198
17. Spontaneously Broken Global Symmetry
where the second line follows from
ˆ
[P
μ ˆ
, Q] = 0
(17.12)
ˆ
ˆ
since Q is an internal symmetry. But the vacuum is an eigenstate of P
μ with
eigenvalue zero, and so
‹ | ˆ 0
ˆ
ˆ
0 j (x)Q|0› = ‹0| ˆ j
0 (0)Q|0›
(17.13)
which states that the matrix element we started from is in fact independent
ˆ
of x. Now consider the norm of Q|0›:
∫
‹0| ˆ ˆ
QQ|0› =
d
3
x‹0| ˆ j
0
ˆ
(x)Q|0›
(17.14)
∫
=
d
3
x‹0| ˆ j
0
ˆ
(0)Q|0›,
(17.15)
ˆ
which must diverge in the infinite volume limit, unless Q|0› = 0. Thus either
ˆ
Q|0›
ˆ
= 0 or Q|0› has infinite norm. The foregoing can be easily generalized
ˆ
to non-Abelian symmetry operators T i .
Remarkably enough, the argument can also, in a sense, be reversed. Coleman (1966) proved that if an operator
∫
ˆ
Q(t) = d
3
x ˆ j
0 (x)
(17.16)
is the spatial integral of the μ = 0 component of a 4-vector (but not assumed
to be conserved), and if it annihilates the vacuum
ˆ
Q(t)|0› = 0,
(17.17)
then in fact ∂ ˆ
μ j
μ
ˆ
= 0, Q is independent of t, and the symmetry is unitarily
ˆ
ˆ
implementable by operators U = exp(iλQ).
We might now simply proceed to the chiral symmetry application. We
believe, however, that the concept of spontaneous symmetry breaking is so
important to particle physics that a more extended discussion is amply justified. In particular, there are crucial insights to be gained by considering the
analogous phenomenon in condensed matter physics. After a brief look at the
ferromagnet, we shall describe the Bogoliubov model for the ground state of a
superfluid, which provides an important physical example of a spontaneously
broken global Abelian U(1) symmetry. We shall see that the excitations away
from the ground state are massless modes and we shall learn, via Goldstone’s
theorem, that such modes are an inevitable result of spontaneously breaking a
global symmetry. Next, we shall introduce the ‘Goldstone model’ which is the
simplest example of a spontaneously broken global U(1) symmetry, involving
just one complex scalar field. The generalization of this to the non-Abelian
case will draw us in the direction of the Higgs sector of the Standard Model.
17. Spontaneously Broken Global Symmetry
where the second line follows from
ˆ
[P
μ ˆ
, Q] = 0
(17.12)
ˆ
ˆ
since Q is an internal symmetry. But the vacuum is an eigenstate of P
μ with
eigenvalue zero, and so
‹ | ˆ 0
ˆ
ˆ
0 j (x)Q|0› = ‹0| ˆ j
0 (0)Q|0›
(17.13)
which states that the matrix element we started from is in fact independent
ˆ
of x. Now consider the norm of Q|0›:
∫
‹0| ˆ ˆ
QQ|0› =
d
3
x‹0| ˆ j
0
ˆ
(x)Q|0›
(17.14)
∫
=
d
3
x‹0| ˆ j
0
ˆ
(0)Q|0›,
(17.15)
ˆ
which must diverge in the infinite volume limit, unless Q|0› = 0. Thus either
ˆ
Q|0›
ˆ
= 0 or Q|0› has infinite norm. The foregoing can be easily generalized
ˆ
to non-Abelian symmetry operators T i .
Remarkably enough, the argument can also, in a sense, be reversed. Coleman (1966) proved that if an operator
∫
ˆ
Q(t) = d
3
x ˆ j
0 (x)
(17.16)
is the spatial integral of the μ = 0 component of a 4-vector (but not assumed
to be conserved), and if it annihilates the vacuum
ˆ
Q(t)|0› = 0,
(17.17)
then in fact ∂ ˆ
μ j
μ
ˆ
= 0, Q is independent of t, and the symmetry is unitarily
ˆ
ˆ
implementable by operators U = exp(iλQ).
We might now simply proceed to the chiral symmetry application. We
believe, however, that the concept of spontaneous symmetry breaking is so
important to particle physics that a more extended discussion is amply justified. In particular, there are crucial insights to be gained by considering the
analogous phenomenon in condensed matter physics. After a brief look at the
ferromagnet, we shall describe the Bogoliubov model for the ground state of a
superfluid, which provides an important physical example of a spontaneously
broken global Abelian U(1) symmetry. We shall see that the excitations away
from the ground state are massless modes and we shall learn, via Goldstone’s
theorem, that such modes are an inevitable result of spontaneously breaking a
global symmetry. Next, we shall introduce the ‘Goldstone model’ which is the
simplest example of a spontaneously broken global U(1) symmetry, involving
just one complex scalar field. The generalization of this to the non-Abelian
case will draw us in the direction of the Higgs sector of the Standard Model.