On Quantum-Electrodynamics and The Magnetic Moment of The Electron

This paper presents a new formulation of quantum electrodynamics that addresses divergences in previous calculations of radiative corrections to electron phenomena. The new Hamiltonian isolates aspects involving high energies from those involving moderate energies. It defines the experimental electron mass as a combination of the mechanical mass and electromagnetic self-energy. This accounts for divergences in the electromagnetic mass while maintaining the weak interaction assumption. The theory predicts a small additional magnetic moment for the electron and is confirmed by precision measurements of hydrogen hyperfine splitting and sodium/gallium g-factors.

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On Quantum-Electrodynamics and The Magnetic Moment of The Electron

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Leonardo Bossi

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J. Schwinger, Phys. Rev.

, 73, 416 1948

On Quantum-Electrodynamics and the Magnetic

Moment of the Electron

J. Schwinger
Harvard University, Cambridge, Massachusetts
December 30. 1947

—— ♦ ——
Reprinted in “Quantum Electrodynamics”, edited by Julian Schwinger
—— ♦ ——

Attempts to evaluate radiative corrections to electron phenomena have

heretofore been beset by divergence difficulties, attributable to self-energy
and vacuum polarization effects. Electrodynamics unquestionably requires
revision at ultra-relativistic energies, but is presumably accurate at mod-
erate relativistic energies. It would be desirable, therefore, to isolate those
aspects of the current theory that essentially involve high energies, and are
subject to modification by a more satisfactory theory, from aspects that
involve only moderate energies and are thus relatively trustworthy. This
goal has been achieved by transforming the Hamiltonian of current hole
theory electrodynamics to exhibit explicitly the logarithmically divergent
self-energy of a free electron, which arises from the virtual emission and ab-
sorption of light quanta. The electromagnetic self-energy of a free electron
can be ascribed to an electromagnetic mass, which must be added to the
mechanical mass of the electron. Indeed, the only meaningful statements
of the theory involve this combination of masses, which is the experimental
mass of a free electron. It might appear, from this point of view, that the
divergence of the electromagnetic mass is unobjectionable, since the indi-
vidual contributions to the experimental mass are unobservable. However,
the transformation of the Hamiltonian is based on the assumption of a weak

1
interaction between matter and radiation, which requires that the electro-
magnetic mass be a small correction (∼ (e2 /hc)m0 ) to the mechanical mass
m0 .
The new Hamiltonian is superior to the original one in essentially three
ways: it involves the experimental electron mass, rather than the unobserv-
able mechanical mass; an electron now interacts with the radiation field only
in the presence of an external field, that is, only an accelerated electron can
emit or absorb a light quantum;1 the interaction energy of an electron with
an external field is now subject to a finite radiative correction. In connec-
tion with the last point, it is important to note that the inclusion of the
electromagnetic mass with the mechanical mass does not avoid all diver-
gences; the polarization of the vacuum produces a logarithmically divergent
term proportional to the interaction energy of the electron in an external
field. However, it has long been recognized that such a term is equivalent
to altering the value of the electron charge by a constant factor, only the
final value being properly identified with the experimental charge. Thus the
interaction between matter and radiation produces a renormalization of the
electron charge and mass, all divergences being contained in the renormal-
ization factors.
The simplest example of a radiative correction is that for the energy of
an electron in an external magnetic field. The detailed application of the
theory shows that the radiative correction to the magnetic interaction energy
corresponds to an additional magnetic moment associated with the electron
spin, of magnitude δµ/µ = (1/2π)e2 /hc = 0.001162. It is indeed gratifying
that recently acquired experimental data confirm this prediction. Measure-
ments on the hyperfine splitting of the ground states of atomic hydrogen
and deuterium2 have yielded values that are definitely larger than those to
be expected from the directly measured nuclear moments and an electron
moment of one Bohr magneton. These discrepancies can be accounted for
by a small additional electron spin magnetic moment.3 Recalling that the
nuclear moments have been calibrated in terms of the electron moment, we
find the additional moment necessary to account for the measured hydro-
1
A classical non-relativistic theory of this type was discussed by H. A. Kramers at the
Shelter Island Conference, held in June 1947 under the auspices of the National Academy
of Sciences.
2
J. E. Nafe, E. B. Nelson, and I. I. Rabi, Phys. Rev. 71, 914 (1947); D. E. Nagel, R.
S. Julian, and J. R. Zachariad, Phys, Rev. 72, 971 (1947).
3
G. Breit, Phys. Rev. 71, 984 (1947). However, Breit has not correctly drawn the
consequences of his empirical hypothesis. The effects of a nuclear magnetic field and a
constant magnetic held do not involve different combinations of µ and δµ.

2
gen and deuterium hyperfine structures to be δµ/µ = 0.00126 ± 0.00019 and
δµ/µ = 0.00131±0.00025, respectively. These values are not in disagreement
with the theoretical prediction. More precise conformation is provided by
measurement of the g values for the 2 S 1 ,2 P 1 , and 2 P3/2 states of sodium and
2 2
gallium.4 To account for these results, it is necessary to ascribe the following
additional spin magnetic moment to the electron, δµ/µ = 0.00118±0.00003.
The radiative correction to the energy of an electron in a Coulomb field
will produce a shift in the energy levels of hydrogen-like atoms, and modify
the scattering of electrons in a Coulomb field. Such energy level displace-
ments have recently been observed in the fine structures of hydrogen, 5 deu-
terium, and ionized helium.6 The values yielded by our theory differ only
slightly from those conjectured by Bethe7 on the basis of a non-relativistic
calculation, and are, thus, in good accord with experiment. Finally, the
finite radiative correction to the elastic scattering of electrons by a Coulomb
field provides a satisfactory termination to a subject that has been beset
with much confusion.
A paper dealing with the details of this theory and its applications is in
course of preparation.

4
P. Kusch and H. M. Foley, Phys. Rev. 72, 1256 (1947), and further unpublished
work.
5
W. E. Lamb. Jr. and R. C. Retherford, Phys. Rev. 72, 241 (1947).
6
J. E. Mack and N. Austern, Phys. Rev. 72, 972 (1947)
7
H. A, Bethe. Phys. Rev. 72, 339 (1947).