Asymptotically isochronous systems

Abstract

Mechanisms are elucidated underlying the existence of dynamical systems whose generic solutions approach asymptotically (at large time) isochronous evolutions: all their dependent variables tend asymptotically to functions periodic with the same fixed period. We focus on two such mechanisms, emphasizing their generality and illustrating each of them via a representative example. The first example belongs to a recently discovered class of integrable indeed solvable many-body problems. The second example consists of a broad class of (generally nonintegrable) models obtained by deforming appropriately the well-known (integrable and isochronous) many-body problem with inverse-cube two-body forces and a one-body linear (“harmonic oscillator”) force.     

Examples of asymptotically abelian quantum systems

A class of examples of quantum systems having a property of asymptotic Abelians is given.     

Duality transformations for general abelian systems

We describe the general structure of duality transformations for a very broad set of abelian statistical and field theoretic systems. This includes theories with many different types of fields and a large variety of kinds of interactions including, but not limited to nearest neighbor, next nearest neighbor, multi-spin interactions, etc. We find that the dual form of a theory does not depend directly on the dimensionality of the theory, but rather on the number of fields and number of different kinds of interactions. The dual forms we find have a generalized gauge symmetry and possess the usual property of having a temperature (or coupling constant) which is inverted from that of the original theory. Our results reduce to the well-known results in those particular cases that have heretofore been studied. Our procedure also suggests variations capable of generating other forms of the dual theory which may be useful in various specific cases.     

An ergodic abelian skeleton for quantum systems

Under the assumption that there exists an optimal stationary coupling of a dynamical quantum system with a dynamical classical system, we prove that the quantum system contains an ergodic classical system.     

Asymptotically optimal quantum circuits for d-level systems

Scalability of a quantum computation requires that the information be processed on multiple subsystems. However, it is unclear how the complexity of a quantum algorithm, quantified by the number of entangling gates, depends on the subsystem size. We examine the quantum circuit complexity for exactly universal computation on many d-level systems (qudits). Both a lower bound and a constructive upper bound on the number of two-qudit gates result, proving a sharp asymptotic of theta(d(2n)) gates. This closes the complexity question for all d-level systems (d finite). The optimal asymptotic applies to systems with locality constraints, e.g., nearest neighbor interactions.     

Derivations commuting with abelian gauge actions on lattice systems

Let be an action of a compact abelian groupG on aC*-algebraA, and assume that the fixed-point subalgebraA is an AF-algebra. We show that if is a closed *-derivation onA commuting with , and the restriction of toA generates a one-parameter group of *-automorphisms, then itself is a generator. In particular, the result applies if is an infinite product action ofG on a UHF algebra. Furthermore, if in this situation ₁ and ₂ are two derivations both satisfying the hypotheses on , and ₁ and ₂ have the same restriction toA , then there exists a one-parameter subgroup of the action with generator ₀ such thatD(₁)D(₂)D(₀) is a joint core for the three derivations, and ₂=₁+₀ on this core.     

Dynamics of multiparticle systems with non–abelian symmetry

We consider the dynamics governing the evolution of a many body system constrained by a nonabelian local symmetry. We obtain explicit forms of the global macroscopic condition assuring that at the microscopic level the evolution respects the overall symmetry constraint. We demonstrate the constraint mechanisms for the case of SU(2) system comprising particles in fundamental, and adjoint representations (‘nucleons’ and ‘pions’). Received: 10 March 2000 / Revised version: 7 July 2000/ Published online: 8 December 2000     

Massless phases and symmetry restoration in abelian gauge theories and spin systems

We give a new, elementary proof for the existence of a deconfining transition to a massless (QED) phase in the four-dimensionalU(1) lattice gauge theory and of an intermediate QED phase, accompanied by dynamical restoration of localU(1) invariance, in the four dimensional _N models, withN large. Our methods can also be used to prove the existence of a phase transition in theXY model in three or more dimensions, in three- and four-dimensional abelian Higgs models, and in more general models admitting some local, abelian gauge invariance.Work supported in part by the NSF under grant DMR 81-00417     

An algorithm for detecting abelian monopoles inSU 2-valued lattice gauge-Higgs systems

We present an algorithm which calculates the monopole number of anSU ₂-valued lattice gauge field, together with a lattice Higgs field, on a simplicial lattice of dimension ≧3. The calculation is gauge invariant. The expected value of the monopole density (for a fixed Higgs field) does not depend on the Higgs field. Partially supported by NSF grants DMS 8607168 and DMS 8907753 Partially supported by PSC-CUNY and by NSF grant DMS 8805485     

Asymptotically normal dynamical semigroups

A definition and a characterization of asymptotically normal dynamical systems are given. In particular, a theorem concerning the return to equilibrium is presented.     

