Fermionic bilinear charge algebra ($[Q_A,Q_B]=Q_{[A,B]},Q_A=\int d^{3}x:{\psi }^{†}A\psi :$)

For constant internal matrices acting on Dirac fields, the equal-time canonical anticommutation relations givewith the corresponding spatial delta functions for fields. The quartic terms cancel after fermion reordering. Integrating proves the displayed algebra for well-defined global Noether charges, with normal ordering and a regulator that preserves the nonanomalous internal symmetry. Orthogonal chiral projectors make the left/right mixed commutators zero. This reduces current-charge closure to a finite matrix calculation.

For normalized triplets $(E^{+},n,e{)}_L$ and $(E^{+},N,e{)}_R$, let the raising matrix have ones in entries $(1,2)$ and $(2,3)$. Its commutator with its adjoint is $\mathrm{diag}(1,0,-1)$. Neutral mixing in $n=c_{\alpha }\nu +s_{\alpha }N$ leaves this result unchanged because $c_{\alpha }^2+s_{\alpha }^2=1$. With current normalization using $1\mp {\gamma }^5=2P_{L,R}$, fermionic bilinear charge algebra gives the displayed proportionality to the leptonic electric charge $Q_{\ell }=e\int :(E^{+†}{E}^{+}-e^{†}e):$. Other current normalizations change the numerical prefactor, not closure. Equal-time algebra does not itself imply conservation of a fermionic weak charge in a symmetry-broken theory.