Stars as Laboratories for Fundamental Physics - MPP Theory Group

316 Chapter 9
(Dolgov 1981)
ρ ij (p) = ⟨ a † j(p)a i (p) ⟩ and ρ ij (p) = ⟨ b † i(p)b j (p) ⟩ , (9.3)
where a i (p) and a † i(p) are the destruction and creation operators for
neutrinos of flavor i in mode p while b is for antineutrinos which otherwise
are referred to by overbarred quantities. The reversed order of the
flavor indices in the definition of ρ(p) guarantees that both matrices
transform in the same way under a unitary transformation in flavor
space. Also, for brevity ⟨ | . . . | ⟩ is always written as ⟨. . .⟩.
The diagonal elements of ρ p and ρ p are the usual occupation numbers
while the off-diagonal ones represent relative phase information.
In the nondegenerate limit, up to a normalization ρ p plays the role of
the previously defined single-particle density matrix. Therefore, the
ρ p ’s and ρ p ’s are well suited to account simultaneously for oscillations
and collisions. In fact, one can argue that a homogeneous neutrino
ensemble is completely characterized by these “matrices of densities”
(Sigl and Raffelt 1993). It remains to derive an equation of motion
which in the appropriate limits should reduce to the previous precession
equation, to a Boltzmann collision equation, and to Stodolsky’s
damping equation (9.1), respectively.
9.2.3 Free Evolution: Flavor Oscillations
The creation and annihilation operators which appear in the definition
of ρ p are the time-dependent coefficients of a spatial Fourier expansion
of the neutrino field [notation dp ≡ d 3 p/(2π) 3 ]
∫
Ψ(t, x) = dp [ a p (t)u p + b † ]
−p(t)v −p e ip·x . (9.4)
More precisely, a p is an annihilation operator for negative-helicity neutrinos
of momentum p while b † p is a creation operator for positivehelicity
antineutrinos. The Dirac spinors u p and v p refer to massless
negative-helicity particles and positive-helicity antiparticles, respectively;
the spinor normalization is taken to be unity. For n flavors,
a p and b † p are column vectors of components a i (p) and b † i(p), respectively.
They satisfy the anticommutation relations {a i (p), a † j(p ′ )} =
{b i (p), b † j(p ′ )} = δ ij (2π) 3 δ (3) (p − p ′ ).
In the massless limit and when only left-handed (l.h.) interactions
are present one may ignore the right-handed (r.h.) field entirely. However,
in order to include flavor mixing one needs to introduce a n × n