In the cases where mathematical expressions appeared in the essays, I have used TeX notation.

Prof. Gottlieb

Now, there can never be two fermions in the same state, so that if we take an operator of emission of a fermion, applied twice over, we must get zero. Calling these emission operators $\eta$, $\eta^2$ is equal to zero. Similarly, the absorption operator, $\tilde\eta$ let us say, is again an operator that we cannot apply twice over without getting zero; so $\tilde\eta^2=0$. It is a bit strange to have these operators with zero squares, but there is nothing wrong with it mathematically.

We have relationships connecting the $\eta$'s with $\tilde\eta$'s that are formally very similar to the corresponding relationships between boson emission and absorption operators. The only difference is in the sign of some of the terms in the equations that express the basic commutation relations.

When I first heard about this work of Jordan and Wigner, I did not like it. The reason was that in the case of the bosons we had our operators that were closely connected with the dynamical variables that describe oscillators. We had operators that had classical analogues. In the case of the Jordan-Wigner operators, they had no classical analogues at all and were very strange from the classical point of view. The square of each of them was zero. I did not like that situation. But it was wrong of me not to like it, because, actually, the formalism for fermions was just as good as the formalism I had worked out for bosons.

I had to adapt myself to a rather different way of thinking. It was not so important always to have classical analogues for everything. What we really needed was to have dynamical variables satisfying commutation relations or anticommutation relations that would be consistent with one another. In terms of such dynamical variables, one can proceed to build up a reasonable quantum theory quite independently of whether or not there is a classical analogue. If there is a classical analogue, so much the better. One can picture the relationships more easily. But if there is no classical analogue, one can still proceed quite definitely with the mathematics. There were several times when I went seriously wrong in my ideas in the development of quantum mechanics, and I had to adjust them.

There is, then, the possibility that holes may appear in the sea. Such holes would be places where there is an extra energy, because one would need a negative energy to make such a hole disappear. Also, such a hole would move as though it had a positive charge. It has an absence of negative charge; so in that respect, also, it appears as a positive charge. Thus the holes appear as particles with positive energy and positive charge.

When I first got this idea, it seemed to me that there ought to be symmetry between the holes and the ordinary electrons, but the only positively charged particles known at that time were the protons; so it seemed to me that these holes had to be protons. I lacked the courage to propose a new kind of particle. I should say that there were good grounds for belief at that time that there were only two particles, two basic charged particles--electrons and protons. There were just two kinds of electricity, positive and negative, and one needed one particle for each kind of electricity. In those days the climate of opinion was very much against the idea of proposing new particles. I certainly did not dare to do it; so I published my idea as a theory of electrons and protons, and I believed that maybe the difference in mass between the electrons and protons would come about in some way from the interaction between the electrons. But I realized the difficulties were enormous because the difference in mass was so great.

I was soon assailed by other physicists on the grounds that there could not be this difference between the mass of the new particles, the holes, and the mass of ordinary electrons. The person who most definitely came out against it was Hermann Weyl; he was essentially a mathematician and was not so much disturbed by physical realities but was very much dominated by mathematical symmetries. He said quite categorically that the new particles formed by these holes would have to have the same mass as the electrons, and I came around to that point of view. We all know the consequences of that. The new particles were given the name of antielectrons and were afterward discovered by the experimenters. The first was Carl Anderson one of the chief physicists whom we have to thank for that. Thus this question has now been resolved.

I should mention that it was, first of all. Louis de Broglie who got the idea that particles in general should be associated with waves. Einstein had already made this association in the connection between photons and waves of light. He had done this in 1905, long before, and de Broglie extended it to particles of all kinds. De Broelie's idea came entirely from the beauty of the mathematics that one got by setting up the equations in relativistic form.

De Broglie's ideas were for a free electron by itself, and Schrodinger extended them to apply to an electron moving in an electromagnetic field. As soon as he got his general equation. he applied it to the hydrogen atom. The result that he obtained was not in agreement with experiment, because Schroedinger did not know at that time about the spin of the eiectron. He was extremely disappointed by this failure. He told me about it many years later. He believed that the whole idea of his wave equation was wrong. He was terribly dejected, and he abandoned it altogether. Then it was some months later that he recovered from his depression sufficiently to go back to this work, look over it again, and to see that if he did it in a nonrelativistic approximation, so far as a nonrelativistic system was concerned, his theory was in agreement with observation. He published his equation then as a nonrelativistic equation.

