(e) an infinite vacuum polarizability;
(f) the familiar frequency-dependent finite polarizability for external electromagnetic fields;
(g) emission and absorption probabilities equivalent to those of the Dirac theory for an electron in an external e.m. field;
(h) new reactive corrections of order e2 to the effective charge and current distribution of an electron, which correspond to vanishing total supplementary charge, and to currents of the order e3/ħc distributed over dimensions of the order ħ/mc, and which include the supplementary potential I, and the supplementary magnetic moment
as special (non-relativistic) limiting cases.
Were such calculations to be carried further, to higher order in e, they would lead to still further renormalizations of charge and mass, to the successive elimination of all “virtual” interactions, and to reactive corrections, in the form of an expansion in powers of e2/ħc, to the probabilities of transitions: pair production, collisions, scattering, etc. Nevertheless, before such a program could be undertaken, or the physically interesting new terms (h) above be taken as correct, a new development is required. The reason for this is the following: the results (h) are not in general independent of gauge and Lorentz frame. Historically this was first discovered by comparison of the supplementary magnetic interaction energy in a uniform magnetostatic field H
with the supplementary (imaginary) electric dipole interaction which appeared with an electron in a homogeneous electric field E derived from a static scalar potential
a manifestly non-covariant result.
Now it is true that the fundamental equations of quantum-electrodynamics are gauge and Lorentz covariant. But they have in a strict sense no solutions expansible in powers of e. If one wishes to explore these solutions, bearing in mind that certain infinite terms will, in a later theory, no longer be infinite, one needs a covariant way of identifying these terms; and for that, not merely the field equations themselves, but the whole method of approximation and solution must at all stages preserve covariance. This means that the familiar Hamiltonian methods, which imply a fixed Lorentz frame t = constant, must be renounced; neither Lorentz frame nor gauge can be specified until after, in a given order in e, all terms have been identified, and those bearing on the definition of charge and mass recognized and relegated; then of course, in the actual calculation of transition probabilities and the reactive corrections to them, or in the determination of stationary states in fields which can be treated as static, and in the reactive corrections thereto, the introduction of a definite coordinate system and gauge for these no longer singular and completely well-defined terms can lead to no difficulty.
It is probable that, at least to order e2, more than one covariant formalism can be developed. Thus Stueckelberg’s four-dimensional perturbation theory(26) would seem to offer a suitable starting point, as also do the related algorithms of Feynman.(27) But a method originally suggested by Tomonaga,(28) and independently developed and applied by Schwinger,(22) would seem, apart from its practicality, to have the advantage of very great generality and a complete conceptual consistency. It has been shown by Dyson(29) how Feynman’s algorithms can be derived from the Tomonaga equations.
The easiest way to come to this is to start with the equations of motion of the coupled Dirac and Maxwell field. These are gauge and Lorentz covariants. The commutation laws, through which the typical quantum features are introduced, can readily be rewritten in covariant form to show: (1) at points outside the light cone from each other, all field quantities commute; and (2) the integral over an arbitrary space-like hypersurface yields a simple finite value for the commutator of a field variable at a variable point on the hypersurface, and that of another field variable at a fixed point on the hypersurface.
In this Heisenberg representation, the state vector is of course constant; commutators of field quantities separated by time-like intervals, depending on the solution of the coupled equation of motion, can not be known a priori; and no direct progress at either a rigorous or an approximate solution in powers of e has been made.lr But a simple change to a mixed representation, that introduced by Tomonaga and called by Schwinger the “interaction representation,” makes it possible to carry out the covariant analogue of the power series contact transformation of the Hamiltonian theory.
The change of representation involved is a contact transformation to a system in which the state vector is no longer constant, but in which it would be constant if there were no coupling between the fields, i.e., if the elementary charge e = 0. The basis of this representation is the solution of the uncoupled field equations, which, together with their commutators at all relative positions, are of course well known. This transformation leads directly to the Tomonaga equation for the variation of the state vector ψ:
III.
