  We revisit an antiferromagnetic quantum phase transition with $\bm{Q} = 2
\bm{k}_{F}$, where $\bm{Q}$ is an ordering wave vector and $\bm{k}_{F}$ is a
Fermi momentum. Reformulating the Hertz-Moriya-Millis theory within the strong
coupling approach to diagonalize the spin-fermion coupling term and performing
the scaling analysis for an effective field theory with quantum corrections in
the Eliashberg approximation, we propose a novel interacting fixed point for
this antiferromagnetic quantum phase transition, where antiferromagnetic spin
fluctuations become locally critical to interact with renormalized electrons.
The emergence of local quantum criticality suggests a mechanism of $\omega/T$
scaling for antiferromagnetic quantum criticality, generally forbidden in the
context of the Hertz-Moriya-Millis theory.
