Neutrino in the presence of gravitational fields: Separation of variables

Abstract

It is well known that the most complete information about single-particle states is contained in its wave function. For spin-1/2 particles this means that it is necessary to have exact solutions of the Dirac equation. In particular, in the case of neutrinos in the presence of gravity, it is necessary to solve the covariant Dirac equation. At present, the existence of neither massless neutrinos (electron neutrinos) nor massive neutrinos (muon and τ neutrinos) cannot be excluded. Since for massless neutrinos any solution of the Dirac equation is also a solution of the Weyl equation, there exists the possibility of studying, from a unified point of view, massive as well as massless neutrinos by means of the Dirac equation. In the search of exact solutions of systems of partial differential equations one can proceed as follows: (a) separation of variables and (b) solution of the corresponding ordinary differential equations. In the present paper, a complete analysis of the separation of variables in the Dirac equation for massive as well as for massless neutrinos is carried out by means of the algebraic method [J. Math. Phys. 30, 2132 (1989)]. It is found that for the massless neutrinos, there are further possibilities of separation of variables, not valid for the massive case.