Showing papers in "Journal of Mathematical Physics in 1998"

Relativistic spin networks and quantum gravity

Abstract: Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2)×SU(2). Relativistic quantum spins are related to the geometry of the two-dimensional faces of a 4-simplex. This extends the idea of Ponzano and Regge that SU(2) spins are related to the geometry of the edges of a 3-simplex. This leads us to suggest that there may be a four-dimensional state sum model for quantum gravity based on relativistic spin networks that parallels the construction of three-dimensional quantum gravity from ordinary spin networks.

A length operator for canonical quantum gravity

Abstract: We construct an operator that measures the length of a curve in four-dimensional Lorentzian vacuum quantum gravity. We work in a representation in which an SU(2) connection is diagonal and it is therefore surprising that the operator obtained after regularization is densely defined, does not suffer from factor ordering singularities, and does not require any renormalization. We show that the length operator admits self-adjoint extensions and compute part of its spectrum which, like its companions, the volume and area operators already constructed in the literature, is purely discrete and roughly quantized in units of the Planck length. The length operator contains full and direct information about all the components of the metric tensor which facilitates the construction of so-called weave states which approximate a given classical three-geometry.

An exactly soluble Schrodinger equation with smooth position-dependent mass

Abstract: The one-dimensional generalized Schrodinger equation for a system with smooth potential and mass step is resolved exactly. The wave function depends on the Heun’s function, which is a solution of a second-order Fuchsian equation with four singularities. The behavior of the transmission coefficient as a function of energy is compared to that of the case of an abrupt potential and mass step. Two limiting cases are also studied: when the width of the mass step is vanishing, and when the smooth potential and mass step tend to an abrupt potential and mass step.

Random cascades on wavelet dyadic trees

Abstract: We introduce a new class of random fractal functions using the orthogonal wavelet transform These functions are built recursively in the space-scale half-plane of the orthogonal wavelet transform, “cascading” from an arbitrary given large scale towards small scales To each random fractal function corresponds a random cascading process (referred to as a W-cascade) on the dyadic tree of its orthogonal wavelet coefficients We discuss the convergence of these cascades and the regularity of the so-obtained random functions by studying the support of their singularity spectra Then, we show that very different statistical quantities such as correlation functions on the wavelet coefficients or the wavelet-based multifractal formalism partition functions can be used to characterize very precisely the underlying cascading process We illustrate all our results on various numerical examples

Cohomology of Lie superalgebras and their generalizations

Abstract: The cohomology groups of Lie superalgebras and, more generally, of e Lie algebras, are introduced and investigated. The main emphasis is on the case where the module of coefficients is nontrivial. Two general propositions are proved, which help to calculate the cohomology groups. Several examples are included to show the peculiarities of the super case. For L=sl(1|2), the cohomology groups H1(L,V) and H2(L,V), with V a finite-dimensional simple graded L-module, are determined, and the result is used to show that H2(L,U(L)) [with U(L) the enveloping algebra of L] is trivial. This implies that the superalgebra U(L) does not admit any nontrivial formal deformations (in the sense of Gerstenhaber). Garland’s theory of universal central extensions of Lie algebras is generalized to the case of e Lie algebras.

Closed formula for the matrix elements of the volume operator in canonical quantum gravity

Abstract: We derive a closed formula for the matrix elements of the volume operator for canonical Lorentzian quantum gravity in four space–time dimensions in the continuum in a spin-network basis. We also display a new technique of regularization which is state dependent but we are forced to it in order to maintain diffeomorphism covariance and in that sense it is natural. We arrive naturally at the expression for the volume operator as defined by Ashtekar and Lewandowski up to a state-independent factor.

Information gain within nonextensive thermostatistics

Abstract: We discuss the information theoretical foundations of the Kullback information gain, recently generalized within a nonextensive thermostatistical formalism. General properties are studied and, in particular, a consistent test for measuring the degree of correlation between random variables is proposed. In addition, minimum entropy distributions are discussed and the H-theorem is proved within the generalized context.

