The Dreams That Stuff is Made of

by Stephen Hawking

  (8.16)

  This is, however, the known expression for H in Maxwell’s theory; (L = − H). Therefore (8.15) holds as well in the sphere S(0) as in R. One has

  (8.17)

  We introduce the notation

  (8.18)

  and have for the action principle

  (8.19)

  The integral

  (8.20)

  in (8.17) gives zero, because

  (8.21)

  If we bear in mind that in the co-ordinate system, where the point charge is at rest, ∫ H(0) dv is proportional to the mass, we have:

  (8.22)

  where v is the velocity of the centre of the electron. In the second integral of (8.17) we have

  (8.23)

  and by partial integration we find, using :

  (8.24)

  The additional surface integral over the infinitely large surface can be omitted, because it gives no contribution to the variation (8.19).

  The result is:

  (8.25)

  We can write (8.25) in the space-vector form:

  (8.25A)

  An electron behaves therefore like a mechanical systemln with the rest mass m0, acted on by the external field .lo

  If the external potential is essentially constant in a region surrounding the electron considered, the diameter of which is large compared with r0, one gets instead of (8.25):

  (8.26)

  and this is entirely equivalent to Lorentz’s equations of motion. But our formula (8.25) holds also for fields which are not constant. Any field can be split up in Fourier components or elementary waves; we may consider each of those separately, and choosing the Z-axis parallel to the propagation of the wave, we can assume that is proportional to e2πi z/λ. Then we see that this Fourier component gives a contribution to the integral (8.25) of the form (8.26), where is now the amplitude of this component and e has to be replaced by an “effective” charge ē, given by

  Using the expression of ρ given by (7.12), and putting z = r cos ϑ,dv = r2 sin ϑ dϑ dφ dr,

  one has

  The ϑ integration can be performed, and one can write

  For waves long compared with r0 one has ē = e, because g (0) = 1. But for decreasing wave-lengths the effective charge diminishes, as the little table for g (x) shows:—

  Table of g (x).lp

  The decrease begins to become remarkable where x ∼ 1, or λ ∼ 2π r0. For large x one has g(x) ≅ 2/x2.

  If we introduce the quantum energy corresponding to the wavelength λ by E = hc/λ, then using (8.6) one has

  x = 1 corresponds to a quantum energy of about 100 m0c2 = 5·107 e. volt. For energies larger than this the interaction of electrons with other electrons (or light waves excited by those) should become smaller than that calculated by the accepted theories. This consequence seems to be confirmed by the astonishingly high penetrating power of the cosmic rays.lq

  Summary.

  The new field theory can be considered as a revival of the old idea of the electromagnetic origin of mass. The field equations can be derived from the postulate that there exists an “absolute field” b which is the natural unit for all field components and the upper limit of a purely electric field. From the standpoint of relativity transformations the theory can be founded on the assumption that the field is represented by a non-symmetrical tensor akl , and that the Lagrangian is the square root of its determinant; the symmetrical part gkl of akl represents the metric field, the antisymmetrical part fkl the electromagnetic field. The field equations have the form of Maxwell’s equations for a po-larizable medium for which the dielectric constant and the magnetic susceptibility are special functions of the field components. The conservation laws of energy and momentum can be derived. The static solution with spherical symmetry corresponds to an electron with finite energy (or mass); the true charge can be considered as concentrated in a point, but it is also possible to introduce a free charge with a spatial distribution law. The motion of the electron in an external field obeys a law of the Lorentz type where the force is the integral of the product of the field and the free charge density. From this follows a decrease of the force for alternating fields of short wavelengths (of the order of the electronic radius), in agreement with the observations of the penetrating power of high frequency (cosmic) rays.

  ELECTRON THEORY

  BY

  J. ROBERT OPPENHEIMER

  From Rapports du 8 Conseil de Physique, Solvay, p. 269 (1950)

  In this report I shall try to give an account of the developments of the last year in electrodynamics. It will not be useful to give a complete presentation of the formalism; rather I shall try to pick out the essential logical points of the development, and raise at least some of the questions which may be open, and which bear on an evaluation of the scope of the recent developments, and their place in physical theory. I shall divide the report into three sections: (1) a brief summary of related past work in electrodynamics; (2) an account of the logical and procedural aspects of the recent developments; and (3) a series of remarks and questions on applications of these developments to nuclear problems and on the question of the closure of electrodynamics.

