  This paper deals with a general method for the reduction of quantum systems
with symmetry. For a Riemannian manifold M admitting a compact Lie group G as
an isometry group, the quotient space Q = M/G is not a smooth manifold in
general but stratified into a collection of smooth manifolds of various
dimensions. If the action of the compact group G is free, M is made into a
principal fiber bundle with structure group G. In this case, reduced quantum
systems are set up as quantum systems on the associated vector bundles over Q =
M/G. This idea of reduction fails, if the action of G on M is not free.
However, the Peter-Weyl theorem works well for reducing quantum systems on M.
When applied to the space of wave functions on M, the Peter-Weyl theorem
provides the decomposition of the space of wave functions into spaces of
equivariant functions on M, which are interpreted as Hilbert spaces for reduced
quantum systems on Q. The concept of connection on a principal fiber bundle is
generalized to be defined well on the stratified manifold M. Then the reduced
Laplacian is well defined as a self-adjoint operator with the boundary
conditions on singular sets of lower dimensions. Application to quantum
molecular mechanics is also discussed in detail. In fact, the reduction of
quantum systems studied in this paper stems from molecular mechanics. If one
wishes to consider the molecule which is allowed to lie in a line when it is in
motion, the reduction method presented in this paper works well.