Here σ is an arbitrary space-like surface through the point P. δψ is the variation in ψ when a small variation is made in σ, localized near the point P; δσ is the four-volume between varied and unvaried sur faces; is the operator of the four-vector electromagnetic potential at P; jµ(P). is the (charge-symmetrized) operator of electron-positron four-vector current density at the same point.
It may be of interest, in judging the range of applicability of these methods, to note that in the theory of the charged particle of zero spin (the scalar and not Dirac pair field), the Tomonaga equation does not have the simple form III; the operator on Ψ on the right involves explicitly an arbitrary time-like unit vector.(30)
Schwinger’s program is then to eliminate the terms of order e, e2, and so, in so far as possible, from the right-hand side of III. As before, only the “virtual” transitions can be eliminated by contact transformation; the real transitions of course remain, but with transition amplitudes eventually themselves modified by reactive corrections.
Apart from the obvious resulting covariance of mass and charge corrections, a new point appears for the light quantum self-energy, which now appears in the form of a product of a factor which must be zero on invariance grounds, and an infinite factor. As long as this term is identifiable, it must of course be zero in any gauge and Lorentz invariant formulation; in these calculations for the first time it is possible to make it zero. Yet even here, if one attempts to evaluate directly the product of zero factor and infinite integral, indeterminate, infinite, or even finite(31) values may result. A somewhat similar situation obtains in the problem, so much studied by Pais, of the direct evaluation of the stress in the electron’s rest system, where a direct calculation yields the value (−e2/2πħc)mc2, instead of the value zero which follows at once as the limit of the zero value holding uniformly, in this order e2, for the theory rendered convergent by the f-quantum hypothesis, even for arbitrarily high f-quantum mass. These examples, far from casting doubt on the usefulness of the formalism, may just serve to emphasize the importance of identifying and evaluating such terms without any specialization of coordinate system, and utilizing throughout the covariance of the theory.
To order e2, one again finds the terms (a) to (h) listed above; the covariance of the new reactive terms is now apparent; and they exhibit themselves again but more clearly as supplementary currents, corresponding to charge distribution of order e3/ħc (but vanishing total charge) extended throughout the interior of the light cones about the electron’s position, and of spatial dimensions ∼ħ/mc; inversely, they may also be interpreted as corrections of relative order e2/ħc and static range ħ/mc to the external fields. The supplementary currents immediately make possible simple treatments of the electron in external fields (where neither the electron’s velocity, nor the derivatives of the fields need be treated small), and so give corrections for emission, absorption and scattering processes to the extent at least in which the fields may be classically described(32); the reactive corrections to the Møller interaction and to pair production can probably not be derived without carrying the contact transformation to order e4, since for these typical exchange effects, not included in the classica
l description of fields, must be expected to appear.
At the moment, to my best present knowledge, the reactive corrections agree with the S level displacements of H to about 1%, the present limit of experimental accuracy. For ionized helium, and for the correction to the electron’s g-value, the agreement is again within experimental precision, which in this case, however, is not yet so high.
3. QUESTIONS
Even this brief summary of developments will lead us to ask a number of questions:(1) Can the development be carried further, to higher powers of e, (a) with finite results, (b) with unique results, (c) with results in agreement with experiment?
(2) Can the procedure be freed of the expansion in e, and carried out rigorously?
(3) How general is the circumstance that the only quantities which are not, in this theory, finite, are those like the electromagnetic inertia of electrons, and the polarization effects of charge, which cannot directly be measured within the framework of the theory? Will this hold for charged particles of other spin?
(4) Can these methods be applied to the Yukawa-meson fields of nucleons? Does the resulting power series in the coupling constant converge at all? Do the corrections improve agreement with experience? Can one expect that when the coupling is large there is any valid content to the Maxwell-Yukawa analogy?
(5) In what sense, or to what extent, is electrodynamics—the theory of Dirac pairs and the e.m. field—“closed”?
There is very little experience to draw on for answering this battery of questions. So far there has not yet been a complete treatment of the electron problem in order higher than e2, although preliminary study(33) indicates that here too the physically interesting corrections will be finite.
