Showing papers in "Journal of Mathematical Physics in 1997"

Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering: Random matrix approach for systems with broken time-reversal invariance

Abstract: Assuming the validity of random matrices for describing the statistics of a closed chaotic quantum system, we study analytically some statistical properties of the S-matrix characterizing scattering in its open counterpart. In the first part of the paper we attempt to expose systematically ideas underlying the so-called stochastic (Heidelberg) approach to chaotic quantum scattering. Then we concentrate on systems with broken time-reversal invariance coupled to continua via Mopen channels; a=1,2,…,M. A physical realization of this case corresponds to the chaotic scattering in ballistic microstructures pierced by a strong enough magnetic flux. By using the supersymmetry method we derive an explicit expression for the density of S-matrix poles (resonances) in the complex energy plane. When all scattering channels are considered to be equivalent our expression describes a crossover from the χ2 distribution of resonance widths (regime of isolated resonances) to a broad power-like distribution typical for the regime of overlapping resonances. The first moment is found to reproduce exactly the Moldauer–Simonius relation between the mean resonance width and the transmission coefficient. Under the same assumptions we derive an explicit expression for the parametric correlation function of densities of eigenphases θa of the S-matrix (taken modulo 2π). We use it to find the distribution of derivatives τa=∂θa/∂E of these eigenphases with respect to the energy (“partial delay times”) as well as with respect to an arbitrary external parameter. We also find the parametric correlations of the Wigner–Smith time delay τw(E)=(1/M)∑a ∂θa/∂E at two different energies E−Ω/2 and E+Ω/2 as well as at two different values of the external parameter. The relation between our results and those following from the semiclassical approach as well as the relevance to experiments are briefly discussed.

Infinitely many Lax pairs and symmetry constraints of the KP equation

Abstract: Starting from a known Lax pair, one can get some infinitely many coupled Lax pairs, infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways. In this paper, taking the well known Kadomtsev–Petviashvili (KP) equation as a special example, we show that infinitely many nonhomogeneous linear Lax pairs can be obtained by using infinitely many symmetries, differentiating the spectral functions with respect to the inner parameters. Using a known Lax pair and the Darboux transformations (DT), infinitely many nonhomogeneous nonlinear Lax pairs can also be obtained. By means of the infinitely many Lax pairs, DT and the conformal invariance of the Schwartz form of the KP equation, infinitely many new nonlocal symmetries can be obtained naturally. Infinitely many integrable models in (1+1)-dimensions, (2+1)-dimensions, (3+1)-dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the KP equation related to the infinitely many Lax pairs.

First order analytic difference equations and integrable quantum systems

Abstract: We present a new solution method for a class of first order analytic difference equations. The method yields explicit “minimal” solutions that are essentially unique. Special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic, trigonometric and elliptic type—Euler’s gamma function being of rational type. We study these generalized gamma functions in considerable detail. The scattering and weight functions (u- and w-functions) associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions. We obtain detailed information on these u- and w-functions, exploiting the difference equations they satisfy.

Exact semiclassical expansions for one-dimensional quantum oscillators

Abstract: A set of rules is given for dealing with WKB expansions in the one-dimensional analytic case, whereby such expansions are not considered as approximations but as exact encodings of wave functions, thus allowing for analytic continuation with respect to whichever parameters the potential function depends on, with an exact control of small exponential effects. These rules, which include also the case when there are double turning points, are illustrated on various examples, and applied to the study of bound state or resonance spectra. In the case of simple oscillators, it is thus shown that the Rayleigh–Schrodinger series is Borel resummable, yielding the exact energy levels. In the case of the symmetrical anharmonic oscillator, one gets a simple and rigorous justification of the Zinn-Justin quantization condition, and of its solution in terms of “multi-instanton expansions.”

Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras, and unipotent varieties

Abstract: We introduce a new family of symmetric functions, which are q analogs of products of Schur functions, defined in terms of ribbon tableaux. These functions can be interpreted in terms of the Fock space representation Fq of Uq(sln), and are related to Hall–Littlewood functions via the geometry of flag varieties. We present a series of conjectures, and prove them in special cases. The essential step in proving that these functions are actually symmetric consists in the calculation of a basis of highest weight vectors of Fq using ribbon tableaux.

Nonrelativistic conformal groups

Abstract: In this work a systematic study of finite-dimensional nonrelativistic conformal groups is carried out under two complementary points of view. First, the conformal Killing equation is solved to obtain a whole family of finite-dimensional conformal algebras corresponding to each of the Galilei and Newton–Hooke kinematical groups. Some of their algebraic and geometrical properties are studied in a second step. Among the groups included in these families one can identify, for example, the contraction of the Minkowski conformal group, the analog for a nonrelativistic de Sitter space, or the nonextended Schrodinger group.

