europhosycs BULLETIN OF THE EUROPEAN PHYSICAL SOCIETY news J.A. Volume 8 Number 10 October 1977 The Relativistic Electron Wave Equation P. A. M. Dirac
An abridged version of the lecture celebrating both the 75th. birthday of Professor Dirac and the 50th. anniversary of the publication of this fundamental equation of Nature, given on July 4 at the Conference on Particles Physics in Budapest. The full text will be published in the Conference Proceedings. Before the period that I want to The outline of Heisenberg’s method I was thinking over Heisenberg’s talk about, that is to say in the early was to set up a theory dealing with ideas, concentrating on the non-com 1920s, we had a period of frustration. only observable quantities. These mutation, and it occured to me rather We had the theory of the Bohr orbits observable quantities fitted into ma by accident that there was really a which worked very well for some sim trices, so he was led to consider ma great similarity between the commu ple problems, essentially for those pro trices, and he had the idea of consi tator of two quantities that do not com blems where only one electron was dering the matrix as a whole instead mute and the Poisson bracket which playing an important role. People of just dealing with particular matrix we have in classical mechanics. As a were trying to extend the theory to elements. Dealing with matrices one result of this similarity, the equations deal with several electrons, for exam is then directed to non-commutative in the new mechanics with non-com ple to the spectrum of helium, where algebra. mutation appeared as analogous to two electrons are concerned, but they Now it was really very difficult for the equations in the old mechanics of did not know how to do it. There were physicists to accept non-commutative Newton, when these old equations basic ambiguities in applying the rules algebra In those days. Heisenberg were expressed in the Hamiltonian of quantization and people did not himself had very grave doubts when form. On the strength of this analo know what to do. They could only pro he first noticed that his algebra was gy one immediately had a general ceed by making various artificial actually non-commutative, and he won connection between the old mecha assumptions and these assumptions dered very much whether he would nics and the new mechanics of were not very successful. not have to abandon the whole idea Heisenberg. Now this frustration is something because of the non-commutation. But That was the start of my work. It that one can understand again very still he found that It was unavoidable gave me a rather different handle well at the present time, because we and he had to accept it. from Heisenberg because I had the have a similar situation with regard I learned about this theory of non-commutation as the essential new to the relativistic quantum theory for Heisenberg in early September, 1925. feature. dealing with high energy particles. It was very difficult for me to appre Again we have this feeling that we ciate it at first, then I suddenly reali don’t know the basic rules. We know zed that the non-commutation was some rules which work only with a actually the most important idea that Contents limited degree of success and essen was introduced by Heisenberg. It was tially we are in a similar situation the one drastic new idea which would Relativistic Electron Wave where we don’t know what the correct provide the whole basis of any new Equation...... 1 basic assumptions are that we can theory which one was going to cons Precision Determination of hold fast to. truct. Working with his matrices, Atomic Masses...... 5 In 1925, the whole situation was Heisenberg was led to a new equation 9th European Conference on suddenly changed by Heisenberg who of motion for them, namely. Atomic Spectroscopy . . . 8 had a really brilliant idea. He was led Quantum Transport in to introduce the idea of non-commuta ih du/dt = uH — Hu, (1) Crystalline Semiconductors . 9 ting algebra into physics. This idea where u is some dynamical variable Public Attitudes to Science . . 11 was most startling and most unexpec and H is a diagonal matrix which re Society News...... 12 ted. presents the energy. turophysics News is published monthly by the European Physical Society. © 1977. Reproduction rights reserved.
