17.4. Goldstone’s theorem
209
contradiction. Instead, however, knowing that |ground› B is not an eigenˆ
ˆ
state of N , we can regard U
−1
α |ground› B as an ‘alternative ground state’
|ground, α› B such that
B ‹ground, α| ˆ
φ|
ˆ
ground, α› B = e
−iα
B ‹ground|φ B |ground› B ,
(17.57)
the original choice (17.52) corresponding to α = 0. There are infinitely many
such ground states since α is a continuous parameter. No physical consequence
follows from choosing one rather than another, but we do have to choose one,
thus ‘spontaneously’ breaking the symmetry. In choosing say α = 0, we are
ˆ
deciding (arbitrarily) to pick the ground state such that B ‹ground|φ|ground› B
is aligned in the ‘real’ direction. By hypothesis, a similar situation obtains
for the true ground state. None of the states |ground, α› is an eigenstate for
ˆ
N : instead, they are certain coherent superpositions of states with different
ˆ
eigenvalues N , such that the expectation value of φ has a definite phase.
17.4 Goldstone’s theorem
We return to quantum field theory proper, and show following Goldstone
(1961) (see also Goldstone, Salam and Weinberg 1962) how in case (b) of the
Fabri–Picasso theorem massless particles will necessarily be present. Whether
these particles will actually be observable depends, however, on whether the
theory also contains gauge fields. In this chapter we are concerned solely with
global symmetries, and gauge fields are absent; the local symmetry case is
treated in chapter 19.
ˆ
Suppose, then, that we have a Lagrangian
ˆ
L with a continuous symmetry
generated by a charge Q, which is independent of time, and is the space
integral of the μ = 0 component of a conserved Noether current:
∫
ˆ
Q = ˆ j 0 (x) d
3
x.
(17.58)
We consider the case in which the vacuum of this theory is not invariant, i.e.
ˆ
is not annihilated by Q.
ˆ
Suppose φ(y) is some field operator which is not invariant under the continuous symmetry in question, and consider the vacuum expectation value
‹0| ˆ ˆ
[Q, φ(y)]|0›.
(17.59)
Just as in equation (17.13), translation invariance implies that this vev is, in
ˆ
fact, independent of y, and we may set y = 0. If Q were to annihilate |0›, the
expression (17.18) would clearly vanish: we investigate the consequences of it
ˆ
ˆ
not vanishing. Since φ is not invariant under Q, the commutator in (17.59) will
ˆ
give some other field, call it φ
′ (y); thus the hallmark of the hidden symmetry
209
contradiction. Instead, however, knowing that |ground› B is not an eigenˆ
ˆ
state of N , we can regard U
−1
α |ground› B as an ‘alternative ground state’
|ground, α› B such that
B ‹ground, α| ˆ
φ|
ˆ
ground, α› B = e
−iα
B ‹ground|φ B |ground› B ,
(17.57)
the original choice (17.52) corresponding to α = 0. There are infinitely many
such ground states since α is a continuous parameter. No physical consequence
follows from choosing one rather than another, but we do have to choose one,
thus ‘spontaneously’ breaking the symmetry. In choosing say α = 0, we are
ˆ
deciding (arbitrarily) to pick the ground state such that B ‹ground|φ|ground› B
is aligned in the ‘real’ direction. By hypothesis, a similar situation obtains
for the true ground state. None of the states |ground, α› is an eigenstate for
ˆ
N : instead, they are certain coherent superpositions of states with different
ˆ
eigenvalues N , such that the expectation value of φ has a definite phase.
17.4 Goldstone’s theorem
We return to quantum field theory proper, and show following Goldstone
(1961) (see also Goldstone, Salam and Weinberg 1962) how in case (b) of the
Fabri–Picasso theorem massless particles will necessarily be present. Whether
these particles will actually be observable depends, however, on whether the
theory also contains gauge fields. In this chapter we are concerned solely with
global symmetries, and gauge fields are absent; the local symmetry case is
treated in chapter 19.
ˆ
Suppose, then, that we have a Lagrangian
ˆ
L with a continuous symmetry
generated by a charge Q, which is independent of time, and is the space
integral of the μ = 0 component of a conserved Noether current:
∫
ˆ
Q = ˆ j 0 (x) d
3
x.
(17.58)
We consider the case in which the vacuum of this theory is not invariant, i.e.
ˆ
is not annihilated by Q.
ˆ
Suppose φ(y) is some field operator which is not invariant under the continuous symmetry in question, and consider the vacuum expectation value
‹0| ˆ ˆ
[Q, φ(y)]|0›.
(17.59)
Just as in equation (17.13), translation invariance implies that this vev is, in
ˆ
fact, independent of y, and we may set y = 0. If Q were to annihilate |0›, the
expression (17.18) would clearly vanish: we investigate the consequences of it
ˆ
ˆ
not vanishing. Since φ is not invariant under Q, the commutator in (17.59) will
ˆ
give some other field, call it φ
′ (y); thus the hallmark of the hidden symmetry