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

= − 1

/

4

/

−

/

2

−

/

+

†

/

−

+

2 †

− 1

/

2

†

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

,

=

0 −

0

.

40

By using the Noether techniques I described earlier, we find

the conserved current,

=

/

+

·

,

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

/

/

−

/

=

2

1

/

/

−

.

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

. It is

convenient to use the Lorenz gauge

/

=0. Equation

can then be written as

/

/

=

2

− 1

/

/

.

43

We define a new field

by

=

− 1

/

/

.

44

From the field equations, this

, unlike the previous ex-

ample beginning with Eq.

obeys the equation of an

uncoupled zero mass boson—the Goldstone boson,

/

/

= 0.

45

If we make the substitution of Eq.

into Eq.

and use

Eq.

we obtain

/

/

=

2

.

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

. 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 ·

=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

.

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

interacts with the electric field

with a coupling of the

form

. 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

for the “nucleons,” the neutron and proton, with

a current

for the “leptons,” the electron and neutrino, that

is,

. 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

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