Covariant Wigner Function Approach to the Boltzmann Equation for Mixing Fermions in Curved Spacetime

Based on the covariant Wigner function approach we derive the quantum Boltzmann equation for fermions with flavor mixing in general curved spacetime. This work gives a rigorous theoretical framework to investigate the flavor oscillation phenomena taking the gravitational effect into account. It is shown that the Boltzmann equation of the lowest order of the expansion with respect to $\hbar$ reproduces the previous result which was derived in the relativistic limit on the Minkowski background spacetime. It is demonstrated that the familiar formula for the vacuum neutrino oscillation can be obtained by solving the Boltzmann equation. Higher order effects of the $\hbar$-expansion is also briefly discussed.