The first people to use the gauge loophole were Peter
Higgs
Francois Englert, and Robert Brout
Many of the
points were later clarified by G. S. Guralnik, C. R. Hagen,
and T. W. B. Kibble
I will stay within the confines of the
electrodynamics of charged scalar particles for the moment
so as to use the work we have already done.
We can write the Lagrangian as
L
= − 1
/
4
/
x A
v
x
−
/
x
v
A x
2
−
/
x
+
ieA x
†
x
/
x
−
ieA x x
+
m
2 †
x x
− 1
/
2
†
x x
2
.
38
The first part of the Lagrangian is the free electromagnetic
part, and the last part is the bosonic part we have already
seen. The middle part is the coupling term.
As before, it is convenient to split
into its real and
imaginary parts and use the two-dimensional notation
=
1
,
2
.
39
To simplify the notation, we shall introduce the 2 2 matrix
q
,
q
=
0 −
i
i
0
.
40
By using the Noether techniques I described earlier, we find
the conserved current,
J
=
i
/
x x q x
+
e x
·
x A x
,
41
whose charges generate global gauge transformations. We
can now break this invariance spontaneously and make the
same choice of vacuums as before. That is, we minimize the
scalar boson potential as we have done in our previous ex-
amples. What is new here is the vector potential, which did
not enter our previous discussion. The result is
/
x
v
/
x A
v
x
−
/
x
v
A x
=
m
2
1
/
m
/
x x
−
A x
.
42
By this point, the reader may wonder where is all this
formalism leading? Behold, you are about to witness a
miracle.
It is at this point we must choose a gauge for
A x
. It is
convenient to use the Lorenz gauge
/
x A x
=0. Equation
can then be written as
/
x
v
/
x
v
A x
=
m
2
A x
− 1
/
m
/
x x
.
43
We define a new field
B x
by
B
=
A x
− 1
/
m
/
x x
.
44
From the field equations, this
x
, unlike the previous ex-
ample beginning with Eq.
obeys the equation of an
uncoupled zero mass boson—the Goldstone boson,
/
x
v
/
x
v
x
= 0.
45
If we make the substitution of Eq.
into Eq.
and use
Eq.
we obtain
/
x
v
/
x
v
B
=
m
2
B
.
46
The “photon” has morphed into a vector meson with mass.
Let us summarize what we have done. The electrodynam-
ics of a charged boson with a spontaneously broken gauge
symmetry in the manifestly covariant Lorenz gauge yields
results consistent with a Goldstone theorem. We obtain an
uncoupled massless Goldstone scalar boson , a massive sca-
lar boson , and a massive vector meson
B
. Because these
masses have the same origin, there is a relation between
them. Because the Goldstone particle is uncoupled, it is also
unobservable and can be ignored.
What happens in the Coulomb gauge where ·
A
=0? I
won’t go though the steps but summarize the results. There is
no Goldstone theorem because the gauge is not explicitly
covariant and no Goldstone boson. There is a massive scalar
boson and a massive vector meson
B
.
The first person to make full use of these ideas was Steven
Weinberg in 1967
To appreciate what he did, we must set
the context. In 1934, Fermi produced the first modern theory
of -decay. He was an expert in quantum electrodynamics,
and hence it was natural for him to use it as a template. In
quantum electrodynamics the current of charged particles
J
interacts with the electric field
A
with a coupling of the
form
J A
. Thus charged currents do not act directly with
each other but only by the exchange of photons. Because
there was apparently no equivalent of the photon for the
weak interactions such as -decay, Fermi directly coupled a
current
J
N
for the “nucleons,” the neutron and proton, with
a current
J
L
for the “leptons,” the electron and neutrino, that
is,
J
N
J
L
. This phenomenological theory worked very well.
One could use it to calculate, for example, the energy spec-
trum of the electrons emitted in -decay. But it came to seem
anomalous. The “strong” interaction between nucleons, as
Yukawa proposed in the prequark days, took place with the
exchange of mesons, the electromagnetic interactions with
photons, and presumably gravitation with gravitons. There
was a suggestion of using the same meson that produced the
strong interactions to produce the weak ones. This idea was
abandoned. But in the 1950s, it was suggested that one or
more weak heavy photons might do the trick. There were
two problems. None had been observed, and the theory that
was being proposed did not make any sense.
The former difficulty was easily disposed of. Because the
contact theory with the currents coupled directly to each
other worked well phenomenologically, it had to be that
these weak mesons were very massive—too massive, it was
argued, for the generation of accelerators that then existed to
produce them. When they finally were produced, it turned
out that their masses were about a hundred times greater than
the nucleon masses. The second difficulty was qualitatively
different. In the theories that were then being proposed, the
weak mesons were being put in “by hand.” They were just
massive particles whose masses had no particular origin. If
one tried to calculate anything beyond the lowest order phe-
nomenology, we obtained terrible infinities. These infinities
were much worse than those in quantum electrodynamics,
which could be swept under the rug by renormalization. In
short, the theory did not make any sense. Theorists were left
grasping for straws. Then came Weinberg.
29
29
Am. J. Phys., Vol. 79, No. 1, January 2011
Jeremy Bernstein
