188
26 ∗ The Goldstone Theorem at Non-zero Temperature
satisfies (with b defined in (25.29))
i lim
k→0
[ ˜
W(k, ω) − ˜
W(−k, −ω)] = (2π)
−1 b δ(ω),
(26.1)
as a distribution in ω, and
lim
k→0
[ ˜
W(k, ω) + ˜
W(−k, −ω)] = 2i ((2π)
−1 b/β) δ
(ω)
(26.2)
as a distribution in ω on test functions g(ω) ∈ (1 − e
−β ω
) S(R), in particular on
antisymmetric test functions of compact support.
Proof. The first part follows as in Theorem 25.3; by the same arguments one has
˜
J (ω) ≡ lim
k→0
˜
J (k, ω) = i lim
k→0
[ ˜
W(k, ω) − ˜
W(−k, −ω)] =
lim
k→0
−2(2π)
2 Im < j 0 ( f ) d E(ω) d E(k) A >= (2π)
−1 b δ(ω),
(26.3)
as a distribution in ω.
Now, the KMS condition gives
˜
J (k, ω) = i (1 − e
−β ω
) ˜
W(k, ω)
(26.4)
and, by the reality of J (x, t),
˜
J (k, ω) = ˜
J (−k, −ω) = −i (1 − e
β ω
) ˜
W(−k, −ω).
(26.5)
By adding (26.4) to (26.5) times e
−βω , one gets
(1 + e
−β ω
) ˜
J (k, ω) = i (1 − e
−β ω
) [ ˜
W(k, ω) + ˜
W(−k, −ω)].
(26.6)
The charge (commutator) integrability condition implies that the right-hand side is
a continuous function of k, as a distribution in ω, so that, on test functions g(ω) ∈
(1 − e
−β ω
) S(R), also the term in square brackets on the r.h.s. of (26.6) has a limit
for k → 0.
Then (26.3), (26.6) imply (26.2). Clearly (1 − e
−β ω
) S(R) contains all the antisymmetric test functions of compact support.
The occurrence of the δ
should not appear strange; by the unitarity of the space and
time translations ˜
W(k, ω) is a measure in (k, ω), but in general it is not a measure
in ω for k fixed; in particular, it need not be a measure in the limit k → 0. By
the charge integrability condition, ˜
J (k, ω) and therefore (1 − e
−β ω
) ˜
W(k, ω) is a
continuous function of k as a distribution in ω, but this does not imply that ˜
W(k, ω)