Correct $\Delta m^2_{ij}$ Dependence for Neutrino Oscillation Formulae

Abstract

The time translation operator for neutrino mass states is often taken to be $e^{-iEt/\hbar}$. This is not relativistically invariant. In kaon mixing, physicists use $e^{-imc^2\tau/\hbar}$ where $\tau$ is the proper time of the kaon state. The factor $mc^2\tau$ is the rest frame value of the four vector product $p_\mu x^\mu$ which is an invariant quantity. If $-i p_\mu x^\mu$ is used in neutrino oscillation formulae instead of $-iEt$, the scale of the $\Delta m^2_{ij}$ is reduced by a factor of two.