on that minimizes the potential for all

. We take the

derivative with respect to and set it equal to zero. Thus

2

−

2

= 0.

26

Equation

has three solutions,

= 0,

2

/

.

27

Equation

corresponds to the values of the potential at 0,

which is a local maximum and 1

/

4

2

/

, two distinct

minima with the same energy. If we pick the one with the

positive minimum for , then for this vacuum

0 0 =

2

/

.

28

Equation

shows that

does not have the usual particle

interpretation and suggests that we introduce a new field to

describe the fluctuations of away from its constant vacuum

value

. We have

= −

.

29

In terms of , the Lagrangian becomes

= 1

/

2

−

2 2

−

3

− 1

/

4

4

+

4

/

4 , 30

where

= 2

2

.

31

This choice of vacuum has produced an , a particle with

a nonzero mass and some peculiar self interactions. But note

that the

→

− symmetry of the original Lagrangian in Eq.

has been broken spontaneously. There is no trace of it in

the transformed Lagrangian. The last term, which is a con-

stant, also deserves further comment. If we were considering

a Hamiltonian, we could add a constant term with no mis-

givings. But as I have mentioned, the Lagrangian is different.

From it, we define the action

. If we add a constant to

the Lagrangian, it adds a term proportional to the time dif-

ference in the action. We had better eliminate this term if we

want a sensible theory.

With this example in mind, we now return to the complex

fields with the continuous global gauge transformation in-

variance. As we shall see, this invariance brings in something

new. The way to deal with this case is to write

= 1

/

2

1

+

2

,

32

where

1,2

are real fields. In terms of these fields, the La-

grangian becomes

= 1

/

2

1

2

+ 1

/

2

2

2

+ 1

/

2

2

1

2

+

2

2

− 1

/

4

1

2

+

2

2 2

.

33

The minima are given by the condition that

1

2

+

2

2

=

2

/

2

.

34

The phase is undetermined. We choose the phase so that at

the minimum,

1

=

2

/

,

2

= 0.

35

We can then displace

1

by its vacuum expectation value in

the vacuum defined by this choice of phase and thus write

= 1

/

2

+

+

.

36

We can rewrite the Lagrangian in terms of these fields.

There will be self-interaction terms of , as well as inter-

actions between them and the additive constant. What inter-

ests us is the “kinetic” term

,

= 1

/

2

2

+ 1

/

2

2

−

2 2

,

37

which shows that the new field is massless and the field

has mass

.

Let us review what we have done. We began with a La-

grangian for a complex field of zero mass, which was glo-

bally gauge invariant. We broke this gauge invariance spon-

taneously and found two interacting real scalar fields. One of

these fields has mass zero, and the other has acquired a mass.

Is this result some freakish artifact of this Lagrangian, or are

we in the presence of a more general phenomenon? The an-

swer is the latter. We have found a realization of what is

known as the Goldstone theorem.

I will not try to give a detailed proof of this theorem here

but only state what it is. There are fine points that I will

discuss shortly. Suppose you have a theory with a certain

number of conserved currents, and these currents give rise to

conserved charges that generate some set of gauge transfor-

mations. If one of these charges has a nonvanishing expec-

tation value so that the gauge symmetry is broken spontane-

ously, then necessarily it will give rise to a mass zero, spin

zero particle—the in the example we have discussed. On its

face, this result would appear to rule out theories of this kind

in elementary particle physics because there are no such par-

ticles. However, there is a loophole, and through it we will

drive a truck. The loop hole is Lorentz invariance.

Needless to say, we want all our theories to be Lorentz

invariant, but they need not be “manifestly” Lorentz invari-

ant. A case in point is electrodynamics. This theory is cer-

tainly Lorentz invariant. When Einstein had to choose be-

tween Newtonian mechanics and electromagnetism, he chose

the latter precisely because it was relativistic. But electro-

magnetism is not manifestly Lorentz invariant in the follow-

ing sense. The photon field

is not well-defined. The

theory is invariant under gauge transformations of the form

→

+ , where is a function of the spacetime point

. This invariance precludes terms such as

in the La-

grangian, and thus the photon has no mass.

To define the theory, we must select a gauge. Two popular

gauges are the Lorenz gauge

with

=0, and the Cou-

lomb gauge with ·

=0. The Lorenz gauge condition is

manifestly Lorentz invariant, and the Coulomb gauge is not.

You can use either gauge to carry out calculations. You will

get the same answers for any physical quantity, and these

answers will be Lorentz covariant.

The proof of the Goldstone theorem that most clearly

makes use of the manifest Lorentz covariance is due to

Walter Gilbert

Gilbert has an interesting history. He got his

Ph.D. in physics from Abdus Salam and then switched into

biology. In 1980, he won the Nobel Prize for chemistry. It

was during his physics period when he published this proof.

For details, an interested reader can read my 1974 review

article

28

28

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

Jeremy Bernstein

SEO Version