Dynamical entropy of quasi-local algebras in quantum statistical mechanics

We study the dynamical entropy in the sense of Connes, Narnhofer, and Thirring of automorphisms on quasi-local algebras in quantum statistical mechanics. We extend their Kolmogorov-Sinai type theorem for AF-algebras to quasi-local algebras which are not necessarily AF-algebras.Work supported in part by Korean Science Foundation and the Basic Science Research Program, Ministry of Education, 1991     

Embeddings ofU(1)-current algebras in non-commutative algebras of classical statistical mechanics

Motivated by multiplicativeK-homology, and understanding critical phenomena in some classical statistical mechanical models, we construct actions ofGL() on the operator algebras of V. Jones and Ocneanu, and analyse these in terms of embeddings ofU(1)-current algebras.Partially supported by the Science and Engineering Research Council     

Dynamical symmetries in mechanics

The object of this review is to discuss methods that enable one to trace the origin of symmetries and conservation laws in mechanics to geometrical symmetries of space-time. Starting with the basic Newtonian assumptions on absolute space and time classical mechanics is developed in configuration space and phase space independently together with the related structures such as force-less mechanics. Heuristic considerations on geometric symmetries in configuration space reveal their intimate relation to conservation laws. Using the methods of differential geometry this relationship is put on a formal footing and symmetry groups of all spherically symmetric single term potentials are classified. The method of infinitesimal canonical transformations is presented as an alternative method of deducing dynamical symmetries of an arbitrary system in phase space. These methods also apply to non-relativistic quantum theory. Possible extension to special and general relatively is also discussed.     

Supersymmetric classical mechanics

The purpose of the paper is to construct a supersymmetric Lagrangian within the framework of classical mechanics which would be regarded as a candidate for passage to supersymmetric quantum mechanics. The authors felicitate Prof. D S Kothari on his eightieth birthday and dedicate this paper to him on this occasion.     

Generalized classical mechanics

Generalized classical mechanics has been introduced and developed as a classical counterpart of the fractional quantum mechanics. The Lagrangian of generalized classical mechanics has been introduced, and equation of motion has been obtained. Lagrange, Hamilton and Hamilton-Jacobi frameworks have been implemented. Oscillator model has been launched and solved in 1D case. A new equation for the period of oscillations of generalized classical oscillator has been found. The interplay between the energy dependency of the period of classical oscillations and the non-equidistant distribution of the energy levels for fractional quantum oscillator has been discussed. We discuss as well, the relationships between new equations of generalized classical mechanics and the well-known fundamental equations of classical mechanics.     

BRST cohomology in classical mechanics

The classical (non-quantum) cohomology of the Becchi-Rouet-Stora-Tyutin (BRST) symmetry in phase space is defined and worked out. No group action for the gauge transformations is assumed. The results cover, therefore, the general case of an open algebra and are valid off-shell. Each cohomology class contains all BRST invariant functions with fixed ghost number (an integer) which differ from each other by a BRST variation. If the ghost number is negative there is only the trivial class whose elements are equivalent to zero. If the ghost number is positive or zero there is a bijective correspondence between the BRST classes and those of the exterior derivative along the gauge orbits. These gauge orbits lie in the phase space surface on which the gauge generators are constrained to vanish. The BRST invariant functions of ghost numberp are then related to closedp-forms along the orbits. The addition of a BRST variation corresponds to the addition of an exact form. Some comments about the quantum case are included. The physical meaning of the classes with ghost number greater than zero is not discussed.Chercheur qualifié du Fonds National de la Recherche Scientifique (Belgium)     

经典力学中加速度相关的Lagrange函数

研究了加速度线性相关的Lagrange函数,在加速度项系数对称的条件下,Lagrange方程保持为二阶微分方程;给出了从运动方程构造加速度相关的Lagrange函数的方法;研究同一系统的加速度相关和加速度无关的Lagrange函数之间的关系.举例说明结果的应用. 关键词： Lagrange方程 加速度相关的Lagrange函数 广义力学 Lagrange函数的规范变换     

