Volume 38, Issue 11, November 1997
Index of content:

Quantum Yang–Mills theory in two dimensions: A complete solution
View Description Hide DescriptionA closed expression of the Euclidean Wilsonloop functionals is derived for pure Yang–Mills continuum theories with gauge groups and and spacetime topologies and (For the theory, we also consider the topology.) The treatment is rigorous, manifestly gauge invariant, manifestly invariant under area preserving diffeomorphisms and handles all (piecewise analytic) loops in one stroke. Equivalence between the resulting Euclidean theory and and the Hamiltonian framework is then established. Finally, an extension of the Osterwalder–Schrader axioms for gauge theories is proposed. These axioms are satisfied in the present model.

Open perturbation and the Riccati equation: Algebraic determination of the quartic anharmonic oscillator energies and eigenfunctions
View Description Hide DescriptionAn algebraic procedure is proposed for the analytical solution of Schrödinger equations that can be viewed as a factorizable equation with an additional potential Once has been expanded in a series of suitable basis functions which are specific to each factorization type, the solution of the Riccati equation associated with the given equation is performed by means of an open perturbation technique, i.e., at each order of the perturbation, an additional balance dependent term is introduced so that the resulting equation becomes solvable. Since the unperturbed potential involves the whole given potential and since the balance term is expected to be small, improved results are expected at low orders of the perturbation, even at the zeroth order. The procedure, well adapted to the use of computer algebra, is applied to the solution of the anharmonic oscillator equation: by means of very simple algebraic manipulations, the trend of the exact values of the energies is rather well reproduced for a large range of values of the coupling constant ( to ).

Reconstruction of secular polynomials for Hubbard model from energy perturbation series for weak and strong coupling cases
View Description Hide DescriptionAn interpolation technique which is based on the strongly and weakly correlated cases of the onedimensional Hubbard model is proposed. The input information consists of the perturbation expansions which are obtained from the Lieb–Wu equations in both limits. The Hubbard model is used to describe cyclic polyene rings, and for the case of sites, this would correspond to benzene. The technique has been applied to several symmetries of the model. It has been shown that the exact secular problem can be reconstructed for these symmetries, and the results for agree exactly with the results which have been obtained in a different way, that is, by using a full configuration interaction calculation.

The operator algebra of the quantum relativistic oscillator
View Description Hide DescriptionThe operator algebras of a new family of relativistic geometricmodels of the relativistic oscillator [I. I. Cotăescu, Int. J. Mod. Phys. A 12, 3545 (1997)] are studied. It is shown that, generally, the operator of number of quanta and the pair of shift operators of each model are the generators of a nonunitary representation of the so(1,2) algebra, except for a special case when this algebra becomes the standard of the nonrelativistic harmonic oscillator.

covariance of the Weyl–Wigner–Groenewold–Moyal quantization
View Description Hide DescriptionThe differential structure of operator bases used in various forms of the Weyl–Wigner–Groenewold–Moyal (WWGM) quantization is analyzed and a derivativebased approach, alternative to the conventional integralbased one is developed. Thus the fundamental quantum relations follow in a simpler and unified manner. An explicit formula for the ordered products of the Heisenberg–Weyl algebra is obtained. The covariance of the WWGMquantization in its most general form is established. It is shown that the group action of that is realized in the classical phase space induces on bases operators in the corresponding Hilbert space a similarity transformation generated by the corresponding quantum which provides a projective representation of the former Explicit expressions for the algebra generators in the classical phase space and in the Hilbert space are given. It is made manifest that this covariance of the WWGMquantization is a genuine property of the operator bases.

Geometrostochastically quantized fields with internal spin variables
View Description Hide DescriptionThe use of internal variables for the description of relativistic particles with arbitrary mass and spin in terms of scalar functions is reviewed and applied to the stochastic phase space formulation of quantum mechanics. Following Bacry and Kihlberg a fourdimensional internal spin space is chosen possessing an invariant measure and being able to represent integer as well as half integer spins. is a homogeneous space of the group parametrized in terms of spinors and their complex conjugates . The generalized scalar quantum mechanical wave functions may be reduced to yield irreducible components of definite physical mass and spin , with and , with spin described in terms of the usual component fields. Viewed from the internal space description of spin this reduction amounts to a restriction of the variable to a compact subspace of , i.e., a “spin shell” of radius in . This formulation of single particles or single antiparticles of type is then used to study the geometrostochastic (i.e., quantum) propagation of amplitudes for arbitrary spin on a curved background space–time possessing a metric and axial vector torsion treated as external fields. A Poincaré gauge covariant path integrallike representation for the probability amplitude (generalized wave function) of a particle with arbitrary spin is derived satisfying a second order wave equation on the Hilbert bundle constructed over curved space–time. The implications for the stochastic nature of polarization effects in the presence of gravitation are pointed out and the extension to Fock bundles of bosonic and fermionic type is briefly mentioned.

