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

reaction +

→

+

+

−

. 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

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 .”

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,

Open Court, LaSalle, IL, 1960 , p. 298.

2

P. W. Higgs, “Broken symmetries and the masses of gauge bosons,” Phys.

Rev. Lett.

, 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.

, 704–729 2008 for a brief biogra-

phy of Lorenz.

4

W. Gilbert, “Broken symmetries and massless particles,” Phys. Rev. Lett.

, 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.

, 1–48 1974 .

6

F. Englert and R. Brout, “Broken symmetries and the mass of gauge

vector mesons,” Phys. Rev. Lett.

, 321–323 1964 .

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Am. J. Phys., Vol. 79, No. 1, January 2011

Jeremy Bernstein

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