Conditions for quantum and classical tomogram-like functions to describe system states and to retain normalization during evolution

TL;DR

This paper investigates the conditions under which quantum and classical tomogram-like functions maintain normalization during evolution, identifying explicit criteria for quantum cases and general conditions for classical functions to represent physical states.

Contribution

It explicitly derives conditions for quantum tomograms to preserve normalization and discusses criteria for classical tomogram-like functions to be valid physical state representations.

Findings

Quantum tomograms preserve normalization only under specific conditions.

Classical Liouville, Moyal, and Husimi equations preserve normalization for any initially normalized functions.

Conditions for optical and symplectic tomograms to represent physical states are provided.

Abstract

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