Hendrik Casimir - Session II

Kuhn:

When we quit yesterday, you were toward the end of talking about Pauli and Zurich; I thought that before we went back, which I hope we will do, let me ask you just a little bit more about that particular period in terms of some of the technical concerns. You had spoken of Pauli’s recently having finished the second Handbuch and looking for topics and the concern with relativity, the five-dimensional formulations, and so on. What can you tell me about Pauli’s feeling at that point about field theory and its state and the extent to which it was likely to work out?

Casimir:

It is rather hard to say. I dont think Pauli really believed very strongly in this sort of five-dimensional theory; he liked it mathematically, and occasionally he thought there might really be good physics in it, but I don't think he really expected that it would solve fundamental difficulties. Also, I think, he was extremely well aware of the difficulties of theory of radiation and radiation field and so on. I told you that we were speaking of various things about radiation formula and such, and about what the limitations would be; so he was well aware that the theories you had at that moment would not be sufficient to explain everything, but in those days, there were very few physical effects that would require, so to say, this next order of magnitude. I do not remember any clear-cut idea, but I remember working with him, and that is perhaps characteristic, because he rather suggested certain problems to me. The first question was a purely mathematical one, that is, of proving the complete reducibility of a useful representation, which I did. I worked on this five-dimensional theory. Then I did something on theory of radiation and radiative damping. You remember [Rosenfeld] we had some correspondence there; what was called “korrespondenz-maessiges Verfahren”, a kind of makeshift procedure for getting intensities of radiation and that sort of thing, and there were certain paradoxes there which I did some work on. I think at the end of that paper I made a certain remark that corrections to formulae might arise which would be outside the scope of the theory of radiation of those days; let’s say, in any case, that these remarks were approved by Pauli and must in a way have expressed also some of his feelings about the situation. It might be worthwhile to look at that and see exactly. I remember writing something there which is fairly cryptic but which in a way gives this sort of idea, in the Zs. fur Physik. Then the next problem was one on scattering, Compton scattering and Klein-Nishina scattering with bound electrons which I have talked about, where the question arose as to how to possibly formulate in a relativistically invariant way this scattering by electrons in a certain state of movement and so on. I think Stueckelberg went a little bit further along these lines later. There was already something which came a little bit in all these things, which, I would say, pointed in the direction of renormalization theories and that it might work a little bit along those lines;however, I think that Pauli hoped that you would have a more important breakthrough with a more consistent formalism, but he didn’t know how to tackle that.

Kuhn:

Would you think that the five-dimensional attempts, through perhaps without great optimism, were nevertheless attempts in this direction, attempts to see whether they might provide a key?

Casimir:

A little bit, perhaps, but Pauli was not a very happy man in that particular winter; his first marriage had just broken up, he was drinking rather heavily, and he felt neither happy about life nor about physics, I would say. So it wasn't one of his most productive periods. I remember one remark he made when we came home from a drinking party; he was slightly tipsy and be started to speak about all kinds of problems, and among things, he said, “Wir leben in einer fuerchterlichen Zeit, in einer kulturlosen Zeit. Ich weiss genau, was jetzt kommen muss, aber das sage ich nicht. Wenn ich es sagen wuerde, wuerden die Leute sagen, ich bin verrueckt. Dann mache ich eben lieber generale Relativitätstheorie, aber ich weiss genau was kommt. Ich weiss es genau. Das sage ich Ihnen, dann vieleicht ein anderes mal. Gute Nacht.” He has never told me, but one had almost the feeling that he had very definite ideas. I would have liked to know what they were like. “Ganz genau, dass es schon kommt.” I asked him later, I think, but he never told me.

Kuhn:

You take it that when he said he knew just what was coming he meant what was coming in physics, or —?

Casimir:

No, no. That related to the world in general, but since he didn't want to pose as a prophet about mankind in general, he thought he might just as well play around with these formulae of general relativity.

Kuhn:

This is really addressed to both of you. I think I have noticed a great many quite different attitudes among different people about the relationship of relativistic to Newtonian formulations. For some people, knowing relativity meant that you could use Newtonian things only as an approximation, but if you really wanted to do fundamental work it had to be relativistic. I think this is most pronounced among the people who were influenced by the English at Cambridge. One the other hand, you get another group of people who were perfectly willing to think of these as sort of equa1ly fundamental but applicable to different sorts of problems; they were not bothered by the fact that there are things you can do in non-relativistic quantum mechanical formulations that you simply cannot do with the same power in the relativistic formulations, and who will therefore think of these as somewhat on a par and will use sometime one and then the other. In that sense, from one point of view, these people would be said to be inconsistent in their mental commitments in the field.

Rosenfeld:

Yes, surely at that time, one was very conscious of the fact that the methods of quantum mechanics which one tried to extend to field theory were essentially non-relativistic and so one had to prove explicitly every time that the result was nevertheless covariant. It was very clumsy, so from that point of view, that is the great progress made by the new methods. However, I must point out there that Weiss, I think it was in '36, under the inspiration of Dirac, had produced essentially the kind of relativistic manifestly covariant formulation that we have now; so the problem was recognized, but it was not regarded, from what I can see, as a fundamental —. One was convinced that the whole thing could be formu- lated relativistically so that was not the main problem. But one was worried all the time, especially when one was dealing with divergences, just about separating out what could be the result of an essential physi- cal divergence and what could be just the result of clumsy mathematics.

Casimir:

But I think that also those people who were quite willing at first to work with formalisms that were non-relativistically invariant realized that it would be necessary in some way or another to extend the formalism so as to become relativistically invariant; sometimes it was done in a rather clumsy way and then later it was done in a more elegant way. I don’t think there was ever anything (to do with) commitments to fundamental ideas, that is, as far as special relativity is concerned. It is a different story when it comes to general relativity. There I remember that at one time Ehrenfest and I entertained some hope that another version of Einstein, the (Vierbein) theory, might have something to do with the Dirac theory of the electron, because the (Vierbein) theory and the spinning electrons seemed to go well together. And at one time, one had the feeling that it would be difficult to write the Dirac equations in such a form that they were covariant in general relativity, and then some sort of very vague idea of the spin of an electron being a kind of gyro-compass for the (Vierbein) floated around among many people. I don’t know whether they got much into print, but I know that several people entertained ideas along those lines. I think it was the Russian Fock who then showed that it was possible to write the Dirac equation in a form that was covariant also for general linear transformations. But, of course, I would say that, even today, there is not much connection between the general relativity theory of rotation and quantum theory. Well, you [Leon Rosenfeld] had been amusing yourself quantizing gravitational fields at that time.

Rosenfeld:

But at that time that was for another purpose.

Casimir:

I know. But as far as special relativity goes, I would say that there is not a difference in commitments, but only a difference in attitude. Some people say, “Well we'll let it go for the time being and see what we can do.”

