  To understand recent works on classical and quantum spin equations and their
topological classification, we develop a unified mathematical framework for
bosonic BdG systems and associated classical wave equations; it applies not
just to equations that describe quantized spin excitations in magnonic crystals
but more broadly to other systems that are described by a BdG hamiltonian.
Because here the generator of dynamics, the analog of the hamiltonian, is
para-, aka Krein-, hermitian but not hermitian, the theory of Krein spaces
plays a crucial role. For systems which are thermodynamically stable, the
classical equations can be expressed as a Schr\"odinger equation with a
hermitian hamiltonian. We then proceed to apply the Cartan-Altland-Zirnbauer
classification scheme: to properly understand what topological class these
equations belong to, we need to conceptually distinguish between symmetries and
constraints. Complex conjugation enters as a particle-hole constraint (as
opposed to a symmetry), since classical waves are necessarily real-valued.
Because of this distinction only commuting symmetries enter in the topological
classification. Our arguments show that the equations for spin waves in
magnonic crystals are a system of class A, the same topological class as
quantum hamiltonians describing the Integer Quantum Hall Effect. Consequently,
the magnonic edge modes first predicted by Shindou et al. are indeed analogs of
the Quantum Hall Effect, and their net number is topologically protected.
