Phase-space formulation of quantum mechanics and quantum-state reconstruction for physical systems with Lie-group symmetries

We present a detailed discussion of a general theory of phase-space distrib utions, introduced recently by the authors [J. Phys. A 31, L9 (1998)]. This theory provides a unified phase-space formulation of quantum mechanics for physical systems possessing Lie-group symmetries. The concept of generaliz ed coherent states and the method of harmonic analysis are used to construc t explicitly a family of phase-space functions which are postulated to sati sfy the Stratonovich-Weyl correspondence with a generalized tracing conditi on. The symbol calculus for the phase-space functions is given by means of the generalized twisted product. The phase-space formalism is used to study the problem of the reconstruction of quantum states. In particular, we con sider the reconstruction method based on measurements of displaced projecto rs, which comprises a number of recently proposed quantum-optical schemes a nd is also related to the standard methods of signal processing. A general group-theoretic description of this method is developed using the technique of harmonic expansions on the phase space. [S1050-2947(99)00702-7].