  1. HISTORY

  The problems with which we are concerned go back to the very beginnings of the quantum electrodynamics of Dirac, of Heisenberg and Pauli.(1) This theory, which strove to explore the consequences of complementarity for the electromagnetic field and its interactions with matter, led to great success in the understanding of emission, absorption and scattering processes, and led as well to a harmonious synthesis of the description of static fields and of light quantum phenomena. But it also led, as was almost at once recognized,(2) to paradoxical results, of which the infinite displacement of spectral terms and lines was an example. One recognized an analogy between these results and the infinite electromagnetic inertia of a point electron in classical theory, according to which electrons moving with different mean velocity should have energies infinitely displaced. Yet no attempt at a quantitative interpretation was made, nor was the question raised in a serious way of isolating from the infinite displacements new and typical finite parts clearly separable from the inertial effects. In fact such a program could hardly have been carried through before the discovery of pair production, and an understanding of the far-reaching differences in the actual problem of the singularities of quantum electrodynamics from the classical analogue of a point electron interacting with its field. In the former, the field and charge fluctuations of the vacuum—which clearly have no such classical counterpart—play a decisive part; whereas on the other hand the very phenomena of pair production, which so seriously limit the usefulness of a point model of the electron for distances small compared to its Compton wave length h/mc, in some measure ameliorate, though they do not resolve, the problems of the infinite electromagnetic inertia and of the instability of the electron’s charge distribution. These last points first were made clear by the self-energy calculations of Weisskopf,(3) and were still further emphasized by the finding, by Pais,(4) and by Sakata,(5) that to the order e2 (and to this limitation we shall have repeatedly to return) the electron’s self-energy could be made finite, and indeed small, and its stability insured, by introducing forces of small magnitude and essentially arbitrarily small range, corresponding to a new field, and quanta of arbitrarily high rest mass.(6)

  On the other hand the decisive, if classically unfamiliar, role of vacuum fluctuations was perhaps first shown—albeit in a highly academic situation—by Rosenfeld’s calculation(7) of the (infinite) gravitational energy of the light quantum, and came prominently into view with the discovery of the problem of the self-energy of the photon due to the current fluctuations of the electron-positron field, and the related problems of the (infinite) polarizability of that field. Here for the first time the notion of renormalization was introduced. The infinite polarization of vacuum refers in fact just to situations in which a classical definition of charge should be possible (weak, slowly varying fields
); if the polarization were finite, the linear constant term could not be measured directly, nor measured in any classically interpretable experiment; only the sum of “true” and induced charge could be measured. Thus it seemed natural to ignore the infinite linear constant polarizability of vacuum, but to attach significance to the finite deviations from this polarization in rapidly varying and in strong fields.(8) Direct attempts to measure these deviations were not successful; they are in any case intimately related to those which do describe the Lamb-Retherford level shift,(9) but are too small and of wrong sign to account for the bulk of this observation.(10) But the renormalization procedure and philosophy here applied to charge, was to prove, in its obvious extension to the electron’s mass, the starting point for new developments.

  In their application to level shifts, these developments, which could have been carried out at any time during the last fifteen years, required the impetus of experiment to stimulate and verify. Nevertheless, in other closely related problems, results were obtained essentially identical with those required to understand the Lamb-Retherford shift and the Schwinger corrections to the electron’s gyromagnetic ratio.

  Thus there is the problem—first studied by Bloch, Nordsieck,(11) Pauli and Fierz,(12) of the radiative corrections to the scattering of a slow electron (of velocity v) by a static potential V. The contribution of electromagnetic inertia is readily eliminated in non-relativistic calculations, and involves some subtlety in relativistic treatment only in the case of spin ½ (rather than spin zero) charges.(13) It was even pointed out (14) that the new effects of radiation could be summarized by a small supplementary potential

  I.

  (where e, ħ, m, c have their customary meaning). This of course gives the essential explanation of the Lamb shift.

  On the other hand the anomalous g-value of the electron was foreshadowed by the remark,(15) that in meson theory, and even for neutral mesons, the coupling of nucleon spin and meson fluctuations would give to the sum of neutron and proton moments a value different from (and in non-relativistic estimates less than) the nuclear magneton.

  Yet until the advent of reliable experiments on the electron’s interaction, these points hardly attracted serious attention; and interest attached rather to exploring the possibilities of a consistent and reasonable modification of electrodynamics, which should preserve its agreement with experience, and yet, for high fields or short wave lengths, introduce such alterations as to make self-energies finite and the electron stable. In this it has proved decisive that it is not sufficient to develop a satisfactory classical analogue; rather one must cope directly with the specific quantum phenomena of fluctuation and pair production.(6) Within the framework of a continuum theory, with the point interactions of what Dirac(16) calls a “localizable” theory—no such satisfactory theory has been found; one may doubt whether, within this framework, such a theory can be formed that is expansible in powers of the electron’s charge e. On the other hand, as mentioned earlier, many families of theories are possible which give satisfactory and consistent results to the order e2.