1 The idea of bringing in non-commu physical importance. There I was was then just a question of a mathe tation proved to be the key to deve wrong. Schrödinger did take these matical transformation to pass from loping a new mechanics, which ena waves seriously. He thought that they the Heisenberg theory to the Schrö bled one to escape from the frustra really would be associated with the dinger theory. They were two mathe tion that had been holding us up motion of an electron in an atom, but matically equivalent theories for the during the previous years. The result one would have to modify the wave same underlying physics. That under was a period of great activity among equation somewhat to take into ac lying physics is what we now call theoretical physicists at that time, count the electromagnetic field in quantum mechanics. great excitement together with great which the electron was moving. We then had a satisfactory situa activity. There was so much work to He tried to guess a good way to tion of one good theory. The result of do developing the new ideas and modify this equation (2) of de Broglie Schrodinger’s work was to introduce a seeing how the equations of the old keeping to the requirements of relati new concept, the wave function Ψ, mechanics could be translated into vity. Well, he was able to guess this which was a great help for the physical the new theory. One could get new equation : interpretation of the theory. It was results very easily and one had great found that if you take Ψ and suitably confidence that one was really getting normalize it, then |Ψ2| gives the proba somewhere. One had the possibility bility of finding the particle in any of developing the new theory in a place. general way and also of applying in to (3) One had to get used to the idea examples and working out equations. that the new mechanics only gave one These equations involved non-com- This equation reduces to the pre probabilities and did not give one the mutating quantities and there was the vious equation (2) when you put the determinism of the previous classical problem of getting some physical electromagnetic potentials A equal mechanics. That was a feature which a interpretation for the results that were to zero. So far as I know it was guess lot of physicists found very hard to obtained with the new equations. This work of Schrodinger to obtain this accept, but which turned out to be problem of getting the interpretation equation from de Broglie’s equation. unavoidable when one had more proved to be rather more difficult than Now when Schrodinger had that power for understanding the results of just working out the equations. It was equation, the first thing he did, of calculations with the non-commutative not completely solved until two or algebra. three years after the original idea of course, was to apply it to the electron non-commutation was introduced. in the hydrogen atom. He worked out I was working on this and conside I don’t think it had ever happened the energy levels of hydrogen, and ring the problem of getting the proba in physics previously, that one had he got a wrong result because his bility for other dynamical variables to equations before one knew the equation did not take into account the have specified values. I worked out a general way to interpret them. But that spin of the electron. He went back to general theory for these probabilities. is what happened in this case. it a few months later and then noticed This general theroy enabled one to In the early examples one just had that if he was less ambitious and just transform the Schrodinger wave func special rules for interpretation. For wrote his equation in a non-relativistic tion to other forms. One then had the example, one had a matrix to repre way and then applied it, he got results possibility of calculating the probabi sent the energy, a diagonal matrix, in agreement with observation apart lity of any dynamical variable having and one said its diagonal elements from the fine structure of the hydrogen a specified value, or of several vari were the energy levels. That was just spectrum, which depends on the rela ables simultaneously having specified a special assumption giving us the tivistic corrections. In the non-relati values, provided they commute with energy levels, and it worked. vistic approximation Schrodinger’s each other. The method was to trans To get a general interpretation one equation reads like this, in the form the Schrödinger function to refer was helped by some other work that absence of a magnetic field : to these variables that one is intere was done independently by Schro- sted in, and again, to form the square dinger. Schrodinger was working quite of its modulus. independently of Heisenberg, and to I was able to work out this general begin with he knew nothing about (4) transformation theory and I felt very Heisenberg’s work. Schrodinger was After this limited success of Schro pleased with it. I think that is the working from an equation of de dinger, one had two quantum theories, piece of work which has most pleased Broglie. This was the wave equation. one based on the wave equation me of all the work that I have done in of Schrodinger and the other due to my life. It pleased me because it did (2) Heisenberg. not come from some lucky accident ; I know when I first heard about it came from logical thinking step by these two quantum theories, I felt a step, seeing each step giving rather De Broglie had proposed this wave bit annoyed. If we have one good theo more detailed knowledge and leading equation simply because he noticed ry, that is all we really want. This was on to the next question to examine and that there was an interesting connec rather too much, an excess of rich resolve. And in this step by step way I tion between its solutions and the rela ness. But it was very soon shown by was able to pass to a general theory. tivistic motion of a particle. If you as Schrödinger that the two theories There was just one bad feature of sume that pr stands for the three com are really equivalent to one another. this new theory ; it was not relativistic. ponents of momentum with po = W/C, You may write the Schrödinger equa It would not apply to particles moving then p Ψ corresponds to ih∂Ψ∂xµ. tion. with speeds comparable with the velo I had read this paper of de Broglie, city of light, because it was based on but did not take the waves seriously. (5) the Newtonian pre-relativity mecha I thought these waves were just a and then this H corresponds to the nics. The operator on the right hand mathematical curiosity without any matrix H in the Heisenberg theory. It side of (4) corresponds to the energy
