  We present a semiclassical formalism for antiferromagnetic (AFM) magnonics
which promotes the central ingredient of spin wave chirality, encoded in a
quantity called magnonic isospin, to a first-class citizen of the theory. We
use this formalism to unify results of interest from the field under a single
chirality-centric formulation. Our main result is that the isospin is governed
by unitary time evolution, through a Hamiltonian projected down from the full
spin wave dynamics. Because isospin is SU(2)-valued, its dynamics on the Bloch
sphere are precisely rotations - which, in general, do not commute.
Consequently, the induced group of operations on AFM spin waves is nonabelian.
This is a paradigmatic departure from ferromagnetic magnonics, which operates
purely within the abelian group generated by spin wave phase and amplitude. Our
investigation of this nonabelian magnonics in AFM insulators focuses on
studying several simple gate operations, and offering in broad strokes a
program of study for interesting new logic families in antiferromagnetic spin
wave systems