structures in string theories
View Description Hide DescriptionThe BRST cohomology of any topological conformal field theory admits the structure of a Batalin–Vilkovisky algebra, and string theories are no exception. Loosely speaking, we say that two topological conformal field theories are cohomologically equivalent if their BRST cohomologies are isomorphic as Batalin–Vilkovisky algebras. In this paper we argue that any string theory (regardless of the matter background) is cohomologically equivalent to some twisted superconformal field theory. We discuss three string theories in detail: the bosonic string, the NSR string and the string. In each case the way the cohomological equivalence is constructed can be understood as coupling the topological conformal field theory to topological gravity. These results lend further supporting evidence to the conjecture that any topological conformal field theory is cohomologically equivalent to some topologically twisted superconformal field theory. We end the paper with some speculative comments on Massey products in topological conformal field theories.

SU() monopoles and Platonic symmetry
View Description Hide DescriptionWe discuss the ADHMN construction for SU monopoles and show that a particular simplification arises in studying charge monopoles with minimal symmetry breaking. Using this we construct families of tetrahedrally symmetric SU and SU monopoles. In the moduli space approximation, the SU oneparameter family describes a novel dynamics where the monopoles never separate, but rather, a tetrahedron deforms to its dual. We find a twoparameter family of SU tetrahedral monopoles and compute some geodesics in this submanifold numerically. The dynamics is rich, with the monopoles scattering either once or twice through octahedrally symmetric configurations.

Can a quantum process be oblivious to observation?
View Description Hide DescriptionIn this paper time is indexed by a “time group” that may be either the real line or the integers. A quantum process is a homomorphic representation of via unitary operators on a Hilbert space with the usual continuity restrictions if time is indexed by the reals. Consider a quantum process subject to nonselective observation at all points in time via an observation operator with purely discrete eigenvalues and corresponding simple eigenvectors. Suppose that the process evolves in such a way that the results of observation at a given time are the same, whether or not there have been previous observations. Suppose further that is spanned by the set of vectors invariant modulo phase under the quantum evolution. If the time group is the integers, then the process is observed to evolve in a classical manner. If the time group is the real line, then the process is never observed to undergo a change of state. The results of this paper are essentially a corollary of von Neumann’s mean ergodic theorem.

Quantized Kronecker flows and almost periodic quantum field theory
View Description Hide DescriptionWe define and study the properties of the infinitedimensional quantized Kronecker flow. This C^{*}dynamical system arises as a quantization of the corresponding flow on an infinitedimensional torus. We prove an ergodic theorem for a class of quantized Kronecker flows. We also study the closely related, almost periodic quantum field theory of bosonic, fermionic, and supersymmetric particles. We prove the existence and uniqueness of KMS and superKMS states for the C^{*}algebras of observables arising in these theories.

The Kirkwood–Buckingham variational method and the boundary value problems for the molecular Schrödinger equation
View Description Hide DescriptionThe approach based on the multiplicative form of a trial wave function within the framework of the variational method, initially proposed by Kirkwood and Buckingham, is shown to be an effective analytical tool in the quantum mechanical study of atoms and molecules. As an example, the elementary proof is given to the fact that the ground state energy of a molecular system placed into the box with walls of finite height goes to the corresponding eigenvalue of the Dirichlet boundary value problem when the height of the walls is growing up to infinity.

Transformation bracket for 2D hyperspherical harmonics and its applications to fewanyon problems
View Description Hide DescriptionThe transformation bracket of hyperspherical harmonics with different sets of hyperspherical coordinates as arguments has been derived for an arbitrary number of particles with arbitrary masses in two space dimensions. The solution of the fouranyon problem is given as an example to illustrate its applications.

On tracial operator representations of quantum decoherence functionals
View Description Hide DescriptionA general quantum history theory can be characterized by the space of histories and by the space of decoherence functionals. In this note we consider the situation where the space of histories is given by the lattice of projection operators on an infinite dimensional Hilbert space. We study operator representations for decoherence functionals on this space of histories. We first give necessary and sufficient conditions for a decoherence functional being representable by a trace class operator on , an infinite dimensional analogue of the Isham–Linden–Schreckenberg representation for finite dimensions. Since this excludes many decoherence functionals of physical interest, we then identify the large and physically important class of decoherence functionals which can be represented, canonically, by bounded operators on .

