space in eight dimensions? It is hard even in three. What do you have in the way of getting a feel for spinor space?

Right, right, I will be glad to talk about that. I talked to a nice fellow from Germany once about those kinds of problems. Some people have trouble driving home; they have trouble in two dimensions. Most of the people in this room do pretty well in three dimensions. But there has not been any evolutionary pressure to think in four dimensions. And so I do not think anyone can do it. I certainly do not know. I cannot do it. And five, six, seven, eight is no better. It is not something that I know how to do or that you know how to do in general. So you have to think very formally about it. There is no other way.

Q19: There are basically eight orthogonal dimensions.

It is essentially eight orthogonal dimensions. [avoiding the issues of pseudo-metrics, or null directions]

Q20: And eight is just as hard to visualize as five.

Well eight is more than five, but once you have started to think formally, it is just a little piece at a time. I do not know how to do it. I have plenty of trouble with four, and five is very difficult. I did a lot of work in five.

Q21: It's a mathematical issue, of just how do you handle these eight independent . . .

That's how it has to be done. I cannot make it any simpler.

Q22: That's the only way to do it.

I do not know of any other way, absolutely not.

Q23: I have a question . . . this language. You mentioned a two electron state in spinor space.

. . . Uh huh.

Q24: The electrons sit side by side.

. . .

Q25: Twice times eight instead of eight times eight.

Right

Q26: Normally you would have eight times eight -- an eight dimensional spinor for one particle and an eight dimensional spinor for the other particle. And the product of those is a sixty four dimensional space.

The product is required by the Fock space interpretation, right. And it is required that its anti-symmetrization is generated by the Pauli principle.

Q27: But you get away with having, in your spinor space, you say the two electrons are side by side, so you have twice times eight.

. . . exactly

Q28: Just factors of n instead of powers of n. How do you get away with it?

How do I get away with it? If the particles are disjoint, then the Fock space factorization splits, they really have no point in common. The interactions do not mix the state, eh? When you do the anti-symmetrization, one pair, after the anti-symmetrization, is zero over here and the other pair is zero over there. So everything is split, there is no double wave function, if they are disjoint. And that is true in spinor space too. So that the only place where they are not disjoint is the precise point where they meet. As long as there is anti-symmetry there, it satisfies the conditions of . . . the Pauli principle, you do not need any more. The anti-symmetrization occurs on a space of lower dimensionality. The argument is that the NP requirement, enforced by Fock space, is no longer true. There are no examples worked out yet. It is an interesting question and that is why I was uncertain of whether I should say anything. It has been two years now, no one has said anything. I am going to say it again. I do not have a problem with that. I will take the chance and see if someone comes by and clobbers me. I would like to understand it better.

Q29: earlier on you talked about a Riemannian space.

Right.

Q30: So I presume that there is some metric.

Yes, when you go into spinor space, there is a fixed metric that is constant over the whole space. All of the properties of the Einstein metric and the five dimensional metric are contained in the coordinate transformation. The fixed metric is basically diagonal (1,1,1,1,-1,-1-,1-,1) and does not change. That is typical of spinor space. People have used that kind of metric in spinor space since the days of VanderWaerden. If you what to go over it we can spend some time, but I think that the epistemology is going to win on that. It is not a problem.

Q31: But you said that gravity also works in . . . So in what sense does gravity appear.

The gravity appears in the transformation to the spinor space. What happens is that you only take one particle at a time into the spinor space. So there is a separate metric for each particle. Now, you have to do that in quantum mechanics, anyway, because there are situations where you have it that the two electrons are on top of each other. So you cannot use one metric because you cannot separate the force of particle A on particle B from the force of particle B on particle A. You get into trouble anyway. So you have to use multiple gravitational metrics. The approximation of the universal gravitational metric is alright in classical mechanics where you have point particles because and they never have any point in common so that you can always squeeze a little bit in there to fix it. When the particles are on top of each other you cannot do that any more. You have to have multiple metrics. And so for each metric you have a set of gamma matrices, which you calculate from that metric, and those gamma matrices carry the gravitational field of that particle. We can talk about it some more sometime; it is a detailed mathematical thing. It is a very hard question.

Q32: Dan, I think Feynman in his book says, the elementary book, that the Pauli exclusion principle is the thing that we perceive as atoms not being able to interpenetrate. How does that relate to things . . .?

Right, it is like the little diagram that I put up there for the two oxygen nuclei. If the phases are wrong, it ceases to exist. The atoms that would interpenetrate cannot exist any more. Quantum mechanics says that it is not there. If the phases cancel out and it goes away, and the probability is all of a sudden zero. And that is why it is such an abstruse concept.

Q33: But this is almost a one dimensional kind of problem--putting one atom on top of another. So you have this eight dimensional space that has one dimensional consequences.

