Oskar Klein – Session VI

Kuhn:

We had your background up to this point yesterday. If it’s agreeable to you, I’d like to go now to perhaps ‘26-’27, and work somewhat more slowly than yesterday on certain of the points that we talked through from that point on. Tell me first: just how much of the time in ‘26 and ‘27 were you in Copenhagen?

Klein:

I came there in the beginning of March ‘26, and then I stayed on until about the same time in ‘31.

Kuhn:

So you were really here quite steadily throughout that time. The point I would particularly like to have you speak to is, I think the earlier parts of what happened in ‘26—particularly the reception of the Schrodinger equation which we’ve already talked about somewhat. The question is about the reaction here in Copenhagen and your own reaction—always both of these, you and others you talked with—to the Dirac paper on transformation theory and the very-similar-in-conclusions earlier paper by Jordan on the neue Begrundrung der Quantenmechanik which began with probablistic axioms and then built to a system very like Dirac’s.

Klein:

Now, Dirac was closer to us than Jordan, because he came to Copenhagen in the fall of ‘27 and spent his time there; and then rather soon he began to—.

Kuhn:

Wasn’t it the fall of ‘26 he came here?

Klein:

No, excuse me, you are right. I confused the years. It was in the fall of ‘26. I think I came to know, just a little before that, this Fermi-Dirac statistics; and he also told us a little about that. I didn’t know Fermi’s work, I think until about the same time. Then he began to tell us a little bit and I think we began to study his papers on q numbers. Perhaps I had begun it a little earlier. But then gradually he came into this work on the transformation theory; and it took us some time to understand the things because he gave some lectures and he wrote all the figures on the blackboard very nicely and he said a few words to them, but they were very, very hard to get so I cannot remember quite when I began to understand it. By and by I studied his papers very carefully—I think it must have been later. So I think at that time I got more the general ideas of it. But I know that Heisenberg followed his work closely and he had followed Jordan’s work closely so that I think he got it earlier. I remember be used Jordan’s methods in his paper on the uncertainty relations and then I remember also that he told me on some occasion during that winter that Bohr joked a little bit about some British physicists who had some competition over who was the greatest—I think that was Darwin and Fowler. Then Heisenberg said that be thought Dirac was really it. Then he told me a little about Dirac’s and Jordan’s work on the transformation theory, with great admiration for both. I and I think, Bohr, knew very little of it at that time. It must have been in the winter of ‘27 or rather in the fall of ‘26, because in the winter Dirac left for Gottingen.

Kuhn:

I think both of those papers actually appeared before the first of 1927; I think they appear late in ‘26, both the Jordan and the Dirac.

Klein:

Then, of course, when one came to know them, they were regarded then as really the end of that fight between matrix and wave mechanics, because they covered the whole thing and showed that they were just different points of view.

Kuhn:

Had that fight been at all active in Copenhagen—the fight between matrix and wave mechanics?

Klein:

Oh yes. That was very interesting in early ‘26 when Schrodinger came to Copenhagen. Schrodinger believed that he could make a wave mechanics analogous to wave optics and in the same relation to classical mechanics as wave optics was to geometrical optics. There were very intense discussions where Bohr and Heisenberg took a very strong position against Schrodinger. You know, Schrodinger wanted to interpret that which Bohr interpreted as probability density as really the density of the electrons. Then they showed that if that was so and if by means of his currents and densities he coupled the thing to the electromagnetic field, then the probability for spontaneous emission would, be proportional to the number of atoms in the upper state multiplied by the number in the lower state. It would be quite against anything one knew.

Kuhn:

But if one leaves Schrodinger’s own interpretation of the wave equation out of the picture, still, there were people who felt more at home with, or liked the Schrodinger approach even though they didn’t agree with Schrodinger—the sort of thing that was represented let, us say, later, when Born does the Elementare Quantenmechanik in the matrix form and Pauli reviews it and says this is terrible to make everything so difficult, why not use the Schrodinger approach? Now, was that sort of quarrel between people, none of whom would have agreed with Schrodinger, also an issue in Copenhagen?

Klein:

That was also present, but did I tell you about these things with Schrodinger earlier? I may have told you something about this fight with Schrodinger because it was rather interesting. Did I tell you about it already?

Heilbron:

The argument about the transition probability you mentioned before.

Klein:

Yes, I had mentioned it before; so I won’t repeat it.

Heilbron:

Who was responsible for that argument about the transition formula, do you remember? Was that Bohr or Heisenberg?

Klein:

I don’t remember who started it. After that, I took it up a little bit in a special way, but then the general argument had come out in the discussion—whether Bohr or Heisenberg I don’t remember who particularly. I took it up about that time just to learn about it and showed that from that same argument one could prove the Rayleigh radiation formula instead of the black body formula.

Heilbron:

Did Schrodinger find any of these arguments particularly convincing?

Klein:

I think that Schrodinger was gradually convinced by Bohr and Heisenberg, but as you know, he came back to that thing in later years again. In answer to your question, very soon, I think, we in Copenhagen learned that these were complementary views as they cane out then in Dirac’s and Jordan’s theory.

Kuhn:

Do you mean complementary—?

Klein:

I mean that matrices had their natural place and the waves their natural place. It didn’t take a very long time for that to come out. I remember that Darwin had some nice statements where he said that he felt it hard to understand the matrix because he was brought up in the waves. And then he said, what Dirac calls an example I call a general theorem.

Kuhn:

I ask this particularly because of one very interesting but undeveloped point that Hund made when I talked with him. He came up also in ‘26 and he had just begun to get deeply involved with the molecular spectra problems. It was while he was up here that he either wrote or finished the first of that famous series of five papers on molecular spectra. He says he showed the manuscript to Bohr; and at the beginning of the manuscript he had—he doesn’t remember quite how it stood—presumably the usual Gottingen statement about using the Schrodinger approach but only as a formalism. You know, this was one of the early Gottingen papers that really used the Schrodinger equation; and it was getting to be customary to say “because it’s easier, or better fits this problem, I shall use wave functions, but of course, only for the convenience of the mathematics.” That passage no longer exists. But he says that Bohr said something to him; he doesn’t remember just what Bohr said, but its general bearing was: that really isn’t right. Of course, Schrodinger is wrong about his interpretation of the wave function, but they have some physical meaning, they are physical in some sense that this sort of statement does not admit. And Hund said this was the first time that somebody had really begun to persuade him that the Schrodinger equation was about something that might be different or more than matrix mechanics with a bow. And he changed the paper accordingly. I’m still quite certain, you see, that Heisenberg, who was also there, would not have said this.

Klein:

I may be going into something a little bit personal because it is hard for me to get it out except in that way. When Schrodinger’s paper came, I took up a train of thought which I had been doing before Schrodinger but hadn’t been able really to solve any case of, and wrote a paper of that which was also connected with the five dimensional theory. Then in the summer I was invited to give some lectures in Leiden and worked on them. I think I gave a seminar, and Dirac was also, there, and I showed some of this to Bohr, which was to carry out this Schrodinger point of view also in the relativistic case but with the scalar equation. Bohr then got very interested in that because be didn’t know much about the Schrodinger equation yet, and he used that [discussion with Klein] a little bit to come into it and at the same time he helped me.

Then what was called then the correspondence treatment of the Schrodinger equation came out. At that time Bohr got very deeply into the connection of this with what he knew very thoroughly earlier from Rayleigh and so about the waves and limitation of waves. That was very fruitful then in his work about the uncertainty principle, where he took it up from that point of view, while Heisenberg’s was very abstract at that point. Heisenberg took, so to say, the derivation of the thing from the matrix point of view by means of Jordan’s transformation theory. There was a little bit of competition at that time so that Bohr wanted to have such things about wave groups worked out in detail, and, I think, I started a little bit; but then Darwin came and Bohr talked with him about it and he began to work with that. At the same time, Heisenberg started about the same thing but had Kennard, who came there from Canada, I think, working at it from the matrix point of view.

