an act of faith to see how these objects fitted into some kind
of symmetric structure. But Murray Gell-Mann took the leap.
He proposed a symmetry that was a generalization of isoto-
pic spin, SU 3 , and suggested that if the various mass dif-
ferences were neglected, the known particles could be orga-
nized in multiplet structures. For example, the known scalar
mesons, including a newly discovered particle that was
called the eta, fitted into an octet. The known hyperons fitted
into a tenfold decuplet, but there was one missing. It was
given the name
−
and its properties were predicted. When it
showed up with these properties, the lingering doubts about
this scheme vanished and Gell-Mann was awarded a well-
deserved Nobel Prize. It was a textbook example of a sym-
metry and its breaking.
This kind of symmetry breaking is nearly as old as the
quantum theory itself. Eugene Wigner and Herman Weyl, for
example, studied the role of group theory in quantum me-
chanics. The idea was that the description of a quantum me-
chanical system could be split into two contributions—a
Hamiltonian that exhibits the symmetries of the group plus a
second Hamiltonian that did not. If the later is “small,” then
some aspects of the original symmetry would still be appar-
ent.
As an example, consider the group of rotations in three-
dimensional space called SO 3 . These rotations are gener-
ated by the orbital angular momentum. Suppose one part of
the Hamiltonian contains only a central force. This part is
invariant under rotations, which means that the angular mo-
mentum operators commute with this part of the Hamil-
tonian. The eigenstates are both eigenstates of the energy and
the angular momentum. If, for example, the nuclear force
that binds the neutron and proton together is represented by a
central force, then the ground state of the deuteron would be
an S-state. But it isn’t. It has a small percentage of the D
state, which manifests itself in the fact that the deuteron has
a quadrupole moment. The rotational symmetry is broken in
this case by adding a tensor force. The total angular momen-
tum, which includes the spin, is conserved, but the purely
orbital part is not. Nonetheless, it is still useful to expand the
wave functions in eigenfunctions of the angular momentum.
In the isotopic spin example, the neutron-proton system in
the absence of electromagnetism shows symmetries under
the group of special unitary transformations SU 2 . Once
electromagnetism is included, the symmetry is broken, but
nonetheless, there are still some useful manifestations. Like-
wise, the elementary particles in the absence of symmetry
breaking are invariant under the special unitary group SU 3 .
If this symmetry is broken, it is still possible to derive rela-
tions among the masses, but their origin was still unex-
plained. However, in the early 1960s, Yoichiro Nambu and
others showed that a second kind of symmetry breaking was
possible in quantum mechanics, which was called spontane-
ous symmetry breaking. To see what it means, we consider
an example that has nothing to do with quantum mechanics.
Consider the equation
d
2
f x
/
dx
2
=
x
2
+
cx
.
1
If
c
=0, Eq.
is symmetric under
x
inversion:
x
→
−
x
. We
drop the integration constants and obtain
f x
= 1
/
12
x
4
+
c
/
6
x
3
.
2
The solution with
c
0 is not
x
inversion symmetric. This,
lack of symmetry is in the spirit of Wigner-Weyl symmetry
breaking. But consider
d
2
f x
/
dx
2
=
x
2
.
3
Equation
is
x
-inversion symmetric. The solution is
f x
= 1
/
12
x
4
+
Bx
+
C
.
4
Unless
B
is zero, the solution does not have the same sym-
metry as Eq.
The symmetry has been “spontaneously
broken” by the choice of solution, determined by the initial
conditions.
To see how spontaneous symmetry breaking works in
quantum mechanics and to understand the consequences, we
consider an example, the self-interactions of a complex sca-
lar field,
x
,
t
=
1
x
,
t
+
i
2
x
,
t
, where
1,2
are real
fields. This field describes a charged spinless particle. It is
the simplest example that I know, and it is the one that was
first considered historically. See, for example, Ref.
It will
lead us to the Higgs mechanism.
We begin by exhibiting the Lagrangian of a free complex
scalar classical field, which corresponds to a particle with a
mass
m
. To make the notation more compact, I will employ
the usual convention of setting
c
=1. Thus
L
=
x
†
x
−
m
2
x
†
x
=
/
t
†
/
t
−
†
−
m
2 †
,
5a
and the corresponding Hamiltonian is
H
=
/
t
†
/
t
+
†
•
+
m
2 †
.
5b
We shall be interested in minimizing the energy. The kinetic
terms are always positive definite, and therefore to minimize
the energy associated with
H
, we must take the fields to be
constants in spacetime.
The Lagrangian in Eq.
yields the field equation
2
t
−
2
−
m
2
x
= 0,
6
where I have simplified the notation by using
x
for
x
,
t
. The
solutions of the field equations for individual particle and
antiparticle states with momentum
p
have energies
p
2
+
m
2
,
which establishes the interpretation of
m
as a particle mass.
The Lagrangian is invariant under the global gauge trans-
formation
→
exp
i
, where is a real number. It is not
invariant under the spacetime dependent or local gauge
transformation
→
exp
i x
. Under an infinitesimal
transformation, a term is added to the Lagrangian of the
form, with
,
standing for the derivative with respect to the
th coordinate
L
=
i
,
†
−
†
,
J
.
7
If we think of as a dynamical variable, then we can write
down the Euler–Lagrange equation for as
L
/
, =
L
/
.
8
The term on the left is the divergence of the current gener-
ated by the local gauge transformation, and the term on the
26
26
Am. J. Phys., Vol. 79, No. 1, January 2011
Jeremy Bernstein
