The effect of a change of gauge on the propagators is studied systematically for quantum electrodynamics. Various gauges are considered, among them the Coulomb, the Landau, the Feynman, and the Yennie gauges. The equivalence of the various formulations of the theory is demonstrated. For the relativistic gauges, the transformation of the wave function renormalization constant is described.

Topics
Quantum electrodynamics
1.
The Coulomb gauge has been used recently by Schwinger and Johnson (Kenneth Johnson, Massachusetts Institute of Technology, private communication, 1959). They have arrived independently at several of the results described in the present note.
2.
J.
 
Schwinger
,
Proc. Natl. Acad. Sci. U.S.
 
37
,
452
(
1951
).
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For a more detailed discussion see, e.g.,
K.
 
Symanzik
,
Z. Naturforsch.
 
9a
,
809
(
1954
)
and
E. S.
 
Fradkin
,
Doklady Akad. Nauk S.S.S.R.
 
98
,
47
(
1954
)
and
100
,
897
(
1955
).,
Dokl. Akad. Nauk SSSR
3.
L. D.
Landau
,
A. A.
Abrikosov
, and
I. M.
Khalatnikov
,
Doklady Akad. Nauk S.S.S.R.
95
,
773
(
1954
).
4.
The M transformation has been given first by
L. D.
 
Landau
 and
I. M.
 
Khalatnikov
,
J. Exptl. Theoret. Phys. (U.S.S.R.)
 
29
,
89
(
1955
);
English translation in
Soviet Phys. JETP
2
,
69
(
1956
). Their derivation, however, is based on an operator gauge transformation, the validity of which appears rather questionable.
5.
H. M.
Fried
 and
D. R.
Yennie
,
Phys. Rev.
112
,
1391
(
1958
).
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6.
K.
Johnson
 and
B.
Zumino
,
Phys. Rev. Letters
3
,
351
(
1959
).
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7.
The transverse part of a vector D is of course defined by
Dtr = D−∇∇−2 divD
.
8.
In order to avoid formal difficulties in connection with the use of the anticommuting spinor sources η and η̄, it is best not to interpret the bar as a relation of hermitian conjugation between η and η̄. Rather, one should consider η̄(x) and η(x) as independent anticommuting symbols and carry out all the formal operations from this point of view. In the final expression one always sets η = η̄ = 0, or more correctly, one takes that part of the expression which is independent of η and of η̄.
9.
A covariant expression for aμ corresponding to the Coulomb gauge in the Lorentz frame characterized by the unit time‐like vector nμ is
aμ = −∂μ+nμ(n⋅∂)∂2+(n⋅∂)2
.
10.
For η = η̄ = 0, Eq. (33) gives the conservation of the vacuum currents.
11.
It has been pointed out by Schwinger that the analogous covariance test fails if an anomalous Pauli moment is introduced into the theory. Schwinger has also shown how the covariance of the theory can be saved by the further introduction of a term describing the self‐interaction of the magnetic moment density. The author would like to thank Dr. Glashow for an illuminating correspondence on this question of covariance.
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© 1960 The American Institute of Physics.
1960
The American Institute of Physics
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