There was a little bit of competition there so that the one thought that the other was the right way in some way, but that was only for a short time. It must have been the spring of ‘27, about the same time that Bohr began to work out complementarity. I think that what you mentioned from Hund had to do with this: that as soon as you had problems where space and time were important, then you could, according to the transformation theory, do it with something which was a generalization of matrices. But then wave functions were, in some way, a very natural way. On the other hand, one can say that Hund’s point of view that they were kind of auxiliary could also be defended and you could say the same about the matrices. Then I think it became common; for me, it became very soon a natural thing then to use matrices sometimes and to use waves sometimes as convenient. That came especially after Dirac’s theory of the electron where matrices were very helpful but also in other problems like the spinning top and things of that sort. The matrix method became very natural. We learned very much from Dirac in that respect, of course.

Kuhn:

Let me be clearer on what it was the Darwin was working on for Bohr. Was it wave packets?

Klein:

Wave packets, both free wave packets and especially also bound wave packets. He took up one thing also which, I think, I had suggested first during those discussions between Bohr and Heisenberg and Schrodinger where Heisenberg had in opposition to Schrodinger said that, if he made a wave packet corresponding to a given direction of the magnetic moment, that would just move the one way; and if had the opposite, then it moved the other way. That would not correspond to the Stern-Gerlach experiment where it is split up. But at that time I made the remark which Bohr came back to in later years—it must have impressed him although it was a very simple remark—namely that having the magnetic field would probably be similar to having an optical medium which was doubly refracting and that would just give that splitting. But I didn’t work it out, mainly because at that time I didn’t like the Schrodinger equation in the first place, and I couldn’t see how it could be done in ordinary space. But then I think that Bohr asked Darwin.

Darwin wanted to have something because he was rather new in these things and wanted to help him. Then Bohr asked me if I had anything against it and I hadn’t, of course. Darwin worked it out rather nicely, not so easily that he had no problem—it was work. Then he also worked out the motion of a wave group in the hydrogen atom ease, which was, of course, a very nice thing also to see how it could in the higher states move in the elliptic orbits, and then gradually spread out. In principle, these things were made very early, I think, by Lorentz, the earliest. I heard about it because I was in Leiden in June of ‘26. He had remarked that one could do so with such wave groups. I think he was thinking of the harmonic oscillator and he took it and he said the general case would be that they would spread so that he would take it against the literal use of wave mechanics. But then Schrodinger wrote a paper not much after that where he took the harmonic oscillator as an example of the opposite, you know, because that keeps together there, you know, and that was very effective. You must tell me if I stray too far away from the subject.

Kuhn:

No, this is very much of interest to me. Now, if Darwin was working particularly on wave packets and following them, and you say Heisenberg was working on similar problems—.

Klein:

He had asked Kennard, who also came and wanted to have a problem. Heisenberg put him to work on about the same problems, but from this Jordan’s transformation theory point of view. That was meant to be then an application of the matrix mechanics. Of course, there was really no very great difference, but there was a difference.

Kuhn:

How did one do the wave packet problem from a matrix point of view?

Klein:

Oh, I think you can say that you solve the matrix equation of motion first, and then also you solve the similar thing for the square of the coordinate. I think matrix first and then you can solve the same thing for the probability of the position being that and that. Then you get to the wave function in reality so—. It was so much in the beginning, although, Heisenberg must have known in principle, if one had asked him, that there couldn’t be much difference. Still, I think, that he was not so familiar with the wave point of view at that time and he was a bit partial to the matrix point of view which he had started, and he thought perhaps that it was deeper physically. Now, of course, one very strong reason was this misinterpretation of wave mechanics, where one was right to be against it. Very soon, I think, that one felt that that was really one theory.

Kuhn:

There was, I take it, not simply this competition, but as time went on, there were actually real disagreements about issues that it is hard now to actually recreate about these two approaches?

Klein:

Yes. Do you mean not only between such a point of view as Schrodinger’s and the others—?

Kuhn:

I simply mean leaving Schrodinger’s a point of view out for the moment. I assume that in Copenhagen was deeply sympathetic with that. Still, you have told us and others have told us, that at least by the spring of 1927, and also very possibly already in the fall of ‘26, there was very real tension between Bohr and Heisenberg over the work that leads into the uncertainty principle on the one hand and complementarity. This must relate to some extent already to what is going on and this doing the same problem by wave packets on the one hand and the matrix probabilistic technique on the other.

Klein:

Oh, I see. Yes, there were; I think you are right: There was some connection. The main difference there between Bohr and Heisenberg was, in fact, not so deep, because it had a little to do, of course, with more personal things—all are human beings—but Bohr had been thinking very deeply about these problems and had discussed them with Heisenberg and hoped to do something with them. He was very strained in those years because there came so much that was new and so many discussions that it was hard for him really to follow those things. Then, when he came back from a vacation in Norway, it took some time before he was, so to say, “up with” the situation. When he came back, Heisenberg had written the papers. He showed him his manuscript, and then Bohr was first very enthusiastic about this beautiful formulation. At the same time he may have felt a little bad—perhaps it was a bit of things which he had in his mind and had not quite finished, but that was not important at all at that time, because his main feeling was that now Heisenberg had really solved a very important problem.

Then he began to look at it more in detail, especially those thought experiments. He discovered that they were mostly wrong. I think afterwards one would not rightly use the word “wrong” because apparently (almost all of us do) things which in a strict sense are wrong because they are the first. You see, Heisenberg discussed this maybe so that in order to observe in what he called the gamma ray microscope the position, there would be change of the state, there would be a Compton effect. But when Bohr began to look closely at this, he saw that the Compton effect in itself would, not create an uncertainty. It was the indeterminacy of the Compton effect because one could not know, when one wanted to have the place, the direction of the rays; that must be arranged. That was, of course, a very important complement to the thing. Then Bohr got rather upset about that. He was always very eager when there were things of that kind, so in the first moments there came some tension in between. I think that tension contributed a bit to Heisenberg’s trying just to stop these things from his own side, so to say. Bohr then got very naturally into using such Rayleigh ideas about the limitations. They came out very naturally there. Of course, Bohr got more deeply into the matter than Heisenberg did. It had to do a little bit with this wave mechanic-matrix mechanics competition—but it was more the technical point that Heisenberg had not really discussed these thought-experiments correctly. That is very hard, of course. I don’t think that anyone was able to do these things except Bohr or those who were in very close connections with Bohr. Bohr and Rosenfeld did some very beautiful work, you know, on the electrodynamics. There were very many erroneous papers about these things also. In later years I always found that I would not try to mix into these things. They are so difficult not to get into, you understand. I mean it’s very easy to get an uncertainty, but the question is to get the minimum uncertainty—the best way to measure things.

Kuhn:

I have had a tentative feeling from some of the things I have read and people have said that very likely there is also one element, at least, that is deeper in this argument, discussion, tension between Bohr and Heisenberg. It’s hard to know how to phrase it without saying too much, because one could easily say too much by putting it in lighter terms. It relates to the issue as to whether one sees duality in this situation or not—and I think Bohr very clearly does. I don’t want to say complementarity, but, I think, this must also relate to this earlier statement to Hund—that, after all, Schrodinger maybe is all wrong, but there is more to the wave function that is simply a calculating technique.

Klein:

Yes, certainly; and that had to do with the space-time because the space-time did not come out naturally from the matrix side. You could get it then by and by just through the transformation theory, it came out by using also singular functions; but then you could ask for the probability, that a coordinate, taken as a q number, had an expectation value of that and that. That would be the same as using waves in the quantum theory in a sense. By and by, the waves made there were of very great importance, and it would probably have been hard to discover the rest without the waves. Both Jordan and Dirac, of course, knew the work of Schrodinger and, I think, were strongly influenced by that when they put it out. But, logically, of course, one could have gotten it without Schrodinger.

Kuhn:

Do you remember things that were said in this discussion that pointed toward duality, the wave-like behavior of matter, that sort of thing—that really entered into the arguments themselves?

