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

Abstract

The time translation operator for neutrino mass states is often taken to be $e^{-iEt/\hslash }$. This is not relativistically invariant. In kaon mixing, physicists use $e^{-im{c}^2\tau /\hslash }$ where $\tau$ is the proper time of the kaon state. The factor $m{c}^2\tau$ 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 $\Delta m_{ij}^2$ is reduced by a factor of two.

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