Intrinsic characterization of the variable separation in the Hamilton–Jacobi equation

Abstract: The nonorthogonal separation of variables in the Hamilton–Jacobi equation corresponding to a natural Hamiltonian H=12gijpipj+V, with a metric tensor of any signature, is intrinsically characterized by geometrical objects on the Riemannian configuration manifold: Killing vectors, Killing tensors, and Killing webs. Comparisons with previous characterizations and some illustrative examples are given.

On quantum field theory with nonzero minimal uncertainties in positions and momenta

Abstract: We continue studies on quantum field theories on noncommutative geometric spaces, focusing on classes of noncommutative geometries which imply ultraviolet and infrared modifications in the form of nonzero minimal uncertainties in positions and momenta. The case of the ultraviolet modified uncertainty relation which has appeared from string theory and quantum gravity is covered. The example of Euclidean φ4-theory is studied in detail and in this example we can now show ultraviolet and infrared regularization of all graphs.

The finite-band solution of the Jaulent–Miodek equation

Abstract: A method to construct the finite-band solution to the soliton equation is presented. We take the Jaulent–Miodek equation as an example. With the use of the nonlinearization of Lax pair, the Jaulent–Miodek equation is decomposed into two finite-dimensional integrable systems, whose properties are studied from the view of an r-matrix. Then through solving these finite-dimensional integrable systems, the finite-band solution of the Jaulent–Miodek equation is obtained.

Conditions for the existence of higher symmetries of evolutionary equations on the lattice

Abstract: In this paper we construct a set of five conditions necessary for the existence of generalized symmetries for a class of differential-difference equations depending only on nearest neighboring interaction. These conditions are applied to prove the existence of new integrable equations belonging to this class.

Hypersymmetry: A Z3-graded generalization of supersymmetry

Abstract: We propose a generalization of non-commutative geometry and gauge theories based on ternary Z3-graded structures. In the new algebraic structures we define, all products of two entities are left free, the only constraining relations being imposed on ternary products. These relations reflect the action of the Z3-group, which may be either trivial, i.e., abc=bca=cab, generalizing the usual commutativity, or non-trivial, i.e., abc=jbca, with j=e(2πi)/3. The usual Z2-graded structures such as Grassmann, Lie, and Clifford algebras are generalized to the Z3-graded case. Certain suggestions concerning the eventual use of these new structures in physics of elementary particles and fields are exposed.

Superintegrable n=2 systems, quadratic constants of motion, and potentials of Drach

Abstract: The properties of superintegrable systems in two degrees of freedom, possessing three independent globally defined constants of motion, are studied using as an approach, the existence of hidden symmetries and the generalized Noether’s theorem. The potentials are obtained as solution of a system of two partial differential equations. First the case of standard Lagrangians is studied and then the method is applied to the case of Lagrangians with a pseudo-Euclidean kinetic term. Finally, the results are related with other approaches and with a family of potentials admitting a second integral of motion cubic in the velocities obtained by Drach.

Generalization of Shannon’s theorem for Tsallis entropy

Abstract: By using the assumptions that the entropy must (i) be a continuous function of the probabilities {pi}(pi∈(0,1)∀i), only; (ii) be a monotonic increasing function of the number of states W, in the case of equiprobability; (iii) satisfy the pseudoadditivity relation Sq(A+B)/k=Sq(A)/k+Sq(B)/k+(1−q)Sq(A)Sq(B)/k2 (A and B being two independent systems, q∈R and k a positive constant), and (iv) satisfy the relation Sq({pi})=Sq(pL,pM)+pL qSq({pi/pL})+pM qSq({pi/pM}), where pL+pM=1(pL=∑i=1WLpi and pM=∑i=WL+1Wpi), we prove, along Shannon’s lines, that the unique function that satisfies all these properties is the generalized Tsallis entropy Sq=k(1−∑i=1Wpiq)/(q−1).

Homogeneous scalar field cosmologies with an exponential potential

Abstract: We shall study spatially homogeneous cosmological models containing a self-interacting scalar field with an exponential potential of the form V(φ)=Λekφ. The asymptotic properties of these models are discussed. In particular, their possible isotropization and inflation are investigated for all values of the parameter k. A particular class of models is analyzed qualitatively using the theory of dynamical systems, illustrating the general asymptotic behavior.

