Paul Dirac

Electrons and the Vacuum

Category: Lectures

Date: 27 June 1956

Duration: 38 min

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Subtitles: EN

Paul Dirac (1956) - Electrons and the Vacuum

This is the earliest lecture by Paul Dirac available in the Mediatheque. He also participated in the first physics meeting 1953, with a lecture entitled “Quantenmechanik und der Äther”, however, there is no tape recording of that lecture. Instead, it can be found in the journal Naturwissenschaftlische Rundschau, volume 6, page 441, 1953

Yesterday Professor Heisenberg and Professor Yukawa spoke to us about fundamental problems in atomic physics. They told us how there are now many particles that are known. Some of them have been discovered only very recently. And the problem of the theoretical physicist is to make up a theory that will account for these particles and explain their properties. So far very little progress has been made with regard to most of these particles. Some of them have properties which are completely not understood. But among these particles there is one which is pretty well understood, namely the electron. We know much more about the electron than about any of the other particles. We have a theory of the electron which is really pretty good. It enables us to calculate for example the energy levels of the hydrogen spectrum to an extremely high degree of accuracy. So high that one can determine very small corrections in the simple formula, corrections known as the Lamb shift. And these corrections as worked out by the theory are in good agreement with experimental results. In spite of this the theory of the electron is not really complete. It is not complete because it is not mathematically self-consistent. The theory is really a surprising mixture of good qualities and bad qualities. So that there is still a problem for the theoretical physicist to get a better theory of the electron. There are many physicists who believe that the problem of getting a better understanding of elementary particles can come about only through setting up a new theory which will explain all these particles. That I believe was the basis of Heisenberg’s theory which he spoke about yesterday. Of course it is quite possible that this is the case. But I am inclined to believe that to get an understanding of the new particles one will need not just one new idea but a whole succession of new ideas. I feel that there are several difficulties facing us and each difficulty will require a new idea to solve it. One cannot hope to be able to get very many good ideas simultaneously. One can only hope to get them one at a time. There may be an interval of many years from one to the next. The whole advance of physics through the course of history has been on the lines of people getting one idea at a time. Each idea just explains one difficulty and leaves the other difficulties untouched. And I feel that the future will be similar to that. And that what we ought to do at present is to concentrate our attention on trying to get one idea which will remove some of the difficulties, perhaps just one difficulty and leave the others untouched. For that reason I have been concentrating my attention on the electron. I feel that the electron being the particle which we understand best is the one where it is most likely that the next important step will be made. And I feel that therefore it is much more likely that we shall be able to improve the theory of the electron in the near future, than that we should for example be able to explain the strangeness of some of the new particles or ideas like that. So for my talk today I want to tell you about my recent work on the electron. The electron carries an electric charge. And this charge is the reason why the electron interacts with the electromagnetic field. The charge produces an electric field around the electron. This electric field is subject to Coulomb's inverse square law of force. That means that if we assume that the charge on the electron is all concentrated at one point, the electric force would tend to infinity as one approaches this point. Now physicists are inclined to believe that an infinite field of any kind can never occur in nature. Nature always seems to deal only with finite quantities. And that must mean either that the charge on the electron is not concentrated at a point. Or else there is some failure in the Coulomb law at very small distances. People have tried to set up theories of the electron for which the charge is not concentrated at a point. Professor Born has worked a good deal on that subject and solved some problems. But still the theories where the charge is not concentrated at a point are all very complicated. And people have not advanced very far with them. All the successes of electron theory, big successes such as the Lamb shift are obtained by working from a point model. There we see the beginning of fundamental difficulties in the theory of the electron. These theories which people work with nowadays are based on the concept of a bare electron. A bare electron is a fictitious thing, an electron without its Coulomb field but still having the proper spin value and also a mass which might be some, which might involve some departure from the actual observed mass. The theory which is in current use involves starting with bare electrons, setting up a theory of bare electrons as a zero order approximation and then introducing the charge on the electron and thus introducing the Coulomb field around the electron as a perturbation. The physical electrons with the Coulomb field around them thus appear only at a later stage in the electron, in the calculations. Now a bare electron is something which is very foreign to nature. And I feel that it might very well be misleading to start off with bare electrons and then just introduce the Coulomb field around the electrons later on as a perturbation. It would be preferable to build up the theory without using the concept of a bare electron at all, to work only with physical electrons. Now there is one rather natural way for doing so which I would like to explain to you. We must consider electrons in interaction with the electromagnetic field. And for describing the electromagnetic