P. A. M. Dirac - Session V

Kuhn:

This would have been at Bristol still, the suggestion to look at quaternion’s.

Dirac:

Yes, that was at Bristol, I think one of my teachers recommended me to it and said I’d be interested in it but I can’t remember which one it was,

Kuhn:

One thing that particularly struck me in your article in the Scientific American in describing the various schemes that occurred to you as possible approaches to remaining problems, you constantly use pictorial models, I wondered to what extent with earlier papers you’d also had pictorial models in mind in advance and had done the mathematics to fit them,

Dirac:

Well, what example are you thinking of anyway?

Kuhn:

You mean what example in the article?

Dirac:

No, in the earlier —.

Kuhn:

Well, .1 would say it’s a little bit hard thinking over the earlier things to see how you could have had pictures and the pictures certainly don’t show at least in any very explicit form in the final papers, so I wondered whether this represented a later stage or the fact that perhaps these ideas are amenable to pictures as the earlier ones were not,

Dirac:

I think it represents the kind of work which I do when I’m not just stimulated by a paper and trying to improve a paper.

Kuhn:

Well, take the case of the electron wave equation. Now here you were not simply stimulated by a paper and trying to improve it, though you clearly had a problem —

Dirac:

I was trying to improve the Schrödinger equation.

Kuhn:

I take it that the quantization of the Faraday field, the single lines of force, is actually one that goes quite far back. This seems to go with your paper on quantized-singularities in the electromagnetic field,

Dirac:

I didn’t have it so far back as that,

Kuhn:

You didn’t, But you do talk there in that paper about lines connecting magnetic poles the possibility of complete lines off to infinity that join or broken lines that have charge at the end, It seems to me a somewhat similar -.

Dirac:

I didn’t have the same idea then that all the Faraday lines of force should be discrete I just mentioned in lectures from time to time that I don’t see how to develop them mathematically without getting into the usual kind of infinities. [Short pause to adjust the microphone]

Kuhn:

You think in the case of the older papers, the ones that we’ve particularly talked about so far that they all take off from an existing set of papers.

Dirac:

I think they rather do and that I tried to form a picture of that paper and getting a clearer picture of that paper enables me to write my own paper.

Kuhn:

Take the Schrödinger equation or the Klein-Gordon relativistic version of it: what sort of picture would you work with for that in trying to improve it? What sort of a picture would be the intermediary between the Schrödinger equation and your own relativistic equation?

Dirac:

I guess I’d be thinking of wave functions and having them multiplied up, forming some kind of density which one would picture spread about in space.

Kuhn:

In that sense, your own wave equation doesn’t much modify the picture, does it? It certainly modifies the equation and the mathematics.

Dirac:

Well, bringing in the four components — that is something which one is really led to from the mathematics, and that’s not something which is suggested from the picture.

Kuhn:

So in this case the mathematics really modifies the picture, rather than the picture suggesting —.

Dirac:

Yes. : Well, they will do both things, influence one another. So I am all the time trying to get better pictures. The renormalization theory I’m not very happy about because of the absence of pictures. One can set up quite good pictures for the renormalization of mass. You would have the Wentzel field or a lambda limiting process. But I haven’t found that corresponding picture for the renormalization of charge.

Kuhn:

When you speak of “picture”, you mean something that interacts with the mathematics, but that is not simply the mathematics. You’re not simply seeing the symbols with a life of their own.

Dirac:

Someway which enables you to understand the equations independently of the approximate method of solving the equations. One ought to have well-defined equations before one starts to apply a perturbation method to solve them, and that is what one doesn’t have with renormalization theory. But you can have it with respect to renormalization ‘of charge if you use some such idea as the Wentzel field.

Kuhn:

The renormalization of mass?

Dirac:

Yes, yes, the renormalization of mass.

Kuhn:

You don’t though in particular think of pictures which represented early attempts on one or another of the —. For example, did anything like picturing go with q-numbers?

Dirac:

I suppose I pictured the q-numbers as some kind of mysterious numbers which represented physical things. It was quite a good time before I appreciated that they were merely operators in Hilbert space. When I did appreciate that I dropped the terminology q-numbers. Q-numbers was something which was imperfectly understood, physical variables satisfying non-commutative algebra.

Kuhn:

When you thought of them as physical things, do you think that thinking of them in that way provided you guidance in their manipulation over and above that given by their formal mathematical property and their direct classical analogues?

Dirac:

I think it does to some extent. When an infinity turns up in the equations one can represent it by a finite quantity which approximates to it, I might often imagine that a series is cut off somewhere at distances which are much smaller than anything which is physically important. That would be quite permissible. That is one way of setting up a picture which will get rid of the infinities. Then one can try to figure out whether such a picture is relativistic or not. One can do that much better than if one worked with the mathematical infinities.

Kuhn:

All of these examples that you give are relatively late ones in terms of the period that we’ve been mostly concerned with so far.

Dirac:

Yes. Well, they’re more fresh in my mind. The delta function came in just from picturing the infinities as something which approximates to them.

Kuhn:

Do you know when they came in? You first introduce them explicitly in the transformation theory paper but there are hints of them in two earlier papers.

Dirac:

I think I published them soon after I had the idea clearly.

Kuhn:

In the paper in which you first develop the Schrödinger theory, which we talked about last time, there is a rather cryptic footnote to the —. You developed a wave function as an expansion of the form CY in a sum or an integral, and then there’s a footnote which says that there are some cases in which you cannot represent the new wave function as but as the limit of a series of such functions and then in the very next paper which is the one on the Compton. effect treated by wave mechanics you use a delta prime function though you don’t call it that and then the delta function as such then comes out very clearly and explicitly in the transformation theory paper.

Dirac:

Well, those earlier papers were really showing the need for it, I noticed the need for it very much in a certain place in Eddington’s book. ..

Kuhn:

How early did you really see the need —? I guess I should ask which book -.

Dirac:

I could find it in a moment.

Kuhn:

I‘d be very glad. The book then is Eddington’s Mathematical Theory of Relativity, 1923 edition, and the footnote is at the bottom of page 190. This would be something you had noticed then really before the delta function began to be directly involved in your own work.

Dirac:

Yes, yes.

Kuhn:

That’s very interesting. It’s just the sort of hint that I’m particularly glad to pick up. You refer in the transformation theory paper, very early in it, to a paper of Lanczos, published in 1926. I think it’s written in 1925 and published in 1926. You say the transformation theory can in a sense, be regarded as a generalization of Lanczos’ theory. Lanczos’ theory is the paper which reformulates matrix mechanics in terms of integral equations and Kernels and then has a continuous matrix. Was that paper an important one to you? Was that one of the ones you studied quite carefully?

Dirac:

I don’t think it was, no, as far I remember.

Kuhn:

He also comes within an ace of introducing the delta function in that paper. That is, he needs a form, a kernel which will serve as a unit matrix.

Dirac:

Well, I wonder why people didn’t introduce it because it’s just a question of notation.

Kuhn:

Well, I think many people must have been uncomfortable with the idea of a function that was that badly behaved. :.

Dirac:

The mathematicians would be.

Kuhn:

The physicists’ functions were generally not even better behaved often than the mathematicians’ functions, although the physicists apparently didn’t worry too much about it.

Dirac:

Well, the physicists could understand it at once as a function which is narrower than any distance which’ he is ever concerned with, and the physicist doesn’t ‘t need to go to the limit, he can just retain it as a function like that. I would always picture it in that way.

Kuhn:

So you have a picture with the sort of physical justification of it that it’s an approximation function; all I need to do is make it narrow enough so that the distances involved are small compared with any I’m concerned with and then let it be a finite distance..

Dirac:

Yes, I would picture it in that way. I would do all my work with such a picture. So I have never worked with Schwartz’s distributions because I find this picture quite adequate for all physical purposes.