  A further general point which emerged from the study of electrodynamics is that—although the singularities occurring in solutions indicate that it is not a completed consistent theory, the structure of the theory itself gives no indication of a field strength, a maximum frequency of minimum length, beyond which it can no longer consistently be supposed to apply. This last remark holds in particular for the actual electron—for the theory of the Dirac electron-positron field coupled to the Maxwell field. For particles of lower and higher spin, some rough and necessarily ambiguous indications of limiting frequencies and fields do occur.

  To these purely theoretical findings, there is a counterpart in experience. No credible evidence, despite much searching, indicates any departure, in the behaviour of electrons and gamma rays, from the expectations of theory. There are, it is true, the extremely weak couplings of β decay; there are the weak electromagnetic interactions of gamma rays, and electrons, with the mesons and nuclear matter. Yet none of these should give appreciable corrections to the present theory in its characteristic domains of application; they serve merely to suggest that for very small (nuclear) distances, and very high energies, electron theory and electrodynamics will no longer be so clearly separable from other atomic phenomena. In the theory of the electron and the electromagnetic field, we have to do with an almost closed, almost complete system, in which however we look precisely to the absence of complete closure to bring us away from the paradoxes that still inhere in it.

  2. PROCEDURES

  The problem then is to see to what extent one can isolate, recognize and postpone the consideration of those quantities, like the electron’s mass and charge, for which the present theory gives infinite results—results which, if finite, could hardly be compared with experience in a world in which arbitrary values of the ratio e2/ħc cannot occur. What one can hope to compare with experience is the totality of other consequences of the coupling of charge and field, consequences of which we need to ask: does theory give for them results which are finite, unambiguous and in agreement with experiment?

  Judged by these criteria the earliest methods must be characterized as encouraging but inadequate. They rested, as have to date all treatments not severely limited throughout by the neglect of relativity, recoil, and pair formation, on an expansion in powers of e, going characteristically to the order e2. One carried out the calculation of the problem in question; (for radiative scattering corrections, Lewis(17); for the Lamb shift, Lamb and Kroll,(18) Weisskopf and French,(19) Bethe(20); for the electron’s g-value, Luttinger(21)); one also calculated to the same order the electron’s electromagnetic mass, its charge, and the charge induced by external fields, and the light quantum mass; finally one asked for the effect of these changes in charge and mass on the problem in question, and sought to delete the corresponding terms from the direct calculation. Such a procedure would no doubt be satisfactory—if cumbersome—were all quantities involved finite and unambiguous. In fact, since mass and charge corrections are in general represented by logarithmically divergent integrals, the above outlined procedure serves to obtain finite, but not necessarily unique or correct, reactive corrections for the behaviour of an electron in an external field; and a special tact is necessary, such as that implicit in Luttinger’s derivation of the electron’s anomalous gyromagnetic ratio, if results are to be, not merely plausible, but unambiguous and sound. Since, in more complex problems, and in calculations carried to higher order in e, this straightforward procedure becomes more and more ambiguous, and the results more dependent on the choice of Lorentz frame and of gauge, more powerful methods are required. Their development has occurred in two steps, the first largely, the second almost wholly, due to Schwinger.(22)

  The first step is to introduce a change in representation, a contact transformation, which seeks, for a single electron not subject to external fields, and in the absence of light quanta, to describe the electron in terms of classically measurable charge e and mass m, and eliminate entirely all “virtual” interaction with the fluctuations of electromagnetic and pair fields. In the non-relativistic limit, as was discussed in connection with Kramer’s report,(23) and as is more fully described in Bethe’s,(24) this transformation can be carried out rigorously to all powers of e, without expansion; in fact, the unitary transformation is given by

  II.

  where Z is the (transverse) Hertz vector of the electromagnetic field minus the quasi-static field of the electron. When this formalism is applied to the problem of an electron in an external field, it yields reactive corrections which do not converge for frequencies v > mc2/ħ, thus indicating the need for a fuller consideration of typical relativistic effects.

  This generalization is in fact straightforward; yet here it would appear essential that the power series expansion in e is no longer avoidable, not only because no such simple solution as II now exists, but because, owing to the possibilities of pai
r creation and annihilation, and of interactions of light quanta with each other, the very definition of states of single electrons or single photons depends essentially on the expansion in question.(25) However that may be, the work has so far been carried out only by treating e2/ħc as small, and essentially only to include corrections of the first order in that quantity.

  In this form, the contact transformation clearly yields:(a) an infinite term in the electron’s electromagnetic inertia;

  (b) an ambiguous light quantum self-energy;

  (c) no other effects for a single electron or photon;

  (d) interactions of order e2 between electrons, positrons, and photons, which in this order, correspond to the familiar Møller interactions and Compton effect and pair production probabilities;