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THEORY OF ELECTRONS AND PROTONS1
P. A. M. Dirac, Cambridge
Foreword
Dirac begins his extremely interesting article with a reference to the relativistic wave equation for the electron that he himself had derived earlier (1). However, the essence of the difficulty he considers with negative energy, as Dirac himself notes, is not connected with the special form of the wave equation; moreover, this difficulty also occurs in pre-quantum theory, being inseparably connected with the fundamental propositions of the theory of relativity.
Let us consider, as an example, a freely moving material particle whose rest mass is equal to \(m\). The kinetic energy of this particle, according to classical mechanics, is equal to \(\frac{mv^{2}}{2}\); according to the theory of relativity, however, its total energy is equal to
\[ W=\frac{mc^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}} . \]
Thus the energy of a particle is, in classical mechanics, a substantially positive quantity (assuming that \(m\) is positive), whereas the sign of the relativistic expression for the energy \(W\) remains indeterminate, since, generally speaking, the sign of the root is indeterminate.
It is true that usually an additional assumption is made tacitly, namely that in the expression for \(W\) only the positive value of the root is to be taken, i.e. that in reality
all particles possess positive energy. This assumption does not lead to internal contradictions in the theory for the following reason. The velocity of a particle \(v\) may take all values from 0 to \(c\), in accordance with which the energy \(W\) may take values from \(+mc^2\) to \(+\infty\) and from \(-mc^2\) to \(-\infty\). Thus the smallest positive value of the energy, \(+mc^2\), and its largest negative value, \(-mc^2\), are separated by a finite interval. Since in the classical (i.e., pre-quantum) theory the energy of a particle can change only continuously, the sign of the energy of each individual particle can indeed not change.
The situation changes essentially in quantum theory, according to which discontinuous (jump-like) changes of state may occur. Thus in quantum theory the assumption of positive energy for all particles will prove free of internal contradictions only if transitions of particles from states of positive energy into states of negative energy prove impossible. Within the framework of the old (Bohr) quantum theory this question cannot be resolved, because that theory in general did not make it possible to calculate the probability of quantum transitions; using the correspondence principle one can estimate the probability only of those transitions to which classically possible changes of state correspond.
According to the new quantum theory, however, transitions from states of positive energy turn out to be possible, and thus the difficulty with negative energy cannot be avoided on the basis of the indicated assumption: even if at some initial moment all particles possessed positive energy, in the course of time some of the particles would have to pass into states of negative energy. Although the probability of these transitions was in fact calculated on the basis of the Dirac equation of the electron, this probability remains finite also when the Dirac equation is replaced by the relativistic form of the Schrödinger equation.
On the other hand, particles of negative energy would have to possess completely anomalous properties: to move against the applied forces, to give up energy when accelerating and to absorb it when their motion is slowed, etc. Thus the difficulties whose consideration led Dirac to the creation of his extremely interesting theory of the proton do indeed have quite fundamental significance.
In conclusion, let us note two difficulties encountered by Dirac’s theory itself, as set forth by him in the present article. First, up to the present time (August 1930) it has not proved possible to confirm Dirac’s suggestion, stated at the end of § 2, that the difference between the masses of the electron and the proton (“hole”) can be explained by the interaction of electrons of negative energy; moreover, certain unpublished considerations of Pauli compel one to doubt the validity of this suggestion, which is of decisive importance for Dirac’s entire theory. Secondly, the probability, accompanying emission, of a spontaneous transition of an electron of positive energy into an unoccupied state of negative energy (“the jump of an electron into a hole,” the neutralization of an electron and a proton) has recently been calculated both by Dirac himself and by other authors,^1 and the calculated probability of these transitions proved to be excessively large, incapable of being reconciled with experimental facts. Dirac hopes that this discrepancy between theory and experiment is explained only by the present impossibility of taking account in the theory of the difference between the masses of the electron and the proton.^2
Ig. Tamm
§ 1. The essence of the difficulties connected with negative energy
The relativistic quantum theory of the motion of an electron in an electromagnetic field successfully predicted the intrinsic rotation of the electron. However, it brought with it certain serious difficulties, indicating the necessity of fundamental changes before it can be regarded as an exact description of nature. The difficulties are connected with the fact that in the wave equation, written in the form:
\[ \left|\frac{W}{c}+\frac{e}{c}A_0+\rho_1\left(\boldsymbol{\sigma}\mathbf{p}+\frac{e}{c}\mathbf{A}\right)+\rho_3mc\right|\Psi=0 \tag{1} \]
^1 P. A. M. Dirac, Proc. Cambr. Phil. Soc. July 1930; Ig. Tamm, Z. Physik 63, 545, 1930. Oppenheimer, who carried out the calculation for a particular case (Phys. Rev., 1930), made a computational error.
