Quantum free-streaming: out of equilibrium expansion for the free scalar fields

The collisionless Boltzmann equation, also called free-streaming, is a convenient approximation. It is rather simple to implement numerically, and and it is effective at reducing the irregularities of rough initial conditions. It can be obtained as a small $\hbar$ limit from a free scalar quantum field. Namely, by neglecting the $\hbar^2$ term in the dynamical evolution of the Wigner distribution, the quantum precursor of the distribution function. In this work it is presented the general form for the exact solutions of the Wigner distribution, for a scalar field undergoing a $(1+1)$-dimensional expansion. Namely, with the symmetry constraint of rotation and translation invariance in the transverse plane. It is very different from the (on-shell) free-streaming of classical particles. It is shown how to recognize the classical, $\hbar\to 0$, limit from the general form. The numerical analysis for a specific example shows very large quantum corrections to the classical free-streaming.