Kuhn:

Heisenberg had told me of his dissatisfaction with Jordan’s additions of a quantized field in the final section of the papers he wrote with Born and also with Born and Heisenberg. Heisenberg was opposed to the addition of relativistic considerations, not because of the odd or difficult mathematics that would have to be used, but because the theory then might be of a totally different nature. Thomas, on the other hand, had no compunction about introducing relativistic considerations, since, so far as he was concerned, relativity had been shown to be correct. In fact, Thomas has told us that in discussions he held at Copenhagen, physicists demurred at his suggestion to tackle problems relativistically, since they held that the results would be more correct by only a few percent. Thomas had no notion that one would get a larger correction, but when he worked out the problem relativisticaily, the factor two dropped out. Now, the question is: since the attitudes of Thomas and Heisenberg seem to be so markedly different on this matter, I wonder whether other physicists, like Pauli, Emrenfest, or Bohr, also held different points of view about the use of relativistic considerations?

Casimir:

Yes. Of course, when I came to Pauli, so much had been done there relativistically, that, in a way, that was settled. Bohr, I would say, was not so much concerned with that question. Ehrenfest, I think, was; he wanted to see the thing relativistically, because this whole question of invariance and formalism counted heavily with him. He was very glad for the Dirac theory, when —. I just remembered yesterday evening that when I had to pass my second examination, my master’s examination, I had to write something like a master’s thesis. I produced something on hsow the Dirac theory could be relativistically invariant, so to say; this question of not being tensors. I didn’t develop this spinor formalism then, which, I think, was just a little bit later, but I worked out something which probably existed already and was very similar to quaternions in three- dimensional space, taking the Dirac matrices and the product of Dirac matrices, to use them for infinitesimal rotations and also for finite rotations and how you could write Lorentz transformations of four- and six-vectors by multiplying them with Dirac matrices and then with the factor before and the factor behind being two-valued representations; I worked out a number of problems that way and showed that you could derive the addition of velocities and so on with this sort of quaternion formalism. He was very much interested in that sort of thing and was always asking how it would be when it was relativistically invariant; when that very idea, as I say, gave rise to Dirac’s equations, that was one thing he was very keenly interested in. When I started to learn about quantum mechanics, some of the main steps had of course been taken already; there was on the one hand always the further extension of the mathematical formalism in which group theory, for instance, had played an important part. Ehrenfest was always very much interested in that and he struggled with it and he had some difficulties in familiarizing himself with that sort of thing; both Pauli and Kramers played an important part. Bohr, of course, was not really very much interested, but since I was then working on my thesis which contained things about rotation groups and so on from time to time, he would very kindly ask me,"Hvordan star dat (???) rotationsvaesen” which I think clearly shows that he did not have a high opinion of this sort of thing, although it was an amusing game. There are a few of these things, for instance, which all of a sudden become common knowledge and where it is not always quite easy to find out who did it first; for instance, this question that the eigenfunctions of a symmetric top are the representations of the three-dimensional group of rotations and that spherical harmonics, when they transform an arbitrary rotation, then the coefficients of transformation are given by the eigeafunctions of a symmetric top, and that these also are spherical harmonics on the three dimensional plane in four dimensional space which is the space of the four Eulerian parameters. One day I knew it and one day everyone knew it; I don’t know where that appeared, but probably, if you looked hard, you would find it in much older literature, but it was never in standard textbooks on spherical harmonics and so on, until suddenly one knew that this was the case. Incidentally, that formed the basis of this work I later did when I was with Pauli on reducible groups because, since these representations are eigenfunctions of a symmetrical top, you can then say that the matrix elements of irreducible representations are eigenfunctions of a certain Hamiltonian, and then you put to yourself the question whether for other classes of groups you can find a similar Hamiltonian of which the matrix elements of irreducible representations are eigenfunctions; and I was able to construct such an operator for any semi-simple group, and this then formed the basis of subsequent proofs of irreducibility and so on. But as I say, Bohr would inquire sometimes about “rotationsvaesen”, but that was the extent of it. Another question, of course, was the rapidly growing application to all branches of physics where Sommerfeld and his school played a predominant role as well as did Heisenberg and his people at Leipzig. England contributed too, to some extent, with Mott and so on. Pauli was not so very much interested in these things, though occasionally some of his people would do very good work along those lines. And in those days Bohr was not particularly interested either, although, I think, he had this idea that the application of wave mechanics to motion of electrons in metals would be important. There must be a letter or inscription of Pauli in connection with Bloch’s first paper on electrons where he says, if I remember rightly, something like, "Aus diesen Abhandlungen von Bloch wirst du sehen, dass dein Sieg Uber die Physiker vollstaendig ist.” or something along similar lines. Where did I see that? Was it in something he wrote on these papers or in a little letter? There must have been some sort of an argument going on, probably in connection with Sommerfeld’s fairly primitive application of Fermi statistics where people said, “This is no good,” and Bohr said, “Well, it is a good beginning and it must be some" Then Bloch’s paper came out and then I think Pauli acknowledged that Bohr had been right. But where I saw that remark? I didn’t invent it myself and I'm quite certain that it must have related to this sort of thing. This, of course, was only natural, since Bohr’s thesis, after all, was on electrons in metals and, though it is very little known, since it is written in Danish, in my opinion, that thesis is a very great masterpiece; it is a very beautiful analysis of many things and you find a lot of things there to which people came back later.

Rosenfeld:

He was keenly aware of the impossibility of explaining those things by classical theory.

Casimir:

Oh, that’s everywhere in his earlier writing, wouldn’t you say?... Then there was in this period when I started this question of the foundations and thinking about complementarity and so on. We were speaking about fields and so on, and later we knew Bohr came back to this question of electromagnetic fields; when I was with him, you might say everything had been done, the uncertainty principle was clear, complementarity to a certain extent for normal one-particle mechanics was clear, but I think Bohr kept going around thinking through these examples, discussing the apparent paradoxes, trying to find different formulations. This was not only a question perhaps of trying to see the thing more clearly or to get more to the bottom of things, but in a way, although there was very little mathematics involved, it was also the elaboration of a certain technique of thinking and of developing really a way of tackling problems without much mathematics; you could see all the things and predict all the things; at this Bohr was extremely ingenious. I think you might say that really in these years he sort of developed that art of making fairly precise qualitative statements about what quantum mechanics would predict without really doing much mathematical work.

Kuhn:

Who followed him closely in that?

