Quantum Liouville's theorem based on Haar measure

Understand

Liouville theorem (LT) reveals robust incompressibility of distribution function in phase space, given arbitrary potentials.

• However, its quantum generalization, Wigner flow, is compressible, i.e., LT is only conditionally true (e.g., for perfect Harmonic potential).
• We develop quantum Liouville theorem (rigorous incompressibility) for arbitrary potentials (interacting or not) in Hamiltonians.
• Haar measure, instead of symplectic measure dp^dq used in Wigner's scheme, plays a central role.