Existence of ground states and KMS states for approximately inner dynamics

A strongly continuous one parameter group of *-automorphisms of aC*-algebra with unit is said to be approximately inner if it can be approximated strongly by inner one parameter groups of *-automorphisms. It is shown that an approximately inner one parameter group of *-automorphisms has a ground state and, if there exists a trace state, a KMS state for all inverse temperatures. It follows that quantum lattice systems have ground states and KMS states. Conditions that a strongly continuous one parameter group of *-automorphisms of a UHF algebra be approximately inner are given in terms of the unbounded derivation which generates the automorphism group.     

Bound entanglement and continuous variables

We introduce the definition of generic bound entanglement for the case of continuous variables. We provide some examples of bound entangled states for that case, and discuss their physical sense in the context of quantum optics. We raise the question of whether the entanglement of these states is generic. As a by-product we obtain a new many parameter family of bound entangled states with positive partial transpose. We also point out that the "entanglement witnesses" and positive maps revealing the corresponding bound entanglement can easily be constructed.     

Modular structure of the local algebras associated with the free massless scalar field theory

The modular structure of the von Neumann algebra of local observables associated with a double cone in the vacuum representation of the free massless scalar field theory of any number of dimensions is described. The modular automorphism group is induced by the unitary implementation of a family of generalized fractional linear transformations on Minkowski space and is a subgroup of the conformal group. The modular conjugation operator is the anti-unitary implementation of a product of time reversal and relativistic ray inversion. The group generated by the modular conjugation operators for the local algebras associated with the family of double cone regions is the group of proper conformal transformations. A theorem is presented asserting the unitary equivalence of local algebras associated with lightcones, double cones, and wedge regions. For the double cone algebras, this provides an explicit realization of spacelike duality and establishes the known typeIII ₁ factor property. It is shown that the timelike duality property of the lightcone algebras does not hold for the double cone algebras. A different definition of the von Neumann algebras associated with a region is introduced which agrees with the standard one for a lightcone or a double cone region but which allows the timelike duality property for the double cone algebras. In the case of one spatial dimension, the standard local algebras associated with the double cone regions satisfy both spacelike and timelike duality.Supported by the National Science Foundation under Grant No. PHY-79-23251Supported in part by C. N. R.     

Algebraic K-systems

Following Emch, we define an algebraic K-system to be an infinite chain of algebras imbedded one into the other and connected by a one-parameter automorphism group. We consider topological consequences as well as convergence properties of states. Finally, we discuss the connection of this concept with Jones' index theory.     

Unitary implementation of automorphism groups on von Neumann algebras

A necessary and sufficient continuity condition is obtained in order that a topological group of automorphisms of a semi-finite von Neumann algebra in standard form is unitarily implemented. The methods used are extended to the study of unitary implementation for a general von Neumann algebra of those automorphism groups that commute with the one-parameter modular automorphism group.This research was partially supported by the National Science Foundation.     

Computing the invariant density of bistable noisy maps

A numerical method is presented allowing the computation of the invariant density of a time-discrete bi- or multistable map perturbed by weak noise. It permits the examination of noise-induced transitions between different stable states in the limit of weak but not amplitude-limited noise. The method is tested by comparing the results with computer experiments. For this purpose a new one-parameter family of bistable maps is introduced. It turns out that the numerics are in good agreement with the experiments. The results suggest the conjecture that in the limit of weak but transition-inducing noise the probability of being in one basin of attraction approaches one. A simple example which can be solved in closed form and which illustrates these findings is discussed.     

Quasistatic Dynamics with Intermittency

We study an intermittent quasistatic dynamical system composed of nonuniformly hyperbolic Pomeau–Manneville maps with time-dependent parameters. We prove an ergodic theorem which shows almost sure convergence of time averages in a certain parameter range, and identify the unique physical family of measures. The theorem also shows convergence in probability in a larger parameter range. In the process, we establish other results that will be useful for further analysis of the statistical properties of the model.     

Universality and self-similarity in the bifurcations of circle maps

The bifurcation structure in a two-parameter family of circle maps is considered. These maps have a (topological) degree that may be different from one. A generalization of the rotation number is given and symmetries of the bifurcations in parameter space are described. Continuity arguments are used to establish the existence of periodic orbits. By plotting the locus of parameter values associated with superstable cycles, self-similar bifurcations are found. These bifurcations are a generalization of the familiar period-doubling cascade in maps with one extrema, to two-parameter maps with two extrema. Finally, a scheme for the global organization of bifurcation in these maps is proposed.     

On the Generator of Massive Modular Groups

The purpose of this paper is to shed more light on the transition from the known massless modular action to the wanted massive one in the case of forward light cones and double cones. The infinitesimal generator δ_m of the modular automorphism group is investigated, in particular, some assumptions on its structure are verified explicitly for two concrete examples.     

Ergodic Properties of a Simple Deterministic Traffic Flow Model

We study statistical properties of a family of maps acting in the space of integer valued sequences, which model dynamics of simple deterministic traffic flows. We obtain asymptotic (as time goes to infinity) properties of trajectories of those maps corresponding to arbitrary initial configurations in terms of statistics of densities of various patterns and describe weak attractors of these systems and the rate of convergence to them. Previously only the so called regular initial configurations (having a density with only finite fluctuations of partial sums around it) in the case of a slow particles model (with the maximal velocity 1) have been studied rigorously. Applying ideas borrowed from substitution dynamics we are able to reduce the analysis of the traffic flow models corresponding to the multi-lane traffic and to the flow with fast particles (with velocities greater than 1) to the simplest case of the flow with the one-lane traffic and slow particles, where the crucial technical step is the derivation of the exact life-time for a given cluster of particles. Applications to the optimal redirection of the multi-lane traffic flow and a model of a pedestrian going in a slowly moving crowd are discussed as well.     