Casimir:

I think there is hardly anyone who really could do these things Bohr did. Also the way he played with orders of magnitude — all these things were very simple. For instance, the thing he taught me and which I found extremely useful in later life was that you always have these different characteristic lengths, the classical radius of the electron, the Compton wave length divided by 2Pi, the Bohr radius, always differing by a factor of 137. In any kind of formula he would always twist things up and down so that you got these things, so he would never write quantities e and c and h with large numbers of 10 to the high powers; he would always combine them so as to get these dimensionless things out. That was a simple art, you might say, but a very definite one. and I would say that it was really an art developed to a high degree of perfection by Bohr. The same thing held with energies: with mc² you have 1/2α² mc² and then you have your fine structures and so on, and the hyperfine structure with the m/M, Then, of course, as regards the question of radiation, I think one of the things Bohr was always aware of and which you find in his first paper on the quantum theory of the hydrogen atom is: if you take a classical orbit of an electron, it would of course radiate, but this radiative damping is only a slight correction because the fine structure constant is so small, you might say, that this radiative damping can be neglected. That was one of the main arguments for him in making a theory of the atom. “These radiative effects,” he said, “are so small and therefore you can, for the time being, use ordinary mechanics and forget about electrodynamics.” I think during the period that I still was at Copenhagen, he said, “But these radiative effects are small, so for the time being, I will first clarify in my mind the particle without its radiative reactions”; and later, of course, he came back to these questions of radiation. You might say it was a little bit late, but even though quantum theory was fairly complete in '28 or ‘29, the number of people who had really mastered it was rather limited. You see it when you look at the literature on other experimental subjects; to the experimentalist it was still very much a closed book. And on the other hand, let’s say, that the group of people who really knew how to manipulate quantum theory didn’t know too much about more pedestrian branches of physics, so it was quite some time until things were worked out and until it was a generally recognized discipline. At Leiden, of course, Ehrenfest worked on it, but it was still regarded as a somewhat esoteric subject and I think this was true in most centers. And, as I say, those people who were proficient in the formalism of quantum theory were not always too familiar with classical physics and with experimental effects that might be interesting to discuss.

Kuhn:

Do you remeither any particular examples of that lack of familiarity with classical physics?

Casimir:

No, not at this moment, but I have the feeling that if I thought hard I could find them without much difficulty. I think that would be fairly easy. For instance, I think Sommerfeld played a very important role there. Another thing, of course, which came in those days was the application to nuclear problems and that, among other things, was something Bohr was very much interested in. We discussed that already, his relations with Gamo his introducing Gamow to Cambridge, and so on. It is perhaps interesting that a problem that Gamow Landau and I discussed frequently in Copenhagen was the question of the electron. In those days there were no neutrons, and so one still thought that the nucleus had in some way to be built of protons and electrons; it was clear that alpha particles and also protons might be treated with wave mechanics applied in a normal way, but it was also evident that you could not trap an electron in a thing as small as a nucleus without getting into terrific difficulties.

Kuhn:

You say this is a problem that you, Gamow, and landau discussed; when would that have been?

Casimir:

That was this last winter when I spent most of my time at Copenhagen, 1930-31, and both Gamow and Landau were there. Gamow, who was writing his book on the structure of the nucleus, marked all passages relating to beta structure and to electrons carefully with beautifully drawn skull and crossbones in his manuscript, and I remember Cambridge University Press asking whether they were permitted to replace the emblem by a simple asterisk! [laughter]

Rosenfeld:

I think he had a stamp made.

Casimir:

Right, he had a stamp made. Gamow wrote a letter saying that we had no objection to the asterisk and I remember that I contributed to that letter the sentence: “It has never been my intention to scare the poor readers more than the text itself will undoubtedly do." [laughter]

Kuhn:

What ways did you see as possible ways out of the problem of the electron in the nucleus in that year?

Casimir:

I would almost say none. On had the feeling that something entirely new would have to come. But another thing which appeared in some of the literature in those days was the question whether, for electrons outside the nucleus also, there would be new factors or forces in its interaction with the nucleus. You said: "in order to trap an electron in a nucleus something extra must happen; what about an electron outside the nucleus?” That question came up in connection with internal conversion of gamma rays and in connection with hyperfine structure; with internal conversion, it was thought at the time that you could not explain the large experimental internal conversion coefficients. I wrote a short note where I had calculated it relativisticaily and found that the conversion coefficients were much too small, and I remember were then bad. The idea that there must be something extra, we didn't know what it is, but it certainly would be proportional to the intensity, to the square of the wave function, at the nucleus, so add something proportional to Ψ_o² and that would be the effect. It was then shown by Mott and his group that you could explain conversion coefficients. I had used an asymptotic expansion which was all right for high energies, but the energies for which things were measured were not as high as all that, and there was the other question that four-poles and magnetic octopoles had much higher conversion coefficients. There is a funny point there; we had thought of that, but I had shown that there is no difference in conversion coefficient for whatever multi-pole you take. That was even right for this asymptotic expansion I had used where essentially you take things in the wave zone when the field behaves like 1/r, but the point is that whereas for dipole radiation, this asymptotic method was not really good, but not very bad either, its convergence was worse and worse as you got to higher poles; and so whereas asymptotically, it would have been right and there would have been no effect, or a factor of two perhaps, when you really worked it out properly for the dipoles, yet there would have been a much larger factor for the higher multipoles because the things converge so badly. This is a little episode which just shows that you must be careful with these things. Another place where it came up was in connection with hyperfine structures where people always thought you could not explain hyperfine structures. And then came Goudsmit pointing to special perturbations, and Fermi and Segre who analyzed certain things further, and you always came to the result that if you did it properly, you did not have to introduce anything but normal electromagnetic interactions in order to explain the interaction of nuclei with outside electrons. I think that’s true even today because the nuclear interactions exist, of course, but they are weaker than the other things so they could give at best only very snail corrections.

Kuhn:

Let me come back to a question which I asked a few minutes ago and which I possibly put badly. In this work that Bohr was doing in connection with the measurement problem, I asked whether anybody else was really following him on this and you said no, in the sense that he was so much the master of this. I was really more interested in finding out who else was interested. You spoke of landau’s reaction yesterday — "That’s not physics.”

Rosenfeld:

But that was before.

Casimir:

Yes, but all the same Landau was interested. I think Pauli was interested. Most people would explain to Bohr what they had been doing and then Bohr would look at it from this point of view; that would help greatly to clarify ideas about the things.

Rosenfeld:

I always had the impression, even in later years, even the last time I saw him in Kiev, that landau always spoke of those points as subtle points, which meant he didn’t want to learn too much about them. But on the other hand, Pauli, and Dirac too, in his own way, were very much interested. Somebody asked Dirac about some point that raised some doubt about the question of complementarity and I remember Dirac’s saying, “That is right.”“How so?” said the questioner. “That is right because Bohr says so and he has thought about it.”

Casimir:

Another thing which has just occurred to me in thinking about these Gamow things and theory of alpha radioactivity is that it is curious that, although this is a very good explanation, a lot of bad papers have been written about it. Gamow's first paper, of course, was mathematically far from perfect. Then wanting to put things right, von Laue wrote a paper which really wasn’t very much better; Born later published an extended paper which was not very good because the solution was not unique.

Rosenfeld:

You set it straight?

Casimir:

There were others who did it right and in Kramers’ book it’s all right, but there were always a number of papers about it and Pauli was always extremely amused when another publication about it appeared. He would say, "Es Gamow’t wieder.” But it’s very curious that a somewhat elementary textbook problem should have given rise to so much difficulty, and that is probably because the whole formalism had been developed very much in connection with Hermitian theory, Hermitian matrices, eigenfunctions and that sort of thing, and therefore, this sort of pseudo-eigenfunction with an exponential decay didn’t quite fit the classical picture of transformation theory, Hermitian matrices, and so on. I think you can find this sort of thing even up to modern days.

