25.1 The Goldstone Theorem
171
Theorem 25.1 (Goldstone) If
I. β
λ , λ ∈ R is a one-parameter internal symmetry group, i.e.
[β
λ
, α x ] = 0, [β
λ
, α t ] = 0, ∀λ ∈ R, x ∈ R
s
, t ∈ R
(25.5)
II. β
λ is locally generated by a charge in the sense of (25.1-4) on a subalgebra A 0
of A, stable under time evolution
III. β
λ is spontaneusly broken in a representation π defined by a translationally
invariant ground state Ψ 0 , i.e. there exists a (self-adjoint) A ∈ A 0 such that
< δ A > 0 = i lim
R→∞
< [Q R , A] > 0 = b = 0,
(25.6)
then, in the subspace generated by the vectors Q R Ψ 0 , R ∈ R, the energy spectrum at zero momentum cannot have a gap (with respect to the ground state
energy).
Proof. Information on the energy momentum spectrum of the state Q R Ψ 0 is provided
by the support of the Fourier transform of the expectations (AΨ 0 , U (x) U (t) Q R Ψ 0 ),
or, in particular, of their imaginary part. This follows from the spectral theorem for
U (x) U (t) (or by inserting a complete set of improper eigenstates of energy and
momentum, see footnote below).
Thus, we are led to analyse the Fourier transform of
J (x, t) ≡ i < [ j 0 (x, t), A] > 0 = 2 Im < A j 0 (x, t) > 0 .
(25.7)
By using the property that β
λ commutes with α t and that it is generated by Q R on
an algebra stable under time translations, we have (Q R (t) = U (t)Q R U (t)
−1
),
i lim
R→∞
< [Q R (t), A] > 0 = i lim
R→∞
< [Q R , α −t (A)] > 0 =< δ(α −t (A)) > 0
=< α −t (δ A) > 0 =< δ A > 0 = i lim
R→∞
< [Q R , A] > 0 = b.
(25.8)
Then, we have
lim
R→∞
|x|≤R
d
s x J(x, t) = b,
namely, by Fourier transforming in x and t,
lim
k→0
˜
J (k, ω) = (2π)
−1 b δ(ω).
(25.9)
171
Theorem 25.1 (Goldstone) If
I. β
λ , λ ∈ R is a one-parameter internal symmetry group, i.e.
[β
λ
, α x ] = 0, [β
λ
, α t ] = 0, ∀λ ∈ R, x ∈ R
s
, t ∈ R
(25.5)
II. β
λ is locally generated by a charge in the sense of (25.1-4) on a subalgebra A 0
of A, stable under time evolution
III. β
λ is spontaneusly broken in a representation π defined by a translationally
invariant ground state Ψ 0 , i.e. there exists a (self-adjoint) A ∈ A 0 such that
< δ A > 0 = i lim
R→∞
< [Q R , A] > 0 = b = 0,
(25.6)
then, in the subspace generated by the vectors Q R Ψ 0 , R ∈ R, the energy spectrum at zero momentum cannot have a gap (with respect to the ground state
energy).
Proof. Information on the energy momentum spectrum of the state Q R Ψ 0 is provided
by the support of the Fourier transform of the expectations (AΨ 0 , U (x) U (t) Q R Ψ 0 ),
or, in particular, of their imaginary part. This follows from the spectral theorem for
U (x) U (t) (or by inserting a complete set of improper eigenstates of energy and
momentum, see footnote below).
Thus, we are led to analyse the Fourier transform of
J (x, t) ≡ i < [ j 0 (x, t), A] > 0 = 2 Im < A j 0 (x, t) > 0 .
(25.7)
By using the property that β
λ commutes with α t and that it is generated by Q R on
an algebra stable under time translations, we have (Q R (t) = U (t)Q R U (t)
−1
),
i lim
R→∞
< [Q R (t), A] > 0 = i lim
R→∞
< [Q R , α −t (A)] > 0 =< δ(α −t (A)) > 0
=< α −t (δ A) > 0 =< δ A > 0 = i lim
R→∞
< [Q R , A] > 0 = b.
(25.8)
Then, we have
lim
R→∞
|x|≤R
d
s x J(x, t) = b,
namely, by Fourier transforming in x and t,
lim
k→0
˜
J (k, ω) = (2π)
−1 b δ(ω).
(25.9)