field we use the electromagnetic potentials. There are 4 of them, A My, My is a subject which is based on 4 values. Now in the next transformation of those potentials one can pass from A My to A My plus the S by the X My, where S is any function of the 4 coordinates which describe Lorentz's spacetime and this transformation does not affect the electromagnetic field. If we accompany this transformation with a suitable transformation of the variables which describe charges, we get what is called a gauge transformation. Now a gauge transformation changes only the mathematical variables which we use for our description of the physical state and does not change anything which is of physical importance. All quantities of physical importance, all quantities that can be measured are gauge invariant. They are unaffected by this transformation. It would seem therefore that it would be reasonable to build up the theory entirely in terms of observable quantities, which would mean building up the theory entirely in terms of gauge invariant dynamical variables. That would mean that whenever some quantity which is not gauge invariant occurs in the equations, it would have to occur combined with certain other quantities in such a way that the whole combination is gauge invariant. Now in quantum mechanics we have electrons jumping about from one state to another. And the way we describe these jumps in our theory is by means of creation and annihilation operators. We suppose that there exist certain operators which cause the creation of an electron, other operators which cause the annihilation of an electron. And when we have an electron jumping from a state here for example to a state here, we can explain that jump by saying that we have the annihilation of an electron here accompanied by the creation of an electron here. These operators of creation and annihilation are all that we need in order to be able to explain electrons jumping about from any state to any other state. The operators of creation and annihilation which naturally appear in the mathematics are operators which refer to bare electrons. And these operators are also not gauge invariant. However, one can modify these operators to make them gauge invariant. One can modify them by introducing a certain complication into them which has the effect of making them gauge invariant. And then if we look into this complication which we have to bring in to make them gauge invariant, we see that it just makes them apply to physical electrons instead of bare electrons. So that the modified operator of creation of an electron means the creation of an electron together with the creation of the Coulomb field around it and similarly for the annihilation operator. So that we can get in that way operators which refer to creation and annihilation of physical electrons and it would seem therefore that we ought to work entirely with these operators and so get a theory which is gauge invariant. And at the same time we have got a theory from which the concept of the bare electron has been eliminated. Well, that part of the work is very satisfactory. We have the mathematics giving us just what the physics needs. But this is not the end of our difficulties. We must also have a good theory of the vacuum. We need to have the vacuum as the very foundation of our theory. And the problem of understanding the vacuum is not at all a simple problem. One can’t say that a vacuum is simply a region of space where there is nothing at all. The complications arise from the negative energy states, which occur in the theory of the electron when one tries to formulate this theory in a relativistic way. The energy of a relativistic electron is defined in terms of its momentum. By the second of those equations on the board, P there is the momentum of the electron, W is the energy, M is the rest mass and C is the velocity of light. That formula which gives the energy in terms of the momentum involves the square root. And therefore it is associated with an ambiguity in sine, one can put plus or minus in front of the square root. There is no way of avoiding this ambiguity of sine when one deals with such a mathematical equation. The effect of this equation is to allow the energy to take on negative values as well as positive values. And you will observe that the positive values are all greater than or equal to m*c^2 and the negative values are all less than or equal to minus m*c^2. If we drew a diagram to show energy levels, then this line stands for zero energy, this line stands for the energy m*c^2, let this line stand for the energy minus m*c^2. Then the actual energy will be represented by a line anywhere above this line for instance m*c^2 to infinity or anywhere below this line, going from minus m*c^2 to minus infinity. Now as long as I teach you classical mechanics, these negative energy levels do not bother them at all because the effects of the mechanics ... (inaudible 17.10) stay continuously and it is therefore impossible for an electron to change from one of these energy values to one of these energy values, which cannot be surmounted in a practical theory. So in a practical theory we may assume that all the electrons are started off in a state of positive energy and they’ll always remain in a state of positive energy and will therefore behave as electrons are observed to behave. In the quantum theory however an electron can jump from one state to another without passing through intermediate states. And so if we suppose the electrons are all started off in states with positive energy, in the quantum theory they may jump to a state of negative energy. And we cannot just disregard the negative energy states. There is one reasonable way of handing these negative energy states in quantum theory. And that involves bringing in Pauli’s exclusion principle. Pauli’s principle tells us that we can never have more than one electron in any state. Now we may assume that all these states of negative energy are occupied with one electron in each of them. And then if we have some further electrons in positive energy states these electrons can never jump into states of negative energy. They are prevented from doing so by Pauli’s principle. They just always stay in states of positive