# Why are neutrino flavour eigenstates expressed in terms of the elements of the complex conjugate of the PMNS matrix?

If we have

$$\begin{pmatrix} \nu_e\\ \nu_{\mu} \\ \nu_{\tau} \end{pmatrix} =\begin{pmatrix} U_{e1} & U_{e2} & U_{e3} \\ U_{\mu1} & U_{\mu2} & U_{\mu3} \\ U_{\tau1} & U_{\tau2} & U_{\tau3} \end{pmatrix} \begin{pmatrix} \nu_1 \\ \nu_2 \\ \nu_3 \end{pmatrix},\tag{1}$$

then why are flavour eigenstates expressed using the complex conjugated elements of the PMNS matrix:

$$|\nu_{\alpha}\rangle=\sum_{i=1}^3U_{\alpha i}^*|\nu_i\rangle,$$ (where $\alpha = e,\ \mu,\ \tau$) rather than $$|\nu_{\alpha}\rangle=\sum_{i=1}^3U_{\alpha i}|\nu_i\rangle,$$ as suggested by $(1)$. Wouldn't it make more sense to define the PMNS matrix as its complex conjugate in the first place?

• What you suggest is done in case of anti-neutrinos. – Yuzuriha Inori Mar 15 '18 at 9:10
• Essential duplicate. – Cosmas Zachos Jul 23 at 14:45