Covariant path integrals on hyperbolic surfaces
View Description Hide DescriptionDeWitt’s covariant formulation of path integration [B. De Witt, “Dynamical theory in curved spaces. I. A review of the classical and quantum action principles,” Rev. Mod. Phys. 29, 377–397 (1957)] has two practical advantages over the traditional methods of “lattice approximations;” there is no ordering problem, and classical symmetries are manifestly preserved at the quantum level. Applying the spectral theorem for unbounded selfadjoint operators, we provide a rigorous proof of the convergence of certain path integrals on Riemann surfaces of constant curvature −1. The Pauli–DeWitt curvature correction term arises, as in DeWitt’s work. Introducing a Fuchsian group Γ of the first kind, and a continuous, bounded, Γautomorphic potential we obtain a Feynman–Kac formula for the automorphic Schrödinger equation on the Riemann surface Γ\H. We analyze the Wick rotation and prove the strong convergence of the socalled Feynman maps [K. D. Elworthy, Path Integration on Manifolds, Mathematical Aspects of Superspace, edited by Seifert, Clarke, and Rosenblum (Reidel, Boston, 1983), pp. 47–90] on a dense set of states. Finally, we give a new proof of some results in C. Grosche and F. Steiner, “The path integral on the Poincare upper half plane and for Liouville quantum mechanics,” Phys. Lett. A 123, 319–328 (1987).

A combinatorial approach to diffeomorphism invariant quantum gauge theories
View Description Hide DescriptionQuantum gauge theory in the connection representation uses functions of holonomies as configuration observables. Physical observables (gauge and diffeomorphism invariant) are represented in the Hilbert space of physical states; physical states are gauge and diffeomorphism invariant distributions on the space of functions of the holonomies of the edges of a certain family of graphs. Then a family of graphs embedded in the space manifold (satisfying certain properties) induces a representation of the algebra of physical observables. We construct a quantum model from the set of piecewise linear graphs on a piecewise linear manifold, and another manifestly combinatorial model from graphs defined on a sequence of increasingly refined simplicial complexes. Even though the two models are different at the kinematical level, they provide unitarily equivalent representations of the algebra of physical observables in separableHilbert spaces of physical states (their sknot basis is countable). Hence, the combinatorial framework is compatible with the usual interpretation of quantum field theory.

Elliptic Ruijsenaars–Schneider model from the cotangent bundle over the twodimensional current group
View Description Hide DescriptionIt is shown that the elliptic Ruijsenaars–Schneider model can be obtained from the cotangent bundle over the twodimensional current group by means of the Hamiltonian reduction procedure.

An exactly solvable twobody problem with retarded interactions and radiation reaction in classical electrodynamics
View Description Hide DescriptionAn exactly solvable twobody problem dealing with the Lorentz–Dirac equation is constructed in this paper. It corresponds to the motion of two identical charges rotating at opposite ends of a diameter, in a fixed circle, at constant angular velocity. The external electromagnetic field that allows this motion consists of a tangential timeindependent electric field with a fixed value over the orbit circle, and a homogeneous timeindependent magnetic field that points orthogonally to the orbit plane. Because of the geometrical symmetries of the charges’ motion, in this case it is possible to obtain the rate of radiation emitted by the charges directly from the equation of motion. The rate of radiation is also calculated by studying the energy flux across a sphere of a very large radius, using the far retarded fields of the charges. Both calculations lead to the same result, in agreement with energy conservation.

Divergences in the solutions of the plasma screening equation
View Description Hide DescriptionClassical kinetic theory using Boltzmann statistics shows that the potential distribution in the screening cloud surrounding a single test charge at rest within a plasma is governed by a threedimensional spherically symmetric plasma screening equation r≠0, where ε=electronic charge, =electron temperature,=ion temperature, and =electron and ion density at large distances from the charge In this paper it is proved rigorously that any nontrivial solution of the screening equation must have the following property: If =potential at a radial distance and then, for any positive integer as either and or and

A class of integrable Hamiltonian systems whose solutions are (perhaps) all completely periodic
View Description Hide DescriptionWe show that the dynamical system characterized by the (complex) equations of motion with is Hamiltonian and integrable, and we conjecture that all its solutions are completely periodic, with a period that is a finite integral multiple of Here is an arbitrary positive integer, Ω is an arbitrary (nonvanishing) real constant, is the Weierstrass function (with arbitrary semiperiods ), and λ,μ are two arbitrary constants; special cases are and of course These findings, as well as the conjecture (which is shown to be true in some of these special cases), are based on the possibility to recast these equations of motion in the modified Lax form with and appropriate matrix functions of the dynamical variables and of their timederivatives

Invariant differential equations and the Adler–Gel’fand–Dikii bracket
View Description Hide DescriptionIn this paper we find an explicit formula for the most general vector evolution of curves on invariant under the projective action of When this formula is applied to the projectivization of solution curves of scalar Lax operators with periodic coefficients, one obtains a corresponding evolution in the space of such operators. We conjecture that the formula we have found gives another alternative definition of the second KdV Hamiltonian evolution under appropriate conditions. In other words, both evolutions are identical provided that the vector differential invariant characterizing the invariant evolution on the space of projectivized curves is identified with the coefficients of the Hamiltonian pseudodifferential operator. We prove the above facts for and further simplify both evolutions in appropriate coordinates so that one can attempt to prove the equivalence for arbitrary