Integrability and the variational formulation of non‐conservative mechanical systems

It is shown that one can obtain canonically‐defined dynamical equations for non‐conservative mechanical systems by starting with a first variation functional, instead of an action functional, and finding their zeroes. The kernel of the first variation functional, as an integral functional, is a 1‐form on the manifold of kinematical states, which then represents the dynamical state of the system. If the 1‐form is exact then the first variation functional is associated with the first variation of an action functional in the usual manner. The dynamical equations then follow from the vanishing of the dual of the Spencer operator that acts on the dynamical state. This operator, in turn, relates to the integrability of the kinematical states. The method is applied to the modeling of damped oscillators.     

Convergence of chaotic attractors due to interaction based on closeness

Exploration of coherence phenomena in ensembles of interacting dynamical systems has been in the centre of research in social, physical, biological and technological systems for decades. But, in most of the studies, either completely percolated time- and space-static networks or temporal connectivities disregarding the systems' own dynamics have been dealt with. In this work, we examine the correlation between structural and dynamical evolution in networks of interacting dynamical systems. We specifically demonstrate the scenario of convergence of a set of chaotic attractors into a single attractor as a result of sufficient interaction based on the closeness of their own states. We characterize this occurrence through different measures, and map the collective states in network parameters' space. We further validate our proposition while exposing the whole scenario for different chaotic systems, namely Lorenz and Rössler oscillators.     

Tensorial Relativistic Quantum Mechanics in (1+1) Dimensions and Boundary Conditions

The tensorial relativistic quantum mechanics in (1+1) dimensions is considered. Its kinematical and dynamical features are reviewed as well as the problem of finding the Dirac spinor for given finite multivectors. For stationary states, the dynamical tensorial equations, equivalent to the Dirac equation, are solved for a free particle, for a particle inside a box, and for a particle in a step potential.     

On ghost fermions

The path integral for ghost fermions, which is heuristically made use of in the Batalin-Fradkin-Vilkovisky approach to quantization of constrained systems, is derived from first principles. The derivation turns out to be rather different from that of physical fermions since the definition of Dirac states for ghost fermions is subtle. With these results at hand, it is then shown that the nonminimal extension of the Becchi-Rouet-Stora-Tyutin operator must be chosen differently from the notorious choice made in the literature in order to avoid the boundary terms that have always plagued earlier treatments. Furthermore it is pointed out that the elimination of states with nonzero ghost number requires the introduction of a thermodynamic potential for ghosts; the reason is that Schwarz's Lefschetz formula for the partition function of the time-evolution operator is not capable, despite claims to the contrary, to get rid of nonzero ghost number states on its own. Finally, we comment on the problems of global topological nature that one faces in the attempt to obtain the solutions of the Dirac condition for physical states in a configuration space of nontrivial geometry; such complications give rise to anomalies that do not obey the Wess-Zumino consistency conditions. Received: 4 May 2001 / Revised version: 10 October 2001 / Published online: 8 February 2002     

Continuity equation for probability as a requirement of inference over paths

Local conservation of probability, expressed as the continuity equation, is a central feature of non-equilibrium Statistical Mechanics. In the existing literature, the continuity equation is always motivated by heuristic arguments with no derivation from first principles. In this work we show that the continuity equation is a logical consequence of the laws of probability and the application of the formalism of inference over paths for dynamical systems. That is, the simple postulate that a system moves continuously through time following paths implies the continuity equation. The translation between the language of dynamical paths to the usual representation in terms of probability densities of states is performed by means of an identity derived from Bayes’ theorem. The formalism presented here is valid independently of the nature of the system studied: it is applicable to physical systems and also to more abstract dynamics such as financial indicators, population dynamics in ecology among others.     