Kuhn:

Reading your Scientific American article reminded me that when I talked with Heisenberg, one of the things he said was that he remembered that at one time he’d had a conversation with you in which emerged a difference of opinion on strategy, You believed that one must handle one problem at a time and he felt that this whole group of problems had to be handled together or not at all.

Dirac:

Yes, yes, I know. A good many people hold Heisenberg’s view, but it’s just too difficult to solve a whole lot of problems together. I think several new ideas will be needed to get out of the difficulties in present-day physics and that these ideas will come one at a time with intervals of several years between them.

Kuhn:

There are two senses in which one can take this. One of them would be that one must in fact be working on a problem and not trying to worry about everything at once if one’s going to make a contribution, that would be in the narrower sense. There would be the broader sense in which one means that we will in fact solve one of these problems by itself without simultaneously solving a number of the others.

Dirac:

I don’t think one should set out to solve one of these problems. One may be working on one of them and get something which contributes to a different problem. So I hardly think it’s right to set out to solve one particular problem. And I’m just making a prediction of how it will go in the future. I think there will be several ideas needed and that they will come at intervals of several years.

Kuhn:

Is this sort of disagreement that’s represented in your discussion with Heisenberg one that you’ve had with other physicists; does the discussion of that problem go back quite a ways?

Dirac:

Yes, I think it does, yes. I think most physicists take the Heisenberg point of view. Perhaps the reason why I don’t take it is that I thought for a time that the one idea of non-commutative algebra would solve everything and then I found that it didn’t so I’ve rather gone to the other extreme of thinking that every idea will meet with only partial success.

Kuhn:

That does seem to me to be a different point. I press it only because I’d like to be sure I’m understanding you. That is, the idea of non-commutatively certainly solved a great many problems together when it came. It didn’t solve all of them.

Dirac:

Well, it’s left a great many. Well, I think people will get other ideas which will each solve a group of problems and leave a lot of problems unsolved.

Kuhn:

One other question out of that paper in the Scientific American, this a narrower one. You speak there quite having gotten the Klein-Gordon equation before he got the non- relativistic equation; I wondered whether he had said that to you or whether you were just taking it for granted that if he got a relativistic form that was the one he got.

Dirac:

He told me he got the relativistic form of the equation before he got the non-relativistic one. I’m not sure which relativistic form he got, but as far as I can see it must be that one, I don’t see what else it could be.

Kuhn:

I certainly don’t have any notion that it is another one. But the straightforward way of getting it, by simply substituting operators, he will not have had access to and’ I’m not sure —. He’s got two different ways of deriving the Schrödinger equation in his first papers -.

Dirac:

You should ask Mrs. Schrödinger if Schrödinger’s left any notes on that.

Kuhn:

We’ve got all the notes he left in Copenhagen now and we’re going over them but there’s very little from that period. I hope we will find something but I’m not optimistic about it. There are some calculations from that period, Stark effect, this sort of thing, but nothing that looks like groping toward the wave equation at least not on our first inspection of it. I think probably one can reconstruct it and I think very likely when one does that one will come out with the Klein-Gordon equation but I’m not sure. I just wondered to what extent you’d taken this picture from things that he’d said and to what extent you’d taken it from —.

Dirac:

Well, all that he said was that he had the relativistic equation for the hydrogen atom before the non-relativistic one and he worked it out and got the wrong spectral terms.

Kuhn:

[showing Dirac] This is a photocopy which was made in Moscow of the Kapitza Club Minute Book. One of the things I particularly noticed looking at it - -.

Dirac:

I just wonder where I left my glasses.

Kuhn:

Heisenberg’s paper of 28 July, 1925, to the Kapitza. Club, is given one meeting [the 94th]; the very next meeting which is 4 August, you give your first paper to the Kapitza Club. Would this early August meeting have been the first meeting at which you were a member; did you speak at the first meeting?

Dirac:

Can I just look (at it)? [Discussion of interpretation of photocopy and whether Heisenberg and Dirac did indeed speak at successive meetings].

Kuhn:

But for example, if the Kapitza Club elects new members on an academic year basis with the end of July doing this; or you could have missed the meeting.

Dirac:

I think they elected any time.

Kuhn:

They don’t necessarily begin their membership by presenting a paper?

Dirac:

No, I think that would be unusual. It could have been that I had been elected previously and then missed that meeting. But I’m a bit surprised that there isn’t a reference to Heisenberg if he actually talked at the meeting.

Kuhn:

Well, the standard thing appears to be for the speaker to write down the title of his talk and his own name. I rather suspect that this was pasted in, in fact it looks as though he may have had a spare passport picture with him and then he simply wrote this in and signed the picture. Because this is clearly two papers. Isn’t it? Yes, this is Babcock’s paper and I think both the handwriting and the subject are distinct from this and that I think is the signature that goes with [it].

Dirac:

Yes, it may be that Heisenberg gave a talk on this German subject here.

Kuhn:

You are quite clear though that you were not at that meeting.

Dirac:

I didn’t hear anything about Heisenberg’s matrices until I got that reprint which Fowler sent me. It wasn’t a reprint, it was a copy of the proofs. I wonder if that copy of the proofs still exists? I had it in my possession for a good many years and I gave it to Fowler .

Kuhn:

I have not been able yet to get information about where Fowler’s papers might be. Unless something has come in Copenhagen since I left which is possible. Those would be particularly good, of course, to find if we can.

Dirac:

It does not look like my writing ‘this meeting here. I don’t quite know why. There are some nice pictures in it from time to time.

Kuhn:

There are some quite interesting subjects also. This I must return to Sir John Cockcroft but I have the microfilm from which this was printed and we shall copy it. You give a paper just after this in, I think, December, 1925, that I wanted to ask you about. Yes, “Light Quantum Theory of Diffraction,” which came in the 15th of December, 1925.

Dirac:

I think that was just to get —. If you have a light diffracted by a grating and you assume that the momentum that each light quantum receives is quantized you get the proper diffraction law.

Kuhn:

Is this the old Duane theory in which you would assume something like an infinite grating and a periodicity over the grating spacing?

Dirac:

I think it must have been that, yes.

Kuhn:

How long in advance would one generally sign up or agree to give a paper to the Kapitza Club? This is quite an old theory and it’s an odd thing for you to be speaking on at this time because by this time you would already have gotten the commutation relations, which you come right over, in fact,. and do on the 19th of January. Would this have., been scheduled for a long time?

Dirac:

As a rule at each meeting they arranged the speaker for the next meeting, but if the lecture finished short and there was some time left over, then anybody might be asked to speak. This is a second talk. ‘I was just called upon to speak just to fill in some time that was left over.

Kuhn:

You would be called upon from the chair to perform without any advance notice or would you have volunteered that ‘if there were time you could describe ‘the following.’

Dirac:

They would first of all ask for volunteers from the chair, and maybe this was something I had just read about and thought it would be good to talk about.

Kuhn:

Were you at all impressed with that theory?

Dirac:

Well, before the discovery of quantum mechanics, I was.

Kuhn:

It would be interesting to know whether this is a date before the discovery of quantum mechanics or not — the 15th of December, 1925.

Dirac:

That is afterwards. I suppose the paper of Duane was published before

Kuhn:

Very long before, yes. I would say by this time that work was more than ten years old. There had been some further work on it by Epstein and Ehrenfest.

Dirac:

Well, the new quantum theory was pretty immature and not especially settled as a subject for talking about it. Did you see the minute book of the De1ta V?

Kuhn:

Yes, and the microfilm is being made. I didn’t sit down and study it but we picked out the first three volumes of the minute book to be microfilmed. [Dirac continues looking at the Kapitza Club Minute Book] If turning pages in that reminds you of anything I would be delighted to hear about it.

Dirac:

Well, I’d have to spend some time on it I think.

Kuhn:

Why don’t I leave it here and if you have a chance and are interested you might look it over.

Dirac:

Oh, this is Cockcroft’s copy, is it?