Global effects due to a chiral cone

Abstract: Scalar and spinor quantum particles are considered in the background space–time generated by a chiral cosmic string. It is shown that in this chiral conical space–time, the wave functions, the energy spectra, and the scattering amplitude associated with a quantum scalar particle depend on the global features of this space–time. In the case of a spinor particle, we study this dependence on the wave functions, energy, and current. These dependences represent gravitational analogues of the well-known Aharonov–Bohm effect in electrodynamics. Using the Hamilton–Jacobi equations, we also look into the motion of light rays and show how the effective potential depends on the global features of the chiral conical space–time.

Nonrelativistic conformal groups. II. Further developments and physical applications

Abstract: The finite-dimensional conformal groups associated with the Galilei and (oscillating or expanding) Newton–Hooke space–time manifolds was characterized by the present authors in a recent work. Three isomorphic group families, one for each nonrelativistic kinematics, were obtained, whose members are labeled by a half-integer number l. Since the action of these groups on their corresponding space–time manifolds is only local, a linearization is introduced here such that the corresponding action is well defined everywhere. In particular, the (l=1)-conformal cases that can be obtained by contraction from the well-known Minkowskian conformal group are treated in more detail. As an application of physical interest, the conformal invariance of the Galilean electromagnetism is studied. In order to achieve it, the pertinent local representations of the Galilean conformal algebras are derived.

Propagation of light in the stationary field of multipole gravitational lens

Abstract: A rigorous mathematical formalism for calculating the propagation of light rays in the stationary post-Newtonian field of an isolated celestial body (or system of bodies) considered as a gravitational lens having a complex multipole structure is developed. Symmetric trace-free tensors are used in the definition of gravitational multipoles instead of the less convenient (in general situations) scalar and vector spherical harmonics. Two types of perturbations of light rays, caused correspondingly by the mass and spin multipoles, are analyzed in full detail. A new simple method of integration for the equations of light propagation is proposed. This method enables us for the first time to obtain complete expressions both for the relativistic time delay and for the angle of the total deflection of light in any order of multipole perturbations without restriction. The results thus obtained can be applied to the interpretation of the secondary weak gravitational lens effects produced by the solar system bodies, ...

Non-Abelian topological mass generation in four dimensions

Abstract: We study the topological mass generation in the 4 dimensional non-Abelian gauge theory, which is the extension of Allen et al.’s work in the Abelian theory. It is crucial to introduce a one form auxiliary field in constructing the gauge invariant non-Abelian action which contains both the one form vector gauge field A and the two form antisymmetric tensor field B. As in the Abelian case, the topological coupling mB∧F, where F is the field strength of A, makes the transmutation among A and B possible, and consequently we see that the gauge field becomes massive. We find the BRST/anti-BRST transformation rule using the horizontality condition, and construct a BRST/anti-BRST invariant quantum action.

Electronic transport properties of quasicrystals

Abstract: We present a review of some results concerning electronic transport properties of quasicrystals. After a short introduction to the basic concepts of quasiperiodicity, we consider the experimental transport properties of electrical conductivity with particular focus on the effect of temperature, magnetic field, and defects. Then, we present some heuristic approaches that tend to give a coherent view of different, and to some extent complementary, transport mechanisms in quasicrystals. Numerical results are also presented and in particular the evaluation of the linear response Kubo–Greenwood formula of conductivity in quasiperiodic systems in the presence of disorder.

Mechanical systems with nonholonomic constraints

Abstract: A geometric setting for the theory of first-order mechanical systems subject to general nonholonomic constraints is presented. Mechanical systems under consideration are not supposed to be Lagrangian systems, and the constraints are not supposed to be of a special form in the velocities (as, e.g., affine or linear). A mechanical system is characterized by a certain equivalence class of 2-forms on the first jet prolongation of a fibered manifold. The nonholonomic constraints are defined to be a submanifold of the first jet prolongation. It is shown that this submanifold is canonically endowed with a distribution—this distribution (resp., its vertical subdistribution) has the meaning of generalized possible (resp., virtual) displacements. The concept of a constraint force is defined, and a geometric version of the principle of virtual work is proposed. From the principle of virtual work a formula for a workless constraint force is obtained. A mechanical system subject nonholonomic constraints is modeled as a deformation of the original (unconstrained) system. A direct characterization of a constrained system by means of a class of 2-forms along the canonical distribution is given, and “constrained equations of motion” in an intrinsic form are found. A geometric definition of regularity for systems under nonholonomic constraints is provided. In particular, the case of Lagrangian systems is discussed. Also systems subject to holonomic constraints and nonholonomic constraints affine in the velocities are investigated within the range of the general scheme.

