Entropy of continuous mixtures and the measure problem

In its continuous version, the entropy functional measuring the information content of a given probability density may be plagued by a "measure" problem that results from improper weighting of phase space. This issue is addressed considering a generic collision process whereby a large number of particles or agents randomly and repeatedly interact in pairs, with prescribed conservation law(s). We find a sufficient condition under which the stationary single-particle distribution function maximizes an entropylike functional, that is free of the measure problem. This condition amounts to a factorization property of the Jacobian associated with the binary collision law, from which the proper weighting of phase space directly follows.     

Quantum mechanics in phase space: I. Unicity of the Wigner distribution function

The joint distribution function in phase space is related to the density matrix by an integral transformation which depends on the rule of correspondence used. All the requirements which can be restrictive for the kernel function defining the transformation are studied. It is shown that the conditions of Galilei invariance, unitarity, reality and normalization lead to the Wigner kernel function in a unique way. Galilei invariance, the requirement that the free particle behaves classically, and the conditions to obtain the correct mixed distributions also lead to the same result.     

Unambiguous Quantization from the Maximum Classical Correspondence that Is Self-consistent: The Slightly Stronger Canonical Commutation Rule Dirac Missed

Dirac’s identification of the quantum analog of the Poisson bracket with the commutator is reviewed, as is the threat of self-inconsistent overdetermination of the quantization of classical dynamical variables which drove him to restrict the assumption of correspondence between quantum and classical Poisson brackets to embrace only the Cartesian components of the phase space vector. Dirac’s canonical commutation rule fails to determine the order of noncommuting factors within quantized classical dynamical variables, but does imply the quantum/classical correspondence of Poisson brackets between any linear function of phase space and the sum of an arbitrary function of only configuration space with one of only momentum space. Since every linear function of phase space is itself such a sum, it is worth checking whether the assumption of quantum/classical correspondence of Poisson brackets for all such sums is still self-consistent. Not only is that so, but this slightly stronger canonical commutation rule also unambiguously determines the order of noncommuting factors within quantized dynamical variables in accord with the 1925 Born-Jordan quantization surmise, thus replicating the results of the Hamiltonian path integral, a fact first realized by E.H. Kerner. Born-Jordan quantization validates the generalized Ehrenfest theorem, but has no inverse, which disallows the disturbing features of the poorly physically motivated invertible Weyl quantization, i.e., its unique deterministic classical “shadow world” which can manifest negative densities in phase space.     

An improvement of image registration based on phase correlation

This paper is dedicated to studying the extended phase correlation method that applies to the sub-pixel evaluation of translational displacement between images. On the basis of experiments, this paper proposes an improved version of classical Hanning window, with involving parameters optimization in the process. We add a window function to the input images in the space domain and a weighting function to the spectrum in the frequency domain. The statistics and distribution of errors prove the applicability of the proposal. The experimental results of the proposed extended phase correlation algorithm show the registration accuracy of 1/100 pixel.     

Fractional generalization of Liouville equations

In this paper fractional generalization of Liouville equation is considered. We derive fractional analog of normalization condition for distribution function. Fractional generalization of the Liouville equation for dissipative and Hamiltonian systems was derived from the fractional normalization condition. This condition is considered as a normalization condition for systems in fractional phase space. The interpretation of the fractional space is discussed.     

A comment on the smoothed Wigner function

The Wigner function is one possible choice, from many, of correspondence between density matrices and quasi-distribution functions on phase space. For pure states, smoothing by convolution with any one of a set of kernels, renders only the Wigner function, from a large set of these quasi-distributions, positive.     

Classical projected phase space density of billiards and its relation to the quantum neumann spectrum

A comparison of classical and quantum evolution usually involves a quasiprobability distribution as a quantum analogue of the classical phase space distribution. In an alternate approach that we adopt here, the classical density is projected on to the configuration space. We show that for billiards, the eigenfunctions of the coarse-grained projected classical evolution operator are identical to a first approximation to the quantum Neumann eigenfunctions. However, even though there exists a correspondence between the respective eigenvalues, their time evolutions differ. This is demonstrated numerically for the stadium and lemon-shaped billiards.     

Quantum Blobs

Quantum blobs are the smallest phase space units of phase space compatible with the uncertainty principle of quantum mechanics and having the symplectic group as group of symmetries. Quantum blobs are in a bijective correspondence with the squeezed coherent states from standard quantum mechanics, of which they are a phase space picture. This allows us to propose a substitute for phase space in quantum mechanics. We study the relationship between quantum blobs with a certain class of level sets defined by Fermi for the purpose of representing geometrically quantum states.     

Scale invariance of the density fluctuations in films and macromolecular aggregates in poly(styrene) solutions

The specific features of the transformation of a polymer solution into a solid state (film) of an amorphous polymer are investigated using electron microscopy. The correspondence between the characteristics of fractal macromolecular aggregates in a solution and the parameters of the spatial distribution of density fluctuations at the surface of the film is established using a linear atactic poly(styrene) as an example. The correspondence exists under the condition that the packing density of coils does not exceed a critical value at the liquid-solid phase transition point and the polymer concentration in the solution provides the formation of a continuous network of entangled macromolecules.     