Relaxation and persistent oscillations of the order parameter in fermionic condensates

We determine the limiting dynamics of a fermionic condensate following a sudden perturbation for various initial conditions. Possible initial states of the condensate fall into two classes. In the first case, the order parameter asymptotes to a constant value. The approach to a constant is oscillatory with an inverse square root decay. This happens, e.g., when the strength of pairing is abruptly changed while the system is in the paired ground state and more generally for any nonequilibrium state that is in the same class as the ground state. In the second case, the order parameter exhibits persistent oscillations with several frequencies. This is realized for nonequilibrium states that belong to the same class as excited stationary states.     

A New Kind of Lax-Oleinik Type Operator with Parameters for Time-Periodic Positive Definite Lagrangian Systems

In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the family of new Lax-Oleinik type operators with an arbitrary \({u \in C(M, \mathbb{R}^1)}\) as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the family of new Lax-Oleinik type operators with an arbitrary \({u \in C(M, \mathbb{R}^1)}\) as initial condition converges to a backward weak KAM solution faster than the Lax-Oleinik semigroup in the time-independent case.     

Long-term memory contribution as applied to the motion of discrete dynamical systems

We consider the evolution of logistic maps under long-term memory. The memory effects are characterized by one parameter, alpha. If it equals to zero, any memory is absent. This leads to the ordinary discrete dynamical systems. For alpha=1 the memory becomes full, and each subsequent state of the corresponding discrete system accumulates all past states with the same weight just as the ordinary integral of first order does in the continuous space. The case with 00.15 the memory effects win over chaos.     

States and automorphism groups of operator algebras

Suppose that a group of automorphisms of a von Neumann algebraM, fixes the center elementwise. We show that if this group commutes with the modular (KMS) automorphism group associated with a normal faithful state onM, then this state is left invariant by the group of automorphisms. As a result we obtain a “noncommutative” ergodic theorem. The discrete spectrum of an abelian unitary group acting as automorphisms ofM is completely characterized by elements inM. We discuss the KMS condition on the CAR algebra with respect to quasi-free automorphisms and gauge invariant generalized free states. We also obtain a necessary and sufficient condition for the CAR algebra and a quasi-free automorphism group to be η-abelian.     

The modular automorphism group of a Poisson manifold

The modular flow of Poisson manifold is a 1-parameter group of automorphisms determined by the choice of a smooth density on the manifold. When the density is changed, the generator of the group changes by a hamiltonian vector field, so one has a 1-parameter group of “outer automorphisms” intrinsically attached to any Poisson manifold. The group is trivial if and only if the manifold admits a measure which is invariant under all hamiltonian flows.

The notion of modular flow in Poisson geometry is a classical limit of the notion of modular automorphism group in the theory of von Neumann algebras. In addition, the modular flow of a Poisson manifold is related to modular cohomology classes for associated Lie algebroids and symplectic groupoids. These objects have recently turned out to be important in Poincaré duality theory for Lie algebroids.     

Universal properties of maps on an interval

We consider itcrates of maps of an interval to itself and their stable periodic orbits. When these maps depend on a parameter, one can observe period doubling bifurcations as the parameter is varied. We investigate rigorously those aspects of these bifurcations which are universal, i.e. independent of the choice of a particular one-parameter family. We point out that this universality extends to many other situations such as certain chaotic regimes. We describe the ergodic properties of the maps for which the parameter value equals the limit of the bifurcation points.     

A Rohlin property for one-parameter automorphism groups

We define a Rohlin property for one-parameter automorphism groups of unital simpleC ^*-algebras and show that for such an automorphism group any cocycle is almost a coboundary. We apply the same method to the single automorphism case and show that if an automorphism of a unital simpleC ^*-algebra with a certain condition has a central sequence of approximate eigen-unitaries for any complex number of modulus one, then any cocycle is almost a coboundary, or the automorphism has the stability. We also show that if a one-parameter automorphism group of a unital separable purely infinite simpleC ^*-algebra has the Rohlin property then the crossed product is simple and purely infinite. Dedicated to: Prof. H. Hasegawa     

Formal Connections for Families of Star Products

We define the notion of a formal connection for a smooth family of star products with fixed underlying symplectic structure. Such a formal connection allows one to relate star products at different points in the family. This generalizes the formal Hitchin connection, which was introduced by the first author. We establish a necessary and sufficient condition that guarantees the existence of a formal connection, and we describe the space of formal connections for a family as an affine space modelled on the formal symplectic vector fields. Moreover, we showthat if the parameter space has trivial first cohomology group, any two flat formal connections are related by an automorphism of the family of star products.     

Field-dependent symmetries in Friedmann–Robertson–Walker models

We consider effective actions of the cosmological Friedmann–Robertson–Walker (FRW) models and discuss their fermionic rigid BRST invariance. Further, we demonstrate the finite field-dependent BRST transformations as a limiting case of continuous field-dependent BRST transformations described in terms of continuous parameter κ$\kappa$. The Jacobian under such finite field-dependent BRST transformations is computed explicitly, which amounts an extra piece in the effective action within functional integral. We show that for a particular choice of a parameter the finite field-dependent BRST transformation maps the generating functional for FRW models from one gauge to another.