Stars as Laboratories for Fundamental Physics - MPP Theory Group

140 Chapter 4
then yields
∫ +∞
−∞
∫
dω
+∞
2π ωS dω
ρ(ω, k) =
−∞
∫ +∞
2π ω −∞
dt e iωt ⟨ρ(t, k)ρ(0, −k)⟩
(4.51)
and a similar expression for S σ . Under the integral, a partial integration
with suitable boundary conditions allows one to absorb the ω factor,
at the expense of ρ(t, k) → ˙ρ(t, k). Because Heisenberg’s equation of
motion informs us that i ˙ρ = [ρ, H] with H the complete Hamiltonian
of the system one finds (Sigl 1995b)
∫ +∞
−∞
∫ +∞
−∞
dω
2π ω S ρ(ω, k) = 1 ⟨[ ] ⟩
ρ(k), H ρ(−k) ,
n B
dω
2π ω S σ(ω, k) = 4 ⟨[ ] ⟩
σ(k), H · σ(−k) . (4.52)
3n B
Here it was used, again, that ∫ dω e iωt = 2πδ(t) and ρ(k) ≡ ρ(0, k) and
σ(k) ≡ σ(0, k).
In order to evaluate this sum rule more explicitly one must assume
a specific form for the interaction Hamiltonian. In the simplest case of
a medium consisting of only one species of nucleons one may assume
that H consists of the kinetic energy for each nucleon, plus a general
nonrelativistic interaction potential between all nucleon pairs which
depends on the relative distance and the nucleon spins, i.e.
H =
N B ∑
i=1
p 2 i
2m N
+ 1 2
N B ∑
i,j=1
i≠j
V (r ij , σ i , σ j ), (4.53)
where again r ij ≡ r i −r j . One can then proceed to evaluate the commutators
in Eq. (4.52). By virtue of the continuity equation for the particle
number one can then show (Pines and Nozières 1966; Sigl 1995b)
∫ +∞
−∞
∫ +∞
−∞
dω
2π ω S ρ(ω, k) =
dω
2π ω S σ(ω, k) =
k2
2m N
,
k2
2m N
+ 4 ⟨
∑ NB
3n B
i,j=1
i≠j
[
σ(k), V (rij , σ i , σ j ) ] · σ(−k)
⟩
. (4.54)
For the density structure function this exact relationship is known as
the f-sum rule.