Phase-space representation of quantum systems in the relative-state formulation

A phase-space representation of quantum systems within the framework of the relative-state formulation is proposed. To this end, relative-position and relative-momentum states are introduced and their properties are investigated in detail. Phase-space functions that represent a quantum state vector are constructed in terms of the relative-position and relative-momentum states, and the quantum dynamics is investigated by using the phase-space functions. Furthermore, probability distributions in phase space are considered by means of the relative-state formulation, and it is shown that the phase-space probability distribution is closely related to the operational probability distribution. The marginal distribution, characteristic function, and operational uncertainty relation are also discussed.