Discrete Quantum Drinfeld–Sokolov Correspondence

We construct a discrete quantum version of the Drinfeld–Sokolov correspondence for the sine-Gordon system. The classical version of this correspondence is a birational Poisson morphism between the phase space of the discrete sine-Gordon system and a Poisson homogeneous space. Under this correspondence, the commuting higher mKdV vector fields correspond to the action of an Abelian Lie algebra. We quantize this picture (1) by quantizing this Poisson homogeneous space, together with the action of the Abelian Lie algebra, (2) by quantizing the sine-Gordon phase space, (3) by computing the quantum analogues of the integrals of motion generating the mKdV vector fields, and (4) by constructing an algebra morphism taking one commuting family of derivations to the other one. Received: 3 July 2001 / Accepted: 14 December 2001     

Star-polarization: A natural link between phase space representation and operator representation of quantum mechanics

The method of *-polarization connects phase space mechanics to the usual operator formulation of quantum theory. A *-polarization is a linear submanifold of the space of C functions on phase space. Elements of a *-polarization are in direct correspondence with the Schroedinger wave functions and this correspondence induces the Weyl correspondence between classical observables and operators. All generalized Moyal algebras admit *-polarizations and a general method is thus available for translating *-quantization into operator language.     

Amplitude phase-space model for quantum mechanics

We show that there is a close relationship between quantum mechanics and ordinary probability theory. The main difference is that in quantum mechanics the probability is computed in terms of an amplitude function, while in probability theory a probability distribution is used. Applying this idea, we then construct an amplitude model for quantum mechanics on phase space. In this model, states are represented by amplitude functions and observables are represented by functions on phase space. If we now postulate a conjugation condition, the model provides the same predictions as conventional quantum mechanics. In particular, we obtain the usual quantum marginal probabilities, conditional probabilities and expectations. The commutation relations and uncertainty principle also follow. Moreover Schrödinger's equation is shown to be an averaged version of Hamilton's equation in classical mechanics.     

Pure state condition for the semi-classical Wigner function

The Wigner function W(p,q) is a symmetrized Fourier transform of the density matrix ρ(q₁,q₂), representing quantum-mechanical states or their statistical mixture in phase space. Identification of these two alternatives in the case of density matrices depends on the projection identity ρ² = ρ; its Wigner correspondence is the pure state condition. This criterion is applied to the Wigner functions obtained from standard semiclassical wave functions, determining as pure states those whose classical invariant tori satisfy the generalized Bohr-Sommerfeld conditions. Superpositions of eigenstates are then examined and it is found that the Wigner function corresponding to Gaussian random wave functions are smoothed out in the manner of mixed-state Wigner functions. Attention is also given to the pure-state condition in the case where an angular coordinate is used.     

双模压缩数态光场的Wigner函数及其特性

借助纠缠态表象及Wigner算符在该表象下的表示,得到双模压缩数态的Wigner函数,数值计算画出相空间中Wigner函数的分布图,并加以分析,发现双模压缩数态两模之间相互关联、相互纠缠,对相空间中Wigner函数分布产生影响. 关键词： 双模压缩数态 Wigner函数 纠缠态表象     

Topological aspects of quantum chaos

Quantized classically chaotic maps on a toroidal two-dimensional phase space are studied. A discrete, topological criterion for phase-space localization is presented. To each eigenfunction is associated an integer, analogous to a quantized Hall conductivity, which tests the way the eigenfunction explores the phase space as some boundary conditions are changed. The correspondence between delocalization and chaotic classical dynamics is discussed, as well as the role of degeneracies of the eigenspectrum in the transition from localized to delocalized states. The general results are illustrated with a particular model.     

两分量玻色-爱因斯坦凝聚系统的自旋压缩

在周期性脉冲撞击的两分量玻色－爱因斯坦凝聚系统中研究了自旋压缩动力学性质,结果表明：原子自旋压缩动力学能够揭示相空间的混沌和规则结构,即当初始相干态处在混沌区域时,自旋压缩在很短时间后消失,而当初始相干态处在规则区域时,自旋压缩则存在很长时间.特别是随着时间的演化,系统的平均自旋方向的分布和摆动也与初态所处的空间结构有着密切的联系.最后,研究了相空间的整体自旋压缩动力学,得到了一种较好的量子-经典对应. 关键词： 玻色-爱因斯坦凝聚 混沌 自旋压缩 平均自旋方向     

Microscopic Reversibility for Nonequilibrium Classical Open Systems

We rigorously show that the probability to have a specific trajectory of an externally perturbed classical open system satisfies a universal symmetry for Hamiltonian dynamics. It connects the ratio between the probabilities of time forward and reversed trajectories to a degree of the time reversal asymmetry of the final phase space distribution in a model-independent framework. Especially, it amounts to a nonequilibrium generalization of the detailed balance between the probabilities of the forward and reversed trajectories under the condition that the initial phase space distribution is described by an equilibrium ensemble. An expression of the microscopic reversibility for the subsystem is also derived based on this relation.     

Simple Approach for Deriving the Weyl Correspondence Product Formula

By making use of the coherent state representation of the Wigner operator we present a simple approach for deriving the Weyl correspondence product formula in complex phase space. The formula tells us how to express the Weyl classical function corresponding to an operator F = AB in terms of the Weyl functions corresponding to A and B.