Asymptotic completeness for Rayleigh scattering

Abstract: We consider an electron bound by some anharmonic external potential and coupled to the quantized radiation field in the dipole approximation. We prove asymptotic completeness for the photon scattering. This means that an arbitrary initial state has a long time asymptotic, which consists of electron plus radiation field in their coupled ground state and finitely many outgoing photons.

A generalized formula for integrable classes of surfaces in Lie algebras

Abstract: We discuss relations between the approach of Fokas and Gelfand to immersions on Lie algebras and the theory of soliton surfaces of Sym. We show that many results concerning immersions on Lie algebras can be reduced to or interpreted within the soliton surfaces approach. We present also some new results, including a generalization of the Fokas–Gelfand formula for integrable classes of surfaces in Lie algebras [and, in particular, in (pseudo)-Euclidean n-dim. spaces]. The generalized formula is used to formulate a method of constructing integrable classes of surfaces. As an example we discuss the class of linear Weingarten surfaces defined by the linear relationship between Gaussian and mean curvatures. We construct explicitly a one-parameter family of linear Weingarten surfaces parallel (equidistant) to a given pseudospherical surface.

Dynamics of generalized Poisson and Nambu–Poisson brackets

Abstract: A unified setting for generalized Poisson and Nambu–Poisson brackets is discussed. It is proved that a Nambu–Poisson bracket of even order is a generalized Poisson bracket. Characterizations of Poisson morphisms and generalized infinitesimal automorphisms are obtained as coisotropic and Lagrangian submanifolds of product and tangent manifolds, respectively.

On the statistical independence of algebras of observables

Abstract: We reexamine various notions of statistical independence presently in use in algebraic quantum theory, establishing alternative characterizations for such independence, some of which are also valid without assuming that the observable algebras mutually commute. In addition, in the context which holds in concrete applications to quantum theory, the equivalence of three major notions of statistical independence is proven.

Superintegrability on the two-dimensional hyperboloid. II

Abstract: This work is devoted to the investigation of the quantum mechanical systems on the two-dimensional hyperboloid which admits separation of variables in at least two coordinate systems. Here we consider two potentials introduced in a paper of C. P. Boyer, E. G. Kalnins, and P. Winternitz [J. Math. Phys. 24, 2022 (1983)], which have not yet been studied. We give an example of an interbasis expansion and work out the structure of the quadratic algebra generated by the integrals of motion.

Form invariance of differential equations in general relativity

Abstract: Einstein equations for several matter sources in Robertson–Walker and Bianchi I type metrics, are shown to reduce to a kind of second-order nonlinear ordinary differential equation y+αf(y)ẏ+βf(y)∫f(y)dy+γf(y)=0. Also, it appears in the generalized statistical mechanics for the most interesting value q=−1. The invariant form of this equation is imposed and the corresponding nonlocal transformation is obtained. The linearization of that equation for any α, β, and γ is presented and for the important case f=byn+k with β=α2(n+1)/(n+2)2 its explicit general solution is found. Moreover, the form invariance is applied to yield exact solutions of some other differential equations.

Topological equivalence of tilings

Abstract: We introduce a notion of equivalence on tilings which is formulated in terms of their local structure. We compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are topologically equivalent whenever their associated groupoids are isomorphic.

Coupling coefficients for Lie algebra representations and addition formulas for special functions

Abstract: Representations of the Lie algebra su(1,1) and of a generalization of the oscillator algebra, b(1), are considered. The paper then introduces polynomials which are related by the coupling (or Clebsch–Gordan) coefficients of the Lie algebra in question; by making a proper choice, these polynomials themselves are related to known special functions. The coupling of two or three representations of the Lie algebra then leads to interesting addition formulas for these special functions. The polynomials appearing here are generalized Laguerre and Jacobi polynomials for the su(1,1) case, and Hermite polynomials for the b(1) algebra.

Debye potentials for Maxwell and Dirac fields from a generalization of the Killing–Yano equation

Abstract: By using conformal Killing–Yano tensors, and their generalizations, we obtain scalar potentials for both the source-free Maxwell and massless Dirac equations. For each of these equations we construct, from conformal Killing–Yano tensors, symmetry operators that map any solution to another.

The ultrarelativistic Reissner-Nordstrom field in the Colombeau algebra

Abstract: The electromagnetic field of the ultrarelativistic Reissner–Nordstro/m solution shows the physically highly unsatisfactory property of a vanishing field tensor but a nonzero, i.e., δ-like, energy density. The aim of this work is to analyze this situation from a mathematical point of view, using the framework of Colombeau’s theory of nonlinear generalized functions. It is shown that the physically unsatisfactory situation is mathematically perfectly defined and that one cannot avoid such situations when dealing with distributional valued field tensors.