Kuhn:

He will also get the microfilm back but I said I’d take it to Copenhagen and copy it there and then return it to him. But Kapitza had these made in response to requests that some sort of filming be done and it makes a handsome volume and an extremely interesting one for us. Let me take you back to Copenhagen which we'd really begun to talk about last time. Where did you live when you were in Copenhagen?

Dirac:

In a pension, (‘Pension Sheck’) it was called, near the town hall.

Kuhn:

Did you do any of your work there or did you work entirely at the institute?

Dirac:

I believe it was entirely at the institute. I 3ust had a bedroom there. I don’t remember having a desk. I don’t think I had a desk there.

Kuhn:

Was it possible that the institute was well enough set up then so that you had a room of your own there?

Dirac:

I just can’t remember whether I had a room of my own or whether I worked in the library.

Kuhn:

Can you give me any idea of what sort of a pattern your life settled into?

Dirac:

Well, I used to walk, as a rule, between my pension and the institute. Took about twenty minutes walking along beside those lakes. Except on Sunday. On Sunday I took a long walk or excursion out into the country, very often alone, but sometimes with Bohr and sometimes with a big group.

Kuhn:

What time did you get to the institute in the morning?

Dirac:

Maybe half past nine. I don’t remember so well.

Kuhn:

Would you be there then for the whole day?

Dirac:

I think I went back for my lunch to the pension. I believe I had all my meals at the pension.

Kuhn:

Then returned afterwards to the institute.

Dirac:

Yes. It meant quite a bit of walking: four times twenty minutes, a day, but that wasn’t too much. I often would think over my work as I was walking around there.

Kuhn:

What would you do in the: evenings? .

Dirac:

I sometimes took walks around the town.

Kuhn:

It’s a good town for that. [Knock on door] .

Dirac:

You asked me ‘what I did in the evenings. One thing was to get in one of those trams and just to go to the terminus; you could go as far as you liked without paying extr4L- 0 I would go to the terminus, see wherever it would take me and then walk back.

Kuhn:

And then walk back. Starting from the center of town you got some very long walks that way ...

Dirac:

Yes. Well, it would fill up the evening, maybe walking a hour or so. I got to know the town pretty well.

Kuhn:

You. were generally alone on these expeditions also?

Dirac:

Yes. I lived very much on my own.

Kuhn:

Was there anyone there whom you particularly remember seeing besides Bohr?

Dirac:

I think Klein was there. I’m not sure Gamow was, there at that time-or later.

Kuhn:

I think later. Actually the question as to who was there we can determine by other means, really more I asked this to see whether there was anyone there whom you may have interacted with more.

Dirac:

Mostly with Bohr. I‘m afraid that interaction just went one way. I mostly listened to him.

Kuhn:

You told me very effectively and movingly of his concern with non- scientific problems. Do you remember the sorts of things he was worrying about in science?

Dirac:

Complementarily. I think it all centered around that.

Kuhn:

But it wasn’t clearly quite complementarily yet. What form did he think of it as taking?

Dirac:

Well, I suppose it was still the correspondence Principle that was’ very much in his mind. He still referred to the Correspondence Principle for some years I think after quantum mechanics really made definite equations which would replace the Correspondence Principle.

Kuhn:

In some ways don’t you think he held on to the Correspondence Principle to the end?

Dirac:

I expect he did. When one gets so absorbed with one• idea one does stick to it always. Just like Einstein thought that non-Euclidian geometry would be the answer to everything.

Kuhn:

But you do feel quite strongly that quantum mechanics, when it developed, displaced the Correspondence Principle.

Dirac:

Yes.

Kuhn:

What about complementarily?

Dirac:

I don’t altogether like it. In the first place it is rather indefinite. It doesn’t provide you with any equations which you didn’t have before and I feel that the last word hasn’t been said yet on the relationship between ‘waves and particles. When it has been said people’s ideas of complementarily will be different.

Kuhn:

I was particularly impressed in this connection with what you say about the Uncertainty Principle, the likelihood that h is a derived constant rather than a fundamental one and that the manner in which this will react back on the Uncertainty Principle.

Dirac:

Yes.

Kuhn:

How did you feel about it at the time — the Uncertainty Principle and complementarily which come pretty much one right after the other.

Dirac:

I was dominated by the equations with non-commutative algebra; qualitative things like the Uncertainty Principle. I felt were of secondary importance.

Kuhn:

When you got the Nobel Prize, Bohr wrote to congratulate you and’ your’ answer to him is on file in Copenhagen. You there speak very wonderfully of your sense of the influence he had had on you and the likelihood that this was really a greater influence than anybody else had had. I wondered to what extent this was just politeness, but also what sort of influence you think he may have had in your work. Did it in any significant way, change its character as a result of contact with him?

Dirac:

I expect I did get a broader outlook. I did have too narrow an outlook previously when I thought action and angle variables were really the answer to everything.

Kuhn:

But that you’d already broken with by the time you got to Copenhagen. Well, possibly not.

Dirac:

Not altogether, no.

Kuhn:

Do you remember anything he said to you in the scientific realm that would have this same impact that it would come back whole, the way the remark about the two gunmen that draw their guns and then don’t fire?

Dirac:

I can’t at the moment, no.

Kuhn:

In this same correspondence there is, I think it’s in the late twenties, after the electron wave equation

Dirac:

Correspondence between whom?

Kuhn:

Between you and Bohr.

Dirac:

Yes.

Kuhn:

In which he is again proposing violation of conservation of energy to solve some problems and you’re saying you think: this will work out in due course.

Dirac:

I was rather prepared to accept it if Bohr proposed it.

Kuhn:

Were you, because on the whole you’re skeptical in your responses.

Dirac:

Well, I wasn’t sufficiently skeptical to propose the neutrino. I did take it as a serious possibility.

Kuhn:

I’ve been interested, in watching Bohr’s work, by the number of times, and I’m not quite sure when it begins, but I think it begins before 1924 and before the Bohr-Kramers-Slater paper, that he suggests that the way to get Out of this one is to relinquish conservation of energy.

Dirac:

I think at that time I. was only too willing to give up, established ideas, just feeling revolutionary. I think I would have been happy if any established ideas were being knocked down.

Kuhn:

When you say this, you refer to what period?

Dirac:

This period when Bohr was proposing non-conservation of energy.

Kuhn:

But with Bohr that doesn’t really specify a period because this happens over and over and over again.

Dirac:

Well, only until the neutrino was settled.

Kuhn:

But you said your own sense that you had ceased to be conservative and would now welcome a new and radical idea extends over the entire period? Now surely still.

Dirac:

Yes. Yes, for instance, one time Jordan proposed that one should go on from non-commutative multiplication to non-associative multiplication I rather liked that idea, but don’t really see how to develop it mathematically. The mathematics of non-associative multiplication seems to be pretty specialized.

Kuhn:

Did you work on the idea?

Dirac:

Yes, I don’t know whether I worked on it much of the time. I have from time to time since then worked on it,

Kuhn:

Do you remember when that suggestion was made?

Dirac:

I think it was made very soon after non-commutative algebra was established. I think Jordan was the first to propose it. Have you interviewed him?

Kuhn:

No, but I shall be seeing him in June. I will see him after the middle of next month. You and he for a while somehow or other, 1 take it without much direct contact, again and again worked out much the same problem.

Dirac:

Who is the “he”?

Kuhn:

Jordan.

Dirac:

Yes.

Kuhn:

I think this was quite independent work and that the relative simultaneity grew out somehow from a similar approach.

Dirac:

Yes. Well, it was just that conditions were ripe for making these developments.

Kuhn:

Did they bring the two of you more together personally or more together in correspondence?

Dirac:

I don’t think so, but I don’t remember what correspondence we did have. I didn’t know German very well at that time and I could only read it with a dictionary.

Kuhn:

You went on to Göttingen from Copenhagen.

Dirac:

Yes.

Kuhn:

Do you remember when you’d have left Copenhagen and gone to Göttingen?

