Correct $Δm^2_{ij}$ Dependence for Neutrino Oscillation Formulae

Abstract

The time translation operator for neutrino mass states is often taken to be e −iEt/h . This is not relativistically invariant. In kaon mixing, physicists use e −imc 2 τ /h where τ is the proper time of the kaon state. The factor mc 2 τ is the rest frame value of the four vector product p µ x µ which is an invariant quantity. If −ip µ x µ is used in neutrino oscillation formulae instead of −iEt, the scale of the ∆m 2 ij is reduced by a factor of two.