The eight dimensional space has a lot of stuff in it, besides what . . . . It has all kinds of things. It is an elucidation. It is incomplete. And so if you start out with something in space time, a couple of Dirac particles, you can put them in to spinor space; and you can see what they do in spinor space; and you can bring them back. The arguments in spinor space come back to space-time. It goes back to his question. How do you visualize it? Yea that works. But actually, part of the problem, with say a polymer chain, is those bonds do not interpenetrate with each other but they are on top of each other. Some of the bonds are two electron bonds; they are on top of each other. And so this funny eight-dimensional space undoes that. Nothing is on top of each other. When nothing is on top of each other, it is much more interesting, much easier to understand and much easier to calculate.

Q34: As long as your skill in thinking in eight dimensions . . . spaces.

No, you cannot be in spaces . . . Let me tell you. If the calculation actually works, if there is a way for a programmer, who forces himself to work out the details, if he can calculate the structure say of carbon, exactly, ab-initio, it will not make any difference how many spaces. It does not matter at all.

Q35: Getting back to that . . . are you saying we can understand it in space time what a given thing is, and when we go through a transformation into spinor space, there is just a set of rules, treat them as abstract rules that there is no intuition related to.

So that is all we have, right now.

Q36: Yea, go into that space, do your calculations, get a result in space-time on the other side. And using that thinking, we could have an initial condition and a final condition and compare the two we are talking about

You could have a result that is useful. And what it says is that in certain cases it is much, much faster. The argument suggests that. We do not know; we do not have any points on the curve. If you can break an NP problem, results are quite significant.

Q37: It is the classic physics curve too about the guy up at the board with all these equations and then a miracle happens and here is the answer.

Q38: Sounds similar.

It is going to be a while. There are lots of things to think about, lots of things to do, lots of sneaky little things that grab your to as you are walking down the sidewalk looking at the sky. It is one step at a time. Someone else may say something about it; otherwise, you will have to wait until I do.

Q39: Has this been applied?

No it is too early for that. There is no application of spinor space anywhere.

Q40: As a related question. We remark that Bose-Einstein particles, no spin, occupy space in a different way than the Fermi-Dirac particles. That is a good question. Some people, the progenitors of quantum field theory, believe that there is some abstract fundamental thing called a Bose particle and an abstract fundamental thing called a Fermi particle.

Q41: That's right

At the present time, there are no stable massive Bose particles known . . . [that are not composite]. There is a certain possibility that the Fermi particles are the fundamental ones and the Bose particles are always derived. We treat them symmetrically because one equation has a plus sign and another equation has a minus sign. The equations for the Bose particle are probably all derived from the other one.

Q42: What do they mean when they talk about a Bose condensed state?

A Bose condensed state means that you have atoms, which are made of Fermi particles. But if you arrange the Fermi particles so that they act like a Bose particle then that collection of atoms behaves differently than if it had a spin one-half. They call that a Bose state. There are still Fermi forces inside those atoms. But if you can approximate the atom in a low, low temperature zone, it always stays in the same state; it acts like the Bose particles. There is a minimum density. The minimum density occurs when all those Bose particles get mashed enough together that you have to eventually start doing Fermi calculations. After a while you do. They do not collapse completely the way you would like them.

Q43: I see so it is really that you have a condensed phase that still occupies a finite volume.

I think that is right. I do no think that the Bose states have zero volume. You put enough particles in there, you have a little ball it starts to be Fermi-like. No?

Q44: The Bose Einstein condensates have a large physical size with a small momentum size so they are macroscopic objects, all particles are in the same momentum state. They don't get close enough to know that they are composed of Fermions.

If they get close together, they hit each other. Then you have to do a Fermi calculation. They are not completely Bose particles anymore, there are other states that happen, and you are done.

Q45: So the Bose-Einstein condensate is another state of matter. . .

Q46: It's the way . . .

You know about this, it is a way to extract money from the government.

Q47: For theoreticians, it would not do me any good at all.

Q48: It's an experimental verification of a fantastic prediction of quantum mechanics.

I agree.

Q49: It's the prediction of the spin-statistics theorem, no matter, we do not know, where it comes from. It's a demonstration that this prediction is true.

That's right.

Q50: Is it the same as saying that each particle is unaware of the other particle?

No, because the two nuclei of the oxygen molecule must know about each other. Otherwise you could get an [odd] integer spin state. They know about each other, but they just treat each other in different ways. It's a very bizarre thing. That's why . . .

Q51: He said something about “they do not interact with each other”.

They do not interact as forces but the wave equation has an internal complexity that is not understood. I am proposing that with the spinor space you might actually be able to do that. That is what I am going after. It offers some understanding, at least for spin one-half particles, like the electron.

Q52: Do I understand you correctly; do I understand you to say that when you do the spectroscopy that half the states are missing?

That is right.

Q53: That is a different thing. Is the state identified with a transition?

No, one way to show that there is a state there is that you have a transition to or from the state.

Q54: Yeah.

When you do spectroscopy, you do differences and you show that the energies from different states add up. And you conclude the existence of the state by looking at a combination of transitions that satisfy the difference formulas. But, there are other ways to look for states. For instance, you can look at how much heat is in a soup of that molecule. The amount of heat that is in a gas of that molecule depends on how many quantum states there are. If half the quantum states are gone, the heat capacity can be half the heat capacity. You can tell if the states are not there and you can tell if they are not accessible by any interaction. . . .