  We identify infinitely many non-invertible generalized global symmetries in
QED and QCD for the real world in the massless limit. In QED, while there is no
conserved Noether current for the $U(1)_\text{A}$ axial symmetry because of the
ABJ anomaly, for every rational angle $2\pi p/N$, we construct a conserved and
gauge-invariant topological symmetry operator. Intuitively, it is a composition
of the axial rotation and a fractional quantum Hall state coupled to the
electromagnetic $U(1)$ gauge field. These conserved symmetry operators do not
obey a group multiplication law, but a non-invertible fusion algebra over TQFT
coefficients. They act invertibly on all local operators as axial rotations,
but non-invertibly on the 't Hooft lines. These non-invertible symmetries lead
to selection rules, which are consistent with the scattering amplitudes in QED.
We further generalize our construction to QCD, and show that the coupling
$\pi^0 F\wedge F$ in the effective pion Lagrangian is necessary to match these
non-invertible symmetries in the UV. Therefore, the conventional argument for
the neutral pion decay using the ABJ anomaly is now rephrased as a matching
condition of a generalized global symmetry.