Rosenfeld:

Yes. It is the problem we discussed recently in the more general context of nuclear reactions and it is a complicated thing because this exponential decay is only part of the phenomenon. There is also the expansion of the wave packet which complicated the thing.

Casimir:

Probably it is true that even in pure mathematics the theory of pseudo-stationary states is not so well developed. You come across the same sort of thing in what you might call a classical problem; if you try to calculate laser modes — an interferometer where you build up a standing wave. But it isn’t really standing because there are always certain diffractions so the waves go off to infinity sideways but only to a limited extent. If you try to do that problem mathematically you run into rather complicated things. There are extremely clumsy and laborious calculations published by people of the Bell Labs, and so on on that sort of thing and I’ve been thinking a little bit about it, but it is not quite easy. But I like that sentence, “Es Gamow't wieder”. Shall we speak a little bit more about Pauli? There are a few more things. He was working on these relativistic things and he was interested in problems of radiation, as you can judge from these things he suggested to me that I might take up; they were partly my own ideas, but be certainly encouraged that sort of thing and there was a little pre-shadowing of a sort of coarse renormalization. He was not too much interested in solid state work, or, in any case, he disliked it, feeling it wasn’t sufficiently exact; he had some work that he had done with Guettinger on problems of rotating molecules and similar things in changing fields and also these questions of magnetic atoms in rotating fields with adiabatic transitions or whether the thing flips over — work which, in a way, foreshadowed much work which later became very important in connection with problems of paramagnetic resonance and all that sort of thing. There is quite a bit of his in those early papers which now becomes very relevant. I also remember giving to a rather poor student as a thing to do for his engineering degree a problem which, if the student had been able to do it, would have led to the student’s writing the paper which was later written by [Richard] Becker, [Gerhard] Boiler and [Fritz] Sauter referring to super conductivity. Namely, it was said that in a super conductor the field is zero. Pauli said that this was all nonsense. The argument had been that if the field changed in a super-conductor there would be an electric field and that would lead to an infinitely high current. Pauli said this is not so because there would be an inertia of electrons and you would have to write that mdv/dt is equal to electric charge times electric field. So he put the problem to one of his students to work out the behavior of a body when the electrons behaved along those lines, but this was a rather poor student and nothing came of that. Then the problem was later solved by Becker, Heller and Sauter, which was again a forerunner of london's theory. You can, of course, put london’s theory in a more philosophical form; you can also say that london simply took the Becker, Heller, Sauter theory and arbitrarily put the integration constant equal to zero — which is also an adequate description of london’s theory. So that is remarkable. There were a few other things he did not like. He always thought that studies on gas discharges were an awful part of physics which he didn’t like. I also remember that he was chairman of the physics coilloquium and some people had asked for a lecture on modern types of radio tubes-appreciate that I said ‘tubes’ and not ‘valves’, as I would normally. Anyway, a man prepared the lecture there and I don’t remember it well, but I believe it was a pretty good one. He gave a summary of all kinds of complicated arrangements of grid structures and this and that and Pauli was sitting there, enjoying himself thoroughly: “Das ist aber lustig, das ist aber lustig, Ich verstehe Uberhaupt kein Wort. Das ist aber wunderbar. Verstehen Sie ein Wort? Das ist aber lustig Kein Wort — was sagt er da? Hab ’ich nie gehoert, was er uberhaupt sagt — nichts verstehen!” And it went on that way until finally he said, very politely: “Ich hoffe dass die jenigen unter uns, die sich einen Vortrag ueber diesem Gegenstand gewuenscht haben, jetzt befriedigt sind.” A most remarkable session! He didn’t even think it was bad physics; he just thought it extremely amusing that he, Pauli, should not be able to understand even one single word of what was being said. That was a colloquium on radio tubes. I remember another seminar where I think I reported on the work of the Mott group on internal conversion using higher multipoles and where we got into a violent argument concerning multipole developments of radiation. That was very funny because he wanted—. If you have the radiation of the nucleus, one wants to expand the e to the ikr in powers of r, but there is the other question that you can have the thing without a steady dipole moment which still emits pure dipole radiation, and Pauli for some reason was against that. He thought you shouldn't call that a dipole, and I remember we had quite a discussion.

Rosenfeld:

I had once a terrible ordeal —

Casimir:

later I wrote a short piece on multipoles in Helvetia Acta and in the introduction I mentioned this battle I had with Pauli. Afterwards I went to him and we kept more or less shouting at each other, but finally we agreed and it was also a question of nomenclature. Then in order to make up for it, since he felt he had perhaps gone a little bit far and that this time I was right [he did the following]. He had just gotten his letter from Springer [Verlag] and he said “Schauen wir mal zusammen, was der Springer mir fuer das Handbuch bezahlt”; it was a mark of great confidence that he showed me his earnings there. In a way he almost repeated the famous story he told about Nernst. Pauli loved to tell it and referred to it as “mein beruehmtes Gespaerch mit Nernst”. Somehow he had met Nernst somewhere, I don’t remember exactly where. Nernst said “Herr Pauli: Ja, ich habe damals bei dem Physikertag einen Vortrag von Ihnen gehoert.” I think it was Physikertag, but I'm not quite sure — "Das war an sich ganz gut. Etwas schillerhaft, ganz gut, und wo sind Sie jetzt?” And Pauli very politely, said, “Ich bin jetzt in Zurich.” Nernst: “Ja, so? Sind sie droben beim Herrn Meyer” — da war ja ein Herr Edgar Meyer, ein grosser Physiker.” oder bei Herr Scherrer?" Pauli: “Nein, nein. Ich bin dort als —” Nernst: “Da haben Sie dann ein Institut?” Pauli: “Nein, ich bin Theoretiker.” And then Nernst started to understand that Pauli really had a chair there: “Ach, so Sie sind also Ordinarius.” Then Nernst, feeling that he hadn’t been sufficiently polite, said: “Aber sagen Sie mir mal, lieber Herr Kollege, koennen Sie denn davon leben?” Das war das beruehmte Gespraech von Pauli mit Nernst. This change: “etwas schuelerhaft” to “Lieber Kollege, koennen Sie davon leben?” Also during that year Bhabha came to Zuerich with an introduction by Fowler; Fowler didn’t like Bhabha very much and thought him rather conceited and not so very capable, and he wrote a letter to Pauli — I don’t know whether it still exists-where he said, “You can be as brutal to him as you like.” Pauli liked this at once, so that already put him into sympathy with Bhabha. Bhabha did some work there and, of course, he was quite capable, but be reminded me in those days of the savage in Huxley’s Brave New World, the man who is discovered there and only speaks Shakespearian English Bhabha had studied Dirac’s book and studied it well, and it was the only form of quantum theory he knew, so that was really like this savage out of Huxley. I told him so; I think he got the point and I think it helped him some. Bhabha and Pauli became very good friends.