Dirac:

I think it was, soon after Christmas. I remember stopping in Hamburg on the way to attend a meeting of the “Deutsche Physikalische Gesellschaft”.

Kuhn:

That would date it. It doesn’t date it immediately for me but it gives us a way to determine it.

Dirac:

Then I traveled with a group of students from Hamburg to Göttingen We had a fourth-class carriage to ourselves.

Kuhn:

Who was in the group?

Dirac:

I remember (Mrs. Robertson) was one of them and she had a picture of a monkey she was showing to people. I don’t know if she was married at that time. Do you know? That’s also something one can check up on.

Kuhn:

By the same token let me ask, do you know when you actually got to Copenhagen?

Dirac:

It was in September. Maybe the 10th or 15th of September, I’m not quite sure.

Kuhn:

That’s quite good enough. Did you find Göttingen terribly different from Copenhagen?

Dirac:

Yes. There was no one there whom I got so friendly with as Bohr.

Kuhn:

Did you see much of Born?

Dirac:

A fair amount, but not so much as I did of Bohr. I did meet some other mathematicians. I think Hubert — was he?

Kuhn:

He would have been alive, yes.

Dirac:

Maybe I went to some of his lectures. Maybe Weyl’s.

Kuhn:

I think Weyl would have been in Zurich.

Dirac:

I have attended lectures of Weyl but that might have been later in Princeton.

Kuhn:

Weyl was working quite closely with Schrödinger , in the spring of 1926. It may be that he came to Gottingen in February or something of the sort but certainly Zurich was his regular base at that point.

Dirac:

I met Franck, I remember.

Kuhn:

What about Courant?

Dirac:

I met him also.

Kuhn:

That I would imagine was a more mathematical school than Copenhagen, including the fact that the physics was more mathematical.

Dirac:

Yes. I did meet more mathematicians there and I picked up some mathematics.

Kuhn:

I wanted to ask you about that because it’s terribly hard for me to be sure that it isn’t just the fact that it’s in German and I read German differently from the way I read English, of course not as well; but the paper you submit from Göttingen in German to the Zeitschrift which is the one again on the collision problem in which you get line breadth, seems to me to read a little differently.

Dirac:

Well, my own knowledge of German was rather feeble.

Kuhn:

But outside of this the mathematics looks a little different —, I’m not sure I’m not putting this all in myself. For one thing, you start right in the beginning — and this is something I notice you do again in your later papers - you write the wave equation in the integral form which you haven’t generally done before, treating the Hamiltonian as a Kernel and of course this is the one that makes it look like a matrix equation. I wondered did this have anything to do with the strong preference for matrices in Göttingen. Were you aware or oppressed or otherwise affected by it?

Dirac:

I think I must have been affected by it. But I expect that that was the most uitab1e way of writing the equation for that purpose.

Kuhn:

It certainly does very well. You know the famous story that is told about Born over and over again is someone saying to him after Born- Jordan “Elementare Quantenmechanik” comes out, ‘Why didn’t you put the Schrödinger equation in the book?” and Borns thumbing through the pages and saying, “There’s the Schrödinger equation” If you look at it, yes, I guess it is but the whole approach is so different and I wondered if there’s any — it’s this sort of story that alerted me to this. Did people take different attitudes there toward the sort of mathematics you did?

Dirac:

Well, I would have learned a lot of mathematics and that would have influenced me.

Kuhn:

You use the word “Fourier transform” I think for the first time in that’ paper. I wondered whether you’d known of Fourier transforms before.

Dirac:

Yes, I must have known them.

Kuhn:

You certainly had been using them.

Dirac:

Yes.

Kuhn:

But it could be possible, this is the sort of thing you point out you’ve done, is to be using them without knowing that there was a developed theory.

Dirac:

I knew about that pretty early. I probably learned about it in Bristol.

Kuhn:

It would be a natural thing to go with studying circuitry.

Dirac:

I was thinking of a mathematics course in Bristol.

Kuhn:

Do you have any particular recollections of Göttingen, the physics there? Did you run in there in any form that you directly remember to be opposition to the Schrödinger approach?

Dirac:

I think there wasn’t any opposition. I remember in that meeting in Hamburg I attended quite a lot of lectures and got some understanding of what the experimenters were doing. They were still working very much on the spectroscopy, doublets, multiplet’s, and the relative intensity of the spectra lines and. things like that.

Kuhn:

Was Sommerfeld at that meeting or did you ever really get to know Sommerfeld?

Dirac:

I did meet him sometimes. I can’t remember whether he was at that meeting.

Kuhn:

What about Pauli?

Dirac:

I don’t remember meeting Pauli there.

Kuhn:

Is he someone you saw something of later?

Dirac:

In Copenhagen. He came to Copenhagen one of the times that I was there.

Kuhn:

How did you and he get on in your view of problems?

Dirac:

I think we got along pretty well. I think he understood my point of view pretty quickly. Quicker than most people.

Kuhn:

How would you have compared Pauli and Heisenberg as physicists and with respect to their view of your work and of the field?

Dirac:

Well, maybe Pauli was a bit more sympathetic than Heisenberg. Of course, Heisenberg has really accomplished more than Pauli by starting off the whole thing.

Kuhn:

You think of Heisenberg, as more than anyone else the key figure in these developments?

Dirac:

Yes, yes. Well, he introduced the non-commutation and even if you didn’t like it—

Kuhn:

Where do you place Schrödinger?

Dirac:

I'd put him close behind Heisenberg, although in some ways Schrödinger was a greater brain power than Heisenberg because Heisenberg was helped very much by experimental evidence and Schrödinger just did it all out of his head.

Kuhn:

What about de Broglie?

Dirac:

De Broglie must have had a very good imagination.

Kuhn:

Is that a compliment? .

Dirac:

Well, I meant that as a compliment, yes. But he’s perhaps not sufficiently critical. One needs to combine a strong imagination with a lot of criticism to be a good physicist.

Kuhn:

Oppenheimer indicates that when he was — the two of you were together in Göttingen he thinks you saw as much or more of him than anyone else there.

Dirac:

That is so. We sometimes went for long walks together although I had many walks alone.

Kuhn:

Was there anyone else there besides Oppenheimer with whom you might have done your walking?

Dirac:

Well, for a time there was Tamm the Russian. I met him first in Göttingen. There were a few others also. We once had an expedition in the Harz Mountains.

Kuhn:

Is your athletic interest pretty, much restricted to walking?

Dirac:

I took up rock-climbing a little later.

Kuhn:

Do you do things like tennis, golf, cricket?

Dirac:

No.

Kuhn:

Let me come back to the papers and again I think back to this first’ big paper that, comes in Copenhagen which is the transformation theory paper. Reading that now, it really does two things that are sort of obviously intellectually, closely related but very likely have separate origins, one of them being the transformation theory itself and the other one being the interpretation. I wondered how those things came together for you. It’s hard to know for example how one could be looking for an interpretation, but it’s hard to see how one would be looking for a transformation theory.

Dirac:

Well, I was for a long time trying to find something to replace the action and angle variables and the transformation theory of the classical Hamiltonian theory. I think I was working on that right from the beginning.

Kuhn:

Well, you thought for some time you had a replacement for the Hamiltonian theory in the uniformizing variables.

Dirac:

Yes. I wanted some equations that would correspond, that would have the same power as the transformation theory of classical theory.

Kuhn:

There’s a very curious remark in a paper slightly earlier than this in which you point out that if p,q is a set of canonical variables, q-numbers, then the ones you get by applying the transformation bp b will also be canonical variables and you then say that it doesn’t look as though this is much use. This is curious to me because this comes in a paper in which you cite the Born-Heisenberg-Jordan paper where they develop the powerful if not necessarily very rigorously developed transformation theory out of exactly that.

Dirac:

The perturbation theory, I believe it was.

Kuhn:

The perturbation theory, excuse me. They develop a perturbation theory but they call this thing the parallel to the classical transformation theory. That is they write sps-1 and call s the equivalent of the canonical transformation function, and then they proceed to expand it. They use it to develop a perturbation theory.

