Stars as Laboratories for Fundamental Physics - MPP Theory Group

Oscillations of Trapped Neutrinos 317
mass matrix M which is nondiagonal in the interaction basis. Even
in this case the “wrong” helicity states will be ignored because in the
ultrarelativistic regime spin-flip reactions are suppressed by an approximate
factor (m ν /2E ν ) 2 ≪ 1. In this limit lepton number violating
effects from possible Majorana masses are also ignored.
In the absence of interactions Ψ satisfies the free Dirac equation, implying
a p (t) = a p (0) exp(−iΩ 0 pt) and b p (t) = b p (0) exp(−iΩ 0 pt), where
Ω 0 p ≡ ( p 2 + M 2) 1/2
(9.5)
is a n×n matrix of “vacuum oscillation frequencies.” In the mass basis
it has only diagonal elements which are the energies E i = (p 2 + m 2 i ) 1/2 .
Therefore, one may use
∫
H 0 = dp
n∑
i,j=1
[
a
†
i(p)Ω 0 ij(p)a j (p) + b † j(p)Ω 0 ij(p)b i (p) ] (9.6)
as a free neutrino Hamiltonian.
In order to find the evolution of ρ p and ρ p one needs to study the
equations of motion of the n × n operator matrices
ˆρ ij (p, t) ≡ a † j(p, t)a i (p, t) and ˆρ ij (p, t) ≡ b † i(p, t)b j (p, t) . (9.7)
With a Hamiltonian H their evolution is given by Heisenberg’s equation,
i∂ t ˆρ = [ˆρ, H] , (9.8)
and similar for ˆρ p . With H = H 0 from Eq. (9.6) one finds
i∂ t ˆρ p = [Ω 0 p, ˆρ p ] and i∂ t ˆρ p = −[Ω 0 p, ˆρ p ] . (9.9)
Taking expectation values on both sides yields equations of motion
for ρ p and ρ p . For two flavors one may use the representation Ω 0 p =
ω 0 p + 1 2 V p · σ and ρ p = 1 2 f p(1 + P p · σ). Then Eq. (9.9) leads to the
precession formulas Ṗp = V p × P p and Ṗp = −V p × P p .
9.2.4 Interaction with a Background Medium
Interactions with a medium are introduced by virtue of a general interaction
Hamiltonian H int (B, Ψ) which is a functional of the neutrino
field Ψ and a set B of background fields; specific cases will be discussed
in Sects. 9.3 and 9.4 below. The equation of motion for ˆρ p is found from