Kuhn:

I wondered a bit about that. I wondered a good deal who else. The texture, the flavor and the approach of that book are so different from that of almost anything else that it’s very hard to see its influence.

Casimir:

Which book?

Kuhn:

The Dirac book. I know of almost nobody who does think that way, and I’m much interested to know that Bhabha —. Were there other people?

Casimir:

I would say there always were but in a somewhat mitigated form. You win notice that, for instance, one influence of the book is that people started to use matrix elements in the Dirac way, instead of having subscripts and that, I would say, was almost universally done, so there there was definitely an influence. But as a matter of fact, when I had to calculate this scattering of Compton radiation by bound electrons, I told Pauli that I had to sum over states of positive energy of the Dirac electron, and Pauli said, “Well, there is a method of Dirac to do that which is somewhere in the Cambridge proceedings. Let’s look, it up.” That was the idea of introducing a projection operator, not for the relativisitic case, but be does it there for some other examples. He may also have done it for the relativistic case; that I don’t remember, but in any case I used that in this little paper, saying we would carry out the summation using a method due to Dirac, Cambridge Phil. —. And I then explained the method, but in a little more "down-to-earth” terms whereas in this Dirac publication it had been fairly incomprehensible. People have often ascribed the trick to me, but it is really a method due to Dirac. But Pauli was very good in that way; he was, of course, terrific in all respects, but if you were working on something, he would always say, “Yes, I think you could use there a method Mr. So-and-So must have used.“ “Let’s look at this paper of Dirac” was just one example, but I think it was also true in many other cases that he knew of some way of calculating an integral or some way of solving this or that problem.

Rosenfeld:

Oh, yes.

Kuhn:

I've heard it said that once he read a paper he remembered from then on its content, the journal, the author, the page.

Casimir:

I don’t think that would be right; I don’t think he would remember the page or even all the details, though he could probably reconstruct it if it were mathematics, but he would not have it at his fingertips. He would know the gist of the argument, the author, and roughly where the paper was published. As a matter of fact, his normal lectures on theoretical physics were very bad — or perhaps not very bad, but not good — unless he really prepared them. If he took the trouble to prepare them well, they would be excellent, but very often he didn’t do that, and just before a lecture he would glance through some lecture notes and say, "Das werden wir schon fertig bringen.” But it isn’t as if he had the mathematics at his fingertips in such a way; let’s say that, from that point of view, Fermi was much better. I think if you took. one of the reasonably advanced things in theoretical physics and asked Fermi at short notice, or perhaps even to give a lecture on it, I think you would get an almost perfect lecture with the formula nicely in place, good derivations, and all that; and that was certainly not the case with Pauli. I also remember a lecture he gave on relativity in which he was deriving the red shift, and somehow, which often happened to him, he got the sign wrong so that it became the Violetverschiebung. Then he began thinking about that, not saying very much but just going to and fro, changing a plus into a minus sign and back again,’ making a few gestures, saying a few sentences which had neither head nor tail, and that lasted. for quite some time, Finally, with a few sign shiftings here and there, the desired sign appeared there, and he said, “Ich hoffe also, dass sie jetzt alle deutlich gesehen haben, dass es sich wirklich um eine Rotverehiebung handelt." I remember a curious thing happened there. I spoke about Stern who measured this proton moment, and there was a nice little trick there of which the answer was due to Fermi. In these measurements of Stern you had to correct for a rotational magnetic moment for the two nuclei and the electron cloud, and Stern had asked how large this correction was. Some theoretical people had taken approximate wave functions for the outer electrons, regarded the whole as a solid charge, and then worked out the rotational moment. Then Stern had some more subtle methods to really determine these rotational moments which came out quite differently; this seemed rather incomprehensible, and Pauli couldn’t offer an explanation. Then came a letter from Stern that Fermi had explained it. As he said, “Die Elektronen rutschen,” and the question is: if you have this rotating system, you really have to say that the cloud is not really quite going around with the rotating nuclei, but if you really see what you are doing when transforming to a rotating system of coordinates and so on, you might say that the “Elektronen rutschen”, a backward movement, which meant that this moment was very much smaller than you would expect otherwise. It’s a rather subtle point, and I remember Pauli was rather interested in that because I had written my thesis on the rotating molecules. He said, “Sie haben ja ein Buch darueber geschrieben, und der Stern sagt, 'sie rutsehen’; erklren Sie mir das.” Well, when you knew how it was, it wasn’t so difficult to explain. That was quite an amusing effect in those days. Of course, another thing was that Pauli had learned to drive a car and that gave life a peculiar flavor; I always claimed later, and Pauli did not deny it, that there was a tacit understanding between us that I wouldn’t say anything about his driving as long as he wouldn’t say anything about my physics. Without being unduly proud of my physics, I think I could say without boasting that it was slightly better than Pauli’s driving, which isn’t saying much because Pauli’s driving was just awful in those days. I said that much later when we were together after the war and somebody asked, “Didn’t you have a very difficult time with Pauli?” I said no, because he wouldn’t say much about my physics if I wouldn’t speak about his driving. “Ja, ja,” said Pauli, “das ist richtig, so war es. Ich fahre jetzt kein Auto mehr, ich fahre kein Auto. Ja, ja Sie machen ja auch keine Physik mehr; also die Sache stimmt noch immer.” And, of course, the high spot in this car-driving of Pauli was the Physiker Tagung at Luzern; Pauli had driven us to Luzern with Bloch and David Inglis and on the return journey there were both Bhabha and Elsasser, one of whom had missed the train or something and the other of whom was with us also when we drove to Luzern. Driving to Luzern was all right, and then we’re eating in the evening, and Pauli was drinking orange juice but it was clear that he didn’t like it. He said, “This (evening I’m) all right.” Then all of a sudden Pauli switched and ordered himself a whiskey soda, so we kept an eye on him and when he had ordered a second whiskey soda, and I think, a third one, his passengers consulted and we said, “Now this is getting dangerous, so what are we going to do?” We made a plan to offer him drinks, get him quite drunk, and then Inglis would drive the car home; Inglis was one of those Americans who have been born and bred in cars and, even more American, a very good and competent driver, so we felt quite happy. The first part of the operation went according to plans, Pauli got quite drunk, and we said, “Now wouldn’t it be better if Inglis drove the car home?” Pauli said, “Oh, no, no question of that. Ich fahre, und ich fahre ziemlich gut .“ Well, all trains had gone by that time and also we felt we couldn’t leave him alone. “Ich fahre nach Hause. Wenn Ihr nicht mitfahren wollt koennt Ihr hier bleiben.” To let him go alone, we felt, we couldn’t do anyway, so there we were and, I think, we somehow packed ourselves with four people in the back seat and Pauli driving with Inglis sitting next to him, ready to try to save something if something could be saved. It was really awful. He started by sounding his horn, he skidded against one curb and across the street to the other until he got going, and then he started screeching around corners. And he was a bad driver in the first place Inglis sitting beside him tried to sober him somewhat while Pauli kept saying, “Ich fahre ziemlich gut,” and when Pauli went around corners in a terrifying way Inglis would say very severly, “Das heisst nicht gut fahren.” There was one incident where the full moon came over the top of the hill and Pauli swore at the driver for not dimming his headlights And another one where he said, “Wir machen jetzt einen Abstecher,” and he went along a little trail which ended unexpectedly At the wagonshed of some farmhouse and where he also began swearing, saying, “Was fuer ein Unglueck war das?” and “Ein Wagen schon ueber meinen Abstecher?” We got home somehow, but we never forgot it. I remember seeing Inglis again in '48 or '49 and I said, “Dave, do you remember the drive with Pauli from Luzern to Zurich?” and he said, “Will I ever forget it?” Well, that has nothing to do with physics. But as I said, Pauli wasn’t quite a happy man in those days; I don’t think he really got much fun out of this sort of thing. Well, Kramers —

