Conditions for Quantum and Classical Tomogram-Like Functions to Describe System States and to Retain Normalizations During Time Evolution

Abstract

It is shown that dynamical equations for quantum tomograms retain the normalizations of their solutions during time evolution only if the solutions satisfy a set of special conditions. These conditions are found explicitly. On the contrary, it is also illustrated that the classical Liouville equation, Moyal equation for Wigner function, and evolution equation for Husimi function retain normalizations of any initially normalized and quickly decaying at infinity functions on the phase space. Necessary and sufficient conditions for optical and symplectic tomogram-like functions to be tomograms of physical states are also discussed.