Quantum trajectories

This Letter presents a new approach to phase space trajectories in quantum mechanics. A Moyal description of quantum theory is used, where observables and states are treated as classical functions on a classical phase space. A quantum trajectory being an appropriate solution to quantum Hamiltonian equations is also a function defined on a classical phase space. It results in a deformation of a classical action of a flow on observables and states to an appropriate quantum action. It also leads to a new multiplication rule for any quantum trajectory treated as a one-parameter group of diffeomorphisms. Moreover, several examples are given, presenting the developed formalism for particular quantum systems.