• entanglement
• liouville equation
• path integral
• superoperator
• von neumann equation

In this thesis I focus on what can be called quantum-classical divide. I consider the Liouville equation and the von Neumann equation, for a single massive particle, in a space called Liouville space. We introduce Liouville space in order to describe the dynamics of the density operator more conveniently. In Liouville space the density operator becomes a vector, we can think at it like a wavefunction, which allow us to use powerful methods already developed. Furthermore we can define operators acting on bra-states an ket-states simultaneously. To distinguish they from ordinary Hilbert space operator we call them superoperators. The first advantage appears, the Liouville equation and the von Neumann equation differ only by a characteristic superoperator. This parallel treatment gives powerful tools to study the differences of the quantum and classical evolutions. Therefore, a path integral is formulated in order to calculate the Green's functions of the evolution equations. A different interaction term in the action turns out to be the difference between the two Grenn's function. I study the free evolution of a classical particle and of a gaussian density matrix. In order to study potentials that cannot be solved exactly, perturbation theory is derived. A new type of Feynman diagrams appears. In more detail, i consider the case of an anharmonic potential and calculate, respectively, the quantum and classical Green's functions to first order in the coupling constant of the interaction. Furthermore, choosing an initial density matrix representing a plane wave state, I compare the two different evolutions graphically. Finally, I re-derive the Liouville equation, the path integral, and the perturbation expansion for the case of two particles. New aspects concerning the (dynamically assisted) entanglement (generation) are demonstrated. However, the theory can be extended to multi-particles case, then the new path integral for classical dynamics should be useful to study classical statistical problems.