Dirac:

I expect I was a bit disappointed with the transformation from p to sps1 because it doesn’t have the same power as the classical transformation. I would explain it now by saying that the new variables have the same eigen values as the old ones. There is no such corresponding restriction in the classical theory. .. . .

Kuhn:

You think it likely that the thing which led you to the transformation theory by one route or another was the attempt to find something that possessed the power of the old Hamiltonian technique, transformation technique, that fitted quantum mechanics.

Dirac:

Something that was equally general.

Kuhn:

And you didn’t feel that the Born-Jordan-Heisenberg paper had that generality.

Dirac:

Just going from p to sps-1? That would not have the same generality Because it leaves eigen values invariant.

Kuhn:

Do you have any recollection how these two rather different components of the paper come together? Do you see what I mean by the two rather different components?

Dirac:

Yes. No, I don’t know which of them came first, if that’s what you mean. Probably the ideas came in somewhat the same way they were set down in the paper.

Kuhn:

There’s one part of that paper that I find extraordinarily difficult to read with understanding. At least I’m conscious of a problem. You remember you speak of a function g of two canonical variables and rj you you then talk.—

Dirac:

I think I’d better get the paper.

Kuhn:

O.K. As a matter of fact, before I even open this, why is this the physical interpretation of quantum dynamics whereas everything else has been made mechanics before?

Dirac:

Perhaps I felt I’d gotten closer to classical dynamics. I think that was it.

Kuhn:

The place that it really enters is in physical interpretation of matrices. I think I probably have to go back. You have a function g, the two canonical variables; you point out in the introduction to the paper that you can fix classically the value of both variables; in quantum mechanics you can fix the value of only one variable; the other one will then vary and what you presume — what you’re going to say now is if you fix the values of tj you get the average value of the function g(f,’) for a given value of over the whole of r{-space. Now in the discussion that follows on this, in the first place, it isn’t at all clear to me that I know what you mean when you speak of having an ft-space once the values, of are fixed. And you go over here to — you make use of this sort of notation where g has now become a c-number with particular values although in some sense g is still a q-number variable.

Dirac:

Well, this is the transformation function, isn’t it?

Kuhn:

Yes, here it is, but here it isn’t. That is, this g is presumably still a q-number. So that the delta is a q-number, yet the limits of integration — it works out very nicely.

Dirac:

I’m consistently using the prime to denote the c-numbers and the thing without a prime to denote a q-number. And this then is just the transformation function.

Kuhn:

Here, we’re all right. It’s up here that because of the fact that the values of 4 can’t be fixed to a numerical value with already fixed to a numerical value; yet with the ‘s fixed t has a range of values which somehow or other now becomes c-number values for purposes of the integration yet I can’t see how that enters. I follow what you do.

Dirac:

Isn’t there just a probability distribution for the r? Do I talk about a probability distribution lab that stage?

Kuhn:

You talk about the fact that you’re going to come out with a probability distribution. But you’re still working toward the probability distribution rather than having it. So that you seem here to be thinking of —. I understand this particularly easily if for example I suppose that the ‘S are fixed c-numbers we’re going to think of the ri’s as ranging over c-number variables but that seems on the other hand quite in conflict with the whole approach you’ve been developing. Although it may be the way you think about the problem here.

Dirac:

Well, it’s just a way of leading up to the probability interpretation, isn’t it?

Kuhn:

Surely it is, but it has seemed to me out of keeping with the fundamental approach that you’ve been developing. Because it does seem to demand that the which you insist properly cannot be pinned down once the epsilon is pinned down, still is allowed explicitly to range over the c-number domain whereas1tIe darn thing seems to me ought to be a q-number.

Dirac:

It has a distribution of values which one can understand very well with the probability interpretation. I suppose that was just the first idea for the probability interpretation.

Kuhn:

Let me then simply point to this very last paragraph which presents issues and a statement which without being the Uncertainty Principle leads so closely into it that I just wonder if you remember at all the sort of discussions that led into that.

Dirac:

You are asking what effect this work had —.

Kuhn:

Well, I ask this against the following sort of background, that Heisenberg —. This paper is done in Copenhagen in the fall of ‘26, during ‘26-26, Heisenberg and Bohr by everyone’s testimony are constant worrying and talking together about the interpretation; out of this comes a paper which seems to be drafted in February or thereabouts by Heisenberg on the Uncertainty Principle and which is full of transformation equations and of course Bohr’s complementarily paper is a direct response to that. The issues raised by that last paragraph have to be central to that (whole debate). It’s conceivable, but hard for me to believe, that you were uninvolved with it. I mean this is the first place in the literature where I run head on into the problems.

Dirac:

I was no longer in Copenhagen in January and February.

Kuhn:

No, clearly I take it you were not there when those final papers emerge but there seems to have been a discussion going on the entire year on these issues.

Dirac:

Is your question what effect this paper had on that discussion?

Kuhn:

What effect it had on the discussion or whether looking at this, remembering this, and remembering Copenhagen bring you back any part of that discussion whether this would have an effect or not.

Dirac:

I expect I gave a talk about this at the Colloquium. I'm pretty sure I did. Probably I didn’t present it well enough for them to appreciate what I’d done and they still felt they had to work a good deal on it.

Kuhn:

Well, isn’t there a sense in which the formulation that emerges from this is different from the one you give here in that you can have uncertainty, or probability distributions, for both of the variables, and then contain them within the region h.

Dirac:

Well, here I’m just supposing that one set of variables is well-defined and the other set.

Kuhn:

And you get averages for the other.

Dirac:

Yes. Yes, I see that this is more restricted in that way.

Kuhn:

But you don’t remember delivering it or talking about it.

Dirac:

I don’t, no.

Kuhn:

We’ll go on to the next paper which is the theory of emission and absorption of radiation, which comes very shortly after this one.

Dirac:

Is that with the second quantization? .

Kuhn:

Yes. You don’t call it second quantization. How do you feel about that name?

Dirac:

Other people introduced that terminology.

Kuhn:

Yes: I’m not quite sure when it first comes in.

Dirac:

I remember the origin of that work was just playing about with equations. I was intending to get a theory of radiation at the time. I was Just playing about with equations and —.

Kuhn:

Really. With what equations?

Dirac:

Well, with the Schrödinger equation. I got the idea of applying the quantization to it1 and worked out what it gave and found out it just gave the Bose statistics.

Kuhn:

Presumably you would have been fooling around with the Schrödinger equation and with an external radiation field as a perturbation.

Dirac:

Well, in the first place, just with the Schrödinger equation itself, seeing what happens when you make the wave function into a set of non-commuting quantities, quantities that don’t commute with their conjugate’ complexes..’ I remember that was the approach that I had for that problem.

Kuhn:

But in the paper itself, you do, of course, suppose a perturbation from the start and there are really again two approaches in the paper with which you’re really concerned to contrast with each other. In one of them, a particular radiation field is to be specified by variables N and 0 , which however you get directly out in terms of the Fourier components of the expansion of the field, but there’s no’ second quantization here. In the other one, variables with the same name, N and 9 are to be taken as the most probable number of systems in various different states and you get a close parallelism between the results of these, provided you suppose that your assembly obeys the Bose-Einstein statistics.

Dirac:

I’ll get the paper again.

Kuhn:

Good.

Dirac:

I suppose the main point was to show that this set of oscillators is identical with a set of Bosons. I got it first by playing about with the Schrödinger equation, in that it led to a set of oscillators, one for each point in the domain of the wave function. . . .

Kuhn:

When you say playing with it, you mean first introducing the idea of the non-commutatively —

Dirac:

Non-commutatively between V and g just to see what it would give.

