Quantum Liouville's theorem based on Haar measure

Abstract

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. The argument is based on general measure theory, independent of specific spaces or coordinates. Comparison of classical and quantum is made: for instance, we address why Haar measure and metric preservation do not work in the classical case. Applications of theorems in statistics, topological phase transition, ergodic theory, etc. are discussed.

arXiv comment9 pages, 1 figure

References in corpus (21)

• Topological Insulators with Inversion Symmetry
• Topological Photonics
• Topological Field Theory of Time-Reversal Invariant Insulators
• Classification of topological quantum matter with symmetries
• Ultrafast optical spectroscopy of strongly correlated materials and high-temperature superconductors: a non-equilibrium approach
• Creating stable Floquet-Weyl semimetals by laser-driving of 3D Dirac materials
• Theory of pump-probe photoemission in graphene: Ultrafast tuning of Floquet bands and local pseudospin textures
• Quantization Methods: A Guide for Physicists and Analysts
• Quantum critical point in graphene approached in the limit of infinitely strong Coulomb interaction
• Controlled Population of Floquet-Bloch States via Coupling to Bose and Fermi Baths
• Ultrafast Terahertz Conductivity Probes of Topologically Enhanced Surface Transport Driven by Mid-Infrared Laser Pulses in Bi${}_2$Se${}_3$
• Light-Driven Raman Coherence as a Non-Thermal Route to Ultrafast Topology Switching
• Discovery of Terahertz Second Harmonic Generation from Lightwave Acceleration of Symmetry--Breaking Nonlinear Supercurrents
• Emergence of topological semimetals in gap closing in semiconductors without inversion symmetry
• Wigner flow reveals topological order in quantum phase space dynamics
• Light Quantum Control of Persisting Higgs Modes in Iron-Based Superconductors
• Electronic Origin of High-Tc Maximization and Persistence in Trilayer Cuprate Superconductors
• Generalized Floquet theory: application to dynamical systems with memory and Bloch's theorem for nonlocal potentials
• Competing Orders and Ultrafast Energy Transfer at the Quantum Limit in a Nb${}_3$Sn Superconductor Probed by Terahertz Electrodynamics
• Chirality manipulation of ultrafast phase switchings in a correlated CDW-Weyl semimetal
• Geometric pumping and dephasing at topological phase transition

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