The S-matrix in classical mechanics

In the Hilbert-space version of classical mechanics, scattering theory forN-particle systems is developed in close analogy to the quantum case. Asymptotic completeness is proved for forces of finite range. Infinite-range forces lead to the problem of stability of bound states and can be dealt with only in some simple cases.It is a pleasure to thank Prof.L. Motchane for his kind hospitality at the I.H.E.S., where most of this work was done, and where the author profited from discussions withD. Ruelle andO. E. Lanford.     

Discrete effects in classical mechanics

By utilizing some results of discrete mechanics, effects related to the discretization of the equations of motion of classical mechanics are analyzed. Some examples of periodic motions are developed and it is shown that the discrete frequencies are different from the continuous one. Numerical calculations involving orbits are presented. A numerical algorithm, suggested by discrete mechanics, is compared with conventional methods of second order.     

Dynamical entropy ofC* algebras and von Neumann algebras

The definition of the dynamical entropy is extended for automorphism groups ofC* algebras. As an example, the dynamical entropy of the shift of a lattice algebra is studied, and it is shown that in some cases it coincides with the entropy density.     

Faux-boolean algebras and classical models

A certain substructure of the lattice of projections on a Hilbert space is defined, and it is shown under what conditions such a structure can be said to have a classical model (in a sense made precise). The results have implications for the interpretation of quantum mechanics.     

Constitution of objects in classical mechanics and in quantum mechanics

The constitution of objects is discussed in classical mechanics and in quantum mechanics. The requirement of objectivity and the Galilei invariance of classical and quantum mechanics leads to the postulate of covariance which must be fulfilled by observable quantities. Objects are then considered as carriers of these covariant observables and turn out to be representations of the Galilei group. Individual systems can be defined in classical mechanics by their trajectories in phase space. However, in quantum mechanics the characterization of individuals can only be achieved approximately by means of unsharp observables.     

Dynamical symmetries and Nambu mechanics

It is shown that several Hamiltonian systems possessing dynamical or hidden symmetries can be realized within the framework of Nambu's generalized mechanics. Among such systems are the SU(n)-isotropic harmonic oscillator and the SO(4) Kepler problem. As required by the formulation of Nambu dynamics, the integrals of motion for these systems necessarily become the so-called generalized Hamiltonians. Furthermore, in most of these problems, the definition of these generalized Hamiltonians is not unique.     

Dynamical symmetries inq-deformed quantum mechanics

The dynamical algebra of theq-deformed harmonic oscillator is constructed. As a result, we find the free deformed Hamiltonian as well as the Hamiltonian of the deformed oscillator as a complicated, momentum dependent interaction Hamiltonian in terms of the usual canonical variables. Furthermore we construct a welldefined algebraSU _q(1,1) with consistent conjugation properties and comultiplication. We obtain non lowest weight representations of this algebra.     

Dynamical origin of quantum mechanics

A new quantization method is proposed to obtain the equation of motion for a quantum system as the Lie-Poisson equation for a probability current on the S¹-bundle over a physical space.     

Superposition in quantum and classical mechanics

Using the mathematical notion of an entity to represent states in quantum and classical mechanics, we show that, in a strict sense, proper superpositions are possible in classical mechanics.Dedicated to the Memory of Charles H. Randall.     

Quantum mechanics as a deformation of classical mechanics

Mathematical properties of deformations of the Poisson Lie algebra and of the associative algebra of functions on a symplectic manifold are given. The suggestion to develop quantum mechanics in terms of these deformations is confronted with the mathematical structure of the latter. As examples, spectral properties of the harmonic oscillator and of the hydrogen atom are derived within the new formulation. Further mathematical generalizations and physical applications are proposed.Work supported in part by the National Science Foundation.