Stars as Laboratories for Fundamental Physics - MPP Theory Group

138 Chapter 4
4.6.2 Nonrelativistic Limit
In a nuclear medium one is interested primarily in the nonrelativistic
limit. The vector current is then dominated by V 0 = ρ = χ † χ where χ
is a nucleon two-spinor. Here, ρ is the operator for the nucleon number
density. The spatial component V is suppressed by a nonrelativistic
velocity factor v. The reverse applies to the axial-vector current where
A 0 is suppressed; it is dominated by the spin density s = 1 2 χ† τ χ where
τ is a vector of Pauli matrices.
In Eqs. (4.40) and (4.41) all terms arise from correlators such as
⟨A 0 V i ⟩ which are suppressed by v 2 , except for the first term of S µν
V
which arises from ⟨V 0 V 0 ⟩, and the second term of S µν
A which arises
from ⟨A i A i ⟩. Therefore, the only unsuppressed components are
S 00
V (ω, k) = S ρ (ω, k) and S ij
A (ω, k) = S σ (ω, k) δ ij , (4.44)
where the density and spin-density dynamical structure functions are
S ρ (ω, k) = 1 ∫ +∞
dt e ⟨ iωt ρ(t, k)ρ(0, −k) ⟩ ,
n B −∞
S σ (ω, k) = 4 ∫ +∞
dt e ⟨ iωt s(t, k) · s(0, −k) ⟩ (4.45)
3n B −∞
(Iwamoto and Pethick 1982).
In order to determine the overall normalization in the nonrelativistic
limit consider the static structure function for an ensemble of N B
nucleons enclosed in a large volume V . At a given time the system is
characterized by a wavefunction Ψ which depends on the locations r i
of the nucleons. Then ρ(r) |Ψ⟩ = ∑ N B
i=1 δ(r − r i ) |Ψ⟩ while the Fouriertransformed
operator V −1 ∫ d 3 r e ik·r ρ(r) is
ρ(k)|Ψ⟩ = 1 V
N B ∑
i=1
e ik·r i
|Ψ⟩. (4.46)
Therefore, a given configuration of nucleons yields
⟨Ψ|ρ(k)ρ(−k)|Ψ⟩ = 1 V
∑N B
e ik·r ij
i,j=1
= N B
V
+ 1 V
N B ∑
i,j=1
i≠j
e ik·r ij
, (4.47)
where r ij ≡ r i −r j . When averaged over a thermal ensemble the second
term will disappear if there are no spatial correlations, and n B = N B /V
so that S ρ (k) = 1.