Wavelets on the n-sphere and related manifolds

Abstract: We present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the (n-1)-sphere Sn-1. based on the construction of general coherent states associated to square integrable group representations. The parameter space of the CWT, X similar to SO(n)xR*(+), is embedded into the generalized Lorentz group SO0(n,1) via the Iwasawa decomposition, so that X similar or equal to SO0(n,1)IN, where N similar or equal to Rn-1. Then the CWT on Sn-1 is derived from a suitable unitary representation of SO0(n,1) acting in the space L-2(Sn-1,d mu) of finite energy signals on Sn-1, which turns out to be square integrable over X. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition, which entails all the usual filtering properties of the CWT. Next the Euclidean limit of this CWT on Sn-1 is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R-->infinity, from which one recovers the usual CWT on flat Euclidean space. Finally, we discuss the extension of this construction to the two-sheeted hyperboloid Hn-1SO0(n-1,1)/SO(n-1) and some other Riemannian symmetric spaces. (C) 1998 American Institute of Physics. [S0022-2488(98)00308-9].

Multidimensional classical and quantum cosmology with intersecting p-branes

Abstract: A multidimensional cosmological model describing the evolution of (n+1) Einstein spaces in the theory with several scalar fields and forms is considered. When a (electro-magnetic composite) p-brane ansatz is adopted the field equations are reduced to the equations for the Toda-like system. The Wheeler–De Witt equation is obtained. In the case when n “internal” spaces are Ricci-flat, one space M0 has a non-zero curvature, and all p-branes do not “live” in M0; the classical and quantum solutions are obtained if certain orthogonality relations on parameters are imposed. Spherically symmetric solutions with intersecting non-extremal p-branes are singled out. A non-orthogonal generalization of intersection rules corresponding to (open, closed) Toda lattices is obtained. A chain of bosonic D⩾11 models (that may be related to hypothetical higher dimensional supergravities and F-theories) is suggested.

Determinant formula for the six-vertex model with reflecting end

Abstract: Using the quantum inverse scattering method for the XXZ model with open boundary conditions, we obtained the determinant formula for the six-vertex model with reflecting end.

Hidden algebras of the (super) Calogero and Sutherland models

Abstract: We propose to parametrize the configuration space of one-dimensional quantum systems of N identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates. It is shown that in this parametrization the Hamiltonians of the AN, BCN, BN, CN and DN Calogero and Sutherland models, as well as their supersymmetric generalizations, can be expressed—for arbitrary values of the coupling constants—as quadratic polynomials in the generators of a Borel subalgebra of the Lie algebra gl(N+1) or the Lie superalgebra gl(N+1|N) for the supersymmetric case. These algebras are realized by first order differential operators. This fact establishes the exact solvability of the models according to the general definition given by Turbiner, and implies that the Calogero and Jack–Sutherland polynomials, as well as their supersymmetric generalizations, are related to finite-dimensional irreducible representations of the Lie algebra gl(N+1) and the Lie superalgebra gl(N+1|N).

Aharonov-Bohm Effect with $δ$--type Interaction

Abstract: A quantum particle interacting with a thin solenoid and a magnetic flux is described by a five-parameter family of Hamilton operators, obtained via the method of self-adjoint extensions. One of the parameters, the value of the flux, corresponds to the Aharonov–Bohm effect; the other four parameters correspond to the strength of a singular potential barrier. The spectrum and eigenstates are computed and the scattering problem is solved.

Equivalence between Chain Categories of Representations of Affine sl(2) and N=2 Superconformal Algebras

Abstract: Highest-weight-type representation theories of the affine sl(2) and N=2 superconformal algebras are shown to be equivalent modulo the respective spectral flows.

New integrable generalizations of Calogero–Moser quantum problem

Abstract: A one-parameter deformation of Calogero–Moser quantum problem is introduced. It is shown that corresponding Schrodinger operator is integrable for any value of the parameter and algebraically integrable in case of integer value.