Boson description of collective states: (II). Description of odd vibrational nuclei

By addition of the so-called ideal quasiparticle to the boson space one can represent the odd fermion states in that product space. In such a way one finds various representations of the fermion operators in terms of the boson operators and ideal quasiparticles. From these boson expansions of the fermion operators a finite one is selected by considering non-unitary transformations. Thus, the direct generalization, of the Dyson representation for even systems is given for the case of odd systems. The Hamiltonian can be divided into three parts: the boson term which describes the vibrational motion of the even core, the unperturbed motion of the quasiparticle, and the interaction between the quasiparticle and the bosons. This interaction consists of two terms, one of which agrees with the term used by Kisslinger and Sorensen ²), which is usually called the dynamical interaction, and the additional term is due to the antisymmetrization between the extra particle and the even core. The latter term can be identified as kinematical interaction which is responsible for the anomalous coupling states. For example, it is demonstrated that this term produces qualitatively the same splitting of the one-phonon multiplet as was obtained by Kuriyama et al. ³) for the j-shell. Furthermore, it is shown for the more complicated case of ¹¹⁷Sn that the effect of this additional interaction between phonons and quasiparticle is important when many shells to the states in the odd nucleus are taken into account.     

Stability of Quantum Systems at Three Scales: Passivity, Quantum Weak Energy Inequalities and the Microlocal Spectrum Condition

Quantum weak energy inequalities have recently been extensively discussed as a condition on the dynamical stability of quantum field states, particularly on curved spacetimes. We formulate the notion of a quantum weak energy inequality for general dynamical systems on static background spacetimes and establish a connection between quantum weak energy inequalities and thermodynamics. Namely, for such a dynamical system, we show that the existence of a class of states satisfying a quantum weak inequality implies that passive states (e.g., mixtures of ground- and thermal equilibrium states) exist for the time-evolution of the system and, therefore, that the second law of thermodynamics holds. As a model system, we consider the free scalar quantum field on a static spacetime. Although the Weyl algebra does not satisfy our general assumptions, our abstract results do apply to a related algebra which we construct, following a general method which we carefully describe, in Hilbert-space representations induced by quasifree Hadamard states. We discuss the problem of reconstructing states on the Weyl algebra from states on the new algebra and give conditions under which this may be accomplished. Previous results for linear quantum fields show that, on one hand, quantum weak energy inequalities follow from the Hadamard condition (or microlocal spectrum condition) imposed on the states, and on the other hand, that the existence of passive states implies that there is a class of states fulfilling the microlocal spectrum condition. Thus, the results of this paper indicate that these three conditions of dynamical stability are essentially equivalent. This observation is significant because the three conditions become effective at different length scales: The microlocal spectrum condition constrains the short-distance behaviour of quantum states (microscopic stability), quantum weak energy inequalities impose conditions at finite distance (mesoscopic stability), and the existence of passive states is a statement on the global thermodynamic stability of the system (macroscopic stability).Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany. verch@mis.mpg.de     

Generally covariant quantum field theory and scaling limits

The formulation of a generally covariant quantum field theory is described. It demands the elimination of global features and a characterization of the theory in terms of the allowed germs of families of states. A simple application is the computation of counting rates of accelerated idealized detectors. As a first orientation we discuss here the consequences of the assumption that the states have a short distance scaling limit. The scaling limit at a point gives a reduction of the theory to tangent space. It contains kinematical information but not the full dynamical laws. The reduced theory will, under rather general conditions, be invariant under translations and under a proper subgroup of the linear transformations in tangent space. One interesting possibility is that it is invariant under SLR(4). Then the macroscopic metric must evolve as a cooperative effect in finite size regions. The other natural possibility is that each family (coherent folium) of states defines a microscopic metric by the scaling limit and the tangent space theory reduces to a theory of free massless fields in a Minkowski space. Irrespective of the assumption of a scaling limit we show that the theory can be constructed from strictly local information.     

