Relating quantum mechanics and kinetics of neutrino oscillations

Abstract

Simultaneous treatment of neutrino oscillations and collisions in astrophysical environments requires the use of (quantum) kinetic equations. Despite major advances in the field of quantum kinetics, structure of the kinetic equations and their consistency with the uncertainty principle are still debated. The goals of the present work are threefold. First, it clarifies structure of the Liouville term. Second, it aims at demonstrating that the kinetic equation is in accord with the uncertainty principle and accounts for neutrino wave packet separation. Finally, we derive kinetic equation for neutrinos propagating in an advective medium and show that in the relativistic limit it reverts to the one derived by Sigl and Raffelt. The obtained results speak in favor of applying kinetic equations for analysis of neutrino propagation in exploding supernovae where neutrino oscillations and collisions, as well as the phenomenon of wave packet separation, might be equally important.

Accounting for the Heisenberg and Pauli principles in the kinetic approach to neutrino oscillations

“…From equation (2.4) it then follows that U(t, t P , p ± 1 2 ∆) ≈ U(t, t P , p)e ∓ i 2 ∆vp(t−t P ) . The resulting approximate Wigner function factorizes into a product a phase and a shape factors [22],…”

“…shape factor is determined (in the ultrarelativistic limit, see reference [22] for an analysis of subleading corrections) only by the initial conditions. For the initial conditions specified in equation (2.5) the flavor-basis counterpart of the shape factor, g αβ = U αi g ij U † jβ , reads [22]…”

“…An evolution equation for the Wigner function of a neutrino propagating in a general potential can be derived directly from the Schrödinger equation as well. Conceptually, the derivation is similar to that presented in reference [22], but here we work directly with the Wigner function and use the relativistic approximation, to facilitate comparison with the kinetic equation for the matrix of densities addressed in section 4. For ij (t, x) the momentum-representation counterpart of the Schrödinger equation (2.1) reads…”

“…(2.46) Performing the first-order gradient expansion, i.e. keeping at most the first-order derivatives with respect to x or p, we can recast equation (2.46) in the familiar form [22],…”

“…In the approximation ∂ p H ≈ v p used above the first anticommutator on its left-hand side simplifies to v p ∂ x (t, x, p) thereby reproducing the velocity term of the Liouville operator in equation (2.37). A detailed analysis of the Liouville term can be found in reference [22]. Let us emphasize, that the presented derivation of the evolution equation does not rely in any way on coarse-graining of the Wigner function [44].…”