The Quantum Liouville Equation is non-Liouvillian

TL;DR

This paper demonstrates that the quantum flow in phase space, described by the Wigner flow, is not incompressible like classical Hamiltonian flow, revealing fundamental differences in quantum phase space dynamics.

Contribution

It shows that the quantum-Liouville equation's flow is non-Liouvillian, introducing the Wigner phase space velocity field with unbounded divergence, unlike classical systems.

Findings

Wigner flow is not incompressible.

Divergence of Wigner flow can be unbounded.

Unbounded divergence lines coincide with zeros of the Wigner function.

Abstract

The Hamiltonian flow of a classical, time-independent, conservative system is incompressible, it is Liouvillian. The analog of Hamilton's equations of motion for a quantum-mechanical system is the quantum-Liouville equation. It is shown that its associated quantum flow in phase space, Wigner flow, is not incompressible. It gives rise to a quantum analog of classical Hamiltonian vector fields: the Wigner phase space velocity field~$\bm w$, the divergence of which can be unbounded. The loci of such unbounded divergence form lines in phase space which coincide with the lines of zero of the Wigner function. Along these lines exist characteristic pinch points which coincide with stagnation points of the Wigner flow.