Quantum Liouville theorem based on Haar measure

Song, Boqun Q.; Smith, J. D. H.; Luo, L.; Wang, J.

doi:10.1103/physrevb.109.144301

Quantum Liouville theorem based on Haar measure

Journal Article · Thu Apr 04 00:00:00 EDT 2024 · Physical Review. B

DOI:https://doi.org/10.1103/physrevb.109.144301· OSTI ID:2337492

^[1]; Smith, J. D. H. ^[1]; Luo, L. ^[1]; Wang, J. ^[1]

Ames Lab., and Iowa State Univ., Ames, IA (United States)

Liouville theorem (L theorem) reveals robust incompressibility of the distribution function in phase space, given arbitrary potentials. However, its quantum generalization, Wigner flow, is compressible, i.e., L theorem is only conditionally true (e.g., for perfect Harmonic potential). Here, we develop quantum L theorem (rigorous incompressibility) for arbitrary potentials (interacting or not) in Hamiltonians. Haar measure, instead of symplectic measure dp$$\bigwedge$$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, here we address why Haar measure and metric preservation do not work in the classical case. Applications of the theorems in statistics, topological phase transition, ergodic theory, etc., are discussed.

View Accepted Manuscript (DOE)

Research Organization:: Ames Laboratory (AMES), Ames, IA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES); USDOE Office of Science (SC), Basic Energy Sciences (BES). Materials Sciences & Engineering Division (MSE)

Grant/Contract Number:: AC02-07CH11358

OSTI ID:: 2337492

Alternate ID(s):: OSTI ID: 2340149
OSTI ID: 2473754

Report Number(s):: IS-J--11,297; IS-J--11,311

Journal Information:: Physical Review. B, Journal Name: Physical Review. B Journal Issue: 14 Vol. 109; ISSN 2469-9950

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

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