Volume 39, Issue 6, June 1998
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Relativistic Gamow vectors
View Description Hide DescriptionThe Friedrichs model has often been used in order to obtain explicit formulas for eigenvectors associated to complex eigenvalues corresponding to lifetimes. Such eigenvectors are called Gamow vectors and they acquire meaning in extensions of the conventional Hilbert space of quantum theory to the socalled rigged Hilbert space. In this paper, Gamow vectors are constructed for a solvable model of an unstable relativistic field. As a result, we obtain a time asymmetric relativistic extension of the Fock space. This extension leads to two distinct Poincaré semigroups. The time reversal transformation maps one semigroup to the other. As a result, the usual PCT invariance should be extended. We show that irreversibility as expressed by dynamical semigroups is compatible with the requirements of relativity.

Quantum martingale measures and stochastic partial differential equations in Fock space
View Description Hide DescriptionA concept of quantum martingale measure is introduced and examples are constructed as quantum stochastic spectral integrals in Fock space. These are then utilized as space–time noise to drive a parabolic stochastic partial differential equation (spde). We establish the existence and uniqueness of the solutions as families of densely defined closable operators in Fock space that are jointly continuous in time and space variables and satisfy a Markov property.

Interaction energy of generalized Abelian Higgs vortices
View Description Hide DescriptionThe interaction energy between vortices of the generalized Abelian Higgs vortices is studied numerically. These are the solitons of the th member of the hierarchy of MaxwellHiggs systems, the member of which is the usual Abelian Higgs model. It is found that for all these vortices have both attractive and repulsive phases, the strengths of these interactions decreasing successively for increasing A detailed comparison of the and cases is made quantitatively, and an extended model, composed of the and members, is studied in some detail and shown to exhibit attractive and repulsive phases.

A relativistic supersymmetry and some algebraic consequences
View Description Hide DescriptionWe analyze the supersymmetry and the shape invariance of the potentials which appear in the Klein–Gordon equations of the relativistic Pöschl–Teller oscillators we have recently proposed. We point out how the main operators depend on the value of the unique parameter involved in the shape invariance of the supersymmetric partner potentials, and we propose a unitary transformation that changes its value.

Strong asymptotics of Laguerre polynomials and information entropies of twodimensional harmonic oscillator and onedimensional Coulomb potentials
View Description Hide DescriptionThe informationentropies of the twodimensional harmonic oscillator, and the onedimensional hydrogen atom, 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 informationentropies of the aforementioned quantummechanical 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.

The modified Newton–Sabatier method for the coupled channel inverse scattering problem with charged particles at fixed energy
View Description Hide DescriptionThe modified Newton–Sabatier method was recently developed for inelastic inverse problems for the scattering of neutral particles. In this paper, the method is extended to the solution of the inelastic inverse scattering problem with charged particles. The bases are radial Schrödinger equations coupled by a local potential matrix known from a certain distance on. The inversion procedure starts with the matrix fixing the asymptotic wave functions of the charged scattering system. This matrix can be transformed to another one for asymptotic constant potentials instead of Coulomb potentials. The new matrix belongs to the same inner potential matrix as the original one and is solved by the modified Newton–Sabatier method similar to the case of neutral particles. An application with a given matrix, belonging to a coupled squarewell potential matrix with an outer Coulomb potential yields good agreement between the inverted potential matrix and the original one.

A nonAbelian square root of Abelian vertex operators
View Description Hide DescriptionThe quantum fieldtheoretical formulation of Kadanoff’s “correlations along a line” in the critical twodimensional Ising model is given in terms of products of Wightman fields. It provides a quadratic factorization, in the sense of operatorvalued distributions, of Abelian chiral vertex operators into nonAbelian exchange fields. This basic result implies polynomial relations between fields from various conformal models in two dimensions; notably the twodimensional local fields with chiral symmetry at level 2 are shown to share the nontrivial peculiarities of the Ising model order and disorder fields. These examples have interesting implications for the properties of products in general.

