The “electroweak” theory of Weinberg plays on the
themes we have discussed but at a higher register. The un-
derlying Lagrangian consists of massless vector mesons,
three of which—the two charged ones and the neutral one—
are coupled to the scalar bosons. In addition, there is the
photon, which is not coupled to these bosons. Then there are
the bosons themselves, which are self-coupled as well. This
Lagrangian has a global gauge symmetry, but the symmetry
group is non-Abelian. In the examples that I have discussed,
the effect of the global gauge transformation is to multiply
the fields by a numerical phase. These examples are Abelian,
and therefore it does not matter in which order two of these
transformations are performed. In the non-Abelian case, it
does matter. The latter case complicates the formalism but
does not change the underlying methodology. Once again the
gauge symmetry is broken spontaneously. The coupled vec-
tor mesons acquire masses, while the photon remains mass-
less and there are massive scalar bosons. Apart from the fact
that this method unifies two otherwise disparate interactions,
it also cures the nonrenormalization issue.
There was always a sort of canary in the mineshaft inter-
action. It was the
v
reaction +
→
W
+
+
W
−
. A neutrino and
an antineutrino interact and produce a pair of weak vector
mesons. No one proposed to measure this reaction, but its
calculation should nonetheless make sense. When this calcu-
lation is done in the conventional theory with no scalar
bosons and the masses being put in by hand, the cross sec-
tion increases without limit as the neutrino momentum ap-
proaches infinity. Here there are no issues of infinities caused
by going to higher order, but rather there is a violation in the
limit that quantum mechanics imposes on the magnitude of
such cross sections due to the conservation of probability.
Weinberg observed that there is a contribution in the elec-
troweak theory from the scalar bosons to this process, which
cancels the terms that violated the quantum limit and renders
the cross sections reasonable. He conjectured that the theory
was renormalizable, which was proven in detail by Martinus
Veltman and his student Gerhard t’Hooft.
The alert reader will notice that something is missing in
this discussion. All the leptons have mass including the neu-
trinos, to say nothing of the masses of the neutrons and pro-
tons. What is the origin of their masses? Hopefully, the
reader will indulge me in a bit of personal reminiscence. For
two years in the late 1950s I was a postdoc at Harvard. Julian
Schwinger was the leading light in theoretical physics at the
time. We, the postdocs and junior faculty, audited whatever
course he happened to be teaching. The material was always
original. The lectures were on Wednesdays, and afterward
the small group of us would have lunch with Schwinger at
Chez Dreyfus
in Cambridge. We would be joined by another
small group from MIT that included Vikki Weisskopf. If
Schwinger had any new ideas, he would try them out on
Wiesskopf. As it happened on this occasion, he had devel-
oped a “theory of everything.” Some of this theory survives
in the work of other people. In 1962, he published a paper on
“Gauge invariance and mass.
In it he raised the question of
whether one could have a massive vector meson in a theory
that had an underlying gauge invariance. This possibility is
not exactly what we have been discussing, but it inspired P.
W. Anderson to use these ideas in condensed matter
physics
Anderson used language in a nonrelativistic con-
text, which is very similar to what we have been discussing.
I remember a lunch in which Schwinger began by saying
to Weisskopf, “Now I will make you a world.” The “world”
was written down on a few paper napkins, one of which I
saved. In any event, one of the things that he said, which has
stuck with me ever since, was that scalar particles were the
only ones that could have nonvanishing vacuum expectation
values. He then went on to say that if you couple one of
these to a fermion by a coupling of the form , then
this vacuum expectation value would act like a fermion
mass. This sort of coupling is how mass generation is done in
principle for the fermions. All particles in this picture would
acquire their masses from the vacuum. We are a long way
from Newton.
I have avoided so far the use of the term Higgs boson—the
analog in the electroweak theory of the . Certainly, Higgs
deserves the credit for first exhibiting the mechanism in the
context of scalar electrodynamics. But as I have tried to
show, it took other people to make it work. The Higgs boson
is what is being looked for at CERN. If they find it, we shall
all be happy and relieved. And if not? I am reminded of a
story about Einstein. He had just received a telegram with
the news that the eclipse expeditions had confirmed his gen-
eral relativity prediction about the Sun bending starlight. He
was very pleased with himself and showed the telegram to
one of his students, Ilse Rosenthal-Schneider. She asked him
what he would have done if the telegram had contained the
news that the experiments disagreed with the theory. He re-
plied, “Da könt mir halt der lieber Gott leid tun-die theorie
stimmt doch. Then I would have been sorry for the dear
Lord. The theory is right .”
ACKNOWLEDGMENTS
The author is grateful to Elihu Abrahams and Roman
Jackiw for helpful communications and to an anonymous
referee for a very careful reading of the paper.
a
Electronic mail:
1
This translation from the Latin can be found in Ernst Mach,
The Science
of Mechanics
Open Court, LaSalle, IL, 1960 , p. 298.
2
P. W. Higgs, “Broken symmetries and the masses of gauge bosons,” Phys.
Rev. Lett.
13
, 508–509 1964 .
3
This is not a misprint. Ludvig Valentin Lorenz was a 19th century Danish
physicist to whom we owe this choice of gauge. I am grateful to Wolf-
gang Rindler for making the fact that this choice is the Lorenz gauge and
not the Lorentz gauge clear so the use of “Lorenz” and not “Lorentz” is
not a misprint. See J. D. Jackson, “Examples of the zeroth theorem of the
history of science,” Am. J. Phys.
76
, 704–729 2008 for a brief biogra-
phy of Lorenz.
4
W. Gilbert, “Broken symmetries and massless particles,” Phys. Rev. Lett.
12
, 713–714 1964 .
5
For a discussion of the boundary question see Jeremy Bernstein, “Spon-
taneous, symmetry breaking, gauge theories, the Higgs mechanism and
all that,” Rev. Mod. Phys.
46
, 1–48 1974 .
6
F. Englert and R. Brout, “Broken symmetries and the mass of gauge
vector mesons,” Phys. Rev. Lett.
13
, 321–323 1964 .
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Am. J. Phys., Vol. 79, No. 1, January 2011
Jeremy Bernstein