Discrete spectral triples and their symmetries

Abstract: We classify zero-dimensional spectral triples over complex and real algebras and provide some general statements about their differential structure. We investigate also whether such spectral triples admit a symmetry arising from the Hopf algebra structure of the finite algebra. We discuss examples of commutative algebras and group algebras.

A geometrical angle on Feynman integrals

Abstract: A direct link between a one-loop N-point Feynman diagram and a geometrical representation based on the N-dimensional simplex is established by relating the Feynman parametric representations to the integrals over contents of (N−1)-dimensional simplices in non-Euclidean geometry of constant curvature. In particular, the four-point function in four dimensions is proportional to the volume of a three-dimensional spherical (or hyperbolic) tetrahedron which can be calculated by splitting into birectangular ones. It is also shown that the known formula of reduction of the N-point function in (N−1) dimensions corresponds to splitting the related N-dimensional simplex into N rectangular ones.

The Maxwell–Vlasov equations in Euler–Poincaré form

Abstract: Low's well-known action principle for the Maxwell–Vlasov equations of ideal plasma dynamics was originally expressed in terms of a mixture of Eulerian and Lagrangian variables. By imposing suitable constraints on the variations and analyzing invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we first transform this action principle into purely Eulerian variables. Hamilton's principle for the Eulerian description of Low's action principle then casts the Maxwell–Vlasov equations into Euler–Poincare form for right invariant motion on the diffeomorphism group of position-velocity phase space, [openface R]6. Legendre transforming the Eulerian form of Low's action principle produces the Hamiltonian formulation of these equations in the Eulerian description. Since it arises from Euler–Poincare equations, this Hamiltonian formulation can be written in terms of a Poisson structure that contains the Lie–Poisson bracket on the dual of a semidirect product Lie algebra. Because of degeneracies in the Lagrangian, the Legendre transform is dealt with using the Dirac theory of constraints. Another Maxwell–Vlasov Poisson structure is known, whose ingredients are the Lie–Poisson bracket on the dual of the Lie algebra of symplectomorphisms of phase space and the Born–Infeld brackets for the Maxwell field. We discuss the relationship between these two Hamiltonian formulations. We also discuss the general Kelvin–Noether theorem for Euler–Poincare equations and its meaning in the plasma context.

Continuous Time and Consistent Histories

Abstract: We discuss the use of histories labelled by a continuous time in the approach to consistent-histories quantum theory in which propositions about the history of the system are represented by projection operators on a Hilbert space. This extends earlier work by two of us [C. J. Isham and N. Linden, J. Math. Phys. 36, 5392–5408 (1995)] where we showed how a continuous time parameter leads to a history algebra that is isomorphic to the canonical algebra of a quantum field theory. We describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about the time average of the energy. We also show that the history description of quantum mechanics contains an operator corresponding to velocity that is quite distinct from the momentum operator. Finally, the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism.

Geodesics and geodesic deviation for impulsive gravitational waves

Abstract: The geometry of impulsive pp-waves is explored via the analysis of the geodesic and geodesic deviation equation using the distributional form of the metric. The geodesic equation involves formally ill-defined products of distributions due to the nonlinearity of the equations and the presence of the Dirac δ-distribution in the space–time metric. Thus, strictly speaking, it cannot be treated within Schwartz’s linear theory of distributions. To cope with this difficulty we proceed by first regularizing the δ-singularity, then solving the regularized equation within classical smooth functions and, finally, obtaining a distributional limit as solution to the original problem. Furthermore, it is shown that this limit is independent of the regularization without requiring any additional condition, thereby confirming earlier results in a mathematically rigorous fashion. We also treat the Jacobi equation which, despite being linear in the deviation vector field, involves even more delicate singular expressions, like the “square” of the Dirac δ-distribution. Again the same regularization procedure provides us with a perfectly well behaved smooth regularization and a regularization-independent distributional limit. Hence it is concluded that the geometry of impulsive pp-waves can be described consistently using distributions as long as careful regularization procedures are used to handle the ill-defined products.