Constrained dynamics for quantum mechanics. I. Restricting a particle to a surface
View Description Hide DescriptionWe analyze constrained quantum systems where the dynamics do not preserve the constraints. This is done, in particular, for the restriction of a quantum particle in to a curved submanifold, and we propose a method of constraining and dynamics adjustment that produces the right Hamiltonian on the submanifold when tested on known examples. This method will be the germ of a “Dirac algorithm for quantum constraints.” We generalize it to the situation where the constraint is a general selfadjoint operator with some additional structures.

de Rham cohomology of and some related manifolds by supersymmetric quantum mechanics
View Description Hide DescriptionWe study supersymmetric quantum mechanics on and U(2) to examine Witten’s Morse theory concretely. We confirm the simple instanton picture of the de Rham cohomology that has been given in a previous paper. We use a reflection symmetry of each theory to select the true vacuums. The number of selected vacuums agrees with the de Rham cohomology for each of the above manifolds.

Quantization of minisuperspaces as ordinary gauge systems
View Description Hide DescriptionSimple cosmological models are used to show that gravitation can be quantized as an ordinary gauge system if the Hamilton–Jacobi equation for the model under consideration is separable. In this situation, a canonical transformation can be performed such that in terms of the new variables the model has a linear and homogeneous constraint, and therefore canonical gauges are admissible in the path integral. This has the additional practical advantage that gauge conditions that do not generate Gribov copies are then easy to choose.
 Top

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


The Maxwell–Vlasov equations in Euler–Poincaré form
View Description Hide DescriptionLow’s wellknown action principle for the Maxwell–Vlasovequations 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–Vlasovequations into Euler–Poincaré form for right invariant motion on the diffeomorphism group of positionvelocity phase space, 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–Poincaré equations, this Hamiltonian formulation can be written in terms of a Poissonstructure 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–VlasovPoissonstructure 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–Poincaré equations and its meaning in the plasma context.

Maxwell equations and the optical geometry
View Description Hide DescriptionWe show that, in conformally static spacetimes,Maxwell equations take their simplest form when written in terms of the socalled “optical metric.”
 Top

 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


Microscopic dynamics from a coarsely defined solution to the protein folding problem
View Description Hide DescriptionIn this work we introduce a leastaction formulation of the protein folding problem casted within a coarse description of the soft mode dynamics of the peptide chain. Ultimately, we show that this coarse variational approach can be lifted to yield the microscopic longtime torsional dynamics responsible for the actual folding process. As a first step, a binary coding of local topological constraints associated to each structural motif is introduced to coarsely mimic the longtime dynamics.Folding pathways are initially resolved as transitions between patterns of locally encoded structural signals. Our variational approach is aimed at identifying the most economic pathway with respect to the stepwise cost in conformational freedom. Our treatment allows us to account for the expediency of the process in proteins effectively capable of in vitro renaturation. We identify the dominant pathway by introducing a coarse version of Lagrangian microscopic dynamics. The coarse folding pathways are generated by a parallel search for structural patterns in a matrix of local topological constraints (LTM) of the chain. Each local topological constraint represents a coarse description of a local torsional state and each pattern is evaluated, translated, and finally recorded as a contact matrix (CM), an operation that is subject to a renormalization feedback loop. The renormalization operation periodically introduces longrange correlations on the LTM according to the latest CM generated by translation. Local topological constraints may form consensus regions in portions of the chain that translate as secondary structure motifs or tertiary interactions. Nucleation steps and cooperative effects are accounted for by means of the renormalization operation, which warrants the persistence of seeding patterns upon successive LTM evaluations. Relevant folding time scales beyond the realm of molecular dynamics simulations become accessible through the coarsely codified representation of local torsional constraints. The validity of our approach is tested vis à vis experimentally probed folding pathways generating tertiary interactions in proteins that may recover their active structure under in vitro renaturation conditions. We focus on determining significant folding intermediates and the late kinetic bottlenecks that occur within the first of the renaturation process. After the computational accessibility of this coarse solution of the folding problem becomes apparent, we show how to lift our variational problem to microscopic dynamics of the peptide chain. The consistency of our approach is revealed by an actual generation of Newton’s laws at the microscopic level through an inverse projection of the coarse dynamics originally generated through the pattern recognition computation.