2 in Newtonian mechanics and not ac mentum values. You could not do that cording to Einstein. This expression at all with the Klein-Gordon equation. The Faculty of Philosophy has to be modified for particles mo Similarly for other dynamical variables, and Natural Sciences of ving with high speeds. you cannot get any information at all the University of Bern, Switzerland, According to Einstein a theory about their probabilities. would like to fill a vacancy should be basically symmetrical bet So I continued to worry about this in the position of a : ween the time and the three space question till the end of 1927, and Full Professor coordinates. Now, you see that we do eventually the solution came rather by for Theoretical Physics becoming effective October 1, 1978. not have that symmetry. In (1) we accident, just by playing with the The working field can be defined have δ/δ t, but no corresponding δ/δxi, mathematics. I noticed that if you within general limits as either : δ/δx2, δ/δx3. in the Schö dinger equa take the matrices σ1, σ2, σ3 describing Physics of Elementary Particles, or tion (4) or (5) we again have δ/δt and the three components of spin for a Relativistic Astrophysics. no corresponding operators of differ spin of half a quantum as described Participation in the lecturing pro entiation with respect to the space by the general transformation theory, gramme at all levels in German is coordinates. So we had the problem to then if you form mandatory, although a reasonable modify the theory to make it relativis (σ1p1 + σ2p2 + σ3p3)2, period of adaptation could be con tic. sidered. The way most physicists tackled you get a very interesting result, just Candidates with corresponding ex that was to go back to equation (3), perience in research and teaching the extended de Broglie equation. This are invited to send their application is a relativistic equation. It was first (Curriculum vitae, list of publica discovered by Schrodinger and was You had thus a sort of square root tions, etc.) by November 1, 1977 to not published by him because it gave the : results not in agreement with obser Now I needed a corresponding Erziehungsdirektion vations for the hydrogen sepctrum. It expression for the square root of the des Kantons Bern was rediscovered independently by sum of four squares. One had to have Sulgeneckstrasse 70, CH-3005 Bern. Klein and Gordon and they did publish the sum of these three squares plus it undeterred by its disagreement with a mass term. One could not get an ral requirements that one could apply observation. So this equation is now expression for the square root of the the transformation theory to it, and known as the Klein-Gordon equation. sum of four squares just by working the requirements of relativity. It turned Now, this is a relativistic equation with these three σ matrices, (which out that the simplest particle satisfying and one can develop it relativistically. are called the Pauli matrices because those requirements is a particle with One can set up the expression he had built up the theory of electron a spin of a half. That was a great spin in terms of them). That was a surprise to me, I thought that simplest serious difficulty for me for some particle would naturally have a zero conjugate complex weeks, until I noticed that there is really no need to keep to two-by-two spin, and that a spin of a half would and can interpret it as the charge matrices like the σ’ s. One can go to have to be brought in later as a com density associated with any solution four-by-four matrices, and then one plication, after one had solved the of the wave equation. And one can put can easily get an expression for the problem of a particle with no spin. But down corresponding expressions for square root of the sum of four squares. it turned out otherwise. the current density to satisfy the re That led me to a new wave equation: I applied this equation to the elec quirements of relativity, and one finds tron in a hydrogen atom in the first that charge is conserved. Further, one approximation and got results in can put down expressions for the agreement with observation. This energy density and momentum density (6) equation automatically gives the cor and for the stress. These expressions rect magnetic moment, and that is why are all relativistic and in agreement involving these α’s, which are four-by- with the conservation laws. four matrices. They are required to Now most physicists were very satisfy certain algebraic conditions, as Swiss Federal Institute happy with this development of the a result of which the square of this of Technology Klein-Golden equation. They said, here operator is just (Lausanne-Switzerland) you have a good relativistic quantum One can modify this equation (6) to Two post-doctoral research posi theory. But I was most unhappy with bring in the electromagnetic field in the tions, one in theory and one in it, because you cannot apply the trans same way that Schrödinger brought experiment, will be available at the beginning of 1978 or earlier, for formation theory to it. For the trans in the electromagnetic field to the physicists interested in working on formation theory you need this equa de Broglie equation (2). The result is the properties of small clusters of tion (5) of Schrodinger, involving just an equation for the electron moving atoms and in interacting with Ph.D. the operator δ/5f, and not the square in the electromagnetic field, in agree students. For the experimental posi tion, preference will be given to of this operator, such as occurs in ment with the basic requirements of applicants having a good know (3). relativity and quantum mechanics. ledge of pulsed NMR or molecular I just had to worry over the problem It was found that this equation gave beam techniques. A two years of getting a relativistic theory which the particle a spin of a half a quantum. appointment is considered, with possible renewal. should be linear in δ/δt the operator δ/δt. And also gave it a magnetic moment. Applicants should submit résumé, The linearity in 8/ was absolutely It gave just the properties that one publication list and names of two essential for me, I just could not face needed for an electron. That was referees to : giving up the transformation theory. really an unexpected bonus for me, Service du personnel de l’Ecole You see with the transformation theory completely unexpected. Polytechnique Fédérale de Lausanne you could work out also the probabi At that time I only wanted a quantum 33, avenue de Cour lity of the particle having given mo theory which would satisfy the gene- CH-1007 LAUSANNE 3 it did not have the error which the It was pointed out by Pauli and drawn is that it is a bad theory. That I Klein-Gordon equation had in giving Weisskopf that one could get a similar have insisted on all along, but most the wrong results for the spectrum theory of several particles by working physicists are inclined