Theory of nonequilibrium first-order phase transitions for stochastic dynamics

Abstract: A dynamic definition of a first-order phase transition is given. It is based on a master equation description of the time evolution of a system. When the operator generating that time evolution has an isolated near degeneracy there is a first-order phase transition. Conversely, when phenomena describable as first-order phase transitions occur in a system, the corresponding operator has near degeneracy. Estimates relating degree of degeneracy and degree of phase separation are given. This approach harks back to early ideas on phase transitions and degeneracy, but now enjoys greater generality because it involves an operator present in a wide variety of systems. Our definition is applicable to what have intuitively been considered phase transitions in nonequilibrium systems and to problematic near equilibrium cases, such as metastability.

Onset of superconductivity in decreasing fields for general domains

Abstract: Ginzburg–Landau theory has provided an effective method for understanding the onset of superconductivity in the presence of an external magnetic field. In this paper we examine the instability of the normal state to superconductivity with decreasing magnetic field for a closed smooth cylindrical region of arbitrary cross-section subject to a vertical magnetic field. We examine the problem asymptotically in the boundary layer limit (i.e., when the Ginzburg–Landau parameter, k, is large). We demonstrate that instability first occurs in a region exponentially localized near the point of maximum curvature on the boundary. The transition occurs at a value of the magnetic field associated with the half-plane at leading order, with a small positive correction due to the curvature (which agrees with the transition problem for the disc), and a smaller correction due to the second derivative of the curvature at the maximum.

Integrable semi-discretization of the coupled modified KdV equations

Abstract: The discrete version of the inverse scattering method proposed by Ablowitz and Ladik is extended to solve multi-component systems. The extension enables one to solve the initial value problem, which proves directly the complete integrability of a semi-discrete version of the coupled modified Korteweg–de Vries (KdV) equations and their hierarchy. It also provides a procedure to obtain conservation laws and multi-soliton solutions of the hierarchy.

Analytic-bilinear approach to integrable hierarchies. II. Multicomponent KP and 2D Toda lattice hierarchies

Abstract: An analytic-bilinear approach for the construction and study of integrable hierarchies is discussed. Generalized multicomponent KP and 2D Toda lattice hierarchies are considered. This approach allows us to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. A generalized hierarchy incorporates basic hierarchy, modified hierarchy, singularity manifold equation hierarchy, and corresponding linear problems. Different levels of generalized hierarchy are connected via invariants of Combescure symmetry transformation. The resolution of functional equations also leads to the τ function and addition formulas to it.

Snapback repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy injection at the middle of the span

Abstract: A wave equation on a one-dimensional interval I has a van der Pol type nonlinear boundary condition at the right end. At the left end, the boundary condition is fixed. At exactly the midpoint of the interval I, energy is injected into the system through a pair of transmission conditions in the feedback form of anti-damping. We wish to study chaotic wave propagation in the system. A cause of chaos by snapback repellers has been identified. These snapback repellers are repelling fixed points possessing homoclinic orbits of the non-invertible map in 2D corresponding to wave reflections and transmissions at, respectively, the boundary and the middle-of-the-span points. Existing literature [F. R. Marotto, J. Math. Anal. Appl. 63, 199–223 (1978)] on snapback repellers contains an error. We clarify the error and give a refined theorem that snapback repellers imply chaos. Numerical simulations of chaotic vibration are also illustrated.

Finite dimensional irreducible representations of twisted Yangians

Abstract: We study quantized enveloping algebras called twisted Yangians. They are analogs of the Yangian Y(gl(N)) for the classical Lie algebras of B, C, and D series. The twisted Yangians are subalgebras in Y(gl(N)) and coideals with respect to the coproduct in Y(gl(N)). We give a complete description of their finite-dimensional irreducible representations. Every such representation is highest weight and we give necessary and sufficient conditions for an irreducible highest weight representation to be finite dimensional. The result is analogous to Drinfeld’s theorem for the ordinary Yangians. Its detailed proof for the A series is also reproduced. For the simplest twisted Yangians we construct an explicit realization for each finite-dimensional irreducible representation in tensor products of representations of the corresponding Lie algebras.

