Stars as Laboratories for Fundamental Physics - MPP Theory Group

146 Chapter 4
4.6.6 Classical vs. Quantum Result
In order to make contact with the quantum calculation that led to the
axion emission rate in Eq. (4.6) it is useful to juxtapose it with the
classical result in terms of the spin-density structure function. The
classical calculation led to
S σ (ω) = Γ σ
ω 2 × ω 2
ω 2 + Γ 2 σ/4
while the quantum result is
S σ (ω) = Γ σ
ω 2 × s(ω/T ) × { 1 for ω > 0,
e ω/T for ω < 0.
(4.70)
(4.71)
The function s(x) has the property s(0) = 1, it is even, and according
to the f-sum rule must decrease for large x. In Fig. 4.7 the classical and
quantum results are shown as dotted and dashed lines, respectively,
where for the purpose of illustration s(x) = (1 + x 2 /4) −1/4 has been
assumed.
Fig. 4.7. Classical, quantum, and compound spin-density structure function
in the nondegenerate limit according to Eq. (4.70) and Eq. (4.71) with
Γ σ /T = 0.2 and s(x) = (1 + x 2 /4) −1/4 .
This comparison highlights an important weakness of the classical
result: it does not obey the detailed-balance requirement S σ (ω) =
S σ (−ω) e ω/T . The classical correlators are invariant under time reflection
t → −t and thus symmetric under ω → −ω. Again, the classical
result is adequate only for |ω| ∼ < T .