Klein:

Of course, in the beginning he said maybe more symbolic things; but gradually things became clearer when he tried to discuss real measurements. Then he, Bohr, used the waves in the sense of the transformation theory. He did that so that he used waves but always with the knowledge that these were not waves in the literal sense, and he pointed out very strongly also that in the literal sense one might say that electrons are particles and that the electromagnetic waves are waves. By this, he meant that [it] could not be the electrons had a macroscopic wave, but that had to do, of course, with the electric charge and so it was perhaps not so deep as it sounded in the beginning. Now we know there is also the Pauli principle which makes it different—perhaps the main difference. But you could think of other waves; for instance, pi mesons you might regard a little more as waves. But then in the electromagnetic case he stressed then that with one equation you could never know the phase. If you could know the energy there, you could not know the phase. If you had macroscopic waves consisting of very many quanta, then you could know the phase, but then you never knew exactly how many quanta there were. This kind of complementarity began to play a very great role. He didn’t call it complementarity yet. I’ve forgotten exactly when he began that. I think he began to use the word about ‘27, then he abandoned it for a little time and called it reciprocity, and then he came back to it.

Kuhn:

Do you remember when he really began to talk about this problem of the electromagnetic field—that if you know the energy, you don’t know the phase; and you can know the phase, if you don’t know the number of quanta, that sort of argument?

Klein:

I think that started in ‘27. I don’t remember exactly when, but there was a whole period when he worked incessantly on this which began in the spring of ‘27 and then went on through the summer and then continued through the autumn afterwards and finally led to these papers be sent to Naturwissenschaften. So that it was among points there. One of the early things I remember from that spring was that he discussed also problems of measuring velocities and charges of particles in the electromagnetic fields and how the uncertainties came in there and what one could measure. I remember he started that in the spring.

Kuhn:

Would that be after he returned and had—?

Klein:

That was all after he returned and after he read Heisenberg’s paper, and so that was all the reaction to Heisenberg’s paper, of course, and he quite admitted that. Heisenberg’s paper was a very important progress and a very strong source of inspiration for him. The tension that came had a little to do with things happening in time and it takes a little time to have a rational reaction—all things happened so fast. As I mentioned last time, he had almost unbounded admiration for Heisenberg, and that was an extra shock when Heisenberg did not see through these things. I mentioned yesterday his first saying about Heisenberg—”He understands everything.” So, that made him very much upset there. But, of course, if you look at it afterwards, when you know the history of science a bit then it’s very natural that the pioneer doesn’t do the things so correctly, although, so to say, he has come into the main trend of ideas which will lead to the things. I believe Bohr would have said so in later years.

Kuhn:

You said that Bohr initially spoke in more symbolic terms. Do you remember the sorts of things he then said?

Klein:

Yes, some of the things were from a much earlier date. Did I mention here that even before quantum theory, he told me, he had begun already in his early student years to think about such philosophical problems as the paradox of [free] will, and then these mathematical analogies? Then he went into functions and connected that also with the way of adiabatic transformations of an atomic system from one stationary state to another. But then in this later period he often used such words as “quantum means a knot in the existence,” so to say, “and this knot cannot be removed; its place may be shifted.” I remember especially he often said this. There may be some words also which I do not remember. But then he tried also to point out that waves and particles were, so to say, different aspects, and he had something with a match box. I don’t remember how, but he did it that way and did it that way. He later had such a box made. I think probably Rosenfeld remembers how it was made. It was so that when you put it on one side, one thing came up, and on the other side, something else came up.

Kuhn:

Was it one of those wooden match boxes that you could slide out either in one direction or the other?

Klein:

Yes, a way of expression when you looked at things from one side or the other. But he was very much against an idea which Eddington wrote a little later when he wanted to call them wavicles or something like that, a combination of waves and particles, because he stressed very strongly that neither the word “wave” nor the word “particle” can be taken very literally but are just short expressions for the experience of having this feature of indivisibility, of individuality, he often called it, and at the same time, you had the other feature, which is symbolized by the waves—that you have the many possibilities and you can tell the probability of this; but in the end every observation contains a feature of individuality. I remember the old philosopher Hoffding, who was already a close friend of Bohr’s father, took this word “individuality of the electron” with a little other meaning—that every electron has its individuality, while Bohr used it in the literal sense of indivisible. He had talked with Bohr about it, and Bohr told me that he did this.

Heilbron:

Perhaps one can ask this sort of question from the other way around. You have told us that when you came back to Copenhagen, before Schrodinger, Bohr had told you about de Broglie’s work?

Klein:

Yes, he had already told me about that in the summer of ‘25, and I had seen a little of it earlier also, but not his thesis.

Heilbron:

What did he say about that?

Klein:

I don’t think he was very interested at that time. I was interested because I had such things in my head then and tried to tell him about it, and then he said that do Broglie had also similar ideas. I had seen some of it, but not his thesis. He was not interested in it then. I tried to interest him in it, but he was not.

Heilbron:

He did get quite interested in your five-dimensional wave—was that before Schrodinger wrote?

Klein:

He was interested in the wave side, but that was after Schrodinger that he got interested. I tried to talk a little to him before, and then I had it also mixed with the five-dimensional thing which he was not very interested in then. In earlier years he had himself—and that played a role for me myself—said that he thought that since you cannot get a connected picture of quantum phenomena and four dimensions that maybe you could in a higher number of dimensions. When I came to this five dimensions from another point of view, then I thought that that might perhaps lead to this; and it took me about a year from when I had learned Schrodinger’s work to develop this idea. I had never believed that one could connect it in space and time like Schrodinger, but I tried to believe that one could do it with a high dimension. Then in working with the many-body problem, just as a continuation of these discussions with Schrodinger, I saw that one couldn’t get anything reasonable. And then I read Dirac’s first paper on radiation, very carefully then. So then it came out very nicely, and that led to this work with Jordan, who had come to the same thing independently. I had got stuck on one point, namely, this which had to do with the self-action of the electron on itself. I had proved all things except for this, which I didn’t know how to do because I wouldn’t (???). So I left the paper unpublished; it was from the spring. I showed it to Heisenberg; he was quite interested in it. Then Jordan came to Copenhagen and talked to Bohr, and Bohr mentioned that I had tried such things. Then Bohr got us together, and Jordan told me what he had done. He had never finished the calculation; but he had started the same ideas; and he had also already the ideas about the antisymmetric quantization at that time, which I didn’t have at all.

Kuhn:

Now I want to go back for a minute and do this one quite carefully. There is one question which we’ve just been talking about that I’d still like to try. Even before Heisenberg’s uncertainty principle as a relatively formal uncertainty principle and without complementarity, both in the Dirac transformation theory paper and in this Jordan neue Begrundung paper, one comes quite firmly to the conclusion that if you specified precisely the coordinates, then all values of the momenta are equally probable; and could, in the Jordan paper, lead also to the converse of this. Now there is still no notion of letting them both be uncertain, but that work comes out appreciably before the Heisenberg uncertainty principle is known, and must itself have led to discussion. I wonder—?

Klein:

Now, apart from Heisenberg I think, none of us followed these formal developments closely so early. Bohr read Dirac’s papers, I think, when he gave them to him for publication, but I don’t think he ever really got into the formal things there. Nor did I. So this was only very slightly known to us when Heisenberg’s paper came, but Heisenberg was really studied.

Kuhn:

So you think a remark of Bohr’s like the wave-particle being a knot in existence also comes after the Heisenberg paper and not before?

Klein:

It came after, yes.

Kuhn:

Certain parts of this could clearly be based on this problem in the Dirac and Jordan papers without yet having the fuller and more general formulation which Heisenberg did?

Klein:

Yes. Heisenberg got it so from Jordan. He got it directly from Jordan, and he knew Dirac’s also, I think. As far as I remember, I mentioned that he spoke to me with great admiration both of Dirac and of Jordan, but I think that he at that time was probably the only one who had really studied that formal side of it.

Kuhn:

But there is, in addition to the formal side, the transformation side of it; there is this clear, let’s say qualitative, or physical, or in any case, not formal conclusion—.

Klein:

But I think that was really at that time a very important thing which really Heisenberg found—that from that point of view, which was very abstract, one could have this much less abstract thing… That was really Heisenberg’s work. I don’t think that either Jordan or Dirac thought of that.