^2 General questions connected with Dirac’s conceptions will be touched upon in connection with another article by Dirac—“The Proton”—to be printed in the next issue of Uspekhi. Ed.
In addition to the expected solutions, for which the kinetic energy of the electron is positive, there is the same number of unexpected solutions with negative kinetic energy of the electron, which would seem to be devoid of any physical meaning.
Thus, for example, in the case of a constant electromagnetic field, equation (1) admits a periodic solution of the form
\[ \Psi = u e^{-\frac{iEt}{h}}, \tag{2} \]
where \(u\) does not depend on \(t\), indicating a stationary state; \(E\) is the total energy of the position, including the expression \(mc^{2}\), required by the theory of relativity.
Solutions (2) with negative values of \(E\) exist just as with positive ones. In fact, if we take the matrix expression of the operators \(\rho_{1}\sigma_{1}, \rho_{2}\sigma_{2}, \rho_{3}\sigma_{3}\) with real matrix elements, then the expression conjugate to any solution of equation (1) will be a solution of the wave equation obtained from (1) by changing the sign of the potential \(A\), and either the original wave function or its conjugate must give a negative \(E\).
This difficulty is not peculiar to the quantum theory of the electron alone, as such, but is general, since it occurs throughout the theory of relativity just as in the classical theory. It arises as a result of the fundamental circumstance that in the relativistic Hamiltonian equation of classical theory:
\[ \left(W+\frac{e}{c}A_{0}\right)^{2} -\left(\mathbf{p}+\frac{e}{c}\mathbf{A}\right)^{2} -m^{2}c^{2}=0 \tag{3} \]
the sign of \(W\), or rather of \(W+eA_{0}\), remains undetermined. Although the operations on the wave function in (1) are linear with respect to \(W\), they are nevertheless, roughly speaking, equivalent to the left-hand side of (3), and the indeterminacy of the sign is preserved. This difficulty is not essential for the classical theory, since its dynamical variables must always change continuously, as a result of which there is here a sharp boundary
between those solutions of the equations of motion for which \(W+eA_0>mc^2\), and those for which \(W+eA_0<-mc^2\), so that the latter may simply be ignored.
In quantum theory, however, we cannot overcome this difficulty so easily. True, in the case of an invariable electromagnetic field, one may separate the solutions of equation (1) of the form (2) with positive \(E\) from the solutions with negative \(E\), asserting that only the former have physical meaning (as is done when applying the theory to the determination of the energy levels in the hydrogen atom); but this is not possible if the system is subjected to perturbations and can pass from one state to another.
In the general case of an arbitrarily varying electromagnetic field we cannot once and for all separate the solutions of the wave equation that give positive energy from the rest. Furthermore, in an exact quantum theory, where the electromagnetic field is also subject to quantum laws, such transitions may occur in which the energy of the electron changes from a positive to a negative value, even in the absence of any external field. The excess energy, ultimately equal to \(2mc^2\), is emitted in the form of radiation (the law of conservation of energy and momentum requires at least two light quanta, formed simultaneously in this process).
Let us now consider somewhat more closely the wave functions representing a state of negative energy of the electron. If we superpose a certain number of wave functions in such a way as to obtain a “wave packet,” then its motion will take place along the classical trajectory determined by Hamiltonian (3) with negative \(W+eA_0\). Such a trajectory, as is easy to see, is a possible trajectory for an ordinary electron with positive energy, moving in an electromagnetic field of the opposite sign, or for an electron with charge \(+e\) (and positive energy) moving in an electromagnetic field of the same sign. \(^1\)
\(^1\) See, for example, H. Weyl, Z. Physik 56, 332, 1929.