Kuhn:

You’re probably dead right. I have to say my initial reaction is one of skepticism because of the -a. That sounds terribly to me what one would say after this gets to be called second quantization and one really begins to think of these as non-commuting wave’ functions. But in practice in this paper, the things that don’t commute never really emerge yet as wave functions. This gets to be what is called second quantization, but, there’s an important sense in which I want to say it isn’t yet second quantization in the paper.. The things that don’t commute are initially the a(t) in an expansion of a time dependent wave function. They’re expansion coefficients, rather than the wave function itself. Now it’s perfectly true that the wave function can be represented ‘by the’ sum of these. . .. .

Dirac:

I think that’s just to, have a discrete set of numbers instead of a continuous range• I think the only purpose of the expansion is to make the, degrees of freedom discrete.

Kuhn:

Let’s just see what happens if we try letting the thing not commute with its conjugate?

Dirac:

Yes, I’m pretty sure that was the starting point of this work, and the application to radiation was just what turned out, without its being expected.

Kuhn:

Well, one’s natural inclination I think in first looking at this idea of something’s not commuting with its conjugate, is that a complex number after all determines its complex conjugate, and that therefore the sort of uncertainty implied in non-commutatively oughtn’t exist in that case. Now, in your book later you’re very careful about introducing the sense in which you can’t treat a wave function as having a determined real .and having; a determined imaginary part, but nothing of that sort is introduced here. What may be implied is some sort of worry about the role of the complex nature, the complexness of the wave function or a concern of that sort.

Dirac:

I don’t think I was worrying about that complexity. I was just making the wave function into q-numbers. I had by then grown very familiar with the possibility of passing from c-numbers to q-numbers.

Kuhn:

Were you concerned about the physical meaning of the undetermined phase?

Dirac:

I did worry about it for a bit, I believe.

Kuhn:

You again exploit that in the paper on quantized singularities and I wondered whether this was an older —. Do you know when you worried about that?

Dirac:

Probably from the start of the transformation theory. The phase was always something more unphysical than the amplitude of the oscillations

Kuhn:

Well, does it make any sense to suggest to you – I’ve got no notion whether this works out or not, but again that seems to me to be a root — that the whole indeterminateness of the phase might very well suggest non-commutatively of the wave function and its complex conjugate

Dirac:

No, I don’t think so, I don’t think so.

Kuhn:

No? O.K. Thinking of the wave function as a q-number doesn’t to me do anything to suggest that it might not commute with its complex conjugate.

Dirac:

You have to find something it doesn’t commute with. If it commutes with everything then it’s no good counting it as a q-number.

Kuhn:

I think you. That’s very helpful. How did this fit with what you said earlier about when —. No, I’m sorry, I’m — as soon as I remembered what you said earlier it was at the Hilbert-space point that you give up the notion of q-numbers. So you would presumably still have it here. Some of the papers that come right after yours by other people suggest that this whole process of second quantization is very much like quantization of the electromagnetic field, Here you’re quantizing a -fie1d and one may succeed in getting the electron out of this process rather as one gets the photon out of field quantization. Did you have these ideas at all at the time

Dirac:

I think I had the idea that they could be applied to the electromagnetic field but I can’t remember having definite ideas about the electron. It was always a Coulomb field which doesn’t lend itself to this sort of treatment. And still doesn’t,

Kuhn:

This paper is really the first one in which you do quantize the field itself — the electromagnetic field here

Dirac:

This one, yes.

Kuhn:

I’m curious to know — you said when you talked about the earlier paper, the Compton effect paper, in which you don’t quantize it that probably at that point you didn’t have it in mind to take a next step, which would somehow be a more full quantization, by quantizing the field. Do you have any notion at what point you would set the idea that these quantities must be also treated as q-numbers?

Dirac:

I can’t remember just when, no. It must have happened sometime intermediate between those two papers.

Kuhn:

I realize that this was a long time ago; I keep asking questions because with some of them things come back. The fact that they don ‘t always is something I know from past experience is bound to happen. You speak at the end of this paper of conversations with Bohr as it was going on. In some papers you acknowledge help with writing and here you acknowledge discussion. What sort of a participant in discussions of this paper is Bohr likely to have been or do you remember his having been? In some ways this seems to me to be a particularly un-Bohrian paper.

Dirac:

He probably asked questions about it which I answered and he talked over certain things. I don’t think it went farther than that.

Kuhn:

Do you remember at all how he reacted to it?

Dirac:

Probably with further questions.

Kuhn:

Were the sort of questions he as1dd with respect to a paper of this sort searching questions that helped you formulate it and helped you with your thinking about it?

Dirac:

They could be, yes. They would b searching questions, I think.

Kuhn:

I ask that only because this paper is so mathematical, so much of what happens here is in terms of the mathematics.

Dirac:

Yes.

Kuhn:

It doesn’t lend itself to physical visualization easily and I —.

Dirac:

I shouldn’t think Bohr would ask much about the mathematics.

Kuhn:

But then with respect to this paper what do you suppose he would ask about?

Dirac:

I can’t remember. This one on the electrodynamics he might ask about the fluctuations in the field, or something like that. He might very well take something which I hadn’t thought of at all and ask what bearing it would have on that subject.

Kuhn:

I’d be terribly interested to know if you remember any questions being asked as early as this about a problem that emerges in the literature only a full year or more later. This is the question ‘of whether spin has any meaning for a free electron. I think this question only gets published by Nevill Mott who gives in an appendix arguments which he attributes to Bohr and this is in a paper that comes after your electron wave equation paper. . .

Dirac:

I remember Bohr was very much concerned with that question.

Kuhn:

Was he concerned with it before your relativistic wave equation paper?

Dirac:

I believe mainly afterwards. He was wondering how real, one should think of this thing as being.

Kuhn:

Clearly that’s a question that poses itself very strongly after you get spin without putting spin in. So that the question I mean to be asking is whether that really existed as a question before —.

Dirac:

I can’t remember that being a question beforehand. He was always very much concerned with the possibility of measuring things, and if it turned out very difficult to measure he would wonder whether it really exists.

Kuhn:

It’s quite a long time between this paper and the dispersion theory paper which is an immediate follow-up and application of this to the paper with the relativistic wave equation. Had you been working toward the relativistic wave equation all the way through? It’s pretty much an intervening year.

Dirac:

I don’t think I was working on it all that time.

Kuhn:

You told me before, I believe, that what you were really working toward was a linear form, one that would preserve transformation theory.

Dirac:

One that would avoid negative probabilities and preserve transformation theory.

Kuhn:

Tell me what you mean by negative probabilities. I wonder where the Klein-Gordon equation gave them.

Dirac:

Well, if you use it for just a single particle and you say the probability of the particle being in a certain place is given by - p sf’, there you have something which can be negative.

Kuhn:

I don’t know that part of the literature as well as I should, but I haven’t seen that referred to.

Dirac:

People nowadays try to get over it by saying that this quantity is the charge density rather than the probability of the particle. being at a place. If you’ve just got one particle present, then of course the charge density is just a measure of the probability of the particle being there.

Kuhn:

Unless you suppose the particle changes its sign. But the whole notion of getting spin into the picture was not part of that effort,

Dirac:

It wasn’t a primary part of the effort.

Kuhn:

Had you realized in advance, had you conceived in advance, that one might have spin dropping out in the way it does?

Dirac:

Not very much in advance. Perhaps when I was beginning to play with this scalar product of sigma with p. I was thinking of spin then.

Kuhn:

I take it that at a fairly early point you reach the conclusion that for the relativistic wave equation you wanted a linear form, before you’d gotten to the point of playing with sigma dot p.

Dirac:

I don’t know that I was as positive as that, I just felt that the quadratic form was unsuitable,

Kuhn:

The decision to take up this direction is I take it then quite unrelated to any thought at that point that you may get spin coming there.

Dirac:

Once one is certain that one has to have a linear equation, the whole thing is trivial.

Kuhn:

Was it? I take it that having had the Pauli theory, the Pauli formulation before makes it much easier to allow yourself a multi- component wave function,

Dirac:

Well, it suggests a two-component wave function; it doesn’t suggest going beyond that.