energy. And they behave like electrons are observed to behave. So with this trick of filling up all the negative energy states we get the basis for the reasonable theory in which the negative energy states will not bother us. This theory however goes a bit further because we may suppose that by some disturbance we take away one of these electrons in negative energy state and bring it up to a state of positive energy. Then we should have an electron suddenly appearing where there wasn’t previously an electron. And at the same time we should have a hole theory in this sea of negative-energy electrons. The hole can be interpreted as a positron. This is a positively charged thing because it’s in a region where there is a lack of negative charge. And also it has a positive energy because it is a place where there is a lack of negative energy. So it is quite reasonable to interpret the hole as a positron. And then you see we have a theory in which we have the possibility of some disturbance creating an electron and positron pair. An electron and a positron are simultaneously created. So that this theory gives us also a theory of positrons and they seem to roughly agree with experiments. That therefore is a possible starting point for getting an explanation of the negative energy states. And you see that it gives us a very different picture of the vacuum from just empty space. The vacuum is now to be looked upon as, well, a vacuum of course is a region of space where there are no electrons and no positrons, no ordinary electrons and also no positrons. And that means that in the vacuum there must be no electrons in positive energy states and no holes in the distribution of negative energy states. So the vacuum will consist of just all the negative energy states occupied by electrons. In the vacuum there is this sea of negative energy electrons, it is a bottomless sea. Well, you might at first think that that’s going to be very disturbing because we have so many electrons in the vacuum. But there is a regularity about these electrons. I first put forward this theory in 1930 and at that time I was not very much worried about this distribution of negative energy electrons. What I said then was that the negative energy electrons in the vacuum form a completely uniform distribution of electrons over the whole of space. And therefore they are unobservable. That one can only expect to be able to observe departures from uniformity. And the departures from uniformity would occur only when we have electrons in positive energy states or holes among the negative energy distribution. And then we should have physical electrons or positrons. However, that simple argument is not really correct. If one examines this distribution of negative energy electrons, one can calculate according to the laws of quantum mechanics, what the density of these electrons is at any place. And one finds that this density is not constant. It is an infinite thing of course but even so it is not a constant infinity. It is subject to violent fluctuations. Now from what I said earlier we should never think of bare electrons. We should suppose that our electrons are always accompanied by Coulomb fields around them. And therefore if we have violent fluctuations in the density of the electrons, there will be violent fluctuations in the Coulomb field. So that our picture gives us a vacuum which is very far from the placid region of the vacuum which one usually thinks of. A vacuum is subject to violent fluctuations in electric density and in Coulomb fields. These violent fluctuations are not a reason for immediately saying that the theory is a bad theory. The vacuum should be the state of lowest energy in which space can exist. And it might very well be that this state of lowest energy is a state in which there are violent fluctuations in some of the physical variables. But these fluctuations would not be observable until one brings in some extra energy. One can compare the situation of the vacuum to the state of lowest energy of some molecule. Suppose we just consider some molecule in its state of lowest energy. We have there a number of electrons moving around in their orbitals. These electrons will of course be accompanied by Coulomb fields so that we have quite a lot of motion going on. And we have quite strong fields present. Even though the molecule is in its state of lowest energy, the fields of the electrons moving around, the electrons themselves are not observable so long as we keep to the state of lowest energy. We have to bring in some additional energy to make them observable. So that with that analogy one can see that there is no immediate reason for rejecting this picture of the vacuum which has these strong fields present. However, when one looks into things more closely one sees that this picture will not really work. One thing that we certainly know about the vacuum is that it is a stationary state. It is a state which does not change in time. And if we examine this picture of the vacuum which I have been giving here with all these negative energy states filled up, we see that this state is not a stationary state. The Coulomb fields which arise from these disturbances of the electric density, these Coulomb fields will themselves disturb the electrons. And they will give rise to the possibility of pair creation. That is to say we might start off with this state where all the negative energy, electron states are occupied. But after a short time we shall no longer be in this state. We shall have a probability of one the negative energy electrons being jerked up to a state of positive energy. And that would be interpreted as a probability for a pair creation. Now it is quite certain that we don’t have pair creation occurring spontaneously in the vacuum. And therefore there is some error in this picture of the vacuum. I have been working on this problem quite recently and have been trying to get a better picture of the vacuum. We must treat the vacuum according to the laws of quantum mechanics. So that it has to be represented by a wave function. A state of anything in quantum theory has to be represented by a wave function. So we must try to find a wave function which will represent the vacuum and which will be a