Maslov Complex Germ Method for Systems with First-Class Constraints

We develop the semiclassical mechanics of systems with first-class constraints. A convenient quantization method is the method based on modifying the inner product used in the theory. We consider semiclassical states of the wave-packet type (with small indeterminacies in both coordinates and momenta) that appear in the theory of the Maslov complex germ at a point. We show that these states have a nonzero norm only if the classical coordinates and momenta lie on the constraint surface. The set of semiclassical states of the wave-packet type forms a (“semiclassical”) bundle whose base is the set of admissible classical states and whose fibers are function spaces determining the form of the wave packet. In some cases, the difference between two semiclassical states has a zero norm; it is therefore possible to introduce the gauge equivalence relation. The semiclassical gauge transformations that are automorphisms of the semiclassical bundle form a Batalin quasigroup. We also study the action of semiclassical observables and of semiclassical evolution transformations. We show that they preserve the norm and the gauge equivalence relation and that the observables coinciding on the constraint surface act on semiclassical states similarly up to the gauge invariance.