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arXiv:2312.04778 · Physical review. B./Physical review. B

Quantum Liouville theorem based on Haar measure

B. Q. Song, Jonathan D. H. Smith, Liang Luo, Jigang Wang

Published Apr 4, 2024• 6 citations
arXivHTMLPDF

Abstract

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). We develop quantum L theorem (rigorous incompressibility) for arbitrary potentials (interacting or not) in Hamiltonians. Haar measure, instead of symplectic measure $dp\ensuremath{\wedge}dq$ 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 the theorems in statistics, topological phase transition, ergodic theory, etc., are discussed.

DOI
10.1103/physrevb.109.144301

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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). We develop quantum L theorem (rigorous incompressibility) for arbitrary potentials (interacting or not) in Hamiltonians. Haar measure, instead of symplectic measure dp\ensuremath∧dqdp\ensuremath{\wedge}dqdp\ensuremath∧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 the theorems in statistics, topological phase transition, ergodic theory, etc., are discussed.
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~w\bm ww, 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.
Poincaré's recurrence theorem, which states that every Hamiltonian dynamics enclosed in a finite volume returns to its initial position as close as one wishes, is a mathematical basis of statistical mechanics. It is Liouville's theorem that guarantees that the dynamics preserves the volume on the state space. A quantum version of Poincaré's theorem was obtained in the middle of the 20th century without any volume structures of the state space (Hilbert space). One of our aims in this paper is to establish such properties of quantum dynamics from an analog of Liouville's theorem, namely, we will construct a natural probability measure on the Hilbert space from a Hamiltonian defined on the space. Then we will show that the measure is invariant under the corresponding Schrödinger flow. Moreover, we show that the dynamics naturally causes an infinite-dimensional Weyl transformation. It also enables us to discuss the ergodic properties of such dynamics.
Wigner functions play a central role in the phase space formulation of quantum mechanics. Although closely related to classical Liouville densities, Wigner functions are not positive definite and may take negative values on subregions of phase space. We investigate the accumulation of these negative values by studying bounds on the integral of an arbitrary Wigner function over noncompact subregions of the phase plane with hyperbolic boundaries. We show using symmetry techniques that this problem reduces to computing the bounds on the spectrum associated with an exactly-solvable eigenvalue problem and that the bounds differ from those on classical Liouville distributions. In particular, we show that the total ``quasiprobability'' on such a region can be greater than 1 or less than zero.
Correct use of the Wigner representation of quantum mechanics, which is realized with joint distributions of quasiprobabilities in phase space, requires the use of certain specific rules and attention to a number of properties which distinguish the quasiprobability from a true probability. This paper is mainly concerned with these problems. In the Wigner representation the quantum Liouville equation appears instead of the Schroedinger equation. The solution may have no physical meaning unless it is subjected to a necessary and sufficient condition which selects an allowed class of distributions which describe quantum-mechanical pure states. This condition contains Planck's constant and imposes, besides the uncertainty relations, severe restrictions on the possible form of the Wigner function. When this condition is satisfied, one can reconstruct the wave function from the Wigner function. In the case of an oscillator, the quantization condition for the energies of the stationary states does not follow from the Liouville equation, but from this supplementary condition. Unlike the true probability density, any Wigner function (except Gaussian ones) of a pure state takes on negative values. Another important peculiarity is that the quasiprobability is not concentrated on certain hypersurfaces, but is smeared out over the entire phase space. These and other features considered in this paper should be kept in mind when using the Wigner representation in quantum-mechanical problems.
The chrono-topology, as introduced axiomatically in a different context, is also supported by Liouville's theorem of statistical mechanics. It is shown that, if time is quantized, the distribution function (d.f.) becomes real. An elementary solution, g, of the classical Liouville equation has been found in phase-space and time, which can be used to construct any differentiable d.f, F(g), satisfying the same Liouville equation. The conditions imposed on F(g) are reality and additivity. The reality requirement, {Im F(g)=0) quantizes: (i) F(g) and makes it time-independent, (ii). The time variable, (iii) The energy. As a verification of chronotopology, the Planck constant h has been calculated on the basis of the time quantization. The d.f. F(g) becomes, after the time quantization, a real generalized Maxwell-Boltzmann d.f, F(g) = exp[g(p, g; l1,l2,..,lN)], depending on Ν quantum numbers. These facts are significant for quantum theory, because they uncover an intrinsic relationship between Liouville's theorem and quantum mechanics.
A numerical technique for the solution of the Quantum Liouville equation is discussed which is expected to be useful for mixed states describing highly excited many body systems.;The equation is represented in phase-space, where the density operator becomes the Wigner function. The Liouville operator which determines its time evolution, separates into a classical and quantum term. When the Wigner function varies slowly in momentum, the quasi-classical approximation is valid in which one keeps only the first order quantum term in the Liouville equation.;The Wigner function is represented by a sample set of gaussian smeared points. In an iterative time development scheme, the points are propagated first along a classical phase-space trajectory, which is then followed by a quantum development. The latter is represented by a stochastic process incorporated into the numerical procedure as a generalized Markovian process.;This method is numerically tested for an anharmonic quartic potential. The contours of the Wigner function at different time intervals are compared with the purely classical and exact results. It is seen that it provides a definite improvement over the purely classical approximation. Various averages are also compared and sources of error discussed.;In addition it was discovered that intermediate mixed states show a 'quantum focusing' effect, wherein the function peaks to values greater than the original maximum. This quantum-mechanical effect did not appear in the pure case, and disappears because of Liouville's theorem for highly mixed states which are essentially classical. There are other numerical schemes, which treat the points as moving in an effective potential, wherein the points retain their identity. These again by Liouville's theorem cannot show this focusing effect.