Kuhn:

Jordan has at least this important piece of it: he said that…for a specified value of q you get equally probable values for p and so on. That is a qualitative, a physical, a measuring conclusion that really comes months before the Heisenberg uncertainty principle—generalization of it?

Klein:

Yes, that was Heisenberg’s background.

Kuhn:

Yes, but was that point also discussed prior to Heisenberg’s paper in Copenhagen?

Klein:

No, I don’t think so, because, you see, I came rather slowly into these things, because I was working with, the kind of ideas that I had before and I tried to learn as much as I needed, so I read Schrodinger’s first papers carefully. But I hadn’t read anything of this this carefully, and I don’t think that Bohr had either, because he was beginning also to learn more then from the wave side. Probably Heisenberg had told us, and especially Bohr, about these things; but one never understands quite things which one isn’t working with. Bohr would have understood it if he had gotten it in the way you have just expressed it now, but these expressions were not so clear. I don’t remember how it is in Jordan’s paper. And in Dirac’s paper, as I read it very carefully then, they were very abstractly written. Jordan’s paper I never read so carefully.

So I think it was really the achievement of Heisenberg to get such more concrete points definitely out of that. Some of it may not have been so new to them—to Jordan and Dirac themselves. I would be interested to ask them about how they reacted to Heisenberg’s work—if they thought that they knew it almost or not. To me, what Heisenberg said was very new; and to Bohr, I think, it was also new, apart from the fact that he had general ideas, as you mentioned last time, about energy and time and those about momentum and space would also have been rather near to him. I don’t think that he had taken that up. At least I don’t remember that he mentioned it. It would be nice to see if anything was in his correspondence about this. I don’t remember anything. What I heard of such discussions of the paradox—the quantum paradox—was also in the more general and symbolic way I tried to mention. But he had very many discussions with Heisenberg when I wasn’t present. They used to ride in the mornings, and then, I think, they discussed such things. I don’t believe that either Bohr or Heisenberg had any clear idea about this until Heisenberg began, and that was, as far as I know, when Bohr was absent to spend three weeks or so in Norway. I think that it was a rather sudden idea with Heisenberg that he could treat these problems so. I think they probably had discussed the problems of how to introduce space and time and all those things into these very abstract formulas. I suppose that Heisenberg got that. But then Heisenberg took up a thing which I would not be astonished if they had discussed some before, but I never heard anything of it. Heisenberg didn’t tell you anything about the origin of his work on the indeterminacy principle? He must remember what he and Bohr were talking of.

Kuhn:

He remembers, in fact, very little of this. We have gotten scraps of it. He has quite a good memory, though not nearly so good a memory for detail as yours. But it’s also clear that the tension had helped to suppress certain details of memory for this period with him.

Klein:

Yes, but I think one must say that Heisenberg made a very important contribution to this which then was a very great inspiration to Bohr, and that by a little ‘phase differences’ there came some tension, which could contain something of this you mentioned. That there was a tension between their points of view could have done much to bring out the complementarity point of view.

Kuhn:

Could we talk a bit about electron spin?

Klein:

I was absent when the main history of the spin happened so that I know it only by what—.

Kuhn:

We’ve talked, I think, about the development up to the point where it is at least fairly generally accepted. There were a few questions in its later history that I think probably you could help us with.

Klein:

There maybe I could help because I came to Copenhagen about the time when Thomas had found the factor 1/2.

Kuhn:

You mentioned earlier a bet between Dirac and Heisenberg. Were you here when it was made?

Klein:

I think Heisenberg told me about it and I think it happened while Dirac was in Copenhagen. I don’t think I was present when it was made.

Kuhn:

What was the subject of the bet?

Klein:

If one talks of it from the point of view of wave equations as I think one could just as well do, it was to find, the wave equation which contained the spin of the electron—a correct relativistic wave equation containing the spin. That means to have an interpretation of the spin from the point of view of quantum dynamics.

Kuhn:

Now, here is something that really puzzles me. One could think of a correct relativistic wave equation for the spin as being a Klein-Gordon equation that now included spin terms; a relativistic version, if you will, of the work that Pauli had done with the spin matrices in the non-relativistic case.

Klein:

Yes, Pauli’s work, I think, was known when they made that bet. In fact, Kramers told me later, it must have been in ‘28 when Dirac’s work was quite new, that he and Kronig had got a wave equation of the second order with spin terms in it which was really the second order equation which one had derived from Dirac’s theory. They had got that before Dirac’s theory came out, but I think, they didn’t know quite how to use it; and there was a very abort time in between, so they never came to publish it. It is clear that that would not have been an impossible way to get it. It was very ingenious the way Dirac got it, which was not so near.

Kuhn:

That, you see, points to the problem I am particularly interested in because when Dirac gets this, he doesn’t ‘add terms.’ He sets up a relativistic wave equation which explains there is a spin terra, rather than adding the spin term as an extra term.

Klein:

That came, of course, as a kind, of wonder.

Kuhn:

You see, Dirac states at the beginning of that paper that one of the problems has been to understand why nature should build a spinning electron. Now that problem is not a problem that Pauli is worrying about when be adds spin terms to the Schrodinger equation, if you will. It’s not clear that one is worrying about it if one asks how to add spin terms or puts them into a relativistic wave equation, which is, I take it, what Kramers and Kronig were doing.

Klein:

Yes.

Kuhn:

Were people looking for something like the Dirac equation? Not in the sense that they wanted a relativistic wave equation, which they clearly did, but in the sense that they wanted an equation which would enable them to derive the spin matrices?

Klein:

I suppose that that was in Dirac’s mind and probably also in Heisenberg’s when they made the bet; but I think the bet was made so that Dirac said that he would have it in three months, and Heisenberg believed that it would take at least a year or something like that. I guess that it must have been at least in Dirac’s mind, wouldn’t you say, when the bet was made?

Kuhn:

I press this point because it isn’t at all clear to me that it was in Dirac’s mind, and that’s what makes the bet so important.

Klein:

Now I may again speak a little personally because it’s hard to answer such a question generally. I remember how I thought myself about it. You know, I puzzled always with this five-dimensional point of view, and I thought that spin would have to appear a little like double refraction and polarization in optics. In that connection, not to take this complicated tensor equation, I was puzzling with the vector equation which Proca then treated and which may play a role now for certain intermediate bosons in weak interactions. I didn’t get anywhere there, but the idea was that with such an equation one would be able to have this kind, of extra angular momenta. I cannot quite reconstruct how clear that was at that time because I came back to it much later when I had much more background.

The only thing I can say is that I had a general idea of the comparison between spin and polarization in a doubly refracting medium, and that I was puzzling with the vector equation—that of a four component equation. I don’t think I could say more than this rather vague statement for this time. Later on I came back to the vector equation and saw that in fact one got spin terms. I think Proca had already shown that, but I came back in a little different connection; I’ve forgotten now, it was time and these things. I tell this to show that it was not so strange a problem to try to see that there should be a natural way of spin coming out—that one should not just add it to an equation, but that it should come out more intrinsically. I suppose that that way of looking at it must have been in many people’s minds.

This particular way of Dirac was very ingenious and was a great surprise to everybody, I think. I remember I was so strongly impressed by it that when I learned this then I gave up this whole five-dimensional point of view completely; and I even gave up doing these things of formulating the equation in a general relativistic way which I was very near to because I thought that Dirac had shown that the right way was quite a different way. I gave that up for many years, and there were very special reasons for my sometimes coming back to it later. In this respect I always tried to keep to Darwin’s maxim: not to have a pet hypothesis. So at that time I thought there were no reasons for this five dimensional theory, and I gave it up. I remember that in the spring of ‘28 after I had been at Cambridge, Pauli was in Copenhagen and we had him to dinner. We had a bottle of wine and we drank to the death of the fifth dimension.

Kuhn:

Now that was really more because of the Dirac electron paper than because of the Dirac radiation theory?

Klein:

No, no, it was the Dirac electron paper because the radiation theory showed clearly that you had to go so on the quantum field theory and not try to get that out in some other way. That was not against Heisenberg’s theory, because then I wrote a paper where I took this not as a classical but as a quantum field. But this that Dirac did made me abandon it completely for several years.