Kuhn:

When did you first meet him?

Casimir:

I mentioned this first lecture that he gave when I was a very young student. And then later when I worked

Kuhn:

You think that he had probably just come to Utrecht then?

Casimir:

Yes, I think so. Then later I suddenly knew that he was at Utrecht when I started to work seriously with Ehrenfest in the autumn of ‘28; he and Ittmann came regularly to Leiden from time to time.

Kuhn:

Then before Kramers I gather that there had not been an awful lot of contact between Utrecht and Leiden?

Casimir:

No.

Kuhn:

But there was fairly regular contact then?

Casimir:

There was then contact, but it was not very regular. Then I saw him off and on, and I saw more of him later in ‘34, of course, when I was his assistant at Leiden, and in ‘33 in that intermediate period after Ehrenfest’s death. Of course, the early days of quantum mechanics, so to say, coincided with his settling in at Utrecht, which may, in a certain way, have slightly hampered his own work. But I have always felt that Kramers was definitely one of the outstanding masters of the art and had an understanding of the theory and a mathematical skill that were quite outstanding. I would say that in virtuosity in solving certain problems, I think, he could compete with Pauli; he was extremely good at that sort of thing. And I have always felt that his contributions in those days were not quite on a par with his understanding of the theory and its principles and his command of mathematical methods. Why is that? He devoted a lot of time, for instance, to this problem of the asymmetric rotator and to developing the theory of Lame functions and so on, which were all very clever but not very useful. It was very difficult work and well done, but you might ask if it was entirely worthwhile. There is some rather complicated work on paramagnetism of oxygen where some group theoretical methods are being used and where you have the feeling, “Is it worth all the trouble he is giving to it at this moment?” One thing that is evident is that he wasn’t out for cheap success, or, in other words, to solve problems that he felt were easy to solve was a thing that didn’t amuse him. He picked things that were difficult, and I would say that he sometimes picked them almost for their mathematical difficulty rather than for their physical content or importance. He was almost a bit quixotic there, I would say.

Rosenfeld:

Yes, I quite agree with that.

Casimir:

In Leiden they had some beautiful large barrel-organs, the kind of thing they also have in Amsterdam, and that was the one thing Kramers loathed — this sort of easy, simple, and mechanical music. I was always amused by it, finding it so crazy when some of the classics were perverted on such a (“piriment”?) , as they call it in Amsterdam. But Kramers really suffered when he heard such things. I sometimes said to him about his physics too, ‘The trouble with your physics is that you don’t like draaiorgeltje.”

Kuhn:

What would he respond when you talked about his physics in that way?

Casimir:

He would just smile. Of course, I wouldn’t really criticize it. For instance, if you would conare Kramers and Bethe, the output of Bethe in those years was very much higher, but I would, say really that the mathematical skill of Kramers, though be did not have Bethe’s speed and fluency, when he did solve a problem, was really more ingenious, and his physics was definitely more subtle and more profound.

Rosenfeld:

I remember one case in which he seemed to have doubts about what he was doing; that was in the work on order-disorder.

Casimir:

Yes, he also put a terrific effort into the order-disorder problem.

Rosenfeld:

I remember that he once asked me if I knew something about some recondite part of mathematics because he had just reduced the order-disorder problem to one special question involving, I think, number theory. I said nothing, and then he said, “Don’t you think the problem is worth putting this mathematical effort into it?” I had not said anrthing at all, I had the impression that he himself felt doubts about pursuing that and that he was a bit conscious of treating the problem as a kind of challenge to his mathematical subtlety.

Casimir:

He tried to draw me into that work and I didn’t respond much; for one thing, it was too difficult for me. This type of mathematics is not the sort of thing I can do, and somehow, I wasn’t very much fascinated. I said, "well, here we have the system of these equations, we have a good approximation at high temperatures, a good approximation at low temperatures, and somehow, they must be tied together.” But of course, the interesting thing for him was to see whether you could find exactly by what kind of singularity; since these solutions were quite different, it was also fairly evident that there must be some sort of singularity. I was not so much interested in this, unless one could see a very elegant way of doing it, with a very new type of mathematics or something; but to do a lot of struggling with complicated and ever-more complicated formulae, to try to get the thing was something I was not much attracted to, and that was, of course, what he did. Then he put quite some effort into developing his own brand of group theory, so to say, his (psi-eta) methods, which is good formalism but it was really only used by his pupil [Hendrik] Brinkman and perhaps by (Wolfe) and one or two other people. But no one really used this formalism, though he did very clever things with it, for instance, the coupling between two vectors when it is not a Lande coupling, but also a higher order coupling, let’s say, with second or third harmonic of the mutual angle between the things. Heisenberg, in one of his papers, uses one of these higher interactions and writes down a general formula which is all wrong and which he then applies only in one special case where the result is right! That’s characteristic of Heisenberg.

Kuhn:

To what extent do you suppose this characteristic of Kramers’ work at Utrecht and Leiden may relate to a perpetuation of the division of functions that had come when he worked with Bohr? That is, he was, I take it, the person who did the mathematics in his work with Bohr, and I wonder whether he may have to some extent drifted in as being the mathematician.

Rosenfeld:

I think he had a bias toward mathematics from the start. I remember Bohr telling me that when he came here and asked whether he could work with Bohr, Bohr called his brother Harald who was the appriser, the practical man who made the practical decisions. “He told me such learned things that I could hardly understand,” Niels said to Harald, “and what can we do with such a mathematician?” Then of course Harald said, “Well, take him.” But it was characteristic that Kramers came to Bohr with some ideas of his own of applying mathematical techniques to quantum theory.

