on that minimizes the potential for all
x
. We take the
derivative with respect to and set it equal to zero. Thus
m
2
−
2
= 0.
26
Equation
has three solutions,
= 0,
m
2
/
.
27
Equation
corresponds to the values of the potential at 0,
which is a local maximum and 1
/
4
m
2
/
, two distinct
minima with the same energy. If we pick the one with the
positive minimum for , then for this vacuum
0 0 =
m
2
/
v
.
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
v
. We have
= −
v
.
29
In terms of , the Lagrangian becomes
L
= 1
/
2
−
m
2 2
−
v
3
− 1
/
4
4
+
m
4
/
4 , 30
where
m
= 2
m
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
t
t
Ldt
. 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
+
i
2
,
32
where
1,2
are real fields. In terms of these fields, the La-
grangian becomes
L
= 1
/
2
1
2
+ 1
/
2
2
2
+ 1
/
2
m
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
=
m
2
/
v
2
.
34
The phase is undetermined. We choose the phase so that at
the minimum,
1
=
m
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
x
= 1
/
2
v
+
x
+
i x
.
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
L
K
,
L
k
= 1
/
2
2
+ 1
/
2
2
−
m
2 2
,
37
which shows that the new field is massless and the field
has mass
m
.
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
A
is not well-defined. The
theory is invariant under gauge transformations of the form
A
→
A
+ , where is a function of the spacetime point
x
. This invariance precludes terms such as
A A
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
A
=0, and the Cou-
lomb gauge with ·
A
=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