Heilbron:

This bet occurred in ‘26, you say? When was the bet made?

Klein:

The bet must have been made probably at the end of ‘26 before Dirac went to Gottingen. I think it took Dirac one year instead of three months, Heisenberg may have said three years and Dirac may have said three months, but as far as I remember, it took Dirac one year.

Heilbron:

The problem here is that Pauli’s spin paper is in ‘27.

Klein:

But I think the manuscript could have come at the end of ‘26, couldn’t it? Or the beginning of ‘27. I don’t remember exactly when Dirac left Copenhagen. Do you know?

Kuhn:

Not much after Christmas, I’m quite certain.

Klein:

I think so too, so therefore I thought the bet might have been later. Could they have met in Gottingen? But there were two papers on this. First there came a paper by Darwin and his wife in and then Pauli’s paper came.

Kuhn:

The Pauli paper is not submitted until May, 1927.

Klein:

I see. But then they must not have known Paul’s paper. Of course, the whole attempt to get the spin was much earlier. I remember and I was telling you of my own attempts. They were earlier because I remember when Darwin’s paper came, I thought that he had done something in the line which I was aiming at, Darwin’s paper was earlier than Pauli’s, so I think my attempts were already from before ‘26. The bet might very well have been then, and Dirac’s paper was published so that it came out early in ‘28, I think. But there was a manuscript which came to Copenhagen earlier—not finished, only a short summary manuscript which Bohr lent me. That must have been either the end of ‘27 or the beginning of ‘28, so that if you take one year from that, I think you will get some time at the end of ‘26; and then Dirac and Heisenberg were both in Copenhagen.

Heilbron:

Whose position would you have shared? Would you have preferred Heisenberg’s end of the bet? Would the people at the institute—?

Klein:

I don’t know. If the bet had been made for myself, I would think that it would have taken me a very long time; but for Dirac I might have said it wouldn’t because he was very swift—I don’t know. But I don’t remember if we heard about the bet at the time or if Heisenberg told me of it later. Heisenberg told me another thing much later which I may mention before I forget it because it was very funny: it was when they both got the Nobel prize. Have I mentioned that to you? When they got the Nobel prize, Heisenberg told me, in Stockholm he asked Dirac if he believed in his theory when the positive electron was discovered. He answered in his very exact way that, “A year before the positive electron was discovered I had abandoned the theory.” That was the hole theory, not the spin.

Kuhn:

The problem comes up at some point in Copenhagen of the question as to whether you can measure the spin of a free electron. Do you remember when that problem starts? Is it only after the Dirac paper?

Klein:

Yes, that was in the autumn of ‘28. Mott and Hartree were present that year and Kronig was there for a time. There were very strong discussions then, when Bohr took that point of view which was in its way correct, but I took another point of view which was in its way correct also, because I knew of the possibility of polarization from that [i.e. Klein’s own attempts to incorporate spin] and in the beginning I didn’t quite get Bohr’s point and he didn’t get my point; so that we had very strong discussions then—sometimes alone, sometimes in the presence of other physicists. He then opposed me very strongly. I never quite got his point, and he didn’t get my point so that we were arguing about things which were not the same. There were observable things which had to do with spin, namely the wave polarization; that was really my point. But he meant that they could not be observed in the classical way and that, of course, was a very important and correct point, and gradually there was agreement on it. I think that already at that time Mott took up the question quantitatively.

Kuhn:

In fact, it was in Mott’s paper that the Bohr arguments first appear; there’s no question. It’s an appendix and he attributes them to Bohr.

Klein:

Oh, yes; I’d forgotten that. I remember also that as a nice case of complementarity, but there none of us saw the complementarity in the beginning.

Kuhn:

But that really is not a problem that exists in ‘26 or in ‘27; it’s only after the Dirac electron—.

Klein:

Yes. There one could really prove it easily in many ways. For instance: one way, I think—maybe Bohr mentioned—was that if you wanted to write the classical motion, then you can do it by means of the Hamilton-Jacobi equation. But then the h would fall out, and the spin depended on the h, so that—.

Kuhn:

There was another bet that I have heard about. I still don’t quite know what it’s about, and Heisenberg who told me about it did not remember it in any detail. He said that he and Pauli had a bet that somehow was over the Klein-Nishina formula, but perhaps it was over the Klein-Nishina formula in some form prior to its publication. His recollections were very vague, but what I understood him to say was that at some point when he knew of it, very likely before publication, there was some term that was totally out of line with observation, or something of the sort, some term or some prediction, some consequence of the formula at some stage of its development that was very far out of line with existing observation. He said his conclusion from this was that this showed that you can’t handle these problems this way; and Pauli’s was, “No, it’s too difficult, the mathematics is too difficult, and you will see whether that will be worked out later and it will all be all right.” And they had a bet which Heisenberg lost; Pauli won the bet. That difficulty was solved. But I’m not clear what the difficulty was.

Klein:

I never heard about the bet because, you see, Nishina and I were in Denmark and neither Heisenberg nor Pauli were in Denmark at that time. They must have made that after the thing was published, I suppose. I could only guess what points it could have been on. First, there were measurements that came out by Lise Meitner and I think Hupfeld and there came, I think, a nice agreement with not-too-high frequencies. Then in such elements as lead, there came deviations at the higher frequencies, and they had to really to do, it was found out, with pair creation. That happened then in the strong field around the lead nucleus, so that that was not against the pure Compton effect formula; but there was a very important contribution, of course, in those cases which came out of Dirac’s hole theory. They could not be derived at that time and so were not included in our formula. It may have been the background for that bet where there was some disagreement. I remember that I had some correspondence with Lise Meitner in which she asked me about it.

I didn’t know the explanation, but I think that came after that. But then, another side of the bet might be that we made a derivation, not by using the Dirac radiation theory, which we could have done, but from the correspondence point of view, since I was more familiar with the correspondence point of view and I knew that in that order of approximation would give correct results. That may have been the point which Heisenberg took up—that that might not give the correct results. There were doubts of that kind. Perhaps I told you about when Dirac had come to Gottingen and he had started that series of papers on the radiation theory. When he had come to the second paper and I think before it was published, then he wrote to me that he had got another dispersion formula than the one I had derived from correspondence theory. It agreed with the one Heisenberg had arrived at earlier in his paper from the matrix point of view. Dirac’s idea was that Heisenberg had not used the vector potential and had used a radiation theory which was not correct, but that they together gave a correct result; and I had used the vector potential and the wrong quantum theory and therefore I got the wrong result. But it was very funny because I looked and found the formulae were identical and one could prove it by using the commutation relation. It was clear that just using the one potential or the other could not be important. One could derive the equations of motion, and they only contained the electric field vector, or magnetic field, and what potential one used could not be important in that case. That was such a very, very rare case that Dirac was wrong that I was quite pleased.

Heilbron:

He accused you of using the wrong theory that was the correspondence theory?

Klein:

I used the correspondence theory, yes. I mention that just because that might have been Heisenberg’s point of view. But then, Heisenberg himself wrote a paper somewhat later where he made what was very near to a proof that the correspondence theory in that approximation is a consequence of the radiation theory. Perhaps it would be wrong to say it was a proof, but it was very near a proof. And Pauli treated that thing very thoroughly in his book then some years later. So I can understand that Pauli took the other point of view and Heisenberg the one point of view. The background might have been these experiments by Lise Meitner and Hupfeld. If you have some occasion, you might ask Heisenberg; he may know.

Kuhn:

I think he does not in detail. He remembers that there was a bet, but he’s lost—.

Klein:

Yes, because I never knew of the bet, they must have done that somewhere else. I think Pauli was still in Hamburg at that time. He may still have met Heisenberg.

