MULTICOMPONENT WENTZEL-KRAMERS-BRILLOUIN APPROXIMATION ON ARBITRARY SYMPLECTIC-MANIFOLDS - A STAR PRODUCT APPROACH

It is known that in the Wentzel-Kramers-Brillouin approximation of mul ticomponent systems like the Dirac equation or Born-Oppenheimer approx imation, an additional phase appears apart from the Berry phase. So fa r, this phase was only examined in special cases, or under certain res trictive assumptions, namely, that the eigenspaces of the matrix or en domorphism valued symbol of the Hamiltonian form trivial bundles. We g ive a completely global derivation of this phase which does not depend on any choice of local trivializing sections. This is achieved using a star product approach to quantization. Furthermore, we give a system atic and global approach to a reduction of the problem to a problem de fined completely on the different ''polarizations.'' Finally, we discu ss to what extent it is actually possible to reduce the problem to a r eally scalar one, and make some comments on obstructions to the existe nce of global quasiclassical states. (C) 1998 American Institute of Ph ysics. [S0022-2488(98)03106-5].