Casimir:

Of course, later he became interested in a lot of things relating to more practical work, too; for instance, work on magnetism and the work he did with Becquerel on the theory of magneto-rotations, and later also, in advanced kinetic theory, there is his work with Kistemaker which is very beautiful work. But also there, I would say, he was interested in the theory of non-uniform gases, the whole Enskog-Chapman story and so on, from which he derived new effects; his approach was always a little bit mathematical, I would say, and he liked to create mathematical methods of his own, as he did in playing with invariants where he worked out these new formulae and so on. He did not always find followers for his particular way of doing it. Some of the things are very elegant; for instance, there is a beautiful little paper on the theory of bands in one-dimensional periodic structures to prove that if you have an arbitrary periodic stcucture in one dimension, you always get bands of energy levels separated by empty states. This he proved by very general methods and very elegantly. Also in his book, of course, there are a lot of things which you don’t find elsewhere and where he gives very elegant proofs, very beautiful ways of calculating normalization integrals and so on; he put a lot of time into writing that book in later years and in thinking over once more the basic principles of theory. Thinking about that and writing the book brought him really to the first steps in renormalization theory. But his own way of tackling it, by trying to separate the internal and external fields, did not turn out to be very fruitful. I think you might say that in some ways he was more a mathematician than a physicist, but then really that is saying too much; but let’s say that he put a rather great emphasis on mathematical theory. He was afraid of over-simplification, but that sometimes led to his making things a little bit complicated. He was not tempted by easy or cheap success but always wanted to do difficult and beautiful problems. And sometimes it may have been a bit of a question of good luck or bad luck in certain of the things he tackled.

Kuhn:

Did he talk at all about his time in Copenhagen?

Casimir:

I had not talked with him much about that, though we spoke sometimes about personal circumstances, recoilections and so on. Of course, his wife always felt that he had been insufficiently recognized, but then she was that sort of woman.

Rosenfeld:

He did not give the impression of holding a grudge on that score?

Casimir:

Oh, no. Not at all, not in any way. But he wasn’t that sort of man, anyway.

Kuhn:

Something you said yesterday made me think that perhaps there were other things that you had to say about the earlier period, which is why I —

Casimir:

I don’t think so. Who was it who told that the difference between Kramers and Heisenberg, when they were both here in the days of the dispersion formula,was that Kramers always had the door of his office open and Heisenberg always closed?

Rosenfeld:

I hadn’t heard that.

Kuhn:

I hadn’t either.

Casimir:

I heard that somewhere. So Kramers really spent a lot of his time helping other people, and he was extremely well liked by the whole Scientific community, who thought he was a great help in many ways, whereas Heisenberg worked more for himself. I heard that remark somewhere and there may be some truth in it. He certainly must have given a lot of his time to the problems of experimental physicists and mechanics and so on.

Rosenfeld:

And Christiansen. He was a great friend of Christiansen. He was also traveling around through the country giving popular lectures on physics. There are some gibes about it in Pauli's letters.

Casimir:

Also in those days of silent movies he worked on explaining a film that had been made on Einstein’s theory of relativity; in those days a film was accompanied by an “explicateur” and Kramers did that to earn some extra money. In those days you also had to have an orchestra, so first you had to have a musical introduction, and Kramers said, “We selected the ‘Ouverture Egmont’ to introduce that film.” I asked him why Egmont and he said, “It’s a good introduction to relativity.” I have not quite been able to understand why, but probably that is due to my lack of musical feeling; anyway, Egmont was chosen, perhaps because the orchestra probably had a rather limited repertoire. Kramers later said he could never hear Egmont being played without thinking of aspects of relativity, and when the last chord crashed, he would have to step forward and say “Mine Darner og Herrer" and begin to speak about relativity. He also told me that that was the occasion upon which he acquired some of the professional’s attitude when you know you can give the same performance any number of times, which we usually don’t reach as physicists.

Rosenfeld:

There used to be a huge piece of quartz crystal on the mantelpiece in Bohr’s office, and he explained that this actually had been stolen from the mineralogical museum; when Krarners went around lecturing about physics, he carried it in his coat pocket to show what nature could do with atoms.

Casimir:

There is still the whole question of the relations between Ehrenfest and Pauli, which are well known and have often been told, of course. That was very curious. Ehrenfest had great admiration as well as great liking for Pauli, but at the same time he felt slightly ill at ease with him. The result was that they were always making nasty remarks to one another, some of which have become quite famous. One of the more funny things is this, and I think the piece of writing is worth reading. When Ehrenfest presented the Lorentz medal to Pauli at the Academy of Science in Amsterdam he made the presentation and made a beautiful address to Pauli about Pauli’s work and so on, This probably is in Ehrenfest’s collected papers, but there is one little thing which is not there, This being a rather formal occasion, Ehrenfest had said it was rather inportant that Pauli come in a nice black suit or something suitable for the occasion and this Pauli at first refused to do, but then said, “Well, I’m wining to do that but you will have to mention this in your presentation address.” And Ehrenfest did, but it is not in the printed version, but it is almost. [leafing through Ehrenfest’ s Collected Scientific Papers to p.619. After explaining that it is because of the exclusion principle that a piece of metal is so voluminous, Ehrenfest says “Sie muessen zugeben, Herr Pauli: Durch eine partielle Aufhebung Ihres Verbotes koennten Sie uns von gar vielen Sorgen des Alltags befreien; zum Beispiel vom Verkehrsproblem unserer Strassen.” And then he said in his presentation, “Und auch von dem Problem, wie man die Kosten fuer schoene neue schwarze Festkleidung etwas herabsetzen koennte.” Pauli, of course, had been waiting for it. That was quite interesting, and interesting also is this paper he wrote quite later on, “Einige die Quantenmechanik betreffenden Erkundigungsfragen” to which Pauli also wrote an answer; I think that answer is also in a way quite significant, because you would see some of Pauli’s attitudes there. Well, that’s about Kramers. Ehrenfest, of course, was in a way a magnificent lecturer. I wouldn’t call it a shortcoming, but I have mentioned that he stressed the logical content and the structure of the theory rather than its quantitative applications; still, he always looked for very striking, graphical ways of expressing himself, illustrating things by means of simple models, etc.

Kuhn:

To what extent do you suppose the problems of converting to quantum mechanics contributed to the discontents that ultimately led to his suicide? I gather the conversion would have been difficult for anybody but perhaps more difficult for Ehrenfest’s sort of mind than it would have been for some others because of the mathematics it involved.