to be rather of hydrogen. from the Klein-Gordon equation and satisfied with it and to work with it. There was one further difficulty left taking the expression for the energy There is some justification for doing with this equation, namely, it was quite density, which is : so, because at the present day one possible for the particle to have states does not have a better theory. of negative energy. I was well aware People have done an enormous of this negative energy difficulty right amount of work with this quantum at the beginning, but I thought it was electrodynamics, as it is called. They a less serious difficulty than the have noticed that, although attempts others, less serious than our not being to solve the wave equation always lead able to apply the transformations of to infinites, those infinites can be man the general transformation theory. Pauli and Weisskopf had the idea aged in a certain way. In particular, it This negative energy difficulty was of changing the Ψ and Ψ here into was shown by Lamb that the infini solved a little while later by my idea dynamical variables referring to emis tes could be removed by a process of bringing in the exclusion principle sion and absorption of particles, and of renormalization. Renormalization of Pauli for electrons, the principle using the total energy, which is means that you assume that your para that one cannot have more than one the integral of this expression over meters e and m occurring in the orig electron in any state, and making the three dimensional space, as the Hamil inal equations are not the same as rather bold assumption that all the tonian, and then putting down the the physically observed quantities. negative energy states in the vacuum standard Schrödinger equation in The general idea of renormalization are filled up, and when there is a hole terms of a big Ψ referring to the whole is quite sensible physically, but the in the negative energy states it ap assembly of particles. With this devel way it is applied here is not sensible, pears as a physical particle. It would opment of the Klein-Gordon equation because the factor connecting the be a particle with a spin similar to one has a theory referring to several original parameters with the new ones that of the electron and it would have particles which all have positive en is infinitely great. It is then not a a positive charge instead of the nega ergy, and which have to be bosons mathematically sensible process at all! tive charge of the electron, and it now, not fermions as one had pre But still, people have worked with would have a positive energy. viously. This theory is also relativistic it, in particular Lamb. The surprising I did not dare to postulate a new and in agreement with the transform thing is that with the infinities dis particle. The whole climate of opi ation theory. carded by these artificial renormaliz nion in those day was against pos Thus there were two possible the ation rules, you get results in agree tulating new particles, quite diffe ories for particles, both relativistic, ment with observation. The agreement rent from what it is now. So I pu both in agreement with the require holds to a very high degree of accur blished my work as a theory of elec ments of the transformation theory, acy. trons and protons, hoping that in some one of them for particles of zero spin Most physicists are very satisfied unexplained way the Coulomb inter satisfying the Bose statistics, the other with that result. They say that all that action between the particles would for particles of spin 1/2, satisfying the a physicist needs is to have some lead to the big difference in mass Fermi statistics. These theories were theory giving results in agreement between the electron and the proton. in a sense equally good. The Fermi with observation. I say, that is not all Of course I was quite wrong there theory applies to electrons and to that a physicists needs. A physicist and the mathematicians soon pointed other particles of spin 1/2, like pro needs that his equations should be out that it was impossible to have such tons. The Klein-Gordon theory may mathematically sound, that in working a dissymmetry between the positive apply to certain kinds of mesons with with his equations he should not negl and negative energy states. It was zero spin. ect quantities unless they are small. Weyl who first published a categorical For both of these theories we have Well, here again I find myself in statement that the new particle would the electromagnetic potentials coming disagreement with the great body of have to have the same mass as the in. These electromagnetic potentials theoretical physicists. They are compl electron. The theory with equal masses have to refer to an external field. Now, acent about the difficulties of quantum was confirmed a little later by obser the next step which we should like to electrodynamics, and I feel that kind vation when the positron was disco do would be to make these potentials of complacency is similar to the com vered by Anderson. into quantum variables satisfying suit placency which people at one time At that stage we had a satisfactory able commutation relations, so as to had with the original Klein-Gordon theory, not for a single particle, but refer to a quantized field of radiation equation. It is a complacency which really for several particles, because interacting with the assembly of parti blocks further progress. with this theory one could have elec cles. Any substantial further progress, I trons jumping between positive and Now, when you do that, you get feel, must come from some drastic negative energy states and such jumps into trouble. You can put down a changes in the basic equations. Just would correspond either to the simul Schrôdinger equation for the whole where they should be I do not know, taneous annihilation of an electron and assembly, particles and electromagne but I feel that this change will be a positron or the simultaneous cre tic field. When you try to solve that rather similar to the change that Heis ation of an electron and a positron. Schrôdinger equation, you find you enberg introduced in 1925. It is a The number of particles was no longer cannot do it. You can apply standard change which people will probably conserved. This was a physical devel perturbation methods and you then come to eventually only by an indirect opment which was quite acceptable at run into infinities. You cannot find any route. The only feature of the new that time and the final result was a solution. You cannot even get a simple theory which one can be sure of, is theory in agreement with the transfor solution referring to the vacuum state. that it must be based on sound and mation laws and with relativity. The only sensible conclusion to be beautiful mathematics. 4