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

2

/

2

=

2

+

.

1

If

=0, Eq.

is symmetric under

inversion:

→

−

. We

drop the integration constants and obtain

= 1

/

12

4

+

/

6

3

.

2

The solution with

0 is not

inversion symmetric. This,

lack of symmetry is in the spirit of Wigner-Weyl symmetry

breaking. But consider

2

/

2

=

2

.

3

Equation

is

-inversion symmetric. The solution is

= 1

/

12

4

+

+

.

4

Unless

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,

,

=

1

,

+

2

,

, 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

. To make the notation more compact, I will employ

the usual convention of setting

=1. Thus

=

†

−

2

†

=

/

†

/

−

†

−

2 †

,

5a

and the corresponding Hamiltonian is

=

/

†

/

+

†

•

+

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

, we must take the fields to be

constants in spacetime.

The Lagrangian in Eq.

yields the field equation

2

−

2

−

2

= 0,

6

where I have simplified the notation by using

for

,

. The

solutions of the field equations for individual particle and

antiparticle states with momentum

have energies

2

+

2

,

which establishes the interpretation of

as a particle mass.

The Lagrangian is invariant under the global gauge trans-

formation

→

exp

, where is a real number. It is not

invariant under the spacetime dependent or local gauge

transformation

→

exp

. 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

=

,

†

−

†

,

.

7

If we think of as a dynamical variable, then we can write

down the Euler–Lagrange equation for as

/

, =

/

.

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

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