Quantum space-time and classical gravity

Abstract: A method has been recently proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example a version of quantized space–time is considered here. It is found that there is a natural differential calculus over this version of quantized space–time using which the only possible torsion-free, metric-compatible, linear connection has zero curvature. It is then the noncummutative version of Minkowski space–time. Perturbations of this calculus are shown to give rise to nontrivial gravitational fields.

Hamiltonian time-dependent mechanics

Abstract: The usual formulation of time-dependent mechanics implies a given splitting Y=R×M of an event space Y. This splitting, however, is broken by any time-dependent transformation, including transformations between inertial frames. The goal is the frame-covariant formulation of time-dependent mechanics on a bundle Y→R, whose fibration Y→M is not fixed. Its phase space is the vertical cotangent bundle V*Y, provided with the canonical 3-form and the corresponding canonical Poisson structure. An event space of relativistic mechanics is a manifold Z whose fibration Z→R is not fixed.

Infinite and finite Gleason’s theorems and the logic of indeterminacy

Abstract: In the first half of the paper I prove Gleason’s lemma: Every non-negative frame function on the set of rays in R3 is continuous. This is the central and most difficult part of Gleason’s theorem. The proof is a reconstruction of Gleason’s idea in terms of orthogonality graphs. The result is a demonstration that this theorem is actually combinatorial in nature. It depends only on a finite graph structure. In the second half of the paper I use the graph construction to obtain results about probability distributions (non-negative frame functions with weight one) on finite sets of rays. For example, given any two distinct nonorthogonal rays a and b, I construct a finite set of rays Γ that contains them, and has the following property: No probability distribution on Γ assigns both a and b a truth value (probability zero or one) unless they are both false. Thus the principle of indeterminacy turns into a theorem of propositional quantum logic (or partial Boolean algebras).

Strong asymptotics of Laguerre polynomials and information entropies of two-dimensional harmonic oscillator and one-dimensional Coulomb potentials

Abstract: The information entropies of the two-dimensional harmonic oscillator, V(x,y)=1/2λ(x2+y2), and the one-dimensional hydrogen atom, V(x)=−1/|x|, can be expressed by means of some entropy integrals of Laguerre polynomials whose values have not yet been analytically determined. Here, we first study the asymptotical behavior of these integrals in detail by extensive use of strong asymptotics of Laguerre polynomials. Then, this result (which is also important by itself in a context of both approximation theory and potential theory) is employed to analyze the information entropies of the aforementioned quantum-mechanical potentials for the very excited states in both position and momentum spaces. It is observed, in particular, that the sum of position and momentum entropies has a logarithmic growth with respect to the main quantum number which characterizes the corresponding physical state. Finally, the rate of convergence of the entropies is numerically examined.

Point vortices on a sphere: Stability of relative equilibria

Abstract: In this paper we analyze the dynamics of N point vortices moving on a sphere from the point of view of geometric mechanics. The formalism is developed for the general case of N vortices, and the details are worked out for the (integrable) case of three vortices. The system under consideration is SO(3) invariant; the associated momentum map generated by this SO(3) symmetry is equivariant and corresponds to the moment of vorticity. Poisson reduction corresponding to this symmetry is performed; the quotient space is constructed and its Poisson bracket structure and symplectic leaves are found explicitly. The stability of relative equilibria is analyzed by the energy-momentum method. Explicit criteria for stability of different configurations with generic and nongeneric momenta are obtained. In each case a group of transformations is specified, modulo which one has stability in the original (unreduced) phase space. Special attention is given to the distinction between the cases when the relative equilibrium is a nongreat circle equilateral triangle and when the vortices line up on a great circle.