stationary state, a wave function which will not correspond to any physical changes taking place. I have not been able to solve this problem accurately. But I have obtained a better approximation than the usual one. I have obtained a wave function to represent the vacuum. For which these disturbances do not give rise to any probability for the occurrence of an electron positron pair. But they give rise only to the probability for the simultaneous occurrence of 2 electron pairs. This of course is still a bad thing, to have 2 electron pairs appearing simultaneously. But it is less disturbing in the calculations than having just a single electron pair appearing. In both cases the probability for the occurrence of these transitions is infinitely great when one works it out. So that we are still a long way from getting an accurate description of the vacuum. I feel that this is really the central problem in theoretical physics at the present time. To understand what the vacuum is. If we don’t know what the vacuum is we simply cannot hope to understand what particles are. Because the particles are necessarily something more complicated than the vacuum itself. And the particles must certainly be always disturbed by any disturbances which occur in the vacuum state. I feel that theoretical physicists have been rather lax in not studying the problem of the vacuum. The reason for it is very understandable, you can’t expect to get anything very exciting just by studying the vacuum. What you want to find is nought as a result of most of your calculations. And that’s not really so exciting as the results which you might expect when you study individual particles. But still we must realise that our present theory of the vacuum is such that it is just wrong, the theory often gives the result infinity when we want the result nought. And until this problem of the vacuum is put in order I feel that it is really rather hopeless to try to get a good theory of any particles. And then I spoke to you about the possibility of reviving the ether. I told you that the ether is not really in contradiction with the theory of relativity when one takes into account the laws of quantum mechanics. Now you may be wondering what is the present situation with regards to the ether. The present situation is still quite indefinite. It is tied up with this question of the description of the vacuum. And until one has a suitable description of the vacuum it will not be possible to say definitely whether there is an ether or not. This picture of the vacuum which I have been describing here does not require the existence of an ether. One can set up all these electrons in negative energy states. And one can do that in a Lorentz invariant way without any reference to an ether. And the result is that the present electrodynamics is built up without an ether. But the present electrodynamics is built up on the concept of the bare electron. This sea of negative energy electrons which we have as the foundation of our theory is a sea of negative energy bare electrons. And we cannot change these electrons, these bare electrons into physical electrons at the present time without getting these big disturbances in the vacuum which we do not know how to handle. People are inclined to assume that the passage from bare electrons to physical electrons is not really a very drastic change. And that since one can deal with the bare electrons very well without the ether, one will also be able to deal with the physical electrons without the ether. But I feel that this argument is not at all a sound argument. Because I feel very strongly that the concept of a bare electron is a bad concept. And any argument based on bare electrons is therefore unreliable. And it might very well be that to pass from the bare electrons to the physical electrons is a process which is very far from being a trivial process and it might involve bringing in an ether. So one should bear this in mind that there is always the possibility that one might have to introduce an ether. One ought to be completely neutral about it. One shouldn't wan to introduce an ether. And at the same time one should not be unhappy if one finds one needs an ether. One ought to be completely unbiased with regard to this question of the ether. Well I have spoken to you about the difficulties that we have with our present theory. And these difficulties are so great. We have infinities appearing where we ought to have zero. These difficulties are so great that one feels that it is very likely that some quite new idea is needed. Professor Heisenberg yesterday was telling us about one possible approach for getting an entirely new starting point. I very much welcome any attempt to get a new starting point. I feel that the present theory basically is worked out. And that people should look for new starting points. There is one possible starting point which I have been considering recently which is in some ways a very attractive one. And I would like to tell you about it. It is based on an idea of the electric field which was current in the last century, the idea of Faraday lines of force. Faraday put forward the idea of describing an electric field by means of a number of lines of force which will originate from the charged body. If we have an electric charge like this, then Faraday proposed that there were many lines of force radiating out from it in all directions. These lines of force may extend to a charge of opposite signs, this is a plus sign, the M on a charge is a minus sign or alternatively they may go to infinity. One can get the complete description of the electric field in terms of these lines of force. The direction of the lines of force at any point shows us the direction of the electric field and the length of the lines of force, their closeness to one another is an indication of the strength of the electric field. So that in this picture of lines of force we can describe anything whatever, any electric field, whatever. We may have just electromagnetic radiation like when a charge is present at all and that field would be described by… (inaudible 37.30) lines of force… (inaudible 37.35) or else lines of force going to infinity in both directions. We have these lines of force moving about...


