Remarks on flavor-neutrino propagators and oscillation formulas

Abstract

We examine the general structure of the formulas of neutrino oscillations proposed by Blasone and Vitiello (BV). Reconstructing their formulas with the retarded propagators of the flavor-neutrino fields for the case of many flavors, we can get easily the formulas which satisfy the suitable boundary conditions and are independent of arbitrary mass parameters ${{\ensuremath{\mu}}_{\ensuremath{\rho}}},$ as obtained by BV for the case of two flavors. In this two-flavor case, our formulas reduce to those obtained by BV under a T-invariance condition. Furthermore, the reconstructed probabilities are shown to coincide with those derived with recourse to the mass Hilbert space ${\mathcal{H}}_{m}$ which is unitarily inequivalent to the flavor Hilbert space ${\mathcal{H}}_{f}.$ Such a situation is not found in the corresponding construction in the manner of BV. Then the new factors in BV's formulas, which modify the usual oscillation formulas, are not the trace of the flavor Hilbert space construction, but come from Bogolyubov transformation among the operators of spin-$\frac{1}{2}$ neutrinos with different masses.