Free energy and correlations of the numbertheoretical spin chain
View Description Hide DescriptionWe analyze the numbertheoretical spin chain with partition function =ζ(β−1)/ζ(β) using the polymermodel technique. The finite (grand) canonical chains give bounds for the limit free energy and internal energy. The correlation functions for inverse temperature are products of twopoint functions. A combinatorial result for general interval graphs is derived.
 Top

 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Generalized similarity solutions for the type D fluid in fivedimensional flat space
View Description Hide DescriptionThe partial differential equation corresponding to the metric function in a fivedimensional flat space describing the perfect fluid distribution has been examined for generalized symmetries via a new approach. It is shown that the approach utilized here has, besides providing generalized symmetries, for some particular values of the parameters involved, yielded new metrics and Petrov type D solutions. Further, for the general case the problem is reduced to the solution of an Abel’s equation of second kind, which obviously generalizes the scope for new metrics and new Petrov type D solutions.

On separability of biHamiltonian chain with degenerated Poisson structures
View Description Hide DescriptionSeparability of biHamiltonian finitedimensional chains with two degenerated Poissontensors, which have Pfaffian quasibiHamiltonian representation, is proved.

An application of Lax’s formalism to a class of curvilinear axially symmetric coordinates in threedimensional space
View Description Hide DescriptionSolutions of a nonlinear system of differential partial equations in twodimensional space are studied. We prove that the considered system is fully integrable and that its solutions determine an infinite class of mappings of the space of curvilinear axially symmetric coordinates onto the space of cylindric coordinates. The corresponding family of reverse transformations is found in a closed form. An infinite number of the first integrals of the nonlinear system is obtained. Symmetries of a class of electrostatic systems associated with the considered family of curvilinear coordinates are discussed.

The potential constraints of the complex dimensional soliton systems and related Hamiltonian equations
View Description Hide DescriptionPotential complete constraints (symmetric and nonsymmetric) of the complex KP and MKP systems are derived by using the general theory for a nonlinear equation to be a Hamiltonian system. All the complex Hamiltonian equations corresponding to this kind of constraint are obtained, which contain the famous nonlinear Schrödinger equations and several new Hamiltonian equations. In addition, the classical Kaup–Newell system and the Heisenberg system are also led by a new reduction approach from the MKP system.

Thermal conduction before relaxation in slowly rotating fluids
View Description Hide DescriptionFor slowly rotating fluids, we establish the existence of a critical point similar to the one found for nonrotating systems. As the fluid approaches the critical point, the effective inertial mass of any fluid element decreases, vanishing at that point and changing sign beyond it. This result implies that the firstorder perturbative method is not always reliable to study dissipative processes occurring before relaxation. Physical consequences that might follow from this effect are commented on.

On different integrable systems sharing the same nondynamical matrix
View Description Hide DescriptionIn a recent paper [Zhijun Qiao and Ruguang Zhou, Phys. Lett. A 235, 35 (1997)], the amazing fact was reported that a discrete and a continuous integrable system share the same matrix with the interesting property of being nondynamical. Now, we present three further pairs of different continuous integrable systems sharing the same matrix again being nondynamical. The first pair is the finitedimensional constrained system (FDCS) of the famous AKNS hierarchy and the Dirac hierarchy; the second pair is the FDCS of the wellknown geodesic flows on the ellipsoid and the Heisenberg spin chain hierarchy; and the third pair is the FDCS of one hierarchy studied by Xianguo Geng [Phys. Lett. A 162, 375 (1992)] and another hierarchy proposed by Zhijun Qiao [Phys. Lett. A 192, 316 (1994)]. All those FDCSs possess Lax representations and from the viewpoint of matrix can be shown to be completely integrable in Liouville’s sense.