This is the earliest lecture by Paul Dirac available in the Mediatheque. He also participated in the first physics meeting 1953, with a lecture entitled “Quantenmechanik und der Äther”, however, there is no tape recording of that lecture. Instead, it can be found in the journal Naturwissenschaftlische Rundschau, volume 6, page 441, 1953. Together with the eight more tape-recorded Dirac lectures, which will eventually be published in the Mediatheque, this gives a unique picture of Dirac’s varying interests over a period of 30 years. Among the participants of the 1956 meeting were Born, Heisenberg and Yukawa. The latter two gave their lectures the day before Dirac and he makes reference to all three. Heisenberg appears over and over again in Dirac’s lecture, because Heisenberg had lectured on a unified theory of elementary particles, a theory that would solve many problems for many particles. Dirac’s line of attack was different, he advocated taking one problem for one particle at the time. So his lecture is quite different from Heisenberg’s, but both are extremely interesting. Speaking slowly in a very clear English, Dirac critizizes the then current technique of first making a theory of the naked electron and then adding its dress, the electromagnetic field, as a perturbation. Instead he wants to construct the theory directly for a physical electron and goes through a number of steps in this direction. He then raises the old problem of the vacuum, the negative energy states and the fluctuations and reports some partial findings. He also refers to his 1953 lecture and speculates that by introducing an ether, the theory might be able to avoid producing naked electrons and positrons by pair creation. It seems that the tape recording was made with a stationary microphone, since when Dirac is using the blackboard, there are difficulties hearing his voice. But one can definitely hear his footsteps as he walks to the blackboard and back!

Anders Bárány