You may wonder how it appears that Schroedinger's early papers were all nonrelativistic, although they were insplred by de Broglie waves and the de Broglie waves were built up from relativistic ideas. It was in this indirect way that it came about. Schroedinger lacked courage to publish an equation that gave results in disagreement with observation. He should have had that courage; he would then have published a second-order equation in $\partial/\partial t$, an equation that was later to be known as the Klein and Gordon equation, although it was discovered by Schroedinger before Oskar Klein and Walter Gordon and was the first wave equation that he worked with. But Schroedinger would only publish something that was not in direct disagreement with observation. People were rather timid in those days, I suppose, and it was left to Klein and Gordon to publish an equation which is now accepted as the correct equation for a charged particle without spin.

When this theory of Schroedinger's appeared. I was a bit annoyed by it because I did not want to be disturbed at all from the development of Heisenberg's ideas and from following up the analogy between Heisenberg's mechanics and Newtonian mechanics. It was wrong of me to have this hostility because Schroedinger's theory was shown by Schroedinger himself and others to be equivalent in its mathematical consequences to the Heisenberg theory, and Schroedinger's theory did provide new insights. It provided new directions for development which one would not have thought of just keeping to the Heisenberg theory.

There were some people thinking about electron spin in those days, but there was a lot of basic opposition to such an idea. One of the first was Ralph de Laer Kronig. He got the idea that the electron should have a spin in addition to its orbital motion. He was working with Wolfgang Pauli at the time, and he told his idea to Pauli. Pauli said, "No, it's quite impossible." Pauli completely crushed Kronig.

Then the idea occurred quite independently to two Young Dutch physicists, George Uhlenbeck and Samuel Goudsmit. They were working in Leiden with Professor Paul Ehrenfest, and they wrote up a little paper about it and took it to Ehrenfest. Ehrenfest liked the idea very much. He suggested to Uhlenbeck and Goudsmit that they should go and talk it over with Hendrik Lorentz, who lived close by in Haarlem. They did go and talk it over with Lorentz. Lorentz said, "No, it's quite impossible for the electron to have a spin. I have thought of that myself, and if the electron did have a spin, the speed of the surface of the electron would be greater than the velocity of light. So, it's quite impossible." Uhlenbeck and Goudsmit went back to Ehrenfest and said they would like to withdraw the paper that they had given to him. Ehrenfest said, "No, it's too late; I have already sent it in for publication "

That is how the idea of electron spin got publicized to the world. We really owe it to Ehrenfest's impetuosity and to his not allowing the younger people to be put off by the older ones. The idea of the electron having two states of spin provided a perfect answer to the duplexity.

All of this took place during my graduate studies. For us, the Dirac equation was a great puzzle, and we had difficulty grasping the significance of the successes mentioned earlier. It was much more mysterious to us than Dirac's transformation theory and his theory of radiation. Imagine a student who had just gone through the conceptual problems of ordinary quantum mechanics and who begins to feel not at ease but barely capable of dealing with Schrodinger wave functions and Heisenberg noncommuting matrices suddenly facing wave functions with four components and with strange transformation properties of which he has never heard before. It was somewhat discouraging.

Today it is hard to realize the excitement, the skepticism. and the enthusiasm aroused in the early years by the development of all the new insights that emerged from the Dirac equation. A great deal more was hidden in the Dirac equation than the author had expected when he wrote it down in 1928. Dirac himself remarked in one of his talks that his equation was more intelligent than its author. It should be added, however, that it was Dirac who found most of the additional insights.

Some new relativistic equations are needed; new kinds of interactions must be brought into play. When these new equations and new interactions are thought out, the problems that are now bewildering to us will get automatically explained, and we should no longer have to make use of such illogical processes as infinite renormalization. This is quite nonsense physically, and I have alwavs been opposed to it. It is just a rule of thumb that gives results. In spite of its successes, one should be prepared to abandon it completely and look on all the successes that have been obtained by using the usual forms of quantum electrodynamics with the infinities removed by artiticial processes as just accidents when they give the right answers, in the same way as the successes of the Bohr theory are considered merely as accidents when they turn out to be correct.