The experience in the meson fields is still very limited. With the pseudo-scalar theory, Case(34) has indeed shown that the magnetic moment of the neutron is finite (this has nothing to do with the present technical developments), and that the sum of neutron and proton moments, minus the nuclear magneton (which is the analogue of the electron’s anomalous g-value) is of the same order as the neutron moment, finite, and in disagreement with experience. The proton-neutron mass difference is infinite and of the wrong sign; the reactive corrections to nuclear forces, formally analogous to the corrections to the Moller interaction, have not been evaluated. Despite these discouragements, it would seem premature to evaluate the prospects without further evidence.
Yet it is tempting to suppose that these new successes of electrodynamics, which extend its range very considerably beyond what had earlier been believed possible, can themselves be traced to a rather simple general feature. As we have noted, both from the formal and from the physical side, electrodynamics is an almost closed subject; changes limited to very small distances, and having little effect even in the typical relativistic domain E∼mc2, could suffice to make a consistent theory; in fact, only weak and remote interactions appear to carry us out of the domain of electrodynamics, into that of the mesons, the nuclei, and the other elementary particles. Similar successes could perhaps be expected for those mesons (which may well also be described by Dirac-fields), which also show only weak non-electromagnetic interactions. But for mesons and nucleons generally, we are in a quite new world, where the special features of almost complete closure that characterizes electrodynamics are quite absent. That electrodynamics is also not quite closed is indicated, not alone by the fact that for finite e2/ħc the present theory is not after all self-consistent, but equally by the existence of those small interactions with other forms of matter to which we must in the end look for a clue, both for consistency, and for the actual value of the electron’s charge.
I hope that even these speculations may suffice as a stimulus and an introduction to further discussion.
REFERENCES
1 Heisenberg and Pauli, Zeits. f. Physik., 56, 1, 1929.
2 J. R. Oppenheimer, Phys. Rev., 35, 461, 1930.
3 V. Weisskopf, Zeits. f. Physik., 90, 817, 1934.
4 A. Pais, Verhandelingen Roy. Ac., Amsterdam, 19, 1, 1946.
5 Sakata and Hara, Progr. Theor, Phys., 2, 30, 1947.
6 For a recent summary of the state of theory, see A. Pais, Developments in the Theory of the Electron, Princeton University Press, 1948.
7 L. Rosenfeld, Zeits. f. Physik., 65, 589, 1930.
8 General treatments: R. Serber, Phys. Rev., 48, 49, 1938, and V. Weisskopf, Kgl. Dansk. Vidensk. Selskab. Math.-fys. Medd., 14, 6, 1936.
9 Lamb and Retherford, Phys. Rev., 72, 241, 1947.
10 E. Uehling, Phys. Rev., 48, 55, 1935.
11 Bloch and Nordsieck, Phys. Rev., 52, 54, 1937.
12 Pauli and Fierz, Il Nuovo Cimento, 15, 167, 1938.
13 S. Dancoff, Phys. Rev., 55, 959, 1939; H. Lewis, Phys. Rev., 73, 173, 1948.
14 Shelter Island Conference, June, 1947.
15 Fröhlich, Heitler and Kemmer, Proc. Roy. Soc., A 166, 154, 1938
16 P. Dirac, Phys. Rev., 73, 1092, 1948.
17 H. Lewis, Phys. Rev., 73, 173, 1948.
18 Lamb and Kroll, Phys. Rev., in press.
19 Weisskopf and French, Phys. Rev., in press.
20 H. Bethe, Phys. Rev., 72, 339, 1947.
21 P. Luttinger, Phys. Rev., 74, 893, 1948.
22 J. Schwinger, Phys. Rev., 74, 1439, 1948, and in press.
23 Report to the 8th Solvay Conference.
24 Report to the 8th Solvay Conference.
25 This may be seen very strikingly in writing down an explicit solution for the Tomonaga equation III below. Formally it is:
In order to define the “exp”, we have at present no other resort than to approximate by a power series, where the ordering of the non-commuting factors for jµAµ at different points of space-time can be simply prescribed (e.g., the later factor to the left). Cf. especially F. J. Dyson, Phys. Rev., in press.