Casimir:

I don’t think that the questions of the interpretation of quantum mechanics and so on contributed to it; what certainly did contribute to it is the feeling that he could no longer master the most advanced stages of the theory. He felt that some of the mathematics, some of the more difficult developments of group theory, some of the field quantizations and all that sort of thing, be didn’t really understand. Either that he didn’t understand that in the way that he wanted to understand things, or — [Break to answer phone] About Ehrenfest’s last years. There was suddenly this feeling which I had, so to say, seen coming, of no longer being able to grasp modern developments, no longer being entirely on top of his subject, getting a little bit out of the development; that was a thing he found difficult to accept. After all, he still knew a lot of things, he could still lecture on a lot of things, he could still criticize a number of things, he could even have done a lot of things himself, but he always had had this wish to be able to understand the latest developments of physics, to try to bring them into a very clear-cut form. He was willing to leave some of the mathematical techniques and all that to others, but he felt that things were escaping him. Einstein once wrote somewhere a necrology where he put this as the essential point. He said this was a man who felt he had to be a great teacher and when he felt that his powers as a teacher of the most modern things were declining, be went out of life. I don’t think it was a simple as all that, but this feeling, which I suppose every one has occasionally and which everyone has around 50 or 55, suddenly played a role; that was one thing. I think the whole question of National Socialism in Germany certainly certainly must have played a role, too. He was very much attached to Goettingen, always went there and took his students there, and to see that all that type of German cultural life was being destroyed was certainly something he took very much to heart. His whole attitude to the Jewish question was a remarkable one; he was in a way very sensitive there. Whether he believed it or not, he always pretended that he really thought that being of Jewish descent was almost indispensible to doing anything worthwhile, in science at least. That was a very remarkable trait of his; I hadn’t seen it so pronounced in anyone else. I remember my friend Rutgers telling me that he found Ehrenfest looking and looking again at a photograph of Lorentz and discovering, to his satisfaction, something in his face which he thought more Jewish characteristics. He said to Rutgers, “Ja, ja. Das war doch einer von uns. Das muss einer von uns gewesen sein.” To me, he used to say, "Wenn du nur ein Viertel juedisches Blut haettest. Und wenn es nur ein Achtel waere, da koennte noch etwas aus dir werden. Du bist aber viel zu Arisch.” In my family it’s really a rather difficult thing, even with a lot of good will, to discover any Jewish blood. In this Frisian peasant stock of mine it was a little bit hard even for Ehrenfest. Very remarkable — not quite funny perhaps, not really. But you can understand under those circumstances what the whole of Hitler and National Socialism must have meant to him, so that was the second question. Then there was the whole question of his marital situation and his not being able to make a decision as to whether he would go away from his wife, to whom he was very much attached, or not. That situation may well have given him the final blow; he was somewhat entangled at that time with another lady who in a way was quite a nice woman. I once had a heart-to-heart talk with her for a whole evening. She had a lot of Ehrenfest’s papers which he left with her and they are now with Martin Klein. She was quite a nice and well-bred person in a way, but —

Kuhn:

Did he talk to you at all about his feelings about Einstein and particularly about Einstein’s failure to follow the new interpretations?

Casimir:

No, not in particular. He was, of course, at the Solvay conference when these things were going on, and there, I would think, his great admiration and friendship for Einstein notwithstanding, he sided with Bohr and he thought Bohr came out on top. On the other hand, he took Einstein’s objections more seriously than many other people did. I told you about this colloquium where Einstein spoke about these things and said, “Die Sache ist schon widerspruchsfrei, aber hat innnerhin ein gewisse Haerte.” Well, the younger generation hardly felt ‘diese Haerte’ and I think Ehrenfest recognized that there was a bit of ‘Haerte’ there, that it wasn’t quite so nice to have to accept these things. I don’t think that that was a thing which unduly upset him in any way, but he did feel, of course as you see from the difragen, a little bit unhappy about some of the aspects of quantum theory. You could also see in Ehrenfest’s later years that he tried to maintain his own style and that it became an effort, that it was no longer quite natural. You must have noticed that in reading. He was forcing himself to his old vivacity and vivid ways of expressing himself while he was really rather miserable in many ways. A curious thing just before his suicide was that the man who realized most clearly that something was very wrong with him was, of all people, Dirac. I think Mrs. Bohr told me that. Ehrenfest was at Copenhagen. very shortly before his return to Holland and the disaster, and I think, Dirac said, “This man is mentally very ill and one shouldn’t let him go alone.” I think Mrs. Bohr told me that. That it was Dirac is very curious, though, of course, he had known him and had been at Leiden with him, but you would perhaps at first sight not expect him to have such insight into human nature. I think what finally triggered the thing, as this mistress of his told me, was that he couldn’t make up his mind whether he would do the one or the other. She said that finally he was relying on her more and more and yet was in a way so independent mentally that she couldn’t stand it any longer and said, “Decide one way or another, but stop this sort of “zwaaien zo” as we say in Dutch. Then he didn’t know what to do; but this I think was only secondary, although it triggered the thing. Well, that’s that; I don’t think I know much more that would be relevant. [Another break] In Gottingen in those days people were amusing themselves with the following game: all normal mathematical symbols, plus and minus, a sigma, a product sigma perhaps, and a factorial are allowed; try to write all numbers using the figure '2' four times. For instance, write 1 as equal to 2 plus 2 over 2 plus 2; 2 is equal to two over two plus two over two, and so on. Some people had gotten up to 20, some to 24, and some to 30 and Dirac had been thinking about that problem quite a bit. They asked him whether he had done anything with that and he said, “Yes,” so they asked him how far he had gotten. “I can write the number n with [four] figure ‘2’s’,” Dirac said, and produced the following formula: which was much admired but which took all the fun out of the game. No one played it after that

Kuhn:

For the recorder, it’s minus log to the base two of log to the base two of the nth square root of two. When was this that this game was played?

Casimir:

I have been in Goettingen twice and this must have been the second time I was there, so that was summer ‘29. But it was strictly within the rules of the game.

Kuhn:

Is it? The use of the n on the right hand side —

Casimir:

But you don’t write the n really, so if you ask “Would you please write 935,” you would just take 935 roots. Definitely, square root was allowed, and it came in very handy sometimes. I do remember the letter quite well; there was something about the “Ritterschaft” and that sort of thing with which Ehrenfest was thoroughly amused. I had been here for some time and I had just completed the thesis, which I defended in October, so it was just the question that Ehrenfest didn’t have many collaborators at that time and he would have liked me to have been back at Leiden, perhaps slightly in connection with this fact that be was beginning to feel just a little bit uncertain about himself, because the thing repeated itself later in ‘32. I came back to Leiden then, was there that winter, and I had thought that I would have liked to stay on a little bit in Copenhagen, but Ehrenfest said, “No, I need you here at Leiden.” So I spent that winter at Leiden. I then got this offer to come to Pauli and Ehrenfest, I think, wasn’t quite happy about the work I had been doing at Leiden that winter, and rightly so probably. So I went to Zurich and again that question arose; Pauli wanted to keep me at Zurich one more year whereas Ehrenfest again wrote a rather emphatic letter saying that he wanted me back at Leiden. There, I think, he was already thinking about taking his own life and he also knew that it would take some time before a successor was appointed, so he wanted to have someone there in the theoretical physics department. That was another thing when he insisted on my coming back, so this happened twice, once when I was at Copenhagen and once when I was at Zuerich. But in a way, it’s quite understandable since he felt that most of his people had gone away and that he would like to have some assistance and,as I say, he was already feeling a little bit uncertain of himself anyway.

Rosenfeld:

Did that letter from Klein help?

Casimir:

No, he laughed a lot and thought it was very amusing, but I went back to Leiden all right; I think it straightened things out. There was a little bit of ill feeling here and there and then some arrangent was made. But that was that question.