Kuhn:

I think by the time the bet took place, Pauli was already in Zurich. That would make it after the publication of your paper. I’d like now to come back to the question of the Dirac radiation theory and then the transition from that to the wave quantization, your own work and Jordan’s work. You, I think, and a number of other people look back to the Dirac radiation theory paper as being a paper in some sense already on wave quantization, on what gets to be called second quantization, which we agree is a bad name. Yet, I feel quite sure, and some other people who read and were deeply influenced by the Dirac paper also think that this was their own point of view at the tine, that when Dirac did that paper that was, not a paper on quantization of wave functions or anything of this sort—this was a transformation approach, and these were using the n’s and theta’s as variables with a neat transformation.

Klein:

It was rather a paper about photons, going from photons and particles to something which would in some way be equivalent to the quantization of the Maxwell field. But Dirac started with photons—that’s what you mean, Dirac started really from photons. He took an ensemble of Bose particles.

Kuhn:

He doesn’t remember, and it certainly is not likely ever now to become clear, but it’s my guess it’s this. He does two things in that paper. He treats the Maxwell field in terms of a set of variables. He takes the Fourier development of the Maxwell field, substitutes oscillators, and then lets the amplitudes of the oscillators appear as q numbers. And he also works around to the other approach in which one thinks directly of the particles. It’s not at all clear which of these comes first. My own guess would be that it’s the oscillators with their amplitudes treated as numbers that comes first, but I don’t really know this.

Klein:

Did you ask him at Cambridge?

Kuhn:

Yes. Now his memory for this sort of thing is extremely bad. He remembers the results, but the process is very much erased.

Klein:

It was not so very long ago that I looked at the paper in connection with some lectures, but I’m not quite certain either which was the beginning. I wonder if he started with the photons as particles and then took the other?

Kuhn:

In the paper itself, he definitely starts the other way. He has both ways, and the fact that the paper opens with the oscillator approach would not prove that in his own development of the paper the other had not come first, so I think the point is open. I talked with Jordan. Jordan said to me that just as soon as he learned the Schrodinger equation he felt quite strongly that now one knew that one must also take this wave equation and quantize these waves in the way he had previously quantized the electromagnetic waves, and then one would get particles out of it; only he did not altogether know how to do it. But as soon as he saw the Dirac paper, he said, “Ah, here is the technique for quantizing the Schrodinger formula.” As you know, he was the person who put into the early matrix mechanics papers all the material on the quantization of the electromagnetic field. The others were somewhat skeptical about this.

Klein:

He must have started at about the same time as I did, then?

Kuhn:

My view of this is that, as many other people read the Dirac paper, although one may look at the Dirac paper as the quantization of the Schrodinger formula, I don’t think Dirac was looking at it that way.

Klein:

Very probably not.

Kuhn:

Therefore there is a very important transition, almost a transition in “Gestalt”. Jordan sees things in that paper as a formalism which, in effect, were not explicit in Dirac’s mind. Did you also?

Klein:

Yes, you see, it must have been with Jordan as it was with me. This came very well into the trend of thought which we had before; in fact, this gave a very strong inspiration; but then, I think, either of us continued on his own train of thought after this impact had come. That might very well have been different from Dirac’s own.

Kuhn:

Tell me then, in your own case, where and how early had this notion of actually quantizing a matter wave function come from?

Klein:

Yes, I can very easily toll this because when I came to Copenhagen I began to learn about Heisenberg’s work and so on. I had been ill the whole time so I know very little of it. Kramers had written me about the first part, I think, sometime in November or December and said that “now there is a new way; Heisenberg had found a way where you had no quantum conditions but you have only one equation.” I think be wrote something like pq-qp, and I didn’t understand a word of it. So I hardly knew anything of Heisenberg’s work until I came to Copenhagen. Then my head was full of those field theoretical attempts with the five-dimensional theory which was then after a time more stressed by the Schrodinger equation. But I think already before Schrodinger’s work came or at about the same time—I don’t remember quite—I had made up in my mind that one had to make a choice if one would be able to base quantum theory on the five-dimension approach, which would be a kind of causal theory in the fifth dimension, or if that were not possible, I would try that as long as I could, but if it were not possible then there would have to be a similar treatment of the fields as the quantization in mechanics. So very early it was clear that it would be possible, but I didn’t make any attempt with it. I think that I would have found it difficult because I wasn’t in the mathematics, but the idea was very clear as a possible way. I think that Jordan’s first paper and then Pauli and Jordan’s paper appeared not very long after that; I’ve forgotten the dates.

Kuhn:

The Pauli-Jordan paper comes after your own, I think. Jordan’s first.

Klein:

Oh, it came after, but the one from Jordan was rather early. But, you see, I tried to neglect these things and this whole point of view as long as possible because I wanted to see if one couldn’t come at it without it. I knew that there might be that necessity to have it. Dirac’s paper came for me at the right moment because then I had begun after I had spent the autumn discussing the beginning of my paper very much with Bohr. I had many other parts of this paper which I hadn’t been able to get ready and wanted to work out. Then it was clear to me that a very important thing still missing was understanding anything of the many-particle problem. I tried that, using developments in the five-dimensional theory literally. I knew that it couldn’t be done in space and time as Schrodinger wanted it. I saw that that led only to confusion, and then Dirac’s paper came, so I tried to see how that would do; and I wanted to see in detail. You know, Dirac’s first paper was the paper without interaction, but I wanted to see how that would work when there is interaction, because that was for me the main problem.

I wanted to have the interaction in space and time, and I didn’t like this generalization in the coordinate space of Schrodinger at all. Now I have no objections to Schrodinger; it’s just the same; but at that time I didn’t like it at all. I wanted to have something in space and time. Then Dirac’s paper gave just the right technique to it, and rather soon, I was able to show that the general term in the coordinate space and the other were really identical. But then I got into troubles with this which really was the self-energy. On the whole I had the tendency not to publish fast, but I showed the thing to Heisenberg, and he was rather interested that one could do it this way from both sides. Earlier he was very much against the fifth dimension; he said, “Why should you have five? Why should you not have an infinity of dimensions?” But when I came to this, he got interested; and also because he saw that I was no longer a heretic. The thing was never published in the spring, and then came that very intense period with Bohr’s complementarity lasting for the whole summer there. Then Jordan came in the autumn, and I must have told Bohr something about it, because Bohr introduced me to Jordan. Jordan had told Bohr that he had also started a similar thing. Then we got together and Jordan very soon saw that my difficulties had to do with the self energy, and he just said, “Let’s just change the order and then we’ll get the self-energy subtracted.” Then we published a paper. I think I wrote most of the first part of it, and Jordan added something of the self energy. The first part he had done also, I think; I never saw what he had done in detail.

Kuhn:

But you, at least, even before you had met Jordan, had worked out the first part?

Klein:

I had worked out the first part of it. Then be added also a generalization of it to other kinds of interactions besides just two particle interactions—a very general scheme. Then we published it. But already at that time he told me of the beginnings of the antisymmetric quantization.

Kuhn:

He had published a paper on that already. It had a mathematical error; the signs were wrong. He may not have published it yet, but it comes out very soon, in immediate response to this interaction—.

Klein:

Yes. I don’t remember if he had published it, or showed it to me, or had it in proofs; but I remember he showed me the thing. I remember that impressed me very much because for a year or longer I had had in my head such vague ideas—that every eigen vibration, corresponds to a particle. I tried to speak to Bohr also about this, but it was so vague that it means that either there is a particle or there is not a particle; and of course that was no theory. Then when Jordan came I saw that that did really give what I wished to have but couldn’t find, namely that you can develop with respect to one particle, eigen functions. Then you have a q number and you can get a number out of that which is one or zero. That gives a very nice way to represent Pauli’s principle. I was very much more satisfied with that, and it corresponded more to my general dreams which I couldn’t at all connect to a theory until I saw what Jordan had done, and it satisfied me completely. Bohr was a little astonished, when I told him that, because he must have thought that I wanted something mere classical. But I was really very satisfied there.

Kuhn:

Your own “dreams,” though, came after the initial Dirac paper; is that right?