Tracking controlled chaos: Theoretical foundations and applications

Tracking controlled states over a large range of accessible parameters is a process which allows for the experimental continuation of unstable states in both chaotic and non-chaotic parameter regions of interest. In algorithmic form, tracking allows experimentalists to examine many of the unstable states responsible for much of the observed nonlinear dynamic phenomena. Here we present a theoretical foundation for tracking controlled states from both dynamical systems as well as control theoretic viewpoints. The theory is constructive and shows explicitly how to track a curve of unstable states as a parameter is changed. Applications of the theory to various forms of control currently used in dynamical system experiments are discussed. Examples from both numerical and physical experiments are given to illustrate the wide range of tracking applications. (c) 1997 American Institute of Physics.     

Cartesian and Lagrangian Momentum

Historical, physical, and geometrical relations between two different momenta, characterized here as Cartesian and Lagrangian, are explored. Cartesian momentum is determined by the mass tensor, and gives rise to a kinematical geometry. Lagrangian momentum, which is more general, is given by the fiber derivative, and produces a dynamical geometry. This differs from the kinematical in the presence of a velocity-dependent potential. The relation between trajectories and level surfaces in Hamilton-Jacobi theory can also be Cartesian and kinematical or, more generally, Lagrangian and dynamical.     

Epistemic Entanglement due to Non-generating Partitions of Classical Dynamical Systems

Quantum entanglement relies on the fact that pure quantum states are dispersive and often inseparable. Since pure classical states are dispersion-free they are always separable and cannot be entangled. However, entanglement is possible for epistemic, dispersive classical states. We show how such epistemic entanglement arises for epistemic states of classical dynamical systems based on phase space partitions that are not generating. We compute epistemically entangled states for two coupled harmonic oscillators.     

费米子Regge轨迹的解析性和阈行为

从π介子核子散射振幅出发,根据Mandelstam表示和么正条件,对于奇异量子数为零、重子数为1的费米子Regge轨迹,求得了下列结果:(1)位置参数α(s)的解析性,只有右方物理割而无左方动力割,但是左方有从s=0到-∞的运动学割。(2)当s<0时α(s)取复数值,并且还有一条同它成复共轭(在s<0范围内)的Regge轨迹。这一对轨迹它们的J-宇称相同,但是空间宇称相反。02时,α(s)为实数。不同宇称的轨迹对s的依赖不同。(3)讨论了费米子Regge迹在阈能(W₀=M+μ)附近的行为。凡是与共振态、束缚态有关的Regge轨迹,α(W₀)≠0。求得了在阈附近的表示,定性方面同玻色子轨迹相同。(4)还有一大类Regge轨迹,它们同共振态、束缚态无关,α(W₀)=0。这类轨迹在能量趋近阈能时,有无穷多个极点趋于ReJ=0(即我们在费米子情况下得到Gribov-Pomeranchuk极点凝聚现象),这类轨迹当能量由阈下趋于阈时,从J平面左半平面共轭成对趋于ReJ=0;当能量从阈上趋于阈时,成对地从第一和第三象限趋于ReJ=0。所有这些极点实数部分Reα→0比虚数部分Imα→0快一或二个数量级。J平面原点是一个凝聚点,这类“非动力”的Regge轨迹阈行为,无论对玻色子、费米子,以至非相对论势散射,定性都相似(只是凝聚的轴线和点不同)。它们实际上是s矩阵在阈的普遍性质的反映。     

Off-mass-shell dynamics in flat spacetime

In the covariant Hamiltonian mechanics with action-at-a-distance, we compare the proper time and dynamical time representations of the coordinate space world line using the differential geometry of nongeodesic curves in 3+1 Minkowski spacetime. The covariant generalization of the Serret-Frenet equations for the point particle with interaction are derived using the arc length representation. A set of invariant point particle kinematical properties are derived which are equivalent to the solutions of the equations of motion in coordinate space and which are functions of either the proper time or the dynamical time. Expressions for the quantities are given for the example of the covariant harmonic oscillator and comments are offered regarding the measurability of the dynamical time.     