Kuhn:

No, but I suppose in some sense that it’s a bigger step from one to two than from two to four. It gives you an index.

Dirac:

But one doesn’t see the need for these extra components when one is just trying to solve the problem of getting rid of the negative probabilities.

Kuhn:

When you started working on it as a linear equation, you say once you see the need for that the rest is trivial. Is it?

Dirac:

Well, if you see it has to be linear in the components of momentum as well as in the energy, then it’s trivial. You just put in arbitrary coefficients and see what properties the coefficients have to satisfy.

Kuhn:

But did the manipulation of the coefficients and relating them, finding convenient forms for them and relating them to the Pauli spin-matrices, come pretty easily?

Dirac:

Well, finding a need for them, for these coefficients, to anti- commute, that came quite easily. That was an immediate, generalization of the anti-commutation of the three sigma’s. And you always have to have their squares equal to 1. So you’ve got all the algebraic properties of those coefficients.

Kuhn:

And from there to the actual matrices, where you choose the representations, that came

Dirac:

Well, you just have to accept the idea that four components are necessary.

Kuhn:

But that was all comparatively quick and easy.

Dirac:

Yes.

Kuhn:

Do you remember at all other approaches that you had taken before you resolved to write the thing down in a linear form?

Dirac:

No, I don’t think so.

Kuhn:

There is something you do in the first derivation of the equation that I wondered whether you had worried about at all, You set the thing up in the absence of a field and you make use of the homogeneity of space to say that the coefficients will be independent of space and time coordinates, and then from there. you derive the anti-commutative properties and so on. You then assume that they will hold their form when you put a field on. I mean this is just taken for granted without comment, whereas it at least be open to you to worry about whether the presence of the field would not destroy the homogeneity of space which you’ve used in the older argument?

Dirac:

I suppose the only way to answer that was by trying an example and I told you how anxious I was when I first tried it on the hydrogen atom.

Kuhn:

But was that a particular source of anxiety? See you also talk about these new -.

Dirac:

The source of anxiety was whether it would be possible to bring in the field in such a trivial way, just following the classical analogues. It might have been a completely wrong way to bring in the field.

Kuhn:

Here is a place where you could have had a picture, a model. I mean you talk about the alphas as having to do with an internal degree of freedom of the electron. Or at least this possibility. Did internal degree of freedom have any pictorial meaning for you or did you try to give it one?

Dirac:

I didn’t at that time try to figure out the properties of the alphas. I think that was first done by Schrödinger, sometime later, when he got the Zitterbewegung. My first concern then was whether this was the correct way to bring in the electromagnetic field. That was the more important problem.

Kuhn:

There the guidance from classical theory is overwhelmingly strong. You simply do the substitution that one’s been doing — well, not for so long. But you thought if it would break down it would break down at that point.

Dirac:

It might very well break down there, yes.

Kuhn:

That’s interesting. Because that’s no longer the bothersome step,

Dirac:

Well, if you have a theory which works all right in the absence of a field, it’s a very useless theory if it doesn’t work with a field also. Of course, one had no idea whether it would give even an approximate right answer, because other things were so different. I suppose you don’t altogether appreciate the anxiety one has when one is trying out a new idea.

Kuhn:

I think, as a matter of fact I do. I recently published a book that is still creating anxiety. Let me say immediately that I don’t mean to compare the product, but the anxiety can be invoked by —.

Dirac:

You’re bringing out a new idea, are you?

Kuhn:

I think. Yes. It certainly has felt new to me repeatedly in the course of —.

Dirac:

You must be pretty well confident in it if you put it in a book,

Kuhn:

Well, I ‘m working in a field in which the book often has ‘a role that the article will take in physics, because it’s harder to cut things up into article length pieces. But ,that’s a function really of difference in field. Do you remember any reactions to this paper on the electron equation. ‘Was this hard for people to —?

Dirac:

No, I think they took it up very quickly. It was probably not so hard as the physical interpretation.

Kuhn:

Had people complained about the physical interpretation?

Dirac:

I don’t think they complained. You were saying how people worked on it a good deal after it was published and published their own versions and if they felt the need to publish their own versions it meant they were not satisfied’ with mine. While here with the electron they were all satisfied with my version.

Kuhn:

What about the negative energy problem? How did that evolve from here? Well, that was a problem right from the beginning.

Kuhn:

Your next contribution to it is two years later.

Dirac:

I felt that writing this paper on the electron was not so difficult as writing the paper on the physical interpretation. That really needed more concentrated and sustained thought to get the ideas clear. This paper on the electron was more accidental and once you got the right road it jumps out at you without ‘any effort, It didn’t need a sustained effort which the physical interpretation needed; gradually changing it from one form to a clearer form and then to a still clearer form. The negative energies were a problem right from the beginning but one just couldn’t do anything about them.

Kuhn:

Does that partly account for the fact that after this paper, you turn over to apparently quite different problems for a while. The papers that come after this are for one thing back in that category directly in taking up a problem in the literature and doing it more carefully or better. I think the next one is the quantum statistics, the basis of statistical quantum mechanics and then you do the quantum mechanics of many-electron systems.

Dirac:

Yes, I just felt I couldn’t make any progress with this. Other people were thinking about the negative energies. Oppenheimer put forward the idea that nature just arranged it in such a way that negative energies were always occupied. No, that came later. Yes, of course, one had to get the statistical ideas clear before one could find the proper interpretation for the negative energy states.

Kuhn:

Has that got anything to do with the fact that you turned to statistics yourself? It’s perfectly clear when you come back to the negative energy states, you use some of the statistics you and other people have been doing in the interim.

Dirac:

Well, it was just that I couldn’t make any progress with this equation.

Kuhn:

But you went to that as another problem, not as one you thought likely to help you with this one.

Dirac:

No, another problem.

Kuhn:

The sense in which you use that material somewhat when you come back is a nice dividend but not one you had calculated on.

Dirac:

Yes, yes.

Kuhn:

At the start of the paper on ‘Professor Dirac’s group theory’, the many electron paper [Number 23] — you describe what has been happening in the field and talk then about the quite elaborate group theoretical papers that have been appearing and the problem that is presented if you have to learn all that abstract mathematics in order to do it. Then you make the remark which I think actually I probably copied -.

Dirac:

Shall I get the paper? Will that help?

Kuhn:

All right, fine. We’ll take it out of that. This is not a deep or technical point but it —. [Examining paper] The remark I’m after is the one beginning with the word “now”, [reading]”. . . group theory is just a theory of certain quantities that do not satisfy the commutative law of multiplication and should thus form a part of quantum mechanics which is the general theory of all quantities that do not satisfy the commutative law of multiplication.”

Dirac:

I could have been more specific by saying that the group theories have certain special quantities which satisfy other special properties besides the non-commutative law of multiplication.

Kuhn:

What strikes me as odd in this statement and therefore may make it particularly illuminating —. I would have conceived of group theory as a mathematical theory and of quantum mechanics as a physical theory.

Dirac:

Well, there’s a close connection in the equations. They use the same kind of equations.

Kuhn:

But in this sense I would think of quantum mechanics as the more special; in so far as they’re mathematical, the extra definiteness of the quantum mechanical equations, such as the use of the Hamiltonian, would make this the more special.

Dirac:

Well, the group theory deals with certain special quantities which have reciprocals. A good many things in quantum theory don’t have reciprocals, those that have the zero eigen-value.

Kuhn:

Excuse me. I totally missed that point. This is what you had in mind in putting the point this way.

Dirac:

Probably. The group theory does deal with a special kind of non- commuting quantity.

Kuhn:

Am I right in supposing that the idea for negative energy states as holes comes pretty close to the time when you submit the paper on it? [Paper 24]

Dirac:

Yes. I don’t think there would be any cause for delaying it.

Kuhn:

That paper starts by pointing out among other things that Weyl has suggested a possible relation of the proton and you then talk about the reasons you can’t just say that it is a proton, the negative energy state is a proton.

