We propose a new derivation of Wigner's function based on the property of positivity of its integrals along straight lines in phase space. Identifying the values of these marginalizations with densities pertaining to invariant observables, we are able to reconstruct Wigner's pseudo-distribution from its slices.

The dynamical equations of quantum mechanics are rewritten in the form of dynamical equations for the measurable, positive marginal distribution of the shifted, rotated, and squeezed quadrature introduced in the so-called “symplectic tomography”. Then the possibility of a purely classical description of a quantum system as well as a reinterpretation of the quantum measurement theory is discussed and a comparison with the well-known quasi-probabilities approach is given. Furthermore, an analysis of the properties of this marginal distribution, which contains all the (...) quantum information, is performed in the framework of classical probability theory. Finally, examples of the harmonic oscillator's states dynamics are treated. (shrink)