Torres-Vega distribution function in the extended phase space

Classical physics gives us our everyday perception of the world. However, the break down of this classical intuition could be explained by quantum mechanics. Hence, quantum corrections to the classical statistical mechanics have always remained an important issue, especially, in optics and laser physics for which the correspondence between the classical limit of a large number of photons and the quantum regime is of most importance. We could, therefore, employ distribution functions to find this relation between the quantum and classical solutions. One of these distribution functions which is also of interest in quantum optics and ultrafast laser physics is Wigner distribution function. Interestingly, it could be transferred to the other probability distributions such as the positive Torres-Vega distribution function. In this paper, for the first time, we obtain the Torres-Vega distribution function directly from the extended phase space. We also by means of a set of unitary transformations in the phase space extract the extended Hamiltonian for the Torres-Vega distribution.