  We propose a new unified theoretical framework to construct equivalent
representations of the multi-state Hamiltonian operator and present several
approaches for the mapping onto the Cartesian phase space. After mapping an
F-dimensional Hamiltonian onto an F+1- dimensional space, creation and
annihilation operators are defined such that the F+1 dimensional space is
complete for any combined excitations. Commutation and anti-commutation
relations are then naturally derived, which show that the underlying degrees of
freedom are neither bosons nor fermions. This sets the scene for developing
equivalent expressions of the Hamiltonian operator in quantum mechanics and
their classical/semiclassical counterparts. Six mapping models are presented as
examples. The framework also offers a novel way to derive such as the
well-known Meyer-Miller model.