Asymptotically Flat Conformal Structures

In the first part of this paper we revisit the theory of weighted spinors on conformal manifolds. In the second part we introduce the notions of asymptotically flat Weyl structures and of associated mass, and we prove a conformal version of the positive mass theorem on conformal spin manifolds.     

Asymptotically FRW black holes

Special solutions of the LTB family representing collapsing over-dense regions corresponding to asymptotically closed, open, or flat FRW models are found. These solutions may be considered as representing dynamical mass condensations leading to black holes immersed in a FRW universe. We study the dynamics of the collapsing region, and its density profile. The question of the strength of the central singularity and its nakedness, as well as the existence of an apparent horizon and an event horizon is dealt with in detail, shedding light to the notion of cosmological black holes. Differences to the Schwarzschild black hole are addressed.     

Asymptotically safe phantom cosmology

New lifetime limits on the charge non-conserving (CNC) electron capture with excitation of the 417.9 keV nuclear level in the ¹²⁷I are established by using the coincidence technique. The analysed exposure is 0.87 ton × yr, collected deep underground at the Gran Sasso National Laboratory of the INFN by the highly radiopure DAMA/LIBRA setup (??250 kg of highly radiopure NaI(Tl)). The new limit on the mean life is ??>1.2×10²⁴?yr (90?% C.L.), about one order of magnitude larger than those previously available for CNC electron capture involving nuclear level excitations of ¹²⁷I and of the same order of magnitude than those achieved for analogous processes in ¹²⁹Xe.     

Asymptotically anti-de Sitter spaces

Asymptotically anti-de Sitter spaces are defined by boundary conditions on the gravitational field which obey the following criteria: (i) they are O(3, 2) invariant; (ii) they make the O(3, 2) surface integral charges finite; (iii) they include the Kerr-anti-de Sitter metric. An explicit expression of the O(3, 2) charges in terms of the canonical variables is given. These charges are shown to close in the Dirac brackets according to the anti-de Sitter algebra. The results are extended to the case ofN=1 supergravity. The coupling to gravity of a third-rank, completely antisymmetric, abelian gauge field is also considered. That coupling makes it possible to vary the cosmological constant and to compare the various anti-de Sitter spaces which are shown to have the same energy.On leave from Département de Physique, Université Libre de Bruxelles, BelgiumChercheur qualifié du Fonds National Belge de la Recherche Scientifique     

Rigidity of Asymptotically Hyperbolic Manifolds

In this paper, we prove a rigidity theorem of asymptotically hyperbolic manifolds only under the assumptions on curvature. Its proof is based on analyzing asymptotic structures of such manifolds at infinity and a volume comparison theorem.The first author’s research is partially supported by NSF grant of China.The second author’s research is partially supported by an NSF grant and a Simon fund.     

Asymptotically free two-dimensional chiral models

The geometrical part of a one-loop counter term of an arbitrary G-invariant chiral theory on the homogeneous space G/H of the Lie group G is calculated. It is shown that the class of asymptotically free theories is wider than the symmetric spaces.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 92–95, July, 1985.     

Asymptotically adapted three-body molecular states

For the first time a formal theory for three-body rearrangement scattering processes in the molecular-state approach is formulated removing difficulties with unphysical long-range couplings.     

Ultraviolet renormalons in abelian gauge theories

We analyze the large-order behaviour in perturbation theory of classes of diagrams with an arbitrary number of chains (i.e. photon lines, dressed by vacuum polarization insertions). We derive explicit formulae for the leading and subleading divergence as n, the order in perturbation theory, tends to infinity, and a complete result for the vacuum polarization at the next-to-leading order in an expansion in l / N f, where N f is the number of fermion species. In general, diagrams with more chains yield stronger divergence. We define an analogue of the familiar diagrammatic R-operation, which extracts ultraviolet renormalon counterterms as insertions of higher-dimension operators. We then use renormalization group equations to sum the leading (in n/ N f ) k corrections to all orders in l INf and find the asymptotic behaviour in n up to a constant that must be calculated explicitly order by order in 1/N_f.