Wigner Trajectory Characteristics in Phase Space and Field Theory

Exact characteristic trajectories are specified for the time-propagating Wigner phase-space distribution function. They are especially simple---indeed, classical---for the quantized simple harmonic oscillator, which serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase space. Applications to duality transformations in field theory are discussed.Comment: 9 pages, LaTex2

On the concepts of radial and angular kinetic energies

We consider a general central-field system in D dimensions and show that the division of the kinetic energy into radial and angular parts proceeds differently in the wavefunction picture and the Weyl-Wigner phase-space picture. Thus, the radial and angular kinetic energies are different quantities in the two pictures, containing different physical information, but the relation between them is well defined. We discuss this relation and illustrate its nature by examples referring to a free particle and to a ground-state hydrogen atom.Comment: 10 pages, 2 figures, accepted by Phys. Rev.

Closedness of star products and cohomologies

We first review the introduction of star products in connection with deformations of Poisson brackets and the various cohomologies that are related to them. Then we concentrate on what we have called ``closed star products" and their relations with cyclic cohomology and index theorems. Finally we shall explain how quantum groups, especially in their recent topological form, are in essence examples of star products.Comment: 16 page

The Moyal-Lie Theory of Phase Space Quantum Mechanics

A Lie algebraic approach to the unitary transformations in Weyl quantization is discussed. This approach, being formally equivalent to the $\star$-quantization, is an extension of the classical Poisson-Lie formalism which can be used as an efficient tool in the quantum phase space transformation theory.Comment: 15 pages, no figures, to appear in J. Phys. A (2001

Husimi Transform of an Operator Product

It is shown that the series derived by Mizrahi, giving the Husimi transform (or covariant symbol) of an operator product, is absolutely convergent for a large class of operators. In particular, the generalized Liouville equation, describing the time evolution of the Husimi function, is absolutely convergent for a large class of Hamiltonians. By contrast, the series derived by Groenewold, giving the Weyl transform of an operator product, is often only asymptotic, or even undefined. The result is used to derive an alternative way of expressing expectation values in terms of the Husimi function. The advantage of this formula is that it applies in many of the cases where the anti-Husimi transform (or contravariant symbol) is so highly singular that it fails to exist as a tempered distribution.Comment: AMS-Latex, 13 page

A finite model of two-dimensional ideal hydrodynamics

A finite-dimensional su($N$) Lie algebra equation is discussed that in the infinite $N$ limit (giving the area preserving diffeomorphism group) tends to the two-dimensional, inviscid vorticity equation on the torus. The equation is numerically integrated, for various values of $N$, and the time evolution of an (interpolated) stream function is compared with that obtained from a simple mode truncation of the continuum equation. The time averaged vorticity moments and correlation functions are compared with canonical ensemble averages.Comment: (25 p., 7 figures, not included. MUTP/92/1

Noncommutative Burgers Equation

We present a noncommutative version of the Burgers equation which possesses the Lax representation and discuss the integrability in detail. We find a noncommutative version of the Cole-Hopf transformation and succeed in the linearization of it. The linearized equation is the (noncommutative) diffusion equation and exactly solved. We also discuss the properties of some exact solutions. The result shows that the noncommutative Burgers equation is completely integrable even though it contains infinite number of time derivatives. Furthermore, we derive the noncommutative Burgers equation from the noncommutative (anti-)self-dual Yang-Mills equation by reduction, which is an evidence for the noncommutative Ward conjecture. Finally, we present a noncommutative version of the Burgers hierarchy by both the Lax-pair generating technique and the Sato's approach.Comment: 24 pages, LaTeX, 1 figure; v2: discussions on Ward conjecture, Sato theory and the integrability added, references added, version to appear in J. Phys.

Quantum Fluctuation Theorems

Recent advances in experimental techniques allow one to measure and control systems at the level of single molecules and atoms. Here gaining information about fluctuating thermodynamic quantities is crucial for understanding nonequilibrium thermodynamic behavior of small systems. To achieve this aim, stochastic thermodynamics offers a theoretical framework, and nonequilibrium equalities such as Jarzynski equality and fluctuation theorems provide key information about the fluctuating thermodynamic quantities. We review the recent progress in quantum fluctuation theorems, including the studies of Maxwell's demon which plays a crucial role in connecting thermodynamics with information.Comment: As a chapter of: F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (eds.), "Thermodynamics in the quantum regime - Fundamental Aspects and New Directions", (Springer International Publishing, 2018

Quantum Mechanics as an Approximation to Classical Mechanics in Hilbert Space

Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket, and a quasidensity operator. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Classical mechanics can now be viewed as a deformation of quantum mechanics. The forms of semiquantum approximations to classical mechanics are indicated.Comment: 10 pages, Latex2e file, references added, minor clarifications mad

Noncommutative geometry and physics: a review of selected recent results

This review is based on two lectures given at the 2000 TMR school in Torino. We discuss two main themes: i) Moyal-type deformations of gauge theories, as emerging from M-theory and open string theories, and ii) the noncommutative geometry of finite groups, with the explicit example of Z_2, and its application to Kaluza-Klein gauge theories on discrete internal spaces.Comment: Based on lectures given at the TMR School on contemporary string theory and brane physics, Jan 26- Feb 2, 2000, Torino, Italy. To be published in Class. Quant. Grav. 17 (2000). 3 ref.s added, typos corrected, formula on exterior product of n left-invariant one-forms corrected, small changes in the Sect. on integratio