Klein:

No, they came before; but they came all during the time when I hoped that one could get something out of the five-dimensional theory but I couldn’t see how, and especially this—that either you had it or you had it not; and I couldn’t see how that came into classical theory. But that didn’t make me decide to abandon the classical idea. I abandoned that only after the Dirac paper, after I had seen that one couldn’t get anything decent out of the many-body problem by using the five-dimensional theory in the literal way and that one could get it so nicely in the other way. The last was, of course, very important because, you know, that when one cannot do a thing, one can always have wishful thinking that perhaps it could be done. You remember Einstein had that almost all his life. But that one could do it so nicely the other way made me quite abandon that way forever; I never came back to it after that—it must have been sometime in March of ‘27. These are the vague ideas I had before, but I couldn’t make anything of them. So I was very happy when Jordan showed me this, that that would be the representation of the Pauli principle I really would like, while this representation with wave functions and the coordinate space and symmetry and non-symmetry, while I saw that it functioned, didn’t satisfy me. Now I know that it’s again complementarity so that both ways are possible. But I will tell you one thing, if I haven’t done it, before I forget it. Many years later I had some conversation with Einstein in l929 at Princeton. Did I tell you about that, about the second quantization? Yes.

Kuhn:

That was very interesting to us.

Klein:

The background for me was this; but then Einstein just joked: “Zweite Quantelung, das ist Sunde im Quadrat.”

Kuhn:

I think I’ve asked as many questions on second quantization as we have and we might as well go back now and talk more about certain of your own five-dimensional ideas—the early parts of it.

Klein:

One word about this. Jordan had there, of course, only the outline, so I think there was a very important contribution which Wigner gave later.

Kuhn:

Oh, yes. I think that’s very clear.

Heilbron:

There were one or two details remaining over from our talk about work—the five-dimensional theory. You say that you had reached an equation for the harmonic oscillator from your work before Schrodinger’s work was published?

Klein:

I had the general wave equation, I had that quite a time earlier; but I tried just before Schrodinger’s work to take a simple case out to see if I could solve it and was puzzling on the harmonic oscillator. But I never could solve it.

Kuhn:

Before the first Schrodinger equation? A four dimension or a five dimension?

Klein:

That was before. Oh, I took that up, I think it was when the Schrodinger equation came.

Heilbron:

And what sort of conditions did you impose on this?

Klein:

I didn’t get far at all. I only mentioned that as an example to show that if I had known more mathematics, perhaps I would have been able to do more. The main trend was to regard the Hamilton-Jacobi equation—I think I told you that—as a wave front equation, and see if I could make a transition there. And I kept the fifth dimension because I wanted such a thing as an electron to have a wave equation which was a general wave equation for all cases. I couldn’t see how one could get the wave equation with the different velocities, and I wanted to say there is a wave equation in a higher numbered dimension, and this is a projection of it in the four dimensional space.

Kuhn:

Do you remember what that harmonic oscillator wave equation looked like?

Klein:

Oh, I think that was easy to write dawn; I’ve forgotten. I mentioned just casually that just before Schrodinger’s work came I was sitting in the library in Copenhagen and puzzling to see if I could solve that simple case; but then I long had known that there would, be a whole class of wave equations corresponding to the Hamilton-Jacobi equation for the motion of an electron in the general electromagnetic and gravitational field. But I was not certain that one should have the superposition principle; and that was, of course, due to the same kind of ideas which Einstein had, namely that one should explain that the particles and waves in some united wave, though then one couldn’t do that. It was a lack of understanding of the real quantum theory. It also contributed to my hesitation that I knew, of course, that one could make a linear equation of it but the same Hamilton-Jacobi equation would come from a non-linear equation where the second order terms were the same. That was a point of indefiniteness.

The harmonic oscillator came in because I wanted to look at a special example, and then I took the simplest. I took the linear, of course. I only wanted this to see how it functioned. I think I said it in connection with the fact that Schrodinger took up the do Broglie’s idea for the hydrogen atom, and he was able to solve the static problem there, and that gave the Balmer terms, and that gave him courage to continue. If I had been able to solve that oscillator question, it might have encouraged me. Things are very difficult as long as you have no result to build on, and I had only such very general results as I knew how to write down. An infinite number of wave equations corresponded to that definite Hamilton-Jacobi equation, and that definite Hamilton-Jacobi equation I wanted to write in the five-dimensional form. I learned that when I gave a lecture on electromagnetism at Ann Arbor, then I wrote the Hamilton-Jacobi equation for this motion, and then I saw that the potentials entered in a very similar way as gik coefficients in general relativity. Then I had already the fifth dimension in my mind in that general way and also in connection Bohr’s remark that perhaps one could get the quantum theory out. But the main thing was that I wanted to have a constant velocity of propagation and I couldn’t see that that would be reconciled with the different velocities one had for an electron.

Heilbron:

What was it that Bohr found so attractive in the five-dimensional approach after Schrodinger?

Klein:

He didn’t find that very attractive at all, I think. That was a very early remark where the stress was that you would not be able to make a connected theory in four dimensions; and then he said perhaps you could if you had one dimension more. I don’t know if he thought any more about it. To me, who had those other ideas in my head, that contributed an inspiration, but I think for Bohr it was just a casual remark with the stress on the fact that you could not do it in four dimensions.

Heilbron:

I had understood you to say previously that Bohr had taken a great interest in the five-dimensional theory after Schrodinger and just before you went to Leiden.

Klein:

No, I think I must have expressed myself badly so that you mistook me. I remember when I got that invitation to Leiden, Bohr was rather skeptical and thought that because I had the invitation I would do propaganda work for the wrong theory, so I had the opposite impression then. He didn’t say it very clearly to me, but I remember once when I told him and Heisenberg at the blackboard in the lecture room about my general ideas before I went to Leiden, so that he was very kind, in his way, and wanted to hear what I intended to lecture on. But I think none of them were really interested; I think they were rather skeptical. I wrote a little note to Nature about this periodicity in the fifth dimension and the quantization of charge; and Bohr wrote something when he sent it to the editor saying that in spite of its abstruse character, it might have something of interest or something of that kind. While he wrote this wanting to be as kind, as possible, he wanted to show that he was rather skeptical.

Heilbron:

But you were invited to Leiden specifically to lecture on this?

Klein:

Yes, did I tell you how that came about?

Kuhn:

I think you did, but I’ve forgotten it already.

Klein:

It was rather funny. Thomas took interest in it, and he asked for a copy of my paper. I had some copies so I gave it to him. Then he brought that to Leiden on his way back to Cambridge, and Ehrenfest asked him to give a lecture on it and he did so. Ehrenfest got very interested, and Ehrenfest asked Lorentz to invite me so I got a very great surprise to get this kind letter from Lorentz inviting me to Leiden. There I was very, very nicely treated when I gave those lectures, and Ehrenfest was rather enthusiastic at that time and I, myself also was. But then, by and by, of course, the difficulties came, which led to that which I—.

Heilbron:

What was Ehrenfest’s great interest in the waves in the five-dimensional—?

Klein:

He believed that would be a way to get the frame for quantum theory. I think at that time for a very, very short time, he also believed that perhaps the quantum riddle night be got—. Of course, he was very deep in classical theory, so nobody was very far from that frame of mind which then Einstein kept—that it would be a good thing if one could get some kind of a real causal background for this strange quantum theory. That was Ehrenfest’s real interest in it; he shared that belief for a short time. But he didn’t share it for long, so I don’t exactly know when he abandoned it because I don’t think I saw him. There was some correspondence. I remember that he wrote me later, it must have been in the fall of ‘27, that Einstein had come to Leiden; and he wanted me to answer some questions that had come up in that connection in discussions with Einstein. So I think he still had some interest then, I suppose that he must have abandoned it either earlier or about the same time as I myself abandoned the belief that one should be able to get quantum theory out of it.

Kuhn:

There is just one thing on this abandonment that I am misunderstanding. You say you basically gave it up when you discovered that the Dirac way gave you a way to handle the many-particle problem?

Klein:

Then at the moment I gave up the whole five-dimensional point of view. But then in the spring of ‘27 I gave up having this aspect as the explanation of quantum theory but I wanted to keep it as a quantum field theory.

Kuhn:

Ah, so it’s that that you give up in ‘28 with the electron?