26 Stueckelberg, An/n. der Phys., 21, 367, 1934.
27 R. Feynman, Phys. Rev., 74, 1430, 1948.
28 S. Tomonaga, Progr. Theor. Phys., 1, 27, 109, 1946.
29 F. Dyson, Phys. Rev., in press.
30 Kanesawa and Tomonaga, Progr. Theor. Phys., 3, 1, 107, 1948.
31 G. Wentzel, Phys. Rev., 74, 1070, 1948.
32 See for instance results reported to this conference by Pauli on corrections to the Compton effect for long wave lengths.
33 F. Dyson, Phys. Rev., in press.*Author’s note, 1956. Questions 1(a) and 1(b) were indeed answered by Dyson, Phys. Rev., 75, 1736, 1949.
34 K. Case, Phys. Rev., 74, 1884, 1948.
Chapter Seven
In 1947 Willis Lamb and Robert Rutherford published a paper entitled “Fine Structure of the Hydrogen Atom by a Microwave Method” in which they described using microwave electromagnetic radiation to demonstrate a small shift in energy among two hydrogen atom states labeled 2S1/2 and 2P1/2. In Dirac’s relativistic quantum theory, electrons in these two states ought to have identical energies. The fact that they do not is a clear indication that the theory is incomplete and demonstrates the need for quantum electrodynamics. Lamb and Rutherford showed that the 2S1/2 level has an energy slightly higher than the 2P1/2 level. The difference in energy corresponds to a photon frequency of about 1,058 Mhz or a wavelength of 28 cm and is now known as the Lamb shift.
Within days of Lamb and Rutherford’s publication, Hans Bethe published an explanation. In the paper “The Electromagnetic Shift of Energy Levels” he showed that the Lamb shift was due to the electron bound in the hydrogen atom interacting with its own electromagnetic field. In today’s terms we would say that the electron emits a photon and then quickly reabsorbs it. This has the net effect of slightly altering the electron’s position, which perturbs the Coulomb force and slightly shifts the energy state. In the Dirac and Heisenberg-Pauli quantum electrodynamic theories, the calculation of the electron interacting with its own electric field is one of the terms that is infinite, as mentioned in the last chapter. Bethe suggested that it may be possible to remove the infinite result by considering that a
free electron already has this infinity included in its measured rest mass. By subtracting the divergent free electron expression, from the divergent bound electron expression the infinity could be removed and a finite answer calculated. Bethe did this and arrived at a predicted value for the Lamb shift that closely matched the observed value. The idea of how to remove infinite energy from calculations has formed the basis of renormalized quantum theory, which we will be presented in the next chapter. Measuring the Lamb shift was the first experiment that required an explanation beyond Dirac’s relativistic quantum theory, and it set the stage for the completion of quantum electrodynamics by Sin-Itiro Tomanaga, Julian Schwinger and Richard Feynman in the following years.
FINE STRUCTURE OF THE HYDROGEN ATOM BY A MICROWAVE METHOD
BY
WILLIS E. LAMB JR. AND ROBERT C. RUTHERFORD
THE spectrum of the simplest atom, hydrogen, has a fine structure ls which according to the Dirac wave equation for an electron moving in a Coulomb field is due to the combined effects of relativistic variation of mass with velocity and spin-orbit coupling. It has been considered one of the great triumphs of Dirac’s theory that it gave the “right” fine structure of the energy levels. However, the experimental attempts to obtain a really detailed confirmation through a study of the Balmer lines have been frustrated by the large Doppler effect of the lines in comparison to the small splitting of the lower or n = 2 states. The various spectroscopic workers have alternated between finding confirmationlt of the theory and discrepancieslu of as much as eight percent. More accurate information would clearly provide a delicate test of the form of the correct relativistic wave equation, as well as information on the possibility of line shifts due to coupling of the atom with the radiation field and clues to the nature of any non-Coulombic interaction between the elementary particles: electron and proton.
The Dreams That Stuff is Made of Page 62