  We first recall a fact which is well-known among mathematical physicists
although lesser-known among theoretical physicists that the standard quantum
mechanics over a complex Hilbert space, is a Hamiltonian mechanics, regarding
the Hilbert space as a linear real manifold equipped with its canonical
symplectic form and restricting only to the expectation-value functions of
Hermitian operators. Then in this framework, we reformulate the structure of
quantum mechanics in the language of symplectic manifolds and avoid linear
structure of Hilbert space in such a way that the results can be stated for an
arbitrary symplectic manifold.