Klein:

, Yes. I wrote the paper, I think it was the end of the summer of ‘27. I tried to treat this as quantum field theory, and then Pauli and Heisenberg for a time were interested in that. Pauli told me that for a little time they intended to base their treatment of quantum electrodynamics on that, but then they gave it up. I’m glad I mentioned this because there must have been a “not” missing just as in that Dirac paper, you remember, that Delbruck was telling you about.

Heilbron:

You mentioned also a little diagram Ehrenfest liked to draw on the blackboard to show the relationship between your waves and de Broglie’s waves—some little sketch.

Klein:

Did I mention that? I remember he did such a thing, and I may have mentioned it; but I don’t remember what it was. I remember he always liked to make diagrams and he made some such diagram, but I have forgotten it. Did I actually mention it, because my memory is very vague of it? I must have said it in this vague way then.

Heilbron:

Well, it was just brought out by something else. There’s another thing in your second paper in the ‘27 paper on the five-dimensional approach—in the introduction to it you made some curious remarks about the sort of casual nature of the conservation laws in the classical physics.

Klein:

Do you mean “causal”?

Heilbron:

I mean “casual” in the sense that one could have had other first integrals; there is just a number of possible first integrals of the laws of motion.

Klein:

What I remember is that I had come then to understand something which was known to many people before, namely, the very important relationship—which Bohr stressed also in his work on complementarity—between energy and time, and between momentum and space coordinates. I took that as the background for saying that now we have also a very rigorous conservation theorem for electric charge and that it would be natural also to introduce a conjugate variable to the charge.

Heilbron:

But you say more. You say, I think, that if you look, in a classical theory, at the conservation laws, they come in sort of a casual way from the laws of motion, as first integrals. One sees the relationship between the space-time description and these particular conservation laws. On the other hand, you continue to say, since there is this casual nature of these particular conservation laws, then one can see that the space-time description also is rather casual and could be relaxed in the quantum theory.

Klein:

I don’t remember what words I used, but I think the idea was that one might define space and time as conjugate variables to energy and momentum; and since energy and momentum are continuous, one might make wave groups which are as small as you like; and therefore, one might define accurate space points and accurate times. I think that was the way I meant it and, therefore, I meant that with the fifth dimension one might take that as a conjugate to charge. But since charge is quantized, one could not hope to have the same way of arbitrarily getting wave packages. Now, in using complete periodicity one could again—but that depends a little. Do you have this paper?

Heilbron:

I thought it was quite curious because I got the impression that what you were saying was: If one had chosen first integrals other than conservation of momentum and conservation of energy, then one would have a description other than that in space-time. I guess I misunderstood that.

Klein:

I think the idea was that there was a similarity between introducing conjugates to the charge and introducing space and time as conjugates to momentum and energy. I think that must have been it. You don’t have the paper?

Kuhn:

I had only the reprint, which can’t be gotten at the moment.

Klein:

It was in that summer of ‘27 I sent it in. But these things have, of course, a certain historical interest, but it’s, of course, very doubtful if they have any real application. In later years I’ve sometimes tried to use this as a kind, of model which should be taken still much less literally than in this ‘27 paper; but I have been a little bit fluctuating there. Still I sometimes believe that this could be a useful model, but only if it is taken very casually and not literally. Of course, we miss a way of really interpreting charge and understanding the transformations to it so that this could help perhaps, or it may not, I would not be certain at all. But there is some relationship between these transformations and those which one can see—I suspect both.

Kuhn:

Mr. Heilbron is having trouble finding the paper.

Heilbron:

I have the box, but I can’t seem to find it. We have the reprints.

Klein:

There might be the Zeitschrift fur Physik if you have the journal.

Kuhn:

Oh, no, we don’t have the journals. They’re in the library in Copenhagen.

Heilbron:

If I can read my translation of that, it’s something to the effect that, “If one thinks of the rather chance role of the conservation laws in classical mechanics, they being just four of the many possible first integrals, it would then appear from the above point of view that the space-time quantities so important for our everyday experience would be relegated to the background in a mathematical foundation of the future quantum theory—.”

Klein:

Do you remember the German word I used which you translated by “chance”?

Heilbron:

No, but I should think it’s “Zufall”, something like that; I don’t know.

Klein:

No, because I don’t recognize this. I remember that I tried to stress the point at that time that energy and momentum may be in some way the background for space and time and that space and time may not have quite the classical meaning so that perhaps, I would say now, I was going too far to the other side, while I had been turned to the first side a little before. But I don’t remember that exactly. I will look it up when I am at home where I have the—. Is that from—?

Heilbron:

That’s from the second one, in ‘27.

Klein:

Yes. I remember very well that I pointed out that. And I think the main reason for pointing out this relation between space and time was that I wanted to have an argument for introducing a coordinate corresponding to the charge. But that coordinate again must have no complete analogy because the charge was quantized. I’m very curious: I shall look it up when I get home.

Kuhn:

Do you have other questions on the five dimensional aspects? You will gather that Mr. Heilbron is our expert on five-dimensional theory. Let me only say, I have not read any of the five-dimensional papers, either yours or others, and I have left what we’ve been able to do with these pretty much entirely to John.

Klein:

Yes. But you’re interested in it. I should look it up. Now it takes some time before I can get it because I am going to the country where we are now, and I haven’t taken it with me; but I’ll be glad to write you if I can make any comment on it.

Heilbron:

Thank you.

Kuhn:

I really only wanted to ask a little bit about the later, but not very much later, development of electrodynamics— ‘29, ‘30, ‘31. It’s hard to know what questions to ask, but one that is obviously of key importance for the subsequent development is the question of the point at which the infinities get to be taken as really basic difficulties. In your paper with Jordan you get rid of these infinities by changing the order in the Hamiltonian. Already in ‘29 in the Heisenberg-Pauli paper they can’t get rid of the infinities yet, but maybe somebody else will be able to. What happens to that problem? Do you remember?

Klein:

I really didn’t work at it, but, of course, I couldn’t help thinking of it. I remember that, in the discussions I had with Jordan then, he was very happy for this formal way of getting rid of it. I was very glad to have that as a provisional way, but I didn’t feel that that should be the solution to the problem. I remember he said that he didn’t believe that the electron had any inside; but I rather believed that if one would have a real quantum field theory, one should take up these problems and I didn’t know the solutions of them. So when the infinities came out, I thought that they meant that there was something which we still don’t know about the field theory. I think it was Wailer who gave the first example of the infinity. And Heitler, I think, also.

But then there came this very nice calculation of Weisskopf on the Dirac theory where this infinity got so very much smaller, only logarithmic. My reaction to it was that now we have got a physically more realistic theory, the infinity gets smaller. My hope had been then the whole time that when they get nearer to a physically realistic theory, then you would get rid of that difficulty; so I regarded it as a real difficulty—perhaps not so basic as those many people did who compared it to the situation in 1925 before one had any quantum mechanics.

I remember I had a discussion with Pauli in rather late years when he took this analogy and I tried to say something of this kind. In ‘25 one had a good classical theory, that is to say, in the words of quantum field theory, one had a good Lagrangian as far as it went, as far as experience went there; but one had no strict way of translating this into quantum theory, only the correspondence way which was only half quantitative. While now, we have what we believe is a good general quantum theoretical scheme, that is to say, a way to formulate answers to different questions. The answers diverge. We have, so to say, the formalism, though we miss, probably, the correct Lagrangian. So therefore I thought that there was no very great analogy.

Somebody in the discussion said something that was rather similar to this but not quite the same. I didn’t want to say anything because there were so many people there and it would take up too much time. This had to do a little with my private ideas that perhaps general relativity would help on the divergence problem, although I don’t think that it would help on all problems. There is very much still missing of other things also. I would rather say that that part of the theory which means to get a correct Lagrangian is at least a part of what is missing. I’m not so certain that the process of quantization is so wrong. Of course, we cannot know, but I think we know that, of course, it’s true that there is much about the Lagrangian which—there are many fields now which we are very unclear about, and we have not taken account of the action of the gravitational field. So that was a bit of my own reaction to these infinities. They are important but perhaps no more important than the further indication that there’s very much we don’t know.