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My book on QFT says that neutrinos are well described by left-handed Weyl spinor. The classical Lorentz-invariant Lagrangian density for that field is:

$$ \mathcal{L} = i\psi^{\dagger}_L\overline\sigma^\mu\partial_\mu\psi_L $$

that clearly has a $U(1)$ symmetry for field transformations such $\psi_L \to e^{i\theta}\psi_L $, with $\theta$ constant.

In the case of Dirac fields, this symmetry is linked to the electric charge of the field quanta, but neutrinos have no charge. Then my question is the following: what is the physical explanation of the Noether charge associated to this symmetry?

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The charge associated to the $U(1)$ symmetry you mention is called weak hypercharge. The relation with the electric charge is the following $Q= T_3+Y/2$ where $T_3$ is the third generator of the $U(2)$ symmetry representing the weak isospin and $Q$ the electric charge. This relation holds for all leptons. Neutrinos have weak isospin $+1/2$ and weak hypercarge $-1$ so that $Q=0$

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