Multicomponent WKB and Quantization

Hamiltonians whose symbols are not simply real valued, but matrix or, more generally, endomorphism valued functions appear in many places in physics, examples being the Dirac equation, multicomponent wave equations like electrodynamics in media, and Yang-Mills theories, and the Born-Oppenheimer approximation in molecular physics. Whereas the semiclassical and WKB approximation of scalar systems is well understood, this is not the case to the same extent for Hamiltonians with matrix valued symbols. In particular, semiclassical states in the scalar case have a nice geometric interpretation as half densities on Lagrangean submanifolds invariant under the Hamiltonian flow, and discrete spectra may be computed using the Bohr-Sommerfeld condition, taking into account the Maslov correction [14, 1, 7, 10]. In the multicomponent case, the analogous structures are not known. The usual WKB ansatz Ψ(x) = a(x, ~) exp(iS(x)/~)