Dirac:

I don’t remember that remark of Weyl.

Kuhn:

Well, I’m not quite sure where it is, but you say — I think I’ve got it almost verbatim -e that this characteristic that the negative energy state can be looked at as an inversion of the sign of charge has led Weyl and others to relate it to the proton. Then you proceed to say that we can’t simply say it is the proton because this introduces paradoxical properties that you mention.

Dirac:

I don’t remember Weyl coming in at that stage.

Kuhn:

Well, he comes in a little later in the second edition of Gruppentheorie I know. But apparently whether he’s come in in a way that helps form your own ideas or whether he’s come in only after you’ve already played with the proton idea ee had you played with the proton idea significantly by yourself before you went to the hole idea?

Dirac:

No. No I hadn’t gotten the connection between the proton and the electron.

Kuhn:

But you know that paper starts out by saying it’s been proposed and it won’t work and then —.

Dirac:

Did I say that someone had proposed it? Or did I say that someone might propose it?

Kuhn:

That paper I might have with me.

Dirac:

I’ve got it here. Yes I see I give a reference.

Kuhn:

Yes, that’s right.

Dirac:

Well, one could just look up that reference and —

Kuhn:

There’s no problem about looking up the reference and seeing what Weyl said; I’m really wondering how this suggestion struck you, and what role it had on the road to the hole theory.

Dirac:

Well, the suggestion by itself wouldn’t lead to anything. There were several people I suppose that had the idea that there, was some connection but it just didn’t lead anywhere. One had to get some quite different idea bringing in the statistics in order to get a workable theory.

Kuhn:

What do you mean in this case by bringing in the statistics?

Dirac:

Bringing in the Exclusion Principle.

Kuhn:

Right. Granting that that’s quite a different theory, the whole manner in which it develops, it does look rather from this paper the notion of getting the proton in somehow or other had had an appeal; but there had been these problems that had to be gotten through if one was going to interpret this at all in terms of another particle.

Dirac:

.At that time I was more concerned with getting a satisfactory theory of the electron. than in bringing in the proton. I would have been quite happy have a satisfactory theory of the electron by itself.

Kuhn:

Wouldn’t it have been awfully natural then to take Oppenheimer’s idea and just fill all those states?

Dirac:

But then you disturb them and they don’t stay filled. You can start off with them all filled but they don’t stay all filled.

Kuhn:

How did you feel about this idea?

Dirac:

Well, the great worry then was the difference in the masses. At that time I thought that a positive particle with the mass of the electron just couldn’t exist, otherwise experimenters would certainly have discovered it. If my theory predicted such a particle, then my theory was wrong. In order to save the theory, it would be necessary to find some cause for the extra mass of the positive particle. I was hoping that in some way the Coulomb interaction might lead to such an extra mass but I couldn’t see how it could be brought about.

Kuhn:

Well, now you also compute annihilation probability in a paper just after this one and you get up to an infinite annihilation probability depending upon relative velocity.

Dirac:

Yes. That was a further difficulty.

Kuhn:

And then very quickly you get the Weyl proof that the masses must be the same.

Dirac:

Yes, I really felt that it should be the same but I didn’t like to admit it to myself. Maybe that was my attitude.

Kuhn:

Well, does this mean you had pretty well decided that this was a bad idea?

Dirac:

I thought it was rather sick, yes. I believe I told you. I wouldn’t say it had to be given up altogether, but it was rather sick.

Kuhn:

Somebody, and I can’t remember who, said he once asked you how you felt about the theory in the period just before the positron was discovered and you said you’d, about given it up by then.

Dirac:

I didn’t see any chance of making further progress.

Kuhn:

Did you still feel there was something fundamentally right about it? I mean that this notion of holes was the direction —?

Dirac:

Well, I think it was better than saying nothing at all about the negative energy states, which one would have had to do if one hadn’t got this idea of the holes. It was a small step forward.

Kuhn:

In the paper on quantized-singularities [Paper 29, 1931] which is about something else, you use this theory in the introduction as an example of an approach through the mathematics, rather than through observation, and there you do say that we’ve got to give up the notion that it’s a proton, but maybe it is a so-far unobserved particle.

Dirac:

Was I so explicit?

Kuhn:

Yes.

Dirac:

I don’t remember that.

Kuhn:

I’ve got that one here. The later papers I have not been through before I got here. So those I’ve got along with me.[reading] “A hole, if there, were one, would be a new kind of particle unknown to’ experimental physics, having the same pass and opposite charge to an electron. We may call such a, particle an anti-electron. We should not expect to find any of them in nature on account of their rapid rate of re-combination with electrons but if they could be produced experimentally in high vacuum, they would be quite stable and amenable to observation. An encounter between two hard gamma rays of energy, of at least half a million volts could lead to the creation simultaneously of an electron and an anti-electron, the, probability of occurrence of this process being of the same order of magnitude as that of the collision of the two gamma rays on the assumption they are spheres of the same size as classical electrons. This probability is negligible, however, with the intensities of gamma rays at present available.”

Dirac:

Yes. I hadn’t realized that the probability was very much greater if you just have one gamma ray hitting a nucleus, hitting a coulomb field.

Kuhn:

Does that sort of statement, so far as you know, bear any close connection, to Blackett’s work? Pretty clearly it doesn’t to Anderson’s.

Dirac:

I don’t think so. I think the experimental people go on their way, pretty well independently. But you could ask Blackett about that.

Kuhn:

Did you see anything of him?

Dirac:

Yes. He was in Cambridge at that time. I saw him quite a bit.

Kuhn:

He was, among the experimentalists, one of the people who paid relatively more attention to theory.

Dirac:

Yes, but I think he would agree that the probability of observing the results of a collision between two gamma rays would be too small to be worth bothering about.

Kuhn:

No, I don’t suggest that he set up the experiment for this reason, but there’s surely a preparation in this remark for one’s stopping talking about these things falling back into the source where you see a pair of divergent tracks.

Dirac:

Well, you could ask Blackett about it.

Kuhn:

In fact Heilbron has. I have not seen the transcript of that talk It often helps, particularly when one is dealing with things this far from the event, to get reactions from more people than the one who was himself most directly involved. How did the news of the positron get to you?

Dirac:

Probably Blackett told me. Blackett would have heard of it at once from Anderson, and I expect he told me.

Kuhn:

Was it a great vindication?

Dirac:

Yes.

Kuhn:

Does that sort of event generate great immediate excitement and satisfaction?

Dirac:

I don’t think it generated so much satisfaction as getting the equation to fit.

Kuhn:

Here, in a sense, you had been deprived of that satisfaction by the non-existence of the particle previously. You were able to get it back when the particle finally appeared.

Dirac:

Yes. I think I really got more satisfaction from the transformation theory.

Kuhn:

Which of these papers is your own favorite? I mean you don’t have to pick just one.

Dirac:

I think the transformation theory was the best one in the sense that I was really working for the result. In other cases I got results unexpectedly, starting off with something different, with different ideas. But in the transformation theory I was working for the result.

Kuhn:

And you think the sense of working for that result goes back to the beginning of the idea of getting this into a fully Hamiltonian form.

Dirac:

I suppose that it goes back to my first reading Heisenberg’s paper.

Kuhn:

And it’s part of that same effort that leads you to put that in Hamiltonian form. This would seem to be a good place for us to stop.

Dirac:

Just one thing — about that Kapitza Club. I was wondering whether maybe I did speak at my first meeting and maybe they asked me to come to the meeting for the purpose of speaking on that subject. That’s a possibility. There’s a possibility that I got my membership by earning it, so to speak, by giving them a talk in the first place.

Kuhn:

That could perfectly well be. I’ve got no notion what that represents, but I wanted to show you this close juxtaposition and just see whether it brought anything out. Unfortunately, although there are annual lists of members, the members for a given year, there’s no indication of when people are elected.