In a separate direction, the idea of mapping variables is proposed to represent quantum transitions among discrete electronic states as classical-like motion of continuous phase space variables, thus treating all the DOFs on an equal footing. Historically, one of the most successful mapping theories in physical chemistry is the Meyer-Miller-Stock-Thoss (MMST) mapping formalism, which maps a N -level system onto N singly excited harmonic oscillators, and thus can be viewed as a generalization of the Schwinger's bosonization approach. The mapping variables of the MMST formalism are conjugate position and momentum of each mapping oscillator. This method, despite its great success and broad applications, has known flaws. This is because the MMST mapping operators belong to a larger Hilbert space that contains other states outside the singly excited oscillator (SEO) subspace of the mapping oscillators, whereas the MMST mapping procedure tries to map the original electronic subspace onto the SEO subspace. It thus requires a projection back to the singly excited mapping subspace to obtain accurate results. As a consequence, the identity operator is not preserved through the MMST mapping and there is an ambiguity of how to evaluate it. Related to the problem of the non-conserving identity, the nonadiabatic dynamics is sensitive to the separation between the state-dependent and the state-independent Hamiltonian. Moreover, projecting the Hilbert space to the SEO subspace also ruins the original simple commutation relations among the MMST mapping operators in their full Hilbert space, making the mapping Hamiltonian potentially containing more terms that require additional approximations to parameterize. Mathematically, the idea of mapping relation is referred to as the generalized Weyl correspondence, in which Lie groups and Lie algebras are the central components. Lie algebras are formed by commutation relations among generators with given structure constants; the elements of a connected matrix Lie group can be expressed as the exponential of the Lie algebra generators, i.e., the exponential map. On the other hand, a Lie group is also a differentiable manifold, which is a phase space with continuous variables. The dual identity of Lie groups naturally construct a bijective map between operators described by the generators represented in the Hilbert space and continuous functions on the differentiable manifold.