Thus, an electron with negative energy moves in an external field as though it possessed a positive charge. This result compelled some to suppose a connection between the electron with negative energy and the proton, i.e., the nucleus of the hydrogen atom. However, one cannot simply assert that an electron with negative energy is a proton, since this leads to the following paradoxes:
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The transition of an electron from a state with positive energy to a state with negative energy would have to be interpreted as the transition of an electron into a proton, which violates the law of conservation of electric charges.
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Although an electron with negative energy moves in an external field as though it possessed a positive charge, nevertheless, on the basis of the law of conservation of momenta it is easy to show that the field created by the electron itself will be the same as if the electron possessed a negative charge. Hence, for example, it follows that an electron with negative energy will repel an ordinary electron with positive energy, although it will itself be attracted by it.
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An electron with negative energy must have the less energy the faster it moves, and, in order to come to a state of rest, it would have to absorb energy.
Particles of this kind have never been observed. A detailed consideration of the conditions which, according to our conceptions, exist in the real world makes it possible to understand that the connection between the proton and the electron with negative energy rests on some other basis, one that permits all the above-mentioned difficulties to be removed.
§ 2. Removal of the Difficulties Connected with Negative Energy
The most probable states of the electron are those whose energy is minimal, i.e., states with negative energy and very great velocity. All the electrons of the world tend to enter this state with emission
radiation. However, as always, here the Pauli exclusion principle comes into force, forbidding more than one electron to occupy one and the same state. Let us now suppose that there are so many electrons in the world that all the most probable states are already occupied by them, or, more precisely, that all states with negative energy are occupied, with the possible exception of some with small velocity. In this case electrons with positive energy will have very little chance of “jumping” into states with negative energy and therefore will behave like ordinary electrons observed in laboratories. We may regard the number of electrons with negative energy as infinitely large, and even infinitely large per unit volume throughout the whole world. But if their distribution is exactly homogeneous, then we must expect that they will be completely imperceptible to us. Only small deviations from perfect homogeneity, caused by the fact that some states of electrons with negative energy have remained unoccupied, can be detected by us.
Let us consider the properties of the unoccupied places or “holes.” The problem is similar to that which arises in the study of X-ray levels in an atom with many electrons. According to the general theory of X-ray levels, a hole formed when one of the inner electrons of an atom is removed may be described as a certain orbit, namely as the orbit of the absent electron removed from that place. Such a description can be justified by quantum mechanics, provided only that the orbit is considered not in Bohr’s sense, but as something represented by a three-dimensional wave function (not taking rotation into account). Thus a hole, or an unoccupied place in a region usually saturated with electrons, is in many respects similar to a single electron in a region usually devoid of electrons.
In the case of X-rays, holes must be regarded as if they possess negative energy, since in order to bring about the disappearance of one of them (i.e. to fill it), an ordinary electron with positive energy must be added to it. Precisely the converse is meant
place in our distribution of electrons with negative energy. These holes will possess positive energy and in this respect will be similar to ordinary particles. Further, the motion of one of these holes in an external electromagnetic field will be similar to the motion of an electron with negative energy which would have filled it. Thus the hole will, as it were, possess charge \(+e\). Thus, we arrive at the conclusion that the holes in the distribution of electrons with negative energy are protons. If an electron with positive energy falls into a hole and fills it, we shall have the disappearance of a proton and an electron with the emission of radiant energy.
In considering the field formed by the distribution of electrons with negative energy, one difficulty arises. In this case the infinite density of electricity, according to Maxwell’s equation,
\[ \operatorname{div}\mathbf{E}=-4\pi\rho \tag{4} \]
must produce a field of infinite divergence. However, it seems natural to interpret \(\rho\) in Maxwell’s equation (4) as a deviation from the normal state of electrization of the world, by which, according to this theory, we shall understand such a state when all places with negative energy are occupied and those with positive energy are free. Then \(\rho\) will consist of a charge \(-e\), arising from each occupied place with positive energy, and a charge \(+e\), arising from each unoccupied place with negative energy. Thus the field of a proton and the field of the charge \(+e\) will be identical.