Off-mass-shell dynamics in flat spacetime

In the covariant Hamiltonian mechanics with action-at-a-distance, we compare the proper time and dynamical time representations of the coordinate space world line using the differential geometry of nongeodesic curves in 3+1 Minkowski spacetime. The covariant generalization of the Serret-Frenet equations for the point particle with interaction are derived using the arc length representation. A set of invariant point particle kinematical properties are derived which are equivalent to the solutions of the equations of motion in coordinate space and which are functions of either the proper time or the dynamical time. Expressions for the quantities are given for the example of the covariant harmonic oscillator and comments are offered regarding the measurability of the dynamical time.     

Regular-to-chaotic tunneling rates using a fictitious integrable system

We derive a formula predicting dynamical tunneling rates from regular states to the chaotic sea in systems with a mixed phase space. Our approach is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the island. For the standard map and other kicked systems we find agreement with numerical results for all regular states in a regime where resonance-assisted tunneling is not relevant.     

Kinematical problem in Spin-wave theory

A spin-1/2, nearest neighbor Heisenberg Hamiltonian acting on a periodic,d-dimensional lattice is considered. Multi-spin-wave solutions to the Schrödinger equation for a Heisenberg ferromagnet involve an unlimited superposition of spin-reversal operators at sites. This violates the physical restriction that no more than one excitation reside on any one site. This exclusion rule affects spin-wave interaction—the determination of these effects is called the kinematical problem. A general nonperturbative treatment that includes kinematical effects in spin-wave theory is developed along the following lines. Using the property of the Heisenberg Hamiltonian that it does not couple states obeying the single occupation condition at all sites with states that violate the single-occupancy condition at some sites, the unphysical multiply occupied states can be eliminated by a nonunitary transformation of the eigenvalue equation. An overcomplete Hamiltonian matrix is obtained that contains all the physical eigenvalues as a subset of its spectrum. Overcompleteness is shown to be a large part of the kinematical problem and several schemes to handle it are discussed. The remainder of the kinematical problem lies in the nonorthogonality of spin waves. It is shown that a new type of distribution, one that is neither Bose nor Fermi, correctly describes free spin-wave statistics at all temperatures. This formal but nonetheless complete solution to the overcompleteness aspect of the kinematical problem is then carried over,in toto, to the boson formulation of the spin Hamiltonian. Application to the calculation of the partition function and to thermal Green's functions is noted.     

Weave States in Loop Quantum Gravity

Weaves are eigenstates of geometrical operators in nonperturbative quantum gravity, which approximate flat space (or other smooth geometries) at large scales. We describe two such states, which diagonalize the area as well as the volume operators. The existence of such states shows that some earlier worries about the difficulty of realizing kinematical states with non-vanishing volume can be overcome. We also show that the Q operator used in earlier work for extracting geometrical information from quantum states does not capture more information than the area and volume operators.     

Hamiltonian quantum dynamics with separability constraints

Schroedinger equation on a Hilbert space H, represents a linear Hamiltonian dynamical system on the space of quantum pure states, the projective Hilbert space PH. Separable states of a bipartite quantum system form a special submanifold of PH. We analyze the Hamiltonian dynamics that corresponds to the quantum system constrained on the manifold of separable states, using as an important example the system of two interacting qubits. The constraints introduce nonlinearities which render the dynamics nontrivial. We show that the qualitative properties of the constrained dynamics clearly manifest the symmetry of the qubits system. In particular, if the quantum Hamilton’s operator has not enough symmetry, the constrained dynamics is nonintegrable, and displays the typical features of a Hamiltonian dynamical system with mixed phase space. Possible physical realizations of the separability constraints are discussed.     

Vector states for single and multiple-pole resonances

The formulation of quantum mechanics in rigged Hilbert spaces is used to study the vector states for resonance states or Gamow vectors. An important part of the work is devoted to the construction of Gamow vectors for resonances that appear as multiple poles on the analytic continuation of theS-matrix,S(E). The kinematical behavior of these vectors is also studied. This construction allow for generalized spectral decompositions of the Hamiltonian and the evolutionary semigroups, valid on certain locally convex spaces. Also a first attempt is made to define the resonance states as densities in an extension of the Liouville space, here called rigged Liouville space.