In this way we can overcome the three difficulties mentioned at the end of the preceding paragraph. It is necessary to postulate the existence of only one basic kind of particle, instead of the two that were required earlier. The obvious tendency of all particles to pass into states with the least energy leads to the fact that all objects observed in nature have positive energy.
Can the present theory explain the great dissymmetry existing between electrons and protons, expressed in their different masses and in the ability of protons to form heavier atomic nuclei? Evidently, the theory gives to a considerable degree a symmetry between electrons and protons. We can interchange their places and assert that protons are the real particles, while electrons are only holes in a uniform distribution of protons with negative energy. However, the symmetry becomes mathematically imperfect if one takes into account the interaction between electrons. If this interaction is neglected, the Hamiltonian describing the whole system will have the form \(\sum H_a\), where \(H_a\) is the Hamiltonian or energy of an electron in state \(a\), and the summation is over all occupied places. It differs only by a constant (i.e., by something independent of precisely which places are occupied) from the sum \(\sum(-H_a)\), taken over all unoccupied places. Thus we obtain formally the same dynamical system if we assume that the unoccupied states add to the Hamiltonian a term \(-H_a\). On the other hand, taking into account the interaction between electrons, we obtain in the Hamiltonian an additional term of the form \(\sum V_{ab}\), where the summation is over all pairs of occupied places \((a,b)\), which is not equivalent to any sum taken over pairs of unoccupied places. Thus, the interaction of electrons gives an essentially different Hamiltonian if protons are regarded as real particles occupying places.
The consequences of this dissymmetry are not easy to calculate from the relativistic point of view, but we hope that in the future it will lead to an explanation of the difference between the masses of the proton and the electron. It is possible that, before this result can be achieved, a more perfect theory of interaction will be needed, perhaps one based on Eddington’s calculation1 of the fine-structure constant \(\frac{e^2}{\hbar c}\).
§ 3. Scattering Phenomena
As an elementary example illustrating the foregoing, let us consider the problem of the scattering of radiation by a free or bound electron.
According to the theory, the phenomenon of scattering must be regarded as a process of a double transition, consisting, first, in the absorption of a quantum by the electron, which immediately passes into another state, and then in emission with the simultaneous jump of the electron into its final position (or first emission and then absorption). Ultimately we have to consider three states of the whole system: 1) the initial state, with a quantum incident on the electron in its initial position, 2) an intermediate one, when there are two quanta or none at all and the electron is in another position, and 3) the final state with the scattered quantum and the electron in its final state.
The first and the last states of the whole system must possess the same total energy, from which the energy in the intermediate stage, lasting a very short time, may differ substantially.
The question arises as to how this scattering process can be interpreted when the intermediate state is a state with negative energy of the electron. According to the former views, this latter has no physical meaning, and it is doubtful that the phenomenon of scattering arising through its mediation could be included in the formula for the scattering coefficient. This leads to serious difficulties, since in many important practical cases almost all scattering acts proceed from an intermediate state with negative electron energy. Indeed, for a free electron and radiation of low frequency, where the classical formulas are valid, all scattering passes through such an intermediate state.
According to the theory set forth here, the jump of an electron into a state with negative energy is absolutely forbidden by the exclusion principle, so that the process of a double transition
with an intermediate state of negative energy is thereby excluded. We have, however, another kind of double transition, namely when first one of the electrons with negative energy jumps into the required final position of the electron with absorption (or emission) of a quantum, and only then the original electron with positive energy falls into the hole formed as a result of the first transition, with emission (absorption) of a quantum. This process ultimately gives a final state of the whole system indistinguishable from that obtained as the result of the direct process, in which one and the same electron makes two successive jumps. These new processes must lead to the same result as the direct processes, which are excluded as a result of the intermediate state with negative energy of the electron, because the matrix elements determining the transition probabilities are the same in both cases, although they enter in the reverse order.
Thus the old scattering formulas, in which no intermediate states were excluded, must be justified.